2_AIChE Journal, Vol. 51, No. 11, (November 2005).pdf

Heterogeneous Combustion: RecentDevelopments and New Opportunities for

Chemical Engineers

Arvind Varma, Victor Diakov, and Evgeny ShafirovichSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907

DOI 10.1002/aic.10697Published online September 27, 2005 in Wiley InterScience (www.interscience.wiley.com).

Introduction

Combustion is arguably the first chemical process used byhumans. The reason for this is simple: Combustion isgenerally a self-sustained process and, once initiated, does

not require external energy input; on the contrary, it releases heatthat can be put to use in various ways. Since Prometheus times, theapplications of combustion have become prevalent throughout ourlife: car engines, power generation, firearms, cigars and cigarettes,rocket propulsion, etc. While, in some cases, combustion occurs inpremixed homogeneous gaseous mixtures, most of the earlierexamples involve heterogeneous processes.

From a chemical engineer’s viewpoint, the self-sustainabil-ity and relative simplicity are attractive features to use com-bustion in addressing important problems facing society. Novelexperimental techniques are constantly bringing new insightabout processes occurring during combustion of condensedmedia. Based on these, fundamental concepts related to heter-ogeneous combustion are being developed which open newpossibilities for applications. The scope of this article is todescribe recent findings in the field and to discuss how thesecan contribute to the solution of contemporary chemical engi-neering problems. In this context, we focus attention on fourspecific topics of significant current and emerging interest:combustion synthesis of advanced materials, hydrogen produc-tion for fuel cells, carbon sequestration, and in-situ resourceutilization in extraterrestrial environments.

It is widely agreed that energy is a critical problem facinghumanity today,1 so the demand for industrial processes withlow energy consumption will continue to increase. An exampleof such technology is combustion synthesis (CS), which hasbeen applied successfully for production of advanced materi-als, such as nanoscale oxide powders, implants and function-ally graded materials.2,3 The development of CS processesdepends on progress in understanding combustion mechanisms

in heterogeneous media. Recent achievements in this field offerpromise for new exciting applications.

Another energy-related problem is the lack of robust powersources, which inhibits further growth opportunities for porta-ble electronic devices. Heterogeneous combustion is likely toplay an important role in this field, leading particularly to newmethods for hydrogen storage.

The rapid worldwide growth in energy consumption is increas-ing greenhouse gas emissions. To avoid further global warming,carbon sequestration technologies are required. In this direction,along with other CO2 capture methods, novel processes, such aschemical looping combustion offer great potential.

Finally, in future space exploration, the need for propellantsand materials will expand chemical engineering approachesbeyond the bounds of Earth. In this context, heterogeneouscombustion is expected to play an important role.

Combustion Synthesis of Advanced Materials

We discuss two wide classes of CS processes. These are theoriginal solid flame CS or self-propagating high-temperaturesynthesis (SHS)2,4 and the process of combusting a liquid(typically aqueous) solution of reactants.3 The CS of initiallysolid samples can be used to obtain such compounds as bo-rides, carbides, hydrides, intermetallics, nitrides, oxides andsilicides in the form of powders, poreless, porous or function-ally graded materials, while solution combustion synthesis(SCS) is an attractive method for producing nanomaterials,particularly oxides.

When using these methods, in each case some specificproperties of the resulting material are targeted. For example,biocompatibility of CS products is of critical importance infabricating biomaterials. In addition, by fine-tuning the oper-ating pressure, it was possible to synthesize poreless orthopedicimplants.5 The CS technique also proved useful in producingporous functionally graded biomaterials.6

Mechanisms of combustion wave propagation inheterogeneous media

The general idea about movement of combustion waves,suggested more than 100 years ago,7 is that layer-by-layer

Correspondence concerning this article should be addressed to A. Varma [email protected].

© 2005 American Institute of Chemical Engineers

Perspective

2876 AIChE JournalNovember 2005 Vol. 51, No. 11

propagation occurs because of initiation of the reaction be-tween premixed components by heat conducting from a neigh-bor layer in which the exothermic reaction is proceeding. Thus,the propagation is determined by two main factors: kinetics ofthe reaction heat evolution and rate of heat transfer from thereaction zone to the unreacted mixture. In heterogeneous sys-tems, reaction kinetics is the controlling step for dense compactmedia containing fine particles (quasi-homogeneous mecha-nism), while particle-to-particle heat conduction is limiting forloosely packed coarse mixtures (relay-race mechanism).8

Developments in experimental techniques have led to obser-vations of new features and to microheterogeneous modelingapproaches. These are increasing fundamental understandingof the involved processes, and enhance progress toward thesynthesis of advanced materials with tailored microstructuresand properties.

A novel digital high-speed video recording technique wasdeveloped, which allows in situ observation of rapid processesoccurring at the microscopic level.9 Using this technique, sig-nificant information about the microstructure of gasless com-bustion waves was obtained and a basis was created for under-standing the mechanisms of fast chemical reactions occurringin heterogeneous media. It was found that the combustion wavemay exhibit unexpectedly complex dynamics. Although onmacroscopic length and timescales the combustion reactionfront may appear to move in a steady manner, on the micro-scopic level it has a complex character that is related to thereaction mechanism.

Over a wide range of experimental conditions, macroscopi-cally steady reaction fronts in heterogeneous mixtures mayexhibit random microscopic fluctuations in shape and instan-taneous velocity, which are directly related to the microstruc-ture of the reaction mixture (i.e. the initial heterogeneity of thereaction medium).10 The reaction wave can propagate in twoqualitatively different modes. The first mode, quasi-homoge-neous reaction wave (QRW), involves a continuous frontwhich moves uniformly and there is practically no variation ofbehavior along the surface of this front (Figure 1a). Anothertype of combustion wave propagation (Figure 1b) exhibits alower average temperature, but relatively bright spots appearrandomly, indicating local regions of high temperature alongthe front. Thus, the front moves forward only as a consequenceof appearance of the hot spots, and the overall progress of thefront occurs only locally in the vicinity of the spot. Based onthese features, this mode of propagation is called scintillatingreaction wave (SRW).

The classical consideration of combustion in condensed sys-tems11 is based on mass and energy balances at the macro-scopic scale and using average values for reactant concentra-tions and temperature. It appears that in heterogeneous media,the characteristic length of heat conduction may often approachthe scale of particle size,12 leading to the SRW regime. The hotspots appear when the local region reaches the ignition tem-perature. Interestingly, in mixtures where the reaction is initi-ated by the melting of a component (note that the SRW modewas observed only in those systems where at least one reactantmelts), the melting point plays the role of ignition temperaturedue to melt spreading and reaction rate acceleration induced bydramatic increase in the contact area and diffusion coefficients.

To capture the features of the SRW propagation mode, amicro-heterogeneous cell model was developed, involving reg-

ular lattice with random voids.13,14 The model was applied tosimulate combustion wave in heterogeneous media by consid-ering a discrete matrix of reaction cells, of particle size dimen-sion, with physical and transport properties varying due torandom organization of the phases in the reaction mixture. Themodel captures well the observed peculiarities of heteroge-neous combustion wave propagation. Specifically, model pre-dictions agree with experimentally determined threshold be-tween quasi-homogeneous and relay-race combustion modes(see Figure 2).

While the variety of dynamic behavior of homogeneouscombustion processes can itself be overwhelming,15 the fea-tures of the observed SRW9 are certainly related to the heter-ogeneity of the reacting media. Even a quasi-homogeneoussteady-state consideration for combustion of imperfectly mixedsolid-solid systems demonstrates that significant deviations inreaction conversion and combustion temperature may arise.16

Another approach, based on localized reaction sites in a regularlattice for one17- and two18- dimensional cases, allows one tocompute the instantaneous combustion velocity distributionsfor both relay-race and quasi-homogeneous mechanisms.

It is evident from the above that significant progress has beenmade, both experimentally and theoretically, in understandinghow combustion waves travel in heterogeneous media leadingto the CS of advanced materials. The prediction and precisecontrol of product properties based on reactant compositionand microstructure, however, still remains a challenge for thefuture.

Synthesis of nanoscale oxides by solution combustion

Solution combustion synthesis (SCS) is a flexible techniquewhere oxidizing and reducing precursors are mixed on themolecular level and, under unique conditions of rapid high-temperature reactions, nanoscale powders of desired composi-tions can be synthesized in one step. The oxidizer (typicallynitrates or oxalates) and fuel (e.g., hydrazine, glycine, urea) are

Figure 1. Two modes of combustion wave propagation:(a) quasihomogeneous; (b) scintillating (fromVarma et al.9).

AIChE Journal 2877November 2005 Vol. 51, No. 11

mixed in an aqueous solution, and heated until self-ignition,yielding product gas and nanoscale oxide powders. Solutioncombustion is a relatively new topic in CS, and is being widelyused by researchers to obtain materials for a vast variety ofapplications, such as oxygen storage capacitors, support andcatalyst for catalytic converters, ceramic pigments, solid oxidefuel cell electrode and electrolyte materials, magnetic ferrites,dielectric and piezoelectric components.3

The exploration of SCS continues by searching for moresuitable and environmentally friendly fuels. For example,Prakash et al.19 developed an SCS technique leading to theformation of complex metal oxides using hexamethylenetetra-mine, which unlike other fuels (e.g., oxalyldihydrazide, carbo-hydrazide), can be synthesized by condensation of ammoniawith formaldehyde, avoiding potentially harmful hydrazine.

A way to optimize the properties of SCS nanocomposites isby applying fuel mixtures and varying their ratio. Thus, de-pending on the fuel mixture consisting of ammonium acetate,urea and glycine, nanoscale zirconia toughened alumina com-posite was synthesized in various particle sizes.20 The fuel-to-oxidizer ratio variation has also been shown to influence prod-uct chemical composition in the microwave initiated SCS of Niand NiO powders.21 It was demonstrated that Ni particles can

be synthesized directly by SCS in air atmosphere, withoutreducing the product. Further, using the SCS of iron oxidenanopowders as an example, it was shown recently that varyingthe mixture compositions, both for fuels and oxidizers, is anattractive methodology to control the product composition andproperties.22

Studies directed toward understanding the mechanisms ofSCS are important to tailor desired product properties. In thiscontext, experimental data concerning mixture reactivity areimportant and TGA/DTA measurements coupled with mass-spectrometry are useful.19 The data become yet more informa-tive when related to in situ SCS measurements.

Dynamic in situ temperature measurements during SCS ofvarious iron oxides were related to TGA/DTA results.22 In thismanner, the mechanisms of solution ignition and combustionwere addressed in detail. It was shown that specific character-istic phase transformations, which vary with the individualfuel-oxidizer system, are responsible for the observed rapidchemical reactions. Similar experimental studies of iron(III)nitrate interactions with model fuels identified the relativereactivities of different functional groups and led to the formu-lation of reaction mechanisms.23

Interestingly, the variation of fuel-to-oxidizer ratio also af-fects the combustion mode of the mixture, as shown for SCS oflanthanum-strontium chromite perovskites.24 Depending onthis ratio, the maximum reaction temperature varied, and eithersmoldering, volume combustion or SHS occurred, leading todifferent product properties. The rate-controlling step duringSCS was also established.

Based on limited mechanistic studies of SCS conductedto-date, it is clear that reactant composition and reaction con-ditions significantly influence product properties. The relation-ship between these, however, is not straightforward, and itsfurther development is a promising direction for the future.Detailed mechanistic studies are expected to play a key role inthis development.

Hydrogen for Fuel Cells

Hydrogen storage for fuel cells is an important multidisci-plinary area of contemporary research, especially attractive forchemical engineers.25 There are several possible methods forhydrogen storage, such as compressed and liquefied hydrogen,hydrogen adsorption on carbon materials of large surface area,and metal hydrides,26 but all have limitations which preventtheir immediate use in practice. In this context, some promisingheterogeneous combustion based techniques have recentlybeen proposed and are described in this section.

Power sources for portable electronic devices

The number and performance characteristics of portableelectronic devices in consumer, industrial, medical and militarymarkets are increasing continuously. They include mobilephones, notebook computers, PDAs, digital cameras, DVDplayers, RFID and barcode scanners, autonomous robots, hand-held diagnostics, infusion pumps, defibrillators, etc. The totalnumber of such units in use is difficult to evaluate, but evenconservative estimates suggest more than a billion. Simulta-neously with increasing performance, the power demand ofportable electronic devices is also increasing. Rechargeable

Figure 2. Combustion wave propagation with varyingparticle size at constant density; Ti-air system(from Varma et al.14).


batteries are approaching their limits of specific energy (�0.2Wh/g for the most advanced lithium-ion batteries). This situa-tion is well illustrated by the fact that batteries in notebookcomputers typically require recharging after 2 h of operation.

Fuel cells provide much higher specific energy and, for thisreason, are expected to be widely used in the near future. Incontrast to batteries, a fuel cell power system includes separateconversion device (fuel cell itself) and fuel storage/feedingsystem, analogous to internal combustion engine and gasolinetank/pump in an automobile. Thus, the future power source forportable electronics will include a permanent fuel cell unitinside the device and replaceable fuel cartridges which can berefilled, recycled or discarded.

Methanol and hydrogen are the most promising candidatefuels to feed portable fuel cells. They have specific energies 6.3and 39.3 Wh/g, respectively. The assumption that direct meth-anol fuel cells (DMFC) can reach conversion efficiency about25% leads to the conclusion that specific energy of methanolcartridge would be �1.5 Wh/g, which is significantly higherthan for the best batteries (�0.2 Wh/g). Unfortunately, DMFCshave many drawbacks, such as low power density, methanolcrossover, electrode poisoning, methanol toxicity, and catalystcost, which are serious obstacles to their use.

Hydrogen fuel cells provide much higher power density, donot have any methanol-related problems, and operate withconversion efficiency about 50%. The key problem in devel-opment and use of hydrogen fuel cell power sources is hydro-gen storage. High pressure gas containers are too heavy (thus,leading to much lower specific energy than DMFC) and liquidhydrogen (temperature 20 K) is not possible for use in portableapplications. It is generally agreed that chemical methods forhydrogen storage provide the maximum H2 yield and are themost promising for portable fuel cell applications. A simpleestimate shows that 8 wt. % H2 yield results in specific energyof hydrogen fuel cartridge �1.5 Wh/g, the same as for DMFCs.The hydrogen fuel cell with the same specific energy as DMFCwould be advantageous because of much higher power densityand elimination of the methanol-related problems. Thus, thenatural question arises: What chemicals and technology shouldbe used to reach the maximum H2 yield?

Heterogeneous combustion methods for hydrogengeneration

Borohydrides of light metals (Li, Be, Na, Mg, Al) are knownto be excellent sources of hydrogen, which can be obtained byreactions of these compounds with water or oxidizing agents.Sodium borohydride (NaBH4), discovered during World War IIby Nobel laureate Herbert C. Brown of Purdue, is produced forvarious applications at the rate of about 10,000 tons per year.Hydrolysis of sodium borohydride is a well-known process,27

extensively studied for hydrogen generation.28,29 Aqueous al-kaline NaBH4 solutions are stable and their contact with se-lected catalysts (e.g., Ru) leads to the reaction

NaBH4 � 2 H2O 3 NaBO2 � 4 H2. (1)

High theoretical hydrogen yield (10.8 wt. % for stoichiometricmixture) makes this process attractive for applications. Theother reaction product, sodium metaborate, is water-soluble,environmentally benign and can either be discarded or recycled

to generate new sodium borohydride. Unfortunately, the prac-tical strength of aqueous NaBH4 solution is limited to 30 wt. %borohydride, thus decreasing maximum H2 yield to 6.3 wt. %.30

Furthermore, reaction initiation requires introducing the cata-lyst to the mixture, which is difficult particularly for portableapplications. This problem is overcome in combustion-basedapproaches for hydrogen generation, which require only igni-tion and no catalyst.

It should be noted that combustion of solid mixtures iswidely used for gas generation to create thrust in rocket en-gines, inflate air bags in case of car collisions, provide emer-gency oxygen to airplane passengers, etc. The amounts of gasesstored in chemical compounds, for example, oxygen in sodiumchlorate (NaClO3) and hydrogen in sodium borohydride(NaBH4), are comparable with storage abilities of liquid (cryo-genic) O2 and H2, respectively, and much larger than those ofpressurized gas tanks. The gas generating compositions can beeasily and safely stored for years, and ignited when the productgas is required.

Recently, new hydrogen-generating pyrotechnic composi-tions were proposed to feed fuel cells for portable electronicsby researchers from CEA (French Commission on AtomicEnergy) and SNPE (French company for development of en-ergetic materials).31-33 In these mixtures, hydrogen is generatedby combustion reactions of metal borohydride with oxidizersalt, such as NH4ClO4, NaClO4 and Sr(NO3)2. These reactionsare highly exothermic, easily initiated, and do not require anycatalyst, which makes the method cost-effective and attractivefor portable applications. Stoichiometric mixtures of the pro-posed reactants, however, exhibit low hydrogen yield, whilethose with high metal borohydride content do not burn. As aresult, the maximum experimental H2 yield obtained by thismethod was 5.9 wt. %.31

An alternative approach for hydrogen generation, proposedin Russia, uses combustion of nanoscale aluminum with gelledwater.34 Here, water acts as an oxidizer for Al and simulta-neously as the sole source of hydrogen. The adiabatic combus-tion temperature of Al/H2O stoichiometric mixture is close to3,000 K (at 1 atm pressure). Despite this high temperature, forself-sustained combustion, the use of nanoscale Al powder andgelling of water (for example, by adding polyacrylamide) arenecessary. The use of nanopowder decreases the ignition tem-perature of Al, while gelling inhibits water evaporation duringcombustion. Hence, the mixture ignites easily and burns ininert atmosphere, producing hydrogen

Al � 3/2 H2O 3 1/2 Al2O3 � 3/2 H2. (2)

Unfortunately, low hydrogen yield of the Al/H2O system (the-oretical limit 5.6 wt % for the stoichiometric mixture) is adrawback of this method.

A new combustion-based method for hydrogen generationwas recently proposed by us.35 To simultaneously reach highhydrogen yield and combustion efficiency, we use triple so-dium borohydride/aluminum/water mixtures, in which wateracts as an oxidizer for both sodium borohydride and aluminumand also as a source of hydrogen (parallel reactions (1) and (2)in one mixture). Sodium borohydride is an additional hydrogensource, while aluminum increases combustion temperature,eliminating the need for catalyst. Nanoscale aluminum pow-


ders are used to ensure high combustion efficiency. Along withthe three reactants, the mixtures include small quantities of agelling agent (e.g., polyacrylamide) and a stabilizer (e.g.,NaOH) to prevent hydrolysis of borohydride at room temper-ature.

It should be noted that the reaction products include, besideshydrogen, sodium metaborate and alumina. Both compoundsare environmentally benign materials. They can be either dis-carded or, if economically justified, recycled.

Experiments with NaBH4/nanoAl/gelled H2O mixtures showthat35

● they exhibit stable self-sustained combustion (see Fig-ure 3);

● hydrogen yield, measured by gas chromatography, ap-proaches 7 wt %;

● hydrogen release efficiency is 74–77%.The proposed novel technology of hydrogen storage, com-

bustible borohydride/metal/water mixtures, makes it possibleto develop hydrogen fuel cell power systems with high specificenergy, high power density, no catalyst and safe reaction prod-ucts.

Power systems based on this technology can be used aschargers for various electronic devices, and have the potentialto dramatically increase their portability. Larger-scale hydro-gen generators can be used for emergency power supplies andin some specific applications, such as power for spacecraft andundersea vehicles. Note that the Apollo 13 accident was causedby explosion of a liquid oxygen tank, which was used to feeda fuel cell power supply of the spacecraft. The use of chemicaloxygen (also using heterogeneous combustion)36 and hydrogengenerators instead of cryogenic liquids in fuel cell powersystems would eliminate such accidents, while providing com-parable oxygen and hydrogen storage densities. In addition, thecombustion heat release can be converted to electricity (forexample, using turbine cycles or thermoelectric effect). This isof particular interest for apparatus with limited energy sources,such as spacecraft in deep space or autonomous weather sta-tions and buoys in polar regions.37

Carbon Sequestration

Methods based on heterogeneous combustion could alsohelp to solve another important global problem engaging at-tention of chemical engineers: carbon sequestration. Accordingto the President’s Global Climate Change Initiative, by the year2012 the greenhouse gas intensity of the U.S. economy shouldbe decreased by 18%. Carbon sequestration, along with makingenergy systems more efficient and increasing the use of lowcarbon fuels, is the key to accomplishing this goal. Carbonsequestration is the capture and storage of CO2 and othergreenhouse gases that would otherwise be emitted to the atmo-sphere. A clear priority for near-term deployments, amongvarious sequestration possibilities, is to capture CO2 fromlarge-scale emission sources, such as power plants, and store itin underground formations.38

There are three main CO2 capture technologies, classified aspost-combustion, precombustion, and oxyfuels. Post-combus-tion refers to capturing CO2 from a flue gas after a fuel hasbeen burnt in air. The CO2 in flue gas is dilute (3–15 vol %),at low-pressure (0.1–0.2 MPa), and often contaminated withtraces of sulfur and particulate matter. Thus, a very largevolume of flue gas has to be treated, so that equipment is largeand capital costs are high.

Precombustion refers to a process where a hydrocarbon fuelis reacted with oxygen, air or steam to give mainly synthesisgas (CO and H2). The carbon monoxide then reacts with steamin a water-gas shift (WGS) catalytic reactor to give CO2 andmore hydrogen. Finally, the CO2 is removed and the hydrogenis used as a fuel in a turbine cycle. This technology signifi-cantly facilitates CO2 removal because the product gas containsCO2 in high concentration (30 – 50 vol %) and at high pressure(1.5 – 3 MPa). There are, however, few gasification-basedpower systems currently in operation.

Oxyfuel is an approach where a hydrocarbon fuel is burnt inpure or nearly pure oxygen rather than air. The combustionproduct is then a mixture of CO2 and water, from which CO2

can be easily separated. The basic infrastructure for oxygen isavailable in coal gasification plants, but the added oxygencapacity increases capital and operating expenses.

In this article, we focus on an alternative approach foroxygen supply, the so-called chemical looping combustion(CLC). The chemical-looping reaction technology is based onthe cyclic use of solid medium which reacts with gaseousreactants. This technology has been used in many processessuch as catalytic cracking and sulfur removal in coal gasifica-tion. The idea to use chemical looping in combustion-basedpower plants was originally proposed by Richter and Knoche39

in 1983. In recent years, CLC has attracted much attentionbecause of easy separation of CO2 in this method. Ishida inJapan has conducted extensive research of CLC since 1994.40-42

In Europe, CLC has been investigated vigorously for the lastfive years.43-46 To our knowledge, few research efforts havebeen devoted to CLC in the U.S. so far.47

In CLC technology, a metal oxide (MeO) is used as anoxygen carrier, which transfers oxygen from air to the fuel. Theoxygen carrier circulates between two interconnected fluidizedbed reactors (see Figure 4), similar to the VPO-catalyzedbutane to maleic anhydride process commercialized by Du-Pont.48 In one reactor, the MeO is reduced by a gaseous fuel,such as natural gas

Figure 3. Combustion front propagation in borohydride/aluminum/water mixture; mass ratio NaBH4:Al:H2O � 1:2:3 (from Shafirovich et al.35).


MeO � CH4 3 Me � CO2 � H2O. (3)

An almost pure CO2 stream can be obtained after cooling theexit gas and condensing H2O. The particles of reduced carrier(Me) are transferred to the second reactor and oxidized withair, regenerating the metal oxide

Me � O2 3 MeO. (4)

This reactor gives a flue gas containing only N2 and unreactedO2. The oxidized carrier (MeO) is then returned to the firstreactor for a new cycle.

CLC can also be used in power plants based on coal gasifi-cation.42 In this case, synthesis gas produced from the gasifierreacts with metal oxide

MeO � CO � H2 3 Me � CO2 � H2O, (5)

while the other steps are the same as described earlier.The total amount of heat evolved in CLC is the same as for

conventional combustion, where the oxygen is in direct contactwith the fuel (e.g., CH4). The advantage of CLC is that CO2 isnot diluted with N2 but obtained in a separate stream and canbe easily recovered by simple steam condensation.

The necessity to use the same oxygen carrier in many cyclesis a critical problem for CLC technologies. Iron, nickel, cobalt,copper, manganese and cadmium (and their correspondingoxides) have been considered as possible candidates for CLCprocess, and iron and nickel have been shown as most prom-ising. The nickel-based oxygen carriers withstand a higheroperating temperature, while iron has lower cost and its oxidesare environmentally friendly materials. A suitable carrier forCLC exhibits high reaction rate and conversion, coking resis-tance, cycle durability and high mechanical strength. For thisreason, the metal oxides are combined with an inert (such asAl2O3, TiO2, ZrO2, SiO2, MgO), which acts as a porous sup-port providing high surface area for reaction, and as a binderfor increased mechanical strength and attrition resistance.

Methods for preparation of oxygen carriers have been de-veloped and the obtained composite particles tested using TGAand experimental reactors, which identified suitable composi-tions and confirmed the potential of CLC. Much work remains

to be done, however, to advance this concept from laboratoryexperiments and estimates to industrial technology. Experi-mental and modeling methods developed in studies on hetero-geneous combustion of metals, coal, propellants and otherenergetic materials will be useful in this regard. It should benoted that combustion of light metals (such as Al and Mg) ingaseous oxidizers (including air and water vapor) was studiedextensively for propulsion applications,49 see also next section.Similarly, combustion of porous carbon particles in air andoxygen was investigated widely for power generation.50 Thisexperience could be effectively applied for oxidation and re-duction reactions (3–5) involving oxygen carrier particles.Thus, heterogeneous combustion approaches can play an im-portant role in solving carbon sequestration problems.

Extraterrestrial Production of Materials andPropellants

Space exploration opens new opportunities for chemicalengineers. In the near future, missions to the Moon and Marswill require propellants and structural materials to supportextraterrestrial activities there and to return crews and/or sam-ples to Earth. In situ resource utilization (ISRU) would beeconomically attractive as compared to transportation of pro-pellants and materials from earth. For example, a human mis-sion to Mars, designed by NASA in response to the SpaceExploration Initiative (1989), and relying completely on earthresources, required $450 billion, while NASA’s mission designusing ISRU (1994) decreased the price tag to $55 billion.51 Thenew U.S. space policy, Vision for Space Exploration, includesISRU as a core component:“In situ resource utilization willenable the affordable establishment of extraterrestrial explora-tion and operations by minimizing the materials carried fromearth, and by developing advanced, autonomous devices tooptimize the benefits of available in situ resources.”52

Heterogeneous combustion can be used in space for bothproduction of materials and propulsion. First, CS has thepotential to produce materials and net-shape articles in space.Indeed, an important advantage of this method is low externalenergy consumption, which is required only to initiate theself-sustained combustion process. This makes CS especiallyattractive for use in space missions, where energy availabilityis limited. One potential application of CS in space is fabrica-tion of replacement parts in long-duration exploration mis-sions. It would be desirable to make spares of malfunctioneditems aboard the spacecraft, when required, rather than carry areserve of spares for all parts. The CS method makes it possibleto fabricate replacement parts and custom tools on-site witheither in situ materials or reactant powders carried from earth.They can be made to near-net shape, with tight tolerances,reliably, quickly, and safely. Furthermore, CS can be used tojoin different types of materials by compositional and func-tional grading.53 Studies in drop towers, research aircraft andspace stations have demonstrated that a majority of CS tech-nologies developed in normal gravity can be successfullyadopted under microgravity conditions. In fact, many materialsproduced in microgravity possess superior properties as com-pared to those synthesized on earth.54

Besides materials synthesis, another promising applicationof heterogeneous combustion in space is related to propulsion.

Figure 4. Chemical looping combustion.


Specifically, metal fuel could be burnt in propulsion systemswith CO2 of Mars and Venus instead of propellants broughtfrom Earth. To demonstrate the importance of this idea forMars exploration, let us first briefly review the history of MarsISRU concept, which involves much chemical engineering.Most Mars ISRU suggestions are based on use of martianatmosphere which contains 95% CO2. One of the first ideaswas to produce methane and oxygen (which constitute anexcellent rocket bipropellant) from martian CO2 and water in asmall chemical processing plant.55 To obtain water on Mars,however, is much more difficult than CO2. For this reason, ina later modification of this approach, a small feedstock ofliquid H2 is transported along with the chemical plant fromEarth to Mars for use instead of water (Sabatier reaction)56

CO2 � 4 H2 3 CH4 � 2 H2O. (6)

The attractive features of this concept are the relatively simpletechnology and the high performance of the produced CH4/O2

bipropellant. The long-term storage and deep space transport ofliquid H2 is, however, a significant problem. This drawback iseliminated in another methodology, which postulates produc-tion of CO and O2 directly by electrolysis of the Martian CO2

in a zirconia cell.57,58 The resulting CO and O2 are then used asa liquid bipropellant. The zirconia electrolyzers, however, arefragile and require high operating temperature (�1,000 K). Thecommon problem in all Mars ISRU scenarios mentioned pre-viously is the significant power required to produce, liquefy,and store cryogenic propellants.

An alternative, combustion-based approach in the MarsISRU suggests burning CO2 directly as an oxidizer with metalsor metal hydrides as fuel in a rocket or jet engine.59,60 Forrocket applications, CO2 should be collected and liquefied,without any further chemical processing, while the fuel couldbe either delivered from earth or produced on mars. Carbondioxide is not a typical oxidizing agent in industrial practice,but there exist fuels (e.g., magnesium and aluminum) whichcan burn with CO2. Analysis of thermodynamic performancecharacteristics, combustion parameters, and other properties forvarious candidate fuels (Li, Be, B, Mg, Al, Si, Ca, Ti, Zr, theirhydrides and mixtures with hydrogen compounds) has led tothe conclusion that Mg is the most promising for rocket enginesusing CO2 as an oxidizer.60,61 Magnesium particles ignite inCO2 environment at relatively low temperature (�1,000 K) andexhibit vigorous vapor-phase combustion62-64 (see Figure 5).Because of use of in situ oxidizer, Mg-CO2 rocket enginesprovide significant advantages in missions with ballistic flightson Mars, even with Earth-imported Mg fuel.65 To improvemetal-CO2 engine performance, aluminum can be used insteadof magnesium if a method is developed to decrease its ignitiontemperature in CO2 (�2,300 K). One such possibility is to coatAl particles with a thin layer of nickel. Recent studies oncombustion of Ni-clad Al particles have shown that they igniteat much lower temperature (�1,000 K), which is associatedwith the intermetallic reactions between Ni and Al on theparticle surface.66,67

This brief review clearly demonstrates that novel combus-tion-based processes have great potential for applications underextraterrestrial conditions (microgravity, atmosphere of Mars,etc.). They will facilitate the production of materials and powerin future space exploration missions.

Concluding Remarks

In this article, we focused attention on four specific topics ofsignificant contemporary interest to society: combustion syn-thesis of advanced materials, hydrogen production for fuelcells, carbon sequestration, and in-situ resource utilization inextraterrestrial environments. On the basis of our expertise andbecause of space limitations, we did not consider other impor-tant areas, such as catalytic combustion, soot formation, drop-lets and spray combustion, gaseous flame synthesis, coal andexplosives.

All of the earlier are seemingly unrelated topics, yet eachincludes heterogeneous combustion as the central theme. Wehope that this article will attract more chemical engineers tothis fascinating area, which is rich both in fundamental con-cepts and important applications. Furthermore, progress in thisfield will arise from a deeper understanding of the combustionmechanisms, and will be aided by development of novel ex-perimental techniques capable of detecting phenomena atshorter time and length scales, along with detailed theoreticaland simulation approaches that quantitatively describe the ex-perimental observations. The prospects of new opportunitiesfor chemical engineers in heterogeneous combustion are indeedglowingly bright.

Acknowledgments

We thank NSF, NASA and ACS-PRF for long-term supportof our research in heterogeneous combustion (current grants,NSF: CTS-0446529; NASA: NNCO4AA36A; ACS-PRF:36711-AC9).

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Effect of Taylor Vortices on Mass Transfer from aRotating Cylinder

R. Srinivasan, S. Jayanti and A. KannanDept. of Chemical Engineering, IIT-Madras, Chennai 600 036, India

DOI 10.1002/aic.10553Published online August 8, 2005 in Wiley InterScience (www.interscience.wiley.com).

Mass transfer from solids, which has important applications in a number of chemicaland pharmaceutical industries, has been studied experimentally and semiempiricallyunder turbulent flow conditions, and correlations are available in the literature tocalculate the mass-transfer coefficients from pellets, rotating cylinders and disks etc.However, mass transfer under laminar flow has not been sufficiently addressed. One of thedifficulties here is the strong Reynolds number dependence of the flow pattern, forexample, due to the onset of Taylor vortices for the case of a rotating cylinder. Thisproblem is circumvented by using a computational fluid dynamics (CFD)-based solutionof the governing equations for the case of a cylinder rotating inside a stationarycylindrical outer vessel filled with liquid. The parameters cover a range of Reynoldsnumber (based on the cylinder diameter, and the tangential speed of the cylinder),Schmidt number and the ratio of the outer to inner cylinder diameters. The results confirmthat the circumferential velocity profile is a strong function of the Reynolds number andvaries from a nearly Couette-type flow at very low Reynolds numbers to a boundarylayer-like profile at high Reynolds numbers. The onset of Taylor vortices has a strongeffect on the flow field and the mass-transfer mode. The calculations show that theSherwood number has a linear dependence on the Reynolds number in the Couette-flowregime, and roughly square-root dependence after the onset of Taylor vortices. Correla-tions have been proposed to calculate the Sherwood number taking account of theseeffects. © 2005 American Institute of Chemical Engineers AIChE J, 51: 2885–2898, 2005Keywords: CFD, solid dissolution, mass transfer, Taylor vortices, rotating cylinder,laminar flow

Introduction

Dissolution of solid materials in liquids finds wide-spread ap-plications in the chemical process industries. Knowledge of themass-transfer rates between a solid and an agitated liquid in avessel is of considerable importance in the design of units, such ascrystallizers, stirred-tank reactors for diffusion controlled solid-fluid reactions, and units involving physical dissolution. Someindustrial examples of solid dissolution process are:

● Dissolution of metals, such as tin, lead and zinc in heatexchange media, such as mercury and sodium at high-temper-atures, and their deposition at lower temperatures.1

● Limestone dissolution in an aqueous solution which is aphenomenon of interest in liming of acid lakes, flue gas desul-furization.

● The sustained drug delivery process in pharmaceuticalapplications which is determined by the rate of solid dissolu-tion in the liquid medium.

● Mass transfer from a rotating cylinder which is of practicalimportance in electrochemical processes, such as electrodepo-sition, coating and metal recovery.2

● Recent application of mass transfer from rotating cylin-

Correspondence concerning this article should be addressed to S. Jayanti [email protected].


FLUID MECHANICS AND TRANSPORT PHENOMENA


ders in filtration and reverse osmosis processes where Taylorvortices (see later) are induced to enhance the mass-transferrates.

Dissolution of solids has been extensively studied from boththeoretical and experimental fronts. Three idealized configura-tions of the solid, namely, suspended pellets, rotating disk androtating cylinder, have been considered in the literature. Eisen-berg et al.3 reported on one of the first systematic studies ofsolid dissolution from a rotating cylinder over a wide range ofReynolds and Schmidt numbers. Johnson and Huang4 studiedthe rates of dissolution of organic solid from a flat surface intoa turbulent liquid in an agitated vessel. Bennett and Lewis1

made a comparison between the dissolution of lead and tin inmercury with the dissolution of benzoic acid in water, andfound that the metal dissolution rates were transport controlled.Sherwood and Ryan5 studied experimentally the mass transferwith and without chemical reaction from rotating benzoic acidcylinders. Marangozis and Johnson6 reviewed solid dissolutiondata of suspended pellets, flat disks and rotating cylinders, andrecommended cube-root dependency of the Sherwood numberon the Schmidt number. Studies on solid-liquid mass transferhave been reviewed extensively by Pangarkar et al.7

There have been a number of more recent studies of masstransfer from a rotating cylinder with and without axialflow.8,9,10,11,12,13,14,15 The configurations studied include caseswith inner cylinder alone rotating and outer cylinder alonerotating, and are spread over large ranges of Reynolds andTaylor and Schmidt numbers. Most of these have been exper-imental studies and do not include detailed measurement of theflow field. An important aspect of mass transfer from a rotatingcylinder is the possibility of occurrence of Taylor vortices.16,17

The rotational motion of the inner cylinder (Taylor vortices donot occur when the outer cylinder alone rotates) induces aprimarily circumferential flow in the annular region betweenthe inner and the outer cylinders. If the rotational speed issufficiently high (but not too high), then large-scale secondaryflows, typically of the size of the gap width as shown in Figure1a, are formed. It is to be expected that the presence of theseTaylor vortices has a strong effect on the mass-transfer ratefrom the cylinder. While this flow situation has been widelyinvestigated for very small gap widths in connection with thestability of the flow, there have been relatively few studies ofthe mass transfer, and also of the flow field for large gap widthstypically used in mass transfer applications. In a recent study,Baier et al.15 used computational fluid dynamics (CFD) simu-lations to obtain the velocity field for the case of the innercylinder rotating, and used a boundary layer approximation tostudy the mass transfer aspects for small values of Taylornumbers. They found that the mass-transfer coefficient in thepresence of the Taylor vortices varied as Sh � Re0.46, whichagreed with the experimental results of Coeuret and Legrand,11

Holeschovsky and Cooney14 who studied for the case of noaxial flow and inner cylinder rotating, and also with the resultsof Kataoka et al.9 who investigated the case of the outercylinder rotating. Earlier, Kawase and Ulbrecht13 used an anal-ogy with the secondary flow in helical coils to arrive at thesame result. Experiments at high Taylor numbers3,12 show astronger dependence of the Sherwood number on the Reynoldsnumber with the exponent being in the range of 0.6 to 0.72.

Thus, there is considerable literature on the mass transferfrom rotating cylinders. However, a detailed description of the

flow field within the annular region is absent in these studies,presumably due to the nontrivial dependence of the velocityfield on the Reynolds number and the radii of the inner andouter cylinders, among other parameters. Baier et al.15 whoincluded some of these effects in their computation of thevelocity field, used a boundary layer analogy for mass transfer.Thus, their study is not applicable for very low Taylor num-bers. In this article, we take advantage of the accuracy—forlaminar flow simulations—offered by computational fluid dy-namics (CFD) techniques to simulate the complete problem,that is, calculate both the velocity field and the scalar concen-tration field, for a range of flow conditions including very lowTaylor numbers where Taylor vortices are not present. Thesimulations cover a range of Reynolds numbers, diameter ra-tios and height to diameter ratios of the cylinders. Results fromthese simulations show clearly the marked change in the mass-transfer coefficient upon the onset of Taylor vortices. Details ofthe calculation methodology, and the results obtained are dis-cussed later.

Problem Formulation and CalculationMethodologyGoverning equations for fluid flow

The equations solved for a CFD solution are the continuityand the momentum equation, which can be written as followsusing vector notation:

Continuity equation: for an incompressible Newtonian fluid

� � u � 0 (1)

Momentum equation

�u�t

� �� uu� � �1

��p � ��2u (2)

Here u is the velocity; p is the static pressure, � is the density,and � is the kinematic viscosity. Since the fluid under consid-eration is a liquid and the velocities are small, a constant-property fluid assumption is made.

The mass concentration of the dissolving substance is cal-culated by solving the following generalized scalar transportequation

�

�tYA � �� uYA� �

1

�� AB�YA� (3)

where YA is the mass fraction of the scalar A, that is, the massof the scalar per total mass in the control volume. �AB is themolecular diffusivity (in kg/ms) of the scalar in the liquid B,defined as �AB � �DAB, where DAB is the kinematic moleculardiffusivity in m2/s. This approach has previously been used byone of the authors,18 to calculate the mixing time in jet-mixedvessels. The scalar is assumed to be a neutrally buoyant scalarso that the density and the viscosity of the fluid are unaffectedby the scalar concentration. Since benzoic acid, the solid underconsideration, is only sparingly soluble in water, the fluid underconsideration, this is a reasonable assumption.


Flow domain and boundary conditions

The flow domain consists of an inner rotating cylinder andan outer stationary cylindrical vessel filled with a liquid up toa given height. The inner solid cylinder is concentric with theaxis of the vessel and rotates at a constant angular velocity.There are no internal parts, such as baffles in the vessel, and theproblem is treated as being axisymmetric for the low Taylornumbers studied in this work. The flow domain consists of theliquid in the annular region between the inner rotating cylinderand the outer stationary cylinder, and is bounded at the bottomby a stationary wall and at the top by a planar shear-freeinterface, as shown in Figures 1b and 1c. The problem isformulated in cylindrical coordinates, the axial, radial andtangential velocity components being u, �, and w, respectively.No-slip boundary conditions are imposed on all walls. Thus, aconstant tangential velocity is specified at the inner cylinder,

and a zero velocity is specified on the outer wall and the bottomwall. A symmetry boundary condition is specified on the topwall to make it a zero-shear surface. Since the major flow is inthe circumferential direction, all the three velocity componentshave to be resolved. For this reason, the flow in a 45° sector ofthe cylinder is simulated, and periodicity condition is imposedon the two bounding circumferential planes. For the massfraction equation, a constant wall mass fraction, which is basedon the solubility limit of benzoic acid in water, estimated to be4.067 � 10�3 at a temperature of 30°C, is applied on thesurface of the inner cylinder and a Neumann boundary condi-tion (zero normal gradient) is applied on the other boundaries.

The solution of the coupled mass, momentum and mass-transfer equations has been done in two stages. In the firststage, the mass and the momentum equations are solved to-gether under steady conditions to obtain the steady-state ve-

Figure 1. (a) Taylor vortices in the annular gap width ofinner cylinder rotating and outer stationary, (b)geometry of the computational domain, and (c)boundary conditions.


locity and pressure fields. In the second stage, the time-depen-dent mass fraction equation, which is coupled in a one-waymode to the mass and momentum equations, is solved with thevelocity field obtained from the steady-state calculations as theinitial condition for the velocity field. The mass fraction in theinterior of the domain at the beginning of the calculation isassumed to be zero throughout.

Details of Numerical SolutionComputer code

All the calculations reported here have been carried outusing the commercially available CFX 4.4 computer codedeveloped by AEA Technology, UK. CFX is a general purposecomputer program, which uses a finite volume method-baseddiscretization of the governing partial-differential equations ona nonstaggered, structured, body-fitted grid. The chequer-boardtype of oscillations of pressure and velocity that are associatedwith the use of a nonstaggered grid are eliminated using theRhie-Chow interpolation scheme19 to estimate the face veloc-ities. The pressure-velocity coupling for incompressible flowsis effected using the SIMPLE family of schemes20 adapted fora nonstaggered body-fitted grid.

Discretization schemes

The computer program has different discretization schemesfor the user to choose from for a particular problem. In thepresent case, the convection terms in the governing flow equa-tions have been discretized using the third-order accurateQUICK scheme,21 while the diffusion terms have been dis-cretized, using the second-order accurate central scheme. Forthe solution of the mass-fraction equation, it was found thatusing the QUICK scheme gave negative values of the massfraction; hence, the SUPERBEE scheme,22 which incorporatesthe total variation diminishing (TVD) property to eliminateoscillations associated with high order differencing schemes,has been used. The overall accuracy of the discretization can,therefore, be said to be of second-order.

Grids

The structured, orthogonal grid for the two-dimensional(2-D) flow domain consisted typically of 9,600 cells. Forexample, in the case with a cylinder height of 0.12 m, and a gapwidth of 0.035 m, a 100 � 96 nonuniform grid was used todiscretize the 2-D flow domain in the x-r plane, x being axialdirection, and r being radial direction. Since the precise loca-tion of the Taylor vortices was not known a priori, uniformgrid spacing was used in the axial direction. In order to resolvethe strong velocity and mass fraction gradients in the radialdirection, an expanding grid was used. Preliminary calculationswere made with different grids to check for grid independenceof the results. The computed velocity profile near the innercylinder was compared for three radial grids in which the gridspacing near the wall was successively reduced by a factor oftwo. Similarly, the grid independence in the axial direction wasalso verified by repeating the calculations with a grid spacingwhich is halved. The temporal discretization was also verifiedby calculating the mass-fraction equation with a series ofdecreasing time steps.

Formulation of the mass transfer problem

The problem under consideration is the determination of themass-transfer coefficient from a rotating cylinder made of thesolid to be dissolved. The specific case being studied is that ofthe dissolution of benzoic acid in water in which the former issparingly soluble. Experimentally this is done by measuring theconcentration of the benzoic acid in the solution as a functionof time. It can be shown4 that the cup mean concentration withtime is given by the following expression

�ln�1 �CA

CAW� � �kcS/V�t (4)

where kc is the mass-transfer coefficient, S is the surface areaof the inner cylinder, V is the volume of the liquid, CA is thebulk mean concentration at time t, CAW is the solubility con-centration of the solid in liquid. The mass-transfer coefficient isobtained from the slope by plotting LHS vs. time. In the presentcalculations, this is simulated by the numerical solution of thetime-dependent transport equation of a passive, neutrally buoy-ant scalar (Eq. 3) with the appropriate diffusivity value andsubject to initial and boundary conditions. (It is assumed thatthe dissolution rate is small enough that one does not have toconsider the radial movement of the cylinder surface due todissolution.) The mass-transfer coefficient can now be obtainedin one of two ways: (1) by plotting the temporal variation of thecup mean concentration in the vessel, which gives the time-averaged mass-transfer coefficient, and (2) from the slope atthe wall of the radial profile of the instantaneous mass fraction.It was found that the two gave nearly the same results; how-ever, the computed temporal variation was not exactly linear,especially in the initial stages of the calculation. Hence, thesecond method was followed to determine the mass-transfercoefficient. Here, two additional complications arise in theevaluation of the mass-transfer coefficient. First, the mass-transfer rate is a function of the local velocity field and,therefore, shows axial variation when Taylor vortices arepresent. An axial average of the computed wall mass flux was,therefore, taken to smooth out these variations. Second, inmany cases, especially at low rotational speeds, the mass-fraction profile did not have a clearly defined bulk concentra-tion. Therefore a cup mean mass fraction, defined as follows

Yc �

�0

L

�Ri

R0

w�r, x�Y�r, x�2�rdrdx

�0

L

�Ri

R0

w�r, x�2�rdrdx

(5)

was used as the bulk mass fraction. The mass-transfer coeffi-cient was then calculated as

kc �jA

��Y* � Yc�(6)


where jA is the axially-averaged mass flux at the inner cylinderat a given time wall which is directly obtained from thecomputer program, Y* is the wall mass fraction obtained fromthe solubility concentration of the solid in liquid and Yc is thecomputed cup mean mass fraction in the vessel at that time.This is then converted into a Sherwood number based onhydraulic diameter as follows

ShDh �kc�2d�

DAB(7)

where d is the gap width.

Validation

It can be seen from the description of the earlier calculationmethodology that the exact equations governing the flow arebeing solved. The major assumptions are the flow is laminar,axisymmetric, Newtonian with the fluid having constant phys-ical properties, which are reasonable for the problem underconsideration. Thus, unlike in turbulent flow calculations, thereare no modeling assumptions as far as the equations are con-cerned. The principal sources of error, therefore, are the dis-cretization of the equations and the boundary conditions. Theformer can be reduced by employing a fine grid, which hasbeen verified by the grid independence tests. The no-slipboundary conditions on the walls require no validation; theimposition of a shear-free interface condition at the top surfacelimits the application to low-rotational speeds so that the in-terface remains flat. Finally, the assumption of axisymmetry isalso a limiting condition, because it is expected that as therotational speed increases, the axisymmetric Taylor vorticesmay become unstable giving rise to a wavy type of Taylorvortices.23

To demonstrate the validity of the simulations, results ob-tained from the present study are compared with theoreticalresults for the onset of Taylor vortices. This case is consideredby a number of researchers. Among these, DiPrima et al24

considered the effect of the ratio of the inner and outer cylinderradii on the stability of Couette flow to axisymmetric andnonaxisymmetric perturbations for the case where the outercylinder is at rest. They found that the ratio of the inner to outerradii �, had a significant influence on the critical Reynolds andTaylor numbers, defined respectively as

Recrit �Rid

�(8)

Tacrit �2�1 � �� Recrit

2

�1 � ��(9)

for the Couette flow to Taylor vortex flow transition to occur.In this study, this theoretical dependence on the radius ratio istaken as a benchmark for the validation of the present calcu-lation methodology. Accordingly, simulations were conductedfor three radius ratios, � namely, 0.632, 0.462 and 0.300, overa range of Reynolds numbers (here, the outer radius is changedwhile keeping the inner cylinder radius constant). Typically,the results are as shown in Figure 2 where the streamfunction

contours are plotted in the annular region for different Reyn-olds numbers, based on inner cylinder diameter ReD for thecase of a radius ratio of 0.462. It can be seen that until the onsetof the instability, no secondary flow is observed in the bulk ofthe annular region. (Since the bottom surface is a wall, thisgives rise to a small amount of recirculation which cannot beassociated with the Taylor vortex phenomenon; this recircula-tion is expected to disappear if the bottom surface too is madea shear-free surface.) Above a certain Reynolds number (orrotational speed for a given geometry), Taylor vortices appearall along the height of the annular region (see discussion on theeffect of the height of the cylinder). This pattern is sustained athigher Reynolds numbers. Thus, for each radius ratio, a fairlydemarcated Reynolds (or Taylor) number exists below which aCouette type of flow without significant secondary flow andabove which Taylor cells are found over the entire domain. Itshould be noted that these flow patterns have been obtained bysolving the steady state governing equations and, hence, theydo not actually mimic the transition. The transitional Taylornumbers obtained from the present CFD simulations are com-pared in Figure 3, with the theoretical results obtained byDiPrima et al24 for the three radius ratios. It can be seen thatexcellent agreement is obtained between the two.

Results from Numerical Simulations

Calculations of the solid dissolution process in a rotatingcylinder have been made for several cases to investigate theeffect of Reynolds number, Schmidt number and radius ratio.The diameter of the inner cylinder (Di) is kept constant at0.03 m in all cases, and three different outer diameters (Do)0.0475, 0.065 and 0.100 m were used to obtain radius ratios (�)of 0.632, 0.462 and 0.300, respectively. The height of thecylinder (L) was kept at 0.12 m in a majority of the cases; butsome cases were repeated varied with twice the height to check

Figure 2. Streamfunction contours plotted in the annularregion for different Reynolds number (ReD) �25, 60, 86, 127, 203, 456, 760 for the case ofradius ratio (�) of 0.462.


that the results were not dependent on the height of the domain.For the mass transfer cases, the base case corresponded to theestimated Schmidt number, Sc, of 877 for a benzoic acid-watersystem. Additional calculations were conducted by varying Scin the range of 100 to 50,000 to determine the Schmidt numbereffect. The Reynolds number, based on the inner cylinderdiameter and its surface speed, varied between 1 and 1,000.The results from these calculations are discussed later.

Typical velocity and mass fraction profiles

Owing to rotation of the inner cylinder, two distinct patternsof flow arise in the annular region in the range of parametersstudied here. The changeover between these can be attributedto the onset of Taylor vortices. On the basis of the nature of thevelocity field, these can be called as “Couette flow (shortenedas Couette flow)” pattern and “Taylor vortex (shortened asTaylor flow) flow” pattern. Typical circumferential velocity

profiles at midheight under these two conditions are shown inFigure 4a, for a radius ratio of 0.462, and a cylinder height toinner diameter of 4. The rotational speed of the inner cylinderis such that the Reynolds number, based on the inner cylinderdiameter is 86 for the Couette flow case and 453 for the Taylorflow case. The velocity profiles are plotted in dimensionlessform in which the circumferential velocity is divided by thesurface speed (�Ri, where Ri is the radius of inner cylinder,and is the angular velocity), and the radial distance from thewall of the inner cylinder is divided by the gap width. It can beseen that under Couette flow conditions, the velocity profilevariation is nearly linear while under Taylor flow conditions, amore boundary layer-like velocity profile is obtained near thecylinder wall. The mass fractions corresponding to these twocases, for a Schmidt number of 877, are shown in Figure 4b (ata time corresponding to the same number of revolutions of theinner cylinder) in dimensionless form, in which the massfraction is divided by the imposed mass fraction at the wall.Once again, a distinct, boundary layer type of variation is foundunder Taylor flow conditions while a nearly linear variationprevails under Couette flow conditions.

Taylor vortices introduce a convective element into thediffusion of the species from the dissolving wall into the bulkof the fluid. However, this convection is highly structured andhelps only to some extent in the overall diffusion and dissolu-tion of the substance. The final process of mixing and disso-lution is still governed by molecular diffusion to the dissolvingfront and is a long process. This is summarized in Figure 5a,wherein the contours of mass fraction of the solute are shownin a single Taylor cell as a function of time for a radius ratio of0.3, Reynolds number of 760 and Schmidt number of 877. Inthe initial period, there is rapid transport of the solute from theinner surface to the outer surface and back towards the innersurface, in a manner consistent with the streamline pattern.However, there is very little solute concentration in the “eye”of the vortex and further mixing of the solute in the core fluid

Figure 3. Transitional Taylor numbers obtained from thepresent CFD simulations compared with thetheoretical results obtained by DiPrima et al24

for the three radius ratios (�).

Figure 4. (a) Typical circumferential velocity profiles at midheight for a radius ratio of 0.462. Reynolds number (ReD)is 86 for the Couette flow case, and 453 for the Taylor flow case. (b) Typical mass fraction profiles atmidheight for a radius ratio of 0.462. Reynolds number (ReD) is 86 for the Couette flow case, and 453 for theTaylor flow case.


is governed by the slow molecular diffusion process. This isillustrated in the gradual shrinking with time of the zero con-centration region at the center of the vortex. The radial varia-tion of the mass fraction along the centerline of the Taylor cellis shown in Figure 5b at different times for this case. Theconvective transport of the solute is effective in increasing thesolute mass fraction near the inner and the outer walls. How-ever, the concentration in the central part of the channel re-mains close to zero. Large mass fraction gradients exists be-tween the central core and the well-mixed layer outside in thepath of the convection currents. From this, it is clear that thereis a relatively thin layer of well-mixed zone, and a large area ofthe unmixed core associated with each cell. Transport of thesolute between these two zones is essentially by moleculardiffusion and is, therefore, very slow.

Parametric Effects of Re, �, L/D and ScEffect of Reynolds number on Taylor vortices

The effect of Reynolds number variation on the overall flowpattern in the annular region can be seen in Figure 2, where thestreamfunction contours are shown at different Reynolds num-bers for a gap width of 0.0175 m. As mentioned earlier, Taylorvortices are formed only after a critical Taylor number. Furtherincrease in the Taylor number leads to some distortion of theTaylor vortices; however, their number remains constant. Acomparison of the radial profiles of the tangential velocitycomponent at midheight is shown in Figure 6a in dimensionlessform. Until the onset of Taylor vortices, all dimensionlessprofiles collapse onto a single curve (Figure 6a shows thevelocity profiles for Re of 25 and 86 falling on the same, nearlyCouette-type profile). As Re increases further, there is a grad-ual deviation from this, and a boundary layer type of variation

is established at a Reynolds number of about 200. Furtherincrease in Re makes the boundary layer-type variation morepronounced. It should be noted that the plots are all at mid-height, and that there is some asymmetry in the Taylor vortexpattern, presumably due to the asymmetric boundary condi-tions at the top and the bottom of the annular region. Thus, themidheight location does not correspond to the same locationrelative to a Taylor vortex, and this adds a bit of uncertainty tothe comparison of the velocity profiles at different ReD. (Thisuncertainty is not in locating the midheight of the cylinder, butin being able to impose the conditions that the midheight of thecylinder always coincides with the midheight of the Taylor cellfor all Reynolds numbers.) However, the transition to a bound-ary layer type velocity profile at high Re is unmistakable. Themass fraction profiles for different Reynolds numbers at aSchmidt number of 877 are shown in Figure 6b. These alsoshow the Couette-type and boundary layer-type variations atlow and high Reynolds numbers, respectively.

Effect of radius ratio

The effect of radius ratio on the dimensionless velocityprofiles is summarized in Figure 7a and Figure 7b. Here thedimensionless tangential velocity is plotted in terms of dimen-sionless radial distance for three radius ratios, namely, 0.632,0.462, 0.300 at a Reynolds number of 76 (Figure 7a), whichcorresponds to the Couette flow condition, and at a Reynoldsnumber of 760 (Figure 7b), which corresponds to the Taylorflow condition. From Figure 7a, we note that, for small gapwidths or high values of �, the radial profile of the circumfer-ential velocity is nearly linear. As � decreases, the velocityprofile becomes nonlinear, though monotonically decreasing.In the Taylor flow condition also (Figure 7b) the radius ratio

Figure 5. (a) Mass fraction contours within a single Taylor vortex cell, and (b) mass fraction profiles at midheight ofthe Taylor vortex cell at different times (s) for a radius ratio of 0.3, ReD of 760 and Sc of 877.


has a significant effect on the tangential velocity profile. It is,therefore, to be expected that the radius ratio will have someparametric influence on the mass-transfer coefficient. Manyexisting correlations1,6,25 for mass-transfer coefficient do notconsider this factor.

Effect of the Schmidt number

The effect of Schmidt number on the mass fraction profilesis investigated by carrying out calculations in the Sc numberrange of 100 to 50,000. The Sc for the reference case ofbenzoic acid-water system is 877. It varies between 600 and1,500 for other typical systems such as salicylic acid-water and2-naphtol-water systems.25 For metals, such as tin, lead andzinc dissolving in liquid metals, such as mercury, the Schmidtnumber is in the range1 of 50 to 100, while for systems such asbenzoic acid-sucrose solution, and benzoic acid-aqueous glyc-erol solution, it can vary in the range8 of 3,000 to 40,000. Theeffect of this large variation of Sc on the mass fraction profileis summarized in Figure 8, where the concentration profiles are

shown for different Sc at a Reynolds number of 76 (Figure 8a),and 760 (Figure 8b). The effect of increasing Sc is to decreasethe rate of diffusion of the wall-imposed mass fraction. Thiseffect predominates at low Re, and the Schmidt number is seento have a large effect on the mass fraction profile. At high Re,the large convective diffusion caused by the Taylor vorticesrestricts the mass fraction variation to close to the walls (Figure8b (1)). However, on a magnified scale (the profiles are re-drawn in Figure 8b (2) over a dimensionless radial distance of0.01), the effect of Sc on the mass fraction profiles is nearly thesame as at low Re. Thus, the effect of Sc on the mass-transfercoefficient scales independently of Reynolds number, as willbe shown later.

Effect of L/D

It is obvious that the height of the cylinder has little effect onthe overall mass-transfer coefficient under Couette flow condi-tions as there is nearly no axial variation of the mass flux.However, the situation is different under Taylor flow condi-

Figure 6. (a) Circumferential velocity profiles at midheight for a radius ratio of 0.462 for different Reynolds number(ReD). (b) Mass fraction profiles at midheight for a radius ratio of 0.462 for different Reynolds number (ReD).

Figure 7. (a) Tangential velocity profiles for three different radius ratios (�) at ReD � 76 (Couette flow condition). (b)Tangential velocity profiles for three radius ratios at ReD � 760 (Taylor flow condition).


tions. Typically, a number of Taylor cells are set up if thecylinder height is long enough. This can be seen in Figure 9a,where the streamfunction contours for a radius ratio of 0.30 areshown at a Reynolds number of 760. It can be seen that anumber of Taylor cells, four in this case, are established andthat the flow pattern is fairly regular. The circulation within oneTaylor cell is shown in Figure 9b in the form of vector plot.The corresponding wall mass flux variation along the height isshown in Figure 9c, for a Schmidt number of 877 for twodifferent heights. (Such strong periodic variation in the localmass-transfer coefficient has also been recorded experimentallyby Kataoka et al9). This shows that except close to the bottomwall, a periodic mass flux variation, consistent with the Taylorcell pattern, is obtained on the inner cylinder surface. Thus, fora long enough height, a repeating pattern is obtained. However,if the cylinder height is less, as in the base case of a height of0.12 m, and an inner cylinder dia. of 0.03 m (both thesedimensions correspond to the experimental conditions of Sh-erwood and Ryan5), it is possible that the L/D ratio has aneffect on the Taylor cell pattern. In order to investigate this,some calculations have been for twice the base case cylinderheight of 0.12 m. The resulting axial mass flux variation iscompared in Figure 8c for a Reynolds number of 760, and a

Schmidt number of 877. It can be seen that the mass fluxvariation for the shorter cylinder case overlaps that for thelonger cylinder case. Calculations with a wider gap show,however, that the predicted pattern may be affected if the gapwidth is increased by a factor of two while keeping the sameheight. These calculations show that the minimum height of thecylinder should be so as to allow a couple of Taylor cells to beresolved. Since the cell height is typically about the same as thegap width (see Figure 2), it can be concluded that the effect ofliquid height would not be significant if L/d 2.

Correlation for Mass-Transfer Coefficient

The calculation of the mass concentration profile enables thedetermination of the mass flux from the surface of the innercylinder as

jA � ��AB

dYA

dr�

r�Ri

(10)

The mass-transfer coefficient kc, can then be evaluated as

Figure 8. (a) Effect of Sc on the mass fraction profile at a ReD � 76 (Couette flow condition). (b) Effect of Sc on themass fraction profile at a ReD � 760 (Taylor flow condition).


kc �jA

��Y* � Yc�(11)

where Y* is the imposed mass fraction at the inner cylinder,and Yc is the mixed cup mean mass fraction defined in Eq. 5.The introduction of Yc, the cup mean value of the mass frac-tion, is done for two reasons. First, because of the finite size ofthe annular region, the concentration at large r is not zero andincreases with time. Second, Yc is the desired parameter froma practical point of view as one can use it to determine howlong the rotating cylinder has to be kept immersed in thesolution for a specified amount of dissolution.

The presence of Taylor vortices makes the mass flux vary

axially, typically as shown in Figure 9c. An axially averagedvalue of the mass flux is, therefore, taken to compute themass-transfer coefficient. This is then converted into a Sher-wood number, based on the hydraulic diameter as in Eq. 7. Itshould be noted that the gap width d here is one half of thehydraulic diameter for the same configuration. Thus, otherdefinitions of Sherwood number, for example, based on theinner cylinder diameter or gap width are possible. The presentdefinition is justified by the fact the Taylor number and theReynolds number describing the flow are better with hydraulicdiameter than on the diameter.

Typical variation of the computed Sherwood number withthe Reynolds number is plotted in Figure 10 for the three

Figure 9. (a) Streamfunction contours for � � 0.30 at ReD � 760. (b) Circulation within one Taylor cell in the form ofvector plot. (c) Axial wall mass flux variation for two different heights at Sc � 877 and ReD � 760.

Figure 10. Sherwood number (ShDh) variation (a) with Reynolds number (ReDh) for three radius ratios (�), and (b) withSchmidt number at ReD � 76 (Couette flow region) and 760 (Taylor flow region).


different radius ratios under consideration. The three curvesmore or less merge in the Taylor flow region, but differ signif-icantly in the Couette flow region. Closer examination of theresults shows that the variation of Sh with Re is typically linearin the Couette flow region, and that it exhibits a square-rootvariation in the Taylor flow region, that is, Sh � Re0.5. Whilethe variation in the Couette flow region has not been reportedhitherto, a number of authors have reported a near-square rootvariation with Reynolds number, as mentioned earlier. Thereason for this different behavior lies in the flow field. In theCouette flow region, the Couette-flow type of flow field givesrise to a linear variation. In the Taylor flow region, the domi-nant feature of the flow is the boundary layer type variationnear the wall. Since the boundary layer thickness in laminarflow typically varies as Re�0.5, the Sherwood number alsovaries as Re1/ 2. The effect of Schmidt number on the Sher-wood number is shown in Figure 10c at a Reynolds number of76 and 760, corresponding, respectively, to the Couette flowand Taylor flow regions. It is seen that over the entire of Sc ofbetween 100 and 50,000, Sh varies typically as Sc1/3 in boththe regions.

Finally, the limiting value of the Sherwood number at verylow Reynolds numbers is of interest. For mass transfer from astationary sphere, it can be shown that the Sherwood numberunder stagnant conditions26 is 2. A similar analysis can becarried out for a stationary cylinder. The mass flux through acylindrical plane at a radial distance of r under steady condi-tions, can be written as

�1

r

�

�r�rNAR� � 0 (12)

where NAR is the mass flux per unit area. Integrating Eq. 12, weget the radial variation of concentration as

CA � C1ln r � C2 (13)

Applying the boundary conditions that at r � Ri, CA � CAW,and that at r � Ro, CA � 0, the two constants can be evaluatedas

C1 �CAW

ln�Ri/Ro�, C2 � � CAW

ln Ro

ln�Ri/Ro�(14)

The mass flux at the cylinder surface can now be evaluated as

NAR�r�Ri � �DAB

�CA

�r�

r�Ri

(15)

The mass-transfer coefficient and the Sherwood number, basedon inner cylinder diameter can now be calculated as

ShD �2

ln�Ro/Ri�(16)

It can be shown for the same boundary conditions that theSherwood number based on the hydraulic diameter

ShDh �kc�2d�

DAB(17)

is for the limiting case is given by

ShDh �2

ln�Ro/Ri��2d

Di� (18)

The computed variation of ShDh with ReDh for very lowrotational speeds is illustrated in Figure 11a for the base case.Here, the speed is varied down to a Reynolds number based onthe inner cylinder dia. of 0.5. It is indeed seen that as theReynolds number is reduced, a limiting value of ShDh isreached. The numerically obtained limiting values are com-

Figure 11. (a) Computed variation of ShDh with ReDh for very low rotational speeds for � � 0.3. (b) Comparison ofnumerically obtained Sherwood number limiting values with that of CFD.


pared in Figure 11b, with that obtained from Eq. 18 earlier fordifferent radius ratios. As expected, the Sh (based on thehydraulic diameter) decreases as the radius ratio increases.However, there is a factor of three difference between thepredictions and that given by the theoretical expression. Part ofthis may be attributed to the discrepancy in the definition of theSherwood number. In the theory, it is based on the concentra-tion difference between the wall and the bulk value, which istaken to be zero. In the CFD simulations, the bulk concentra-tion is based on the cup mean value and takes account ofvelocity and concentration profiles. The corresponding limitingSh values are shown in Figure 11b with filled squares. Thelimiting Sh values obtained by taking the mass fraction value atthe outer radius as the bulk concentration are shown here asopen squares. It can be seen that these show a much closeragreement with the theoretical value for a stationary cylinder.The discrepancy that lies between the two, even with a con-sistent definition of the Sherwood number, may be attributed tothe fact that the limiting case considered here is that of arotating cylinder instead of a translating cylinder for which Eq.18 is applicable. From a practical point of view, a Sherwoodnumber definition based on the cup mean value is more useful,and the limiting value is taken as

Shlim �12

ln�Ro/Ri�

d

Di(19)

Using the numerical “data” obtained from the CFD simula-tions, the following correlations for the Sherwood number havebeen developed for the Couette flow and the Taylor flow re-gimes

ShDh �12

ln�Ro/Ri�

d

Di� 0.0048ReDhSc1/3�Di

Do��1.25

(20)

ShDh �12

ln�Ro/Ri�

d

Di� 0.46ReDh

1/ 2Sc1/3�Di

Do��0.1

(21)

Using the theoretical analysis of DiPrima et al.,24 the followingcorrelation is developed to determine the critical Taylor num-ber at which the transition to Taylor vortex regime occurs

Tacrit �1555

�1.05 (22)

If the Taylor number for a given situation is less than Tacrit,then the Couette flow correlation, that is, Eq. 20, is used todetermine Sherwood number. Otherwise, the Taylor flow cor-relation, that is, Eq. 21, is used. The range of parameters forthese correlations to be valid is

0.3 � 0.7

1 ReD 2000

1 Sc 50000

A comparison between the Sherwood number correlation(Eq. 20 and Eq. 21), and the CFD data on which it is based isgiven in Figure 12a (Couette flow region) and Figure 12b(Taylor flow region) which shows, not surprisingly, very goodagreement (within ten percent error) between the two over therange of parameters. The predictions of the correlation with theexperimental data of Holman and Ashar8 is shown in Figures13a and 13b for benzoic acid-water and benzoic acid-glycerolsystems, respectively. It should be noted that Holman andAshar8 use a rather high value of Sc of 10,000 for the firstsystem; this has been adjusted to a more representative value of877 (which agrees with the estimates of Bennett and Lewis,1

Marangosis and Johnson6 and Tripathi et al25 among others) inthe computation of the experimental Sh in Figure 13a. Thereported Sc of 43,200 for the second system is in agreementwith data reported for aqueous glycerol solutions of differentconcentrations, and the same value of Sc is used in the com-parison in Figure 13b. Calculations using Eq. 22 show that

Figure 12. Comparison between the Sherwood number correlation and CFD data for (a) Couette flow region, and (b)Taylor flow region.


these data lie in the Taylor flow regime. Although there is somescatter, there is good qualitative and quantitative agreementbetween the correlation and the data for both the systems.

Conclusions

CFD simulations are done for the case of a cylinder rotatinginside a stationary cylindrical outer vessel filled with liquid.The parameters covered a range of Re, Sc, L/D ratio andradius ratio. The results confirm that the circumferential veloc-ity profile is a strong function of Re and varies from a nearlyCouette-type flow at very low Reynolds number to a boundarylayer like profile at high Reynolds number. The onset of Taylorvortices has a strong effect on the flow field and the masstransfer mode. Results show that Sherwood number has a lineardependence on the Reynolds number in the Couette-flow re-gime, and has linear dependence after the onset of Taylorvortices. A correlation based on two regions in the flow fieldfor the Sherwood number has been developed in the parameterrange 0.3 � � � 0.7, 1 � ReD � 2000 and 1 � Sc �50,000. Good agreement was found between the experimentaland proposed correlation.

AcknowledgmentThe calculations reported here have been done using the computational

facilities of the CFD Centre, IIT-Madras, India.

Notation

CA � bulk mean concentration at time tCAW � solubility concentration of the solid in liquid

d � gap widthD � cylinder dia.

DAB � kinematic diffusivityi, o � inner and outer cylinder

jA � average-mass fluxL � cylinder height

kc � mass-transfer coefficientNAR � mass flux

r � radial distance from the wall of the inner cylinderR � cylinder radius, m

Recrit � critical Reynolds number, Rid/�ReD � Reynolds number based on inner cylinder dia., RiDi/�

ReDh � Reynolds number based on hydraulic dia., Ri(2d)/�S � surface area of the solid

Sc � Schmidt number, �/DAB

ShD � Sherwood number based on inner cylinder diameter, kcDi/DAB

ShDh � Sherwood number based on hydraulic diameter, kc(2d)/DAB

Shlim � Sherwood number limiting, 12(d/Di)/[ln(Ro/Ri)]t � time

Tacrit � critical Taylor number, 2(1 � �) Re2/(1 � �)u, v, w � velocity components

V � volume of the liquidWtip � surface speed of the inner cylinder, Ri

YA � mass fractionYc � cup mean mass fractionY* � imposed wall mass fraction

Greek letters

� angular velocity (in radians per second)� � density� � kinematic viscosity

�AB � molecular diffusivity� � radius ratio, Ri/Ro

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8. Holman KL, Ashar ST. Mass transfer in concentric rotating cylinderswith surface chemical reaction in the presence of taylor vortexes.Chem Eng Sci. 1971;26:1817–1831.

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10. Legrand J, Dumargue P, Coeuret F. Overall mass transfer to the

Figure 13. Predictions of the correlation with the experimental data of Holman and Ashar8 for (a) benzoic acid—water,and (b) benzoic acid-glycerol systems.


rotating inner electrode of a concentric cylindrical reactor with axialflow. Electrochimica Acta. 1980;25:669–673.

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Manuscript received Oct. 26, 2004, and revision received Mar. 9, 2005.


Study on Shrinkage Characteristics of HeatedFalling Liquid FilmsFeng Zhang, Zhibing Zhang, and Jiao Geng

Dept. of Chemical Engineering, Nanjing University, Nanjing, 210093, China


A liquid film flowing down along a heated solid surface may be substantially influencedby the Marangoni effect, which arises from the existence of a surface tension gradient thatis induced by the variations of surface temperature in the transverse direction of the film,causing appreciable contraction of the film. In this work, the governing equations forlaminar falling liquid films over a heated plate with constant surface temperature weresolved and the surface temperature profile of the film was obtained. By considering thesurface tension gradient and shrinkage characteristics in the brim of the heated film, ashrinkage model that gives the film interfacial area was derived based on the experimentalinfrared images. A comparison of the model with experimental data shows that the modelcan satisfactorily describe the flow characteristics of falling heated films. The model isexpected to be useful for the design and operation of falling film equipments. © 2005American Institute of Chemical Engineers AIChE J, 51: 2899–2907, 2005Keywords: heat transfer, Marangoni effect, falling films, shrinkage model

Introduction

Falling films are usually used as the heat- and mass-transfermedia in industrial equipment such as vertical condensers, filmevaporators, and absorption towers.1 In the processes of heatedfalling films, heat transfer occurs and interacts with film flow,leading to the variations of the liquid properties, especiallyviscosity and surface tension. Several researchers2,3 indicatedthat the variations of liquid viscosity (thermoviscosity effects)would affect the film flow, whereas the nonuniformity of thesurface tension at the liquid–gas interface, usually referred toas the Marangoni effect, could strongly influence the flowdynamics and thus the heat transfer.4

In recent years, interfacial instabilities of heated films suchas the formation of surface waves, breaking of a stream intorivulets, and evaporation/termination of the liquid layer at acontact line, have been observed and theoretically analyzed.5-10

Joo et al. studied the interfacial instability mechanism of heatedfalling films and derived a long-wave evolution equation to

describe both the surface wave and the thermocapillary insta-bility that were caused by either the gravity or the surfacetension gradient. They concluded that the coupled temporalinstabilities could create surface deformation and lead to anarray of rivulets aligned with flow at moderate flow and heat-transfer rates.10 In 1997, by completely solving the governingequations for the heated falling film, they further performed thethree-dimensional simulation of the instabilities and rivuletformation. It was found that the spontaneous rupture and riv-ulet formation occurred even in a thin film under the combinedinfluences of thermocapillary and surface-wave instabilities.5

The Kabov research group4,11-13 successfully performed exper-iments of heated falling films and developed a theoreticalmodel to describe the influence of the Marangoni effect thatwas induced by the temperature and concentration gradientsalong the liquid–vapor interface. Very useful correlations forheat-transfer coefficient and film breakdown (including surfacetension effect) were obtained, and the instabilities of a thinfalling film, resulting from the Marangoni effect, were indi-cated to have a pattern of “regular horseshoe-like structures.”This pattern of instable flow caused a decrease of the heat-transfer coefficient with an increase of the Reynolds number.4

Skotheim et al.6 also worked on the instability of the heated

Correspondence concerning this article should be addressed to Z. Zhang [email protected].



falling films and showed that the surface tension gradientsinduced by the temperature difference, curvature pressure,gravity, and heat conduction could all influence the shape andstability of the steady flow profile. A model based on thelong-wave evolution theory was proposed to describe the sta-tionary curvature of the film and its stability, which were ingood agreement with experimental data.

There are several studies on the dry patch forming in liquidfilms flowing over a heated surface. Some researchers14-16

reported that the Marangoni effect induced by the surfacetension gradient could substantially affect the dry patch forma-tion. Zuber and Staub14 observed the effects of the thermocap-illary, the vapor thrust forces, as well as the forces arising fromflow and surface wetting, on the dry patch. They thus presenteda reasonable analysis for the prediction of the conditions thatwould permit a dry patch to form and remain stationary in aliquid film flowing over a heated surface. Shi and Zhang16

analyzed the breakage of subcooled liquid film flowing downonto a heated plate with a simplified transverse deformationmodel and a dry patch model. The length of deformation areaand critical film thickness were expressed as functions ofsurface tension coefficient and heat flux in their models, vali-dating their analysis.

As a liquid film flows down over a heated surface, thevariation of surface temperature occurs in both the streamwiseand the transverse directions. The streamwise case arises fromheat transfer and surface wave, whereas the transverse casearises from different flow rates in the brim of the film. Suchtemperature variations generate surface tension gradient andthermocapillary phenomena that are effective on the film flow.However, most of the work mentioned above took into accountvariations of surface temperature only in the streamwise direc-tion. The variation in the transverse direction was usuallyignored. In fact, the temperature gradient in the transversedirection is usually much higher than that in the streamwisedirection (Figure 1) and might be more effective to heat trans-

fer and film flow because the gravity-driven flow dominating inthe streamwise direction is absent in the lateral direction.5 Thisneglect may cause the correlative theory to stray largely fromthe experimental observation. In 2001, using water as theworking fluid, our research group17 performed experiments forthe heat transfer of falling films (under no-evaporation condi-tion) on a heated/cooled smooth stainless steel plate. It wasfound that the heated film was obviously contracted to form aninverse trapezoid, whereas the cooled film was expanded toform a right trapezoid. The width of the falling film decreasedwith increasing heating temperature, and vice versa. Actually,the shrinkage of heated film or expansion of cooled film wasconcluded to arise from the surface tension gradient in thetransverse direction. For a heated film, temperature in the brimwas higher than that in the center because of the slower flowand the sufficient heat transfer in the brim. This resulted inmuch smaller surface tension in the film margin than that in thecenter. The contraction of the film occurred under the effect ofsuch surface tension gradient from brim to center of the film. Inthe case of the cooled film, the higher surface tension in themargin than that in the center leads to expansion of the film,which is completely opposite to the results obtained for theheated film. An infrared image is shown in Figure 1 to illustratethe surface temperature field of the heated liquid film that hasinitial uniform temperature. Different colors in the thermogra-phy represent different temperature magnitudes: the darker thecolor, the lower the temperature. The horizontal black solid lineindicates the upper position of heater edge. It can be clearlyseen that the heated film below the black solid line could bedivided into two sections: the uniform central part, with analmost constant temperature, and the rim parts between dashedlines and boundaries of the film. The dashed lines are identifiedas temperature sidelines for the heated film. The transversetemperature curve of the heated film, located at about 0.05 mdown from the upper edge of the heater, was recorded andshown in Figure 2 to have much higher temperature at theedges (the two peaks in the figure). Therefore, the evidentsurface tension gradient (shown by the arrow in Figure 1)attributed to the temperature gradient in the rim part apparently

Figure 1. Shrinkage of the heated film.

Figure 2. Transverse surface temperature of the heatedfilm.


contracted the film, resulting in a substantial reduction of theinterfacial transfer area.

Because of the absence of fundamental information concern-ing the hydrodynamics of the liquid film, mass- and heat-transfer analyses for industrial falling film equipment are stillbased on empirical methods in general. In a wetting wallcolumn or a packed column the partial pressure driving force isdetermined by the conditions in the gas phase and the equilib-rium concentrations of the solutes in the solvent. The remain-ing factors affecting the mass-transfer rate are the mass-transfercoefficient and the interfacial transfer area. The informationthat would permit separation of those two factors was notavailable, leading to the use of a combined coefficient, K�,where K is the mass-transfer coefficient and � is the mass-transfer area per unit volume. Because K and � are bothindependent functions of the liquid conditions, use of a com-bined coefficient makes a rational analysis difficult.18 As forheat-transfer equipment, the heat-transfer rate is determined bythe interfacial area and the temperature difference betweenworking fluid and heater. The interfacial area, usually regardedas the mechanical area of the heater, is overestimated becausethe heated liquid film could not completely cover the solidsurface. The breakage of the liquid film and the dry patchesmay be frequently encountered in various common industrialheat- and mass-transfer equipment.

Therefore, for both heat- and mass-transfer processes, anunderstanding of the liquid film characteristics, especially theinterfacial transfer area, is quite important. Based on the ex-perimental infrared images done in this study, a governingequation for a heated film flowing down a vertical plate ofconstant wall temperature was presented to calculate the tem-perature profile of the falling film. Then, a shrinkage modelwill be further developed by taking into account the effect ofthe transverse temperature gradient on the film flow and Ma-rangoni flow and be validated by designing and carrying out theexperiments of heated falling films.

TheoreticalModel of heated liquid film flow

As shown in Figure 3, the surface of a thin liquid filmflowing down a vertical plate under the gravity emerges with

small waves. Compared with the average film thickness �, theheights of these waves are negligibly small, so that the filmsurface can be regarded as smooth, as shown in Figure 4.Saouli19 theoretically investigated heat transfer of a laminarfalling film along an inclined plated with constant heat flux tosuccessfully obtain the velocity and temperature profiles. Inthis work, a two-dimensional model for a fully developedlaminar falling film over a vertical plate of constant walltemperature would be presented to obtain the film temperatureprofile in the central part.

Because the inertia term in the momentum equation is neg-ligibly small compared to the body force term, the momentumequation in the direction x of gravity is simplified as

��2ux

� y2 � �g � 0 (1)

where ux is the velocity component in the x-direction; �, �, andg represent the dynamic viscosity, liquid density, and gravita-tional acceleration, respectively. Because the temperature dif-ferences in both the x- and z-directions are very small com-pared to the temperature difference in the y-direction of thefilm, and flow rate in the z-direction is also negligible com-pared with ux, heat transfers by convection in the z-directionand by conduction in the x- and z-directions are all neglected.Then, the governing energy equation can be expressed usingthe following equation

ux

�T� x, y�

� x� a

�2T

� y2 (2)

where a � �/�cp represents for the thermal diffusivity.The boundary conditions of no slip at the wall and no stress

in the film surface are used for the momentum equation

y � 0 ux � 0 (3a)

y � ��ux

� y� 0 (3b)

The energy boundary conditions are as follows:

Figure 3. Heated falling film.

Figure 4. Physical model and coordinate system.


Initial Temperature of the Film

T�0, y� � T0 (4a)

Constant Temperature of the Plate Surface

T� x, 0� � Tw (4b)

Adiabatic Film Surface

�T� x, ��

� y� 0 (4c)

Integrating Eq. 1 leads to the expression of the velocityprofile of the film

ux � um�1 �1 y

��2� (5)

where

um �1

2 ��g

� ��2 (6)

The liquid mass flow rate per unit perimeter is

Q � �0

�

�uxd y (7)

Substituting Eq. 5 in Eq. 7 generates the expression of theliquid film thickness

� � �3�Q

�2g � 1/3

(8)

Thus, by combining Eqs. 2 and 5, the energy equation can berewritten as follows

um�1 �1 y

��2� �T� x, y�

� x� a

�2T

� y2 (9)

The following dimensionless variables were defined

Y �y

�X �

ax

um�2 U�Y� �u� y�

um

��X, Y� �T� x, y� T0

Tw T0(10)

By substituting Eq. 10 in Eqs. 9 and 4, neglecting the entranceeffects, separating the variables, and integrating Eq. 9, we havethe following expression

��X, Y� �12

5X �

6

5�1 Y�2

1

5�1 Y�4 (11)

Equation 11 is the working expression for the temperatureprofile of the liquid film.

Model for the shrinkage of heated film

Consideration of the surface tension gradient in the rim partand its effect on the heated falling film, suggests a model, asillustrated in Figure 5, for the expression of the film shrinkagein the initial section. The film is postulated to be symmetric tothe central line (x-axis) of the film so that it is sufficient toanalyze only one side of the film. In Figure 5, the z-direction isnormal to the x–y plane as shown in Figure 4. For distinguish-ing the two parts of the film flow, a dashed line is drawnbetween the uniform central part and the rim part. The film inthe uniform central part is assumed to be of constant thickness� and constant flow rate Q, whereas the liquid within the rimpart is regarded as flowing along the channel regulated by thedashed line and the contact line AB. Moreover, D stands for thewidth of the rim part and (x � wcos �)/D is the surfacetension gradient in the rim. x and w represent the surfacetension at the temperature of T(x, �) and Tw, respectively. T(x,�) represents the surface temperature of the liquid film in theuniform central part, and Tw the temperature of the platesurface. By neglecting the effect of gravity and inertia forcethat are negligibly small in the z-direction, the momentumequation in the z-direction of the rim becomes

��2uz

� y2 � 0 (12)

The boundary conditions are as follows:At the Wall

uz � 0 (12a)

At the Film Surface

��uz

� y�

�

� zcos � (12b)

where uz is the velocity component of the liquid flow in thez-direction and � is the angle of the tangent to the contact linewith the x-direction.

By integrating Eq. 12 across the film using the above bound-

Figure 5. Shrinkage of a heated film.


ary conditions and incorporating the Marangoni flow rate in theliquid layer as given by Kabov,12 the velocity in the z-directionis derived to be

�uz

��

�x wcos ��

4Dcos � (13)

where � is the contact angle along the contact line.According to the illustration in Figure 5 we have uz/um � tg�

and cos � � um/�um2 � uz

2. Then Eq. 13 becomes

uz ��x wcos ��

4�D

um

�um2 � uz

2 (14)

Defining

C ��

4�D�x wcos �� (15)

and rearranging Eq. 14, lead to the following expression

uz ��2

2um��1 �

4C2

um2 � 1/ 2

1� 1/ 2

(16)

As the liquid film flowing down x distance on the heatedplate, the shrinkage width �Z on one side of the film is

�Z � �0

x/um

uzdt (17)

where t represents the time and x/um is the time duration for thefilm flowing over x distance.

Substituting Eq. 16 into Eq. 17 produces the followingexpression

�Z ��2

2 �0

x ��1 �4C2

um2 � 1/ 2

1� 1/ 2

dx (18)

It is assumed that the surface tension decreases linearlywith the temperature: � 0 � (T � T0); then Eq. 15becomes

C ��

4�D�x wcos ��

�

4�Dw�1 cos ��

� �T Tw� (19)

where T � T(x, �). When y � �, Y � 1, Eq. 11 is reduced to

��X, 1� �12

5X (20)

By substituting Eq. 10 in Eq. 19, the following equation isobtained

T� x, �� T0

T� T0�

12

5

ax

um�2 (21)

The temperature of the surface without stress can thus be givenas

T� x, �� T0 �12

5�T

ax

um�2 (22)

where �T � T� � T0.Substituting Eq. 22 in Eq. 19 gives

C ��

4�D �w�1 cos �� T�12

5

ax

um�2 1�� A0 � Kx

(23)

where

A0 ��

4�D�0 wcos �� (24)

K �� T

4�D

12

5

a

um�2 �3 �Ta

5�Dum�(25)

By combining Eq. 23 with Eq. 18 and then integrating, theshrinkage width of one side of the film can be finally expressedas

�Z ��2

6

um

K�Sx

3/ 2 S03/ 2� 3�Sx

1/ 2 S01/ 2�

x � 0, xe (26)

where

Sx � ��2 A0 � 2Kx

um� 2

� 1� 1/ 2

� 1

S0 � ��2 A0

um� 2

� 1� 1/ 2

� 1

A0

um�

0

�um

�

4D �1 w

0cos ��

K

um�

3

5

1

D

Ma0

Pe2

x � xe when T�x, �� Tw

where xe represents the length of the contracted film; �um/0 isdenoted as a dimensionless parameter called the Capillarynumber; Ma0 � �T�/�a stands for Marangoni number, whichis a nondimensional measure of the surface tension gradient;and Pe � um�/� is denoted as the Peclet number. Equation 26provides a new way to numerically calculate the shrinkage


width of a liquid film flowing down a heated vertical plate. Itimplies that the shrinkage width is affected by the flow rate, thephysical properties of liquid, the temperature difference of (Tw

� T0), and the initial width of the liquid film (contained in thethree nondimensional parameters �um/0, Ma0, and Pe). D, thewidth of the rim part, is also a very important parameter. Thetransverse temperature gradient and surface tension gradientare enhanced as D decreases, resulting in dramatic film con-traction.

ExperimentalExperimental setup

To determine the unknown variable D and to obtain theshrinkage width of the heated film, an experimental setup asshown in Figure 6 was designed. The test section [900 �300 � 6 mm (length � width � thickness)] of the stainlesssteel plate was carefully polished. The experiment was ex-pected to allow the liquid film to flow down at a scheduled flowrate over the vertical plate, being indirectly heated by hot waterat a preset temperature. The surface temperature of the platecould be maintained by the hot water at a given temperature.The cool water source, used as the experiment media, was alsomaintained by a storage tank at a constant temperature. Dis-tilled water in the storage tank was pumped to an upper tank

and then flowed down to the test section at a scheduled flowrate. The flow rate ranging from 0.03 to 0.36 kg m�1 s�1 wasmeasured with a rotameter. The liquid from the upper tankentered a fluid distributor installed in the upper position of thetest section. The liquid exit gap could be adjusted manually tocreate liquid films of different thicknesses according to theneeds of the experiments. The liquid flowing down the platewas collected by a vessel and recycled to the storage tank. FivePt-100 thermal resistances were also installed to measure thetemperature values of the film inlet and outlet, storage tank,upper tank, and the air near the film, respectively. The uncer-tainty of these temperature values was within �0.1°C. Thesurface temperature and the shape of the film were determinedby a highly sensitive infrared camera, Therma CAM ™SC3000 (FLIR Systems, Inc., Portland, OR). Because the filmedge was very thin, temperature at the edge of the film wasregarded as the surface temperature of the wall, Tw.

The initial temperature of the falling water film was set andmaintained at several constant values during the experiments(T0 � 20–35°C). The temperature of plate surface Tw wascontrolled as the temperature of hot water ranging from 30 to70°C. The physical characteristics of working fluid and basicexperiment parameters are given in Table 1.

Characteristics of heated falling liquid films

The experiment shows that the flow of the heated fallingliquid film is mainly affected by the temperature difference (Th

� T0) and the film flow rate. As (Th � T0) increases, the liquidfilm apparently contracts and becomes very narrow. The reasonis that the surface tension gradient in the transverse direction ofthe film is enhanced. The effect of the flow rate on the shrink-age of the heated film flow is quite complex. On one hand, afast film flow rate is beneficial to the lateral expansion of thefilm. On the other hand, both film temperature and the rimwidth D decrease with an increase of flow rate, and the trans-verse surface temperature gradient correspondingly increasesin the rim of the film. Therefore, the surface tension gradient isalso increased, thus apparently causing the film to contract. Forexample, as shown in Figure 7, the heated film at a flow rate0.03 kg m�1 s�1 is quite narrow, but then increases appreciablywith liquid flow rate, increasing from 0.03 to 0.12 kg m�1 s�1.However, when the flow rate is 0.12 kg m�1 s�1, the filmwidth starts to increase slowly. Interestingly, the quick liquidcontraction is found only near the entrance, but the width tendsto keep constant to the end after the liquid flows beyond acertain distance. This phenomenon may imply that the effect ofviscous force and the lateral expansion effect of the liquid floware finally balanced with Marangoni effect in the transversedirection of the film.

Figure 6. Experimental system.Legend: 1: test section; 2: liquid collection vessel; 3: liquidstorage tank; 4: pump; 5: upper tank; 6: rotameter; 7: fluiddistributor; 8: liquid exit gap; 9: infrared camera; 10: com-puter.

Table 1. Physical Characteristics of Working Fluid and Operation Parameters

Physical Characteristics Operation Parameters

Working fluid: distilled water Flow rate, Q: 0.03–0.36 kg m�1 s�1

Initial temperature, T0: 20–30°C Temperature of hot water, Th: 30–70°CDensity, �: 995.3–998.5 kg/m3 Reynolds number, Re � 4Q/�: 137–2024Viscosity, �: 7.16–9.90 � 10�4 Pa � s Prandtl number, Pr � CP�/�: 4.79–6.91Surface tension, : 70.6–72.9 dyn/cmHeat conductivity, �: 0.5997–0.6241 W m�1 k�1


Results and DiscussionCalculation of the rim width

As mentioned above, D represents the width of the rim partand (x � wcos �)/D stands for the surface tension gradient inthe rim. With a decrease of D, (x � wcos �)/D consequentlyincreases and the Marangoni effect in the transverse directionof film becomes intensive. According to Figure 2, the rim partof the heated film is not flat but protuberant in the cross section.This bulgy structure is considered to result from the combina-tion of the central film flow and the shrinkage of the film.Generally, the cross section of the rim part can be assumed to

be semicircular, as illustrated in Figure 8, where � in the figurestands for the thickness of the uniform central film. �p repre-sents the peak of bulge and � is the contact angle of the liquidfilm on the plate. By using the geometrical analysis, the widthof the rim part can be expressed as a function of �, �, and �p

D �� sin �

�1 cos �� 2�

�1 cos �� 1� �� 1�2 (27)

where

�p � �� 1 (28)

where �, �, and the factor � are, respectively, functions ofphysical properties of the liquid, the temperature differencebetween Tw and T(x, �), and film flow rate. It is obvious that Dincreases with increasing film thickness �. � is given by Eq. 8when the temperature is set to the average value of the inlet andoutlet of the film. As an example for computing thickness ofheated films at the temperature of T0 � 20°C and Th � 70°C,variations of D with � and � at a flow rate of 0.24 kg m�1 s�1

are illustrated in Figure 9. As can be seen, D decreases notice-ably with an increase of �. In fact, the wettability of the liquidon the solid is highly related to the contact angle �: the largerthe contact angle, the less the wettability of the solid, leadingto the smaller value of D. In addition, an increase of D with thefactor � can be found in Figure 9, which is clearly understand-

Figure 8. Cross section view of the rim part.B, boundary of the film; P, peak of the rim; T, transitionalpoint from the rim to central part of the film.

Figure 9. Variations of D with � and �.

Figure 10. Variations of �Z vs. x.

Figure 7. Shapes of heated films at different flow rates.T0 � 20°C, Tw � 70°C, x*� �x. Units are in kg m�1 s�1: (1)0.03; (2) 0.06; (3) 0.09; (4) 0.12; (5) 0.15; (6) 0.18; (7) 0.24;(8) 0.30; (9) 0.36.


able because of the semicircular structure of the cross sectionof the rim in Figure 8. Moreover, D varies slightly when � 50° (Figure 9). � � 50° can be regarded as the upper limit ofthe contact angle of the liquid on the plate, which is consistentwith the fact that the receding contact angle of the distilledwater on stainless steel flat plate ranges from 12 to 46°.20 Inaddition, it should be noted that the width of the rim Dincreases along the downstream and finally reaches a maximum

(Figure 1). Therefore, in the actual calculation of the shrinkagemodel, the average D over the contracted film was used insteadof the D at a certain distance of x.

Results of calculation

The shrinkage width �Z of the heated film (T0 � 20°C, Th �70°C) was calculated from Eq. 26. As shown in Figure 10, �Zincreases with the increase of the distance x of film for all theliquid flow rates. �Z also increases with decreasing liquid flowrate at the same x. Such phenomena are in accordance with thevariation of the film boundary vs. flow rate, as shown in Figure7. Additionally, the value of �Z at two flow rates of 0.06 and0.09 kg m�1 s�1 differs substantially from values at other flowrates, indicating the insufficient coverage of the film for theplate.

If Z0 is assumed to be the initial width of the film, the widthZ at distance x can be calculated as

Z � Z0 2�Z (29)

As shown in Figure 11, Z decreases with decreasing flow rateand is dramatically reduced as the flow rate is lower than acertain value.

Figure 11. Variations of Z with x and liquid flow rate.

Figure 12. Comparisons of experiments with calculations.T0 � 20°C, Th � 70°C.


Comparison of the model with the experiments

Comparison of the calculated results by the shrinkage modelwith the experimental data is made and shown in Figure 12.The symbols represent the measured film boundaries, whereasthe single lines are results calculated from Eq. 26. The calcu-lations were in good agreement with the experiments at higherflow rates of 0.24 and 0.18 kg m�1 s�1, with deviations of 3.8and �3.3%, respectively. With decreasing liquid flow rate, thedeviations increase to �5.6 and �10.9% at flow rates of 0.09and 0.06 kg m�1 s�1, which are still acceptable from anindustrial perspective. This implies that the model can accu-rately describe the shrinkage of heated falling films and thusaccurately present the interfacial area. In addition, the modelcan be extended to a more realistic level by including theeffects of evaporation at the film edge, the effect of temperatureon viscosity variation, and other dynamic parameters. This willbe the focus of future work.

Conclusions

The characteristics of heated falling films over a verticalstainless steel plate have been investigated both experimentallyand theoretically, and the following conclusions were made:

(1) From infrared thermographic experiments of heated filmflowing down a vertical stainless steel plate, it is shown that theliquid film is heated and contracted from the beginning stage ofthe entrance to form the shape of an inverse trapezoid. The filmshrinkage and the noticeable transfer area reduction are attrib-uted to the surface tension gradient that results from the vari-ations of temperature in the transverse direction of the film.This indicates that the Marangoni effect arising from the exis-tence of surface tension gradient seriously affects the fallingfilm process.

(2) The governing equations for a laminar falling liquid filmover a vertical heated plate of constant surface temperaturewere solved and the temperature distribution was obtained forthe exact expression of Marangoni effect in the heated film.

(3) A model describing the characteristics for the shrinkageof heated falling film was established with consideration of thefilm flow over a vertical plate and the surface tension gradienteffect in the transverse direction of the film. Comparison withthe experiments demonstrated that the model could accuratelydescribe the shrinkage characteristics of the process.

AcknowledgmentsThe authors are grateful for the financial support from National 985

Project of China and the National Natural Science Foundation of China(No. 20576050). We also thank Professor Youting Wu for valuable help.

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flowing down an inclined plane with accompanying heat transfer andinterfacial shear. AIChE J. 1978;24:803-810.

2. Wilson SK, Duffy BR. On the gravity-driven draining of a rivulet offluid with temperature-dependent viscosity down a uniformly heated orcooled substrate. J Eng Math. 2002;42:359-372.

3. Goussis DA, Kelly RE. Effects of viscosity variation on the stability ofa liquid film flow down heated or cooled inclined surfaces: Finitewavelength analysis. Phys Fluids. 1987;30:974-982.

4. Kabov OA, Scheid B, Sharina IA, Legros JC. Heat transfer and rivuletstructures formation in a falling thin liquid film locally heated. Int JTherm Sci. 2002;427:664-672.

5. Ramaswamy B, Krishnamoorthy S, Joo SW. Three-dimensional sim-ulation of instabilities and rivulet formation in heated falling films.J Comput Phys. 1997;131:70-88.

6. Skotheim JM, Thiele U, Scheid B. On the instability of a falling filmdue to localized heating. J Fluid Mech. 2003;475:1-19.

7. Schatz FM, Neitzel PG. Experiments on thermocapillary instabilities.Annu Rev Fluid Mech. 2001;33:93-127.

8. Wang KH, Ludviksson V, Lightfoot EN. Hydrodynamic stability ofMarangoni films. AIChE J. 1971;17:1402-1408.

9. Ludviksson V, Lightfoot EN. Hydrodynamic stability of Marangonifilms. AIChE J. 1968;14:620-626.

10. Joo SW, Davis SH, Bankoff SG. A mechanism for rivulet formation inheated falling films. J Fluid Mech. 1996;321:279-298.

11. Kabov OA. Breakdown of a liquid film flowing over the surface witha local heat source. Thermophys Aeromech. 2000;7:513-520.

12. Kabov OA, Chinnov EA. Heat transfer from a local heat source tosubcooled liquid film. High Temp. 2001;39:703-713.

13. Zaitsev DV, Kabov OA, Evseev AR. Measurement of locally heatedliquid film thickness by a double-fiber optical probe. Exp Fluids.2003;34:748-754.

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17. Geng J. Influence of Marangoni Effect on Distillation in PackingTower. PhD Dissertation. Nanjing, China: Nanjing University; 2002.

18. Dukler AE, Bergelin OP. Characteristics of flow in falling liquid films.Chem Eng Process. 1952;48:557-563.

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Manuscript received Nov. 21, 2004, revision received Mar. 31, 2005, and finalrevision received Jun. 5, 2005.


Visualizations of Boger Fluid Flows in a 4:1Square–Square Contraction

M. A. AlvesDepto. de Engenharia Quımica, CEFT, Faculdade de Engenharia da Universidade do Porto, 4200-465 Porto, Portugal

F. T. PinhoCentro de Estudos de Fenomenos de Transporte, Depto. de Engenharia Mecanica, Universidade do Minho, Campus de

Azurem, 4800-058 Guimaraes, Portugal

P. J. OliveiraDepto. de Engenharia Electromecanica, Unidade Materiais Texteis e Papeleiros, Universidade da Beira Interior, 6201-001

Covilha, Portugal


Visualizations of the 3-D flow in a 4:1 square–square sudden contraction for twoviscoelastic Boger fluids and two Newtonian fluids were carried out at low Reynoldsnumbers. In these creeping flow conditions, the vortex length remained unchanged forNewtonian fluids, whereas a nonmonotonic variation with flow rate was observed for theBoger fluids. Initially, the corner vortex slightly increased with flow rate to a local peakat a Deborah number of De2 � 6, before decreasing significantly to a minimum at De2 �15 (De2 is based on downstream characteristics). Finally, for Deborah numbers � 20there was intense vortex enhancement until a periodic flow was established at higher flowrates (De2 � 45–52). The strong elastic vortex enhancement was preceded by theappearance of diverging streamlines on the approach flow and, for the Boger fluid withhigher polymer concentration, vortex enhancement took place through a lip vortexmechanism. © 2005 American Institute of Chemical Engineers AIChE J, 51: 2908–2922, 2005Keywords: viscoelastic flow, Boger fluid, diverging flow, visualization, 3D contractionflow

Introduction

Flow visualizations have always been important in fluidmechanics and motivated a wealth of research in the field, asillustrated in Van Dyke’s classic work An Album of FluidMotion.1 They provide a clear insight of many phenomena andreinforce elaborate mathematics. Non-Newtonian fluids areusually more viscous than common Newtonian fluids and theirflows are frequently investigated by flow visualization because

they tend to take place in the laminar regime. A compilation offlow visualization studies specifically for non-Newtonian flu-ids, was written by Boger and Walters in 1993,2 and constitutesa representative sample of works published in this importantsubfield of fluid mechanics.

Sudden contraction flows are classical benchmark problemsused in computational rheology,3 with the explicit assumptionof two-dimensional (2-D) flow to simplify the simulations, anda large number of visualization studies in planar and axisym-metric contractions have been published in the literature. Theflow behavior of non-Newtonian fluids in these simple config-urations can be very surprising; there can be substantiallydifferent flow patterns for fluids with apparently similar

Correspondence concerning this article should be addressed to M. A. Alves [email protected].



rheological characteristics, and distinct patterns also arise incomparable configurations as a result of geometric differences(such as planar vs. axisymmetric).

The following paragraphs provide a review of relevantworks in sudden contraction flows of axisymmetric, planar, andsquare–square type, and a summary of the main conclusions ispresented in Table 1.

Axisymmetric contractions

For axisymmetric contractions, pioneering visualizationstudies with viscoelastic fluids were carried out by Cable andBoger4-6 and Nguyen and Boger,7 who also reviewed previouswork in the field. At low flow rates they observed that thesalient corner vortex was controlled by viscous forces and anormalized length of 0.18 was measured, with the normaliza-tion based on the upstream pipe diameter. At higher flow ratesa dramatic vortex growth was reported for all geometries (cf.Table 1) and for both shear-thinning and constant viscosityelastic fluids. At even higher Deborah numbers the flow be-came asymmetric and time dependent. At intermediate condi-tions Cable and Boger observed the onset of diverging flowupstream of the contraction plane and the simultaneous reduc-tion of the salient vortex length, a phenomenon partially attrib-uted to inertia. Fluid elasticity was considered the cause for theflow instabilities because the Reynolds numbers were wellbelow those associated with transitions arising from inertialeffects.

From the mid-1980s the 4:1 contraction became the focus ofa variety of experimental and numerical work. The experimen-tal work concentrated on the investigation of low Reynoldsnumber flow transitions and elastic instabilities and relied onseveral techniques, as in McKinley et al.8 At that time thenumerical methods were usually unable to predict accuratelythe subcritical steady flows and the emphasis was focused onimproving the numerical techniques.

In 1986 Boger et al.9 investigated the behavior of twodifferent Boger fluids having similar shear properties and yetfound quite different vortex dynamics; for a polyisobutylene inpolybutene fluid (PIB/PB) vortex enhancement was precededby the growth of a lip vortex, whereas for polyacrylamidedissolved in corner syrup (PAA/CS) the lip vortex was absent.It was concluded that an additional measure of extensionalproperties had to be taken into account and in his 1987 reviewpaper10 Boger suggested that parameter to be the extensionalviscosity, and described in more detail the flow dynamics in thesudden contraction with increasing Deborah number. For somefluids only a convex corner vortex exists, growing in size andbecoming concave as elasticity increases. For other fluids thecorner vortex extends to the reentrant corner, and a lip vortexis formed. For high contraction ratios, the two vortices areinitially separated, but similar distinct vortices were also seenby McKinley et al.8 in small contraction-ratio experiments. Asthe elasticity increases, the strength of the lip vortex grows atthe expense of the corner vortex, whereas the length of thelatter remains unchanged. Eventually, the lip vortex occupiesthe whole contraction plane region and a further increase in theDeborah number leads to a larger concave-shaped vortex. Ateven higher Deborah numbers a small pulsating lip vortex nowappears and leads to unsteady behavior. All these features have

been observed in axisymmetric contractions under conditionsof negligible inertia.

Recent experimental investigations by Rothstein and Mc-Kinley11,12 clearly demonstrate the important role of exten-sional viscosity on the dynamics of vortex growth and on theassociated enhanced pressure drop in contraction flows. Theyconjectured that the extra pressure drop results from an extradissipative contribution to the elastic stress arising from astress-conformation hysteresis in the nonhomogeneous exten-sional flow generated near the contraction plane. Such conjec-ture has implications on constitutive-equation modeling and ledto the recent numerical simulations of Phillips et al.13 with aclosed form of the adaptive length scale (ALS) model of Ghoshet al.,14 which accounts for hysteresis of the conformationtensor. These authors were able to qualitatively predict largepressure drop enhancements, although there were still discrep-ancies in comparison with experiments.

Planar contractions

Investigations in planar contraction flows began soon afterthose in axisymmetric contractions and, although they con-firmed some of the findings observed for the circular geome-tries, some notable discrepancies also emerged. Walters andWebster15 found no significant vortex activity for Boger fluidsin the 4:1 case, in marked contrast to observations in 4.4:1circular contractions. However, for shear-thinning fluids vortexgrowth was observed in both planar and axisymmetric geom-etries. To help clarify these differences, Evans and Walters16

studied the flows of shear-thinning and constant viscosity elas-tic fluids (aqueous solutions of polyacrylamide) through planarcontractions (contraction ratios of 4:1, 16:1, and 80:1) andalways found strong vortex enhancement for the shear-thinningfluids, even for the smaller contraction ratio, and difficulties inobserving vortex activity for Boger fluids. They also reportedthat both the contraction ratio and fluid elasticity contributed tovortex enhancement with shear-thinning fluids. However, forthe larger contraction ratios a lip vortex was seen and a growthmechanism similar to that previously found for circular dieswas observed, although in the 4:1 planar contraction there wasno sign of such a lip vortex. In a subsequent paper Evans andWalters17 looked at the behavior of shear-thinning fluids in thesmaller contraction ratio of 4:1 to investigate whether a lipvortex mechanism of vortex growth was still at work. Theyfound that for the less viscous/concentrated polymer solutionssuch lip vortex could be generated and that inertia played acritical role in separating the corner and lip vortices. Indepen-dent lip and corner vortices were also found in the numericalsimulations of creeping flow by Alves et al.18,19 for a 4:1contraction with Boger fluids represented by either the upperconvected Maxwell (UCM) or the Oldroyd-B models.

The relevance of extensional viscosity in planar contractionflows was emphasized in the experimental investigations ofWhite and Baird,20,21 who used two polymer melts: polystyrene(PS) and low-density polyethylene (LDPE). Whereas a vortexwas observed for the LDPE, it was absent from PS flows andthe difference was attributed to the distinct extensional viscos-ities, given their similar weak shear-thinning behavior. Thiswas corroborated when the same authors later used a constitu-tive equation that represented accurately the measured exten-sional viscosity of both fluids,22 and were able to predict


Table 1. Relevant Investigations of Viscoelastic Fluid Flow in Axisymmetric, Planar, and Square Contractions*

Ref. Authors (year)Flow Conditions and Test

Fluids Main Conclusions

4–6 Cable and Boger(1978, 1979)

Circular, � � 2, 4Shear-thinning fluids

Vortex flow regime with vortex growth at low flow rates.Divergent flow regime at intermediate flow rates and

inertial effects.Unstable flow regime at large flow rates.

7 Nguyen and Boger(1979)

Circular, 4 � � � 14.6Boger and shear-thinning

fluids

Sequence of flow regimes referring to lip vortexbehavior:

(1) vortex growth; (2) asymmetric flow; (3) rotatingflow; (4) helical flow (pulsating).

15 Walters and Webster(1982)

Planar and circular, � � 4Boger and shear-thinning

fluids

Boger fluids: substantial vortex activity in circularcontraction and virtually none in planar case.

Shear-thinning fluid: substantial vortex activity in circularand planar contractions.

Rounding the corner dramatically changes flow.24 Walters and Rawlinson

(1982)Planar, � � 13.3SQ/SQ, � � 13.3Boger fluids

No noticeable vortex activity for the 13.3:1 planarcontraction.

Significant vortices observed for the 13.3:1 3-D SQ/SQcontraction.

Vortex enhancement with increasing flow rate.Slight asymmetry at highest flow rates.

9 Boger et al. (1986) Circular, 4.4 � � � 16.5Boger fluids

Simultaneous lip and corner vortices, with lip vortexdominating as De increases:

De � 3 no influence of � on lip vortex;De � 3 influence of � on lip vortex.

16 Evans and Walters(1986)

Planar, � � 4, 16, 80Boger and shear-thinning

fluidsSQ/SQ, � � 16Boger fluids

Planar:Boger fluids: difficulty in observing vortex activity.Shear-thinning fluids: substantial vortex activity in all

cases.Large �: vortex growth associated with lip vortex.Small �: vortex growth is not associated with lip

vortex.SQ/SQ:

Strong vortex enhancement for Boger fluids.20 White and Baird

(1986)Planar, � � 5.9Weakly shear-thinning

melts

Vortex growth for LDPE due to early onset of shear-thinning and strain hardening extensional viscosity.

No vortex growth for PS which has no strain hardeningextensional viscosity.

Relevance of extensional viscosity.21 White and Baird

(1988)Planar, � � 4, 8Weakly shear thinning

polymer melts (LDPE,PS)

Confirmation of the observations of White and Baird20

for lower and higher contraction ratios.

22 White and Baird(1988)

Planar, � � 4, 8Numerical, Full PTT, � �

0

Simulations confirm experiments of White and Baird21 interms of streamlines and birefringence.

Size and intensity of vortex determined by parameter �,which controls extensional viscosity in the PTTmodel.

17 Evans and Walters(1989)

Planar, � � 4Shear-thinning fluids

For some fluids there are lip and corner vortices, with theformer responsible for the vortex enhancementmechanisms, as for large �.

For mobile fluids inertia separates corner and lip vortices.8 McKinley et al. (1991) Circular, 2 � � � 8

Boger fluidSequence of flow regimes near the lip vortex: (1) stable;

(2) time periodic; (3) quasi-periodic; (4) aperiodic.Time-dependent 3-D flow for 2 � � � 5 at high flow

rates.Rounding corner delays transitions.

46 Boger and Binington(1994)

Circular, � � 4Sharp and rounded

reentrant cornerBoger fluids

Each of the two fluids studied react differently to therounded corner: dramatic changes are observed forthe polyacrylamide based Boger fluid, in deepcontrast to less dramatic changes observed for thepolyisobutylene based fluid.

47 Purnode and Crochet(1996)

Numerical simulationswith a FENE-P model

Planar, � � 4, 16, 80Shear-thinning fluids

Predictions qualitatively match the experimental resultsof Evans and Walters.16,17

Lip vortex not associated with inertia.

26 Mompean and Deville(1997)

Numerical, Oldroyd-Bmodel

2-D, � � 43-D planar � � 4

2-D: no prediction of lip vortex; decrease of cornervortex size with Deborah number.

3-D: Corner vortex in center plane always smaller thanin 2D case.


numerically, at least qualitatively, the different vortex patternsobserved.

The surprisingly different behavior of Boger fluids in circu-lar and planar contractions has been further studied by Nigenand Walters.23 Experiments in both types of contraction wereconducted with Newtonian and Boger fluids having identicalshear viscosities and it was demonstrated that, although therewere higher extensional strain rates in the planar geometry thanin the circular die, virtually no elastic vortex enhancement wasfound in the former geometry in contrast to the latter. Giventhat the vortex dynamics is a complex function of strain rates,strains, and molecular conformation histories,12 these differ-ences are not so surprising.

Square–square contractions

Experiments with flows through square–square contractionsare scarce, partly because they are more complex but alsobecause of their inadequacy to be used for comparison with2-D simulations. In their 1982 experiments, Walters and Raw-linson24 reported similarities between the flows through circu-lar and square–square contractions. In particular, the differ-

ences found between flows of Boger fluids in planar andcircular contractions were also seen for planar and 13.3:1square–square contractions and at intermediate Deborah num-bers a diverging flow was seen upstream of the contraction asoriginally observed by Cable and Boger4 for circular geome-tries.

In the square duct contraction, and in contrast to the 2-Dplanar case, isopressure lines can close in the cross-streamdirections, as in the axisymmetric geometry, but there are nownormal stress imbalances leading to secondary flows. The dy-namics of the vortices in combination with these new second-ary flow structures and fluid elasticity are still to be wellcharacterized and understood.

In view of the above, it comes as no surprise that flowpredictions of square–square contraction flows are virtuallyinexistent except for the work of Xue et al.,25 to be discussedbelow. Most of the existing three-dimensional (3-D) viscoelas-tic flow simulations in contractions basically considered the2-D planar contraction extending along the spanwise directionand so the 3-D effects were attributed only to the presence ofend walls. Some computations of such planar 3-D contraction

Table 1. Relevant Investigations of Viscoelastic Fluid Flow in Axisymmetric, Planar, and Square Contractions* (Continued)

Ref. Authors (year)Flow Conditions and Test

Fluids Main Conclusions

27 Xue et al. (1998) Numerical2-D: UCM3-D planar: Boger and

shear-thinning fluids(UCM, PTT with � �0)

2-D: Lip and corner vortex growth mechanismsdepending on elasticity and Deborah numbers.Appearance of lip vortex depends on polymerviscosity and is promoted by inertia.

3-D planar� UCM: confirms presence of lip vortex for some flow

conditions.� PTT: qualitative agreement with experiments of

Quinzani et al.48

25 Xue et al. (1998) Numerical2D and 3D SQ/SQ, � � 4Boger and shear-thinning

fluids: UCM, PTT (� �0)

Relevance of transient extensional behavior alongcenterline. In 3-D vortex activity correlates wellwith steady extensional properties, but not in 2-D.

11 Rothstein andMcKinley (1999)

Circular, � � 4Boger fluid

Measurement of enhanced pressure drop. It is suggestedthat this is attributable to an additional dissipativecontribution of polymer stress manifested in stressconformation hysteresis.

184243

Oliveira and Pinho(1999); Alves et al.(2000)

Planar, numerical� � 4UCM and PTT fluids

Lip vortex growth mechanism for UCM fluid.No lip vortex for PTT fluid.Streamline divergence for high Deborah flows, enhanced

by inertial effects (UCM).Importance of mesh refinement and discretization

schemes for numerical accuracy.12 Rothstein and

McKinley (2001)Circular, 2 � � � 8Various curvatures of

cornerBoger fluid

� � 2: steady elastic lip vortex4 � � � 8: no lip vortex observed, only corner vortex.Rounding the reentrant corner leads to shifts in the onset

of flow transitions to larger Deborah numbers.The role of contraction ratio on vortex growth dynamics

is rationalized by using a dimensionless ratio of theelastic normal stress difference in steady shear flowto that in transient uniaxial extension.

23 Nigen and Walters(2002)

Planar and circular2 � � � 32Boger fluids

Planar case: no sign of steady vortex enhancementCircular: lip mechanism responsible for vortex growth

and enhanced pressure drop.19 Alves et al. (2003) Planar, numerical; � � 4

Highly refined meshOldroyd-B and PTT(linear and exponential)models

Lip vortex enhancement for Oldroyd-B fluidaccompanied by decrease in corner vortex.

Intense corner vortex growth for linear PTT and no lipvortex.

High Deborah number simulations for PTT models.

*� represents the contraction ratio. Works are presented in chronological order.


flows (in fact, quasi-2-D) were presented by Mompean andDeville26 and Xue et al.27; their intention was mainly to illus-trate the capabilities of their finite-volume methods in 3-Dsimulations, so they did not comment on the results of theircalculations in relation to 3-D effects and in fact the detailed3-D flow structure was not well resolved. The only 3-D effectmentioned by Xue et al.27 was that the average velocity in a3-D channel differs from the average velocity in a 2-D channel,and therefore the Deborah number should be modified accord-ingly.

In a second article, Xue et al.25 tackled the actual 3-Dsquare–square contraction using the Phan-Thien–Tanner model(PTT) (with a nonzero second normal stress difference coeffi-cient) and UCM constitutive equations. They found strongvortex enhancement for both fluid models, in contrast to theresults for the planar contraction where only the PTT fluid(shear-thinning fluid) exhibited vortex enhancement. Thesecontrasting behaviors mirror those observed experimentally foraxisymmetric and planar contraction flows, and are explainedby the 3-D nature of circular and square–square contractions.The numerical investigations of Xue et al.25 related the calcu-lated flow dynamics with the transient extensional viscosity ofthe fluids, but nothing was reported regarding unexpected sec-ondary flows in the contraction region or the onset of flowinstabilities; they did mention, however, the existence of thewell-known secondary flow in the fully developed straightupstream and downstream noncircular ducts.

In conclusion, the vast majority of experimental work oncontraction flows of viscoelastic liquids has been on axisym-metric and (approximately) planar configurations, with theexpectation that 2-D numerical simulations would allow foradequate predictions. A square–square contraction arrange-ment is a good compromise between geometric simplicity andcomplex 3-D flow structure, and appears as a good candidatefor a prototype 3-D test case that is necessary for validatingnumerical 3-D codes. However, before embarking on the heavytask of performing full 3-D simulations of such viscoelasticflows, both under steady and transient conditions, a thoroughexperimental investigation is required. This is the motivationfor the present experimental work, which concentrates on char-acterizing the flow patterns in square–square contractions ofelastic fluids having a constant shear viscosity.

Experimental Setup

The experimental apparatus is depicted in Figure 1. The rigconsisted of two consecutive square ducts (length: 1000 and300 mm) and a vessel. The sides of the square ducts were 2H1

� 24.0 mm and 2H2 � 6.0 mm, respectively, thus defining the4:1 contraction ratio. The flow rate was set by an adequatecontrol of applied pressure on the upper duct and frictionallosses in the long coiled pipe (T in Figure 1) located betweenthe smaller duct and the vessel, at the bottom of the rig. Toachieve low flow rates the coiled pipe was 8 m long and had adiameter of 4 mm, whereas for higher flow rates the diameterof the coiled pipe was 6 mm. Applied pressure was keptbetween 0.5 and 4 bar, and the dashed lines in Figure 1represent the pressurized air lines. This flow arrangement ad-equately controlled the flow rate without significant constric-tions, such as valves, that would eventually degrade the poly-

mer molecules. The flow rate was measured by a stopwatch andthe passage of the liquid-free surface at two marks in the upperduct. In all tests the fluid temperature was measured and thefluid properties were taken from the rheometric master curvesshown in the next section. To improve the quality of thevisualizations the rig was placed inside a dark room.

A 10 mW He–Ne laser light source was used to visualize theflow patterns. The laser beam passed through a cylindrical lensto generate a light sheet illuminating highly reflective tracerparticles suspended in the fluid. These were 10 �m PVCparticles (at a concentration of about 15 mg/kg fluid) addedduring the preparation of the fluid. The path lines formed wererecorded using long time exposure photography with a con-ventional camera (Canon EOS300 with a macro EF100 mmf/2.8 lens), as sketched in Figure 2, which includes a schematicrepresentation of the test section. The terminal velocity of thePVC particles was negligible: assuming Stokes flow conditionsit was estimated to be 0.15 �m/s in the less viscous Newtonianfluid used (N85), which is three orders of magnitude smallerthan the minimum flow bulk velocity.

The shear viscosity (�) and the first normal stress differencecoefficient (�1) in steady shear flow, and the storage and lossmoduli (G�, G�) in dynamic shear flow were used to charac-terize the rheology of the Boger fluids. These properties weremeasured with an AR2000 rheometer from TA Instruments,using a cone–plate setup with 40 mm diameter and 2° angle. Afalling ball viscometer from Gilmont Instruments (model GV-2200) was also used for some viscosity measurements with theNewtonian fluid N85.

Figure 1. Flow rig.PR, pressure regulator; V1 to V6, ball valves; R, reservoir;CL, cylindrical lens; T, long coiled pipe; S, free surface; C,contraction plane.


Rheological Characterization of the Fluids

Four fluids based on mixtures of glycerin and water wereinvestigated: two viscous Newtonian fluids (N85 and N91) andtwo viscoelastic Boger fluids (PAA100 and PAA300). Theircompositions, densities, and zero-shear rate viscosities arelisted in Table 2. The Boger fluids were prepared by dissolvinga small amount of polyacrylamide (PAA; Separan AP30 pro-duced by SNF Floerger) in the Newtonian solvent N91. Tominimize the intensity of shear thinning a small amount ofNaCl was added, as described in detail in Stokes.28 To reducebacteriological degradation of the solutions the biocide KathonLXE, produced by Rohm and Haas, was also added. The fluiddensities were measured at 21.2°C using a picnometer.

Newtonian fluids

For the N85 fluid the measured shear viscosity was � �0.125 Pa�s at 18°C, the temperature at which the visualizationstook place. For the N91 fluid the rheometer was used tomeasure the shear viscosity at temperatures ranging from 15.9to 25.0°C. This second Newtonian fluid, which served assolvent for the PAA solutions, was also used to evaluate theaccuracy of the rheometer and to estimate the base noise levelin dynamic tests (represented as the dashed baseline of 2G� inFigures 3b and 4b). For N1, the measurements at � � 100 s1

indicated a zero reading within experimental uncertainty, asthey should. The uncertainty in measuring N1 was of the orderof 10 Pa, in agreement with the specifications of the manu-facturer for the cone-plate geometry used. For � � 100 s1

inertial effects became important, and negative values of N1

were measured. This inertial effect for Newtonian fluids waswell predicted by the following equation29:

N1,inertia � 0.152R2 (1)

where and R represent the angular velocity and the radius ofthe cone, respectively.

The effect of temperature on the shear viscosity for the N91fluid is well described by an Arrhenius equation, defining ashift factor aT of the form

ln�aT� � ln��T�

��T0�� H

R �1

T�

1

T0�� (2)

Figure 2. Flow visualization technique and details of thetest section.

Table 2. Composition and Properties of the Fluids (Mass Concentrations)

DesignationPAA(ppm)

Glycerin(%)

Water(%)

NaCl(%)

Kathon(ppm)

(kg/m3) �0 (Pa � s)

N85 — 84.99 15.01 — 25 1221 0.125 (18°C)N91 — 90.99 7.51 1.50 25 1250 0.367 (20°C)PAA100 100 90.99 7.50 1.50 25 1249 0.487 (20°C)PAA300 300 90.97 7.50 1.50 25 1247 0.735 (20°C)

Figure 3. Material parameters in steady and dynamicshear flow for the PAA100 fluid.(a) Open symbols for shear viscosity and first normal stressdifference coefficient; solid symbols for G� and ��; (b) com-parison between G� and G� data (symbols) and fitting bythree-mode Oldroyd-B model (solid lines).


where �(T0) represents the viscosity at the reference absolutetemperature T0, here taken as T0 � 293.15 K. Fitting thisequation to the experimental data gave H/R � 6860 K and�(T0) � 0.367 Pa�s (more details can be found in Alves30).

Generally, the shift factor aT is defined as31

aT ��0�T�

�0�T0�

T0

T

0

(3)

where �0(T0) designates the zero shear-rate viscosity at thereference temperature T0, and and 0 are the fluid densities attemperatures T and T0, respectively. When the range of tem-peratures is limited the definition of the shift factor can besimplified by dropping the effects of density and temperatureratios31 and thus Eq. 2 is the form adopted in this work, with�(T) � �0(T) and �(T0) � �0(T0).

Boger fluids

For the Boger fluids the steady shear properties were mea-sured at temperatures ranging from 15.3 to 30.0°C. The time–temperature superposition principle was found to be valid forboth fluids and was used to build master curves. The value of

H/R obtained for the solvent was found to be adequate for thetwo Boger fluids. The reduced rheological quantities (subscriptr) thus become as defined by Eqs. 4a–4d32

r � �T0� � aT�T� �r � ��T0� � aT��T� (4a)

G�r � G��T0� � G��T� G �r � G��T0� � G��T� (4b)

��r � ��T0� � ��T�/aT � �r � ��T0� � ��T�/aT (4c)

�r � ��T0� � ��T�/aT �1r � �1�T0� � �1�T�/aT2 (4d)

The reduced steady shear viscosity and first normal-stresscoefficient data (�r, �1r vs. �r) are plotted in Figures 3a and 4afor the PAA100 and PAA300 fluids, respectively. Both figuresalso include dynamic shear data in appropriate form (��r, 2G�r/r

2 vs. r) to compare the corresponding limiting behaviors atvanishing deformations, but note their different ordinate scales.The dynamic shear data are plotted separately in Figures 3b and4b, which include predictions of G� and G� (solid lines), usinga three-mode Maxwell model plus a Newtonian solvent con-tribution. The parameters of these multimode models are listedin Table 3, at reference temperature T0.

The data in Figures 3a and 4a clearly show the limitingbehavior of the measured properties in steady and dynamicshear flows. The reduced shear viscosity is approximatelyconstant at reduced shear rates in the range 0.3 to 50 s1 for thePAA100, whereas for the PAA300 it decreases approximately10% per decade of reduced shear rate. At �r � 54 and 28 s1

(for the PAA100 and PAA300, respectively) there is an abruptgrowth in �r and �1,r and this is accompanied by a slightreduction in reduced shear rate (with the rheometer operating in“controlled stress mode”). This phenomenon results from anelastic instability leading to 3D flow, which is frequentlyobserved with Boger fluids in cone–plate and plate–plategeometries, as investigated previously by Phan-Thien33 andMcKinley et al.34 More details of this instability for these fluidscan be found in Alves.30

Figures 3b and 4b show that the three-mode Oldroyd-Bmodel is accurate enough to predict G� and G� within themeasured range, whereas a single-mode model was found to beunable to give accurate predictions of G� and G� over the wholerange of frequencies. Data at low and high frequencies, leadingto values of G� close to the base line (noise level), wereexcluded. This baseline, determined as a function of angularspeed for deformations of 0.10 and 0.50 using the Newtonianfluids N85 and N91 and deionized water, represents the sen-sitivity of the rheometer and can be used to estimate theexperimental uncertainty in G� for the viscoelastic fluids.

The relaxation spectra will be useful later to quantify the

Figure 4. Material parameters in steady and dynamicshear flow for the PAA300 fluid.(a) Open symbols for shear viscosity and first normal stressdifference coefficient; solid symbols for G� and ��; (b) com-parison between G� and G� data (symbols) and fitting bythree-mode Oldroyd-B model (solid lines).

Table 3. Linear Viscoelastic Spectra for Boger Fluids atReference Temperature (T0 � 293.15 K)

Mode k �k (s)

�k (Pa � s)

PAA100 PAA300

1 3.0 0.075 0.232 0.3 0.027 0.0903 0.03 0.018 0.048Solvent — 0.367 0.367


Deborah number and for future numerical simulations of thisflow. The Deborah number is used here to quantify elasticeffects and is defined identically to the Weissenberg numberreported in other works in the literature, based on downstreamquantities.

Flow Visualization Results

Newtonian fluidsFlow visualizations were carried out first with the Newtonian

fluids to assess the effect of inertia on the flow structure and toserve as a reference for comparison against the results for theelastic Boger fluids. For the 2-D planar and axisymmetricgeometries a fair amount of knowledge has been established onthe behavior of Newtonian fluids,10 although such informationis missing for square–square contractions.

Figure 5 shows photographs of stream traces of the flow ofthe Newtonian fluids in the middle plane of the 3-D suddencontraction and Figure 6 plots the variation of the normalizedvortex length (xR/2H1) with Reynolds number, here defined as

Re2 �2H2U2

�(5)

where H2 and U2 designate the downstream duct half-width andbulk velocity, respectively.

As expected, inertia leads to a reduction of the corner vortex,especially noticeable for Reynolds numbers � 1. At low Reyn-olds numbers, inertial effects are negligible and xR/2H1 asymp-totes to 0.163. It is interesting to note that, under creeping flowconditions, the vortex size for a 4:1 circular contraction asymp-totes to exactly the same value, xR/2H1 3 0.163, whereas fora 4:1 2-D planar contraction the size is somewhat different,xR/2H1 3 0.1875 (values based on numerical simulations; seeAlves et al.30,35). For the circular contraction, Boger10 quotes avalue of xR/(2H1) � 0.17 0.01 based on both experimentsand numerical predictions, in close agreement with our simu-lations.

For Newtonian fluids, all the experimental flow features arewell captured by numerical simulations, shown on the rightcolumn of Figure 5; in addition, the predicted variation of thevortex length with the Reynolds number, shown by the curve inFigure 6, quantitatively matches the measured data. Thesenumerical simulations were obtained with a finite-volumemethodology developed by the authors.36

Figure 5 also highlights the point that, even though at firstsight the flow inside the vortex looks 2-D, in reality it is 3-D.In contrast with the planar and axisymmetric sudden contrac-tion flows, none of the recirculations is ever closed in thesquare contraction, as revealed by a careful inspection of thestreaklines. This was also confirmed by visual inspection andespecially by the numerical calculations from which the flowdescription sketched in Figure 7 was outlined. Figure 7 showsstreaklines from fluid particles starting at two different planesin the upstream duct: the middle plane perpendicular to the wall(ABCD), always seen in the photos, and the second plane(EFGH) at 45° to the wall passing through opposite corners ofthe square cross section (hereafter referred to as corner plane).

Figure 5. Experimental (left column) and numerical (rightcolumn) streaklines for the flow of Newtonianfluids N85 and N91 in the middle plane of asquare–square sudden contraction at T �18.0 � 0.2°C.

Figure 6. Influence of the Reynolds number on the vor-tex length at the middle plane of a square–square sudden contraction for Newtonian flu-ids N85 and N91.Comparison between experiments and numerical predictions.


Particles flowing near the walls along the corner plane enter thecorner-plane vortex, rotate toward its center, and then drifttoward the middle plane along the eye of the recirculatingregion, in this way entering the vortex located at the middleplane. Once here the particles now rotate toward the peripheryof the middle-plane vortex and exit the vortex at the reentrantcorner going into the downstream duct.

Open 3-D recirculations were also predicted by other authorswho simulated Newtonian flows in 3-D sudden expansions37,38

(quasi-2-D as the expansion took place in one dimension only)and in 3-D (quasi-2-D) backward-facing steps.39,40 The inho-mogeneous flow conditions experienced by fluid particles com-ing either along the upstream square–duct walls or along itscorners, into the contraction plane region, result in differentvelocity gradients, thus leading to stress gradients and complexsecondary flows.

Boger fluids

To quantify the strength of elastic effects with Boger fluidsit is convenient to use a single relaxation time in the definitionof the Deborah number, which can be either based on upstream(De1) or downstream flow conditions (De2). Most studies ofcontraction flows define the Deborah number with a deforma-tion rate characteristic of downstream flow conditions (leadingto De2). Alves et al.41 showed that lip vortex activity (whenpresent) scales with De2, independently of the contraction ratio(�), whereas dimensionless corner vortex size (xR/2H1) andintensity depend on De2/�. Thus, hereafter the results arepresented with Deborah number based on downstream quanti-ties, De2. Nevertheless, conversion between upstream anddownstream Reynolds and Deborah numbers is straightfor-ward: Re2 � 4Re1; De2 � 64De1.

The downstream Deborah number is defined as

De2 ��p�T�U2

H2�

aT�p�T0�U2

H2(6)

where �p is an equivalent relaxation time based on the Old-royd-B model, and calculated from the linear viscoelastic spec-tra using the following equations

�p � �k�solvent

�k (7)

�p �1

�p�

k�solvent

�k�k (8)

In this way, it is guaranteed that at low deformation rates(and low angular velocities), the viscoelastic behavior of theequivalent single mode Oldroyd-B model is identical to that ofthe multimode model (say, �1,0 � lim�30 �1 � ¥k 2�k�k �2�p�p). This definition of relaxation time considers only therole of the polymer additive in the absence of solvent toestablish the elasticity of the fluid, but it should be noted thatsome authors prefer to define a Maxwell relaxation time as �0

� �1,0/2�0.Thus, from the data in Table 3 the parameters of the equiv-

alent single-mode Oldroyd-B model are the following: for thePAA100, �0 � �s � �p � 0.487 Pa�s, resulting in a solventviscosity ratio �s/�0 � 0.754, and a relaxation time �p � 1.947s; for the PAA300, �0 � �s � �p � 0.735 Pa�s, �s/�0 � 0.499,and �p � 1.952 s. Clearly, with the relaxation time defined asin Eq. 8 we have almost the same relaxation time for bothBoger fluids. Because these two polymer solutions are dilute,the configurations/behavior of the individual polymer mole-cules in the same solvent should not differ significantly, and sothe relaxation times of the molecules should be similar. How-ever, a higher polymer concentration leads to higher “elastic-ity” because the fluid becomes more viscous and the polymermolecules take longer to relax. This can be quantified in twoalternative ways, either by using the total viscosity of the fluidto calculate the Maxwell relaxation time, as mentioned above(leading to �0 � 0.480 and 0.977 s for PAA100 and PAA300,respectively), or by using the so-called elasticity number (El2),defined by the ratio of the Deborah and Reynolds numbers

El2 �De2

Re2�

�p�0

2H22 (9)

which is independent of flow kinematics and takes into accountthe viscosity of the solution: El2 � 42.2 and 63.9 for thePAA100 and PAA300, respectively. This results in thePAA300 fluid being 1.52 times more elastic than PAA100. Wenote, however, that if the elasticity number is defined as El ��p�p/2H2

2 (��1,0/4H22), to be consistent with the definition

of �p, then the PAA300 fluid is approximately three times more“elastic” than PAA100 (in agreement with the ratio betweenthe zero shear rate values of the first normal stress coefficientmeasured for both fluids), a figure more in line with theincrease in polymer concentration. In any case, whatever thedefinition adopted to measure “elasticity,” the solution ofPAA300 is more elastic than that of PAA100 and this effectexplains the different flow behaviors reported hereafter.

Streak photographies at the middle plane of the contractionare presented in Figure 8 for the flow of the PAA100 fluid forincreasing values of the flow rate. The flow characteristics are

Figure 7. 3-D particle trajectories for a Newtonian fluidunder conditions of negligible inertia.


complex, but are caused only by elastic effects because inertiais not significant (Re2 � 0.95). At relatively low values of theDeborah number (De2 � 2.65), viscous effects predominateand the flow pattern is similar to that seen in Figure 5 forNewtonian fluids, with a concave separation streakline and the3-D nature of the recirculating flow in evidence. With increas-ing Deborah number changes progressively occur: first, there isa very slight increase in vortex size while the separation streak-line straightens; then, the corner vortex progressively decreasesin size to about a quarter at De2 � 15, an effect that cannot beattributed to inertia because the Reynolds number remains wellbelow Re2 � 1. McKinley et al.8 reported results in the 4:1axisymmetric contraction bearing some resemblance withthese: negligible elastic effects for De2(�) � 1 (using a shear-rate-dependent Maxwell relaxation time) and a decrease in thecorner vortex size for Deborah numbers up to De2(�) � 3.However, in contrast to McKinley et al. for the axisymmetriccontraction, the formation of a strong lip vortex while thecorner vortex decreases in size is not seen in the presentexperiments, although the higher curvature of the streaklines atthe reentrant corner suggests the possibility that a weak lip

vortex does exist (see photo for De2 � 13.4). In fact, thedecrease of xR at low De2 is compatible with the existence of alip vortex,19 although it has been observed that xR usuallygrows monotonically when such a lip vortex is absent. Inaddition, the fluids used in those references were clearly moreelastic than here: in McKinley et al.8 the elasticity number isevaluated as El2 � 610 and in Rothstein and McKinley12 as El2� 2100.

As the Deborah number further increases the corner vortexstarts to grow, and simultaneously there are changes in thecharacteristics of the streamlines approaching the contractionplane. For De2 24.1, the approach flow streaklines in themiddle plane and close to the contraction are seen to progres-sively diverge with flow elasticity, moving away from thecenterline. This anomalous effect had already been predictedby Oliveira and Pinho42,43 and Alves et al.18 for the flow ofBoger fluids in a 4:1 plane sudden contraction and was alsoobserved in the experiments of McKinley et al.8 for the flow ofa Boger fluid in circular contractions. Attempts to explain thecauses for divergence of streamlines upstream of the contrac-tion for Boger fluids have pointed to a local intense increase inextensional viscosity leading to an increased flow resistance inthe centerline just upstream of the contraction plane as theextension rates grow in a region of predominantly extensionalflow characteristics. Such extensional thickening is a char-acteristic of Boger fluids, but there are not enough data tocorrelate rheological behavior with diverging flow. McKinleyet al.8 discuss this issue and mention different degrees of flowdivergence for fluids with similar extensional viscosity behav-ior, arguing for the relevance of the total Hencky strain to this

Figure 8. Influence of elasticity on the streakline flowpatterns at the middle plane of a 4:1 square–square sudden contraction for PAA100 at T �18.1 � 0.2°C.


flow feature. However, the total Hencky strain is here un-changed because it is a function of the contraction ratio. Alsoof note here, the diverging streamlines with Boger fluids wereobserved in the absence of noticeable lip vortex activity. Thiscontrasts with the visualizations of McKinley et al.8 in theaxisymmetric geometry, but agrees with their conclusions thatdiverging streamlines and lip vortex formation are probablyunrelated.

As discussed more recently by Rothstein and McKinley,12

extensional viscosity alone does not explain the variety of flowfeatures in contraction flows; hysteresis effects on the stress-conformation, normal stress effects in shear and extension, aswell as contraction ratio and transient extensional viscositybehavior are relevant to the existence or not of a lip vortex andto the enhanced pressure drop, so the issue of diverging stream-lines should be at least as complex. In fact, the early experi-ments of Cable and Boger4-6 had already shown strong diverg-ing flow in the presence of combined shear-thinning andinertial effects, suggesting that inhomogeneous shear condi-tions near the wall are equally important. Quite interestingly, inthe experiments of Walters and Rawlinson24 in a 13.3:1 squar-e–square contraction, there is an abrupt expansion of the cen-tral jet as the fluid is just about to enter the smaller duct (cf.their Figure 5 at W � 0.002), but there is already an enhancedvortex and the diverging streamlines may be related in part tothe fluid particles being dragged into the recirculation, some-thing that we did not observe in our experiments.

The growth of the large vortex continues with elasticity, andthe flow remains steady up to De2 � 52. At higher Deborahnumbers, the 3-D nature of the recirculating flow seems to havechanged, as can be assessed by careful comparison of streak-lines in the recirculation at De2 � 50.4 with those taken at lowDeborah numbers. As the Deborah number further increasesthe flow becomes periodic, possibly because of an elasticinstability, and this is observed by the crossings in somestreaklines at De2 � 54.2. The amplitude of the oscillationsincreases with De2, as seen in the three photos of Figure 9 takenat three moments in time within a cycle for two differentsupercritical flow conditions. At even higher flow rates the flowloses its periodicity.

For the more concentrated Boger fluid, PAA300, the streak-line flow visualizations are shown in Figures 10a and 10b forconditions corresponding to two different temperatures, T �21.0 and 17.5°C. The second set was taken 3 months after thefirst, with the same fluid sample (it was stored in a refrigerator),and with the purpose of checking the repeatability of the flowand absence of fluid degradation over time under the testedconditions. In general terms, the influence of Deborah numberis the same as for PAA100, but a couple of important differ-ences are worth mentioning. First, the increase of the cornervortex located on the axial midplane at low Deborah numbersis also observed, but it is now more intense than that with thePAA100 fluid. This is clear in Figure 11, which compares thevariation for both fluids of the vortex size in the midplaneagainst the Deborah number (De2): for PAA300 xR/2H1 peaksby more than 25% relative to the Newtonian value at De2 � 6,followed by a decrease to about 0.07 in the range De2 � 6 to15. A second more important difference in relation to PAA100is that a lip vortex is now clearly seen in the midplane for acertain range of Deborah numbers (see photos at De2 � 8.70and 9.62 in Figure 10). In particular, at De2 � 9.62 when

inertia is still negligible, the lip vortex has similarities to thatfound for Boger fluids in the 4:1 plane contraction calculationsof Alves et al.18 (cf. their Figure 7 at De2 � 4 for UCM fluids).

It is known from experimental and numerical work in the 4:1plane sudden contraction flow with Boger fluids,23,44 that a lipvortex is formed at low to moderate Deborah numbers. This lipvortex grows with elasticity and eventually dominates thecorner vortex, the characteristic flow feature of Newtonianflows and low-Deborah number viscoelastic flows. In contrast,for the axisymmetric sudden contraction flow of Boger fluidsthe corner vortex is normally found to grow with elasticitywithout the presence of any lip vortex35,45; there are exceptionsto this Boger fluid behavior, as in the experiments of Boger etal.9 and McKinley et al.8 with PIB/PB solutions. Boger andBinnington46 also observed a lip vortex mechanism for a PAA-based fluid in a 4:1 circular contraction with rounded corners,in deep contrast with the (usual) strong corner vortex enhance-ment observed in the same 4:1 circular contraction, but with asharp reentrant corner. Rothstein and McKinley12 argued aboutthe competing roles of extensional stresses and shear-inducednormal stresses on vortex growth mechanisms, and concludedthat lip vortices are associated with a domination of shear-

Figure 9. Streaklines for the flow of PAA100 in the mid-dle plane of a square–square contraction atthree different moments within two differentoscillating supercritical flow conditions.


induced elasticity, whereas corner vortex growth is dominatedby extensional stresses, thus predominating at large contractionratios and/or large Deborah numbers.

Given the similarities between the square–square contractionand the axisymmetric geometry in terms of both extensionalstrain rates developed along the centerline and the possibility ofpartial balance of cross-stream/azimuthal pressure and stressgradients, it comes as no surprise that the visualizations ofEvans and Walters16 for a square–square 16:1 contraction donot show any lip vortex and instead illustrate a strong enhance-ment of the middle plane corner vortex (see their Figure 11).The presence of cross-stream secondary flows in the squarecontraction increases shear-induced normal stresses relative tothose found in the corresponding circular contraction flow and,according to the above mechanism of Rothstein and McKinley,this will widen the range of contraction ratios where the lip

Figure 10. Influence of elasticity on the streakline flowpatterns at the middle plane of a 4:1 square–square sudden contraction for PAA300.(a) Visualizations at 21.0°C; (b) visualizations at 17.5°C.


vortex is observed. Such an effect is similar to rounding thereentrant corner in axisymmetric contractions: Rothstein andMcKinley12 report for this case the existence of a lip vortex forthe 4:1 contraction and a delay in corner vortex development,confirming the observations of Boger and Binnington.46

At higher flow rates/elasticity (De2 � 20) there is vortexenhancement through a lip vortex mechanism. A divergingstreakline pattern upstream of the contraction plane is alsoobserved, but now in the presence of a lip vortex. The appear-ance of diverging streaklines in a 3-D square–square contrac-tion flow of Boger fluids, with and without lip vortex, isdocumented here for the first time and confirms the suggestionof McKinley et al.8 that the two phenomena are probablyunrelated. Simulations by Oliveira and Pinho42 for creepingflow of UCM fluid in a 4:1 planar contraction corroborate thispoint, and the numerical results presented in Oliveira andPinho43 for the same flow under conditions where inertia is notnegligible (Re2 � 1) illustrate that inertial effects enhancestreamline divergence for high Deborah number flows.

At even higher Deborah numbers, say for De2 46.7 (De1

0.726), the flow of PAA300 becomes periodic, showing abehavior similar to that reported above for the PAA100 fluid.This periodic flow results from an elastic instability and occursearlier than for the PAA100 solution, at De2 � 45 (De1 � 0.7).Figure 12 shows three photos with the PAA300 solution per-taining to a cycle of events for two different supercritical flow

Figure 11. Variation of normalized vortex length withflow elasticity for Boger fluids PAA100 andPAA300.

Figure 12. Streaklines for the flow of PAA300 in the mid-dle plane of a square–square contraction atthree different moments within oscillating su-percritical flow conditions.(a) De2 � 55.1 and Re2 � 0.577; (b) De2 � 61.5 and Re2 �0.643.


conditions, confirming that the time-average size of the time-dependent vortex continues to grow with Deborah number.

Conclusions

Flow visualizations were carried out in the middle plane ofa 4:1 square–square sudden contraction for Newtonian andBoger fluids under conditions of negligible inertia, usingstreakline photography. The Newtonian flow patterns were ingood agreement with our own numerical results and showinertia to be negligible for Reynolds numbers below Re2 � 1.The flow field was clearly three-dimensional, exhibiting openvortices with both Newtonian and viscoelastic fluids, and in-ertia tended to push the corner vortex toward the contractionplane.

For the two Boger fluids the upstream vortices were seen toincrease slightly at low Deborah numbers, before an intensedecrease took place leading to a minimum vortex size at De2 �15. As the Deborah number further increased, the streaklineson the central region of the approaching flow started to diverge,whereas the vortices grew strongly with elasticity until the flowbecame periodic and eventually chaotic at higher flow rates.

For the more concentrated Boger fluid (PAA300) these ef-fects were more intense because of the corresponding higherelasticity number, and a major difference in flow features wasclearly seen: after the vortex attained its minimum size atintermediate Deborah numbers, 10 � De2 � 15, a lip vortexappeared and grew with elasticity. For the PAA100 solution,and although only corner vortex enhancement was observed,the variation of vortex size with Deborah number was compat-ible with the existence of a weak lip vortex (not captured withour visualization technique).

We speculate that the secondary flow in the cross section ofthe rectangular channel tends to increase the role of shear-induced normal stresses, thus leading to the appearance of a lipvortex at this contraction ratio and consequently delaying thevortex growth and instability to higher Deborah numbers, com-pared to the corresponding situation of a circular contraction.Such flow features, and the existence of a diverging flow wellupstream of the contraction, are reported here for the first timein relation to the square–square contraction flow of Bogerfluids. However, similar flow features have been previouslyreported for circular contractions.

AcknowledgmentsThe authors thank Prof. M. P. Goncalves and D. Torres for help in

characterizing the rheology of the fluids used in this work. M. A. Alvesthanks Fundacao Calouste Gulbenkian for financial support. Sponsorshipof FEDER by the FCT program POCTI/EME/37711/2001 is gratefullyacknowledged. Finally we are grateful for the relevant comments of one ofthe reviewers whose helpful suggestions significantly improved this article.

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Manuscript received Oct. 8, 2004, and revision received Mar. 12, 2005.


Experimental Approaches for UnderstandingMixing Performance of a Minireactor

G. G. Chen, G. S. Luo, S. W. Li, J. H. Xu, and J. D. WangThe State Key Laboratory of Chemical Engineering, Dept. of Chemical Engineering, Tsinghua University,

Beijing, 100084, China


Three experimental approaches are presented to determine the mixing characteristicsof a new kind of high-efficiency membrane dispersion minireactor in which microfiltrationmembranes are applied as the dispersion media. Residence time distribution (RTD) curveswere measured for determining the macromixing characteristics. A typical Dushmanreaction (iodide–iodate) coupled with a neutralization and precipitation reaction ofBaSO4 were introduced to characterize the micromixing performance of a single-phasemixing process. A dye extraction method was also applied to study the micromixingperformance of a two-liquid phase-mixing system. The RTD result showed that the flowperformance in the minireactor was almost in the plug flow condition. The micromixingperformance was expressed with a segregation index, which could be �0.002. Thesingle-phase micromixing performance reached the desired level. The result of precipi-tation of BaSO4 showed that the mixing performance had a substantial influence on theparticle size and size distribution. It was found that in the single-phase mixing process themixing performance was mainly influenced by the phase flux and the membrane pore size.With decreasing dispersed fluid flux or the membrane pore size, or with increasingcontinuous fluid flux, the micromixing performance was enhanced. The dye extractionmethod can correctly determine the mixing performance of a two-liquid phase-mixingsystem. Unlike the single-phase mixing process, in the two-liquid phase-mixing processthere was a minimum value for the mixing efficiency with the change of the continuous-or dispersed-phase flux, when the membrane pore size was �0.9 �m. © 2005 AmericanInstitute of Chemical Engineers AIChE J, 51: 2923–2929, 2005Keywords: membrane dispersion, micromixing, macromixing

Introduction

Mixing, a widely used unit operation process in the chemicalindustry, has a decisive impact on the overall performance ofthe processing engineering. Mixing can be classified as micro-mixing (mixing at the molecular scale), mesomixing, and mac-romixing. Many methods have been developed over the lastfew years to describe the mixing performance in differentdevices. The most popular of broadly considered methods of

measuring macromixing performance is to elucidate the resi-dence time distribution (RTD). Residence times and residencetime distributions are important characteristics for all chemicalreactors because they provide information about macromixingand the flow behavior in the respective reactors.1,2 Some au-thors3-6 have reviewed the characteristics of micromixing andits available measurement methods. Almost all the developedmethods are based on mixing-sensitive chemical conversions,and can be classified into three main schemes: single reaction(A � B3 R), consecutive competing reactions (A � B3 R;R � B 3 S), and parallel competing reactions (A � B 3 R;C � B3 S). In the latter two schemes, yield and/or selectivity,achieved for specific reaction products, constitute the parame-

Correspondence concerning this article should be addressed to G. S. Luo [email protected].



ters for quantifying the mixing performance, of which the“Villermaux/Dushman method” has gained wide acceptance.The published results show that the micromixing performancecan be determined very well when the volume ratio betweenthe two miscible solutions is very large. On the other hand,most of the methods are designed to describe the single-phasemixing process in which one miscible solution is mixed intoanother solution. For the multiphase-mixing processes onlyfew results can be referenced at the present time. In 2004 a dyeextraction method7 was developed to explain a liquid–liquidmixing process, which is based on the phase transfer of thesolvatochromatic dye Nile red from a water/methanol phase toa heptane phase.

Over the past decades, microstructured devices have trig-gered an explosion of scientific and industrial interest. Therapid development of microchemical systems has led to aconsiderable variety of microstructured devices, especially formicroreaction and micromixing processes. Many researchersfocused on the micromixing performances in different micro-structured micromixers.7-11 The micromixing characteristic isconsidered to be the basis for development of new microchem-ical systems. Ehrfeld and coworkers 8 studied the micromixingperformance in several micromixers with a competing reactionsystem, and compared it with that in the traditional macro-scopic mixing units. Panic et al.7 provided an improved exper-imental protocol for the Dushman reaction by controlling thefinal pH value in the reactors, which supplied more reliableexperimental data than those in Ehrfeld et al.8 They alsodescribed the mixing performance of five micromixers whosemixing principle and internal structure geometry were differ-ent. Two experimental methods of competing parallel reactionand dye extraction were introduced.

In our laboratory a new kind of microstructured device,designated membrane dispersion minireactor, was developed,in which microfiltration membranes are applied as the disper-sion media.12-16 In the reactor, single-phase mixing and mul-tiphase-mixing processes can be successfully performed. Whena liquid–liquid mixing process is carried out, the dispersionphase is dispersed as small droplets into the continuous phasewithin a very short time. Previous works demonstrated that thistype of minireactor, like most of the microstructured devices,has the characteristics of high mass transfer efficiency, largecapacity, low energy cost, and controllability.14 The membranedispersion minireactor has been used in liquid–liquid extractionprocesses, such as oil deacidification,15 extraction of citric acidand succinic acid,14,16 and in the preparation process of nano-particles such as BaSO4

12 and TiO2.13 However, the mixingperformance of the new device has not been tested. To better

understand the mixing performance of the membrane disper-sion minidevice, several experimental approaches have beenintroduced. The macromixing performance was characterizedwith RTD curves. The micromixing performance of the singlephase was detected by a parallel competing reaction system andthe liquid–liquid phase-mixing process was denoted by a dye-extraction method. Moreover, the mixing performance wasfurther tested by the preparation of BaSO4 nanoparticles.

Experimental Equipment and ApproachesMinireactor

An illustration of the membrane dispersion minireactor usedin this work is shown in Figure 1. It can be seen that there arethree main sheets in the minireactor: two stainless steel sheetsand a membrane sheet. The dispersed solution inlet and dis-persed solution chamber are in one sheet. The continuoussolution inlet, mixing chamber, and slurry outlet are in theother sheet. The three sheets are assembled with bolts. In thisminireactor, the dispersed solution (phase) is pressed through amicrofiltration membrane into the mixing chamber to mix withanother miscible solution or immiscible phase. The geometricsize of the mixing chamber was 12 � 4 � 1 mm. The Nimicrofiltration membrane with pore sizes of 0.2 and 0.9 �m,and stainless steel membranes with a pore size of 5 �m wereapplied, respectively.

Measurement of RTD curves

Residence time distribution curves can provide necessaryinformation for better understanding of the macromixing per-formance in chemical reactors.1,2 In this work, a stimulus–response method was applied to determine the residence time,and the experimental setup17,18 is shown in Figure 2. A solutionof methylic orange was selected as the tracer, and the diluteaqueous solution of methylic orange was injected into thereactor with a six-way valve (Waters automated switchingvalve). Two online ultraviolet (UV) spectrometers (8823A,Beijing Institute of New Technology Application, China) wereset at the inlet and outlet to detect the concentration.

Parallel competing reaction for single-phase mixing

Micromixing was studied by means of a parallel competingreaction: the Dushman reaction between iodide and iodatecoupled with a neutralization reaction.3,4 The reaction formulasare shown as follows

H2BO3� � H� 3 H3BO3 �quasi-instantaneous� (1)

5I� � IO3� � 6H� 3 3I2 � 3H2O �fast� (2)

Figure 1. Minireactor.

Figure 2. Experimental setup for measurement of RTDcurve.


I2 � I�7 I3� (3)

Many researchers have studied the kinetics of these reactions.19

The fractional yield of I2 and I3� were measured by spectro-

photometry (UV–vis recording spectrophotometer, UV-8345,Hewlett–Packard) at 353 and 286 nm. A segregation index, XS,is defined to explicitly quantify the micromixing quality,4

which is defined as

XS �Y

YST� �nI2 � nI3

�

nH�,0��6nIO3

�,0 � nH2BO3�,0

3nIO3�,0

��

(CI2�CI3�)V

nH�,0�6CIO3

�,0 � CH2BO3�,0

3CIO3�,0

� (4)

In Eq. 4, Y is the ratio of acid mole number consumed byreaction 2 divided by the total acid mole number injected; YST

is the value of Y in total segregation case when the micromix-ing process is infinitely slow. So the value of XS lies between0 and 1; for perfect micromixing, XS � 0, and in a totallysegregated medium, XS � 1.

The experiment setup is shown in Figure 1. In the single-phase mixing process, two reactant solutions, diluted sulfuricacid solution and a solution containing the other reactants, wereprepared. The acid solution was pumped into the dispersed-phase chamber, and the other solution was pumped directlyinto the mixing chamber. The flow rates were controlled by twometering pumps. The triiodide complex formed in the minire-actor was detected by sampling at the outlet.

Nanoparticle preparation

The precipitation of barium sulfate has been investigated byexperimental method to determine the influence of the mixingperformance in the reactors.20-23 Precipitation experiments withbarium sulfate show some influence of micromixing intensityon the particle size and of macromixing on particle morphol-ogy.23

Two reactant solutions of sodium sulfate and barium chlo-ride were pumped into the minireactor with two meteringpumps, and the two miscible solutions were micromixed in thereactor with sodium sulfate solution as the continuous fluid.Because the reaction system was at supersaturation, nanopar-

ticles of barium sulfate were synthesized. The particles wereseparated by centrifugation. After being washed five times withdistilled water and once more with absolute ethyl alcohol, andthen dried in an oven at 80°C for 12 h, the barium sulfatenanoparticles were obtained. The morphology of the nanopar-ticles was characterized by SEM (JEM-6301F, JEOL, Tokyo,Japan) and TEM (JEM200CX, JEOL) images.

Dye-extraction experiment for liquid–liquid mixing

The micromixing performance of a liquid–liquid mixingprocess in the new minireactor was measured with a dye-extraction method.7 The solvatochromatic dye Nile red wasselected as the indicator, water/methanol as the aqueous phase,and n-heptane as the oil phase. Nile red was extracted from theaqueous phase to the oil phase. In the membrane dispersionminireactor, the dispersed phase was dispersed as microdrop-lets into the continuous phase through the microfiltration mem-brane, which provided a substantial interface area for masstransport in the two-liquid mixing process. The two phaseswere pumped into the minireactor from the continuous-phaseinlet and dispersed-phase inlet, respectively. The mixed oil-in-water or water-in-oil suspension flowed out from the outlet.After quick phase separation, the Nile red concentration in theorganic phase was measured with a spectrophotometer (UV–vis recording spectrophotometer, UV-8345, HP) at 489 nm andthe amount of the extracted dye was calculated.

Results and DiscussionMacromixing performance

Figures 3 and 4 show the stimulus–response curves obtainedunder various dispersion-phase fluxes. The response curves atthe outlet are very similar to those stimulus curves at the inlet,indicating that the fluid passed through the minireactor in anideal plug-flow mode. The results mean that the residence timeis the function of just two phase flow rates. The axial mixingcoefficients are very small, and uneven flow could almost beignored.

Micromixing performance of the single-phase mixingprocess

Equation 4 indicates that Xs is a function of transfer rate ofH�. Xs will increase if the transfer rate of H� decreases, which

Figure 3. Residence time distribution curves (0.9 �mmembrane).

Figure 4. Residence time distribution curves (5 �mmembrane).


means more segregated medium to be reached. The effect ofthe dispersed fluid flux on the segregation index Xs is shown inFigure 5. Xs decreased with an increase of the dispersed fluidflux. With the increase of the dispersed fluid flux, the turbu-lence intensity and the mass transfer efficiency of H� wereimproved. Then, the segregation index Xs decreases, and themixing performance becomes enhanced. The mixing perfor-mance in the minireactor is determined not only by micromix-ing, but also by meso- or macromixing. When the flux ratio issmall, the mixing performance is mainly determined by micro-mixing, although with increasing flux ratio the role of meso- ormacromixing to the reactor’s performance becomes increas-ingly more important. Usually the flux ratio is �0.1 for deter-mining micromixing performance in normal stirring reactors.However, the flux ratio in this work ranged from 0.1 to 10.0,which is always applied in real industry processes. Thereforethe obtained data in this work are much more reliable andcomparable for designing and scaling-up the equipment.

Figure 6 shows the effect of the continuous fluid flux on thesegregation index. As the continuous fluid flux increases, thesegregation index decreases, which indicates that the mixingperformance becomes enhanced. The segregation index couldbe �0.002, which means an almost perfect mixing state. In-creasing continuous fluid flux results in a decreasing residencetime, which does not confer any benefit to the mixing, but

Figure 5. Effect of dispersed fluid flux on segregationindex.

Figure 6. Effect of continuous fluid flux on segregationindex.


rather results in decreasing flux ratio and increasing turbulenceintensity, two phenomena that are of substantial benefit tomicromixing. Meanwhile an increase in continuous fluid fluxcould reduce the mixing size, and the mixing process would bepredominantly governed by the micromixing characteristic. Ifthe micromixing characteristic is robust, the mixing perfor-mance of the minireactor will be closer to the perfect mixingstate.

On the other hand, it can be seen that the segregation indexis in the range of 0.010–0.032, shown in Figures 5 and 6. Thus,it is possible to conclude that the micromixing performance ofthe minireactor is much better than that of normal reactors, andthe micromixing performance can be improved through in-creasing the continuous fluid flux and the dispersed fluid flux.

Figure 7 shows the effect of the membrane pore size on themixing performance, which was substantially influenced by themembrane pore size. When the pore size was 0.9 or 5 �m, themixing performance under the same experimental conditionswas similar. When the pore size was 0.2 �m, the mixingperformance was enhanced to an even greater degree. Obvi-ously the membrane pore size will influence the mixing size.With a decrease of the pore size the mixing size decreases, andthus a better mixing state could be attained. The curves inFigure 7 also show that the micromixing performance can befurther improved by increasing the continuous fluid flux.

Precipitation of nanoparticles

Figure 8 is a SEM microphotograph of BaSO4 nanoparticlesprepared by mixing 100 mol/m3 BaCl2 solution with 400mol/m3 Na2SO4 solution. When the continuous fluid flux andthe dispersed fluid flux were 0.40 and 0.37 cm3/s, respectively,the particles prepared in the membrane dispersion reactor werein the sub–200-nm range, which is much smaller than the sizeof those (range: 1–10 �m) precipitated in the normal Taylor–Couette reactor or other classic stirred tank reactors.20-23 Thesize of the particles is affected by the micromixing perfor-mance.23 The better performance of micromixing results insmaller particles. Thus compared against other reactors, thenew minireactor has a potentially higher micromixing perfor-mance.

In our previous work,12 the relationship between the opera-tion conditions and the average particle size was investigated.The particle size decreases with increasing continuous fluidflux. An increase of the continuous fluid flux results in bothincreasing supersaturation and improvement of the micromix-ing, both of which favor precipitation of nanosized particles.

The influence of the membrane pore size on the average

Figure 7. Effect of membrane pore size on micromixing.

Figure 8. SEM images of BaSO4 particles.BaCl2: 100 mol/m3, 0.37 cm3/s dispersed fluid; Na2SO4: 400mol/m3, 0.40 cm3/s continuous fluid; 0.2 �m membrane.

Figure 9. Influence of membrane pore size on averagesize of BaSO4 nanoparticles.

Figure 10. Influence of membrane pore size on size-dis-tribution of BaSO4 nanoparticles.Dispersed phase: BaCl2, 100 mol/m3, 0.37 cm3/s; continu-ous phase: Na2SO4, 400 mol/m3, 0.40 cm3/s.


sizes and particle size distribution of barium sulfate is shown inFigure 912 and Figure 10. It can be seen that the averageparticle sizes decrease and the particle size distributions be-come narrower with decreasing membrane pore size. As themembrane pore size decreases, the micromixing performanceis vastly improved, resulting in improved morphology and sizedistribution of the nanoparticles.

Micromixing performance of liquid–liquid mixingprocess

The processes of multiphase mixing are always encounteredin unit operation and chemical reaction engineering. To under-stand the mixing performance of multiphase-mixing processes,a liquid–liquid mixing process in the minireactor was carriedout. After phase separation, the absorbance of the organicphase was measured. The results with the oil phase used as thedispersed phase are plotted in Figures 11 and 12. The influ-ences of the membrane pore size are also shown in thesefigures. When the membrane pore size was 0.2 �m, the mixingefficiency of the two-liquid phase-mixing process was similarto that of the single-phase mixing process, and was increasedwith increasing flow fluxes. However, when the membranepore size was 0.9 or 5 �m, unlike the single-phase mixingprocess, the mixing efficiency was initially decreased with anincrease in the continuous-phase flux. After a minimum valuewas reached, it was increased. The results are similar to the

mass-transfer performances of a membrane dispersion extrac-tor 24 and the results in Panic et al.7 and Benz et al.25 In thetwo-liquid phase-mixing process the mixing efficiency is afunction of the mixing condition, mixing size (droplet size),and overall mass-transfer coefficients. When the membranepore size is smaller, the mixing size decreases and the mass-transfer coefficients increase with the continuous-phase flux.The mixing becomes more uniform, and thus the mixing effi-ciency is higher. However, when the membrane pore size islarger, the mixing size becomes larger. Therefore, the mixingconditions deteriorate and mixing efficiency decreases. As thecontinuous-phase flux increases, both the mixing size and theresidence time will decrease, although the mass-transfer coef-ficient will increase. At a lower flux value the mixing conditionbecomes worse, and the mixing efficiency is predominantlycontrolled by the residence time. When the flux was �0.6cm3/s, the mixing condition became more uniform and theoverall mass-transfer coefficient was higher. Thus, the mixingefficiency was increased with the continuous-phase flux.

Figures 13 and 14 show the experimental results of thetwo-liquid phase-mixing process when the aqueous phase wasused as the dispersed phase. The results in Figures 13 and 14are similar to those in Figures 11 and 12. The mixing efficiencyfor using the aqueous phase as the dispersed phase was lowerthan that for using the organic phase as the dispersed phase.The mixing size (droplet size) is affected not only by themembrane pore size, but also by the system’s physical prop-

Figure 11. Absorbance at 489 nm with organic phase asthe dispersed phase.

Figure 12. Extraction efficiency with organic phase asthe dispersed phase.

Figure 13. Absorbance at 489 nm with aqueous phaseas the dispersed phase.

Figure 14. Extraction efficiency with aqueous phase asthe dispersed phase.


erties and the wetting ability of the membrane to the twophases. When the dispersed phase was changed from the oilphase to the aqueous phase, the droplet size became larger as aresult of the stronger wetting ability of the membranes to theaqueous phase. This is the possible reason for the deteriorationof the mixing performance.

Conclusions

The mixing performances of a novel kind of membrane disper-sion minireactor have been investigated with various experimentalapproaches. The RTD result showed that the flow performance inthe minireactor was almost in the plug-flow condition. The single-phase micromixing performance measured with the Dushmanreaction (iodide–iodate) in the minireactor attained the desiredcondition, and the segregation index could be �0.002. The mi-cromixing performance improved with the increasing of the con-tinuous fluid flux and dispersed fluid flux. Moreover, the micro-mixing performance was apparently enhanced by decreasing themembrane pore size. The result of precipitation of BaSO4 showedthat the mixing performance had a substantial influence on theparticle size. The membrane pore size is a major contributingfactor to micromixing performance. With increasing membranepore size, the micromixing performance became better, and theBaSO4 particle size apparently decreased. It was found that thedye-extraction method can be applied to correctly determine themixing performance of a two-liquid phase-mixing system. Likethe single-phase mixing process, the mixing performance of thetwo-liquid phase system was influenced by the membrane poresize and the two-phase flow rate. When the pore size was �0.9�m, however, there was a minimum value for the mixing effi-ciency with the changes of the continuous- or dispersed-phaseflux.

AcknowledgmentsWe gratefully acknowledge the financial support of the National Nature

Science Foundation of China (20476050; 20490200) and SRFDP(20040003032).

Notation

d32 � mean diameter of particles, d32 � ¥ d 3/¥ d 2, nmFc � flux of continuous solution (phase), cm3/sFd � flux of dispersed solution (phase), cm3/sFt � total flux of dispersed solution (phase) and continuous solution

(phase), cm3/sR � flux ratio of dispersed phase and continuous phase

XS � segregation indexu � electric voltage information of absorbance of methyl orange, volt

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Modeling of an IGC Experiment to AnalyzeTernary Polymer–Solvent Systems

Peter K. Davis, J. Larry Duda, and Ronald P. DannerCenter for the Study of Polymer Solvent Systems, Dept. of Chemical Engineering, The Pennsylvania State University,

University Park, PA 16802


A variation of the capillary column inverse gas chromatography (IGC) experiment isproposed to measure the thermodynamics and diffusion coefficients in ternary polymer–solvent–solvent systems, and a theoretical model has been developed for this proposedexperiment. The model has been derived to include both main and mutual cross-diffusioncoefficients. Ternary vapor–liquid equilibria was represented by a coefficient matrix of theisotherm tangents. The governing equations of the model were uncoupled and solved usingboth an analytical Laplace transform technique and a numerical finite-difference tech-nique. Solution of the model shows that in the presence of strong cross-diffusion andthermodynamic coefficients, unusual chromatographic behavior can be observed, sug-gesting that IGC can be a useful technique for measuring thermodynamic and mass-transport properties of ternary polymer–solvent–solvent systems. © 2005 American Instituteof Chemical Engineers AIChE J, 51: 2930–2941, 2005Keywords: inverse gas chromatography (IGC), multicomponent diffusion and thermody-namics, polymer–solvent systems, modeling

Introduction

Capillary column inverse gas chromatography (CCIGC) hasproven to be a useful experimental technique to measure ther-modynamic and mass-transport properties in polymer–solventsystems.1-5 In CCIGC, a thin film of polymer is coated on theinside of a capillary wall. An inert carrier gas transports a smallpulse of solvent through the column and a detector measuresthe elution of solvent from the column. Because the solventabsorbs and diffuses through the thin polymer film, the solventelution can be related to the partition coefficient and diffusioncoefficient by comparison to a theoretical model. Capillarycolumns have a major advantage over packed columns becausethe polymer coating thickness is much more uniform, thus

allowing accurate measurement of diffusion coefficients. In-verse gas chromatography has been used in the past to measurediffusion coefficients for binary and pseudobinary polymer–solvent systems. Danner et al.5 used finite-concentrationCCIGC to measure effective diffusion coefficients of tolueneand methanol in the PVAc–methanol–toluene system.

A true ternary polymer–solvent–solvent system has a matrixof four mutual diffusion coefficients that govern mass transport

�Dp� � �Dp11 Dp12

Dp21 Dp22� (1)

The subscripts indicate solvent 1 and solvent 2. The Dp11 andDp22 are referred to as the main terms of the diffusion coeffi-cient matrix and Dp12 and Dp21 are the cross terms. Thediffusive flux of each solvent, ji

�, in such a system can bedefined in terms of these diffusion coefficients relative to thesystem’s volume average velocity

Correspondence concerning this article should be addressed to J. L. Duda [email protected].



j1� � �Dp11��1 � Dp12��2 (2)

j2� � �Dp21��1 � Dp22��2 (3)

Here, ��i is the gradient in mass concentration of solvent i.Finite-concentration IGC has been used by Joffrion and

Glover6 and Tsotsis et al.7 to measure ternary-phase equilibriaand effective binary diffusivities, but not the cross-diffusivities.Diffusion coefficients in polymer–solvent systems are usuallymeasured by gravimetric-type experiments. Binary gravimetricexperiments are fairly simple. The single binary diffusion co-efficient is determined from the time needed to equilibrate apolymer sample with the vapor of solvent. Gravimetric exper-iments proceed by measuring the weight gain of the polymervs. time. There are several sorption experiments that rely onthis type of measurement, although these experiments fail forternary experiments because the weight gain of the samplearises from sorption of both solvents. Thus, it is not possible todetermine how much of each solvent is in the polymer vs. time.Furthermore, the vapor is no longer pure, and the compositionof this phase must also be known as a function of time. Amodified quartz spring gravimetric experiment has been used tomeasure solubility (not diffusion) of two solvents in a poly-mer.8,9

Although gravimetric experiments are quite common formeasuring diffusion coefficients, they are not the only availabletechniques. Infrared spectroscopy has been used to measure thecross-diffusion coefficients in the PIB–MEK–toluene system.10

Drying experiments are another possibility to measure ternarydiffusion coefficients.11 In such an experiment, a polymer–solvent–solvent solution is devolatilized in a convective oven.Measurement of the effluent solvent concentration in the vaporphase can be used to determine the polymer solution weightloss arising from both solvents vs. time. Such an experimentrelies on extensive modeling because mass and heat transferoccur in both the vapor and polymer phases. In addition, theexperiment is conducted over extremely wide ranges of solventconcentration in the polymer phase. Thus, the experiment reliesheavily on a model for the concentration dependency of thediffusion coefficients. Such models do exist, but have not beentested because of the absence of any ternary diffusion data.12,13

Inverse gas chromatography has a major advantage over dryingbecause experiments are conducted at a single concentration(usually at infinite dilution). However, experiments can beconducted at finite concentrations as shown by several inves-tigators.5,6,14-16

In this communication, a new CCIGC experiment is pro-posed to measure the four member diffusion coefficient matrixand vapor–liquid equilibria at finite concentrations in ternarypolymer–solvent–solvent solutions. Analysis of such an exper-iment requires a model that describes the complex diffusionprocess in the column. A new CCIGC model has been devel-oped capable of analyzing ternary CCIGC data.

Proposed Experimental Procedure

The proposed experiment is an extension of that used by Dan-ner et al.5 to measure properties of pseudobinary systems. Adiagram of this new experimental setup is shown in Figure 1.

A quartz capillary column is coated with a thin film of the

polymer of interest and a ternary mixture of an inert carrier gasand vapor of the two solvents is continuously passed throughthe column. The vapor composition is controlled by the tem-perature of the two saturators containing each of the puresolvents. The effluent streams from the saturators are thenmixed and sent through the column giving a ternary vapormixture of constant concentration. When equilibrium is at-tained, a small amount of one of the solvents is injected into thecolumn. This perturbation of solvent travels through the col-umn and interacts with the polymer–solvent–solvent mixture.A suitable detector measures the vapor concentration of eachsolvent at the exit of the column. The detector must be capableof distinguishing between each solvent while they are eludingsimultaneously. The common thermal conductivity detector(TCD) or flame ionization detector (FID) will not work becausethese will give an average reading for both solvents. Becausethe injection perturbs the vapor composition of both solvents,the experiment will produce elution profiles for both solvents.Analysis of both elution profiles gives information about thethermodynamics and diffusion coefficients of the ternary sys-tem.

Model for Ternary CCIGC Experiments

The original CCIGC model was developed by Macris1 forbinary polymer–solvent systems at infinite dilution. The vaporphase was pure carrier gas containing a small solvent concen-tration from the injection. Here, this model is extended toternary polymer–solvent–solvent systems at finite solvent gasconcentrations. A diagram of the process to be modeled isshow in Figure 2.

In the derivation of the model, the following assumptions aremade:

(1) The entire system is isothermal.(2) The pressure drop in the column is insignificant.(3) The polymer coating thickness is much smaller than the

column radius.(4) Gas-phase diffusion is fast enough to keep the solvent

radially well mixed, making Taylor dispersion insignificantcompared to conventional axial dispersion.

(5) The polymer-phase diffusion coefficients are constant

Figure 1. Proposed experimental setup for a ternaryCCIGC experiment.


within the small concentration range of the solvent perturba-tions.

(6) The vapor-phase diffusion coefficients are constant andthe vapor-phase cross-diffusion coefficients are negligible.

(7) The thermodynamics of bulk absorption can be de-scribed by a coefficient matrix of the isotherm tangent slopes.These coefficients are constant over the small concentrationrange of the perturbation.

(8) The polymer film has a uniform thickness through theentire column.

(9) The only significant diffusion in the polymer phaseoccurs in the radial direction.

(10) The carrier gas is insoluble in the polymer and does notadsorb on any surfaces.

(11) Surface adsorption by the solvents is negligible.(12) No chemical reactions occur.(13) The partial molar volume of the solvents in the polymer

is constant.(14) Swelling of the polymer is insignificant over the con-

centration range of the perturbation.(15) The solvent injection can be modeled by a Dirac delta

function.Based on assumption (2), the pressure in the column is con-stant. Because the total pressure in the column is never muchgreater than atmospheric, it will be assumed that the ideal gaslaw applies. An ideal gas at constant temperature and totalpressure has a constant molar density Cmolar

Cmolar �C1

M1�

C2

M2�

C3

M3�constant� (4)

Here, Ci is the mass concentration of each species in the gasphase and Mi is the molecular weight of each species. Thesubscripts define the following species in the vapor: 1 �solvent 1, 2 � solvent 2, and 3 � inert carrier gas. Further-more, because the solvent perturbation is small compared to theoverall concentration, the overall mass density (C) of the gas isapproximately constant

C � C1 � C2 � C3 � constant (5)

Under this approximation, the species continuity equationsin the gas phase can be expressed relative to the mass averagevelocity (V)

�C1

�t�

�

� z�C1V� � Dg11

�2C1

� z2

�2

R �Dp11

��1

�r �r�R

� Dp12

��2

�r �r�R

� (6)

�C2

�t�

�

� z�C2V� � Dg22

�2C2

� z2

�2

R �Dp21

��1

�r �r�R

� Dp22

��2

�r �r�R

� (7)

�C3

�t�

�

� z�C3V� � ��Dg11

�2C1

�z2 � Dg22

�2C2

�z2 � (8)

Here, �i represents the mass concentrations of the solvents inthe polymer phase. In the polymer phase, the subscript 1denotes solvent 1 and 2 represents solvent 2. In this formula-tion, the cross-diffusion coefficients in the gas phase are as-sumed to be negligible. The appropriate boundary conditionsare

t � 0 �C1

C2

C3

� � �Cb1

Cb2

Cb3

� V � V0 (9)

z � 0 �C1

C2

C3

� � ��t�C01 � Cb1

��t�C02 � Cb2

��t�C03 � Cb3

� (10)

z � �C1

C2

C3

� � �Cb1

Cb2

Cb3

� V � V0 (11)

Here, C0i is the strength of the pulse of species i and Cbi is thebackground (plateau) concentration of species i.

Assuming that the polymer thickness is much smaller thanthe column radius, the two solvent species continuity equationsin matrix notation for the polymer phase are

�

�t ��1

�2� � �Dp11 Dp12

Dp21 Dp22� �2

�r2 ��1

�2� (12)

t � 0 ��1

�2� � ��b1

�b2� � ��0

Cb1

Kp11dC1 � �0

Cb2

Kp12dC2

�0

Cb1

Kp21dC1 � �0

Cb2

Kp22dC2(13)

r � R ��1

�2� � ��0

C1

Kp11dC1 � �0

C2

Kp12dC2

�0

C1

Kp21dC1 � �0

C2

Kp22dC2 (14)

Figure 2. Ternary CCIGC experiment.


r � r � ��

�r ��1

�2� � �0

0� (15)

Here Kpij is the slope of the isotherm tangent and �bi is thebackground (plateau) concentration of species i.

The diffusive fluxes are defined with respect to the volumeaverage velocity in the polymer phase. Equations 6, 7, 8, and12 constitute five coupled partial differential equations thatmust be solved to model the elution of the two solvents fromthe column. These equations were uncoupled using a coordi-nate transformation17 and were solved using the technique ofLaplace transforms. The details of the solution are in AppendixA. In the solution, it is shown that the solvent concentration inthe gas phase at the detector can be expressed in the Laplacedomain

Yi � Hiexp 1

2�i� � 1

4�i2 � i�s�� (16)

Here, Yi is a pseudo-Laplace variable for the gas concentrationof solvent i at the detector. Hi, �i, and i(s) are vectors ofdimensionless parameters. Analytical inversion of Eq. 16 intothe time domain is not possible, so numerical inversion wasused to solve the model by a Fast Fourier Transform (FFT)algorithm.

Model Predictions for Ternary CCIGCExperiments

The purpose of deriving this model is to permit analysis ofexperimental ternary elution profiles with the intent of deter-mining the matrix of diffusion and thermodynamic coefficients.The Dpii and Kpii are often referred to as the main terms,whereas Dpij and Kpij are referred to as the cross terms in thecoefficient matrices.

The aim here is to measure all four diffusion and thermo-dynamic coefficients. It is questionable that all eight uniqueparameters can be obtained from the elution profiles of a singleCCIGC experiment. For this reason, experiments should beconducted with two different columns. The first column shouldbe designed such that there is negligible diffusion resistance inthe polymer phase. This is accomplished by coating a very longcolumn with a sufficiently thin polymer coating. Because therewill be negligible mass transfer resistance, experiments withthis column will produce symmetric elution profiles that areindependent of the diffusion coefficients. These elution profilescan be uniquely related to the thermodynamic coefficients.Once the thermodynamic coefficients are known, a second,shorter column should be made with a thicker coating to givesignificant diffusion resistance. Experiments on this columnwill give asymmetric elution profiles that can be used to obtaina unique set of four mutual diffusion coefficients.

Diffusion coefficient sensitivity

Experiments on the short, thick column will produce elutionprofiles that depend on the diffusion coefficients. It is possiblethat mass transport in the ternary system can be described usingonly the main diffusion coefficients (cross-diffusion coeffi-cients equal to zero). This effectively corresponds to two

pseudobinary diffusion processes because the two solvents willdiffuse independent of each other. Although the proposed ex-periment is capable of measuring such pseudobinary diffusioncoefficients, it is also possible that both main and cross-diffu-sivities are needed to accurately describe the mass-transportprocess. In such a case, the elution profiles of the two solventsmust provide information about all four diffusion coefficients.To demonstrate the effect of the cross-diffusion coefficients onthe elution profile, the model described above was solved overvarying ranges of the cross-diffusion coefficients. Nominalvalues of the other parameters were used in the simulations.The cross thermodynamic coefficients were assumed to bezero. The results of the simulations are shown in Figures 3, 4,and 5. The dimensionless parameters for the simulations aregiven in Table 1. In each of the figures, a different value of theratio of the cross-diffusion coefficient to the main diffusioncoefficient for both of the two solvents was used.

In Figure 3, the cross-diffusion coefficients are 100 timessmaller than the main diffusion coefficients. Under such con-ditions, the cross-diffusion coefficients make essentially nocontribution to the shape of the elution profiles. There was alsoonly a negligible difference when the cross terms were 75%smaller than the main terms (not shown). In these cases, masstransport in the ternary system can be accurately represented astwo pseudobinary systems because diffusion of the two sol-vents can be described by just the main diffusion coefficients.

Figure 3. Simulation of a ternary CCIGC experiment.Solid line is ternary model prediction for solvent 1 and dottedline is that for solvent 2. Symbols indicate ternary modelsimulation for insignificant cross terms (Dpij � 0): solvent 1(Œ), solvent 2 (f).

Table 1. Values of the Dimensionless Groups for theSimulations in Figures 3, 4, and 5*

Parameter Figure 3 Figure 4 Figure 5

11 0.53 0.53 0.5322 0.13 0.13 0.1312 � 21 �11 1.0 1.0 1.0�12 10.0 1.1 1.01�21 10.0 1.1 1.01�22 1.0 1.0 1.0�1 � �2 1.0 10�4 1.0 10�4 1.0 10�4

� 1.0 1.0 1.0

*Dimensionless group definitions are given in Appendix A.


However, the diffusion cross terms used in Figures 4 and 5definitely give elution profiles different from those obtainedwithout cross terms. In these cases, it is not possible to repre-sent the ternary diffusion processes as two pseudobinary sys-tems. Interestingly, the elution profiles give very unusual be-havior in the presence of significant diffusion cross terms.Multiple peaks and solvent concentrations dipping below theirbaseline are observed under such conditions. This unique be-havior can be quite useful when trying to determine the sig-nificance of the cross-diffusion coefficients in polymer–solventsystems. If the cross-diffusion coefficients are negligible, sin-gle peaks and positive dimensionless concentrations will beobserved in a CCIGC experiment. If multiple peaks and neg-ative dimensionless concentrations are observed from the IGC,then the cross-diffusion coefficients are significant and can bemeasured. Performing experiments at different solvent concen-trations can give the concentration dependency of both mainand cross-diffusion coefficients in ternary polymer–solventsystems.

The diffusion coefficients in the matrix are not independent

of each other. They are constrained by material balances, theOnsager reciprocal relationships, and thermodynamic stability.The Onsager relationships and thermodynamic stability specifythat the diffusion coefficient matrix has real-positive eigenval-ues.18 For a ternary system, the necessary constraint to ensurethis is

Dp11 � Dp22 0

Dp11Dp22 � Dp12Dp21 � 0

�Dp11 � Dp22�2 � �Dp11Dp22 � Dp12Dp21� (17)

In all simulations, these expressions were obeyed and nomaterial balances were violated. Material balance violation canbe detected by negative dimensional concentrations.19 Negativeconcentrations can be obtained by solving the continuity equa-tions with unrealistic diffusion coefficients. Although the con-tinuity equations can be mathematically satisfied with negativeconcentrations, a negative mass cannot physically exist. Thus,any negative concentrations violate material balance con-straints. In these simulations, no negative concentrations wereobtained.

Numerical solution

To this point, model simulations were obtained from theanalytical Laplace domain solution. A numerical solution wasalso developed for the model based on an implicit finite-difference method. The numerical model is quite useful be-cause it can be used to verify some of the assumptions neededfor the analytical solution. For example, in the analytical so-lution (see Appendix A), it was assumed that the gas-phasediffusion coefficient matrix was symmetrical (Dg11 � Dg22). Itwas argued that this assumption had no bearing on the elutionprofiles of the solvents. To test this argument, the numericalsolution was generated for Dg11 � Dg22 and for Dg22 � 10 Dg11. The results of the simulation are shown in Figure 6. Even



Figure 6. Numerical simulations to determine the effectof axial dispersion on the elution profile.Solid line is analytical model prediction for solvent 1 anddotted line is that for solvent 2 when Dg11 � Dg22. Symbols:solvent 1 (Œ), and solvent 2 (f), indicate numerical modelsolution when Dg22 � 10Dg11. All other parameters were thesame as those used in Figure 4.


when the gas-phase diffusion coefficients are different by afactor of 10, the elution profiles are still the same. This verifiesthe assumption of a symmetric Dg matrix. More details for thenumerical solution can be found in Appendix B.

Parameter Estimation

In the proposed experiment, the diffusion coefficient andthermodynamic coefficient matrices will be obtained from theelution profiles of the two solvents. Eight unknown parametersare needed to completely describe these elution profiles. Asmentioned, experiments should be carried out using two dif-ferent columns. The first column should be designed to give aninsignificant amount of diffusion resistance in the polymer.Experiments on this column can be used to obtain the thermo-dynamic coefficients Kij. As described, these are the tangents tothe isotherm planes and, although constant during any experi-ment, they will vary with bulk composition changes. After anumber of Kij with sufficient proximity in concentration havebeen measured, the actual bulk solubilities can be obtained bynumerical integration according to Eq. 13 subject to the initialconditions (Ci � 0, �i � 0).

Experiments on a second, shorter, thicker column shouldthen be used to obtain the diffusion coefficients. Inverse gaschromatography parameter estimation is usually carried out bymoment analysis.1-4 In a binary experiment, the first and secondmoments can be related to the partition and diffusion coeffi-cient, respectively. This procedure cannot be used directly forternary analysis because there are more parameters than mo-ments. In addition, the complex Laplace domain expression forYi makes the analytical expressions for the moments difficult toobtain. Thus, it is proposed that parameter estimation proceedby Fourier or time domain fitting. In this procedure, the pa-rameters are adjusted such that the experimental elution pro-files match the model elution profiles for each solvent. Thisprocedure was first introduced by Pawlisch2 and used by otherinvestigators for binary systems.4,5,20,21 In binary experiments,the partition and diffusion coefficient are adjustable parametersused to match the experimental elution to that of the binarymodel.

Caution must be exercised when measuring the diffusioncoefficients at finite concentrations arising from variation in thecoating thickness �. As the solvents swell the polymer coating,the film thickness will increase. In general, this will lead to anunderestimation of the diffusion coefficients. During any par-ticular experiment, the small solvent perturbation does notsignificantly change the coating thickness, but as the bulkconcentration changes from experiment to experiment, swell-ing will become important. However, because the experimentson the column with insignificant diffusion resistance give thethermodynamics of the polymer–solvent–solvent system, thisswelling can be predicted and accurate diffusion coefficientscan be obtained.

Summary and Conclusions

A new capillary column inverse gas chromatography exper-iment has been proposed to measure the four member diffusioncoefficient matrix and the vapor–liquid equilibrium in ternarypolymer–solvent–solvent systems. The proposed experiment iscapable of measuring the diffusion and partition coefficients

over wide ranges of solvent concentration. The uniqueness andpower of this technique is that an experiment is conducted bymeasuring a small solvent perturbation around a fixed compo-sition. Thus, experiments are performed at a given solventconcentration such that the diffusion and thermodynamic co-efficients can be considered constant. A model has been devel-oped that describes the complex diffusion process in the col-umn. Both analytical and numerical solutions have beenobtained. Simulation of the proposed experiment indicates thatCCIGC is quite sensitive to the cross-diffusion coefficientswhen they are of comparable order to the main terms. Thissensitivity makes CCIGC a promising technique to measure thevalues of all four diffusion coefficients and the true vapor–liquid equilibrium in ternary polymer–solvent–solvent systems.

Notation

�i � mass concentration of species i in the polymerphase, kg/m3

�bi � background mass concentration of species i inthe polymer phase, kg/m3

C � mass density of the vapor phase, kg/m3

Ci � mass concentration of species i in the vaporphase, kg/m3

Cbi � background mass concentration of solvent i inthe vapor phase, kg/m3

C0i � strength of the solvent pulse (kg-s/m3) for spe-cies i

Dpij � mutual binary diffusion coefficient matrix forthe polymer phase, m2/s

Dgij � mutual binary diffusion coefficient matrix forthe gas phase, m2/s

Kpij � dimensionless coefficient matrix (isotherm tan-gent slope) describing the thermodynamics be-tween the gas and polymer phases

L � column length, mMi � molecular weight of species i, kg/mol

r � radial coordinate, mR � radius of the gas–polymer interface, mt � time, s

V � carrier gas velocity, m/sV0 � characteristic carrier gas velocity, m/s

z � axial coordinate, m� � polymer coating thickness, m

ij � R/Kpij� � thermodynamic dimensionless group�ij

2 � V0�2/LDpij � polymer-phase diffusion dimensionless group�ij � Dgij/LV0 � axial dispersion dimensionless group

� � C01/C02 � ratio of the inlet pulse strengths

Literature Cited1. Macris A. Measurement of Diffusion and Thermodynamic Interactions

in Polymer–Solvent Systems Using Capillary Column Inverse GasChromatography. PhD Thesis. Amherst, MA: Dept. of Chemical En-gineering, University of Massachusetts; 1979.

2. Pawlisch CA, Laurence RL, Macris A. Solute diffusion in polymers. 1.The use of capillary column inverse gas chromatography. Macromol-ecules. 1988;20:1564.

3. Arnould DD. Capillary Column Inverse Gas Chromatography(CCIGC) for the Study of Diffusion in Polymer Solvent Systems. PhDThesis. Amherst, MA: Dept. of Chemical Engineering, University ofMassachusetts; 1989.

4. Hadj Romdhane I. Polymer–Solvent Diffusion and Equilibrium Pa-rameters by Inverse Gas–Liquid Chromatography. PhD Thesis. StateCollege, PA: Dept. of Chemical Engineering, The Pennsylvania StateUniversity; 1994.

5. Danner RP, Tihminlioglu F, Surana RK, Duda JL. Inverse gas chro-matography applications in polymer–solvent systems. Fluid PhaseEquilibria. 1998;148:171.

6. Joffrion LL, Glover CJ. Vapor–liquid equilibrium of the ternary n-


heptane/isopropyl alcohol/atactic polypropylene mixture from pertur-bation gas chromatography. Macromolecules. 1986;19:1710.

7. Tsotsis TK, Turchi C, Glover CJ. Comparison of multi-component gaschromatography measurements of vapor–liquid equilibrium with staticmeasurements using a polymer/two solvent ternary system. Macro-molecules. 1987;20:2445.

8. Tanbonliong JO, Prausnitz JM. Vapour–liquid equilibria for somebinary and ternary polymer solutions. Polymer. 1997;38:5775.

9. Liu H, Huang Y, Wang K, Hu Y. Vapor–liquid equilibria for mixedsolvents–polymer systems: Measurement and correlation. Fluid PhaseEquilib. 2002;194:1067.

10. Hong SU, Barbari TA, Sloan JM. Multicomponent diffusion of methylethyl ketone and toluene in polyisobutylene from vapor sorption FTIR-ATR spectroscopy. J Polym Sci Part B: Polym Phys. 1998;36:337.

11. Price PE, Wang S, Romdhane IH. Extracting effective diffusion pa-rameters from drying experiments. J Eng Appl Sci. 1997;43:1925.

12. Alsoy S, Duda JL. Modeling of multicomponent drying of polymerfilms. AIChE J. 1999;45:896.

13. Zielinski JM, Hanley BF. Practical friction-based approach to model-ing multicomponent diffusion. AIChE J. 1999;45:1.

14. Brockmeier NF, McCoy RW, Meyer JA. Gas chromatographic deter-mination of thermodynamic properties of polymer solutions. I. Amor-phous polymer systems. Macromolecules. 1972;5:464.

15. Brockmeier NF, McCoy RW, Meyer JA. Gas chromatographic deter-mination of thermodynamic properties of polymer solutions. II. Semi-crystalline polymer systems. Macromolecules. 1972;6:176.

16. Brockmeier NF, McCoy RW, Meyer JA. Gas chromatographic deter-mination of thermodynamic properties of polymer solutions. Macro-molecules. 1972;5:130.

17. Toor HL. Solution of the linearized equations of multicomponent masstransfer: I. AIChE J. 1964;10:448.

18. Kirkaldy JS, Weichert D, Zia-Ul-Haq. Diffusion in multicomponentmetallic systems. VI. Some thermodynamic properties of the D matrixand the corresponding solutions of the diffusion equations. Can J Phys.1963;41:2166.

19. Price PE, Romdhane IH. Multicomponent diffusion theory and itsapplications to polymer–solvent systems. AIChE J. 2003;49:309.

20. Surana RK, Danner RP, Tihminlioglu F, Duda JL. Evaluation ofinverse gas chromatography for prediction and measurement of diffu-sion coefficients. J Polym Sci Part B: Polym Phys. 1997;35:1233.

21. Surana RK. Advances in Diffusion and Partition Measurements inPolymer–Solvent Systems Using Inverse Gas Chromatography. PhDThesis. State College, PA: Dept. of Chemical Engineering, The Penn-sylvania State University; 1997.

22. Vrentas JS, Vrentas CM, Hadj Romdhane I. Analysis of inverse gaschromatography experiments. Macromolecules. 1993;26:6670.

Appendix A: Model Solution

To aid in the model solution, the following dimensionlessvariables are introduced

��1��2� � �

�1 � �1b

V0

LKp11C01

�2 � �2b

V0

LKp22C02

(A1)

�C�1C�2C�3� � �

C1 � Cb1

C01V0

LC2 � Cb2

C02V0

LC3 � Cb3

C03V0

L

(A2)

t� �tV

L(A3)

r� �r � R

�(A4)

z� �z

L(A5)

Here, V0 is the characteristic carrier gas velocity. Substitutingthese variables into polymer species continuity equations (Eqs.12–15) gives

�

�t� ��1��2� � �

1

�112

11

22�122 �

22�

11�212

1

�222

�2

�r�2 ��1��2� (A6)

t� � 0 ��1��2� � �0

0� (A7)

r� � 0 ��1��2� � � 1

11

12�22�

21

1 �C�1C�2� (A8)

r� � 1�

�r� ��1��2� � �0

0� (A9)

Summation of the three species continuity equations (Eqs.6–8) for the gas phase gives the total continuity equation

C�V

� z�

2

R �Dp11

��1

�r �r�R

� Dp12

��2

�r �r�R

��

2

R �Dp21

��1

�r �r�R

� Dp22

��2

�r �r�R

� (A10)

Substituting the dimensionless variables into this equationgives

�V�

� z��

2

11�112 �C01V0

CL � ��1�r�

�r��0

�2

22�122 �C02V0

CL � ��2�r�

�r��0

�2

11�212 �C01V0

CL � ��1�r�

�r��0

�2

22�222 �C02V0

CL � ��2�r�

�r��0

(A11)

For a typical CCIGC experiment, (C0iV0/CL)is on the orderof 10�5, whereas 2/1�21

2 is usually on the order of 1 or 10.Thus, it can be safely assumed that the mass average velocityin the column is constant


�V�

� z�� 0 and V � V0 (A12)

Under this approximation, the solvent species continuity equa-tions simplify to

�C1

�t� V0

�C1

� z� Dg11

�2C1

� z2

�2

R �Dp11

��1

�r �r�R

� Dp12

��2

�r �r�R

� (A13)

�C2

�t� V0

�C2

� z� Dg22

�2C2

� z2

�2

R �Dp21

��1

�r �r�R

� Dp22

��2

�r �r�R

� (A14)

Because the velocity is assumed constant, it is no longernecessary to consider the carrier gas continuity equation. Thesolvent species continuity equations can be expressed in di-mensionless form

�

�t� �C�1C�2� �

�

� z� �C�1C�2� � ��

�2

� z�2 �C�1C�2�

� �D��

�r� ��1��2��

r��0

(A15)

where

�D� � 2 �1

11�112

1

�22�122

�

11�212

1

22�222

and

�� Dg11

V0L0

0Dg22

V0L

In dimensionless form, the necessary boundary conditions are

t� � 0 �C�1C�2� � �0

0� (A16)

z� � 0 �C�1C�2� � ��t��

��t�� (A17)

z� � �C�1C�2� � �0

0� (A18)

Because dimensional analysis showed the gas velocity to beconstant, the governing equations are linear. This makes an

analytical solution possible. Macris1 showed that the equationsfor the binary system can be solved using Laplace transforms.This approach cannot be directly applied because the equationsare coupled. To uncouple the equations a new matrix, [A], isdefined. This matrix is constructed by setting its columns to theeigenvectors of the matrix in Eq. A6

�A� � � 1�2 �

1

�222

22�

11�212

�1 �1

�112

11

22��122

1 (A19)

Here, �1 and �2 are the eigenvalues of the matrix in Eq. A6.They can be found from the quadratic rule to be

�1 �1

2�tr � ��tr�2 � 4 det (A20)

�2 �1

2�tr � ��tr�2 � 4 det (A21)

Here tr is the trace of the matrix

�1

�112

11

22��122

22�

11�212

1

�222

and det is the determinant of the matrix. The reason for creatingthe [A] is that is has a unique mathematical property

�A��1 �1

�112

11

22��122

22�

11�212

1

�222

�A� � � �1 00 �2

� (A22)

Multiplying both sides of Eq. A6 by [A]�1 gives

�

�t��A��1 ��1

��2� � �A��1 �

1

�112

11

22��122

22�

11�212

1

�222

�A�

� �A��1�2

�r�2 ��1��2� (A23)

The identity matrix [I] � [A][A]�1 has been inserted be-tween the matrix and the second derivative. Substituting Eq.A22 into this result gives


�

�t� � ��1��2� � � �1 0

0 �2� �2

�r�2 � ��1��2� (A24)

In this equation, pseudoconcentrations have been introduced

� ��1��2� � �A��1 ��1

��2� (A25)

Equation A24 can be written as two uncoupled partial dif-ferential equations in terms of these pseudocompositions

��1�t�

� �1

�2��1�r�2 (A26)

��2�t�

� �2

�2��2�r�2 (A27)

The boundary conditions are also multiplied by [A]�1 to give

t� � 0 � ��1��2� � �0

0� (A28)

r� � 0 � ��1��2� � �B� � C�1

C�2� (A29)

where

�B� � �A��1 � 111

12�22�

21

1 �A�

r� � 1�

�r� � ��1��2� � �0

0� (A30)

The polymer-phase species continuity equations are now twouncoupled differential equations. These types of equations canbe solved using Laplace transforms. The following Laplacedomain variables are introduced

�Q1�s, r��Q2�s, r�� 1�t�, r��

��2�t�, r�� (A31)

�Y1�s, z��Y2�s, z�� C�1�t�, z��

C�2�t�, z�� (A32)

Using these variables, Eqs. A26 and A27 can be transformedinto the Laplace domain

Q1 � s � ��1�t� � 0� � �1

�2Q1

�r�2 (A33)

Q2 � s � ��2�t� � 0� � �2

�2Q2

�r�2 (A34)

Using the initial condition from Eq. A28

�2Q1

�r�2 �Q1 � s

�1� 0 (A35)

�2Q2

�r�2 �Q2 � s

�2� 0 (A36)

The boundary conditions are also transformed into theLaplace domain

r� � 0 �Q1

Q2� � �B� �Y1

Y2� (A37)

r� � 1�

�r� �Q1

Q2� � �0

0� (A38)

Equations A35 and A36 are second-order linear homoge-neous equations with constant coefficients. Thus, the generalsolutions have the form

Q1 � c1e�s/�1r� � c2e

��s/�2r� (A39)

Q2 � c3e�s/�2r� � c4e

��s/�2r� (A40)

The ci are constants that are determined from the boundaryconditions given in Eqs. A37 and A38. With these constants,the solution reduces to

Q1 � �B11Y1 � B12Y2�

� � 1

1 � e�2 s/�1e�s/�1r� �

e�2 s/�1

1 � e�2 s/�1e��s/�1r�� (A41)

Q2 � �B21Y1 � B22Y2�

� � 1

1 � e�2 s/� 2e�s/� 2r� �

e�2 s/� 2

1 � e�2 s/� 2e��s/� 2r�� (A42)

The gas-phase equation requires the gradient of Qi at thegas–polymer interface. This is found by differentiating Eqs.A41 and A42 with respect to r�

� �

�r��r��0

�Q1

Q2� � �C� �Y1

Y2� (A43)

where

�C� � �� s

�1tanh�� s

�1� 0

0 �� s

�2tanh�� s

�2� �B�

As with the polymer phase, both sides of the gas-phase equa-tions can be multiplied by [A]�1


�A��1�

�t� �C�1C�2� � �A��1

�

� z� �C�1C�2�

� �A��1��A��A��1�2

� z�2 �C�1C�2�

� �A��1�D��A��A��1�

�r� ��1��2��

r��0

(A44)

It will be assumed that Dg11 � Dg22. Although this isprobably not a good approximation, the value of � typically hasan insignificant effect on the shape of the elution profile.22

Thus, using this approximation should not affect the end result.With this assumption, [�] is a symmetric matrix, and Eq. A44and the boundary conditions can be written as

�

�t� � C�1C�2� �

�

� z� � C�1C�2� � ��

�2

� z�2 � C�1C�2�

� �E��

�r� � ��1��2�� r��0

(A45)

t� � 0 � C�1C�2� � �0

0� (A46)

z� � 0 � C�1C�2� � �A��1 ��t��

��t�� (A47)

z� � � C�1C�2� � �0

0� (A48)

In Eq. A45, [E] � [A]�1[D][A]. Using the variables defined inEqs. A31 and A32, these equations can be converted into theLaplace domain

s �Y1

Y2� �

�

� z� �Y1

Y2� � ��

�2

� z�2 �Y1

Y2� � �E�

�

�r� �Q1

Q2��

r��0

(A49)

z� � 0 �Y1

Y2� � �A��1 �1

1� (A50)

z� � �Y1

Y2� � �0

0� (A51)

Substituting for the gradient given by Eq. A43

s �Y1

Y2� �

�

� z� �Y1

Y2� � ��

�2

� z�2 �Y1

Y2� � �F� �Y1

Y2� (A52)

In this equation

�F� � �E� �� s

�1tanh�� s

�1� 0

0 �� s

�2tanh�� s

�2� �B�

Next, the equations must be uncoupled by multiplying bothsides of the equation by another matrix, [G]�1

s�G��1 �Y1

Y2� � �G��1

�

� z� �Y1

Y2� � �G��1��G�

� �G��1�2

� z�2 �Y1

Y2� � �G��1�F��G��G��1 �Y1

Y2� (A53)

Similar to [A] for the polymer phase, [G] is constructed fromthe eigenvectors of [F]

�G� � � 1F2 � F22

F21

F1 � F11

F12

1 (A54)

where F1 and F2 are the eigenvalues of the [F] matrix. Theycan be found from the quadratic rule to be

F1 �1

2�tr � ��tr�2 � 4 det (A55)

F2 �1

2�tr � ��tr�2 � 4 det (A56)

Here tr is the trace of the F matrix and det is the determinantof the G matrix. The reason for creating the G matrix is that ithas a unique mathematical property

�G��1�F��G� � � F1 00 F2

� (A57)

Here, new pseudo-Laplace variables are defined:

� Y1

Y2� � �G��1 �Y1

Y2� (A58)

Substitution of these pseudovariables gives

s � Y1

Y2� �

�

� z� � Y1

Y2� � ��

�2

� z�2 � Y1

Y2� � � F1 0

0 F2� � Y1

Y2�

(A59)


z� � 0 � Y1

Y2� � �H1

H2� (A60)

z� � � Y1

Y2� � �0

0� (A61)

In Eq. A60

�H1

H2� � �G��1�A��1 �1

1�The uncoupled equations can be written as

�2Y1

� z�2 �1

�1

�Y1

� z�� 1�s�Y1 � 0 (A62)

�2Y2

� z�2 �1

�2

�Y2

� z�� 2�s�Y2 � 0 (A63)

where

i�s� �s � Fi

�i

Like the polymer-phase equations, these are second-orderlinear homogeneous differential equations with constant coef-ficients. Thus, the general solutions have the form

Y1 � c1expz�� 1

2�1� � 1

4�12 � 1�s��

� c2expz�� 1

2�1� � 1

4�12 � 1�s�� (A64)

Y2 � c3expz�� 1

2�2� � 1

4�22 � 2�s��

� c4expz�� 1

2�2� � 1

4�22 � 2�s�� (A65)

Here, ci are constants that can be determined from the boundaryconditions. Using Eqs. A60 and A61, the general solution isfound to be

Yi � Hiexpz�� 1

2�i� � 1

4�i2 � i�s�� (A66)

Because the gas concentration at the detector is needed, thisexpression is evaluated at the end of the column (z� � 1).

Yi � Hiexp 1

2�i� � 1

4�i2 � i�s�� (A67)

Appendix B. Details of Numerical Solution

The numerical model is based on an implicit finite-differencetechnique. First, the species continuity equations for the gasand polymer phases are discretized using a Taylor series ex-pansion. For example, the polymer-phase solvent 1 continuityequation given in Eq. A6 is expressed as

��1,i � ��1,i�1

�t�

1

�112 ��1, j�1 � 2��1, j � ��1, j�1

�r2 ��

11

22�122 � ��2, j�1 � 2��2, j � ��2, j�1

�r2 � (B1)

Here, the i and j subscripts represent discretized time andradial positions, �t and �r are the sizes of the time andradial step, respectively. A similar equation can be derivedfor solvent 2. The discretized gas-phase continuity equationfor solvent 1 is

C�1,i � C�1,i�1

�t�

C�1,k � C�1,k�1

�z� �1�C�1,k�1 � 2C�1,k � C�1,k�1

�z2 �� D11��1, j�2 � ��1, j�1

�r � � D12��2, j�2 � ��2, j�1

�r � (B2)

Here, k represents the discrete axial positions in the column and�z is the axial step size. An expression similar to Eq. B2 canbe derived for solvent 2. The discretized polymer-phase bound-ary conditions are

t� � 0 ��1, j

��2, j� � �0

0� (B3)

r� � 0 ��1, j�1,k

��2, j�1,k� � � 1

11

12�22�

21

1 �C�1,k

C�2,k� (B4)

r� � 1 ��1, j�N

��2, j�N� � ��1, j�N�1

��2, j�N�1� (B5)

Here, N is the total number of discrete points in the radialdirection. Similarly, the gas-phase boundary conditions are


t� � 0 �C�1,k

C�2,k� � �0

0� (B6)

z� � 0 �C�1,k�1

C�2,k�1� � �

1

�t1

�t (B7)

z� � �C�1,k�10M

C�2,k�10M� � �0

0� (B8)

Here, M is the total number of discrete points in the axialdirection. The condition at infinite z� was specified at a largevalue of z� (k � 10M). The gas- and polymer-phase speciescontinuity equations were solved with a simple tridiagonalsolver at each time step. A constant step size time integratorwas used to solve all four equations at each time step in amarching fashion.

Manuscript received Aug. 18, 2003, and revision received Mar. 7, 2005.


Generic Crystallizer Model: I. A ModelFramework for a Well-Mixed Compartment

M. J. HounslowDept. of Chemical and Process Engineering, University of Sheffield, Sheffield S10 2TP, U.K.

A. E. LewisChemical Engineering Department, University of Cape Town, Rondebosch 7701, South Africa

S. J. SandersSanders Simulation, Denville, NJ 07834

R. BondyOLI Systems Inc., Morris Plains, NJ 07950


A model framework is described for crystallization of a single solid species in awell-mixed compartment at steady state. The model framework applies to both Type I (thatis, nonhigh yield) crystallization and Type II (high yield) crystallization. The frameworkconsists of population balances incorporating nucleation, growth, aggregation, breakage,classification, and dissolution, coupled with mass and energy balances. The model allowsany number of product streams, any number of feed streams, one vapor product stream,nonrepresentative sampling, but only one solid species. The numerical strategy used tosolve the resulting set of nonlinear integro-differential equations transforms them into amatrix of algebraic equations. Two algorithms for the solution for Type I crystallizationare proposed, both of which consist of solving the material and energy balances sequen-tially with the population balance and iterating around only one variable. Both algorithmsuse an existing material and energy balance solution package, which is linked to thepopulation balance equations. The first solution algorithm solves the population balanceequations using a Newton–Raphson solver with finite-difference approximations for thederivatives, converging around a variable related to the crystal mass and the numberdensity for each interval. The second algorithm solves the population balance equationsusing a successive substitution technique with root bracketing and iterates around thesuspension density. The choice of algorithm depends on the nature of the system to bemodeled. A similar framework is suggested for the solution for Type II crystallization,except that the iteration variable is the growth rate at a fixed supersaturation ratio. © 2005American Institute of Chemical Engineers AIChE J, 51: 2942–2955, 2005

Introduction

This article addresses the specification of a generic model todescribe crystallization of a single solid species in a continuous

stirred tank as shown in Figure 1. The model uses a structureconsisting of input stream data (temperature, pressure, compo-sitions, flow rates, and enthalpies) combined with a functiondescribing the particle size distribution (PSD).

The modeling of a particulate system necessitates the use ofthe population balance equation (PBE), a statement of conti-nuity for particulate systems, first introduced by Hulbert andKatz1 and Randolph and Larson.2,3 The requirements necessary

Correspondence concerning this article should be addressed to M. J. Hounslow [email protected].


PARTICLE TECHNOLOGY AND FLUIDIZATION


to completely determine the formation and dynamics of mul-tidimensional particle distributions are covered in Randolphand Larson,4 Ramkrishna,5,6 and in extensive detail in Houn-slow and Wynn.7

There are various techniques available for solving PBEs (seeRamkrishna5,6 for a general review). In this case, a discretiza-tion method is used, which considers the PSD to consist ofgroups of particles of different sizes, described by a vector ofvalues whose elements correspond to discrete particle sizes.With a given set of input conditions, an appropriate set ofoperating conditions, and information about crystallizationmechanisms, the model can describe the output stream. Thereal strength of a model such as this is that the solution to thePBE can be coupled to a thermodynamic package such as OLISystems Inc.8 or Aspen Plus9 that rigorously calculates thesupersaturation using the comprehensive solution chemistryembodied in such a package.

For this model, there can be Kin streams flowing into thevessel, each containing solid, liquid, or solid and liquid. Sim-ilarly, there are Kout streams flowing out. There is also a streamof vapor leaving and an indirect heat flow.

The model framework allows for nucleation, growth, aggre-gation, breakage, classification, and dissolution. Destruction offines can be treated as two coupled crystallizers with classifi-cation and dissolution of fines in one of them. It is important tonote that this is not an MSMPR (mixed suspension, mixedproduct removal) model because it is possible to have non-mixed product removal.

Population Balance

Consider the population balance equation (PBE) for a con-tinuous stirred tank at steady state. There are Kin streamsflowing in, at volumetric rates Qk

in, bearing solids with numberdensities nk

in(l). Similarly, there are Kout streams flowing out, atvolumetric rates Qk

in, bearing solids with number densitiesnk

out(l). The tank has a volume V and within the tank the numberdensity is everywhere constant at n(l). This is illustrated inFigure 2.

It must be noted that this representation is incomplete ifmore than one solid species is present. For processes excludingaggregation, a simple extension is to have a separate numberdensity for each solid; if aggregation is present, a differentapproach is needed. This latter approach has been developed byHounslow et al.10 for two solid systems and can probably beextended to multiple solids if desired. In the work discussedhere, it is assumed that only one solid is present.

Within the tank, the state of the solution (such variables as

temperature, pressure, and composition) is represented by avector of state variables, s.

The PBE at steady state is

�k�1

Kin

Qkinnk

in�l � � �k�1

Kout

Qkoutnk

out�l � � Vr�n�, l, s� (1)

where r[n�, l, s] is the density functional describing the netrate of destruction of particles. For most purposes the heart ofthe PBE is this term, and, provided that nk

out(l) can be written interms of n(l), the definition of r in terms of n(l) converts Eq. 1into an equation to be solved for n(l).

Mechanisms

The identification of appropriate kinetic equations for thevarious mechanisms included in the population balance equa-tion is core to its predictive capacity. Thus, an overview ispresented here of some of the models available to describe thevarious kinetic processes relevant to crystallization. Althoughthe various crystallization mechanisms and kinetics of nucle-ation and growth, for example, are fundamental to the frame-work, the choice of function relating the kinetics to the super-saturation and other process variables is of much lessimportance.

Nucleation. Nucleation is classified as being primary orsecondary, depending on the mechanisms through which itoccurs.11 Randolph and Larson4 consider the phenomenon ofnucleation and discuss a number of nucleation models. Theserange from the fundamental Arrhenius-type expression for therate of homogeneous nucleation12 to one accounting for heter-ogeneous effects.13 They conclude that an expression based onthe Miers nucleation model14 has been most successful inmatching experimental work. The formulation proposed byRandolph and Larson for the nucleation rate B0, the number ofnuclei formed per volume per second, is

B0 � knucii �c � c*�I (2)

where B0 is the nucleation rate (m�3 s�1), knucii is the nucleation

rate constant, c is the solute concentration (kg m�3), and c* isthe equilibrium solute concentration (kg m�3).

It is presumed that knucii can be a function of temperature but

that I is not.Garside15 proposes an idealization of primary nucleation

based on a kinetic law formally identical to the thermodynamiclaw for homogeneous nucleation (but where the parameters Aand B do not have the same physical meaning).

B0 � Ae��B/�ln2�S�� (3)

Figure 2. Balance region for PBE.

Figure 1. CST.


where S is the supersaturation ratio.Sohnel and Garside16 confirm a similar form for primary

homogeneous nucleation (applicable over a limited range ofsupersaturation)

B0 � knuciii SI � knuc

iv cI (4)

where B0 is the nucleation rate (m�3 s�1), I is the kinetic“order” of nucleation, S is the supersaturation ratio, c is thesolute concentration (kg m�3), and knuc

iIi and knuciv are nucleation

rate constants (m�3 s�1 and kg�1 s�1, respectively).Although the nucleation rate constants knuc

iIi and knuciv have no

physical significance, Eq. 4 is very often used in practice.Secondary nucleation, besides having an obvious depen-

dency on the level of supersaturation, is generally considered tobe related to the suspension density

B0 � knucv �S � 1�IMT

J (5)

where MT is the suspension density (kg m�3) and knucv is the

nucleation rate constant (kg�1 s�1).The nucleation rate constant knuc

v is assumed to be related tothe stirring power and to exhibit a temperature dependencyaccording to an Arrhenius-type relationship.17 The exponent I,which does not depend on temperature, lies between 0.5 and2.5. The exponent J is generally assumed to be of the order of1 and to depend slightly on temperature.18

Gahn and Mersmann19 proposed a new model for secondarynucleation, in which a direct relationship was formulated be-tween the crystallization kinetics and the frequency and energyof particle–impeller collisions depending on a number of ma-terial properties of the crystals. This model has been widelyaccepted and is expressed as follows

B0 � �nuc�S � 1�IMTJNimp

k (6)

where �nuc is the secondary nucleation rate constant (m3 kg�2)and Nimp is stirrer speed (s�1).

To assess the predictions of the Gahn–Mersmann19 model,Bermingham et al.20 evaluated three additional kinetic modelsthat had also been developed for crystallization processes dom-inated by secondary nucleation and growth. In these models,inter alia, the attrition rate is not simply a function of theoverall crystal concentration, but also of the crystal size distri-bution. Bermingham concluded that these models,21-23 althoughlargely empirical, do provide significantly more physical in-sight into the dominant crystallization phenomena than thetraditional power-law models.

Growth. Crystal growth is typically denoted by G, thelinear crystal growth rate, and usually has a supersaturationdependency ranging from first to second order depending onthe rate-controlling step.11 McCabe24 proposed that the linearrate of crystal growth is independent of size, an observationthat is conventionally called the L law. If l is defined as acharacteristic dimension, then the linear growth rate can bedefined as

G �dl

dt(7)

where G is the linear growth rate (m s�1), l is the characteristicdimension (m), t is time (s), and the mass growth rate is definedas

1

A

dm

dt� 3

kv

ka�sG (8)

where A is area of the crystal (m2), m is mass of the crystal(kg), kv is the volume shape factor, ka is the area shape factor,and �s is solid density (kg m�3).

Because crystal growth involves transport of the growthunits from the bulk of the solution to the surface of the crystaland the incorporation of those units into the crystal lattice,growth can be controlled by either diffusion or surface inte-gration. Experimental determination of the growth kineticssupported by microscopic examination of crystals often allowsdetermination of the prevailing growth mechanism.25

For diffusion-controlled growth (crystallization of com-pounds with a high solubility, static crystallization, and crys-tallization from viscous solutions25) the growth rate becomes

1

A

dm

dt� 3

kc

1 � w�c � c*� (9)

where kc is the mass-transfer coefficient (mol m�2 s�1), w isthe mass fraction of solute, and c � c* is the concentrationdriving force (mol m�3).

The mass-transfer coefficient, kc � DAB/�, can be derivedfrom a number of correlations for the Sherwood number, Sh �kcL/D, such as Sh � 2 0.6Re1/2Sc1/3,25 where DAB is thebinary diffusion coefficient (mol m�1 s�1), � is the diffusionlayer thickness (m), Re is the Reynolds number, and Sc is theSchmidt number.

For surface integration controlled growth, Mullin26 identifiedthree possible mechanisms: spiral growth, growth by two-dimensional nucleation, and rough growth, of which spiralgrowth occurs most frequently. The general empirical equationfor surface integration controlled growth is26

G � kgrow�S � 1�I (10)

where G is the linear particle growth rate (m s�1) and kgrow isthe growth rate constant (m s�1).

In the spiral growth mechanism, which occurs at low super-saturation levels, the growth units are incorporated only atkinks on the crystal surface. A defect must first be generated onthe surface, whereafter growth proceeds layerwise on this de-fect. This results in a spiral dislocation, which will continue topropagate as long as growth continues. For spiral growth atvery low supersaturations, the exponent in Eq. 10 is I � 2 and,for high supersaturations, I � 1.25

At relatively higher supersaturations, two-dimensional nu-clei occur on the crystal surface and the necessary kinks forfurther growth are generated. This is the two-dimensional nu-cleation growth mechanism, for which the exponent in Eq. 10becomes I � 2.

At higher supersaturations, growth units can attach them-selves anywhere on the crystal surface and the surface becomesrough. In this case, the exponent in Eq. 10 is I � 1.


Besides the power-law formulation discussed above, othergrowth rate formulations are either a size-dependent lineargrowth rate27-29 or growth rate dispersion.30-32

Aggregation. The first contribution to a model for particleaggregation was from Smoluchowski, who developed an ag-gregation frequency for particles aggregating by Brownianmotion.33 Subsequently, the practice has been to describe aggre-gation in terms of an aggregation kernel, a measure of the fre-quency with which a particle of size l aggregates with one of size�. Implicit in this description is the assumption that all particlecollisions are binary in nature; in other words, the particle con-centration is relatively low. Sastry34 first proposed that the aggre-gation kernel � be viewed as the product of two factors

��L, �� 0f�L, �� (11)

The first factor, �0, is the size-independent portion of the rateconstant and depends on operating conditions such as the localfluid velocity field and chemical environment, whereas thesecond factor, f (L, �), is some function of particle size andoften reflects the mechanism of aggregation. Table 135 summa-rizes a range of aggregation kernels formulated in this way.

Breakage. Birth and death rates of particles arising frombreakage have been defined by Prasher36 and are reformulatedin Nicmanis and Hounslow37 as number-based quantities. In therate expressions, the breakage selection rate constant [S(l, s)] is therate at which a fragment of a particle of size l will be selected tobreak. In the case study by the same workers,37 a binary breakagefunction and size-dependent breakage rate are used.

Ramkrishna5 formulates an ad hoc model for breakage inwhich breakup of particles occurs independently of each other.The breakage frequency S(l, r, s, t) is thus sufficient to char-acterize the rate of destruction of particles of state (l, r) at timet, where r is the location in physical space and t is timedependency. This formulation is consistent with that suggestedby Randolph and Larson4 in a general description of empiricalbirth and death processes. For simplicity of formulation, therate at which particles are selected for breakage can be con-sidered to be independent of spatial position and, at steady state,the time dependency falls away. Hounslow et al.38 use a breakagefunction for which there is an analytical solution. The selectionrate is taken to increase simply with size cubed and the binarybreakage function is formulated as one that gives uniform prob-ability of all fragment sizes on a volume scale. That is

S�l, s� � kbreaklm (12)

where S is the breakage selection rate (s�1), kbreak is thebreakage rate constant (s�1 m�3), l is the particle size (m), and

b�l, x, s� �6x2

l3 (13)

Similarly, Attarakih et al.,39 in their model of droplet break-age, use the following functional form for the breakage selec-tion rate function

S�l � � kbreaklm (14)

where kbreak and m are positive parameters. For the daughterdroplet distribution, b(l, x) can have two functional forms:

(1) Uniform daughter droplet distribution, where it is as-sumed that there is an equal chance to form a daughter dropletof any smaller size when a mother droplet breaks up and thusthe distribution is independent of daughter droplet volume

b� x, l � �2

l(15)

(2) Parabolic daughter droplet distribution, where it is as-sumed that there is a greater or lesser chance to form twodaughter droplets of different sizes upon breakage of themother droplet

b� x, l � �24� x2 � xl � � 6l2

l3 (16)

Classification. A simple approach to classification is tospecify the cut size of fines.4 A more comprehensive approachis to consider segregation as a function of energy dissipationand particle size11 or to include the effect of superficial flowvelocity, flow direction, viscosity, and density difference be-tween the liquid and the solid phases.11

Another approach is one where the selection function is afunction of size and the state of the vessel contents. The formof the function, applied to stream k, is

yk�l, s� � H�l � l0� (17)

where yk(l, s) is the selection function for stream k as a functionof size and solution conditions and H(l � l0) is the Heavisidestep function.

In this case, stream k is a representative sample of the bulkfor sizes greater than some value l0, but containing no particlessmaller than that size. Thus

nkout � H�l � l0�n�l � (18)

where nkout is the number density of stream k out (m�4) and n(l)

is the number density of vessel contents (m�4).The hydrocyclone literature also has some useful size selec-

tivity equations. For example, Plitt and Kawatra40 propose thefollowing equation for the cut size

d50�c� �14.8Dc

0.46Di0.6Do

1.21e0.063V

Du0.71h0.38Q0.45��S � �L�0.5

(19)

Table 1. Aggregation Kernels after Bramley et al.35

Mechanism Kernel �(l, �)

Size independent �0

Brownian motion (Smoluchowski, 1917) �0(l �)(l�1 ��1)Gravitational (Berry, 1967) �0(l �)2(l � �)Shear [Smoluchowski (1917) and Low

(1975)] �0(l �)3

Particle Inertia (Drake, 1972) �0(l �)2(l2 � �2)Thompson kernel, empirical (Thompson,

1968) �0(l3 � �3)/(l3 �3)


where d50(c) is the “corrected” d50 (�m); Dc, Di, Do, and Du

represent the inside diameters of hydrocyclone, inlet, overflow,and underflow, respectively (cm); V is the volumetric percent-age of solids in the feed; h is the distance from the bottom of thevortex finder to the top of the underflow orifice (cm); Q is the flowrate of feed slurry (m3 h�1); and �S and �L represent the density ofsolids and density of liquid, respectively (g cm�1).

The “corrected” d50 is taken from the “corrected” classifi-cation curve. The corrected d50 is calculated by assuming that,in all classifiers, solids of all sizes are entrained in the coarseproduct liquid by short-circuiting in direct proportion to thefraction of feed water reporting to the underflow.41

Lynch and Rao42 used weight percent solids, instead of usingvolume percent solids as in Eq. 19. Their model for calculatingcut size is expressed as

log10d50�c� � 0.0173�w � 0.0695Du

� 0.0130Do � 0.0048Q � K1 (20)

where �w is the weight percent solids in the feed slurry and K1

is a constant.Dissolution. Kramer et al.43 used the method proposed by

Jager44 where the kinetic dissolution rate is calculated as afunction of a mass-transfer coefficient and relative supersatu-ration. In another work, the same authors11 observe that disso-lution is practically always limited by the mass-transfer step.Both the rate of dissolution and the rate of mass transfer limitedgrowth will thus be limited by the local energy dissipation.Bermingham et al.20 used a kinetic expression that has a first-order dependency on the supersaturation to describe the disso-lution rate in their population balance equation. The rate con-stant for dissolution is calculated as follows

kc�l � �DAB

l �2 � 0.8�� l4

L3 �1/5� L

DAB�1/3� (21)

where

� �NeNimp

3 Dimp5

V(22)

and kc is the mass-transfer coefficient (ms�1), DAB is the binarydiffusion coefficient (m2 s�1), l is crystal length (m), is thespecific power input (W kg�1), L is the kinematic viscosity ofthe liquid (m2 s�1), Ne is the power number (also known as theNewton number), Nimp is impeller frequency (s�1), Dimp isimpeller diameter (m), and V is tank volume (m3).

Dissolution can also be incorporated as a size-independentshrinkage rate constant

D�l, s� � D (23)

where D is the shrinkage rate constant (m s�1), or as a masstransfer limited process, as follows

D�l, s� �2kc�c � c*�

�m(24)

where kc is the mass-transfer coefficient (ms�1), c � c* is theconcentration driving force (mol m�3), and �m is the molardensity of particles (mol m�3).

Despite this wealth of information relating to crystallizationmechanisms, reporting of modeling studies in the open litera-ture relating to the solution of PBEs tends to fall into one oftwo categories. In the first category, the studies restrict them-selves to a specific system and its measured kinetics and focuson the validation of models with experimental data. In thesecond category, studies tend to focus on the numerical aspectsof solving the PBE, are usually restricted to the very simplestof generic mechanisms, and neglect to use the formulations thatare considered to most accurately model the phenomena beingconsidered.

In the first category, numerous studies are presented in theliterature. Even the very comprehensive work undertaken byBermingham et al.20 is limited to the ammonium sulfate sys-tem. In the second category, Kumar and Ramkrishna,45 indemonstrating a new discretization method for solving PBEs,use their technique to simulate various combinations of initialconditions, nucleation, growth, and aggregation. The functionsfor all the mechanisms considered are various combinations ofexponential and gamma initial distributions; constant, linear,and size-dependent nucleation rates; constant, linear, and size-dependent growth rates; and constant and sum aggregationkernels. Kiparissides and Alexopoulos46 consider only con-stant, block pulse, and size-dependent and/or surface area–dependent nucleation rates; constant and size-dependentgrowth rates; and a Brownian aggregation rate kernel.

This work, however, is proposed to be truly generic, in thatpractically any chemical system for which there is thermody-namic data may be considered. In addition, the frameworkincorporates a comprehensive selection of kinetic processesrelevant to crystallization and allows the selection of kineticmodels for the processes that are believed to most accuratelymodel the various phenomena.

The overall PBE

Once the exit flow term has been modified for classification,and the terms for net rate of destruction attributed to nucle-ation, growth, dissolution, aggregation, and breakage havebeen incorporated, the maximal form of the PBE is47

�k�1

Kin

Qkinnk

in�l � � n�l � �k�1

Kout

Qkoutyk�l, s� � V��B0�s� fnuc�l, s�

�d

dl�G�l, s�n�l �� D0��l � �

d

dl�D�l, s�n�l ��

�1

2 �0

l

��3�l3 � x3, x, s�� l3�l3 � x3�2

n�3�l3 � x3�n�x�dx

� n�l ��0

��l, x, s�n�x�dx � S�l, s�n�l �

��l

S�x, s�b�l, x, s�n�x�dx (25)


subject possibly to the boundary conditions n( ) � n(0�) � 0,where

yk�l, s� (26)

is the selection function or grade efficiency and is a function ofsize and the state of the vessel contents

�B0�s� fnuc�l, s� (27)

is the rate of destruction of particles attributed to nucleation

d

dl�G�l, s�n�l �� (28)

is the rate of destruction of particles stemming from growth

D0��l � �d

dl�D�l, s�n�l �� (29)

is the rate of destruction of particles attributed to dissolutionand D0�(l), the rate of disappearance, is a bookkeeping term toprevent the occurrence of particles of negative size

�1

2 �0

l

��3�l3 � x3, x, s�� l3�l3 � x3�2

n�3�l3 � x3�n�x�dx

(30)

is the rate of destruction of particles stemming from aggregationand

S�l, s�n�l � ��l

S�x, s�b�l, x, s�n�x�dx (31)

is the rate of destruction of particles stemming from breakage.The moment transformation so often used in the application

of PBEs to process models is not possible for arbitrary sizedependency. It is, however, possible to make a partial trans-formation for the third moment

m3 � �0�

l3n�l �dl (32)

Transformation of Eq. 25 gives

�k�1

Kin

Qkinm3k

in � �k�1

Kout

Qkoutm3k

out�s� � Vr3 (33)

where

r3�n�, s� ��0�

l3r�n�, l, s�dl (34)

is a function of [�B0(s)f nuc(·, s), G(·, s), D(·, s), n(·)] andy�3(s) � �0�

l3y(l, s)dl.Replacing the third moment with the suspension density

MT � �SkVm3 � �SkV �0�

l3n�l �dl (35)

gives

�k�1

in

QkinMTk

in � �k�1

out

QkoutMTk

out � V�SkVr3 (36)

This last equation has a crucial role in coupling the materialand population balances.

Numerical form of the PBE

Analytical solutions for Eq. 25 are scarcely ever available, soit follows that some numerical strategy will be required. Allsuch numerical strategies known to the authors represent thesize distribution at a vector of sizes l, at neq discrete points

l � �l1, l2, . . . , lneq� (37)

by neq discrete values N, related reasonably directly to thenumber density

N � �N1, N2, . . . , Nneq� (38)

In this report, it is assumed that each of the Ni is the number ofparticles, per unit volume of suspension, in the size range (li,li1), that is

Ni � �li

li1

n�l �dl (39)

In the most sophisticated discretization strategies, the valuesfor l are chosen and adjusted adaptively by the algorithm as itsolves for values for N. This approach offers high accuracy andshort execution times at the cost of very complex program-ming, and is not pursued here. A more tractable family ofdiscretization strategies involves fixing the vector l for theentire problem. In this case virtually all strategies adopt ageometric discretization of the size axis so that the ratio

r �li1

li� 21/3q q � J (40)

is a constant.47 Satisfactory accuracy for CST problems isusually achieved with r � 21/3 so that, for a 1000-fold range ofparticle sizes, neq � 30 is required. (For growth-dominatedPFR problems a value of r � 21/12 may be needed in whichcase, for the same size range neq � 120 is needed.)

The discretized form of Eq. 33, the discretized populationbalance equation (DPBE), is


�k�1

Kin

QkinNk,i

in �l � � �k�1

Kout

QkoutNk,i

out�l � � VRi�N, s� i � 1· · ·neq

(41)

where Nk,i is the number of crystals per unit volume in the kthstream and the ith size interval and Ri is the rate of destruction ofparticles in the ith size interval. The classification functions relatethe exit stream size distributions to the size distribution in the bulk

Nk,i � yk�l�i� Ni (42)

where the average size in the ith size interval is

l�i �1 � r

2li (43)

The discrete forms of Eqs. 35 and 36 are

MT � �SkVm3 � �SkV �i�1

neq

l�i3Ni (44)

and

�k�1

Kin

QkinMTk

in � �k�1

Kout

QkoutMTk

out � V�SkV �i�1

neq

l�i3Ri (45)

The task now is to determine expressions for Ri.

Nucleation, growth, and dissolution

To resolve the issue of how to apply the boundary conditionat zero size, it is necessary to consider nucleation, growth, anddissolution (ngd) together. Thus, adapting Hounslow et al.47 toincorporate dissolution

Ringd �

D0 � B0f1nuc �

2

�1 � r�l1��1 �

r2

r2 � 1��D1 � G1�N1 �r

r2 � 1�D2 � G2�N2� i � 1

�B0finuc �

2

�1 � r�li� r

r2 � 1�Di�1 � Gi�1�Ni�1 � �Di � Gi�Ni �

r

r2 � 1�Di1 � Gi1�Ni1� 1 � i � neq

�B0fneq

nuc �2

�1 � r�lneq

� r

r2 � 1�Dneq�1 � Gneq�1�Nneq�1 � �Dneq � Gneq�Nneq� i � neq

(46)

The D0 term, the rate of disappearance, in Eq. 46 is not aconstitutive equation, unlike the nucleation rate constant B0,but just a bookkeeping term to prevent the occurrence ofparticles of negative size. Further evidence is that the mag-nitude of D0 depends on the solution to the PBE, not onconditions in the vessel, such as supersaturation. As shownin Eq. 49, the value of D0 depends on n(0) and cannot bedetermined without knowledge of that value. By contrast thenucleation rate depends on the state of the vessel.

For the continuous system of equations, the boundary con-dition to be applied, n(0�) � 0, results in

D0 � �B0 � �D�0, s� � G�0, s��n�0� fnuc � ��l ��D�0, s� � G�0, s��n�0� fnuc continuous (47)

If the rate of growth at zero size is greater than the rate ofdissolution, the rate of disappearance will be zero, in whichcase we have

D0 � � 0 D�0, s� � G�0, s� � 0B0 � �D�0, s� � G�0, s��n�0� fnuc � ��l �

�D�0, s� � G�0, s��n�0� fnuc continuous(48)

It is probably most sensible to apply this equation on the first sizeinterval and to represent n by a first-order difference, in whichcase

D0 � 0 D1 � G1 � 0

B0 � �D1 � G1�N1

l2 � l1

fnuc � ��l �

�D1 � G1�N1

l2 � l1

fnuc continuous

(49)

AggregationThe full form for arbitrary q, given by Litster et al.48 and

refined by Wynn, is49

Riagg � � �

j�1

i�1

�i�1, jNi�1Nj

2� j�i1�/q

21/q � 1

� �p�2

q �j�i�p�1

i�p

�i�p, jNi�pNj

2� j�i1�/q � 1 � 2�� p�1�/q

21/q � 1


�1

2�i�q,i�qNi�q

2

� �p�1

q�1 �j�i1�p

i1�p1

�i�p, jNi�pNj

�2� j�i�/q � 21/q � 2�p/q

21/q � 1

� �j�1

i�l1

�i, jNiNj

2� j�i�/q

21/q � 1� �

j�i�12

�i, jNiNj (50)

where

p � Int�1 �q ln�1 � 2�p/q�

ln 2 � (51)

and Int[x] subtracts the fractional part (if any) from x. Note thatcare must be taken here when calculating the value of p

because of rounding errors when the integer part of the calcu-lated expression is extracted.

Breakage

The discrete form developed by Hounslow et al.38 is

Ribreak � SiNi � �

j�i

neq

bi, jSjNj (52)

Although this seems a very simple recasting of the continuousform of the equation, it must be understood that the Si and bi,j

cannot be deduced by evaluating the continuous functions atl�i and l�j. Instead, a more complicated process is required (seeEqs. 31 and 34 in Hounslow et al.38).

Solving the DPBE

The maximal form of the DPBE is obtained by combiningEq. 41 with Eqs. 42, 46, 50, and 52

�k�1

Kin

QkinNk,i

in � Ni �k�1

Kout

Qkoutyk,i � VRi�N, s� � V�Ri

ngd � Riagg

� Ribreak� i � 1· · ·neq (53)

For a problem where the feed size distributions are given, Eq.53 provides neq equations in neq unknowns: that is, the vectorof Ni values, N. Solving the DPBE is then a matter of solvingthese neq equations for these neq unknowns.

Irrespective of the sophistication of the method or fineness ofthe discretization used, Eq. 25 is transformed into neq simulta-neous nonlinear algebraic equations

ai � BiN � NTCiN � 0 i � 1, 2, . . . , neq (54)

where ai is a (different) constant for each equation, Bi is a(different) vector of constants (of length neq) for each equation,and Ci is a (different) matrix of size neq � neq for eachequation.

Embedded within this is the discrete equivalent of Eq. 42

Ni,k � yk,iNi (55)

The suspension density is calculated from

MT � �SkV �i�1

neq

l�i3Ni (56)

There are two possible solution strategies for the discretizedpopulation balance equations. The most obvious solution strat-egy is to solve the (neq 1) equations using a simple nonlinearequation solver with numerical estimation of the Jacobian(where the extra equation is Eq. 56). This has been found to bea robust and simple-to-implement technique50 and this is one ofthe strategies that has been adopted here. A Newton–Raphsontechnique has been implemented, with finite-difference approx-imations for the derivatives. The system of equations is“dense” because all equations involve all unknowns. Nonethe-less, robust convergence is obtained even with poor initialestimates of the solution and a numerical Jacobian. However,the expense of evaluating the Jacobian is considerable, and soother approaches are desirable.

The second solution strategy uses the observation that, in theabsence of aggregation, Eq. 53 is in fact linear in N. With thatobservation, it is useful to write Eq. 53, with the results of Eqs.46 and 52 but without Eq. 50 shown explicitly. For example,with 1 � i � neq, this becomes

�k�1

Kin

QkinNk,i

in � Ni �k�1

Kout

Qkoutyk,i � V� � B0fi

nuc

�2

�1 � r�li� r

r2 � 1�Di�1 � Gi�1�Ni�1 � �Di � Gi�Ni

�r

r2 � 1�Di1 � Gi1�Ni1� � SiNi � �

j�1

neq

bi, jSjNj � Riagg

This equation, and the results for i � 1 and i � neq, may bewritten as a set of equations

A � BN � 0 (57)

where A is a column vector of length neq and B is an neq � neq

matrix that can be decomposed into two matrices, E a tridi-agonal matrix and F an upper-right triangular matrix

B � E � F (58)

A � �A1

A2

···

Aneq

� (59)


E � �b1 c1 0 · · · · · · 0a2 b2 c2

···0 a3 b3 c3

······· · ·

· · ·· · ·

······ aneq�1 bneq�1cneq�1

0 · · · · · · · · · aneq bneq

� (60)

F � �F1,1 F1,2 F1,3 · · · · · · F1,neq

0 F2,2 F2,3 · · · · · · F2,neq

0 0 F3,3······

· · ·· · ·

· · ······· 0 Fneq�1,neq�1 Fneq�1,neq

0 · · · · · · · · · 0 Fneq,neq

�(61)

and the elements of A and B are

Ai � � �k�1

Kin

QkinNk,1

in � VB0f1nuc � VD0 � VR1

agg i � 1

� �k�1

Kin

QkinNk,i

in � VB0finuc � VRi

agg 1 � i � neq

(62)

ai �2Vr�Di�1 � Gi�1�

�1 � r��r2 � 1�li1 � i � neq (63)

bi � 2V�D1 � G1�

�1 � r��r2 � 1�l1� VS1 � �

k�1

Kout

Qkoutyk,1 i � 1

2V�Di � Gi�

�1 � r�li� VSi � �

k�1

Kout

Qkoutyk,i 1 � i � neq

(64)

ci � �V2

�1 � r�li

r

r2 � 1�Di1 � Gi1� 1 � i � neq

(65)

Fi, j � �Vbi, jSj 1 � i � neq; j i (66)

It may be seen that, for any one set of conditions for whichthe DPBE is to be solved, B is a constant, from which it followsthat B�1 is also a constant. On the other hand, A depends on theaggregation rate, which in turn depends on N by Eq. 50. Thestrategy is to invert Eq. 57 to give

N � �B�1A�N� (67)

where now A is shown explicitly to be a function of N.Equation 67 may be applied iteratively using the followingalgorithm:1. Assemble B2. Find B�1

3. While N is changing, Do:a. Calculate Ab. Determine N from Eq. 67

4. Exit

The Material and Energy BalancesStream data

Representation of two types of streams is included: solid–liquid streams and vapor-only streams. Within each stream dataspecification, flows are represented as species molar flows foreach phase and some redundant data are included to preventfrequent recalculation. For a system containing J species thestream specifications are as shown in Table 2.

The material balance

The material balance states simply that the molar flow in ofeach species equals the molar flow out of that species

�k�1

Kin

nTk

in � �k�1

Kout

nTk

out � nV � �� (68)

where � is a matrix of stoichiometric coefficients for the Eindependent solution-phase chemical reactions taking placewith reaction rates given by the vector �. Note that the pro-duction of solid does not count as a reaction here, given thatEq. 68 counts, for example, NaCl(aq) with the same speciesnumber as that of NaCl(s). The rate of generation of solid canbe deduced from the PBE, Eq. 25. The mass balance couplesdirectly to the PBE by the suspension density MT, as given inEq. 36, which in turn can be related to the molar flows asfollows

MT,k �

�i

MR,inSk,i

1

�S�

i

MR,inSk,i �1

�L�

i

MR,inLk,i

(69)

where MR,i is the relative molar mass for species i and �S and�L are the mass densities for the solid and liquid phases,respectively.

Table 2. Stream Definitions

Variable Unit Symbol

StreamType

SL V

Temperature °C T ● ●Pressure Pa P ● ●Total flow mol s�1 nT ● ●Total solid flow mol s�1 nST ●Total liquid flow mol s�1 nLT ●Liquid species

flows mol s�1 nL � (nL1, nL2

, . . . , nLJ) ●

Solid species flows mol s�1 nS � (nS1, nS2

, . . . , nSJ) ●

Vapor speciesflows mol s�1 nV � (nV1

, nV2, . . . , nVJ

) ●Volumetric flow m3 s�1 Q ●Specific enthalpy J mol�1 H ● ●Size distribution m�3 N � (N1, N2, . . . , Nneq

) ●


It will usually be the case that the solution composition of allexiting liquid streams will be the same so that, if the molfractions of each species in solution in the vessel are x, then

nLk,i

nLTk

� xi (70)

where nLTkis the total molar flow of liquid in stream k. The

total flow for each stream is

nTk � nSTk � nLTk (71)

Similarly, if the solid has mol fractions xS

nSk,i

nSTk

�xSi (72)

and the vapor mol fractions xvap

nivap

nTvap � xi

vap (73)

The volume flow rate for the solid–liquid streams may bedetermined from

Qk � �i�1

J

MR,i�nSk,i

�S�

nLk,i

�L� (74)

The energy balance

The energy balance can be written most succinctly in termsof the specific enthalpies H. The balance is

�k�1

Kin

nTkin Hk

in � �k�1

Kout

nTkoutHk

out � nTvapHvap � �q (75)

It is to be expected that the specific enthalpy of the solutionand of the solids in each of the exit streams will be the same asthat in the vessel, that is,

nTkHk � nTSkHS � nTLkHL (76)

In addition, it is assumed that all exit streams have the sametemperature and pressure as those of the vessel

Tkout � Tvap � T Pk

out � Pvap � P (77)

Solving the Balance EquationsProblem specification

A standard specification would be● Complete description of the feedstreams● Total flow rates of all but one of the solid–liquid exit

streams● Crystallization kinetic equations: B0(s), f nuc(l, s), G(l, s),

D(l, s), �(l, x, s), S(l, s), b(l, x, s)

● Heat input, q● Vessel size, V● Selection functions yk(l)● Composition of the vapor stream● Vessel operating pressure

Unknowns to be determined

The model must then determine the complete specification ofthe product streams

● Solid liquid streams: (7 2J neq) � Kout variables● Vapor stream: 4 J● Vessel conditions: T, x, HS, HL, MT, X�, N, that is, 3 J

E neq (note that x counts as J � 1 unknowns)● The rate constants, such as B0, G0, D, �, S (that is, 5)

The total number of unknowns is: (7 2J neq) � (Kout 1) 5 E

Equations available

In Table 3, the available equations for a Type I (that is,nonhigh yield) crystallization are presented. So, for example,for a system

● with three cations, three anions, four complexes, and wa-ter, that is, J � 11

● four independent reactions, E � 4● two exit streams, that is, Kout � 2● neq � 30

the total number of equations to be solved is (7 22 30) �(2 1) 5 4 � 186. For a Type II, or high-yield system,one of the kinetics equations would be lost, and an additionalequilibrium equation (for the solid) asserted.

Tearing

There are many commercial thermodynamic modules thatallow the solution of material and energy balance problems,8,9

so it seems sensible to use these and therefore to deconstructthe problem between the thermodynamic calculations on theone hand and the population balances on the other hand. Thesupersaturation ratio is calculated at a fixed T, P, composition,and mass of crystal precipitated. The supersaturation ratio thenallows the calculation of the kinetic parameters to be used inthe solution of the population balances. In fact, this is the realstrength of a model such as this. Coupling the solution to thePBE to a thermodynamic package means that the supersatura-tion is rigorously calculated on the basis of the comprehensivesolution chemistry embodied in such a package.

The solution of the population balances then allows thecalculation of a new vector of number of particles (Ni) and themass of crystal precipitated from the suspension density.

Thus, for Type I systems, the iteration occurs over (neq 1)variables. For Type II systems, the algorithm still applies, butit is necessary to iterate over the growth rate.

Type I Systems: Newton–Raphson Solution Algorithm1. Estimate Mc, the total crystal mass in the crystallizer

produced by crystallization, by varying Mc to match the user-specified estimate of the supersaturation ratio. Compute umc, atransform of Mc, as follows

umc � �ln�1 �Mc

Mcmax� (78)


This transform prevents Mc from exceeding the maximumamount of solute available Mcmax and is used only for thispurpose in the algorithm. For classified product removal, thecalculated value of Mcmax is the same as that for mixed productremoval because it is merely used as the upper numericalbound.

2. Estimate a starting value for Ni such that MT (suspensiondensity) � Mc/Vm.

3. Calculate MT, the suspension density, for the current setof Ni values.

4. Calculate the supersaturation ratio S for the currentvalue of Mc, the mass of crystal precipitated.

5. Calculate the predicted value of the mass of crystalprecipitated, Mc_calc from

Mc_calc �MTVm

1 � �MT

�c� (79)

6. Calculate the kinetic rate constants from the vector s.7. Compute the error for RHS(i) as �¥ {[RH-

S(i) of Eq. 48]2}. Compute the error for RHS(neq 1) � Mc �Mc_calc.

8. Compute the derivatives d(Eq. 53)/dNi using finite dif-ferences.

9. Compute the derivatives d(Eq. 53)/dumc.10. Compute corrections to the Ni and umc from the solution

of

Ax � b

where A is the the Jacobian; b is the RHS vector of errors in thematerial and energy balances and Mc � Mc_calc; x representscorrections to the variables Ni and umc, designated �Ni andumc.

11. Set � � 1.12. Update Ni using Ni(k 1) � Ni(k) �(�Ni) and umc(k

1) � umc(k) � �(umc), where k is the iteration number.

13. Compute Mc from

Mc � Mcmax�1 � exp��umc�� (80)

14. Compute a new error using steps 3–8.15. If the error was reduced, check for convergence. In this

case, the tolerance is 1.0E-6. If converged, exit.16. If the error was reduced but not converged, go to step 3

and calculate another Newton step.17. If the error was not reduced, set � � �/4 and go to step

12 to take a shorter step in the direction of the Newtoncorrection.

Type I Systems: Successive Substitution with Root-Bracket-ing Solution Algorithm. The root–bracketing algorithm be-low is used to solve Eq. 36 for the suspension density MT.

1. Perform a mass and energy balance assuming no solid isprecipitated.a. Denote MT as MT

0.b. Calculate the kinetic rate constants from the vector s.c. Solve the PBE; note the error in Eq. 36 as 0.

2. Perform a mass and energy balance assuming solid is pre-cipitated to form a saturated solution.a. Denote MT as MT

max.b. Calculate the kinetic rate constants from the vector s.c. Solve the PBE; note the error in Eq. 36 as max

3. If 0 � max � 0, exit with error.4. If 0 � 0,

a. then set MT1 � MT

0, MT2 � MT

max, 1 � 0, 2 � max

b. else set MT1 � MT

max, MT2 � MT

0, 1 � max, 2 � 0

5. Set MT � 0.5(MT1 MT

2).a. Solve the material and energy balance with the mass

precipitated corresponding to MT.b. Calculate the kinetic rate constants from the vector s.c. Solve the PBE; note the error in Eq. 36 as .d. If � 1 � 0,

i. then MT1 � MT, 1 �

ii. else MT2 � MT, 2 �

6. Do While � tol. � 0

Table 3. Equations Available

Source Equation Instances

Thermodynamics HL(T, P, x) 1HS(T, P) 1Hvap(T, P, xvap) 1T(P, x) 1Equilibria E

Population balance 54 neq

Kinetics Given 5Classification 55 neq � Kout

Material balance 68 JSolid flow 56, 69 Kout

Definition of suspension density 56 1Solution composition 70 J � Kout

Total flows 71 Kout

Solid composition 72 J � Kout

Energy balance 75 1Steam enthalpies 76 Kout

Equilibration of T and P 77 2(Kout 1)Vapor species flows 73 JExit flows Given Kout � 1Volume flows 74 Kout

Total (7 2J neq) � (Kout 1) 5 E


a. MT � 0.5(MT1 MT

2).b. Solve the material and energy balance with the mass

precipitated corresponding to MT.c. Calculate the kinetic rate constants from the vector s.d. Solve the PBE; note the error in Eq. 36 as .e. If � 1 � 0,

i. then MT1 � MT, 1 �

ii. else MT2 � MT, 2 �

7. Exit.

For type I crystallization, the choice of the most appropriatealgorithm depends on the nature of the system to be modeled.For cases where the index of aggregation (Iagg) is small, whereIagg � 1 � (m0/B0�) � 0.333, it is appropriate to use thesuccessive substitution method because this will converge eco-nomically and rapidly. For cases with higher indices of aggre-gation, the Newton–Raphson solver is more appropriate, giventhat the nonlinearity introduced by the aggregation terms be-comes more significant. This is covered in more detail in asubsequent article.

Type II Systems: Successive Substitution with Root-Bracket-ing Solution Algorithm. The root-bracketing algorithm belowis used to solve Eq. 36 for the size-independent portion of thegrowth rate G0, starting from a pair of first guesses (0, G0

max)1. Perform a mass and energy balance with solid precipitated to

form a saturated solution.2. Solve the PBE with G0 � G0

0 � 0; note the error in Eq. 36as 0.

3. Solve the PBE with G0 � G0max; note the error in Eq. 36 as

max.4. Do While 0 � max � 0

a. G0max � 2G0

max.b. Solve the PBE with G0 � G0

max; note the error in Eq. 36as max.

5. If 0 � 0,a. then set G0

1 � G00, G0

2 � G0max, 1 � 0, 2 � max

b. else set G01 � G0

max, G02 � G0

0, 1 � max, 2 � 0.6. Set G0 � 0.5(G0

1 G02).

a. Solve the PBE; note the error in Eq. 36 as .b. If � 1 � 0,

i. then G01 � G0, 1 �

ii. else G02 � G0, 2 � .

7. Do While � tol. � 0 a. G0 � 0.5(G0

1 G02).

b. Solve the PBE; note the error in Eq. 36 as .c. If � 1 � 0,

i. then G01 � G0, 1 �

ii. else G02 � G0, 2 � .

8. Exit.

Conclusions

The modeling of a single continuous stirred tank crystallizerwith any number (�1) of product streams, any number of feedstreams, one vapor product stream, nonrepresentative sam-pling, and only one solid species has been described.

The mechanisms allowed are nucleation, growth, aggrega-tion, dissolution, breakage, and classification.

The model constitutes coupled material, energy, and popu-lation balance equations.

The population balance equation, Eq. 25, serves two roles:

(1) It links the size distribution from a crystallizer to thekinetics of the processes taking place in that crystallizer.

(2) It couples the kinetics to the mass balance equation, byenabling calculation of the mass of solid produced.Solution of the PBE requires both conversion to a discrete formas well as numerical solution of the discrete form. Two algo-rithms for the solution of the resulting matrix of algebraicequations are proposed: The first solves the population balanceequations using a Newton–Raphson solver with finite-differ-ence approximations for the derivatives, converging around avariable related to the crystal mass and Ni for each size interval.The second uses a successive substitution technique with rootbracketing and iterates around the suspension density. Thechoice of the most appropriate algorithm depends on the natureof the system to be modeled. A similar framework is suggestedfor the solution for Type II crystallization, except that theiteration variable is the growth rate at a fixed supersaturationratio.

AcknowledgmentsThis work was supported by the EPSRC of the United Kingdom, OLI

Systems Inc., and the National Research Foundation of South Africa.

Notation

a � a coefficient, Eq. 54a � component of EA � component of AA � parameter in Eq. 3A � area of crystal, m2

b � component of B and Eb(l, x, s) � breakage function, m�1

B � parameter in Eq. 3B0(s) � nucleation rate, m�3 s�1

c � component of Cc � component of Ec � solute concentration, kg m�3

c* � equilibrium solute concentration, kg m�3

d50(c) � “corrected” d50 for the hydrocyclone (Eq. 19),�m

D � shrinkage rate constant in Eq. 23, m s�1

D(l, s) � linear rate of shrinkage, m s�1

D0 � rate of disappearance, m�3 s�1

DAB � binary diffusion coefficient (m2 s�1) in calculat-ing kc in Eq. 9, mol m�1 s�1

Dimp � impeller diameter, mDc, Di, Do, Du � inside diameters of hydrocyclone, inlet, vortex

finder, and apex, respectively (Eq. 19), cmE � number of independent reactions

f nuc(l, s) � the nuclei size distribution, m�1

finuc � the discrete equivalent of f nuc(l, s)F � component of F

G(l, s) � linear rate of crystal growth, m s�1

G0max � maximum guessed value for growth rate, m s�1

H � specific enthalpy, J mol�1

H � Heaviside step function in Eq. 17h � the distance from the bottom of the vortex finder

to the top of the underflow orifice in the hydro-cyclone (Eq. 19), cm

i � a size classIagg � index of aggregation, Iagg � 1 � (m0/B0�)

J � number of species presentK1 � a constant in Eq. 20

knucii , knuc

iii , knuciv , knuc

v � rate constants for nucleation, kg�1 s�1

k � a stream numberk � iteration number

ka � area shape factorkbreak � rate constant for breakage, m�3 s�1

kc � mass-transfer coefficient, ms�1


kgrow � rate constant for growth, ms�1

kV � volume shape factorK � number of streamsl � particle size, m

l0 � cut size for classification, mm � mass of crystal, kg

m0 � zeroth momentm3 � third momentMc � mass of crystal precipitated, kg

Mc_calc � predicted mass of crystal precipitated, kgMcmax � maximum mass of solute available, kg

MR � relative molar massMT � suspension density, kg m�3

n(l) � number density, m�4

neq � number of size classesnLT � total molar flow of liquid, mol s�1

nST � total molar flow of solid, mol s�1

nT � total molar flow, mol s�1

nj � molar flow of species j, mol s�1

Ne � power number (also known as Newton number)Ni � number of particles per unit volume in size class

i, m�3

Nimp � impeller frequency, s�1

P � pressure, Paq � heat flow, Wq � parameter determining the fineness of the dis-

cretizationQ � flow rate of feed slurry to the hydrocyclone (Eq.

19), m3 h�1

Q � volumetric flow rate (L min�1 in Eq. 20), m3 s�1

r[n�, l, s] � density functional describing rate of destructionof particles, m�4 s�1

r � discretization ratio, Eq. 40r3 � rate of destruction of third moment, s�1

Ri � rate of destruction of particles in interval i; thediscrete equivalent of r[n�, l, s]

Re � Reynolds numberS � supersaturation ratio

Sc � Schmidt numberSh � Sherwood number

S(l, s) � selection rate constant for breakage, s�1

t � time (s) and time dependency for breakage fre-quency

T � temperature, Kumc � transform of Mc in Eq. 78

V � tank volume, m3

V � volumetric percentage of solids in the feed to ahydrocyclone, Eq. 19

Vm � molar volume of the aqueous phase from calcu-lated from supersaturation, m3

w � mass fraction of solutex � mol fractionx � particle size, m

y(l, s) � selection function or grade efficiency for classi-fication

Vectors

A � a vector of coefficients, Eq. 57B � a matrix of coefficients, Eq. 57C � a matrix of coefficients, Eq. 57E � a tridiagonal matrix, Eq. 58F � an upper right triangular matrix, Eq. 58

X� � vector of reaction rates, mol m�3 s�1

l � list of particle sizes, Eq. 37N � a list of particle numbers Eq. 38� � matrix of stoichiometric coefficients

nL � (nL1, nL2

, . . . , nLJ) � species molar flow in the liquid, mol s�1

nS � (nS1, nS2

, . . . , nSJ) � species molar flow in the solid, mol s�1

nV � (nV1, nV2

, . . . , nVJ) � species molar flow in the vapor, mol s�1

r � location in physical space for breakage fre-quency

s � a vector of conditions describing the solu-tion, such as supersaturation, temperature,etc.

x � vector of mol fractions

Greek letters

�0 � size-independent portion of aggregation rate constant, m3 s�1

� � aggregation rate constant, m3 s�1

� � Dirac delta distribution� � diffusion layer thickness, m

p � defined in Eq. 51 � error in Eq. 36 � specific power input, W kg�1

�w � weight percent solids in the feed slurry to a hydrocyclone, Eq. 20� � particle size, m

�nuc � rate constant for secondary nucleation, m3 kg�1

L � kinematic viscosity of the liquid, m3 s�1

�S � solids density, kg m�3

�m � molar density of particles, mol m�3

�L � liquid density, kg m�3

� � density, kg m�3

Subscripts and superscripts

agg � aggregationbreak � breakagegrow � growth

I � exponent for concentration or supersaturation in nucleation andgrowth rate equations

i � size interval numberin � inlet streamsJ � exponent for suspension density in nucleation and growth rate

equationsk � stream numberL � liquidm � exponent in volume-dependent breakage rate equation

ngd � nucleation, growth, and dissolutionnuc � nucleationout � outlet streams

S � solidT � totalV � vapor

vap � vapor

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Manuscript received Nov. 5, 2004, and revision received Feb. 21, 2005.


Scale-up Effect of Riser Reactors: ParticleVelocity and Flow Development

Aijie Yan and Jesse (Jingxu) ZhuDept. of Chemical and Biochemical Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9


The influence of riser diameter on the axial and radial particle velocity profiles and flowdevelopment is studied in a 10 m high twin-riser system (76- and 203-mm ID risers).Cross-sectional average particle velocity is somewhat lower for the larger riser with asteeper radial particle velocity profile. The flow development in the riser center is nearlyinstant with the particle velocity remaining high. There is no significant difference for thetwo risers in the center region. In the wall region, the flow development is significantlyslower and the particle velocity of the smaller riser is higher. The flow development slowsdown in the larger riser. In all locations measured, there was a clear dependency betweenthe local particle velocity and solids concentration of both risers. Gas distribution andparticle aggregation are considered the key factors that influence the local hydrodynamicsin the twin-riser system. © 2005 American Institute of Chemical Engineers AIChE J, 51:2956–2964, 2005Keywords: circulating fluidized bed, particle velocity profile, flow development, riserdiameter, five-fiber optic probe

Introduction

Gas–solid reactions take place in a wide range of chemicalprocesses, such as circulating fluidized bed (CFB) combustionand fluid-catalytic cracking (FCC). Because of their use inpetroleum refineries for FCC, circulating fluidized beds were“reinvented”1,2 in the late 1970s. Since then, intensive studieshave been carried out to improve these industrial processes.FCC units are used in most refineries all over the world toconvert high molecular weight gas oils or residuum chargestocks into lighter hydrocarbon products in a riser reactorwithin a few seconds. However, a shorter and a more uniformcatalyst residence time in the riser reactor would potentiallylead to a better reaction performance (larger amounts of desiredproducts and/or a higher conversion). That is, because cokedeposition rapidly deactivates the catalyst leading to reducedselectivity and some desirable products (gasoline, light olefins,light cycle oil) may further react to cause over-cracking. These

non-beneficial factors could be limited or avoided by havingmore uniform axial and radial particle flow structure in theriser, leading to shorter and more uniform solids as well as gasresidence times.

To apply fundamental knowledge from pilot scale risers intoa commercial riser, it is necessary to achieve a clear under-standing of the scale-up effects. Extensive research has beencarried out on CFB riser reactors.3-17 Although there were someresearches on particle velocity in the literature,3-5,7,9,12,17,18 mostdata were taken in risers of small diameter (�0.15 m) and / orshort length (�7 m). Considering most industrial risers are of1 m or larger in diameter and of 20 m or taller in height, it isnot reliable to base the industrial design on the experimentalresults obtained in small diameter and very short risers sincesmall diameter vessels are dominated by wall effects and theentrance and exit structures have great influence on the shortheight vessels. It is very important to understand the scale-upeffect on the hydrodynamics inside the risers of significantsizes. Unfortunately, no previous studies have been publishedwith respect to gas and particle flow over a wide range ofoperating conditions in risers of different diameters. Therefore,more research on the influence of riser diameter on particle

Correspondence concerning this article should be addressed to J. Zhu [email protected].



velocity distribution and flow development is required in large-scale CFB risers to improve and optimize the existing industrialCFBs and enhance the design for novel applications.

To minimize the possibility of expensive errors in scaling-upto commercial operation, and to optimize and improve thedesigns of existing industrial FCC riser reactors, a good un-derstanding of the scale-up obtained from pilot-scale risers,such as the knowledge of the particle flow distribution in CFBrisers of significant size, is necessary. This work used a twin-riser system, with two 10 m long riser units having internaldiameters (ID) of 76 mm (3 in.) and 203 mm (8 in.) andotherwise having identical geometries (sharing the same down-comer and air supply, using the same particles), over a widerange of operating conditions with the solids circulation rate upto 200 kg m�2 s�1 and superficial gas velocity up to 8.0 m/s.The influence of riser diameter on the axial and radial particlevelocity profiles and flow development is carefully studied bymeasuring the local particle velocity with an optical-fiber probealong the twin-riser system. The information obtained from thisstudy could be beneficial, especially for the FCC industry.

Experimental Apparatus

The experiments were conducted in a twin-riser circulatingfluidized bed system shown schematically in Figure 1, consist-ing of two 10 m long risers, 76- and 203-mm IDs, which usethe same downcomer (storage tank) of ID 0.32 m. FCC catalystwith a mean diameter of 67 �m and a particle density of 1500kg/m3 were used. The total solids inventory was about 350 kg,equivalent to a solids level of nearly 5 m in the downcomer.The humidity level of the air was controlled between 70 and80% to minimize the effect of electrostatics in the system.

The configurations (except for the diameter) are identical forboth risers. The relative dimensions of the inlet and outletopenings to the diameters are equal to the corresponding di-

ameter. After passing through a short inclined pipe section,where a flip valve was installed to control the solids flow rate,the solids from the storage tank entered the riser bottom at aheight of 0.21 m in each riser and were accelerated by air innear-ambient conditions (at the base of the riser, the tempera-ture is about 20°C and absolute pressure is about 105.7 kPa).After initial mixing and accelerating, the gas–solids suspensiontraveled up in the column and passed through a smooth exitinto the primary cyclone for gas–solids separation, and escapedsolids entered into the secondary and tertiary cyclones, with thefinal gas–solids separation carried out by a bag filter. Twoseparate sets of cyclones were used for each riser, although thetwo risers shared the same bag filter. Only one riser operated ata given time. From the bottom of the large-capacity bag filter,collected fine particles could be returned to the downcomer.The solids flow rate measuring device located in the top portionof the downcomer sectioned the column into two halves with acentral vertical plate and with two half-butterfly valves fixed atthe top and the bottom of the two-half section. By appropriatelyflipping over the two valves from one side to the other, solidscirculated through the system can be accumulated in one sideof the measuring section for a given time period to provide thesolids circulation rate.

A five-fiber optic velocity probe was inserted into the col-umn to measure the particle velocity. Each of the five fibers isa silicon optical fiber of diameter 200 �m. The probe consistsof a horizontal cylindrical portion of diameter 2 mm and lengthabout 0.3 m, leading to a 10 mm long tip having an oval-shapedcross section of 0.5 � 1.8 mm (width � height). The five-fiberoptic probe consists of two light-emitting fibers (B and D) andthree light-detecting fibers (A, C, and E) arranged precisely inthe same vertical line. A particle flowing by the center pointbetween any two neighboring fibers will produce a reflectivesignal to a detection fiber. By counting the time differencebetween the two signals from A–B and B–C (or C–D andD–E), the velocity of a particle passing along the array of thefive fibers can be determined. More details of the five-fiberoptic probe have been presented by Zhu et al.19 The particlevelocity was measured at 11 radial positions (r/R � 0.00, 0.16,0.38, 0.50, 0.59, 0.67, 0.74, 0.81, 0.87, 0.92, and 0.98) on eightaxial levels (Z � 1.53, 2.73, 3.96, 5.13, 5.90, 6.34, 8.74, and9.42 m) for the 76-mm ID riser and five axial levels (Z � 1.47,2.69, 3.91, 5.90, and 8.79 m) for the 203-mm ID riser underseveral operating conditions as given in Table 1. At eachmeasurement location the sampling time was typically �30 sand the amount of sampled particles � 2500.

The local solids concentration was measured with a reflec-tive-type fiber-optic concentration probe. The 3.8 mm diameterprobe tip consisted of nearly 8000 emitting and receivingquartz fibers, each having a diameter of 15 �m. The activearea, where the fibers were located, was about 2 � 2 mm. More

Table 1. Operating Conditions

76-mm ID Riser 203-mm ID Riser

Ug (m/s) Gs (kg m�2 s�1) Ug (m/s) Gs (kg m�2 s�1)

5.5 50 5.5 503.5 100 5.5 755.5 100 5.5 1008.0 100 8.0 1005.5 200

Figure 1. Twin-riser circulating fluidized bed apparatus.


details of this probe can be found in Zhang et al.,20 and moredetails of the concentration probe measurements from the sametwin-riser system can be found in Yan and Zhu.21 The solidsconcentration was measured in the same positions and at thesame operating conditions as the particle velocity. Two sam-ples were taken at each location, and the total sampling timewas 60 s. By combining the results from the current study withthose of Yan and Zhu,21 the cross-sectional net solids fluxeswere obtained for each axial elevation. Those were found to bein a good agreement with the results from the solids flow ratemeasuring device (Gs) in the downcomer (within 10–15%).The cross-sectionally mean solids holdups were obtained byaveraging the local solids holdups measured at 10 radial posi-tions (excluding the center point) with the fiber-optic concen-tration probe at each axial elevation.

Results and DiscussionDevelopment of radial profiles of particle velocity

In our study, we used large-scale CFB risers (with diametersup to 0.2 m and bed height of 10 m to ensure that the particleflow is fully developed in the riser) with high gas velocity (upto 8 m/s) to study the influence of bed diameter on radialprofiles of the particle velocities. Figure 2a shows the radialprofiles of particle velocity on eight axial elevations of the76-mm ID riser under five operating conditions. The solidscirculation rates were 50, 100, and 200 kg m�2 s�1 and thesuperficial gas velocities were 3.5, 5.5, and 8.0 m/s. Radialprofiles of particle velocity on five axial elevations of the203-mm ID riser under four operating conditions are shown inFigure 2b. The solids circulation rates were 50, 75, and 100 kgm�2 s�1 and the superficial gas velocities were 5.5 and 8.0 m/s.In general, the particle velocities are more uniform in the uppersection than in the lower section of the riser at all radialpositions and higher in the center than in the wall region of theriser at all axial locations. Figure 2 also shows that the particlevelocity in the center region of the riser remains nearly constantthroughout the riser under each operating condition for bothrisers, and then decreases toward the wall. Figure 2 furthershows that the particle velocity in the center region of the riserdoes not seem to change significantly under the same superfi-cial velocity within the tested range of operating conditions forboth risers. As a result, the flow development is mostly repre-sented by the increase of the particle velocity toward the risertop at r/R from about 0.50 to 1.00. In the riser bottom, theparticle velocity drops significantly toward the wall. Towardthe riser top, the particle velocity in the wall region tends toincrease. Increasing the superficial gas velocity Ug increasesthe particle velocity throughout the riser. The solids circulationrate (overall solids flux) Gs seems to have only a slight influ-ence on the value of particle velocity. In addition, the radialprofile of particle velocity seems more uniform with lowersolids flux. That is, increasing solids flux slows down the flowdevelopment. Similar phenomena were also observed in theradial particle velocity profiles of the same 76-mm ID riserunder higher flux operating conditions as shown in Parssinenand Zhu.17 Because of the limitation of the capacity of thestorage tank, the solids flux in the 203-mm ID riser cannotreach a very high flux. It will be our future work to study higherflux conditions in the 203-mm ID riser after modifying theequipment to increase the capacity of the storage tank.

Figure 2. (a) Radial profiles of local particle velocity un-der five operating conditions on eight axial lev-els of the 76-mm ID riser, showing the flowdevelopment along the riser; (b) radial profilesof local particle velocity under four operatingconditions on five axial levels of the 203-mm IDriser, showing the flow development along theriser.


Figure 2 also shows the flow development with respect to theradial profiles of particle velocity under each operating condi-tion in the 10 m high twin-riser system. At the bottom section(Z � 1.53 and/or 2.73 m), the radial profile shows a flat centerregion, turning then smoothly downward toward the wall, andhaving a fairly wide wall region with a velocity value of �2m/s. This may be referred to as a horizontal “S”-shape. In theupper levels, the solids accelerate more in the radial region ofr/R � 0.0 to 0.4, leading to a combination of linear (but notflat) and parabolic-shaped radial velocity profile. When thesolids reach the exit section, the particles slow down some-what, and the corresponding velocity profiles become scatteredas a result of the exit features as the solids pass through the 90° smooth elbow. Observed from Figure 2, it seems that the flowdevelopment in the 203-mm ID riser is considerably slowerthan that of the 76-mm ID riser. The shape change of the radialprofiles of the 203-mm ID riser also happens at a much higherlevel than that of the 76-mm ID riser. This is because of thescale-up effect, which will be discussed later in more detail.

Figure 3 provides the typical radial profiles of particle ve-locity for both risers at Z � 5.90 m. Figure 3 also shows thatincreasing Ug tends to make the radial profiles more uniform.There is little influence of solids flux on the radial distributionsof particle velocity, which is likely a consequence of thequicker flow development under lower flux, in agreement withthe observations from Figure 2. Similar tendencies are ob-served in the other axial levels. Zhou et al.18 also observed that,under lower flux conditions, increasing solids circulation rateleads to steeper radial profiles for particle velocity.

By comparing the velocity profiles at 1.53, 2.73, and 3.96 m(marked as closed symbols with connecting lines), Figure 4shows that increasing Ug from 5.5 to 8.0 m/s (with Gs � 100

kg m�2 s�1) causes a faster flow development. On the otherhand, increasing Gs seems to slow down the flow developmentslightly under a constant Ug of 5.5 m/s. The flow developmentat higher flux is slower, as observed under higher flux operatingconditions in the same riser as shown in Parssinen and Zhu.17

It is also observed in Figure 4 that the shape change of theradial profiles of the 203-mm ID riser is at a much higher levelthan that of the 76-mm ID riser.

When observing the radial profiles of the riser from thebottom to the top, it is also seen that the flow develops first inthe riser center region, and then gradually and progressivelycloser to the wall as the solids pass through the riser. This canbe seen more clearly in Figure 5, which plots the particlevelocities in three radial regions on eight axial elevations in the76-mm ID riser and five axial elevations in the 203-mm IDriser (the particle velocity in each region is obtained by aver-

Figure 3. Comparison of the typical radial profiles of lo-cal particle velocity in the same five axial levelsfor the risers.

Figure 4. Comparison of the development of radial pro-files of local particle velocities for the risers.

Figure 5. Comparison of the average particle velocitiesin different radial regions along the riser underall three operating conditions for the risers.


aging the local particle velocities in that region). In this figure,the difference in the flow development in the three radialregions is clearly revealed. Furthermore, the scale-up effect onthe flow development is also clearly observed. In the centerregion of the riser (0.0 � r/R � 0.632, 40% of the cross-sectional area), the particles already gained a fairly high ve-locity before the height of 1.5 m, whereas a maximum velocityis finally reached at the height of 3–4 m. There is no significantdifference for the two risers in this region. In the middle region(0.632 � r/R � 0.894, 40% of the cross-sectional area), on theother hand, the particle velocity is increasing throughout theriser, and the increase is more obvious with higher gas velocity.Meanwhile, the particle velocity of the 76-mm ID riser in-creases more sharply than that of the 203-mm ID riser in thismiddle region. That is, increasing riser diameter slows downthe flow development. In the wall region (0.894 � r/R � 1.0,20% of the cross-sectional area), the particle velocity remainslow up to about 4–6 m and then increases slowly toward theriser top. The flow development in this region is significantlyslower than that in the two other regions. Moreover, it is alsoclearly shown that the particle velocity of the 203-mm ID riserincreases much less than that of the 76-mm ID riser in thisregion. From the riser bottom to the top, the radial distributionof the 76-mm ID riser becomes uniform much earlier than thatof the 203-mm ID riser. Thus, the flow development slowsdown with large riser. Figure 5 also indicates that increasinggas velocity significantly enhances the flow development (thatis, to cause particles to accelerate quicker and at lower levels inthe riser). The same tendency was also observed under higherflux operating conditions in the same 76-mm ID riser as shownin Parssinen and Zhu17; however, the flow development ofhigher flux conditions is much slower, especially in the middleregion.

In Figure 5, the particle velocities from the riser bottom tothe top in the center region are almost the same and changeonly slightly along the riser. However, the differences of par-ticle velocities from the riser bottom to the top at the middleand wall regions are much more obvious. Figure 5 clearlyshows that particle acceleration (and therefore flow develop-ment) first starts from the center and then extends to the wall.This dictates the development of the radial profiles of particlevelocity. In the bottom section, only those particles in thecenter region had some acceleration so that the horizontalS-shape appears. In the middle section, particles in the middleradial region start to accelerate so that the particle velocity inthis region increases, resulting in the combination of linear (butnot flat) and parabolic-shape profile. In the exit section, parti-cles in the middle region accelerate further to catch up withthose particles in the center region, and particles in the wallregion also begin to accelerate, leading to a parabolic radialprofile. Because the flow development slows down when theriser diameter is scaled-up, the shape change of the radialprofiles of the 203-mm ID riser happens at a much higher levelthan that of the 76-mm ID riser, as also observed in Figures 2and 4.

Figure 6, which compares the radial profiles of particlevelocity between the 76- and the 203-mm ID risers, indicatesthat there is a slight difference in the center region of radialprofiles of the two risers. However, it is quite different in thewall region. The radial distributions of particle velocity of the76-mm ID riser are comparably more uniform and less sensi-

tive to the change of the axial position than that of the 203-mmID riser. The difference is mainly a result of riser diameterdifference. This may be explained by some secondary effects,such as the introduction of the solids to the side at the base ofthe riser, poor dispersion of solids in the riser, or the shorterL/D in the larger-diameter vessel. Further research is neededfor these secondary effects. Figure 7 compares the radial solidsholdup profiles of the twin-riser system at Z � 5.84 m. It isshown that the solids holdup of the 203-mm ID riser at eachradial position is higher than that of the 76-mm ID riser underthe same operating conditions and at the same axial level,especially in the wall region. In an earlier paper21 by the sameauthors reporting on the axial and radial solids concentrationdistribution, similar results can also be observed in the otheraxial levels. That is, under the same solids circulation rate andgas velocity, a larger riser has a larger solids concentration ateach radial position, especially in the wall region. This showsa much more substantial wall effect, even in the relative mag-nitude, for the larger-diameter riser. A denser concentration ofsolids occupies the wall region of the large-diameter riser andrestricts the gas flow at the wall. As a result, the particlevelocity tends to be lower in the wall region for the larger-diameter riser. To maintain the cross-sectional Ug, the gasvelocity has to be correspondingly higher in the center regionof the large riser. A higher gas velocity in the center could alsolead to higher particle velocities. Again, this verified thescale-up effect of risers on the flow development: increasingriser diameter impedes flow development.

Figure 8a shows the variation of particle velocity in the wallregion with the cross-sectional mean solids holdup, obtained by

Figure 6. Comparison of the radial profiles of local par-ticle velocity for the risers of different diame-ters.


averaging the local solids holdups measured at 10 radial posi-tions (excluding the center point) with the fiber-optic concen-tration probe at each axial elevation, at each axial level of the76-mm ID riser. For this 76-mm ID riser, with a high cross-sectional mean solids concentration (riser bottom), the particlevelocity at the wall region is relatively low. With a lowercross-sectional solids concentration (riser top), the particlevelocities are higher at the wall region. A simple regressionprovides the following correlation between the particle velocityat the wall region and the cross-sectional average solids holdupof the 76-mm ID riser

Vp,wall � �0.98 ln�� s� � 2.23

�0 � �� s � 0.42, Vp � �1.38 m/s� (1)

This correlation is very close to that obtained from the higherflux conditions of the same 76-mm ID riser.17 The particlevelocity in the wall region of the 203-mm ID riser could alsobe related to the cross-sectional solids concentration at eachaxial level, as shown in Figure 8b. However, the relationshipbetween the particle velocity in the wall region and the cross-sectional solids concentration of the 203-mm ID riser is dif-ferent from that of the 76-mm ID riser. As mentioned earlier,because of the slow flow development of the large riser, thecross-sectional solids concentrations of the 203-mm ID riser

are relatively higher, so that the particle velocities near the wallare lower than those of the 76-mm ID riser at the same axiallevel. Therefore, as shown in Figure 8, no data appeared in therange of very low cross-sectional solids holdup and high par-ticle velocity for the wall region of the 203-mm ID riser. Theparticle velocity in the wall region and the cross-sectionalsolids holdup of the 203-mm ID riser are more clusteredtogether than that of the 76-mm ID riser.

Comparison of the axial development of particlevelocities in the twin-riser system

Figure 9 compares the axial profiles of the one-dimensional(cross-sectional average) particle velocities on each axial levelobtained with the fiber-optic velocity (Figure 9a) and with theconcentration (Figure 9b) probes under the same operatingconditions for both risers. In Figure 9a, the local particlevelocity was weighted with the local solids holdup in each

Figure 8. (a) Relationship between the local particle ve-locity near the riser wall (r/R � 0.92 and 0.98)and the cross-sectional mean solids holdup onall axial elevations under all measured operat-ing conditions for the 76-mm ID riser; (b) rela-tionship between the local particle velocitynear the riser wall (r/R � 0.92 and 0.98) and thecross-sectional mean solids holdup on all axialelevations under all measured operating con-ditions for the 203-mm ID riser.

Figure 7. Comparison of radial profiles of local solidsholdup the twin-riser system at Z � 5.84 m.


radial position to give the true cross-sectional average velocity,given that a higher solids population is typically flowing nearthe wall than in the center. In Figure 9b, the cross-sectionallyaveraged solids concentration21 was used to calculate the av-erage particle velocity [�Gs/(�p�� s)]. The excellent agreement(with average deviation of �10%) between Figure 9a andFigure 9b further confirms the reliability of the two optical-fiber probes used in this study.

In Figures 9a and 9b, the cross-sectional average particlevelocity of the 203-mm ID riser is lower than that of the 76-mmID riser under the same operating conditions, although there isno significant change for the shape of the axial profiles. That is,in terms of scale, the cross-sectional average particle velocitydecreases with the riser diameter, but the shape of the axialprofile changes only negligibly with the riser diameter. Thesecorrespond well to the typical axial solids holdup profiles(S-shape) of our previous study.21 Seen from Figures 9a and 9b,increasing gas velocity obviously increases the axial particlevelocities for both risers. However, changing solids fluxesappears to have no obvious influence on the axial particlevelocities for both risers.

As mentioned earlier, at the same solids circulation rate andgas velocity, a larger riser has a larger solids concentration ateach radial position, especially in the wall region. Thus, alarger riser has a larger cross-sectional average solids concen-tration at each radial position. To maintain the cross-sectionalGg, the cross-sectional average particle velocity has to becorrespondingly lower in the large riser.

The large difference between the average particle velocities inthe two risers may not be completely attributed to the scale, butmay also be explained by the steepened radial solids distributionin the larger riser. By closely analyzing the particle velocity ineach cross-section (Figure 6), it is easier to see that there is notmuch difference in the center region between the two risers; the

203-mm ID riser has a much lower particle velocity closer to thewall. Because more particles accumulated at the wall for the largeriser (as shown in Figure 7), the weighted cross-sectional averageparticle velocity becomes much lower in the 203-mm ID riser. Tofurther illustrate the point, the straight mathematic average particlevelocity, without taking the weight of the radial solids distribution,were plotted in Figure 9c for the two risers. Compared with Figure9a, the difference between the mathematic mean particle velocitiesof the two risers shown in Figure 9c is much smaller, suggestingthat the lower local particle velocity near the wall in the larger riseris only part of the reason for the larger riser to have a lowercross-sectional average particle velocity at each axial level at thesame solids circulation rate and gas velocity. In other words, thescale-up effect may not be as large as shown in Figure 9a.

In the study on the axial distributions of solids holdup, Yan andZhu21 reported three axial sections along the twin-riser system: thebottom dense section (Z � 3 m), a middle section, and a top exitsection. Such a three-section structure can also be observed in theaxial profiles of the cross-sectional average particle velocityshown in Figures 9a and 9b: the bottom section (�3–4 m), withparticle velocity � 2 m/s and no obvious solids acceleration; amiddle section; and a top exit section.

Particle velocities vs. solids holdups

The local particle velocity at a given radial position of acertain axial level can also be directly related to the local solidsconcentration at the same position throughout the twin–risersystem. As shown in Figure 10, the local particle velocitydecreases monotonically with the local solids concentrationand such a decrease follows the same pattern in all axialsections for both risers. Such a strong dependency could beunderstood only by considering solids aggregation at each localposition: a higher solids concentration leads to stronger particle

Figure 9. (a) Comparison of axial profiles of one-dimensional particle velocity in the twin-riser system: averaged fromlocal particle velocities (weighed by local solids concentration); (b) comparison of axial profiles of one-dimensional particle velocity in the twin-riser system: deduced from the cross-sectional average solidsconcentration. (c) Comparison of axial profiles of one-dimensional particle velocity in the twin-riser system:averaged from local particle velocities measured at 10 radial positions (excluding the center point).


aggregates, resulting in a lower effective drag between theparticles and gas, and therefore a lower particle velocity undera constant gas velocity. The independence of the above varia-tion with the axial locations shows the inherent connectionbetween the particle concentration and velocity. As a result, asingle correlation works for all axial sections in each riser:

For the 76-mm ID Riser

Vp � �1.85 ln��s� � 3.12

�0 � �s � 0.42, Vp � �1.52 m/s� (2)

For the 203-mm ID Riser

Vp � �2.23 ln��s� � 4.73

�0 � �s � 0.42, Vp � �2.80 m/s� (3)

The difference between the two correlations may result fromthe scale-up effect as discussed above.

The average local particle velocity in the three radial sec-tions (0.0 � r/R � 0.632, 0.632 � r/R � 0.894, and 0.894 �r/R � 1.0) can also be related to the cross-sectional mean solidsholdup at a given axial position for both risers. Figures 11a and11b show the relationship of the average local particle velocityand the cross-sectional mean solids holdup for the 76- and the203-mm ID risers. The trends of the three regions in the203-mm ID riser were also represented as darker lines in Figure11a to compare with those in the 76-mm ID riser. The differ-ence in the flow development in the three radial regions is alsoclearly shown in Figure 11. Similar to the case stated before inFigure 5, the particle velocities were first developed in thecenter region of the riser (0.0 � r/R � 0.632) and the particlevelocities are the highest. In the middle region (0.632 � r/R �0.894), on the other hand, the particle velocity is intermediate.In the wall region (0.894 � r/R � 1.0) the particle velocityremains low all over the riser. This again verified that the flowdevelops first in the center region of the riser, and then grad-ually and progressively closer to the wall as the solids passthrough the riser. Furthermore, the scale-up effect on the flowdevelopment is also clearly observed. There is no significantdifference for the two risers in the center region. In the middleregion, on the other hand, the particle velocity of the 76-mm ID

riser is higher than that of the 203-mm ID riser. In the wallregion, the particle velocity of the 76-mm ID riser is alsohigher than that of the 203-mm ID riser. That is, the flowdevelopment is faster in a smaller riser diameter in these tworegions. Thus, the overall flow development is slower with thelarge riser. These scale-up effects are the same as shownbefore. A clear dependency between the particle velocity andsolids concentration is also shown in these figures.

Conclusions

Numerous measurements were performed in a twin-risersystem to show the dependency of the particle velocity distri-bution and the flow development on riser diameter, using afiber-optic probe on eight axial levels in a 10 m long riser of76-mm and 203-mm IDs. The particle velocity is more uniformin the upper section than in the lower section of the riser at all

Figure 11. (a) Relationship between average particle ve-locity in the three radial regions and cross-sectional mean solids holdup for the 76-mmID riser (dark line: trends for the 203-mm IDriser); (b) relationship between average parti-cle velocity in the three radial regions andcross-sectional mean solids holdup for the203-mm ID riser.

Figure 10. Relationship between local particle velocityand local solids holdup for both risers.


radial positions and is higher in the center than in the wallregion of the riser at all axial locations. In the center region ofthe riser (0.0 � r/R � 0.632), the particle velocity remain highand relatively constant throughout the riser, suggesting veryquick solids flow development in the riser center, and thengradually closer to the wall as the solids pass through the riser.In the middle radial region (0.632 � r/R � 0.894), particlevelocity is increasing and continues throughout the column. Inthe wall region (0.894 � r/R � 1.0), however, the flow devel-opment is significantly slower. Furthermore, there is no signif-icant difference for the two risers in the center region. In themiddle and wall regions, on the other hand, the particle velocityof the 76-mm ID riser is higher than that of the 203-mm IDriser. The radial particle profiles change from an S-shape, in thebottom of the riser, to a combination of linear (but not flat) andparabolic shape in the middle, and eventually, parabolic withsome scattering at the exit. The shape change of the radialparticle profiles of the 203-mm ID riser happens at a muchhigher level than that of the 76-mm ID riser. The results alsoshow at the same solids circulation rate and gas velocity, alarger-diameter riser has a steeper radial particle velocity pro-file at each axial level. Therefore, all the results show that theflow development slows down with larger-diameter riser.

Increasing the superficial gas velocity Ug increases the particlevelocity throughout the riser length. By increasing Ug, the flowdevelopment was shown to become faster. Increasing solids fluxGs, on the other hand, slows down the flow development.

The cross-sectional average particle velocity is somewhat lowerwith the larger-diameter riser resulting from the scale-up effect. Inall locations measured, there was a clear dependency between thelocal particle velocity and concentration for each riser. In addition,it is also because of the scale-up effect that the relationshipbetween the particle velocity in the wall region and the cross-sectional solids concentration, and the correlation of the localparticle velocity and solids concentration of the 203-mm ID riserare different from those of the 76-mm ID riser. More researchmust be done to study the riser diameter effect (D), the particle todiameter effect, a change in the dispersion and flow patternsarising from an effect at the riser inlet, or the effect of thelength/diameter ratio on the development of the acceleration zone.

Gas radial distribution and particle aggregation are consid-ered the key factors that affect the local hydrodynamics in thetwin-riser system.

AcknowledgmentsThe authors gratefully acknowledge the financial support from the Natural

Sciences and Engineering Research Council of Canada and help from J. H.Parssinen, J. Wen, H. Zhang, and X. Zhu during the experiments.

Notation

D � riser diameter, m or inchGs � solids circulation rate (solids flux), kg m�2 s�1

r � radial distance from riser axis, mR � radius of riser, m

Ug � superficial gas velocity, m/sVp � particle velocity, m/s

Vp,ave � average particle velocity in three radial regions, m/sVp,wall � particle velocity in the wall region, m/s

Z � height from the riser bottom, m�� s � cross-sectional mean solids holdup�s � local solids holdup

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cm-diameter circulating fluidized bed. In: Basu P, Large JF, eds.Circulating Fluidized Bed Technology II. Oxford, UK: PergamonPress; 1988:123-137.

4. Glicksman LR. Circulating fluidized bed heat transfer. In: Basu P,Large JF, eds. Circulating Fluidized Bed Technology II. Oxford, UK:Pergamon Press; 1988:13-29.

5. Hartge EU, Rensner D, Werther J. Solids concentration and velocitypatterns in circulating fluidized beds. In: Basu P, Large JF, eds.Circulating Fluidized Bed Technology II. Oxford, UK: PergamonPress; 1988:165-180.

6. Kunii D, Levenspiel O. Entrainment of solids from fluidized beds. I.Hold-up of solids in the freeboard. II. Operation of fast fluidized beds.Powder Technol. 1990;61:193-206.

7. Nowak W, Mineo H, Yamazaki R, Yoshida K. Behavior of particles ina circulating fluidized bed of a mixture of two different sized particles.In: Basu P, Horio M, Hasatani M, eds. Circulating Fluidized BedTechnology III. Oxford, UK: Pergamon Press; 1991:219-224.

8. Bi H, Zhu JX. Static instability analysis of circulating fluidized bedsand concept of high-density risers. AIChE J. 1993;39:1272-1280.

9. Knowlton T. Interaction of pressure and diameter on CFB pressuredrop and holdup. Proc of workshop on Modeling and Control ofFluidized Bed Systems, Hamburg, Germany, May; 1995:22-23.

10. Bai D, Issangya AS, Zhu JX, Grace JR. Analysis of the overallpressure balance around a high-density circulating fluidized bed. IndEng Chem Res. 1997;36:3898-3903.

11. Issangya AS, Bai D, Bi HT, Lim KS, Zhu J, Grace JR. Axial solidsholdup profiles in a high-density circulating fluidized bed riser. In:Kwauk M, Li J, eds. Circulating Fluidized Bed Technology V. Beijing,China: Science Press; 1996:60-65.

12. Issangya AS, Bai D, Grace JR, Lim KS, Zhu J. Flow behaviour in theriser of a high-density circulating fluidized bed. AIChE Symp Ser.1997;93:25-30.

13. Issangya AS, Bai D, Bi HT, Lim KS, Zhu J, Grace JR. Suspensiondensities in a high-density circulating fluidized bed riser. Chem EngSci. 1999;54:5451-5460.

14. Grace JR, Issangya AS, Bai D, Bi HT, Zhu JX. Situating the high-density circulating fluidized bed. AIChE J. 1999;45:2108-2116.

15. Karri SBR, Knowlton TM. A comparison of annulus solids flowdirection and radial solids mass flux profiles at low and high massfluxes in a riser. In: Werther J, ed. Circulating Fluidized Bed Tech-nology VI. Frankfurt, Germany: Dechema; 1999:71-76.

16. Parssinen JH, Zhu JX. Axial and radial solids distribution in a long andhigh-flux CFB riser. AIChE J. 2001;47:2197-2205.

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Manuscript received May 27, 2004, and revision received Mar. 14, 2005.


Experimental and Modeling Study of ProteinAdsorption in Expanded Bed

Ping Li, Guohua Xiu, and Alirio E. RodriguesLaboratory of Separation and Reaction Engineering, Dept. of Chemical Engineering, Faculty of Engineering, University of

Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal

DOI 10.1002/aic.10536Published online July 19, 2005 in Wiley InterScience (www.interscience.wiley.com).

Streamline DEAE is the first-generation adsorbent developed for expanded bed ad-sorption (low-density base matrix with wide particle size distribution and ligand sensitiveto ionic strength and salt concentration), and Streamline direct CST I is the second-generation adsorbent (high-density base matrix with narrow particle size distribution andligand not sensitive to ionic strength and salt concentration). In this paper, experimentswere carried out for bovine serum albumin (BSA) protein adsorption in expanded beds,where a Streamline 50 column was packed either with Streamline DEAE or with Stream-line direct CST I. The hydrodynamics, BSA dynamic binding capacity, and BSA recoveryin the whole expanded bed adsorption process were compared for both adsorbents. Amathematical model, in which intraparticle diffusion, film mass transfer, liquid axialdispersion, solid axial dispersion, particle size axial distribution, and bed voidage axialvariation were taken into account, was developed to predict the breakthrough curves inexpanded bed adsorption. BSA breakthrough curves in expanded bed adsorption weremeasured for both Streamline DEAE and Streamline CST I, and compared with thepredictions from this mathematical model. The effects of intraparticle diffusion, film masstransfer, liquid and solid axial dispersion, particle size axial distribution, and bed voidageaxial variation on the breakthrough curves were evaluated for expanded bed adsorptionwith both adsorbents. © 2005 American Institute of Chemical Engineers AIChE J, 51:2965–2977, 2005

Keywords: expanded bed adsorption, protein adsorption kinetics, modeling, break-through curves, residence time distributions

Introduction

Expanded bed adsorption is a single operation in whichdesired proteins are purified from particulate-containing feed-stocks without the need for separate clarification, concentra-tion, and initial purification. This technology has been widelyapplied to capture proteins directly from crude feedstocks, such

as E. coli homogenate, yeast, fermentation, mammalian cellculture, milk, animal tissue extracts, and other unclarified feed-stocks, and various applications have been reported from lab-scale to pilot-plant and large-scale production.1–9

With specially designed adsorbents and columns, the adsorp-tion behavior in expanded beds is comparable to that in fixedbeds.1 Streamline DEAE and Streamline SP are typical first-generation adsorbents, developed for expanded bed adsorption.The modified Sepharose matrices allow capture of biomol-ecules directly from unclarified feedstocks; the adsorbents havehigh binding capacities and product yields, attributed to stable

Correspondence concerning this article should be addressed to A. E. Rodrigues [email protected].


SEPARATIONS


expanded beds, and long life as the result of high chemical andmechanical stability. Many theoretical and experimental re-searches were reported in the literature10-17 with respect to thehydrodynamics and protein adsorption kinetics in expandedbeds packed with the first-generation adsorbents. Streamlinedirect CST I, a second-generation adsorbent recently developedfor use in expanded bed adsorption, has two special featurescompared with the first-generation adsorbents: a high-densitybase matrix and a salt-tolerant ligand. The high-density matrixmeans minimizing dilution arising from biomass or viscosityand reducing dilution buffer consumption; the ligand’s lack ofsensitivity to ionic strength means there is no need for dilutionof feedstock because of high ionic strength. However, hydro-dynamics and protein adsorption kinetics in expanded bedspacked with the second-generation adsorbent have not beeninvestigated in detail.

In this article, experiments were carried out for protein[bovine serum albumin (BSA)] adsorption in expanded beds,where a Streamline 50 column was packed with 300 mLStreamline DEAE and with 300 mL Streamline direct CST I,respectively. The experimental results are compared for bothadsorbents to give a comprehensive evaluation of the hydro-dynamics, BSA dynamic binding capacity, and BSA recoveryin the whole expanded bed adsorption process.

The hydrodynamics and adsorption kinetics in expandedbeds are more complex than that in fixed beds. The liquid axialdispersion in expanded beds is more significant than that infixed beds; because of the fluidized nature of the expanded bed,adsorbent particle axial dispersion occurs. Moreover, there arevariations of particle size axial distribution and bed voidageaxial in expanded beds for the specially designed adsorbentswith wide particle size distribution.11,15,16,18 Models availablefor fixed beds may be not adequate to describe the hydrody-namic and adsorption behavior in expanded beds.

Wright et al.14 developed a mathematical model to predictthe breakthrough curve for protein adsorption in a fluidizedbed, where intraparticle diffusion resistance, film mass transferresistance, liquid axial dispersion, and adsorbent particle axialdispersion were taken into account. Later, Tong et al.19 andChen et al.20 used this model to predict the breakthrough curvesin the expanded bed adsorption. When capturing proteins in anexpanded bed with a high flow velocity, the slow diffusion rateof proteins results in high intraparticle diffusion resistance,significantly affecting the breakthrough curve. It is argued that,in this case, the particle size, characterizing the diffusion pathin the adsorbent particles, should have a substantial effect onthe breakthrough curves.13 Therefore, simulation results shouldbe improved when the particle size axial distribution and bedvoidage axial variations are taken into account in the model.Tong et al.17 modified the mathematical model by taking intoaccount the particle size axial distribution in expanded beds.Following their experimental research using in-bed monitoringin expanded beds, Bruce et al.12 predicted the in-bed break-through curves in expanded beds by using zonally measuredparameters. Li et al.21 developed a three-zone model to predictin-bed breakthrough curves and confirmed the effect of theparticle size axial distribution and bed voidage axial variationson the breakthrough curves in expanded beds. Recently, Kac-zmaarski et al.22 also analyzed the effects of the axial and localparticle size distribution and bed voidage axial variation on thebreakthrough curves in expanded beds.

Up to now, theoretical and experimental researches havefocused on protein adsorption onto Streamline DEAE orStreamline SP in expanded beds. Streamline DEAE andStreamline SP are first-generation adsorbents; the adsorbentmatrix has low density and large particle diameter with wideparticle size distribution (100–300 �m). The matrix of thesecond-generation adsorbent, Streamline direct CST I, has highdensity and small particle diameter with narrow particle sizedistribution (80–165 �m). In this study, BSA breakthroughcurves in expanded bed adsorption are measured for bothStreamline direct CST I and Streamline DEAE, and a mathe-matical model is developed to predict the breakthrough curvesand to compare with the experimental results. The effects ofintraparticle diffusion resistance, film mass transfer resistance,liquid axial dispersion, solid axial dispersions, adsorbent par-ticle size axial distribution, and bed voidage axial variation onthe breakthrough curves will be evaluated for expanded bedspacked with Streamline DEAE and with Streamline direct CSTI, respectively.

Although the adsorbents and the columns are designed spe-cifically for expanded beds to maintain stable bed expansion,the liquid axial dispersion is more significant than that in fixedbeds. Sometimes, some inadequate operation—such as thecolumn not being in a vertical position, trapped air in thebottom distribution system, clogging of the bottom distributionsystem and pump pulse—would make the liquid axial disper-sion more significant. Therefore, it is necessary to exactlymeasure and predict the liquid axial dispersion in expandedbeds to allow stable bed expansion during protein adsorption.

Usually, the liquid axial dispersion coefficient in expandedbeds is measured from residence time distribution (RTD) byusing the moment method.11,23-25 Based on the experimentaldata of RTD curves, the mean residence time and the varianceof distribution can be calculated. Then, by letting the firstabsolute moment of the dispersion model equal the meanresidence time, and the second central moment of the disper-sion model equal the variance of distribution, the liquid axialdispersion coefficient can be obtained easily. However, thevalidity of the dispersion model cannot be judged by thismoment analytical method because the fitting degree of thecalculated response curves to that measured experimentallycannot be evaluated directly. Sometimes, it will cause a sig-nificant deviation for the estimation of the liquid axial disper-sion coefficient with the baseline drift and baseline fluctuationof experimental RTD curves during measurements. Fernandez-Lahore et al.26 fitted the experimental RTD curves with thetheoretical model in Laplace domain using the expressionoriginally developed by Villermaux et al.27 In this paper, ananalytical solution for diract input mode, in which liquid axialdispersion, tracer intraparticle diffusion, and film mass transferall are taken into account, is used to fit experimental RTDcurves to better estimate the liquid axial dispersion coefficient.

ExperimentalEquipment

A pilot-scale Streamline 50 column (Amersham PharmaciaBiotech, Uppsala, Sweden) was used in all expanded bedexperiments. Masterflex® peristaltic pumps (Cole-Parmer In-strument Co., Vernon Hills, IL) were used for buffer and feedapplication and to raise and lower the hydraulic adaptor of the


Streamline 50 column. A Jasco 7800 UV detector (Tokyo,Japan) equipped with a flowcell was used to monitor onlineBSA effluent concentration from the expanded bed and tracer(acetone) effluent concentration in RTD experiments, and theabsorbance signal was logged using the data-acquisition soft-ware in a personal computer.

Adsorbents

Streamline DEAE and Streamline direct CST I were pur-chased from Amersham Pharmacia Biotech. Streamline DEAEis a weak anion exchanger with an –O–CH2CH2–N�(C2H5)2Hfunctional group, with the following characteristics: its matrixconsists of macroporous crosslinked 6% agarose constrainingcrystalline quartz core materials, with a particle density ofabout 1200 kg/m3, a particle size distribution of 100–300 �m,and a mean particle size of 200 �m. Streamline direct CST I isan ion exchanger with a multimodal functional group, with thefollowing characteristics: its matrix consists of macroporouscrosslinked 4% agarose constraining stainless steel core mate-rials, with a particle density of about 1800 kg/m3, a particle sizedistribution of 80–165 �m, and a mean particle size of 135�m.

Model protein

The model protein, bovine serum albumin (BSA; productnumber A3059, further purified fraction V, �99%), was pur-chased from Sigma (St. Louis, MO). The molecular weight ofBSA is about 65,400 g mol�1, diffusion coefficient at infinitedilution in water is about 6.15 � 10�11 m2/s, radius of gyrationis 29.8 Å, and isoelectric point is about 4.7.

A sample solution of BSA was prepared with the appropriatebuffer; in the case of Streamline DEAE the buffer is 20 mMphosphate buffer (pH � 7.5), a mixture of Na2HPO4/NaH2PO4,and in the case of Streamline direct CST I the buffer is 50 mMacetate buffer (pH � 5), a mixture of acetate sodium and aceticacid. Distilled water was used in all experiments.

Batch adsorption experiments

Before performing adsorption isotherm experiments, the ad-sorbents must be saturated by phosphate buffer or acetatebuffer. Adsorbents, in the amount of 0.5 mL by particle vol-ume, are equilibrated with 30 mL of different concentrations ofBSA solution about 8 h at 25°C on a shaking incubator (about30 rpm); then BSA concentration in supernatant liquid is mea-sured by UV 7800 detector at 280 nm (using a 2-mL quartzcuvette). The adsorption capacity is calculated by mass bal-ance.

For the kinetic experiment 2 mL of adsorbent was mixedwith 100 mL of BSA solution in a flask. The adsorption wascarried out in the shaking incubator at 25°C at 150 rpm. Everyfew minutes, about 2 mL of the liquid phase was aspiratedusing a suction tube to determine protein concentration, and thesample was immediately returned to the vessel. By this proce-dure, the time course of BSA concentration in the batch ad-sorber was determined to estimate the effective pore diffusivityof BSA in Streamline DEAE and Streamline direct CST I.

Residence time distribution (RTD) experiments

A Streamline 50 column is packed either with 300 mLStreamline DEAE or with 300 mL Streamline direct CST I to

maintain a stable expansion. For Streamline direct CST I, thesettled bed height is 15.6 cm and the settled bed voidage is0.39; for Streamline DEAE, the settled bed height is 16.5 cmand the settled bed voidage is 0.4. The liquid axial dispersioncoefficient in expanded beds is measured by the RTD method.Acetone inert tracer is used in all RTD experiments. About50% v/v concentration of acetone (1.5 mL buffer aqueoussolution) is used for the dirac input mode, and input position atthe bottom of the column.

Before carrying out the RTD measurement, the bed is ex-panded about 1 h by 20 mM phosphate buffer (pH � 7.5) forStreamline DEAE, and by 50 mM acetate buffer (pH � 5) forStreamline direct CST I. A dirac input of acetone sample isapplied to the column at the bottom of the bed; the effluentliquid sample passes through the flowcell online, where theacetone concentration is monitored by UV detector at 280-nmwavelength; and the ABS (absorbance) signal of acetone islogged by the data-acquisition software in a personal computer.

Experimental procedures for the whole expanded bedadsorption process

Expansion/Equilibration Stage. First, the equilibrationbuffer is pumped through the column with upward flow to theexpected expansion degree. Second, the expanded bed is al-lowed to stabilize at this degree of expansion for about 30–40min. Then, the liquid axial dispersion in the expanded bed ischecked by the RTD method; the liquid axial dispersion coef-ficient should be as small as possible by avoiding inadequateoperation. The adaptor will be positioned about 0.5 cm abovethe height to which the bed expands, to reduce the dead volumein expanded beds.

Adsorption Stage. When the expanded bed is stable andequilibrated with the appropriate buffer, the process switches tofeedstock application. Because of protein adsorption on adsor-bents, the expanded bed height gradually drops, especially inthe expanded bed of low-density Streamline DEAE; therefore,the liquid flow velocity will be increased gradually during theadsorption process to maintain a constant degree of bed expan-sion. The average liquid velocity was calculated from the ratioof total feed volume supplied to the column to the operationtime in the loading process. The effluent stream from the top ofthe column will pass through the flowcell, where BSA effluentconcentration is monitored online by UV 7800 detector, andthe ABS (absorbance) signal of BSA is logged by the data-acquisition software in a personal computer. The BSA concen-tration in the feed was 2 kg/m3, showing a linear relationbetween the UV detector signal (ABS) and BSA effluent con-centration during the adsorption stage.

Washing Stage. When the measurement of the break-through curve was completed, the process switches to the washbuffer to wash out the excess protein, other loosely boundmaterials, and particulates from the column in the expandedmode until the effluent absorbance reaches a relative stablevalue. In the expanded bed of Streamline direct CST I, becauseof the highly favorable adsorption of BSA, the effluent absor-bance approaches the baseline, meaning almost irreversibleadsorption.

Elution Stage. After washing, the pump is turned off andthe bed is allowed to settle. When the adsorbent has settled, theadaptor is moved down toward the surface until the edge of the


adaptor net touches the bed. Elution buffer is then pumpedthrough the settled bed with a downward flow to elute BSAprotein. Because the BSA concentration in the effluent is veryhigh, the effluent samples are collected by small sample tubes,and the BSA concentration in these samples is then monitoredby UV 7800 detector.

Mathematical Model for the Protein Adsorption inExpanded BedsMathematical model

The mathematical model is developed based on the work ofWright et al.,14 where intraparticle diffusion, film mass trans-fer, liquid axial dispersion, and solid axial dispersion weretaken into account. In addition, the particle size axial distribu-tion and bed voidage axial variation in expanded beds are alsoincluded in the model.

The material balance equation for the liquid bulk phase in anexpanded bed is

DL

�2��B�Z�C

�Z2 � u�C

�Z� �B�Z�

�C

�t�

3kf�Z��1 � �B�Z�

R�Z�

� �C � �c�r�R�Z� � 0 (1)

where DL is the axial dispersion coefficient; C is the concen-tration in the fluid phase; u denotes superficial velocity; c is theconcentration in the adsorbent pore; Z is the axial distance fromcolumn entrance; �B(Z)denotes bed voidage at the axial dis-tance Z of the column, and thus [1 � �B(Z)] denotes thefractional volume taken up by the solid phase; r is the radialcoordinate in the adsorbent particle; R(Z) is the radius of theadsorbent at the axial distance Z of the column; t is the time;and kf (Z) is the film mass transfer coefficient at the axialdistance Z of the column.

Boundary Conditions

DL��C

�Z�Z�0

�u

�B�0��C�Z�0 � C0 (1a)

��C

�Z�Z�H

� 0 (1b)

Initial Condition

t � 0 C�Z, 0� � 0 (1c)

The mass balance for adsorbent bulk phases in the expandedbed column is described as

�1 � �B�Z��q�

�t� Dax,S

�2q�

�Z2 � �1 � �B�Z�3

R�Z�kf�Z�

� �C � �c�r�R�Z� (2)

where the q� is the average adsorbent phase concentration andDax,S is the solid axial dispersion coefficient in the expandedbed.

Boundary Conditions

��q�

�Z�Z�0

� 0 (2a)

��q�

�Z�Z�H

� 0 (2b)

Initial Condition

t � 0 q� �Z, 0� � 0 (2c)

The pore diffusion equation in the adsorbent is described as

�P

�c

�t�

�q

�t� De��2c

�r2 �2

r

�c

�r� (3)

where q is the adsorbed amount concentration in adsorbent, �p

is the particle porosity, and De is the adsorbate effective porediffusivity.

Boundary Conditions

De��c

�r�r�R�Z�

�R�Z� Dax,S

3�1 � �B�Z�

�2q�

�Z2 � kf�Z��C � �c�r�R�Z�

(3a)

��c

�r�r�0

� 0 (3b)

Initial Condition

t � 0 c�r� � 0 q�r� � 0 (3c)

In Eq. 3, the relationship between q and c depends on theadsorption equilibrium of the selected experimental system.Based on the experimental measurements for BSA proteinadsorption on Streamline DEAE and on Streamline direct CSTI, the Langmuir isotherm is assigned as

q �qmc

kd � c(4)

where qm is the adsorption capacity and kd is the dissociationconstant, both of which are determined by experiments.

The correlations recommended to estimate the particle sizeaxial distribution and bed voidage axial variation in expandedbeds, packed with the first-generation adsorbents, are

R�Z� �d� p

2 �1.20 � 0.51Z

H� (5)

�B�Z� � �� B�0.629 � 0.738Z

H� (6)


where the averaged adsorbent particle diameter d� p is an aver-aged value over the wider particle size distribution, and the bedaveraged voidage �� B is estimated as a function of the wholeexpanded bed height28

�� B � 1 � � (1 � �0)H0

H � (7)

where �0 and H0 are the settled bed voidage and the settled bedheight, respectively.

The correlations given by Eqs. 5 and 6 are based on theexperimental data of Bruce et al.,11 where a Streamline 50column was packed with Streamline SP to measure both theparticle size axial distribution and bed voidage axial variationin the expanded bed. Because that system is similar to ourexperimental setup, that is, a Streamline 50 column packedwith Streamline DEAE (the same Streamline matrix), we usedin the modeling the hydrodynamic results obtained by Bruce etal.11 Kaczmarski et al.22 recommended two different equationsfor the particle size axial distribution and bed voidage axialvariation; simulated breakthrough curves using correlationsfrom both groups11,22 are similar.

Model parameters

The effective pore diffusivity De in adsorbents is determinedby independent batch experiments.

The liquid axial dispersion coefficient DL is measured ex-perimentally during the expansion stage by the residence timedistribution method.

The adsorbent axial dispersion coefficient Dax,S is estimatedby the correlation of Van Der Meer et al.,29 using experimentalvalues for superficial velocity u, as follows

Dax,S � 0.04u1.8 m2/s (8)

The Wilson–Geankopolis equation,30 applicable to lowReynolds number (Eq. 9), is used to estimate the film masstransfer coefficient kf (Z) in an expanded bed

Sh �1.09

�B�Z�Re1/3Sc1/3 �0.0015 � Re � 55� (9)

where Re is the Reynolds number [�2R(Z)u/�], Sc is theSchmidt number [��/(Dm)], Sh is the Sherwood number[�2kf (Z)R(Z)/Dm], and Dm is the molecular diffusion coeffi-cient.

Numerical method

The model equations are numerically solved by the orthog-onal collocation method. Equations 1 and 2 are discretized atcollocation points in the axial direction in the column, and Eq.3 is discretized at collocation points in the particle radialdirection, leading to a set of ordinary differential equationswith initial values that are integrated in the time domain usingGear’s stiff variable step integration routine. To obtain a stablenumerical solution for a highly favorable adsorption isotherm,21 bed axial collocation points and 21 particle radial colloca-tion points are used.

Results and DiscussionAdsorption isotherm and BSA effective pore diffusivity

Based on the independent batch adsorption equilibrium ex-periments, as shown in Figure 1, circle points represent theexperimental data at room temperature (25°C). BSA proteinadsorption isotherm on both Streamline DEAE and Streamlinedirect CST I can be approximately described by the Langmuirequation as follows.

BSA Adsorption on Streamline DEAE

q �92.59c

0.065 � c(10a)

BSA Adsorption on Streamline Direct CST I

q �82.15c

0.0109 � c(10b)

According to the Langmuir equation, the separation factor fcan be defined as

f �1

1 � c0/kd(11)

which characterizes the adsorption conditions. For a BSA con-centration of 2 kg/m3, which will be used in the adsorptionkinetics experiments in batch adsorber and in expanded bedadsorption later, the separation factors are 0.032 for BSAadsorption on Streamline DEAE and 0.0055 for BSA adsorp-tion on Streamline direct CST I, respectively, indicating highlyfavorable adsorption for BSA on Streamline adsorbents, espe-cially on Streamline direct CST I (where virtually irreversibleadsorption occurs).

Batch adsorption kinetic experiments are carried out to es-timate BSA effective pore diffusivity in Streamline DEAE andin Streamline direct CST I; Figure 2 shows typical experimen-tal data of the bulk concentration profiles in a batch adsorber(with 100 mL of 2 kg/m3 BSA aqueous solution and 2 mLadsorbent).Then, the experimental data of the bulk concentra-

Figure 1. BSA adsorption isotherms on StreamlineDEAE and on Streamline direct CST I at roomtemperature (�25°C).Circle points: experimental data; lines: the calculated resultsby Langmuir equations, Eqs. 10a and 10b.


tion profile are fitted with the simulation results of the porediffusion model to estimate BSA effective pore diffusion co-efficient in Streamline DEAE as 3.2 � 10�11 m2/s and inStreamline direct CST I as 1.7 � 10�11 m2/s. It should beemphasized that in the pore diffusion model, the particle size isassigned an average particle diameter over the particle sizedistribution (d� p� 200 �m for Streamline DEAE; d� p � 135 �mfor Streamline direct CST I), and the particle porosity is as-signed a value of 0.55, according to the published literature.31

The film mass transfer resistance is not negligible for BSAadsorption on Streamline DEAE; in contrast, the film masstransfer can be neglected for BSA absorption on Streamlinedirect CST I in the batch experimental system. A method toestimate both film mass transfer coefficient and effective porediffusivity from a single bulk concentration–time curve in abatch adsorber has been reported elsewhere.32

Bed expansion and liquid axial dispersion coefficient inexpanded bed

The expansion degree of expanded beds packed either withStreamline DEAE or with Streamline directs CST I is measuredat various superficial flow velocities, as shown in Figure 3,where a Streamline 50 column is packed either with 300 mLStreamline DEAE or with 300 mL Streamline direct CST I; thesettled bed height is 15.6 cm for Streamline direct CST I and16.5 cm for Streamline DEAE. It is apparent that at the same

degree of expansion, the superficial liquid flow velocity for thenew Streamline direct CST I is higher than that for the oldStreamline DEAE, given that the particle density of the “new”adsorbent (�1800 kg/m3) is greater than that of the “old”adsorbent (�1200 kg/m3). When the expansion degree is equalto 2, the superficial flow velocity for Streamline DEAE is only228 cm/h, but for Streamline CST I, it allows a higher velocityof the feedstock (up to 553 cm/h) to pass through its expandedbed.

The residence time distribution method with a dirac tracer(acetone) input mode is used to estimate the liquid axial dis-persion coefficient in expanded beds packed with StreamlineDEAE and packed with Streamline direct CST I, respectively;the experimental data of RTD curves are shown in Figure 4,marked as circle points.

Based on the experimental data of RTD curves, the meanresidence time (t�m) and the variance of distribution (2) arecalculated as

t�m �0

� tCdt

0� Cdt

��¥ tiCi��t

�¥ Ci��t(12)

2 �0

� t2Cdt

0� Cdt

� t�m2 �

�¥ ti2Ci��t

�¥ Ci��t� t�m

2 (13)

If we then let the first absolute moment �1 of the dispersionmodel equal the mean residence time (t�m) and the secondcentral moment �2 of the dispersion model equal the varianceof distribution (2), the liquid axial dispersion coefficient canbe easily obtained. Some common calculation formulas, Eq.14,33 Eq. 15,11 and Eq. 16,34 are summarized as follows:

2

t�m2 �

2�� BDL

uH(14)

2

t�m2 �

2�� BDL/uH� � 3�� BDL/uH�2

1 � 2�� BDL/uH� � �� BDL/uH�2 (15)

2

t�m2 � 2

�� BDL

uH� 2�� BDL

uH � 2

�1 � e�uH/��BDL�� (16)

Figure 3. Relationship between bed expansion degreeand superficial liquid flow velocity in expandedbeds packed with Streamline DEAE and withStreamline CST I.

Figure 2. Estimation of BSA effective pore diffusivity inStreamline DEAE adsorbents and in Stream-line CST I adsorbents in batch adsorber.Circle points: experimental data; solid lines: simulation re-sults of the pore diffusion model with De � 3.2 � 10�11 m2/sand kf � 8.6 � 10�6 m/s for Streamline DEAE (a) and withDe � 1.7 � 10�11 m2/s, and kf � � for Streamline direct CSTI (b).


When considering the tracer intraparticle diffusion resis-tance, film mass-transfer resistance, and liquid axial dispersion,one has

2

t�m2 �

2uR� 2�P2

15H

�1 � �� B�

�� B � �p�1 � �� B�2 � 1

�PDP�

5

kfR��

2�� BDL

uH

(17)

In Eq. 17, estimations of acetone intraparticle diffusivity andparticle porosity for acetone are described in the Appendix.

The results calculated by different formulas are similar.Because of the small size of the tracer molecule, acetone, theeffect of the tracer intraparticle diffusion is negligible. ForStreamline DEAE, when the bed expands to about twice thesettled bed height with 228 cm/h superficial flow velocity, theliquid axial dispersion coefficient is 4.45 � 10�6 m/s; forStreamline direct CST I, when the bed expands to about twicethe settled bed height with 553 cm/h superficial flow velocity,the liquid axial dispersion coefficient increases to 17.2 � 10�6

m/s as a result of the liquid flow velocity increase. In thecalculation, the mean residence time and variance of the resi-dence time distribution of the sampling system and the precol-umn tubing were subtracted from the measured mean residencetime and variance of the overall residence time distribution toyield the corrected mean residence time and variance of theRTD of the expanded bed column. As an example, the mean

residence time of extra-column volume and variance are t�c �38 s and c

2 � 352 s2 at 228 cm/h flow rate and t�c � 18 s andc

2 � 126 s2 for 553 cm/h, respectively.If there is a baseline drift or a baseline fluctuation for

experimental RTD curves, the liquid axial dispersion coeffi-cient estimated by the previous calculation method may deviatefrom the real value of liquid axial dispersion coefficients.Therefore, in Figure 4, the experimental data of RTD curvesare fitted by the analytical solutions with the dirac input ofacetone tracer to confirm the calculation accuracy, and at thesame time, the parametric sensitivity is analyzed. The analyt-ical solution is given in the Appendix for reference, in whichthe tracer intraparticle diffusion resistance, film mass transferresistance, and liquid axial dispersion coefficient are all takeninto account. In Figure 4, the mean residence time has beencorrected, and the variance associated with extra-column vol-umes was negligible compared to the variance associated withthe expanded bed.

BSA protein breakthrough behavior in expanded bedspacked with Streamline direct CST I and withStreamline DEAE

BSA Breakthrough Behavior in Expanded Beds Packed withStreamline Direct CST I. When a given volume (300 mL) ofStreamline direct CST I is packed into a Streamline 50 column,the settled bed height is 15.6 cm. BSA aqueous solution withconcentration 2 kg/m3 prepared with 50 mM acetate buffer(pH � 5) is introduced to the column in upward flow with 181mL/min flow velocity. The experimental data of the BSAbreakthrough curve are shown in Figure 5, marked as circlepoints.

The uniform model, where the model parameters are averagevalues all over the column (average particle diameter andaverage bed voidage), is used to predict the breakthroughcurve, as shown in Figure 5 (solid line). Because of the heavieradsorbent, Streamline direct CST I, with narrower particle sizedistribution (80–165 �m), the effects of the particle size axialdistribution and the bed voidage axial variation on the break-through curves are smaller, so the uniform model predicts thebreakthrough curve in expanded beds reasonably well.

Figure 5. BSA breakthrough curve in expanded bedpacked with Streamline direct CST I.Circle points: experimental data; solid line: uniform model;dashed line: the analytical solution (Eq. 18) with irreversibleadsorption (qm � 82.15 kg/m3). The experimental conditionsand model parameters are summarized in Table 1.

Figure 4. Experimental data of RTD curves are fitted bythe analytical solution with the dirac input ofacetone tracer at expanded beds packed withStreamline DEAE and packed with Streamlinedirect CST I.(a) Settled bed height 16.5 cm, expanded degree as 2, super-ficial liquid flow velocity 228 cm/h, � � Dpt/R� 2 � 0.128t; (b)settled bed height as 15.6 cm, expansion degree as 2, super-ficial liquid flow velocity 553 cm/h, � � Dpt/R� 2 � 0.281t.


The model parameters De, kf, DL, and Dax,S characterize theeffects of intraparticle diffusion resistance, film mass-transferresistance, liquid axial dispersion, and solid axial dispersion onthe breakthrough curves during expanded bed adsorption. Fig-ure 6 demonstrates the individual contribution of each modelparameter (De, kf, DL, or Dax,S) on the breakthrough curveduring expanded bed adsorption. First, by controlling film masstransfer resistance, the liquid and solid axial dispersion must beas small as possible to neglect their effects on the breakthroughcurves (kf increased 10 times, DL and Dax,S decreased 10 times);the effect of the model parameter De on the breakthrough curveis demonstrated in Figure 6, represented by a dashed line. It isapparent that the contribution of BSA effective pore diffusivityto the breakthrough curves is dominant (dashed line). Then, inturn, the effects of film mass transfer resistance, liquid axialdispersion, and solid axial dispersion are considered in themodel; the simulation results are demonstrated in Figure 6,respectively. The dashed–double-dotted line represents thesimulation result when both the intraparticle diffusion resis-tance and the film mass transfer resistance are taken intoaccount in the model. By comparing with the simulation result(dashed line, considering only De), the effect of the film masstransfer coefficient is not negligible for such the highly favor-able protein adsorption isotherm because the breakthrough timeis significantly shortened as the result of the effect of kf. Thedashed-dotted line represents the simulation results where in-traparticle diffusion resistance, film mass transfer resistance,and liquid axial dispersion are taken into account in the model,and the solid line represents the simulation results when intra-particle diffusion resistance, film mass transfer resistance, liq-uid axial dispersion, and solid axial dispersion all are taken intoaccount in the model. It is apparent that the effects of the liquidaxial dispersion coefficient and the solid axial dispersion co-efficient are smaller even at high liquid flow velocity (up to 553cm/h) if the bed expansion is stable.

Based on the simulation results, as shown in Figure 6, theeffects of the liquid axial dispersion and solid axial dispersionon the breakthrough curves are smaller in a stable expandedbed packed with Streamline direct CST I. BSA protein adsorp-

tion on Streamline direct CST I is highly favorable, leading toalmost irreversible adsorption (Figure 1); therefore, instead ofthe numerical solution, a simple analytical solution derived atthe irreversible adsorption isotherm with qm � 82.15 kg/m3

(Eq. 18), which takes into account both the intraparticle diffu-sion and film mass transfer, may be used approximately topredict the breakthrough curves in the expanded bed (as shownin Figure 5, dashed line). The analytical solution is indeed closeto the experimental data in expanded bed adsorption. More-over, the analytical solution (dashed line in Figure 5) is similarto the simulated breakthrough curve from a model where onlythe intraparticle diffusion and film mass transfer are significant(dashed–double-dotted line in Figure 6). This analytical solu-tion was reported by Weber and Chakravorti35 based on theassumption of constant pattern, as follows

��1 � 1� Npore �15

�3tan�1�2� � 1

�3 � �15

2 �ln(1 � � � �2)

�1

3� �5

Bi�ln�1 � �3� � 1 �

5

2�3(18)

where the dimensionless parameters are defined as

�1 �

�ut

H� �� B�

Npore �15�1 � ��B�DeH

uR� 2 � � �1 �C

C0�1/3

��1 � �� B�qm

C0Bi �

kfR�

De

Figure 6. Contribution of each model parameter (De, kf,DL, or Dax,[infi]S) to the breakthrough curve inexpanded bed packed with Streamline directCST I.Dashed line: uniform model with De � 1.7 � 10�11 m2/s(neglect kf, DL, and Dax,S effects); dashed–double-dotted line:uniform model with De � 1.7 � 10�11 m2/s and kf � 10.6 �10�6 m/s (neglect DL and Dax,S effects); dashed–dotted line:uniform model with De � 1.7 � 10�11 m2/s, kf � 10.6 �10�6 m/s, and DL � 17.2 � 10�6 m2/s (neglect Dax,S effect);solid line: uniform model with considering De, kf, DL, andDax,S effects (De � 1.7 � 10�11 m2/s, kf � 10.6 � 10�6 m/s,DL � 17.2 � 10�6 m2/s, and Dax,S � 3.45 � 10�7 m2/s). Theother calculation conditions are the same as in Figure 5.

Table 1. Experimental Conditions and Model ParametersUsed for the Simulation of the Breakthrough Curves in

Expanded Beds

Expanded Bed of StreamlineDirect CST I

Expanded Bed of StreamlineDEAE

H0 � 15.6 cm H0 � 16.5 cm�� B0 � 0.39 �� B0 � 0.4d� p � 135 �m d� p � 200 �m�p � 0.55 �p � 0.55C0 � 2 kg/m3 C0 � 2 kg/m3

u � 15.37 � 10�4 m/s* u � 7.10 � 10�4 m/s*H � 32.0 cm H � 33.5 cmH/H0 � 2.05 H/H0 � 2.03�� B � 0.7025 �� B � 0.7045qm � 82.15 kg/m3 qm � 92.59 kg/m3

kd � 0.0109 kg/m3 kd � 0.065 kg/m3

De � 1.7 � 10�11 m2/s De � 3.2 � 10�11 m2/sDL � 17.2 � 10�6 m2/s DL � 4.45 � 10�6 m2/skf � 10.6 � 10�6 m/s kf � 6.6 � 10�6 m/sDax,S � 3.45 � 10�7 m2/s Dax,S � 8.78 � 10�8 m2/s

*u is the average value during the adsorption stage, with the interval minimum–maximum liquid velocity of 6.68–7.52 � 10�4 m/s for Streamline DEAE, andof 15.20–15.54 � 10�4 m/s for Streamline direct CST I.


It should be emphasized that this formula requires that �1Npore

� 2.5 � 1/Bi.The effects of particle size axial distribution, bed voidage

axial variation, liquid axial dispersion, and solid axial disper-sion on the breakthrough curve are smaller than those ofintraparticle diffusivity and film mass-transfer coefficient forBSA adsorption in the expanded bed packed with streamlinedirect CST I; therefore, the uniform model and the analyticalsolution give a reasonable fit of experimental breakthroughcurves. However, in the uniform model, the effects of theparticle size axial distribution and bed voidage axial variationon the breakthrough curve are neglected, which results inpredicted breakthrough curves slightly broader than the exper-imental data (greater amount of larger-size adsorbent at thebottom of the column and low amount of smaller-size adsor-bent at the top of the column, will make the real breakthroughcurve steeper). In the analytical solution, the effects of theparticle size axial distribution and the bed voidage axial vari-ation on the breakthrough curve (making the breakthroughcurves slightly steeper), and the effects of the liquid axialdispersion and the solid axial dispersion on the breakthroughcurve (making the breakthrough curves slightly broader) are allneglected; it happens that the sharpening effect almost equalsthe broadening effect for our experimental conditions, leadingto an analytical solution that is closer to experimental data.

BSA Breakthrough Behavior in Expanded Beds Packed withStreamline DEAE. When Streamline DEAE (300 mL) ispacked into a Streamline 50 column, the settled bed height is16.5 cm. A 2 kg/m3 BSA aqueous solution, prepared with 20mM phosphate buffer (pH � 7.5) is introduced in upward flowat the column bottom with 83.6 mL/min flow velocity. The bedis expanded to about twice the settled bed height. The exper-imental data of the BSA breakthrough curve are shown inFigure 7, marked as circle points. First, the uniform model isused to predict the breakthrough curve, and the simulationresult (dashed line, in Figure 7) is compared with the experi-mental data. The simulation result does not fit the experimentaldata very well.

Based on the experimental results, at the initial breakthroughstage, the real adsorption behavior in expanded beds is betterthan the predicted results by the uniform model; however, there

is a tailing behavior of the breakthrough curves when theeffluent concentration approaches the feed concentration. Inpreviously published articles, the tailing behavior was ex-plained by the presence of the dimer in BSA samples, ormicroporous diffusion in the macroporous adsorbent, or proteinsteric hindrance on active sites of the surface of the adsorbent.Until now, the explanation is still unclear for the tailing be-havior of the breakthrough curves that often occur for bothmedium and large size protein adsorption. Because the Stream-line adsorbents are macroporous, at the initial adsorption stage,the macroporous diffusion should be considered; that is, thebreakthrough curves may be predicted by the macroporousdiffusion model for the initial breakthrough stage.

For the protein adsorption in a stable expanded bed, the slowdiffusion rate of macromolecular protein in adsorbent signifi-cantly affects the breakthrough behavior in the expanded bed.Therefore, the particle size should have an effect on the break-through curves as the result of particle diameter, characterizingthe diffusion path in the adsorbent particle (small particlediameters having a shorter diffusion path length, leading tolower diffusion resistance than that in larger particle). Com-pared with Streamline direct CST I, Streamline DEAE has awider particle size distribution (100–300 �m); when the bedexpands, the smaller and lighter particles move to positions atthe top of the expanded bed, the larger and heavier particles tothe bottom, and more adsorbents will be present at the bottomof the column. At the top zone of the column there is a smallamount of adsorbent, and thus the real adsorption behavior inexpanded beds is better than the predicted result by the uniformmode. The simulation result by the modified uniform modelwith taking into account the particle size axial distribution (Eq.5) and bed voidage axial variation (Eq. 6) is shown in Figure 7,denoted by the solid line. Compared with the uniform model,the simulation results with the modified uniform model betterdescribe the initial breakthrough. For such a highly favorableadsorption isotherm, BSA protein diffuses by a shrinking coremode in the Streamline DEAE, so the more significant effect ofthe particle size axial distribution and bed voidage axial vari-ation on the breakthrough curves will be observed at the end ofthe breakthrough curves, and not at the initial breakthroughstage. In our previous work21 for the small-size protein (ly-sozyme) adsorption on Streamline SP in expanded beds, wherethe tailing behavior of the breakthrough curves was not ob-served, a significant improvement in the simulation results bythe modified uniform model can be observed.

In Figure 8, we give the detailed comparisons among threesimulation results: the dashed line represents the uniformmodel, the dashed-dotted line represents the modified uniformmodel by taking into account the particle size distribution, andthe solid lines represent the modified uniform model by takinginto account both the particle size distribution and bed voidageaxial variation. From Figure 8, by comparing the simulationresults of the uniform model (dashed line) and modified uni-form model with particle size axial distribution (dashed–dottedline), the effect of the particle size distribution on the break-through curve is significant; however, by comparing the sim-ulation results of the modified uniform model with particle sizeaxial distribution (dashed–dotted line) and the modified uni-form model with both particle size axial distribution and bedvoidage axial variation (solid line), the effect of the bed void-age axial variation on the breakthrough curve is small. It should

Figure 7. BSA breakthrough curve in expanded bedpacked with Streamline DEAE.Circle points: experimental data; dashed line: uniform model;solid line: modified uniform model; dashed–dotted line: mod-ified uniform model with double kf value. The experimentalconditions and model parameters are summarized in Table 1.


be emphasized that there is an effect of bed voidage axialvariation on the breakthrough curve when adsorbents havelarger particle diameter with wider particle size distribution.Kaczmarski and Bellot22 claimed that the bed voidage axialvariation had no effect on breakthrough curves only for thecase of smaller-size adsorbents (d� p values of 50 or 150 �m)with relatively narrow particle size distribution.

It is observed that the real bed adsorption behavior at theinitial breakthrough stage is still better than the predictedresults even if we modify the uniform model by taking intoaccount the particle size axial dispersion and bed voidage axialvariation. Based on the theoretical analysis, as shown in Figure6, the film mass transfer coefficient also has an important effecton the breakthrough time for such a highly favorable adsorp-tion isotherm. It is very important to correctly estimate kf in thesimulation. From the published the papers, the film mass trans-fer coefficient kf is estimated by the correlations for fixed bedsor their revised formula. In expanded beds, the adsorbentparticles are suspended in the liquid phase and fluctuate slightlyup and down, which will favor the film mass transfer; that is,the real kf value should be larger than the estimated value of kf

by the previous correlations. In Figure 7, simulation resultswith double kf value (kf � 13.2 � 10�6 m/s) will more nearlyreflect the real breakthrough time.

Comprehensive evaluations on the whole expanded-bedprotein adsorption process with Streamline DEAE andwith Streamline CST I

Experiments are carried out for the whole expanded-bedBSA protein adsorption process with Streamline direct CST I(Figure 9) and with Streamline DEAE (Figure 10). A Stream-line 50 column is packed either with Streamline direct CST I orStreamline DEAE at the same amount of the adsorbents (300mL). With the same degree of expansion (twice the settled bedheight), 2 kg/m3 BSA aqueous solution is applied to the ex-panded beds and BSA protein is adsorbed by the adsorbents;after the adsorption stage, the bed is washed and BSA protein

recovery proceeds at the elution stage. The detailed operationprocedures have been previously described.

Based on the experimental results shown in Figures 9 and10, the comprehensive evaluations on the hydrodynamics, BSAdynamic adsorption capacity, and BSA recovery in the ex-panded bed adsorption process with Streamline DEAE andwith Streamline direct CST I are summarized as follows:

(1) For the same degree of expansion and the same ex-panded bed height, the high-density Streamline direct CST Iallows a higher feed flow velocity (553 cm/h) to pass throughthe expanded bed; in contrast, a low feed flow velocity (259cm/h) is allowed to pass through the expanded bed packed withlow-density Streamline DEAE. Because of the high flow ve-locity, the liquid axial dispersion coefficient is correspondinglyincreased in expanded beds of Streamline direct CST I. Basedon our experimental and theoretical research on the BSA break-

Figure 8. Effect of the particle size axial distribution andbed voidage axial distribution on the break-through curves in expanded beds.Circle points: experimental data; dashed line: uniform model;dash–dotted line: modified uniform model with consideringparticle size axial distribution; solid line: modified uniformmodel with considering both particle size axial distributionand bed voidage axial distribution. The experimental condi-tions and model parameters are the same as in Figure 7.

Figure 9. Effluent curves of BSA protein during adsorp-tion, washing, and elution stages in expandedbed packed with Streamline direct CST I.At the adsorption stage, 2 kg/m3 BSA aqueous solution,prepared with 50 mM acetate buffer, pH � 5, is applied fromthe bottom of the expanded bed at 181 mL/min flow velocity;at the washing stage, 50 mM acetate buffer, pH � 5, is appliedfrom the bottom of the expanded bed; and at the elution stage,50 mM acetate buffer with 1 M NaCl, pH � 7, is applied fromthe top of the settled bed at 39 mL/min flow rate.

Figure 10. Effluent curves of BSA protein during adsorp-tion, washing, and elution stages in expandedbed packed with Streamline DEAE.At the adsorption stage, 2 kg/m3 BSA aqueous solution,prepared with 20 mM phosphate buffer, pH � 7.5, is appliedfrom the bottom of the expanded bed at 83.6 mL/min flowvelocity; at the washing stage, 20 mM phosphate buffer,pH � 7.5, is applied from the bottom of the expanded bed;and at the elution stage, 20 mM phosphate buffer with 0.5 MNaCl, pH � 7.5, is applied from the top of the settled bed at37 mL/min flow rate.


through behavior in an expanded bed of Streamline direct CSTI, the effect of the liquid axial dispersion on the breakthroughcurve is also slight, even at such a high flow velocity if the bedexpansion is stable.

(2) At 5% BSA breakthrough point during expanded bedadsorption, BSA dynamic binding capacity on Streamline di-rect CST I is 34 mg (BSA)/mL of settled bed volume and BSAdynamic binding capacity on Streamline DEAE is 50 mg(BSA)/mL of settled bed volume. However, BSA bindingcapacity on Streamline direct CST I is not sensitive to ionicstrength in feedstock, which means there is no need for dilutionof feedstock arising from high ionic strength. In contrast, BSAbinding capacity on Streamline DEAE is very sensitive to theionic strength in feedstock; when the ionic strength in feed-stock is increased to 50 mM, the BSA adsorption capacitydecreases by half. BSA dynamic binding capacity, Q5%, iscalculated from the BSA breakthrough curve at expanded bedadsorption stage; the formula is defined as

Q5% �

�0

V5%

�C0 � C�dV

VA(19)

where V is the effluent liquid volume from expanded beds, V5%

is the effluent liquid volume at 5% BSA breakthrough point,and VA is the settled bed volume of adsorbents.

(3) At the washing stage, it is found that BSA effluentconcentration quickly decreases and approaches the baselinefor the expanded bed of Streamline direct CST I, which meansalmost irreversible adsorption for BSA binding to StreamlineCST I. BSA adsorption on Streamline DEAE is reversible;when washing, some BSA can be desorbed from StreamlineDEAE so that the effluent concentration approaches a relativelystable value.

(4) The ligand of Streamline DEAE is very sensitive to saltconcentration in buffer, so BSA adsorbed on Streamline DEAEcan be eluted easily by increasing the salt concentration to 0.5M in 20 mM phosphate buffer, pH � 7.5. BSA recovery in thewhole expanded bed adsorption process can reach 91%. Fromthe BSA effluent concentration curve during the elution stage,it is observed that BSA can be eluted almost completely fromStreamline DEAE; however, as a result of desorption duringthe washing stage, BSA recovery is not up to 100%—a smallamount of the elution buffer is consumed, as shown in Figure10. Streamline direct CST I has a multimodal ligand that is notsensitive to the salt concentration. Therefore, it is very difficultto elute BSA from Streamline CST I in the column only byincreasing the salt concentration in 50 mM acetate buffer,pH � 5. Therefore, to accomplish elution of adsorbed BSAproteins, both salt concentration and pH value in acetate bufferare increased. Here, when the elution buffer is 50 mM acetatebuffer with 1 M NaCl at pH � 7, BSA recovery is up to 87%.From the BSA effluent concentration curve during the elutionstage, it can be noted that BSA cannot be eluted completely,and the amount of the elution buffer consumed is also greater,as shown in Figure 9.

Conclusions

With the specially designed adsorbents (Streamline DEAEand Streamline direct CST I), a stable expanded bed can beformed. At the same degree of expansion, the high-densityStreamline direct CST I allows a higher feed flow velocity topass through the expanded bed; in contrast, a lower feed flowvelocity is allowed to pass through the expanded bed of lower-density Streamline DEAE. With the high feed flow velocity,the liquid axial dispersion is more significant in the expandedbed of Streamline direct CST I than that in the expanded bed ofStreamline DEAE.

In spite of the existence of intraparticle diffusion resistance,film mass transfer resistance, liquid axial dispersion, and solidaxial dispersion during expanded bed adsorption, the contribu-tion of BSA effective pore diffusivity to the breakthroughcurves is dominant. The film mass transfer coefficient has asignificant effect on the initial breakthrough time for the highlyfavorable protein adsorption isotherm; liquid axial dispersionand solid axial dispersion have a lesser effect on the break-through curves, even at high liquid flow velocity (up to 553cm/h for Streamline direct CST I), if the bed expansion isstable.

Because of the narrow particle size distribution of Stream-line direct CST I, the effects of the particle size axial dispersionand the bed voidage axial variation on the breakthrough be-havior in the expanded bed are small, and thus the uniformmodel can be used to predict the breakthrough curves withacceptable accuracy. In contrast, because of the wide particlesize distribution of Streamline DEAE, the effects of the particlesize axial distribution and bed voidage axial variation on thebreakthrough curves in the expanded bed should be taken intoaccount in the model.

Based on the experimental results, at 5% BSA breakthroughpoint during expanded bed adsorption, the BSA dynamic bind-ing capacity on Streamline DEAE is 50 mg (BSA)/mL ofsettled bed volume, larger than that on Streamline direct CSTI (34 mg (BSA)/mL of settled bed volume). However, the BSAbinding capacity on Streamline CST I is not sensitive to ionicstrength in the feedstock, which means there is no need fordilution of feedstock even at high ionic strength. In contrast,the BSA binding capacity on Streamline DEAE is very sensi-tive to the ionic strength in the feedstock; when the ionicstrength in the feedstock is increased to 50 mM, BSA adsorp-tion capacity decreases by half.

The ligand of Streamline DEAE is very sensitive to saltconcentration in the buffer, so BSA adsorbed on StreamlineDEAE can be easily eluted by increasing the salt concentrationto 0.5 M in 20 mM phosphate buffer, pH � 7.5. BSA recoveryin the whole expanded bed adsorption process reaches 91%,and a slight amount of the elution buffer is consumed. Stream-line direct CST I has a multimodal ligand that is not sensitiveto the salt concentration. Both salt concentration and pH valueshould be increased in the elution buffer; for example, 50 mMacetate buffer with 1 M NaCl, pH � 7, is used as the elutionbuffer in this experiment, and BSA recovery in the wholeexpanded bed adsorption process is up to 87%, and the amountof the elution buffer is greater than that consumed for Stream-line DEAE. In addition, from the BSA effluent concentrationcurve during the elution stage, it is observed that BSA cannotbe eluted completely from Streamline CST I.


AcknowledgmentsWe acknowledge financial support from Fundacao para a Ciencia e

Tecnologia (FCT Grant SFRH/BD/6762/2001).

Notation

Bi � Biot numberc � concentration in particle pore, kg/m3

C � concentration in fluid, kg/m3

C0 � inlet concentration in fluid, kg/m3

d� p � average diameter of adsorbent, mDL � liquid axial dispersion coefficient, m2/sDm � molecular diffusion coefficient, m2/sDp � intraparticle pore diffusivity, m2/sDe � intraparticle effective pore diffusivity, m2/s

Dax,S � solid axial dispersion coefficient, m2/sH0 � settled bed height, mH � expanded bed height, mkd � dissociation constant defined by Eq. 4, m3/kgkf � film mass transfer coefficient, m/sn � amount of sample injected, kg

Pe � Peclet numberq � adsorbed concentration in adsorbent, kg/m3 particleq� � averaged adsorbent phase concentration, kg/m3

qm � maximum adsorbed concentration, defined by Eq. 4, kg/m3

particler � radial distance from center of particle, mR � radius of adsorbent, m

Re � Reynolds numberSc � Schmidt numberSh � Sherwood number

t � time, su � superficial liquid flow velocity, m/s

Vc � column volume, m3

Z � axial distance from column entrance, m

Greek letters

�0 � settled bed voidage, m3/m3

�B � bed voidage of expanded bed, m3/m3

�p � adsorbent porosity, m3/m3

� � liquid viscosity, Pa�s � liquid density, kg/m3

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Appendix: Analytical Solution for the ResidenceTime Distribution (RTD) Curve

In the mathematical model, the liquid axial dispersion, in-traparticle diffusion, and film mass transfer resistance are alltaken into account. The material balance equation for fluidphase is

DL

�2C

�Z2 �u

�� B

�C

�Z�

�C

�t�

�1 � �� B�

�� B

3

R��PDp��c

�r�r� �R

� 0

(A1)

The material balance equation in pore particle is

�p

�c

�t� �pDp��2c

�r2 �2

r

�c

�r� �0 � r � R� � (A2)

The initial and boundary conditions for Eqs. A1 and A2 are

C�Z, 0� � 0 (A3)

c�r, Z, 0� � 0 (A4)

C�0, t� � C0tc��t� with dirac input mode (A5)

C��, t� is limited (A6)

��c

�r�r�0

� 0 (A7)

�PDp��c

�r�r� �R

� kf�C � �c�r� �R (A8)

Based on published works,36-38 the analytical solution for theRTD curve at dirac input mode, is

y�� 1

�0

�

exp�Pe�

2� ��a2 � b2 � a

2��

cos�� a2 � b2 � a

2�d� (A9)

where

a � Pe�Pe

4� 3��

Bi2I1 � Bi(I12 � I2

2)

(Bi � I1)2 � I2

2 � (A10)

b � �Pe�� 3�Bi2I2

(Bi � I1)2 � I2

2� (A11)

I1 � ��

2 � sinh�2� � sin�2�

cosh�2� � cos�2�� 1 (A12)

I2 � ��

2 � sinh�2� � sin�2�

cosh�2� � cos�2�� (A13)

The dimensionless variables are

y�� C

C0�c� �

Z

H� � �1 � �� B

�� B��p Bi �

kfR�

�pDp

� �� BHDp

uR� 2 Pe �uH

��BDL� �

Dpt

R� 2

where the reference concentration is the ratio between theamount injected and the fluid volume in the column, C0 �n/�� BVc and �c � Dptc/R�

2 (tc � �� BH/u is the space time).The tracer (acetone) intraparticle pore diffusivity is esti-

mated approximately as Dp � 1.28 � 10�9 m2/s; the film masstransfer coefficient kf is estimated by the correlation Eq. 9; andthe particle porosity for acetone is estimated by first moment�1, equal to the mean resistance time of the experimental RTDcurve, as suggested by Boyer and Hsu.31

Manuscript received Dec. 13, 2004, and revision received Feb. 22, 2005.


The Fiber-Coating Model of BiopharmaceuticalDepth Filtration

Glen R. Bolton, Daniel LaCasse, Matthew J. Lazzara, and Ralf KuriyelMillipore Corporation, Bedford, MA 01730


Depth filters are often used in biotechnology processes to remove cell debris from cellculture solutions. A model of filter plugging was developed to allow predictions ofrequired filter area from limited amounts of initial data. The filter microstructure wasidealized as an assembly of randomly oriented, straight, cylindrical fibers. It was assumedthat filters plug as solids coat the surfaces of fibers, making them thicker and reducingfilter porosity and permeability. The Carman–Kozeny equation was used to allow calcu-lation of filter permeability as a function of filter fiber radius, filter solid fraction, andvolume filtered. An explicit equation for filtrate volume as a function of time duringconstant pressure operation was derived, and an explicit equation for pressure as afunction of time during constant flow rate operation was derived. A second model thataccounted for the combined effects of fiber coating and surface caking was also derived.The models were tested on data from a replica cell culture fluid filtered through a glassfiber based depth filter and data from E. coli lysate filtered through glass fiber anddiatomaceous earth based depth filters. The fiber coating model provided good fits of thevolume vs. time data and good predictions of filter capacity from limited initial data. Thecombined cake-fiber coating model provided improved performance over the fiber coatingmodel. The models are useful tools for sizing of depth filters for fermentation and cellculture applications and may be useful for other liquid and gas filter applications. © 2005American Institute of Chemical Engineers AIChE J, 51: 2978–2987, 2005Keywords: bioseparations, membrane separations, depth filter, fouling, sizing

Introduction

A wide range of processes utilize filters to remove particlesfrom aqueous solutions. Depth filters are often used in biotech-nology processes to remove cell debris and other particulates.The filter area required by these applications can either bedetermined by filter permeability or filter capacity. Filter ca-pacity is an indication of how much volume can be filteredbefore plugging occurs. Capacity during constant pressure op-eration is defined as the amount of fluid per filter area that canbe processed until the flow rate declines to a set fraction of theinitial flow. Capacity under fixed flow rate conditions is deter-

mined when the pressure increases to some set multiple of theinitial value. Fouling of a filter can occur by deposition ofparticles inside or on top of the filter.1,2 There are four mech-anistic models that are typically used to describe fouling.Complete blocking assumes that particles seal off pore en-trances and prevent flow. Intermediate blocking is similar tocomplete blocking but assumes that a portion of the particlesseal off pores and the rest accumulate on top of other depositedparticles. Cake filtration occurs when particles accumulate onthe surface of a filter in a permeable cake of increasing thick-ness that adds resistance to flow. Standard blocking assumesthat particles accumulate inside filters on the walls of straightcylindrical pores. As particles are deposited, the pores becomeconstricted and the permeability of the filter is reduced. Thismodel was developed by Hermans and Bredee3 and has typi-cally been applied to the fouling of microporous filters.4

Correspondence concerning this article should be addressed to G. Bolton [email protected].



The physical assumptions in the classical blocking modelsare not representative of most depth filters and are therefore notwidely used. The straight cylindrical pore microstructure as-sumed in the classical models does not represent the fibrousstructure of most depth filters. Cake can form above depthfilters but the pore blockage and pore constriction models donot accurately describe the mechanisms of depth filter fouling.Depth filters are typically fouled by various mechanisms ofdeposition of particles on the internal surfaces of the media.5

In this work an alternative model that treats filters as randomfiber matrices was developed. The model predicts a decrease infilter permeability as solids from the process stream deposit onthe filter fibers. This deposition increases the effective radius ofthe fibers and decreases the filter void fraction. A combinedmodel that accounts for cake formation and fiber coating oc-curring simultaneously was also developed. The models pro-vided good fits of volume vs. time data during filtration andmade accurate predictions of filter capacities from limitedinitial data.

TheoryFiber coating model

A model of internal filter plugging that treats filters asassemblies of straight cylindrical fibers was developed. It as-sumes that filters plug as solids coat the surfaces of fibers,making them thicker and reducing filter permeability. This isillustrated in Figure 1.

The model was developed by first relating the fiber radius tothe volume of solids removed in a manner similar to that of thestandard blocking model1

LF2�rfdrf � � c

1 � ��AdV (1)

�LF�rf2 � rf 0

2 � � � c

1 � ��AV (2)

where rf is the fiber radius, LF is the total length of all fibers, cis the volume of solids per unit of filtrate volume, � is theporosity of solid when deposited on the fibers, and V is thevolume processed, normalized by filter area A. The filter solidfraction � is related to the fiber radius, filter thickness LM, andother parameters.

�rf 02 LF

LMA� �0 (3)

The subscript “0” is used to indicate an initial value of a givenquantity. After fouling has occurred, the new solid fraction canbe related to the new fiber radius by

� ��rf

2LF

LMA� �0� rf

rf 0� 2

(4)

Although the deposited solids are assumed to be porous in Eq.2, Eq. 4 assumes the solids are nonporous. This assumptionwill be accurate if the flow through the deposited solids isminimal. The term Vmax describes the amount of solutionvolume that can be filtered until the filter void volume iscompletely filled with solids. A new term, the fiber coatingconstant Kf, is equal to the inverse of Vmax. By material balanceof solids in the filter volume, Vmax is calculated as

Vmax �1

Kf� LM�1 � �

c ��1 � �0� (5)

Figure 1. Illustration of a filter composed of randomly oriented fibers before (left) and after (right) plugging.The filters plug as solids coat the surfaces of fibers, increasing fiber thickness and reducing filter permeability.


Rearranging Eq. 3 to an expression for LF and substitutinginto Eq. 2 yields Eq. 6, which allows calculation of rf and �from V and four physical constants: LM, c, �, and �0. Insertionof Eq. 5 reduces this to two physical constants, Kf and �0.

rf2

rf 02 �

�

�0� �1 � � c

1 � �� V

LM�0� � �1 �

(1 � �0)

�0KfV�

(6)

Combining Eq. 6 with a theoretical model that relates hydraulicpermeability to fiber radius allows a filter-plugging model to bederived that relates permeability to filtrate volume.

The Carman–Kozeny model,

�

rf2 �

�1 � ��3

20�2 (7)

where � is filter Darcy permeability, was developed by extend-ing a correlation between Reynolds number and friction factorfor laminar flow through pipes to flow through porous media.6

It has also been used to describe flow through fibrous media.7

The flow rate through a filter (dV/dt) can be calculated fromDarcy’s Law as

dV

dt� �

�P

�LM(8)

where �P is the transmembrane pressure and � is the solutionviscosity.

In this study it was found that the Carman–Kozeny modelwas ideal for use in the fiber coating model of filter plugging.As described in more detail in Appendix A, the Carman–Kozeny model accurately represented both rigorous theoreticalpredictions and a large amount of experimental data over therelevant range of fiber volume fractions.

Fiber coating model: constant pressure

To obtain the equations for the fiber coating model, Eq. 6 canbe substituted into Eqs. 7 and 8 to yield flux as a functionvolume for filtration experiments run at constant pressure

dV

dt� � �P

�LM� rf

2�1 � ��3

20�2 �J0�1 � KfV�3

�1 �(1 � �0)

�0KfV� (9)

where J0 is the initial solvent flux. The equation can be inte-grated to obtain an explicit equation for filtrate volume as afunction of time

V �1

Kf�

1

Kf�1 � �1 �

(1 � �0)� (10)

where is a dimensionless parameter, defined as

� �2KfJ0t � 1�� 0

1 � �0� � 1 (11)

To determine the filtrate volume as a function of percentageflux decline, such as the volume after the flux has declined 75%from its initial value, it is necessary to solve for the roots of Eq.9. The real root of this cubic polynomial can be calculatedusing Cardan’s formula8

V �1

Kf�1 � �3 � �� 3 � � �� (12)

where �, , and � are placeholders defined, respectively, as

� ��1 � �0�

3�0

J

J0

�1

2�0

J

J0

� � �3 � 2

Fiber coating model: constant flow

For filtration experiments run at a constant flux, Eq. 9 can berearranged to obtain an explicit expression for pressure as afunction of time, where V � J0t

�P

�P0�

�1 �(1 � �0)

�0KfV�

�1 � KfV�3 (13)

This model is similar to a model of increasing pressure dropwith deposition of solids on packed spheres.5 To determine thefiltrate volume as a function of relative pressure increase, suchas the volume after the pressure has increased tenfold from theinitial value, Eq. 12 can be used, where the parameters � and will now be defined, respectively, as

� ��1 � �0�

3�0

�P0

�P

�1

2�0

�P0

�P

Equations 10 and 13 were derived assuming that fibers neveroverlapped and all solids deposited evenly along the fibers. It ispossible, however, to have fibers that intersect or have a sig-nificant portion of their surface area in contact with one an-other. In this case the fiber radius will increase with permeatevolume faster than that described by Eq. 6 because the solidswill deposit on fiber surfaces only where there is no contact oroverlap. Equations 10 and 13 are corrected for fiber overlap inAppendix B. In practice filters typically will have characteris-tics that are between the cases of overlapping and nonoverlap-ping fibers. For the glass fibers used in this study there will befiber contacting and but no overlap initially. Once solids aredeposited, however, the coated fibers will overlap at the pointswhere the fibers were initially in contact. This is illustrated inFigure 1. The sizing predictions in this study were not signif-icantly changed by accounting for the effects of fiber overlap.


Combined cake-fiber coating model

Typically foulants accumulate throughout the thickness ofdepth filters. Large foulants such as intact cells, however, canaccumulate as cakes above filters. To account for the effects ofthese cakes, the fiber coating model was modified by addingthe resistance caused by these cakes. The cake resistance R canbe calculated as a function of permeate volume

R

R0� �1 � KcJ0V� (14)

where Kc is the cake filtration constant and R0 is the initial filterresistance.2 For the combined model, the filter resistance in-creases with permeate volume as the fibers become coated withfoulants and, simultaneously, the cake thickness increases withpermeate volume. This is similar to recently developed modelsof the combined effects of the classical fouling modes.9 HigherKf or Kc values physically correspond to a larger reduction infilter permeability for a given permeate volume. The flux canbe calculated as a function of permeate volume using thefollowing equation, where the total resistance is the sum of theresistances of the fibrous filter and the cake

dV

dt�

J0

��1 �(1 � �0)

�0KfV� (1 � KfV)�3 � KcJ0V� (15)

Equation 17 can be integrated to calculate time as a function ofvolume

t �KcV

2

2�

1

2Kf J0��1 �

(1 � �0)

�0(KfV)2�(1 � KfV)�2 � 1� (16)

Equations 17 and 18 reduce to the equations for cake filtra-tion when Kf is small and to the equations for fiber coatingwhen Kc is small. It is difficult to rearrange Eq. 18, a fourth-order polynomial in volume, to calculate volume as a functionof time, although Eq. 18 is still useful for fitting experimentaldata and predicting filter capacities. Typically the experimentalvalues of time are used as inputs to plugging models to calcu-late volumes. The predicted volumes are then compared to theexperimental values to determine the best-fit parameters. Thebest-fit parameters in Eq. 18 can be determined by using theexperimental volume values as inputs and calculating the cor-responding time values. The predicted times can then be com-pared to the experimental values and the parameters Kc and Kf

can de determined.For the case where the filtration experiment is run at a

constant flux, Eq. 17 can be rearranged to obtain Eq. 19, whichallows explicit calculation of pressure as a function of time,where V � J0t

�P

�P0�

�1 �(1 � �0)

�0KfV�

�1 � KfV�3 � KcJ0V (17)

To determine the filtrate volume as a function of relativepressure increase, such as the volume after the pressure hasincreased tenfold from the initial value, it is most convenient tosolve Eq. 19 numerically.

ExperimentalGlass-fiber filter

Filters were composed of glass fibers with a binder resin(AP1504700, Millipore, Bedford, MA). Fiber radii were deter-mined from scanning electron micrographs and filter perme-abilities. Samples were sputter coated using a Au/Pd targetbefore SEM analysis. Images were acquired at a 1000� mag-nification using a Topcon DS-130-C scanning electron micro-scope. The fibers observed in scanning electron micrographshad radii ranging from 0.2 to 5.0 �m. The filter solid fractionwas determined to be 0.07 � 0.01 by measuring the volumedisplacement of the media as follows. Three 47-mm mediadisks were rolled up and submerged in a 20-mL graduatedcylinder containing 18 mL of water. The cylinder was then putin a 1 gallon sonic cleaner (FS30H, Fisher Scientific, Hampton,NH) for 10 min to remove air bubbles and the volume dis-placement was measured. This was done in triplicate. Themedia thickness was determined to be 0.54 mm using calipers.The average hydraulic permeability of the media was 9400 Lm�2 h�1�psi. The effective fiber radius was determined byfitting the initial permeability of the filter using Eq. 16. Theeffective fiber radius was 0.17 microns, which was at the lowend of the range observed in filter SEMs.

E. coli lysate

Plugging experiments were performed with a solution oflysed Escherichia coli BL21 cells obtained from Dr. CarlLawton of the University of Massachusetts, Lowell. The lyastewas obtained frozen at pH 7.0 in a mixture of 50 mM sodiumacetate with 2 mM EDTA lysis buffer and PBS media buffer.

Replica cell culture fluid

A fluid representing a clarified solution from a mammaliancell culture vessel was made up of the following components:13.47 g/L DMEM (D-7777, Sigma, St. Louis, MO), 3.70 g/LNaHCO3 (7412, Mallinckrodt, Hazelwood, MO), 1 g/L Plu-ronic F-68 (P1300, Sigma), 0.0775 g/L DNA(D6898, Sigma),5.0 g/L soy peptone fraction IV (P0521, Sigma), 1.5 g/L dairywhey (W1500, Sigma), and 3.00 mg phosphatidylcholine(P3644, Sigma). The feed solution was used within 2 h afterpreparation to minimize variability.

Filter testing equipment: constant pressure

The plugging solution was contained in a pressure vessel(XX1100000, Millipore). The filter was placed in a 47-mmstainless steel holder (XX4404700, Millipore). An electronicbalance aperture and a collection vessel were used to collectand measure the filtrate weight. The electronic balance con-sisted of a load cell (1042-3-I, Tedea Huntleigh, Covina, CA)that interfaced with a computer through a data-acquisitionboard.

During constant pressure experiments, the system was pres-surized with air and a digital pressure gauge was used


(2798K12, McMaster-Carr, Atlanta, GA). The 47-mm filterdisks were wetted with water and placed in holders. Theholders were then connected to 600-mL feed reservoirs. Thefeed solutions were added to their appropriate chambers andthe four chambers were pressurized to 3 psi. The filter holderswere vented to remove air. The filtrate valves were then openedand data acquisition began. The filter surface area was 1.38 �10�3 m2. Data were collected at 1-s intervals for 1 h.

During constant flow experiments, a three-head peristalticfeed pump (7523-50, 77201-60, Cole Parmer, Vernon Hills, IL)was used with a pressure gauge (XXPXLGAGE, Millipore).The load cell, data acquisition, and filter holder were the sameas those used in the constant pressure experiments.

ResultsConstant pressure: E. coli lysate on glass fiber

The E. coli lysate solution was diluted 10-, 100-, and 1000-fold with pH 7.2 PBS buffer (DF2314-15-0, Fisher) and filteredat a constant pressure through the glass filters. The permeatevolume was measured as a function of time until the flux haddecreased by 95% from its initial value.

The permeate volume and time data were fit using the fibercoating model, Eq. 10, by varying Kf, and using the cake-fibercoating model, Eq. 18, by varying Kf and Kc. The best fit wasdetermined by minimizing the sum of squared residuals (SSR),where the residual was equal to the difference between anexperimental data point and a model prediction. The residualwas calculated from volume values for the fiber coating model,and from time values for the cake-fiber coating model. Thevolume vs. time data are compared to model predictions inFigures 2, 3, and 4 for the different lysate dilutions.

The fiber coating model provided good fits of the E. colilysate data. The solid volume fraction, determined from Eq. 12,increased from the initial value of 0.07 to 0.54 and 0.39 for thestreams diluted 10- and 1000-fold, respectively. The cake-fibercoating model provided better fits than the fiber coating modelalone. The difference between the fit of the two models in-creased with decreasing dilution factor. The combined model

does benefit from the use of two fitted parameters. It is possi-ble, however, that more cake formed above the filters used withthe more concentrated streams, as would be expected, resultingin the improved fit of the combined model. The Kf term wasnear zero, indicating that the cake model alone provided thebest fit for the 10 � diluted stream. Table 1 summarizes theresiduals, an indication of quality of fit, and the fitted param-eters from the two models for the different streams.

Constant flow rate: replica cell culture fluid on glassfiber

The applicability of the models to data from a constant flowexperiment performed with the replica cell culture fluid wastested. The fluid was filtered at a constant flow rate of 2600 Lm�2 h�1 through the AP15 glass media and the pressure wasmeasured. The experiment was performed with the replica cellculture fluid diluted four- and ten fold. The experimental ap-

Figure 2. Volume vs. time data compared to the fibercoating and cake-fiber coating model predic-tions with 10� diluted E. coli lysate run at 3 psithrough AP15 glass-fiber media.

Figure 3. Volume vs. time data compared to the fibercoating and cake-fiber coating model predic-tions with 100� diluted E. coli lysate run at 3psi through AP15 glass-fiber media.

Figure 4. Volume vs. time data compared to the fibercoating and cake-fiber coating model predic-tions with 1000� diluted E. coli lysate run at 3psi through AP15 glass-fiber media.


paratus, filter area, and filter thickness were the same as de-scribed previously. The pressure vs. time data were fit to thefiber coating model, Eq. 13, and the cake-fiber coating model,Eq. 19, by minimizing the SSR values. The initial pressuredrop was too low to measure accurately. It was estimated to be0.3 psi by dividing the flux by the filter permeability. The dataand model predictions are shown in Figures 5 and 6. Table 2summarizes the residuals and fitted parameters from the twomodels.

The fiber coating model provided good fits of the pressureincrease observed in the experiments for both the 4 and 10 �diluted replica cell culture fluid stream. The fits were improvedby using the combined cake-fiber coating model, although thismodel does benefit from the use of two fitted parameters. It ispossible that in these experiments, similar to the observation inthe E. coli lysate experiments, more cake formed above thefilter used with the more concentrated stream, resulting in theimproved fit of the combined model.

Constant flow rate: E. coli lysate on Millistak

The applicability of the models to constant flow data for thefiltration of 10 � diluted E. coli lysate on commercially avail-able fibrous depth filters was tested. The fluid was filtered at aconstant flow rate of 364 L m�2 h�1 through a Millistak A1HC(MA1HC23HH3, Millipore) disposable depth filter device andthe pressure was measured. The filter consisted of about 3 mmof an open grade of cellulose media embedded with diatoma-ceous earth, followed by 3 mm of a tighter grade, followed bya thin layer of microporous membrane. The fibrous, cellulose-based structure of this media makes it ideal for application ofthe fiber coating model. The solid fraction of the first layer wasdetermined to be 27%, and the second layer 23%, from mea-surements of volume displacement using water and a volumet-ric flask. The solid fraction of the overall filter was taken as25%, the thickness-weighted average of the two measurements.The pressure vs. time data were fit to the fiber coating model,Eq. 13, by minimizing the SSR values. The best fit of the

cake-fiber coating model occurred with Kc near zero and wasidentical to the fit of the fiber coating model alone. The fibercoating model provided a good fit of the pressure increaseobserved in the experiment, as shown in Figure 7.

The ability of the fiber coating and cake-fiber coating modelsto accurately predict filter sizing was evaluated by fitting alimited amount of initial pressure data and then comparing thepredicted volume corresponding to a set pressure limit to theexperimental data. This is representative of scale-down studiesused to estimate area requirements for large-scale filter pro-cesses. Generally only limited time and process fluid are avail-able. Extrapolating data from partially plugged filters to higherplugging values using a plugging model can save time, processfluid, and allow more experiments to be performed. Parameterswere fit to the initial data and the volume that would beobtained when the pressure reached the final pressure wascalculated. The predictions are compared to the actual volumesat the respective final pressures in Figure 8 for the fiber coatingmodel (Eq. 13) and Figure 9 for the cake-fiber coating model(Eq. 19).

The fiber coating model predictions in Figure 8 were within20% of the experimental data when more than 15% of the fullpressure range was used for parameter fitting. This was true forboth dilutions of the replica cell culture fluid run on the glassfilter and the E. coli lysate run on the Millistak media. Thisindicates the fiber coating model is a useful tool for makingaccurate filter sizing predictions with heterogeneous depth fil-tration media based on limited amounts of pressure vs. volumedata. Occasionally, fibrous clarification media will be limitedby breakthrough of impurities and not by pressure increase asa result of fouling. In this case other criteria such as permeateturbidity will be more important for sizing.

The combined cake-fiber coating model provided some im-provement in the accuracy of the capacity predictions. Thecake-fiber coating model predictions in Figure 9 were up to16% closer to the actual values than the fiber coating model forthe 10 � diluted replica cell culture fluid, and up to 4% better

Table 1. Error of Fit and Model Parameters for the Fiber Coating and Cake-Fiber Coating Models*

Model Dilution Factor Initial Flux, J0 (m/s) Model Fit Error, SSR Fit Parameter Values

Fiber coating model (Eq. 10) 10 7.83 � 10�3 4.37 � 10�3 Kf � 1.25 � 101

Cake-fiber coating model (Eq. 18) 10 7.83 � 10�3 2.63 � 101 Kf � 1.00 � 10�8

Kc � 7.78 � 104

Fiber coating model (Eq. 10) 100 7.83 � 10�3 2.09 � 104 Kf � 1.39Cake-fiber coating model (Eq. 18) 100 7.83 � 10�3 5.46 � 102 Kf � 9.36 � 10�1

Kc � 3.01 � 103

Fiber coating model (Eq. 10) 1000 7.83 � 10�3 7.80 � 105 Kf � 9.87 � 10�2

Cake-fiber coating model (Eq. 18) 1000 7.83 � 10�3 3.24 � 104 Kf � 8.16 � 10�2

Kc � 1.02 � 102

*Data for 10,100 and 1000� diluted E. coli lysate run at 3 psi through AP15 glass-fiber media.

Table 2. Error of Fit and Model Parameters for the Fiber Coating and Cake-Fiber Coating Model*

Model Dilution Factor Initial Flux, J0 (m/s) Model Fit Error, SSR Fit Parameter Values

Fiber coating model (Eq. 13) 4 7.25 � 10�4 41.6 Kf � 2.92Cake-fiber coating model (Eq. 19) 4 7.25 � 10�4 10.7 Kf � 2.76

Kc � 3.75 � 104

Fiber coating model (Eq. 13) 10 7.25 � 10�4 24.1 Kf � 1.72Cake-fiber coating model (Eq. 19) 10 7.25 � 10�4 3.43 Kf � 1.64

Kc � 1.49 � 104

*Data for 4 and 10� diluted replica cell culture fluid run at 2600 L m�2 h�1 through AP15 glass-fiber media.


for the 4 � diluted replica cell culture fluid. The combinedmodel did not provide better fits of the E. coli lysate data andwas not used for capacity predictions. It is likely that thecake-fiber coating model will be more accurate than the fibercoating model for sizing filters in processes where large fou-lants that accumulate above filters are present.

Conclusions

A model of internal filter plugging that assumed filters werecomposed of straight cylindrical fibers was developed. Themodel relates the volume filtered to the filter solid fraction andfiber thickness by assuming that all plugging solids coated thefibers and increased the filter solid fraction. The Carman–Kozeny equation was used with Darcy’s Law to calculate fluxas a function of volume filtered. The flux equation was inte-

grated to derive an explicit equation for filtrate volume as afunction of time when operating at constant pressure. Anexplicit equation for pressure as a function of time whenoperating at a constant flow rate was also derived. A secondmodel that accounts for the combined effects of fiber coatingand cake formation was also developed.

The models were tested by filtering different dilutions of anE. coli lysate through filters composed of glass fibers at con-stant pressure. The fiber coating model provided good fits ofdata from constant pressure experiments run with E. coli lysateon glass filters. The fiber coating model made accurate filtercapacity predictions using limited pressure vs. time data forconstant flow fouling of glass filters by the replica cell culturefluid and commercially available depth filtration media by E.coli lysate. The cake-fiber coating model provided improved

Figure 5. Pressure vs. time data compared to the fibercoating and cake-fiber coating model predic-tions with 4� diluted replica cell culture fluidrun at 2600 L m�2 h�1 through AP15 glass-fiber media.

Figure 6. Pressure vs. time data compared the fibercoating and cake-fiber coating model predic-tions with 10� diluted replica cell culture fluidrun at 2600 L m�2 h�1 through AP15 glass-fiber media.

Figure 7. Pressure vs. time data compared to fiber coat-ing model predictions for experiment run at364 L m�2 h�1 with 10� diluted E. coli lysate onMillistak A1HC depth filter media.

Figure 8. Accuracy of capacity predictions from the fi-ber coating model.Predicted final volume normalized by actual final volume as afunction of percentage of pressure range used to determineparameter values. Data for the 4 and 10� diluted replica cellculture fluid run at 2600 L m�2 h�1 through AP15 glass-fibermedia, and 10� diluted E. coli lysate on Millistak A1HCdepth filter media run at 364 L m�2 h�1.


volume vs. time fits and improved sizing predictions. Theimprovement was greatest for concentrated streams where it ispossible cakes formed above the filters. The fiber coatingmodel and cake-fiber coating models will be useful tools for thesizing of depth filtration media used in biotechnology andpotentially for other types of fibrous filters used with bothliquids and gases.

AcknowledgmentsThe authors thank Dave Yavorsky, Carl Lawton, Ven Raman, Oleg

Shinkazh, John Royce, and Matthew Tomasko for providing the filters, thereplica cell culture fluid, and the E. coli lysate, and Robin de la Parra fortaking the SEM images.

Notation

A � total filter area, m2

c � volume of solids per unit of filtrate volumeKc � cake filtration constant, s/m2

Kf � fiber coating constant, 1/mLF � total fiber length, mLM � filter thickness, m

P � pressure, kg/ms2

J � solvent flux, m/srf � fiber radius, mt � time, s

V � volume filtered, m3/m2

Vmax � maximum volume filtered, m3/m2

Greek letters

� dimensionless placeholder in Eq. 10� � dimensionless placeholder in Eq. 12 � dimensionless placeholder in Eq. 12� � dimensionless placeholder in Eq. 12� � porosity of deposited solids� � filter solid or fiber volume fraction� � Darcy permeability, m2

� � solution viscosity, kg/ms

Literature Cited1. Grace HP. Structure and performance of filter media. AIChE J. 1956;

2:307-336.2. Hermia J. Constant pressure blocking filtration laws—Application to

power-law non-Newtonian fluids. Trans IChemE. 1982;60:183-187.3. Hermans PH, Bredee HL. Principles of the mathematical treatment of

constant-pressure filtration. J Soc Chem Ind. 1936;55T:1-4.4. Badmington F, Wilkins R, Payne M, Honig ES. Vmax testing for

practical microfiltration train scale-up in biopharmaceutical process-ing. Biopharm. 1995;Sep.:46-52.

5. Tien C. Granular Filtration of Aerosols and Hydrosols. Stoneham,MA: Butterworth; 1989.

6. Carman PC. Fluid flow through granular beds. Trans IChemE. 1937;15:150-166.

7. Sullivan RR, Hertel KL. The flow of air through porous media. J ApplPhys. 1940;11:761-765.

8. Nickalls RWD. A new approach to solving the cubic: Cardan’s solu-tion revealed. Math Gazette. 1993;77:354-359.

9. Bolton GR, LaCasse D, Kuriyel R. Combined models of membranefouling: Development and application to microporous and ultraporousmembranes. J Membr Sci. 2005; in press.

10. Ogston AG. The spaces in a uniform random suspension of fibers.Trans Faraday Soc. 1958;54:1754-1757.

11. Bolton GR, Deen WM. Limitations in the application of fiber-matrixmodels to glomerular basement membrane. In: Layton HE, WeinsteinAM, eds. Membrane Transport and Renal Physiology (IMA Series:Mathematics and Its Application). New York, NY: Springer-Verlag;2001:141-156.

12. Jackson GW, James DF. The permeability of fibrous porous media.Can J Chem Eng. 1986;64:364-374.

13. Higdon JJL, Ford GD. Permeability of three-dimensional models offibrous porous media. J Fluid Mech. 1996;308:341-361.

14. Clague DS, Kandhai BD, Zhang R, Sloot PMA. Hydraulic permeabil-ity of (un)bounded fibrous media using the lattice Boltzmann method.Phys Rev E. 2000;61:616-625.

Appendix A

There are a number of theoretical models that relate filterpermeability to fiber radius and fiber volume fraction.11 Jack-son and James12 modeled the hydraulic permeability of a ran-dom fibrous filter using a cubic array of cylindrical rods. Theresistance to flow was the sum of that arising from the rodsaligned parallel and perpendicular to the direction of flow.Permeability data were reviewed for materials with a widerange of fiber radii and fiber volume fractions. The model fitthe data well for fiber volume fractions 0.2.

Higdon and Ford13 obtained numerical solutions to flowaround three-dimensional fibrous media using the spectralboundary element method. They obtained results for a widerange of filter solid fractions using slender-body theory at lowfiber concentrations and the lubrication approximation at highfiber concentrations. Their results agreed with the data re-viewed in Jackson and James12 for fiber volume fractionsbetween 0.0309 and 0.865.

Clague et al.14 used a Lattice–Boltzman simulation methodto make numerical calculations of the Darcy permeability of adisordered, three-dimensional array of cylinders. Their results(presented in their Figure 8) also fit the data in Jackson andJames12 and agreed with the face-centered cubic fiber latticepredictions of Higdon and Ford13 for fiber volume fractionsbetween 0.05 and 0.7.

The Carman–Kozeny model,

�

rf2 �

�1 � ��3

20�2 (A1)

Figure 9. Accuracy of capacity predictions from thecake-fiber coating model.Predicted final volume normalized by actual final volume as afunction of percentage of pressure range used to determineparameter values. Data for the 4 and 10� diluted replica cellculture fluid run at 2600 L m�2 h�1 through AP15 glass fibermedia.


where � is filter Darcy permeability, was developed by extend-ing a correlation between Reynold’s number and friction factorfor flow through pipes to flow through porous media.5 It hasalso been used to describe flow through fibrous media.6 Theflow through a filter can be calculated from Darcy’s Law:

dV

dt� �

�P

�LM(A2)

In this study the Carman–Kozeny model predictions werecompared to the numerical results of Higdon and Ford12 andClague et al.13 The Carman–Kozeny equation provided goodagreement with the results from the other two studies. Acomparison of the models is shown in Figure A1.

The predictions of the Carman–Kozeny were within 50% ofthe Higdon and Ford predictions for fiber volume fractionsbetween 0.1 and 0.8. This range is typical of the majority offibrous media. For low fiber volume fractions, between 0.05and 0.1, the Carman–Kozeny predictions were higher by lessthan a factor of two.

In Figure A2 the predictions of the Carman–Kozeny modelare compared to the experimental data summarized by Jacksonand James.11 The Carman–Kozeny model fit the majority of theexperimental data. The predictions were within the scatter ofthe summarized data for volume fractions above about 0.05. Atvolume fractions 0.05 the model overpredicted the data. Thisindicates the Carman–Kozeny model can be used as a simplemathematical flux equation that accurately represents both rig-orous theoretical predictions and a large amount of experimen-tal data.

Appendix B

Equations 10 and 13 were derived assuming that fibers neveroverlapped and all solids deposited evenly along the fibers. It ispossible, however, to have filters with fibers that intersect orhave a significant portion of their surface area in contact with

one another. In this case the fiber radius will increase withpermeate volume faster than as described by Eq. 6 because thesolids will deposit on fiber surfaces only where there is nocontact or overlap with another fiber.

The following derivation corrects the fiber coating model forthe effects of fiber overlap both initially and as the fibers growuniformly in radius. The filter is assumed to be composed offibers that are randomly distributed with respect to one another.One consequence of the assumption of randomness is thatfibers in this hypothetical matrix will overlap with one another.Thus, the true solid volume fraction of the filter cannot becalculated directly using the number concentration of fibers andthe volume of an individual fiber. For completely nonintersect-ing fibers, the solid volume fraction varies as rf

2 and � is relatedto �0 as

� � �0� rf

rf 0� 2

(B1)

For any random-fiber matrix with a nominal value of �, thetrue solid volume fraction �* can be calculated by consideringthe partitioning of a point-sized particle in that matrix. As asolute becomes infinitely small, the fraction of the filter volumefrom which it is excluded is equal to the true solid volumefraction of the filter. The partition coefficient () for such asolute is given by

� 1 � �* (B2)

To relate �* to �, we consider now the classical result ofOgston10 for the partitioning of rigid spherical solutes in ran-dom-fiber matrices, wherein no specific accounting for fiberintersection is made. The Ogston result relates to the soluteradius rs, fiber radius rf, and nominal fiber solid volume frac-tion �, as

� exp��1 �rs

rf�2� (B3)

Figure A2. Hydraulic permeability as a function of fibervolume fraction.The permeabilities calculated with the Carman–Kozenymodel are compared to the experimental data summarized inthe Jackson and James review.12

Figure A1. Hydraulic permeability as a function of fibervolume fraction.The permeabilities calculated with the Carman–Kozenymodel are compared to those calculated with the Jacksonand James model12 and with the numerical calculations ofHigdon and Ford13 and Clague et al.14


As rs 3 0, the Ogston result reduces to exp(��). EquatingEqs. B2 and B3,

1 � �* � exp�� (B4)

In the fibrous filter the true solid volume fraction will increaseas solids are deposited according to

V� c

1 � �� LM��* � �*0� (B5)

The maximum volume that can be filtered or Vmax will bedescribed by

Vmax � LM�1 � �

c ��1 � �*0� (B6)

Inserting Eq. B6 into B5 yields

�* � �1 � �*0�V

Vmax� �*0 (B7)

The fiber radius will increase faster with a given change in truesolid volume fraction than an equivalent change in nominalsolid volume fraction. The fiber radius can be related to the truesolid volume fraction by inserting Eq. B4 into Eq. 4.

�

�0�

ln�1 � �*�

ln�1 � �*0�� rf

rf 0�2

(B8)

An equation for flux as a function of true solid volumefraction can be obtained by inserting Eq. B8 for fiber radiusinto the Carman–Kozeny equation:

dV

dt� � �P

�LM� rf

2�1 � �*�3

20�*2 � J0

ln�1 � �*�

ln�1 � �*0��*0

�*�2�1 � �*

1 � �*0�3

(B9)

An equation for flux as a function of volume can be obtainedby inserting the equation relating true solid volume fraction topermeate volume, Eq. B7, into Eq. B9. This can be solvednumerically for volume vs. time at different values of theparameter Vmax.

The fibers will be thicker for a given change in solid volumefraction when accounting for the fiber overlap. Solids willdeposit more on the ends of fibers where there is no overlapthan in centers where the surface area is unavailable. The fluxincreases with the square of fiber radius in Eq. B9. For a singlesolid fraction value, the flux will be higher when the fiberradius is larger. It will thus require a larger amount of solids tocause a similar flux decline when accounting for fiber overlap.

An equation similar to B9 can be derived for constant flowconditions, as follows:

�P

�P0�

ln�1 � �*0�

ln�1 � �*� ��*

�*0�2�1 � �*0

1 � �*�3

(B10)

An explicit equation for calculation of pressure as a function oftime or volume can be obtained by inserting the equationrelating the true fiber volume fraction to permeate volume, Eq.B7, into Eq. B10.

The effect of the overlap on the predicted filter capacities inFigure 8 was determined. The predictions made using Eq. B10were within 1% of the fiber coating model for the 10 � dilutedreplica cell culture fluid run on glass filters and the E. colilysate run on Millistak. The overlap model predictions were upto 4% better for the 4 � diluted replica cell culture fluid run onglass filters. The effect of overlap is not expected to signifi-cantly alter the capacity predictions. In cases where it is im-portant, Eq. B10 can be used to calculate filter capacitiesnumerically.

Manuscript received Sep. 3, 2004, and revision received Mar. 7, 2005.


Three-Bed PVSA Process for High-Purity O2Generation from Ambient AirJeong-Geun Jee, Sang-Jin Lee, Min-Bae Kim, and Chang-Ha Lee

Dept. of Chemical Engineering, Yonsei University, Shinchon-dong, Seodaemun-gu, Seoul, 120-749, Korea


A three-bed PVSA (pressure vacuum swing adsorption) process, combining equi-librium separation with kinetic separation, was developed to overcome the 94% O2

purity restriction inherent to air separation in the adsorption process. To produce97�% and/or 99�% purity O2 directly from air, the PVSA process with two zeolite10X beds and one CMS bed was executed at 33.44 – 45.60 to 253.31 kPa. In addition,the effluent gas from the CMS bed to be used for O2 purification was backfilled to thezeolite 10X bed to improve its purity, recovery, and productivity in bulk separation ofthe air. PVSA I, which made use of a single blowdown/backfill step, produced an O2

product with a purity of 95.4 –97.4% and a recovery of 43.4 – 84.8%, whereas PVSAII, which used two consecutive blowdown/backfill steps, produced O2 with a purity of98.2–99.2% and a recovery of 47.2– 63.6%. Because the primary impurity in the O2

product was Ar, the amounts of N2 contained in the product were in the range of4000 –5000 ppm at PVSA I and several tens of ppm at PVSA II. A nonisothermaldynamic model incorporating mass, energy, and momentum balances was applied topredict the process dynamics. Using the linear driving force (LDF) model withconstant diffusivity for the equilibrium separation bed and a modified LDF model withconcentration dependency of the diffusion rate for the kinetic separation bed, thedynamic model was able to accurately predict the results of the experiment. © 2005American Institute of Chemical Engineers AIChE J, 51: 2988 –2999, 2005Keywords: high-purity O2, PVSA, backfill step, blowdown step, CMS, zeolite 10X

Introduction

In the study of air separation, classical cryogenic distillationhas been widely recognized as one of the most common meth-ods of O2 production. However, this technology is not com-patible with relatively small size [�100 TPDc (tons per daycontained)] O2 plants.1 Recently, the development of highlyselective adsorbents such as LiX, LiAgX, and LiCaX hascontributed to the possibility of manufacturing up to 200 TPDcby vacuum swing adsorption (VSA) process, at a power con-sumption rate competitive with that of cryogenic distillation.2

During the last 30 years, commercial applications for ad-

sorptive O2 generation from ambient air using several molec-ular sieve zeolites have been expanded.3-7 However, becauseair contains a small amount of Ar, which has physical proper-ties similar to those of O2, the product generated from thezeolite bed typically contains a substantial amount of the Arimpurity. Therefore, in the equilibrium separation process us-ing zeolites, O2 purity is limited to about 94% purity. For thisreason, O2 produced by adsorption technology typically has90–93% purity, of which 4–5% is Ar and 2–6% N2, and isgenerally used in many chemical processes such as the biolog-ical treatment of wastewater, steel industries, paper and pulpindustries, and glass-melting furnaces.8,9

In addition to these industries, there is a great demand for O2

with a purity level of 99% or higher in other applications, suchas welding and cutting processes, plasma chemistry, ozonegeneration, medicine, combustion, cylinder filling, and O2 for

Correspondence concerning this article should be addressed to C.-H. Lee [email protected].



breathing at high altitudes in pressurized space suits or invarious laboratory applications.10

Recently, because of the reduction and sequestration of CO2,O2 inhalation combustion processes have been developed ex-tensively for use in, among others, the iron, steel, and inciner-ation industries.11 In these industries, O2 needs to be at least95% pure to be used cost-effectively in fuel combustion be-cause of the carbon tax placed on CO2 capturing. Moreover, therecommended amount of N2 impurity contained in an O2 prod-uct for controlling the amount of NOx released under theallowable exhaust standard is �1%.

As a result of extensive industrial demand for high-purityO2, several adsorption processes have been developed, thepatents for which involve a two-stage process consisting ofzeolite beds for nitrogen removal and carbon molecularsieve (CMS) beds for the removal of argon and nitrogenimpurities.12-15 Knaebel and Kandynin14 posited that an O2

product with 99.6% purity and 25% recovery could beobtained from a five-step, four-bed PSA using two consec-utive blowdown steps. They also noted that a desirable levelof productivity would be about 11 Nm3 m�3 h�1. Hayashi etal.16 observed that an O2 recovery ratio of up to 50% wasobtained during the production of O2 with a purity of 99%through a connection between the zeolite Ca-X bed and theCMS bed, each of which consisted of three beds. Recently,Jee et al.17 reported that a two-bed PSA process with CMScould purify 95% O2 to 99.8% purity with 78% recovery.However, these processes mentioned need to overcome theadditional equipment costs to set up the purification unit andthe significant product loss for applications as a O2 balanceunit with a cryogenic unit and a direct on-site unit.

In this study, a three-bed pressure vacuum swing adsorption(PVSA) process, consisting of two zeolite 10X beds and oneCMS bed, was developed to produce high-purity O2 with �1%N2 impurity from ambient air. The first two zeolite 10X bedsfunctioned to achieve bulk separation of the air. Thereafter, theCMS bed purified the O2 supplied from the zeolite 10X beds.To increase the purity and productivity of the PVSA process,we proposed refluxing the waste from the CMS bed into thezeolite 10X beds by using vacuum pressure in the zeolite bed.As a result, the PVSA process could be operated with oneblower/compressor and one vacuum pump.

A nonisothermal dynamic model incorporating mass, en-ergy, and momentum balances was used to predict dynamicbehavior in the PVSA process, combining equilibrium separa-tion with kinetic separation. The linear driving force (LDF)model with constant diffusivity for the equilibrium separationbed and the concentration-dependent rate model for the kineticseparation bed were used concurrently to the dynamic model.

Description of the PVSA Process

Two different three-bed PVSA cycles were proposed toproduce 97� and 99�% O2 generation from ambient air. Thecyclic sequences of the proposed PVSA processes are repre-sented in Table 1.18

In the case of the PVSA I process, the two zeolite bedswere operated as a PVSA sequence containing pressureequalization and single backfill steps from the CMS bed.Meanwhile, the CMS bed was operated as a PSA sequencewith one blowdown step. As noted in Table 1, when the firstzeolite bed produced highly concentrated oxygen containingimpurities such as Ar and N2, the CMS bed was pressurizedby the product from the zeolite 10X bed until it reached theadsorption pressure of the bed and this pressure was main-tained throughout the adsorption step of the process. Duringthe adsorption step, Ar and N2 impurities were removed bythe kinetic selectivity of CMS, although the concentration ofO2 effluent from the CMS bed was much higher than that ofthe ambient air. The effluents from the adsorption step of theCMS bed were supplied to the second zeolite bed andregenerated by vacuum pressure, as a backfill step to par-tially pressurize the zeolite bed with the concentrated oxy-gen. This step prevented any of the gas supplied by thezeolite bed from being wasted in the CMS bed. During thepressure equalization step in the zeolite bed, the CMS bedunderwent the idle step for the cyclic symmetry of theprocess. The idle step provided the contact time for thediffusion of O2.17 Then, during the pressurization step of thesecond zeolite bed, the CMS bed produced purified oxygenat the blowdown step.

In the case of the PVSA II process, a two-stage blowdownstep of the CMS bed, followed by a fractionation of its oxygenproduct, was implemented to increase O2 purity to 99%. Be-

Table 1. Cyclic Sequences of Three-Bed PVSA Process for Air Separation and O2 Purification*

PVSA I

Step [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ]

1st zeolite bed [ PR 1 ] [ AD 1 ] [ DPE 1 ] [ VU 2 ] [ BF 2 ] [ PPE 2 ]2nd zeolite bed [ VU 2 ] [ BF 2 ] [ PPE 2 ] [ PR 1 ] [ AD 1 ] [ DPE 1 ]CMS bed [ BD 2 ] [ PR 1 ] [ AD 1 ] [ ID ] [ BD 2 ] [ PR 1 ] [ AD 1 ] [ ID ]

product (O2 supply from 1st zeolitebed)

product (O2 supply from2nd zeolite bed)

PVSA II

Step [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ]

1st zeolite bed [ PR 1 ] [ AD 1 ] [ DPE 1 ] [ VU 2 ] [ BF1 2 ] [ PPE 2 ] [ BF2 1 ]2nd zeolite bed [ VU 2 ] [ BF1 2 ] [ PPE 2 ] [ BF2 1 ] [ PR 1 ] [ AD 1 ] [ DPE 1 ] [ VU 2 ]CMS bed [ BD2 2 ] [ PR 1 ] [ AD 1 ] [ ID ] [ BD1 2 ] [ BD2 2 ] [ PR 1 ] [ AD 1 ] [ ID ] [ BD1 2 ]

product (O2 supply from1st zeolite bed)

product (O2 supply from2nd zeolite bed)

*1, Cocurrent flow;2, countercurrent flow; PR, pressurization step; AD, adsorption step; DPE, depressurizing pressure equalization step; VU, vacuum step; BF1,first backfill step; BF2, second backfill step; PPE, pressurizing pressure equalization step; ID, idle step; BD1, first blowdown step; BD2, second blowdown step.


cause Ar and N2 impurities remain in the gas phase and notsolid phase, most of them can be removed at the beginning ofthe blowdown step, but the O2 concentration of the effluent ismuch higher than that of ambient air. Therefore, in the PVSAII process, one more backfill step by the effluent from the firstblowdown step of the CMS bed was added to the zeolite bedcompared to the PVSA I process.

Mathematical Model

To understand the dynamic behaviors of a PVSA process,mathematical models were developed on the basis of the fol-lowing assumptions: (1) the gas phase behaves as an ideal gasmixture, (2) radial concentration and temperature gradients arenegligible, (3) thermal equilibrium between adsorbents andbulk flow is assumed, (4) the flow pattern is described by theaxially dispersed plug-flow model, and (5) the pressure dropalong the bed is measured using Ergun’s equation. In numerousstudies the significance of the radial gradient was neglected andsimilar assumptions in simulating the adsorption processeswere also made.19,20

The component and overall mass balances for the bulk phasein the adsorption column are given by

�DL

�2ci

�z2 ��uci�

�z�

�ci

�t� �p�1 � �

� � �qi

�t� 0 (1)

��uC�

� z�

�C

�t� �p�1 � �

� � �i�1

n�qi

�t� 0 (2)

Another characteristic of the adsorption process is the tem-perature variation caused by the heat of adsorption and desorp-tion. To minimize the deviation in the prediction of high-purityproduct, the energy balance for the gas phase also includes theheat transfer to the column wall, expressed as

�KL

�2T

�z2 � ��gCpg

��uT�

�z� ��t�gCpg � �BCps�

�T

�t

� �B �i�1

n

��Hi��qi

�t�

2hi

RBi�T � Tw� � 0 (3)

where �t is the total void fraction [�� (1 � �) � �], � is theparticle porosity, and �B is the bed density [�(1 � �) � �p].

To consider the heat loss through a wall and the heat accu-mulation in the wall, another energy balance for the wall of theadsorption bed was used

�wCpwAw

�Tw

�t� 2�RBihi�T � Tw� � 2�RBoho�Tw � Tatm� (4)

where Aw � �(RBo2 � RBi

2 ).In this study, the proposed PVSA process was operated by a

combination of two cyclic sequences (Table 1). Therefore, animbalance in the pseudo-cyclic operation among these bedswas inevitable, although the effluents remained systematicallyconnected to each other. In the simulation using gProms soft-

ware, shown in Figure 1a, after retaining the temporal datafrom the effluent stream of the zeolite 10X bed during theadsorption step, these data were used in the pressurization andadsorption steps of the CMS bed. The temporal effluent datafrom the end of the CMS bed during its adsorption step wasused in the first backfill step of the zeolite 10X bed. Thetemporal data from the CMS bed during the blowdown stepwere applied to the second backfill step of the zeolite 10X bed(Figure 1b). The two sets of boundary conditions were appliedto the zeolite bed and CMS bed separately.

The boundary conditions for zeolite two-bed PVSA are asfollows:

Boundary Conditions for Feed Pressurization (PR) and Ad-sorption (AD) Steps

�DL��ci

�z��z�0

� u�ci�z�0� � ci�z�0�� ci

�z��z�L

� 0 (5a)

�KL��T

�z��z�0

� ��gCpgu�T�z�0� � T�z�0�� T

�z��z�L

� 0

(5b)

Boundary Conditions for First Backfill (BF1), Second Back-fill (BF2), and Pressurizing Pressure Equalization (PPE) Step

�DL��ci

�z��z�L

� u�ci�z�L� � ci�z�L�� ci

�z��z�0

� 0 (6a)

�KL��T

�z��z�L

� ��gCpgu�T�z�L� � T�z�L�� T

�z��z�0

� 0

(6b)

Figure 1. Simulation scheme of three-bed PVSA processwith two zeolite beds and one CMS bed.(a) BF1 and (b) BF2.


Boundary Conditions for Depressurizing Pressure Equal-ization (DPE) and Countercurrent Vacuum Regeneration(VU) Steps

��ci

� z��z�0

� ��T

� z��z�0

� 0 ��ci

� z��z�L

� ��T

� z��z�L

� 0

(7)

The boundary conditions for a CMS one-bed PSA are asfollows:

Boundary Conditions for PR and AD steps

�DL��ci

�z��z�0

� u�ci�z�0� � ci�z�0�� ci

�z��z�L

� 0 (8a)

�KL��T

�z��z�0

� ��gCpgu�T�z�0� � T�z�0�� T

�z��z�L

� 0

(8b)

Boundary Conditions for Idle (ID), First Blowdown (BD1),and Second Blowdown (BD2) Steps

��ci

� z��z�0

� ��T

� z��z�0

� 0 ��ci

� z��z�L

� ��T

� z��z�L

� 0

(9)

Initial Condition for Fluid Flow

ci� z, 0� � c*i qi� z, 0� � q*i (10)

In this study, the adsorption beds were saturated with pureoxygen before each run.

Initial Condition for Heat Flow

T� z, 0� � Tatm (11)

The pressure history during a PSA experiment at the feed orbed end was fitted by polynomials and used as a boundarycondition for the overall mass balance.21

To consider the pressure drop effect across the bed, Ergun’sequation was introduced as a momentum balance22

�dP

dz� au � b�u�u� (12a)

a �150

4Rp2

�1 � ��2

�2 b � 1.75�1 � ��

2 Rp�(12b)

where u is the interstitial velocity.The multicomponent adsorption equilibrium was predicted

by the following:Loading Ratio Correlation (LRC) Model

qi �qmiBiPi

ni

1 � ¥j�1n BjPj

nj(13)

where qmi � k1 � k2 � T, Bi � k3exp(k4/T), and ni � k5 � k6/T.In this study, the general LDF model assuming constant

diffusivity was applied to the sorption rate of equilibriumseparation bed (zeolite 10X)

�qi

�t� i�q*i � qi� i �

15Dei

rp2 (14)

In the case of the kinetic separation bed (CMS), the modifiedLDF model with concentration-dependent diffusivity wasused.23,24

�qi

�t� i�q*i � qi� i �

15Dei

rp2

Dei � Ci � Pri0.5�1 � BiPi�

2 (15)

Table 2. Adsorption Isotherm and Diffusion Rate Parameters of N2, O2, and Ar on Zeolite 10X and CMS

Constants

Zeolite 10X CMS

O2 Ar N2 O2 Ar N2

Equilibrium Parameters

For LRC modelk1 (mol/g) 4.568 � 10�3 3.697 2.330 � 10�3 1.527 � 10�2 2.042 � 10�2 2.363 � 10�2

k2 (mol g�1 K�1) �1.071 � 10�5 �5.010 � 10�6 �4.400 � 10�6 �3.230 � 10�5 �5.300 � 10�5 �6.380 � 10�5

k3 (1/kPa) 9.866 � 10�5 1.651 � 10�5 1.674 � 10�8 2.290 � 10�5 23.97 � 10�5 36.10 � 10�5

k4 (K) 578.9 869.7 3586 966.1 324.6 1444k5 3.552 3.184 1.384 1.187 1.646 1.692k6 (K) �797.4 �665.6 �142.1 �106.0 �238.2 �270.0

Heat of adsorption,��H� (kJ/mol) 9.623 10.46 16.32 13.81 14.23 13.39

Adsorption Rate Parameters

For rate modelDci (cm2/s) in Eq. 14 0.0153 0.0147 0.0043Ci (1/s) in Eq. 15 0.024 0.000047 0.000090


Values of the adsorption isotherm and rate parameters of N2,Ar, and O2 on CMS and zeolite 10X, presented in Table 2, werevery similar to the published data.25-27

Experimental

Zeolite 10X (Baylith WE-G 639; Bayer AG, Leverkusen,Germany) and CMS (3A; Takeda Chemicals, Tokyo, Japan)were used as adsorbents in the PVSA experiments. Character-istics of each adsorbent are listed in detail in Table 3. Beforethe experimental runs, the zeolite 10X was regenerated at 613K overnight and the CMS at 423 K.

The zeolite 10X adsorption beds were made of stainless steelaccording to the following dimensions: length, 100 cm; ID,3.44 cm; and wall thickness, 2.67 mm. The dimensions of theCMS bed were: length, 100 cm; ID, 2.2 cm; and wall thickness,1.75 mm. Characteristics of the adsorption beds are listed indetail in Table 3.

The PVSA apparatus is shown in Figure 2. Three resis-tance temperature detectors (RTD, Pt 100 ) were installed10, 50, and 80 cm from the feed end to measure temperaturevariations inside the bed. The flow rate was regulated by amass flow controller and the total amount of feed flow wasmeasured by a wet gas meter (Sinagawa Co., Tokyo, Japan).To keep the pressure of the adsorption bed constant duringthe adsorption step of the zeolite 10X bed, a back-pressureregulator was installed between the zeolite 10X bed and theCMS bed. An additional back-pressure regulator was alsoequipped at the end of the CMS bed to keep the pressure ofthe adsorption bed constant during the adsorption step of theCMS bed. Bed pressure was measured by pressure transduc-ers equipped at the top and bottom of each bed.

The concentration of influent and effluent was analyzed by amass spectrometer (QMG 422; Balzers Instruments, Lichten-stein). This analysis was confirmed by gas chromatography(HP 5890II; Hewlett–Packard, Palo Alto, CA). The system wasfully automated by a personal computer running a controlprogram, and all measurements including flow rate, pressure,temperature, and concentration were saved on the computerthrough the use of an AD converter.

A high-performance vacuum pump (DAH-60; Ulvac KikoInc., Kanagawa, Japan) was applied to bring the vacuum stepof the zeolite 10X bed up to a level of 33.44–45.60 kPa,depending on the applied step time.

Each activated adsorbent bed was filled with pure O2

(99.9�%) to prevent contamination from the outside air. Be-fore running each experiment, the adsorption beds were vacu-umed for 2 h. As an initial condition, the PVSA experimentswere conducted at the bed saturated with pure O2 (99.9�%)under the same level of adsorption pressure as used throughoutthe experiment. The temperatures of the feed, bed, and sur-roundings were kept in the range of 297 to 300 K during theexperiments.

The ternary mixture (N2/O2/Ar; 78:21:1 vol %; DaeSungIndustrial Gas Co., Seoul, Korea) was used as feed gas for thePVSA experiments. Specific operating conditions such as op-erating pressure, feed flow rate, and each cycle/step times canbe found in Table 4.

Results and DiscussionCyclic performance of PVSA I with a single blowdown/backfill step

Because the effluent from the zeolite 10X bed during the ADstep at 253.31 kPa was supplied as feed gas for the PR and ADsteps of the CMS bed, the CMS bed was pressurized to 162.12kPa. The effluent from the AD step of the CMS bed wasbackfilled into the other zeolite 10X bed. Therefore, the pres-sure of the zeolite bed increased slightly after the VU step. It isnoteworthy that the CMS bed underwent two cycles for everyone cycle of the zeolite bed because each of the zeolite bedsand the CMS bed was tied to the same cycle time. To preventconfusion of the number of cycles, the cycle number of the

Figure 2. Apparatus for a three-bed PVSA process.

Table 3. Characteristics of Adsorbent and Adsorption Bed

Characteristic Zeolite 10X CMS

Adsorbent

Type Pellet PelletAverage pellet size, Rp (cm) 0.115 0.28Heat capacity, Cps (J g�1 K�1) 1.13 0.96Pellet density, �p (g/cm3) 1.1 0.90Particle porosity, � 0.36 0.30Bed density, �B (g/cm3) 0.82 0.63

Adsorption Bed

Length, L (cm) 100 100Inside radius, RBi (cm) 1.72 1.1Outside radius, RBo (cm) 1.987 1.275Heat capacity of column, Cpw

(J g�1 K�1) 0.50 0.50Density of column, �w (g/cm3) 7.83 7.83Internal heat transfer coefficient,

hi (J cm�2 K�1 s�1) 3.85 � 10�3 3.85 � 10�3

External heat transfer coefficient, ho

(J cm�2 K�1 s�1) 1.42 � 10�3 1.42 � 10�3


PVSA process was determined by the cycle of the zeolite 10Xbed.

The temperature excursion in the zeolite 10X bed wasaround 5 K, whereas in the CMS bed it was 1 K. Because theexperiment was performed under O2 saturated conditions, thetemperature history of the CMS bed showed an increase duringthe initial cycles. The temperature cyclic steady state for theprocess was reached at 15 cycles.

Figure 3 shows the cyclic variation of the mean concentra-tion produced by the zeolite 10X and CMS beds at Run 1(Table 4). The concentration variation curves also reached acyclic steady state after nearly 15 cycles. In previous studies,the two-bed O2 PSA purifier packed with CMS reached thecyclic steady state after only three cycles17 when the binaryfeed (O2/Ar; 95:5 vol %) was used. However, in this study, thecyclic steady state of the CMS bed was delayed because of thecyclic steady state of the zeolite 10X bed.

As noted in Figure 3, the zeolite 10X bed, as a bulk airseparator, produced 93.5% O2 with 2.5% N2 and 4% Ar im-purities. The product purity was higher than that of conven-tional O2 VSA at similar operating conditions because thezeolite bed was partially pressurized by the BF step with theconcentrated O2 from the AD step of the CMS bed. Becausesuch product quality was supplied to the CMS bed, the purityof O2 purified through the CMS bed was about 96.8% at thecyclic steady state (Figure 3). The amount of N2 impuritysignificantly decreased to the level of 4000–5000 ppm (�0.4–0.5 vol %) and Ar impurity also decreased from 4 to 2.7%. Theresults simulated by the dynamic model corresponded with theresults of the experiment in both the equilibrium and kineticseparation beds.

The simulated concentration profiles of each adsorbate at thegas phase of the zeolite 10X bed are presented in Figures 4a to4c. At the end of the AD step (Figure 4a), the mass-transferzone (MTZ) of each adsorbate became steeper as that of eachgas propagated to the bed end. In addition, the concentration ofAr at the bed end became slightly higher than that of N2. At theend of the BF step after the DPE and VU steps (Figure 4b), theMTZ of N2 was observed retreating to the feed end with abroader shape than it had at the VU step and O2 and Araccumulated in higher concentrations at the bed end. Finally, atthe end of the PPE step (Figure 4c), each MTZ retreated furtherto the feed end bearing the concave shape of N2 MTZ and theconvex shape of O2 and Ar MTZs.

Figures 4d and 4e show the axial concentration profilesalong the CMS bed at the end of each step. As noted inFigure 4d, the O2 concentration profile at the end of the PRstep was formed at the concave shape around the bed endbecause the Ar and N2 in the gas phase were concentrated atthat location of the bed. Moreover, because a higher con-centration of Ar than N2 was supplied to the CMS bed andthe diffusion rate of Ar was much slower than that of N2, theconcentration of Ar at this step was higher than that of N2.At the end of the AD step (Figure 4e), the gas-phase con-centration of each adsorbate was nearly constant at each ofthe axial positions because the adsorbed phase was nearlysaturated with highly concentrated O2. This result meansthat a tangible amount of the impurities in the gas phase wasremoved during the AD step time. In addition, because the

Table 4. Operating Conditions for PVSA for Air Separation and O2 Purification

Run No. Process

AdsorptionPressure of

Zeolite10X Bed

(kPa)

AdsorptionPressure ofCMS Bed

(kPa)

Vacuum Pressureof Zeolite 10X

Bed (kPa)Feed Rate

(LSTP/min)

PVSA I* (BD-PR-AD-ID-BD-PR-AD-ID) PVSA II* (BD2-

PR-AD-ID-BD1-BD2-PR-AD-ID-BD1)

1 PVSA I 253.31 162.12 38.50 2.0 40-20-20-20-40-20-20-20 (Basestep time)

2 174.28 33.44 40-30-10-20-40-30-10-20

3 131.72 45.60 40-10-30-20-40-10-30-20

4 PVSA II 167.19 39.52 30-20-20-20-10-30-20-20-20-10(Base step time)

5 177.32 33.44 30-30-10-20-10-30-30-10-20-10

6 136.79 44.58 30-10-30-20-10-30-10-30-20-10

*The step sequence and time for the CMS bed were presented.

Figure 3. Cyclic variation of O2 purity and Ar/N2 impuri-ties produced from the BD step of CMS bedduring the cyclic operation of PVSA I at baseoperating condition (Run 1).


concentration of O2 during the AD step in the CMS bed wasrelatively high, the effluent BF gas could improve the per-formance of the zeolite bed (Figure 4c). At the end of the IDstep, the MTZs of N2 and Ar at the bed end increasedslightly because the small amount of O2 diffused into thesolid phase. Then, high-purity O2 was produced at the end ofthe BD step (Figure 4f). As shown in Figure 3, a greateramount of Ar than N2 was included in the product becausethe Ar with the slower diffusion rate remained in the gasphase.23,28 In addition, the MTZs of Ar and N2 crossed overat the vicinity of the bed end in the BD step because thedesorption of N2 from the adsorbed phase was quicker thanthat of Ar.

Effect of adsorption step time in the CMS bed onadsorption dynamics in PVSA I

As shown in Figure 4, the AD step in the CMS bed playeda key role in removing the N2 and Ar impurities from thezeolite bed. In addition, because this step was connected to thezeolite bed as a BF step, the AD step time in the CMS bedcontributed to the purity and recovery of the product in theprocess.

Therefore, the effect of the AD step time of the CMS bed onO2 purity and recovery was studied at the fixed AD step timeof the zeolite bed, 40 s, which corresponded to the sum of thePR and AD step times of the CMS bed (Table 1). Although theAD step time of one zeolite bed was fixed to 40 s, the AD step

of the CMS bed was tied to the VU and BF steps of the otherzeolite bed. Therefore, a decrease in the AD step time of theCMS bed caused an increase in the VU step time (highervacuum pressure in Table 4) as well as an decrease in the BFstep time of the zeolite bed.

As shown in Figure 5a, as the AD step time of the CMS beddecreased from 30 to 10 s (the PR step time of the CMS bedincreased from 10 to 30 s), the purity of both the zeolite 10Xand the CMS beds decreased linearly because of the reducedamount of backfill gas in the zeolite bed and the diminishedtime for impurity removal in the CMS bed. The effect of theAD step time of the CMS bed on the purity of each bed wasslightly greater in the zeolite bed than in the CMS bed. Asshown in Figure 5b, with a decrease in the AD step time, O2

recovery substantially increased to 85% because the extendedPR step time of the CMS bed led to increased adsorptionpressure in the CMS bed. Compared to the change of recoveryin the AD step time, the effect of the AD step time of the CMSbed on purity was relatively small because the extended VUstep time in the zeolite bed could compensate somewhat for thereduced BF gas (Table 4).

Figure 6 shows the axial concentration profiles at the ADstep in the CMS bed and at the BF step in the zeolite 10X bed.In Figure 6a, the MTZs of the components did not completelyproceed to the CMS bed end within the AD step time of 10 s.Therefore, this AD step time was not sufficient to remove the

Figure 4. Simulated O2 concentration profiles in the gasphase at (a) AD, (b) BF, and (c) DPE steps ofzeolite 10X bed and (d) PR, (e) AD, and (f) BDsteps of CMS bed during the PVSA I operationat base operating condition (Run 1).

Figure 5. Effect of PR and AD step times of CMS bed on(a) O2 purity and (b) O2 recovery at base ad-sorption pressure and base feed flow rate con-dition (Runs 1 to 3).


impurities of N2 and Ar in the CMS bed. Furthermore, N2 actedas the primary impurity in the product because the concentra-tion of N2 in the bed was higher than that of Ar under that steptime. However, over 20 s, all the MTZs were linear along theCMS bed, showing a lower concentration of N2 than Ar. Theamount of N2 significantly decreased with an increase in ADstep time.

The corresponding step in the zeolite bed is shown inFigure 6b. At the BF step time of 10 s (corresponding to theAD step time of 10 s in the CMS bed), the MTZs of O2 andAr were unfavorable because a relatively large amount of N2

was supplied from the CMS bed. However, as the BF steptime increased, the MTZ of N2 retreated to the feed end andthe MTZs of O2 and Ar became favorable because the highlyconcentrated O2 feed backfilled from the AD step of theCMS bed was supplied from the bed end. Consequently, anincreased AD step time in the CMS bed led to an increase inthe purity of the O2 product through the AD step of thezeolite bed, although the recovery decreased (Figure 5)because a greater amount of the gas supplied from thezeolite bed was used in the BF step.

Cyclic performance of PVSA II with double blowdown/backfill steps

As to be expected, most of the impurities of the product fromthe CMS bed in PVSA I were recorded at the initial period of

production time (Figure 4). To improve the product purity ofO2 in PVSA II, a fractionation step of the product from theCMS bed was added to PVSA I, as shown in Table 1.

Unlike PVSA I, an additional BF step (BF2) was added inthe zeolite 10X bed and two consecutive BD steps (BD1 andBD2) were applied to the CMS bed. The effluents from the ADand BD1 steps in the CMS bed were used to partially pressurizethe zeolite 10X bed at the BF1 and BF2, respectively. Incomparison to PVSA I, PVSA II was operated at the same totalcycle time, adsorption pressure, and feed flow rate (Table 4),all based on the zeolite bed.

Figure 7 shows the representative cyclic behaviors of theCMS and zeolite 10X beds at Run 4, comparing the exper-iment’s effluent O2 purity and Ar/N2 impurities with thecalculated results. Similar to the PVSA I process, the con-centration variation reached a cyclic steady state after nearly17 cycles. However, in Run 4 of PVSA II, O2 purity fromthe CMS bed increased noticeably to 98.8%. Compared withPVSA I’s performance in Figure 3, it is noteworthy that theamount of N2 in the zeolite bed and the amount of Ar in theCMS bed were reduced significantly in PVSA II. In the finalproduct of PVSA II, the mole fraction of Ar impurity con-tained in the product decreased to 1.2%, whereas that of N2

was several tens of ppm. In addition, O2 purity from thezeolite 10X bed exceeded 96% with 3.6% Ar and 0.4% N2,recognized as the theoretical limitation of the adsorptionequilibrium process.

Figures 8a– 8c show the axial concentration profile at theend of each step along the zeolite 10X bed in PVSA II. Asshown in Figures 8a– 8c, the MTZs of all the components inPVSA II were similar to those in PVSA I (Figures 4a– 4c).However, the axial concentration amount of N2 at the bedend was less than that of Ar in the entire PVSA II process,which stands in marked contrast to PVSA I (Figures 4a– 4c).

Figure 7. Cyclic variation of O2 purity and Ar/N2 impuri-ties produced from BD2 step of CMS bed dur-ing cyclic PVSA operation for 99% O2 genera-tion at base operating condition (Run 4).

Figure 6. Axial concentration profiles along the bed inthe gas phase at the end of (a) BF step ofzeolite 10X bed and (b) AD step of CMS bed atbase adsorption pressure and base feed flowrate condition (Runs 1 to 3).


At the end of the PR and AD steps in Figure 8a, the MTZsof N2 in PVSA II propagated to the bed end more slowlythan those in PVSA I. This was mainly caused by the secondbackfill gas, which was composed of highly concentratedO2. In Figure 8b, like Figure 4c, the concave shape of axialN2 concentration at the bed end was formed at the end of thePPE step even though the bed had for the most part beenregenerated through previous regeneration steps such as VUand BF1. This concave shape had a detrimental effect on theproduction of high-purity O2 because the N2 impurity nearthe bed end was contained in the product at the AD step ofthe next cycle. However, the concave shape of N2 MTZ andthe convex shape of O2 MTZ at the bed end disappeared atthe end of the BF2 step (Figure 8c) because the high-purityO2 from the CMS bed was reintroduced at the bed end of thezeolite 10X bed.

Figures 8d–8f show the axial concentration profile at the endof each step along the CMS bed in PVSA II. Compared to theaxial concentration profiles of PVSA I in Figure 4d, the roll-ups of N2 and Ar at the PR step occurred at the bed end asshown in Figure 8d. In addition, because the amount of N2

impurity supplied from the zeolite 10X bed was noticeablysmaller (Figure 8a), the concentration profiles of N2 throughoutthe steps were much lower than those in Figure 4. Therefore, asshown in Figure 8e, the CMS bed in PVSA II was saturatedwith a higher concentration of O2 at the end of the AD step thanwas that in PVSA I. Moreover, by introducing the two-stageblowdown step, considerable amounts of Ar and N2 wereremoved from the CMS bed through the BD1 step, which was

operated under pressures ranging from adsorption pressure tomedium pressure. Simultaneously, the effluent gas at this stepwas supplied to the zeolite bed through the BF2 step (Figure8c). As a result, high-purity O2 of 99% could be producedthrough the BD2 step of the CMS bed (Figure 8f) and the bedwas kept clean for the next step.

Effect of adsorption step time in the CMS bed onadsorption dynamics in PVSA II

The effect of AD step time in the CMS bed on O2 purity andrecovery in PVSA II was studied to compare to the results withthose of PVSA I in Figure 5.

As shown in Figure 9a, when the AD step time of theCMS bed increased from 10 to 30 s, the purity of both thezeolite 10X and CMS beds increased linearly, similar to theresults noted in Figure 5a. Consequently, oxygen with99.2% purity was produced at the AD step time of 30 sbecause a greater amount of impurities had been removedfrom the CMS bed and a greater amount of O2 had beensupplied to the zeolite bed through the BF1 step. Therefore,the O2 purity of the zeolite 10X bed also increased to 97%under this AD step time. However, as shown in Figure 9b,the recovery diminished as the PR step time decreasedbecause the decreased PR step time of the CMS bed led toa lower adsorption pressure in the bed.

Compared with the results of PVSA I, the deviation of the

Figure 9. Effect of PR and AD step times of CMS bed on(a) O2 purity and (b) O2 recovery at base ad-sorption pressure and base feed flow rate con-dition (Runs 4 to 6).

Figure 8. Simulated O2 concentration profiles in the gasphase at (a) AD, (b) PPE, and (c) BF2 steps ofzeolite 10X bed and (d) PR, (e) AD, and (f) BD2steps of CMS bed during the PVSA II operationat base operating condition (Run 4).


simulated recovery in the PVSA II is slightly higher thanthat in the PVSA I. Because one more backfill step and onemore blowdown step were applied to the PVSA II, respec-tively, more errors in these steps, especially velocity, mightbe accumulated in the simulation. It is clear that PVSA IIcan produce a higher purity of O2 than PVSA I under thesame conditions in cyclic time, pressure, and feed flow rate.However, the recoveries of Runs 4 and 5 were about 10 –20% lower than the corresponding Runs 1 and 2 in PVSA Ias a result of the BD1 step in the CMS bed and the BF2 inthe zeolite bed. In the case of Run 6, it is noted that O2 with99.2% purity could be produced at a higher recovery thanthe corresponding Run 3 in PVSA I.

Figure 10 shows the axial concentration profiles of the BF1and BF2 steps in the zeolite bed. Similar to the results in Figure6b, as the BF step time increased, the MTZs of each componentretreated to the feed end (Figure 10a). However, unlike theresults in Figure 6b, the MTZs of O2 and Ar maintained theirfavorable shape even at a BF1 step time of 10 s and the MTZsof Ar crossed near the center of the bed. This was because theBF2 step following from the CMS bed led to a more highlysaturated O2 condition throughout the bed.

When the same BF2 step time was applied to the above threeBF1 conditions, the MTZ of each adsorbate further retreated tothe feed end (Figure 10b). The BF1 step of 30 s showed thelowest MTZ of Ar and the highest MTZ of O2 at the BF2 step.Moreover, because the convex shape of the O2 MTZ and theconcave shape of the N2 MTZ at the end of the PPE step(Figures 4c and 8b) disappeared at the end of the BF2 step, the

zeolite bed had conditions favorable to the effective supplyingof better feed to the CMS bed.

Figure 11 shows the axial concentration profiles of theBD1 and BD2 steps in the CMS bed of PVSA II. At the endof the BD1 step (Figure 11a), the O2 mole fraction in the gasphase was in the range of 97–98% and the N2 mole fractionwas �0.005%. The lower mole fraction of impurities at theend of the BD1 step was achieved with an increase in ADstep time because the AD step in the CMS bed worked toremove the impurities. After the BD2 step, as shown inFigure 11b, the O2 mole fraction exceeded 99.5% and almostthe same concentrations of impurities remained regardlessof AD step time conditions in the CMS bed. Therefore, theCMS bed was ready to purify feed under more favorableconditions in the PVSA II process.

Work is currently under way to optimize the process byusing a parametric study and to assess economical analysis byusing the pilot-plant study.

Conclusions

The three-bed PVSA process was refined to surpass 94% O2

purity from ambient air using adsorption technology. By usingtwo zeolite 10X beds and one CMS bed, the PVSA process

Figure 11. Axial gas phase concentration profilesalong the bed at the end of (a) BD1 and (b)BD2 steps of CMS bed at base adsorptionpressure and base feed flow rate condition(Runs 4 to 6).BD1 and BD2 step times were fixed to 10 and 30 s, respec-tively.

Figure 10. Axial gas phase concentration profilesalong the bed at the end of (a) BF1 and (b)BF2 steps of zeolite 10X bed at base ad-sorption pressure and base feed flow ratecondition (Runs 4 to 6).BF2 step time was fixed to 10 s.


could produce O2 with a purity 94% directly from theambient air. In addition, the nonisothermal dynamic modelsuccessfully predicted the cyclic behaviors in the experiment.

In the PVSA I process, which adopted a single stage of theblowdown/backfill step, the cyclic steady state was reachedafter about 14–15 cycles. In the case of the bulk air separator,the zeolite 10X bed, O2 with a purity of 90.2–93.7% wasproduced, whereas the impurities of N2 and Ar were 2.5–6.3and 3.5–4.2%, respectively. This O2 product was further puri-fied to 95.4–97.4% with a recovery of 43.4–84.8% in the CMSbed. The amount of N2 in the product was 4000–5000 ppmduring the PVSA I process.

The effect of the PR and AD step times in the CMS bed onthe performance of the process was analyzed at the fixed ADstep time of the zeolite 10X bed. As a consequence, theincreased AD step time of the CMS bed contributed to anincrease in O2 purity because the AD step of the CMS bedacted as an impurity-removal step and a greater amount ofbackfill gas was supplied to the zeolite 10X bed. However, theexcessive AD step time of the CMS bed led to a significantdecrease in recovery because the backfill and vacuum steptimes of the zeolite bed increased simultaneously.

In the PVSA II process, two-stage blowdown/backfill stepswere adopted to increase O2 purity by fractionating the O2

product from the BD step of the CMS bed. The zeolite 10X bedproduced O2 with a purity of 95.2–97.1%. This process over-came the purity limit of common adsorption processes usingzeolites because of the two-stage backfill steps supplied fromthe CMS bed. Thus, PVSA II could produce an O2 purity of98.2–99.2% with a recovery of 47.2–63.6% through the CMSbed. Also, the amount of N2 impurity decreased to �100 ppm.Similar to PVSA I, the increased AD step time of the CMS bedled to an increase in O2 purity and a decrease in O2 recovery inthe PVSA II process. However, because more highly concen-trated O2 was supplied from the zeolite bed in PVSA II ratherthan in PVSA I, the CMS bed in PVSA II could be kept fairlyclean after the production step, regardless of the applied ADstep time in the CMS bed. Therefore, the CMS bed had morefavorable conditions for the purification of feed in PVSA II.

In the near future, further results will be presented of theparametric study to directly produce a maximum of 99.8�%O2 from air by using the proposed PVSA processes.

AcknowledgmentsThe financial support of the Carbon Dioxide Reduction & Sequestration

R&D Center (C002-0103-001-1-0-0) is gratefully acknowledged.

Notation

AW � cross-sectional area of the wall, cm2

B � equilibrium parameter for Langmuir—Freundlich model,kPa�1

ci � i component concentration in bulk phase, mol/cm3

Cpg, Cps, Cpw � gas, pellet, and wall heat capacity, respectively, J g�1�K�1

De � effective diffusivity defined by solid diffusion model,cm2/s

DL � axial dispersion coefficient, cm2/shi � internal heat-transfer coefficient, J cm�2 K�1 s�1

ho � external heat-transfer coefficient, J cm�� K�1 s�1

��H� � average heat of adsorption, J/molk � parameter for Langmuir and LRC models

KL � axial thermal conductivity, J cm�1 s�1 K�1

L � bed length, cmn � dimensionless equilibrium parameter for Langmuir–

Freundlich modelP � total pressure, kPaPr � reduced pressure

q, q*, q� � amount adsorbed, equilibrium amount adsorbed, and av-erage amount adsorbed, respectively, mol/g

qm � equilibrium parameter for Langmuir–Freundlich model,mol/g

R � gas constant, J mol�1 K�1

Rp � radius of pellet, cmRBi, RBo � inside and outside radius of the bed, respectively, cm

t � time, sTatm � temperature of atmosphere, K

T, Tw � pellet or bed temperature and wall temperature, respec-tively, K

u � interstitial velocity, cm/syi � mole fraction of species i in gas phasez � axial distance in bed from the inlet, cm

Greek letters

� � particle porosity�, �t � voidage of adsorbent bed and total void fraction, respec-

tively�g, �p, �B, �w � gas density, pellet density, bulk density, and bed wall

density, respectively, g/cm3

� LDF coefficient, s�1

� viscosity, Pa�s

Subscripts

B � bedi � component ip � pelletg � gas phases � solid phase

w � wall

Literature Cited1. Kumar R. Vacuum swing adsorption process for oxygen produc-

tion—A historical perspective. Sep Sci Technol. 1996;31:877-893.2. Rege SU, Yang RT. Limits for air separation by adsorption with LiX

zeolite. Ind Eng Chem Res. 1997;36:5358-5365.3. Rei � G. Status and development of oxygen generation processes on

molecular sieve zeolites. Gas Sep Purif. 1994;8:95-99.4. Kawai M, Kaneko T. Present state of PSA air separation in Japan. Gas

Sep Purif. 1989;3:2-6.5. Wilson SJ, Webley PA. Cyclic steady-state axial temperature profiles

in multilayer, bulk gas PSA—The case of oxygen VSA. Ind Eng ChemRes. 2003;41:2753-2765.

6. Jiang L, Biegler LT, Fox VG. Simulation and optimization ofpressure-swing adsorption systems for air separation. AIChE J.2003;49:1140-1157.

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8. Yang RT. Gas Separation by Adsorption Processes. Boston, MA:Butterworth; 1987.

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10. Zayaraman A, Yang RT, Cho SH, Bhat SG, Choudary VN. Adsorptionof nitrogen, oxygen and argon on Na-CeX zeolites. Adsorption. 2002;8:271-278.

11. Ko D, Siriwardane R, Bigler RT. Optimization of a pressure-swingadsorption process using zeolite 13X for CO2 sequestration. Ind EngChem Res. 2003;42:339-348.

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14. Knaebel KS, Kandynin A. Pressure swing adsorption system to purifyoxygen. U.S. Patent No. 5 226 933; 1993.

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16. Hayashi S, Kawai M, Kaneko T. Dynamics of high purity oxygenPSA. Gas Sep Purif. 1996;10:19-23.

17. Jee JG, Kim MB, Lee CH. Pressure swing adsorption processes topurify oxygen using a carbon molecular sieve. Chem Eng Sci. 2005;60:869-882.

18. Lee CH. Apparatus for producing oxygen and method for controllingthe same. PCT/KR2004/001209; 2004.

19. Jee JG, Lee JS, Lee CH. Air separation by a small-scale two-bedmedical O2 PSA. Ind Eng Chem Res. 2001;40:3647-3658.

20. Kikkinides ES, Yang RT. Effects of bed pressure drop on isothermaland adiabatic adsorber dynamics. Chem Eng Sci. 1993;48:1545-1555.

21. Jee JG, Park H, Haam S, Lee CH. Effects of nonisobaric and isobaricsteps on O2 pressure swing adsorption for an aerator. Ind Eng ChemRes. 2002;41:4383-4392.

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Manuscript received Nov. 23, 2004, and revision received Mar. 10, 2005.


New Numerical Method for Solving the DynamicPopulation Balance Equations

Q. HuV & R Limited, P. O. Box 1149, London, ON N6P 1R2, Canada

S. Rohani and A. JutanDept. of Chemical & Biochemical Engineering, University of Western Ontario, London, ON N6A 5B9, Canada


A new numerical scheme is proposed for solving general dynamic population balanceequations (PBE). The PBE considered can simultaneously include the kinetic processes ofnucleation, growth, aggregation and breakage. Using the features of population balance,this method converts the PBE into a succession of algebraic equations which can besolved easily and accurately. The new method is free from stability and dispersionproblems of general numerical techniques. Some benchmark problems with analyticsolutions were tested. In all cases tested this method gave accurate results with very fewcomputational requirements. For nucleation and size-independent growth without aggre-gation and breakage, the numerical method gives exactly the same result as analyticsolution. © 2005 American Institute of Chemical Engineers AIChE J, 51: 3000–3006, 2005

Introduction

The population balance equation (PBE) has been used tomodel a variety of particulate systems. Analytical solutions tothe PBE can only be obtained for very few special cases.Numerical solution of the PBE remains a considerable chal-lenge due to aggregation and breakage terms. Most of theexisting numerical methods are tailored to handle specificapplications and lack generality (Raphael et al., 1995).

The method of weighted residuals with global functions isone of the most popular methods to solve the PBE(Ramkrishna, 1985). In the method of weighted residuals, thesolution is approximated by a linear combination of a series ofchosen basis functions, whose unknown coefficients are deter-mined by satisfying the PBE to define a residual. The idea ofweighted residuals is to find the coefficients that force theresiduals to be orthogonal to a chosen set of weighting func-tions. The method of moments is equivalent to the method ofweighted residuals if the weighting functions are chosen to be

polynomials. Since polynomials weights are often a poorchoice for population balances on semi-infinite intervals, andnot all PBE formulations can be reduced to moment equations,the method of moments is not recommended. A limitation ofglobal functions is that they cannot always capture the featuresof the solution, especially when there are sharp changes anddiscontinuities in the solution. Discontinuities arise in crystal-lizer systems with product classification or fines destruction.Finite-element methods approximate the solutions with localfunctions which can be tailored to handle discontinuities andsharp changes.

Gelbard and Seinfeld (1978) considered orthogonal colloca-tion and spline collocation on finite elements to solve popula-tion balance equations with nucleation, growth and aggrega-tion. The semi-infinite particle size domain is truncated at somelarge value, and then the finite domain is divided into elements.Lower order polynomials are used to approximate the solutionon each elements. Nicmanis and Hounslow (1998) solved thesteady-state PBE with aggregation, breakage, nucleation andgrowth using collocation and Galerkin methods. The method isbased on an error estimate of the second moment. Raphael et al.(1995) used orthogonal collocation to solve the problem ofisoelectric precipitation of sunflower protein. Bennett and Ro-

Correspondence concerning this article should be addressed to S. Rohani [email protected].


PROCESS SYSTEMS ENGINEERING


hani (2001) solved a PBE by combining Lax-Wendroff andCrank-Nicholson methods. Rigopoulos and Jones (2003) usedlinear elements to represent the solution of dynamic PBE. Themethod is expected to be faster than higher-order finite elementcollocation methods, but it is less accurate.

An alternative numerical technique for solving PBE is dis-cretization method, which divides the particles into discrete butcontiguous size ranges (Baterham et al., 1981; Hounslow et al.,1988; Kostoglou and Karabelas, 1994). This method requires auniform particle distribution, which is unrealistic for processessuch as crystallization or coagulation of aerosols where ag-glomeration exists. Kumar and Ramkrishna (1997) extendedthis method for solving PBE for breakage and aggregation ofparticles. The method combines the features of discretizationtechnique with the method of characteristics. Detailed reviewsof previous work on solving PBEs have been made by Kosto-glou and Karabelas (1994), Vanni (2000), and Lee (2001).

Most of the earlier numerical methods are difficult to imple-ment, computationally demanding, or lacking in accuracy.They usually are tailored to solve individual problems. Theirpredictions of moments are subject to errors. The objective ofthis paper is to present a new technique for solving dynamicPBE for nucleation, growth, breakage and aggregation, allprocesses occurring simultaneously. The approach makes useof the properties of population balance. The new method is freefrom problems due to stability and dispersion of the numericalsolutions. The proposed approach has been tested on a numberof benchmark problems. Comparison between the analytic so-lutions and the numerical solutions indicates the distributionscan be accurately predicted by the proposed method. In addi-tion, the proposed method is easy to implement with fewcomputational requirements.

Population Balances

In this work, we consider the general PBE of the form(Randolph and Larson, 1988)

�n�v, t�

�t�

��G�v�n�v, t��

�v� Bnuc�v� � Bagg�v� � Dagg�v�

� Bbr�v� � Dbr�v� (1)

where n(v, t) is the population density of particles of volumev and time t, G(v) is growth rate for particles of volume v,Bnuc(v) is the nucleation rate of particles of volume v, Bagg(v)and Dagg(v) are the birth and death rates of particles of volumev due to aggregation, and Bbr and Dbr are the birth and deathrates of particles of volume v due to breakage. Bagg(v) andDagg(v) can be written as (Hulburt and Katz, 1964)

Bagg�v� �1

2 �0

v

��v � v�, v��n�v � v�t�n�v�, t�dv� (2)

Dagg�v� � n�v, t� �0

�

��v, v��n�v�, t�dv� (3)

where �(v, v�) is the aggregation kernel. The birth and deathrates of particles due to breakage can be written as (Prasher,1987)

Bbr�v� � �v

�

��v, v��S�v��n�v�, t�dv� (4)

Dbr�v� � S�v�n�v, t� (5)

where �(v, v�) is the breakage function and S(v) is the rate ofbreakage for particles of volume v. Equation 1 is a first orderhyperbolic differential equation for which very few analyticalsolutions have been found.

Formulation of the AlgorithmApproximation of solution

In the absence of aggregation and breakage, a representationof population balance is shown in Figure 1, where populationbalance distributions at time t and t � �t are demonstrated.The particles grow into the volume range [v(t � �t), v ��v(t � �t)] from volume range [v(t), v � �v(t)] over thetime interval �t. n(v(t), t) and n(v(t � �t), t � �t) representthe population density at time t and t � �t, respectively. Thepopulation balance implies (Hu et al., 2004)

n�v�t�, t��v�t� � n�v�t � �t�, t � �t��v�t � �t� (6)

However, in the presence of nucleation, aggregation and break-age, the Eq. 6 is not satisfied anymore. In this case, we assumethat

n�v�t�, t��v�t� � n�v�t � �t�, t � �t��v�t � �t�

� ��v�t��v�t��t (7)

The last term in Eq. 7 is included due to nucleation, aggrega-tion and breakage, and �(v(t)) is a parameter to be determined.

On the basis of the definition of growth rate, we have

v�t � �t� � v�t� � G�v�t��t (8)

which implies

Figure 1. Population balance.


�v�t � �t� � �1 ��G�v�

�v �v�v�t�

�t��v�t� (9)

Using the Taylor series, we have

n�v�t � �t�, t � �t� � n�v�t�, t � �t�

��n�v, t � �t�


�v�t � �t� � v�t�� (10)

With Eq. 8 and Eq. 10 can be written as

n�v�t � �t�, t � �t� � n�v�t�, t � �t�

��n�v, t � �t�


G�v�t��t (11)

Substituting Eq. 9 and Eq. 11 into Eq. 7 yields

n�v�t�, t� � n�v�t�, t � �t� ��n�v, t � �t�


G�v�t��t

� n�v�t�, t � �t��G�v�


�t

��n�v�t�, t � �t�


�G�v�


G�v�t��t�2

� ��v�t��t (12)

which can be written as

n�v�t�, t � �t� � n�v�t�, t�

�t� �G�v�t��

�n�v�t�, t � �t�


� n�v�t�, t � �t��G�v�


� ��v�t��

��n�v, t � �t�


�G�v�


G�v�t��t (13)

If �t 3 0, Eq. 13 becomes

�n�v, t�

�t� G�v�

�n�v, t�

�v� n�v, t�

�G�v�

�v� ��v� (14)

which is equivalent to Eq. 1 if

��v� � Bnuc�v� � Bagg�v� � Dagg�v� � Bbr�v� � Dbr�v� (15)

therefore, we can use Eq. 7 to solve Eq. 1. Using Eq. 8 and Eq.9, Eq. 7 can be changed to

n�v�t � �t�, t � �t� �n�v�t�, t� � ��v�t��t

1 ��G�v�


�t

(16)

Thus, given the population density at time t, the populationdensity at time t � �t can be solved by Eq. 16.

For numerically solving the PBE (Eq. 1), the infinite domainof the particle volume v must be truncated to a finite upperlimit. The error in the ith moment of the solution due to domaintruncation is

Mie � �

vmax

�

vin�v�dv (17)

where vmax is the upper limit of the finite domain. In practice,the population density n(v) tends to zero at sufficiently largeparticle volumes, so vmax can be chosen to be sufficiently largesuch that Mi

e is negligibly small.In this method, the truncated domain v � [0, vmax] is

partitioned into N discrete and contiguous elements, that is, wesubdivide it into subintervals with common endpoints, callednodes. The time step is set to �t. We use (vj,i, nj,i) to representa node on the particle distribution profile on the vn-coordinateplane. The index j denotes the time j�t, and the index i � 0,1, . . . , N indicates the series number of the nodes on theparticle distribution profile. Note that only the initial truncateddomain (at t � 0) needs to be partitioned, and the nodes on thevn-coordinate plane at other time can be approximated succes-sively using Eq. 8 and Eq. 16 which imply

vj�1,i � vj,i � G�vj,i��t (18)

nj�1,i �nj,i � ��vj,i��t

1 ��G�v�

�v �v�vj,i

�t

(19)

for i � 0, 1, . . . , N and j is a nonnegative integer.

Approximation of birth and death terms

In Eq. 19, the term �(vj,i) still needs to be solved. We usecubic spline interpolation to find additional nodes we may needin this section, while (nj,1 � nj,0)/(vj,1 � vj,0), (nj,N �nj,N�1)/(vj,N � vj,N�1) are used as the end slopes for the cubicspline.

Gauss-Legendre quadrature is used to perform integrationsover each element [vj,i, vj,i�1] (Hildebrand, 1956). An approx-imation to the integral

��1

1

f� x�dx � �i�1

N

wN,if� zN,i� (20)

is obtained by sampling f( x) at the N unequally spaced abscis-sas zN,1, zN,2, . . . , zN,N, where the corresponding weights arewN,1, wN,2, . . . , wN,N. The abscissas and weights for Gauss-Legendre quadrature can be computed analytically for small N.For N � 5


��1

1

f� x�dx � w1f� z1� � w2f� z2� � w3f� z3�

� w4f� z4� � w5f� z5� (21)

where

z1 � �1

21�245 � 14�70, w1 �

1

900�322 � 13�70�

z2 � �1

21�245 � 14�70, w2 �

1

900�322 � 13�70�

z3 � 0, w3 �128

225

z4 �1

21�245 � 14�70, w4 �

1

900�322 � 13�70�

z5 �1

21�245 � 14�70, w5 �

1

900�322 � 13�70�

For ease of reading the earlier list the notation zi and wi insteadof zN,i and wN,i, has been used respectively.

To apply the rule over the interval [a, b], use the change ofvariable

t �a � b

2�

b � a

2x and dt �

b � a

2dx

then the relationship ab f(t)dt � �1

1 f((a � b)/ 2 � ((b �a)/ 2) x)((b � a)/ 2)dx is used to obtain the quadrature for-mula

�a

b

f�t�dt �b � a

2 �i�1

N

wN,if�a � b

2�

b � a

2zN,i�

If there is no aggregation and breakage, the solution Eq. 19becomes

nj�1,i �nj,i � Bnuc�vj,i��t

1 ��G�v�

�v �v�vj,i

�t

(22)

then the accuracy of the solution can be increased by simplyselecting a small �t. Furthermore, if the growth rate is volume-independent, that is, �G/�v � 0, (Eq. 22) can be simplified to

nj�1,i � nj,i � Bnuc�vj,i��t (23)

which indicates that accurate solution can be obtained. This isa very useful property for some batch crystallization processes.

Case Studies

The method was tested for six cases of the PBE, whereanalytical solutions are available. The numerical results werecompared with analytic solutions. The CPU time reportedcorresponds to a desktop computer with 1.6 GHz Intel Pentium4 processor.

Aggregation and size-dependent growth

By letting the nucleation rate Bnuc(v), breakage function�(v, v�), and the rate of breakage S(v) to be zero, the PBE foraggregation and size-dependent growth is

�n�v, t�

�t�

��G�v�n�v, t��

�v�

1

2 �0

v

��v � v�, v��n�v

� v�t�n�v�, t�dv� � n�v, t� �0

�

��v, v��n�v�, t�dv� (24)

The initial distribution is

n�v, 0� �N0

v0exp��

vv0� (25)

and size-dependent growth rate is

G�v� � �1v (26)

Case A1. For the size-independent aggregation kernel

��v, v�� 0 (27)

the analytical solution has been derived by Ramabhadran et al.(1976)

n�v, t� �M0

2

M1exp��

M0

M1v� (28)

where M0(t) and M1(t) are the first two moments

M0�t� � �0

�

n�v, t�dv �2N0

2 � �0N0t(29)

M1�t� � �0

�

vn�v, t�dv � N0v0exp��1t� (30)

The constant N0, v0, �1 and �0 were set to 1. The truncationpoint (vmax) was chosen as 10, and the initial truncated domainwas evenly partitioned into 20 discrete elements with commonends. The time step �t was chosen to be 0.02.

The problem (Eq. 24) was solved using the proposedmethod. The particle density distributions for the initial time,t � 0.5, t � 1 and t � 2 are shown in Figure 2. The solid linesrepresent the analytical solution, while the symbols are the


nodal values obtained by the proposed method. The numericaland analytical solutions are in excellent agreement.

Case A2. For the size-dependent aggregation kernel

��v, v�� 1�v � v�� (31)

The analytical solution is (Ramabhadran et al., 1976)

n�v, t� �

M02

M1exp��

M0

M1�2N0

M0� 1�vI1�2�1 �

M0

N0

N0vM1

�M0vM1

�1 �M0

N0

(32)

where I1� is the modified Bessel function of the first kind oforder one, and

M0�t� � N0exp��1N0v0

�1�1 � exp��1t�� (33)

M1�t� � N0v0exp��1t� (34)

The constant �1 was set to 1. The truncation point vmax wasselected to be 30. The domain [0, vmax] was evenly partitionedinto 30 elements. The time interval was 0.02. Comparison ofsimulation with the analytical solution is shown in Figure 3.The solids lines represent the analytical solution, and the sym-bols are the numerical results. Again, the numerical solutionoverlaps the analytical solution excellently.

Case A3. If the initial distribution is

n�v, 0� �N0vv0

2 exp��vv0� (35)

and �(v, v�) and G(v) are the same as in Case A1, then theanalytical solution is (Ramabhadran et al., 1976)

n�v, t� �M0

2

M1

1

�1 � M0/N0

exp��N0vM1

�sinh��1 �M0

N0

N0vM1

�(36)

where M0 and M1 are defined in Eq. 29 and Eq. 30, respec-tively. All constants and truncation are the same as in Case A1.Figure 4 compares the numerical solution with the analyticalsolution. The solids lines represent the analytical solution,while the symbols are the nodal values. The agreement is quitegood.

Errors in the first moment of the numerical solution and CPUtime are recorded in Table 1. The accuracy of the numericalsolutions is satisfactory, as shown in Table 1, despite the use ofa fairly coarse grid.

Figure 2. Aggregation with constant kernel �(v, v�) � 1,and size-dependent growth rate G(v) � v.

Figure 3. Aggregation with sum kernel �(v, v�) � v � v�,and size-dependent growth rate G(v) � v.

Figure 4. Aggregation with constant kernel �(v, v�) � 1,size-dependent growth rate G(v) � v, and initialdistribution (Eq. 35).


Breakage

The PBE for pure breakage is obtained by setting the growthrate G(v), the nucleation rate Bnuc(v), and the aggregationkernel �(v, v�) to zero, that is

�n�v, t�

�t� �S�v�n�v, t� ��

v

�

��v, v��S�v��n�v�, t�dv� (37)

Case B1. If the probability that a particle breaks up intotwo pieces is independent of both the size of the object and ofthe pieces, then

��v, v�� 2

v�(38)

S�v� � v (39)

For the initial distribution

n�v, 0� � exp��v� (40)

the analytical solution is (Ziff and McGrady, 1985)

n�v, t� � �1 � t�2exp�v�1 � t�� (41)

Figure 5 shows the initial distribution and the distributions att � 0.5, and t � 1 for this case. The solid line represents the

analytical solutions, and the symbols represent the numericalsolutions. The numerical solutions match the analytical solu-tions very well.

Case B2. In this case, the rate of breakage is proportionalto the size, and

��v, v�� 2

v�(42)

S�v� � v2 (43)

For the exponential initial distribution (Eq. 40), the analyticalsolution is (Ziff and McGrady, 1985)

n�v, t� � exp��tv2 � v�1 � 2t�1 � v�� (44)

Distributions at different moments for this case are shown inFigure 6. The numerical solutions are in good agreement withthe analytical solutions.

Parameters used for numerical solution, errors in the firstmoment and CPU time were summarized in Table 2.

Nucleation and growth

For simultaneous nucleation and size independent growththe PBE is

�n�v, t�

�t� G

�n�v, t�

�v� B0�v� (45)

Table 1. Aggregation and Size-Dependent Growth Rate

Case Time �tNo. of

Elements Error in M1% CPU (s)

A1 1 0.02 20 0.15 34A2 1 0.02 30 0.46 76A3 1 0.02 20 0.28 34

Figure 5. Breakage with a binary breakage function �(v,v�) � 2/v�, and size-dependent breakage rateS(v) � v.

Figure 6. Breakage with breakage rate proportional tothe particle volume S(v) � v2.

Table 2. Breakage

Case Time �tNo. of

Elements Error in M1% CPU (s)

B1 1 0.02 50 0.50 80B2 1 0.02 50 0.55 80


where B0 and G are constant. The analytical solution for thismodel is (Hounslow et al., 1988)

n�v, t� �B0

Gu� t �

vG� (46)

where u is the unit step function. With G � 1 and B0 � 1,comparison between the numerical solution with the analyticalsolution is shown in Figure 7, where the solid line representsthe analytical solution, and the symbols are for the numericalsolution. It should be noted that the proposed method providesabsolutely accurate solution in this case and is better than anyother existing numerical methods. The CPU time was only0.02 s.

Conclusions

In this work, a new numerical method for solving dynamicPBE was proposed. The new method is able to accuratelypredict the solution of PBE involving any combination ofparticulate system subprocess, that is, nucleation, growth, ag-gregation and breakage, with reasonable computational re-quirement. The proposed approach is free from stability anddispersion problems of other numerical methods such as finite-element method. The numerical scheme requires a partition ofthe particle-size domain, but there are no restrictions on thelocation of nodes when using this method. An evenly spacedpartition was used in this method, although other partition

methods can also be applied. Accuracy of the numerical solu-tion can be simply improved by smaller time step and morepartition elements of particle size domain without any stabilityproblems, which makes this method particularly useful fordeveloping hierarchical solution for particulate processes. Acompromise between computational time and accuracy can beeasily achieved by choosing the node resolution.

Literature CitedBaterham, R. J., J. S. Hall, and G. Barton, “Pelletizing Kinetics and

Simulation of Full Scale Balling Circuits,” in Proc. 3rd Int. Symp. onAgglomeration, Nurnberg, Germany (1981).

Bennett, M., and S. Rohani, “Solution of Population Balance Equation witha New Combined Lax-Wendroff Cranck-Nicholson Method,” Chem.Eng Sci., 56, 6623 (2001).

Gelbard, F., and J. H. Seinfeld, “Numerical Solution of the DynamicEquation for Particulate Systems,” J. Comput. Phys., 28, 357 (1978).

Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-Hill, NewYork (1956).

Hounslow, M. J., R. L. Ryall, and V. R. Marshall, “A Discretized Popu-lation Balance for Nucleation, Growth, and Aggregation,” AIChE J., 34,1821 (1988).

Hu, Q., S. Rohani, D. X. Wang, and A. Jutan, “Nonlinear Kinetic Param-eter Estimation for Batch Cooling Seeded Crystallization,” AIChE J., 50,1786 (2004).

Hulburt, H. M., and S. Katz, “Some Problems in Particle Technology. AStatistical Mechanical Formulation,” Chem. Eng Sci., 19, 555 (1964).

Kostoglou, M., and A. J. Karabelas, “Evaluation of Aero Order Methodsfor Simulating Particle Coagulation,” J. Colloid Interface Sci., 163, 420(1994).

Kumar, S., and D. Ramkrishna, “On the Solution of Population BalanceEquations by Discretization—III. Nucleation, Growth and Aggregationof Particles,” Chem. Eng Sci., 52, 4659 (1997).

Lee, M. H., “A Survey of Numerical Solutions to the Coagulation Equa-tion,” J. Phys. A: Math. Gen., 34, 10219 (2001).

Nicmanis, M., and M. J. Hounslow, “Finite-Element Methods for Steady-State Population Balance Equations,” AIChE J., 44, 2258 (1998).

Prasher, C. L., Crushing and Grinding Process Handbook, Wiley, NewYork (1987).

Ramabhadran, T. E., T. W. Peterson, and J. H. Seinfeld, “Dynamic ofAerosol Coagulation and Condensation,” AIChE J., 22, 840 (1976).

Ramkrishna, D., “The Status of Population Balances,” Rev. Chem. Eng., 3,49 (1985).

Randolph, A. D., and M. A. Larson, Theory of Particulate Processes, 2nded., Academic Press, New York (1988).

Raphael, M., S. Rohani, and F. Sosulski, “Isoelectric Precipitation ofSunflower Protein in a Tubular Precipitator,” Can. J. Chem. Eng., 73,470 (1995).

Rigopoulos, S., and A. G. Jones, “Finite-Element Scheme for Solution ofthe Dynamic Population Balance Equation,” AIChE J., 49, 1127 (2003).

Vanni, M., “Approximate Population Balance Equations for Aggregation-Breakage Processes,” J. Colloid Interface Sci., 221, 143 (2000).

Ziff, R. M., and E. D. McGrady, “The Kinetics of Cluster Fragmentationand Depolymerisation,” J. Phys. A: Math. Gen., 18, 3027 (1985).

Manuscript received Feb. 17, 2004, and revision received Feb. 15, 2005, and finalrevision received May 3, 2005.

Figure 7. Nucleation and growth.


Integration of Data Uncertainty in LinearRegression and Process Optimization

Marco S. Reis and Pedro M. SaraivaGEPSI-PSE Group, Dept. of Chemical Engineering, University of Coimbra, Polo II—Pinhal de Marrocos,

3030-290 Coimbra, Portugal


Data uncertainties provide important information that should be taken into accountalong with the actual data. In fact, with the development of measurement instrumentationmethods and metrology, one is very often able to rigorously specify the uncertaintyassociated with each measured value. The use of this piece of information, together withraw measurements, should—in principle—lead to more sound ways of performing dataanalysis, empirical modeling, and subsequent decision making. In this paper, we addressthe issues of using data uncertainty in the task of model estimation and, when it is alreadyavailable, we show how the integration of measurement and actuation uncertainty can beachieved in the context of process optimization. Within the scope of the first task (modelestimation), we make reference to several methods designed to take into account datauncertainties in linear multivariate regression (multivariate least squares, maximumlikelihood principal component regression), and others whose potential to deal with noisydata is well known (partial least squares, principal component regression, and ridgeregression), as well as modifications of previous methods that we developed, and comparetheir performance. MLPCR2 tends to achieve better predictive performance than all theother tested methods. The potential benefits of including measurement and actuationuncertainties in process optimization are also illustrated. © 2005 American Institute ofChemical Engineers AIChE J, 51: 3007–3019, 2005Keywords: measurement uncertainty, multivariate least squares, maximum likelihoodprincipal component regression, partial least squares, principal component regression,optimization under uncertainty

Introduction

The large amounts of industrial and laboratorial data gener-ated in the chemical process industries and stored in databasesdo have a substantial potential to set the ground for furtherprocess improvement and optimization. Noting that this poten-tial is not always being fully developed and that the goals thatwere present at the conception of such databases are often notbeing achieved, numerous efforts have been made and docu-mented in the literature toward a more effective use of these

information resources, that is, in the fields of process monitor-ing,1,2 fault detection and diagnosis,3 and data mining.4 How-ever, quite often these approaches do not explicitly and quan-titatively take into account data quality, or do so only in animplicit or tacit way. Following the efforts undertaken in themetrology field, with respect to the characterization and quan-tification of measurement uncertainty, in a rigorous and nor-malized approach,5 we believe it is quite appropriate and timelyto develop and apply methods that explicitly and consistentlytake into account this important piece of information.

Measurement uncertainty is a well-defined quantity andthere are well-documented standardized procedures that assistits specification or estimation. Basically, uncertainty is definedas a “parameter associated with the result of a measurement

Correspondence concerning this article should be addressed to M. S. Reis [email protected].



that characterizes the dispersion of the values that could rea-sonably be attributed to the measurand.” 5 The standard uncer-tainty u (to which we will often refer simply as “uncertainty”)should be expressed in terms of a standard deviation of thevalues obtained under the same experimental conditions, andcan be obtained either from the analysis of collected data (theso-called Type A evaluation) or through other adequate means(Type B evaluation). The availability of raw values, along withtheir associated uncertainties, implies that we should not haveonly one data table available for analysis, but in fact two (onewith raw values and another one with the corresponding mea-surement uncertainties). Therefore, with this additional infor-mation at our disposal, we should be able to take advantage ofit through its integration into our data analysis tasks. Forinstance, data reconciliation6-8 is designed to handle noisymeasurements, to adjust raw data in some optimal way, so thatit conforms to conservation laws and other constraints. The factthat the objective function to be minimized consists of qua-dratic terms involving the inverse of variance–covariance ma-trices of measurements7 indicates that uncertainty informationis in fact being considered in data reconciliation. However,there are many application scenarios where no conservationlaws are available to perform preliminary data reconciliation,such as the analysis of spectra, microarray data, and laborato-rial data sets. Furthermore, uncertainty-based methods can beapplied to data sets after reconciliation or filtering.

Sometimes it happens that uncertainty associated with mea-surements is sufficiently small for the techniques that disregardit completely or treat it in a very simplified way (such asassuming homoscedastic behavior), still holding as adequate.However, these are tacit assumptions, quite often not verifiedor clearly stated. The main purpose of this article is to bring theissue of data uncertainty into the priorities for the data analyst,which should explicitly address it in a preliminary phase, aswell as to present, develop, and test procedures that do exploitand take advantage of data uncertainty information.

In particular, we will address the use of uncertainty infor-mation in two different tasks: model estimation and processoptimization. In the next section, we refer several methodolo-gies with the potential of integrating uncertainty in the estima-tion of parameters for a multivariate linear model. This type ofmodel is widely used in the analysis of industrial data sets, andits prediction ability, when parameters are estimated by differ-ent methodologies, is thus an important issue in practicalapplications. In the following section, a complementary situa-tion regarding the use of uncertainties, that is, when a model isconsidered to be known, is illustrated under the context ofprocess optimization. Then, in the fourth section, we presenttwo case studies that provide the ground for comparison amongall the methods referenced in the second section and anothercase study that illustrates the methodology presented in thethird section. We end this paper with a discussion section,where some computational issues are addressed (fifth section)and some final conclusions are drawn. Apart from the compar-ative study undertaken in the case study section, new methods(unc-PLS3, unc-PLS4, and unc-PLS5) are also presented andtested. Methods MLMLS, unc-PLS1, unc-PLS2, MLPCR2,rMLS, and rMLMLS are carefully described elsewhere.9 Theformulations presented in the third section provide also acontribution to the explicit consideration of measurement un-certainties for process optimization.

Measurement Uncertainties in Model Estimation

This section is devoted to the description of four groups ofmultivariate linear regression methods that have the potential toaccommodate measurement noise information, either explicitlyor implicitly. As already mentioned, our focus on multivariatelinear regression methods arises from the quite widespread useof this type of approaches in the development of input/outputmodels for industrial and/or laboratorial applications. The sev-eral methodologies here addressed are combined under fourseparate groups, according to their affinity: ordinary leastsquares (OLS), ridge regression (RR), principal componentregression (PCR), and partial least squares (PLS, also referredto as “projection to latent structures”). These four basic meth-ods do not explicitly incorporate measurement uncertainty in-formation, so that several alternatives already developed arealso presented, as well as other recent modifications that wepropose here and do take uncertainty information explicitlyinto consideration.

OLS group

Ordinary least squares (OLS) and multivariate least squares(MLS)10,11 parameter estimates for a linear regression modelare the solutions of the optimization problems formulated inEqs. 1 and 2 of Table 1.

OLS tacitly assumes a homoscedastic behavior (that is, withconstant variance) for the noise error term in the standard linearregression model. On the other hand, MLS is built on an errorin variables (EIV) functional relationship relating true values ofboth the input and output variables, which are then affected byzero mean random errors with a given covariance structure(presumed to be known). In the denominator of Eq. 2 we canfind a term, se

2(i), that results from the summation of theuncertainties associated with the response to those arising fromthe propagation of uncertainties of the predictors to the re-sponse, according to a formula derived from error propagationtheory10,12:

Table 1. Formulation of the Optimization ProblemsUnderlying OLS and MLS Methods

OLS bOLS � arg minb��b0· · ·bp�T

��i�1

n

[y(i) � y(i)]2� (1)

MLS bMLS � arg minb��b0· · ·bp�T

��i�1

n[y(i) � y(i)]2

se2(i) � (2)

MLMLS bMLMLS � arg maxb��b0· · ·bp�T

��b�

��b� � �1

2n ln�2��

i�1

n

ln��i�

�1

2 �i�1

n �[y(i) � y(i)]2

��i

2 �(3)


se2�i� � uy�i�2 � �

j�1

p

bj2uX�i, j�2

� 2 �j�2

p �k�j�1

p

bjbkcov��j�i�, �k�i�� (4)

where uX(i, j) and uy(i) are the uncertainties associated with theith observation of the jth input and output variables, respec-tively, and �j(i) is the random error affecting the ith measure-ment of variable j; bj represents the coefficient of the linearregression model associated with variable j.

The method whose objective function is presented in Table1, Eq. 3, is derived from the analysis of the Berkson case(controlled regressors with error) within the scope of EIVmodels13,14 and under the assumption of Gaussian errors. Theobjective function arises from the maximization of the resultinglikelihood function, and we included this approach in ourpresent study given both the similarity between the quadraticfunctional part of its objective function and the one underlyingMLS, and its simplicity. Because the solution for the Berksoncase formulation is sometimes similar to MLS,14 we makereference to the above formulation maximum likelihood mul-tivariate least squares (MLMLS), to stress the statistical originof the underlying objective function.

RR group

A well-known characteristic of the OLS method is the factthat the variance of its parameter estimates increases when theinput variables become more correlated. Computational simu-lations showed us that the same applies to MLS. One possibleway to address this issue consists of enforcing an effectiveshrinkage in the coefficients under estimation, following aridge regression (RR) regularization approach. It basically con-sists of adding an extra term to the objective function thatpenalizes large solutions (in a square norm sense). Optimiza-tion formulations underlying RR estimates,15,16 as well as thoseproposed for its counterparts based on MLS and MLMLS,rMLS and rMLMLS, respectively (standing for “ridge MLS”and “ridge MLMLS”), are presented in Table 2.

PCR group

PCR17,18 is another methodology that handles collinearityamong predictor variables. It uses those uncorrelated linearcombinations of the input variables that most explain inputspace variability [from principal components analysis (PCA)]as the new set of predictors, where the response is to be

regressed onto. These predictors are orthogonal and thus thecollinearity problem is overcome if we disregard the linearcombinations with small variability explanation power.19 Afterdeveloping MLPCA, which estimates the PCA subspace in anoptimal maximum likelihood sense, when data are affected bymeasurement errors with a known uncertainty structure,20

Wentzell et al.21 applied it in the context of developing a PCRmethodology that incorporates measurement uncertainties(MLPCR). As in PCR, MLPCR consists of first estimating aPCA model, now using MLPCA, to calculate the scoresthrough nonorthogonal (maximum likelihood) projections tothe estimated MLPCA subspace (instead of the PCA orthogo-nal projections), and then applying OLS to develop a finalpredictive model. This technique makes use of the availableuncertainty information in the former phases (estimation of aMLPCA model and calculation of its scores), but not during thestage at which OLS is applied. Therefore, Martınez et al.10

proposed a modification to the regression phase, to make itconsistent with the efforts of integrating uncertainty informa-tion carried out in the initial stages, which consists of replacingOLS by MLS (we will call this modification MLPCR1). Toimplement MLS in the second phase, estimated score uncer-tainties for the ith observation need to be calculated, beinggiven by the diagonal elements of the following matrix10

Zi � PT�diag�uX�i, :��1P��1 (8)

where diag is an operator that converts a vector into a diagonalmatrix, and P is the matrix of maximum likelihood loads. Inour study, we will compare these algorithms based on OLS andMLS (MLPCR and MLPCR1, respectively), with the one ob-tained when we use the MLMLS algorithm instead of MLS, inthe second phase of MLPCR (MLPCR2).

PLS group

PLS17,18,22-27 is a widely used algorithm in the chemometricscommunity that also adequately handles noisy data with cor-related predictors in the estimation of a linear multivariatemodel. As in PCR, PLS finds a set of uncorrelated linearcombinations of the predictors, belonging to some lower-di-mensional subspace in the X-variables space, where y is to beregressed onto. However, in PLS, this subspace is the one that,while still adequately covering the X-variability, provides agood description of the variability exhibited by the Y-vari-able(s). Here we will make reference to a pair of classes of PLSalgorithms, one implemented from raw data and another basedon covariance matrices.

Table 2. Formulation of the Optimization Problems Underlying RR, rMLS, and rMLMLS

RRbRR � arg min

b��b0· · ·bp�T��

i�1

n

(y(i) � y(i))2 � � �j�1

p

b(j)2� (5)

rMLSbrMLS � arg min

b��b0· · ·bp�T��

i�1

n(y(i) � y(i))2

se2(i)

� � �j�1

p

b(j)2� (6)

rMLMLSbrMLMLS � arg min

b��b0· · ·bp�T��

i�1

n

ln(se(i)) � �i�1

n(y(i) � y(i))2

se2(i)

� � �j�1

p

b(j)2� (7)


Tab

le3.

PL

S1as

aSu

cces

sion

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PLS algorithms implemented directly from raw data

The algorithmic nature of PLS22,26 can be translated into thesolutions of a succession of optimization subproblems,17,18,23 aspresented in the first column of Table 3 for one of its commonversions, relative to the case of a single response variable(PLS1). However, if besides having available raw data, [X � y],we also know their respective uncertainties, [uX � uy], then oneway to incorporate this additional information into a PLSalgorithm is through an adequate reformulation of the optimi-zation subtasks. Therefore, we have modified the objectivefunctions underlying each optimization subproblem to incor-porate measurement uncertainties, but still preserving the suc-cessful algorithmic structure of PLS. Such a sequence of opti-mization subproblems is presented in the second and thirdcolumns of Table 3, where MLS and MLMLS replace OLS inseveral algorithmic stages, giving rise to the uncertainty-basedequivalents unc-PLS1 and unc-PLS2, respectively.

PLS algorithms implemented from covariance matrices

There are several alternative ways to develop a PLS model,most of them leading to very similar or even exactly the sameresults. In fact, Helland25 has shown the equivalence betweentwo of such algorithms (one based on orthogonal scores andanother using orthogonal loadings instead), both of them basedon available raw data matrices for the predictors and responsevariables. Another class of PLS methods that encompasses theso-called SIMPLS, developed by Sijmen de Jong (see Table 4),or the approach presented by Kaspar and Ray,28 built onprevious work from Hoskuldsson,29 consists of algorithms en-tirely based on data covariance or cross-product matrices. Forthe single response case, a SIMPLS solution provides exactlythe same results as Svant Wold’s orthogonalized PLS algo-rithm, leading to only minor differences when several outputsare considered. Matrices S and s in Table 4 do play a centralrole in PLS. Theoretical analysis of this algorithm25,30 leads tothe conclusion that the calculated vector of coefficients, whena latent variables are considered, PLS

a , is given by

PLSa � Va�Va

TSVa��1Va

Ts (9)

where Va � [v1, v2, . . . , va] is any (m � a) matrix whosecolumns span the following Krylov subspace, �a(s; S), that is,the subspace generated by the first a columns of the Krylovsequence, {s, Ss, . . . , Sa�1s}. Thus, matrices S and s define thestructure of the relevant Krylov subspace where the PLS solu-tion will lie. In fact, the columns of the PLS weighting matrixW, which define the subspace of the full predictor space withmaximal covariance with the response, do form an orthogonal

base of �a(s; S). The relevancy of S and s for PLS provided themotivation to direct some efforts toward the incorporation ofuncertainty information in the computation of better estimatesfor both of these matrices. The reason that we have not calledthem estimates so far is explained by the lack of a consistentstatistical population model underlying PLS.24,31,32 However,when we now say that our goal is to calculate “better” covari-ance matrices, this implies that some goodness criteria must beassumed. Therefore, to give a step forward toward the integra-tion of measurement uncertainties in our analysis, one shouldpostulate a statistical model to provide an estimation setting forthe covariance matrices S and s. For the sake of the presentwork, we consider the following latent variable multivariatelinear relationship for Z � [xT � y]T, which has the ability toincorporate heteroscedastic measurement errors with knownuncertainties (these uncertainties are considered by now to beindependent of the true levels for the noiseless measurands)

Z�k� � Z � A � l�k� � �m�k� (10)

where Z is the (m � 1) � 1 vector of measurements, Z is the(m � 1) � 1 mean vector of x, A is the (m � 1) � a matrix ofmodel coefficients, l is the a � 1 vector of latent variables, and�� m is the (m � 1) � 1 vector of measurement noise. This modelis still incomplete because we need to provide it with theprobability density functions assumed for each random com-ponent

l�k� � iid MNa�0, l�, ��m�k� � id MNm�1�0, m�k��

l�k� and ��m� j� are independent @k, j (11)

where MN stands for multivariate normal distribution, i is thecovariance matrix of the latent variables, m(k) is the covari-ance matrix of the measurement noise at time k, given bym(k) � diag[�� m

2 (k)]. Thus, for estimating the covariance ma-trix, we assume a multivariate behavior for Z that can beadequately described by propagation of the underlying varia-tion of p latent variables, plus added noise in the full variablespace. This model and the calculation details associated withthe estimation of the unknown parameters are fully describedelsewhere.33 It can be shown that the probability density func-tion of Z, under the conditions stated above, is a multivariatenormal distribution with the following form

Z�k� � id MNm�1�Z, Z�k�� (12)

where

Z�k� � l � m�k� l � AlAT (13)

With the raw measurements (Z) and the associated uncer-tainties [from which we can calculate m(k)], it is possible toestimate Z and l by maximizing the likelihood function.Matrix Z(k) � l � m(k) is the estimate of the covariancematrix for noisy measurements at time step (k), but becausePLS is based on S and s, it requires single estimates for thepopulation parameters (and not one per time step k). Thus, wemaintain the estimate of the covariance of noiseless data, l,

Table 4. SIMPLS Algorithm43

S � XTXs � XTyfor a�1, . . . ,A

r � 1st left singular vector of sr � r/(rTSr)1/2

R � [R,r]P � [P,Sr]s � [I � P(PTP)�1P•

T]sendT � XR


but average out the heteroscedastic square uncertainties, tocome up with a single term, �m, leading to

Z � l � �m (14)

With the estimate of Z, we can finally calculate the esti-mates for S and s: S � Z(1 : m, 1 : m), s � Z(1 : m, m � 1).The algorithm that consists of implementing the SIMPLS al-gorithm with these matrices as inputs will be referred to here asunc-PLS3. In the present context, we use the full measurementspace to estimate Z (a � m) because we want the relevantsubspace for prediction to be defined by the PLS algorithmitself, and not by a previous estimation step. In the predictionphase, when new values for the predictors become availablealong with their measurement uncertainties, and the goal is topredict what the value of the response variable would be, weadd an additional calculation step before applying the unc-PLS3 regression vector (calculated in the estimation phase).This step consists of projecting the new multivariate observa-tion in the full X-space into the subspace that is relevant forpredictions (that is, the one spanned by the columns of theweighting matrix, W in PLS or R in SIMPLS). The availabilityof the associated uncertainties leads to a generally nonorthogo-nal projection methodology that consists of estimating theprojected points using a maximum likelihood approach, just asthe one adopted in MLPCA.21 In the present study, we alsotested an algorithm that implements the same nonorthogonalprojection operation, but using the weighting matrix providedby PLS (a hybrid version of the classic PLS because it containsa projection step that incorporates measurement uncertainty),herein referred to as unc-PLS4. For the sake of completeness,we also introduced another methodology, based on the sameweighting matrix as unc-PLS3, but that bypasses the non-orthogonal projection step, designated as unc-PLS5.

Measurement Uncertainties in ProcessOptimization

In the previous section we have addressed the explicit in-corporation of measurement uncertainty in statistical modeldevelopment. We now move to a different working scenario,where an appropriate model is already available and our goal isto use it for process optimization, but also taking into accountinformation regarding measurement and actuation uncertain-ties. In particular, we address the problem where one wants tooptimize an objective function (such as maximizing some profitmetric or minimizing a cost function), for a given measurementof the vector of load variables (L), by manipulating another setof variables (M). However, because of the presence of uncer-tainties, the following issues do arise:

● Measured quantities (that is, the loads L and the outputs Y)are affected by measurement noise, with statistical character-istics defined by their associated uncertainty

L � L � �L Y � Y � �Y (15)

with quantities marked with a tilde accent (�) being the valuesactually available, whereas L and Y are the corresponding true,but unknown, values for these quantities (Figure 1).

● Similarly, the set-point that we specify for the manipu-

lated variables (Z) does not correspond to the exact true valueof the manipulation action over the process. In fact, because ofactuation noise, there is also here another uncertainty source tobe taken into account.

Considering that we want to drive the process in such a wayas to minimize some relevant cost function, ��, we proposethe following formulation that incorporates measurement andactuation uncertainties, in the calculation of the adequate val-ues for the manipulated variables to be specified externally,when a given measurement for the load is acquired (L). Asoften happens in the formulation of optimization problemsunder uncertainty, the objective function constitutes an ex-pected value for the performance metric, taken over the spaceof uncertain parameters:

Formulation I

minZ

E��L, Z, Y��

s.t. g�Y, L, Z� � 0

L � L � �L

Z � Z � �Z

Y � Y � �Y (16)

where E�{ � } is the expectation operator,

E��

�� j�� d� (17)

� � [�LT, �Z

T, �YT]T and j(�) provide the joint probability density

function for the uncertain quantities �. The available model isrepresented by g(Y, L, Z) � 0, and we will assume here that theuncertainty associated with its parameters is negligible (if not,such uncertainties can also be incorporated into our problemformulation34).

In Formulation I, we assume that the relevant quantities forevaluation of the performance metric are the values of L and Zthat really affect the process, as well as the measured value ofthe output. We point out that these assumptions do not neces-sarily hold in every situation. For instance, sometimes theperformance metric should be calculated with the “true” valueof the output, Y, instead of Y] (Formulation II, see below), as isthe case when output measurements become available withmuch less uncertainty in a subsequent stage (such as from

Figure 1. Schematic representation of measured quan-tities [as seen by an external operator andmarked with a tilde (�)] and the quantities thatare actually involved in the underlying process.


off-line laboratory tests). Other times, only measured valuesshould be used because no better measurements or reconcilia-tion procedures can be adopted. The correct formulation istherefore case dependent, and should be tailored to each par-ticular situation.

Formulation II

minZ

E��L, Z, Y��

s.t. g�Y, L, Z� � 0

L � L � �L

Z � Z � �Z (18)

In our case studies section we will also present the resultsobtained for the situation where uncertainties are not at alltaken into account, and thus where the manipulated variablevalues are found by solving the following problem

Formulation III

minZ

��L, Z, Y�

s.t. g�Y, L, Z� � 0 (19)

Case Studies

In this section we present the results reached from compar-ative analysis encompassing all the methods mentioned above(PLS, unc-PLS1, unc-PLS2, unc-PLS3, unc-PLS4, unc-PLS5,RR, rMLS, rMLMLS, PCR, MLPCR, MLPCR1, MLPCR2,OLS, MLS, and MLMLS), and illustrate the implementation ofthe approach treated in the third section under a realisticsimulation scenario, using a model estimated from a real paperpulp pilot digester.

Case studies 1 and 2 provide different contexts to set aground for the comparison study among the multivariate linearregression methods. In both of them, a latent variable modelstructure is adopted to generate simulated data, given that thiskind of model structure is quite representative of data collectedfrom many real industrial processes because the number ofinner sources of variability that drives process behavior isusually of a much smaller dimensionality than the number ofmeasured variables.35,36 The latent variable model used has thefollowing form

X � 1n � XT � TP � E

Y � 1n � YT � TQ � F (20)

where X and Y are the m � 1 and k � 1 vectors with thecolumn averages of X and Y; 1n is an n � 1 vector of ones; Xis the n � m matrix of input data; Y is the n � k matrix ofoutput data; T is the n � a matrix of latent variables thatconstitute the inner variability source, structuring both theinput and output data matrices; E and F are n � m and n � kmatrices of random errors; and P and Q are a � m and a � kmatrices of coefficients.

The model used in our simulations consists of five latentvariables (a � 5) that follow a multivariate normal distributionwith zero means and a diagonal covariance (Ia, that is, the

identity matrix of dimension a). The dimension of the inputspace is set equal to 10 and that of the output space equal to 1(m � 10, k � 1). Rows of the P matrix form an a-orthonormalset of vectors with dimension m. The same applies to matrix Q,which consists of an a-orthonormal set of vectors with dimen-sion k.

Each element of matrices E and F of random errors is drawnfrom a normal distribution with zero mean and standard devi-ation given by the uncertainty level associated with that spe-cific variable (column of X or Y) for a particular observation(row). These uncertainties were allowed to vary, and thisvariation is characterized by the heterogeneity level (HLEV),which measures the degree of variation or heterogeneity ofuncertainties from observation to observation: HLEV � 1means a low variation of the noise uncertainty or standarddeviation from observation to observation, whereas HLEV � 2means a highly heteroscedastic behavior for the noise uncer-tainties. More specifically, for variable Xi the uncertaintiesalong the observation index are randomly generated from auniform distribution centered at u� (Xi) (the average uncertaintyfor a given variable), with range given by R(HLEV) �K2(HLEV) � u� (Xi), where K2 � 0.01 (if HLEV � 1; lowheterogeneity level) or K2 � 1 (if HLEV � 2; high heteroge-neity level), that is,

u�Xi�k�� U�u� (Xi) �R(HLEV)

2, u�(Xi) �

R(HLEV)

2 In the present study, u� (Xi) was kept constant at 0.5 times thetheoretical standard deviation calculated for each noiselessvariable.

Case study 1: complete heteroscedastic noise

With the goal of evaluating overall performance of themethods under different uncertainty structures for the measure-ments errors, the following sequence of steps was adopted:

(1) We set the tuning parameters for each method and foreach set of conditions (number of latent dimensions for PLSand PCR methods, and ridge parameter for RR methods).Regarding PLS and PCR methods, we did set a � 5. As forridge methods, we selected our ridge parameter using cross-validation and the generation of a logarithmic grid in the rangeof plausible values (the criterion used in cross-validation isRMSEPW). This procedure is repeated 10 times, and the me-dian of the best values is chosen as the tuning parameter to beused in our simulations. Variables are “auto-scaled” in allmethods, except for OLS, MLS, and MLMLS.

(2) For each scenario of HLEV (1 or 2), two noiseless datasets are generated according to the latent variable model pre-sented above: a training or reference noiseless data set and atest noiseless data set, both with 100 multivariate observations.Furthermore, a random sequence of uncertainties (noise stan-dard deviations) for all the observations belonging to eachvariable is generated according to HLEV.

(3) Zero-mean Gaussian noise, with standard deviationgiven by the uncertainties calculated in (2), is generated andadded to the noiseless training and testing data sets, after whicha model is estimated according to each linear regressionmethod (using the training data set) and its prediction perfor-


mance evaluated (using the test data set). This process of noiseaddition, followed by parameter estimation and prediction, isrepeated 100 times, and the corresponding performance metricssaved for future analysis.

Performance metrics used for prediction assessment are thesquare root of the weighted mean square error of prediction inthe test set (RMSEPW), where the weights are the result ofcombining the predictor and response uncertainties, and themore familiar root mean square error of prediction (RMSEP)

RMSEPW�i� � 1

n �k�1

n [y(k) � y(k)]2

uy(k)2 � [uX(k, :)*2]TB*2 i � 1100

(21)

RMSEP�i� � 1

n �k�1

n

[y(k) � y(k)]2 i � 1100 (22)

where n is the number of observations in the test set.At the end of the simulations, we do have 100 values for the

above metrics available for comparing the performancesachieved by the different methods, under a given noise struc-ture scenario. To take into account both the individual vari-ability of the performance metrics for the different methods, aswell as their mutual correlations, we based the comparisonstrategy in paired t-tests among all the different combinationsof methods. Therefore, for each simulation scenario, pairedt-tests were used to determine whether method A is better thanmethod B (a Win for method A), performs worse (a Loss), orif there is no statistical significant difference between both ofmethods A and B (a Tie), for a given significance level (weused � 0.01). For the sake of simplicity, we will only presenthere the number of wins, losses, and ties that each methodobtained for each simulation scenario.

Figure 2 presents the comparison results for the scenario

HLEV � 1, using RMSEP as performance metric (because thetrends for RMSEP and RMSEPW do not differ significantly,only those for the more familiar RMSEP are presented).

Examining first the performance of the methods belonging tothe same group, we can see the following for this simulationscenario:

● OLS Group. MLS performs worse than OLS andMLMLS shows the best performance among the three meth-ods. In general terms, comparing all the methods where MLSand MLMLS have similar roles (such as unc-PLS1/unc-PLS2,rMLS/rMLMLS, MLPCR1/MLPCR2), the second versionnever resulted in worse results and, as a matter of fact, almostalways significantly improved them.

● RR Group. Both rMLS and rMLMLS conducted to im-proved results with respect to those obtained by RR.

● PCR Group. MLPCR does not improve over PCR predic-tive results, but MLPCR2 leads to an improvement.

● PLS Group. Methods unc-PLS3 and unc-PLS5, both us-ing uncertainty-based estimation of the relevant covariancematrices for PLS, present the best performance. Their similarperformance results can be explained by the fact that, undermild homoscedastic situations and if the variables presentapproximately equal uncertainties associated with them, theorthogonal and nonorthogonal projections almost coincide. Thesame applies for the comparison of PLS and unc-PLS4, bothusing PLS weighting vectors but different projection strategies.Comparing the results obtained for all the methods against eachother, we can see that MLPCR2 is the one that presented thebest overall performance, followed by PCR, MLPCR, unc-PLS3, and unc-PLS5.

Figure 3 summarizes the results obtained for conditionHLEV � 2. A comparison of performances regarding methodswithin the PLS group shows that those methods that estimatethe covariance matrices using uncertainty information (unc-PLS3, unc-PLS5) present better performance then their coun-terparts that use the same projection strategies (unc-PLS4, PLS,respectively). However, looking now to the methods that differ

Figure 2. Results for number of losses, ties, and wins for each method, under the simulation scenario with hetero-geneity level (HLEV) � 1 [using root mean square error of prediction (RMSEP)].


only on the projection methodology, we can see that those thatare based on orthogonal projections achieve better results thatthose based on nonorthogonal maximum likelihood projec-tions. This result is quite interesting and will be further dis-cussed below. In the PCR group we can see that all the methodsperform quite well. As for the remaining groups of methods,the trends mentioned for HLEV � 1 remain roughly valid.MLPCR2 continues to be the method with the best overallperformance, followed by MLPCR and a group of methods thatinclude MLPCR1, PCR, and unc-PLS5.

Case study 2: handling missing data

In this second case study, we analyze the prediction perfor-mance of the several methods when missing data are present(both in model estimation and in prediction), and a very simplestrategy for handling missing data is adopted: mean substitu-tion. For uncertainty-based methods, one also has to specify theassociated uncertainty, and the values we have considered hereare the standard deviations of the respective variables duringnormal operation. Other more sophisticated methodologies formissing data imputation during model estimation are also avail-able for regression methods (especially PLS and PCR37), aswell as methods for handling missing data once we havealready available an estimated model.38 Analogous approachescan also be developed for the uncertainty-based techniques thatrequire only the estimated value and the respective uncertaintyto fill existing blanks. However, the aim of this study is toassess the extent to which one can easily handle missing datain model estimation and prediction (that is, with minimumassumptions regarding missing values and the least modifica-tion over standard procedures), taking advantage of the possi-bility of using uncertainty information. That being the case, wedecided to keep the same replacement strategy among allmethods, so that the real advantage of handling such an addi-tional piece of information, provided by measurement uncer-tainties, can be easily evaluated and compared with the currentalternatives.

Because our focus here is related with the evaluation of themethods regarding prediction when missing data is present, weadopted a simulation structure which is now different from thatof case study 1. For each simulation the following steps arerepeated and the corresponding results saved:

(1) Generate a new latent variable model (matrices Q and P)and noiseless data to be used for model estimation and predic-tion assessment. Also generate measurement uncertainties to beassociated with each nonmissing value, according to the valueof HLEV used in each simulation study.

(2) Generate a new “missing data mask” that removes (onaverage) a chosen percentage of the data matrix [X � Y]. Weused a target percentage of 20%, both for the reference and testdata sets.

(3) Generate and add noise to the noiseless data that werenot removed, according to the measurement uncertainties gen-erated in (1).

(4) Replace missing data with column means for the data setused to estimate the model, and calculate the associated uncer-tainties using the columns standard deviations, for the samedata set.

(5) Estimate models using the data set constructed in (4).(6) For the test data set, do the same operation as in (4).

(using the same values for the input values and uncertainties)and calculate the predicted value for the output variable. Cal-culate overall performance metrics (RMSEPW and RMSEP).

The results obtained with HLEV � 1 are presented in Figure4, where we can see that within the PLS group methodsunc-PLS5 and unc-PLS3 lead to improved predictive perfor-mances, but now with unc-PLS3 presenting better results thatunc-PLS5, that is, the nonorthogonal projection seems to bringsome added value when missing data are present, under ho-moscedastic scenarios. In the PCR group, all MLPCR methodsoutperform the conventional PCR. As for the other groups,results obtained follow the same trends verified when no miss-ing data were present. In global terms, MLPCR2 presents the

Figure 3. Results for number of losses, ties, and wins for each method, under the simulation scenario with HLEV �2 (using RMSEP).


best overall performance, followed by MLPCR1, MLPCR, andunc-PLS3.

By analyzing the results for HLEV � 2 (Figure 5), we canalso see that unc-PLS3 and unc-PLS5 still show the bestpredictive performance within the PLS group, but now withunc-PLS3 presenting lower scores relatively to the previousscenario (HLEV � 1), a result that is consistent with what wasverified in case study 1. In the global comparison, afterMLPCR2 we can find MLPCR1 and MLPCR. Therefore, underthe conditions adopted for this simulation study, we can con-clude that MLPCR methods tend to have the best overallperformance in the presence of missing data.

We point out that when adopting a methodology that inte-grates data uncertainty, one follows the same calculation pro-

cedure adopted for the situation where no data are missing,simply replacing the missing elements with rough estimatesthat will be properly weighted by the algorithms, according totheir associated uncertainties. However, if we do have availablebetter estimates, such as those arising from more sophisticatedimputation techniques, one can also integrate them as well,without any further changes.

Case study 3: process optimization under datauncertainty

This case study illustrates the integration of measurementuncertainties in process optimization decision making. Theproblem we address herein consists of calculating the values

Figure 4. Results for number of losses, ties, and wins for each method, under the simulation scenario with HLEV �1 and 20% of missing data (using RMSEP).

Figure 5. Results for number of losses, ties, and wins for each method, under the simulation scenario with HLEV �2 and 20% of missing data (using RMSEP).


for the manipulated variables to be specified (Z) to minimize acost function, when measurements for the loads become avail-able (L). This particular case study is based on the followingmodel, developed for a batch paper pulp pilot digester39

TY � 55.2 � 0.39 � EA � 324/�EA � log10S� � 92.8

� log10�H�/�EA � log10S� (23)

This model relates pulp total yield (TY) with effective alkali(EA, a measure of the joint concentration of Na2OH and Na2S,the active elements in the cooking liquor), sulfidity (S, thepercentage of Na2S in the cooking liquor), and H factor (H, afunction of the temperature profile across the batch).

We consider the situation where a cost function (L) penalizesdeviations from a target value for TY (52%): the penalty forlower values is attributed to fiber losses, and that for highervalues to deterioration in other pulp properties. Our cost func-tion also considers the cost of S and H (proportional to theirrespective magnitudes). As an example, Figure 6 illustrates theshape of the assumed cost function for S � 20 and H � 1000

L � �100�TYsp

100�

TY

100 �S

4�

H

500d TY � TYsp

752�TYsp

100�

TY

100 2

�S

4�

H

500d TY � TYsp

(24)

In this example, EA is assumed to be a load variable, andthus our optimization goal consists of calculating the S and Hvalues that minimize expected cost in the presence of uncer-tainties for both measurements and process actuations. Formu-lations I, II, and III hold for this example, with L � EA, Z �[S H], and Y � TY (Table 5).

We further assumed that the vector of uncertain quantities,� � [�EA, �S, �H, �TY]T, follows a multivariate normal distri-bution with zero mean and diagonal covariance given by

� � diag��22 22 502 42�� (25)

where diag stands for the operator that converts a vector into adiagonal matrix with its elements along the main diagonal.

To illustrate the implementation of the formulations abovereferred, let us consider that the observed value for EA is 15

�EA). Table 6 summarizes the results obtained for the manip-ulated variables (S and H) and the average cost obtained withthe objective function assumed under formulations I and II,with a third degree specialized cubature being used for estima-tion of expected values.40

From Table 6 we can see that under the simulation condi-tions considered here, and assuming that the relevant objectivefunction is the one associated with formulation I, the optimalsolution obtained when one disregards measurement and actu-ation uncertainties (formulation III) corresponds to an averagecost increased by 136%. If the relevant objective function werethe one corresponding to problem formulation II, the averagecost increase would be 51%. It should also be noticed that thelocation of the optimal solution in the (S, H) decision space,found if one ignores uncertainties, is quite distant from the trueone.

The cost associated with the nonconsideration of these typesof uncertainties decreases when their magnitude becomessmaller. Figure 7 presents the results obtained for three alter-native problem formulations, when the covariance matrix foruncertain quantities is multiplied by a monotonically decreas-

Figure 6. Cost function for deviations of total yield (TY)from its target value (52%), for S � 20 and H �1000.

Table 5. Optimization Formulations I, II, and III as Applied to Case Study 3

Formulation I Formulation II Formulation III

minS,H

E��EA, S, H, TY�� minS,H

E��EA, S, H, TY�� minS,H

��EA, S, H, TY�

s.t. g(TY, EA, S, H) � 0 s.t. g(TY, EA, S, H) � 0 s.t. g�TY, EA, S, H� � 0

EA � EA � �EA EA � EA � �EA

S � S � �S S � S � �S

H � H � �H H � H � �H

TY � TY � �TY

Table 6. Solutions Obtained under Formulations I, II, andIII, and Their Associated Average Costs

Solutions

Average Cost ($)

Formulation I Formulation II

I S� 7.16 10.80 5.93H�1602.0

II S� 7.83 11.16 5.40H�1184.2

III S� 5.38 25.46 8.17H�1274.6


ing shrinkage factor, 0.9i. As expected, the differences arisingfrom the solutions associated with such three optimizationformulations tend to vanish when measurement and actuationuncertainties decrease. Furthermore, the average cost also de-creases because of the improved quality of information ob-tained from measurement devices and the better performanceof final control elements, as one moves across the severalsimulation scenarios considered here.

Discussion

Results presented in the previous section highlight not onlythe potential of using all the information that is available (dataand associated uncertainties), but also the difficulty that such atask may encompass, with respect to model estimation. In fact,we have come across with some unexpected results and rele-vant issues have been identified and merit being discussed here.

First of all, we stress the fact that, even though simulationresults are strictly valid within the conditions established, theycan provide useful guidelines for real processes that presentstructural similarities with them. The fact that classical meth-ods do not make explicit use of uncertainty informationmay not be very relevant if it represents just a small part ofthe global variability exhibited by variables. Therefore, uncer-tainty-based methods presented here are expected to bringpotentially more added value only under contexts where un-certainty is quite high (noisy environments) or experimentshave large variations. In other words, these methods shouldcomplement their classical counterparts, depending on thenoise characteristics that prevail in measured data.

Still regarding model estimation, we have found some con-vergence problems in MLMLS, something that is not unusualin approaches based on numerical optimization of a nonlinearobjective function. However, problems in MLPCR2 arisingfrom the nonconvergence of MLMLS are usually rare. Fromthe experience that we have gathered so far, no limitations werefound regarding the implementation of MLPCR2 in the anal-ysis of real industrial data. The poor performance of MLSunder the scenarios considered here, where predictors are

strongly correlated, may indicate that the inversion operationundertaken at each iteration is interfering with its performance(the matrix to be inverted in this method becomes quite ill-conditioned under collinear situations of the predictors). Re-sults obtained for the ridge regularization of MLS (rMLS)show an effective stabilization of this operation. As for PLSmethods, the extensive solution of small optimization problemscan make unc-PLS1 and unc-PLS2 more prone to numericalconvergence problems than the original PLS method, some-thing that does not occur with the remaining uncertainty-basedPLS methods (unc-PLS3, unc-PLS4, and unc-PLS5), given thatthey are based on the estimation of covariance matrices andprojection operations. Quite interesting is the fact that, whencomparing under heteroscedastic situations (Figure 3) PLSmethods that adopt the same estimation procedure for thecovariance matrices but differ in the projection phase (as hap-pens with pairs PLS/unc-PLS4, unc-PLS3/unc-PLS5), one cansee that the use of uncertainty-based maximum-likelihood non-orthogonal projections seems to be detrimental for predictionwith respect to orthogonal projections. In fact, a separate sim-ulation study showed evidence toward a reduced variance ofthe orthogonal projection scores, when compared to the oneexhibited by maximum likelihood projection scores. Appar-ently, for heteroscedastic scenarios, oscillations in the non-orthogonal projection line may also bring some added variabil-ity to the scores, other than the one strictly arising fromvariability attributed to noise sources. This increased disper-sion in the reduced space of the scores, usually the one relevantfor prediction purposes, can increase prediction uncertaintyarising from poorly estimated models, something that is in linewith the results presented in Figure 3. Finally, there are alsosome approximations considered in the methods that may in-terfere with their predictive performance and should be con-sidered in future developments. That is, methods unc-PLS1 andunc-PLS2 neglect uncertainties in the load vectors andMLPCR1/MLPCR2 do assume the score uncertainties to beindependent.

We emphasize that, although we have focused here onsteady-state applications, our approaches can also be usedunder the context of dynamic models, that is, through theconsideration of lagged variables41-44 (the PLS methods basedon the uncertainty-based estimation of covariance matrices,however, do need some modifications to cope with the noisecorrelations appearing with the use of lagged variables). Forsuch situations, one may also consider uncertainty descriptionsconnected with robust control methodologies, such as H-infin-ity approaches.

Conclusions

In this paper we address the importance of specifying mea-surement uncertainties and how this information can be used intwo distinct tasks: model estimation and process optimization.With respect to model estimation, under the conditions studiedmethod MLPCR2 presented the best overall predictive perfor-mance. In general, those methods based on MLMLS presentimprovements over their counterparts based on MLS. We havealso illustrated the potential advantage of using measurementand actuation uncertainties in process optimization problemformulations and solutions. Our study points out the relevanceof not neglecting measurement/manipulation uncertainties

Figure 7. Behavior of average cost (formulation I), cor-responding to solutions for the three alterna-tive problem formulations, using ¥¥� � 0.9i.


when addressing both on-line and off-line process optimiza-tion.

Future work will address the application of uncertainty-based methods in real industrial contexts, using the guidelinesextracted from the results achieved in our comparative studypresented herein, regarding the most adequate methods to beadopted for a certain noise/data structure scenario.

AcknowledgmentsThe authors gratefully acknowledge FCT (Fundacao para a Ciencia e

Tecnologia, Portugal) for financial support through research projectPOCTI/EQU/47638/2002.

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Manuscript received May 25, 2004, and revision received Mar. 7, 2005.


Design and Testing of a Control System forReverse-Flow Catalytic Afterburners

Miguel A. G. Hevia, Salvador Ordonez, and Fernando V. DıezDepartamento de Ingenierıa Quımica y Tecnologıa del Medio Ambiente, Universidad de Oviedo,

Julian Claverıa 8, 33006 Oviedo, Spain

Davide Fissore and Antonello A. BarresiDipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino,

Corso Duca degli Abruzzi 24, 10129 Torino, Italy


The design and testing of a control system for reverse-flow catalytic afterburners based on atwo-variable bifurcation map is focused on. The combustion of lean methane–air mixtures isconsidered as the test reaction and a bench-scale apparatus, with a special temperature-controlsystem based on dynamic compensation of the thermal losses to achieve adiabatic operation, isused for validation purposes. The aim of the control system is to avoid both catalyst overheatingand reaction extinction when the adiabatic temperature increases and the flow rate of the feedchanges. Stability maps of the reactor are obtained by means of numerical simulations, showingthe values of the operating parameters (switching time), which allows fulfillment of theoperating constraints (catalyst maximum temperature and methane conversion) when theinlet concentration and flow rate change. This system was realized and tested experimen-tally, mainly for inlet concentration changes, proving to be effective in all cases investi-gated. © 2005 American Institute of Chemical Engineers AIChE J, 51: 3020–3027, 2005Keywords: reverse-flow reactors, autothermal combustion, reaction extinction, adiabaticreactor

Introduction

The reverse-flow catalytic reactor (RFR) is a device in whichthe feed flow direction is periodically reversed, thus giving rise tounsteady-state operation. When the combustion of cold and leanvolatile organic compound (VOC) mixtures in air is carried out,the reversal of the flow direction traps the heat of reaction insidethe bed, whose ending parts act as regenerative heat exchangers,thus allowing autothermal combustion and eliminating the needfor auxiliary fuel to sustain the reaction. Extensive investigationsabout the RFR, including both numerical simulations and exper-

imental analysis, were performed in the past 30 years and werereviewed, for example, by Matros and Bunimovich.1

In addition to the intrinsically dynamic behavior of the RFR,one must deal with unexpected external perturbations (in the feedconcentration, composition, temperature, and flow rate), whichmay lead to reactor extinction (and thus later to reactant emis-sions) or catalyst overheating (and thus subsequent deactivation).To avoid these problems it is necessary to implement someclosed-loop control strategy. Few papers have appeared in theliterature concerning this topic. Van de Beld and Westerterp2

discussed different possible solutions to avoid reactant emissionsin a RFR in the case of temporary reduction of the concentrationof the feed, but none of these proved to be effective:

(1) Increasing of the feed temperature requires a high en-ergy input and, for control purposes, it is too slow because theentire bed has to be heated by the hotter feed.

Correspondence concerning this article should be addressed to A. A. Barresi [email protected].



(2) Adding combustibles to the feed seems the most logicalmethod to use because it is simple to implement, although areactive combustible, when added to the feed, will already react inthe inlet part of the catalyst bed and the increase of the maximumtemperature in the center of the bed is initially small. Moreover,the ignition temperatures of the original pollutants and the sup-porting fuel must be almost equal; otherwise, the reactor mayestablish the temperature profile corresponding to the componentwith the lower ignition temperature and will not be able to oxidizethe component with the higher ignition temperature.3-5

(3) An electrical heating device in the center of the reactormay increase the temperature in the hottest zone of the bed,even very rapidly, although the heat must be transferred fromthe heating device to the catalyst, thus introducing an addi-tional heat transport resistance. Experimental results of Cunillet al.6 show that this device is effective, given that the maxi-mum temperature in the reactor is now a function of theelectrical power and almost independent of the mixture com-ponents and composition; thus a mixture of contaminants withvery different ignition temperatures can be burned completely,provided that the electrical power is sufficiently high. If aheating support is required for an extended period of time theelectrical heating device may become too expensive; in thiscase, adding combustibles in the central part of the reactor is abetter choice, even if lack of good distribution of the combus-tible over the entire cross-sectional area may cause difficulties.

(4) Permanent addition of a component difficult to oxidize,fed in a concentration sufficient to ensure the ignition, may bean alternative solution: the temperature level in the reactor willbe high and all other less refractive contaminants will certainlybe converted. The main drawback is that this is a ratherenergy-consuming solution; furthermore, an increase in theinlet concentration may be difficult to handle because thecatalyst may become overheated.

(5) Supply of a hot gas in the center of the reactor is aninefficient way to increase the temperature because the heatcapacity of the gas is low compared to that of the bed.Finally, Barresi and Vanni7 investigated the possibility ofavoiding reaction extinction by acting on the switching time.They considered a one-point control strategy, with a tempera-ture measurement located at either end of the active portion ofthe bed: flow direction is changed when temperature at thecontroller located close to the inlet drops below a certain setpoint. Their simulations provided evidence that at a higher setpoint the maximum temperature is also increased, although—atleast for large portions of inert material—conversion cannot bearbitrarily improved by this strategy; this is explained by theshorter periods necessary for higher set points, resulting in astronger washout effect. If the inlet concentration is lower thanthe minimum adiabatic temperature rise that is required toallow for autothermal operation, the change of the switchingtime may not be sufficient and the system evolves toward theextinction following a Zeno trajectory.8

Besides the problem of reaction extinction, the temperaturein the reactor should not exceed the maximum allowable cat-alyst temperature, which may occur when the concentration ofcontaminants becomes too high during a significant period oftime. A short peak of high concentration will not be a problembecause the heat capacity of the system is high and it will taketime before the maximum temperature exceeds the limit. Fur-thermore, the RFR exhibits some self-control with respect to

the inlet concentration. For a low concentration, the reactiontakes place in the central part of the reactor, so that part of thecatalyst is used to store heat and to preheat the cold feed. If theinlet concentration is increased, the temperature and concen-tration profiles will move in the direction of the inlet and outletof the reactor and a temperature plateau will develop. Forhigher inlet concentrations full conversion is already obtainedin the first layers of the catalyst bed and the length of the “heatexchanger” equals that of the inert sections and remains con-stant. From this point, the maximum temperature will increasevery rapidly with the inlet concentration. Different technicalsolutions have been proposed in the literature9:

(1) Dilution of the feed with additional air has the main draw-back that the total flow rate increases significantly, thus requiringadditional energy costs to overcome the higher pressure drop;moreover, there is the problem of obtaining reliable and inexpen-sive on-line measurements of the inlet concentration.2

(2) Increase of the switching time may lower the maximumtemperature, particularly when high inert fraction is used.3,6

(3) Heat recovery by internal heat exchange was discussedby Grozev and Sapundzhiev10 and by Nieken et al.,3,4 whoindicated that the switching period has to be increased, so thatthe reaction front approaches the center of the reactor before anoticeable influence on the maximum temperature can be no-ticed. However, no remarkable reduction of the maximumtemperature is possible through intermediate cooling, irrespec-tive of the switching period used, although the energy recoveryincreases. Sapundzhiev et al.11 used an internal heat exchangeto preheat the cold feed. External cooling restricts the region ofoperating variables that allows for autothermal behavior ascombustion can now be quenched if the coolant removes moreheat than is generated.12

(4) Cold gas injection into the middle of the packed bedmay have an effect similar to that of intermediate coolingthrough heat exchange, leading to a significant decrease of thetemperature in the middle of the reactor, whereas the maximumtemperature is more or less unaffected.4

(5) Hot gas withdrawal from the center of the reactor allowsfor almost complete energy recovery, but again the maximumtemperature cannot be lowered sufficiently: sidestream with-drawal just shifts the reaction zone toward the center of thereactor.4

(6) Structured fixed bed. If the packed bed is composed ofportions with high effective conductivity and of portions of lowconductivity then it depends on the position of the temperaturefronts whether the maximum temperature is lowered or in-creased. If the fronts are always in the portion with high axialconductivity, the maximum temperature will be low and theefficiency of the heat recovery will be weak; if the fronts stayin the portion with low conductivity, the opposite is true.Because the position of the fronts can be shifted by the amountof the hot gas withdrawn, an interesting possibility exists tooperate the reactor in either of the two regimes. To exploit thispossibility a structured fixed bed is needed, where the packedbed is composed of three portions: an inner portion with higheffective conductivity support and two outer portions with lowconductivity support.4

Budman et al.13 considered the application of a conventionalPID controller with anti-windup: exit concentration, the con-trolled variable, is used to infer the maximum temperature(because thermocouple locations in the bed are fixed, it may be


extremely difficult to evaluate the maximum temperature,given that the temperature profiles move during the operationof the reaction). The same authors also considered a feedfor-ward regulation of the exit concentration through knowledge ofthe inlet concentration. The idea behind this strategy is that ifthe inlet concentration is known, it is possible to evaluate theoptimal combination of switching time and cooling rate toobtain full conversion and safe operation in the reactor. In thisregard, a number of simulations have been performed to iden-tify the parameters region for safe operation.

In this work a control system was used to control thereverse-flow operation in a catalytic afterburner: the switchingtime was varied to avoid both catalyst overheating and reactionextinction. By means of numerical simulations, a stability mapof the reactor is obtained, showing the values of the switchingtime that allow fulfillment of the operating constraints (catalystmaximum temperature and methane conversion) when the inletconcentration and flow rate change.

To apply this control strategy we need to know both the inletflow rate (which is easy to do and inexpensive) and the inletconcentration. Because on-line measurements may be difficultand expensive, a soft sensor based on high-gain techniques canbe used to estimate the inlet concentration from some temper-ature measurements.14

The article is structured as follows: in Section 2 the detailedmodel used in the control system is described. In Section 3 theexperimental apparatus is introduced and the mathematical modelvalidated. In Section 4 the control algorithm is introduced andvalidated by means of simulations and of experiments.

The Model

A one-dimensional heterogeneous model was used to model theRFR. Pressure loss inside the system was neglected and plug flow,with dispersive transport of mass and energy, was assumed for thegas phase; the ideal gas law was used. The transient term wastaken into account in the gas-phase equations as well as in theenergy equation for the solid phase, whereas the solid catalyticsurface was considered in pseudosteady-state condition. The ef-fect of the intraparticle mass transport was included in the modelby means of the effectiveness factor. The effect of the reactor wallon the thermal balance of the system is taken into account bymeans of a further energy balance equation for the wall. Thus, thedynamics of the adiabatic process can be described by the follow-ing set of partial differential-algebraic equations (PDAE):

Continuity Equation for the Gas Phase

�cG

�t�

�

� xcGv � �

i�1

nr kG,iav

�� yS,i � yG,i� (1)

Continuity Equation for Component j in the Gas Phase

� yG, j

�t� Deff

�2yG, j

� x2 � v� yG, j

� x�

kG, jav

cG�� yS, j � yG, j�

� yG, j �i�1

nr kG,iav

cG�� yS,i � yG,i� (2)

with j � 1 . . . (nr � 1).

Energy Balance for the Gas Phase

�TG

�t�

keff

�Gcp,G

�2TG

� x2 � v�TG

� x�

hav�TS � TG�

�Gcp,G�

�4hi

�Gcp,GDR,i�TG � TW� (3)

Mass Balance for the Solid Phase

kG, jav� yS, j � yG, j� � ��S�1 � �� k�1

NR

�k�j,kR�k, j (4)

with j � 1 . . . nr.Energy Balance for the Solid Phase

�TS

�t�

S

�Scp,S

�2TS

� x2 �hav

�Scp,S�1 � ��TS � TG�

�1

cp,S�i�1

nr � �k�1

NR

�k�i,kR�k� ��Hf,i� (5)

Energy Balance for the Reactor Wall

�TW

�t�

W

cp,W�W

�2TW

� x2 �4

cp,W�W�DR,e2 � DR,i

2 �DR,ihi�TG � TW�

(6)

Adiabatic operation is considered because the particulardevice adopted guarantees pseudoadiabatic behavior, as will beshown in the following. For the catalytic part of the reactor, afirst-order rate equation was considered in the mass balance:

R�k, j � kryS, j � kexp��Ea/RTS�yS, j (7)

k � k��S�1 � �� RTG (8)

whereas for the inert sections the reaction rate was set equal tozero. Similarly, the solid physical properties—density, specificheat, and thermal conductivity—were set equal to the valueseither of the catalyst or of the inert sections, depending on theaxial position in the reactor. If the physical and transportproperties of the catalyst and of the inert sections are different,adequate boundary conditions (that is, identity in the heat andmass fluxes) must be specified at the boundary surface.

Conventional Danckwerts boundary conditions were as-sumed in x � 0 and x � L. Initially, the gas-phase temperaturewas considered constant along the reactor and equal to the inletvalue, and the solid temperature was considered constant andequal to the preheating value.

Transport and dispersion parameters were evaluated accord-ing to previous works on the same subject.15 Further details canbe found in a previous publication.16

Table 1 shows the values of the main parameters and oper-ating conditions used in the simulation of the RFR. The influ-


ence of the temperature and composition on the density and onthe specific heat of the gas was taken into account.

The domain of the spatial variable x was discretized on a gridof equally spaced points; 101 points are enough to ensure agrid-independent solution. As a consequence, the PDAE sys-tem (Eqs. 1–6) was transformed into a differential algebraicequations (DAE) problem. The integration in time of the dif-ferential part of the system was performed by implementing theFortran routine LSODES, from the package ODEPACK.17

Both relative and absolute tolerance were set to the square rootof the working machine precision.

After a transient period, the solution of the system evolvestoward a pseudosteady state (PSS), given that the behavior ofthe reactor (temperature and concentration profiles) is the samewithin every cycle.

The Bench-Scale Rig and Model Validation

The design of the experimental reactor to carry out the modelvalidation (and subsequent test of the control system) wasmainly aimed to reproduce as fairly as possible the behavior oflarge-scale industrial reactors. The major drawback foundwhen working on the fulfillment of this goal, apart from theinfluence of the wall thermal inertia that is taken into accountin our model, is the difficulty in obtaining operation regimesclose to adiabaticity with small-sized reactors. This is becausethe higher ratio of external surface to volume, which is char-acteristic of small reactors (low diameters), causes the heatlosses to be relatively more important with regard to the overallthermal balance of the system. Thus, whereas in large-indus-trial reactors adiabatic behavior can be assumed, it is notpossible to do the same with small lab-scale or even pilot-scalereactors and it is necessary to use heat-transfer coefficients totake into account the heat losses.18,19

In this work, this problem is solved using a special devicewith its own control system, which acts over the temperatureoutside the reactor. Such a control system is able to cause suchouter temperature to dynamically follow the axial inner tem-perature in every axial position, thus eliminating the drivingforce to the heat transmission in the radial direction, andtherefore the heat losses. Because axial temperature profiles arenot uniform and change with time as a consequence of un-steady-state operation, the heating system was divided intoseven sections or band heaters, each of them independentlycontrolled by different PID controllers. The central band heateris 0.10 m long and the others are 0.065 m long: this was thebest compromise between efficiency and complexity of the

device. To follow the dynamic of the temperature in the reac-tor, cooling air has to be supplied to cool down reactor sur-roundings when the bed temperature decreases, thus avoidingfurther heating arising from the thermal inertia of the bandheater. This system is currently undergoing a patent applica-tion20 and has been described in detail in a previous paper.16

The reactor is a 316 stainless steel tube with 0.496 m effective(filled with packing) length, an inner diameter of 5.17 10�2 m,and a wall reactor thickness of 1.15 10�3 m. The packingincludes catalyst and inert material. The latter consists of twosections of glass spheres (4 10�3 m diameter) at both ends ofthe bed, each one taking up 24.8% of the effective length. Acommercial catalyst was used composed of a mixture of metaloxides supported on -alumina, forming spheres with an averagediameter of 4 10�3 m. The kinetics of methane combustionover this catalyst was studied in an isothermal fixed-bed reactor,21

from which it was determined that a pseudo-first-order law accu-rately fitted the results. Analysis of the gas streams was carried outby means of an HP 6890 GC equipped with an HP-5 capillarycolumn and flame ionization detector (FID).

Several experiments were carried out, changing the values ofthe main operating parameters, such as switching time, inletconcentration, and surface gas velocity to test the adequacy ofthe model to simulate the behavior of the reactor. Figure 1shows an example of the solid temperature profiles obtained forvarious values of the operating parameters—switching time,inlet concentration, and surface gas velocity—when the PSShas been reached (the temperature profiles at the middle of asemicycle are shown). The experimental values are comparedto those obtained by simulation of the adiabatic reactor, alsoproving that the dynamic compensation of the heat losses iseffective in achieving adiabatic operation and avoiding over-compensation. The agreement between the experimental mea-sured values of the temperature and the simulated values isfairly good. It is important to highlight that no fitting parame-ters were used in the simulation of the reactor so that the modelcan be considered fully predictive.

The Control Algorithm

By means of numerical simulation a stability map of the reactorcan be obtained that delimits regions where maximum tempera-ture (or conversion) is higher or lower than a certain value. Thefull map is a function of three parameters: the switching time(which will be the manipulated variable) and the inlet concentra-tion and surface velocity (which are the disturbances). Figure 2(top graph) shows a section of this map, obtained with a value ofthe surface velocity of the gas equal to 0.14 m s�1, whereas Figure2 (bottom graph) shows a section of the map, obtained with avalue of the inlet concentration equal to 4000 ppmV. Because ofprocess constraints on outlet reactant conversion and maximumtemperature on the solid, a well-defined operating region is ob-tained as a function of the three parameters: the disturbances andthe manipulated variable. In both sections of the stability mapshown in Figure 2 the region corresponding to methane conver-sion � 99.95% and maximum solid temperatures � 650°C isevidenced. In the definition of the operating region it is alsopossible to take into account modeling errors by defining a per-centage uncertainty over these curves.

Until the point corresponding to the operating conditions(inlet concentration, inlet surface velocity, and switching pe-

Table 1. Main Operating Parameters Considered in theSimulations of the RFR

Preheating temperature 650 KPellet diameter 2–4 mmCatalyst density, �S 2170 kg m�3

Catalyst specific heat, cp,S 848.1 J kg�1 K�1

Catalyst porosity 0.519Catalyst tortuosity 2Catalyst thermal conductivity,

eff 0.472 W m�1 K�1

Bed void fraction, � 0.36Frequency factor, k� 1.0886 105 mol kg�1 s�1 Pa�1

Activation energy, Eatt 1.12504 105 J mol�1

Inlet gas temperature 298 K


riod) is in the optimal region no control action is taken; if theinlet concentration increases too much, as well as if the inletsurface velocity increases too much, the control system has twopossibilities: it may increase the switching time (because themaximum temperature decreases slightly when the switchingtime is increased) or, if this is not sufficient, it may dilute thefeed to remain in the optimal working region. If the inletconcentration decreases too much, as well as if the inlet surfacevelocity decreases, a control action is taken just when the outletconversion starts decreasing (because of the thermal inertia ofthe system, the reactor is able to sustain periods of low inletconcentration, with no decrease in the outlet conversion): eitherthe switching time is reduced, if this can increase the conver-sion, or auxiliary fuel is added when the inlet concentration, orthe inlet surface velocity, is lower than the minimum value thatallows adiabatic operation.

In the example considered in Figure 3 (top graph) a reduc-tion of the switching time is sufficient to avoid any conversiondecrease (bottom graph) after the reduction in the methaneconcentration of the feed. The subsequent increase in the inlet

Figure 1. Simulated (lines) axial temperature profilesand actual measurements (symbols) in themiddle of a cycle, when the PSS was reached,for various values of the switching time (graphA, uG,0 � 0.143 m s�1, yCH4,0 � 3500 ppmV), gasvelocity (graph B, yCH4,0 � 3500 ppmV, tc �600 s), and inlet methane concentration (graphC, uG,0 � 0.143 m s�1, tc � 600 s).

Figure 2. Example of stability map of a RFR (top graph:uG,0 � 0.14 m s�1; bottom graph: yCH4,0 � 4000ppmV).The solid line separates the region where extinction occursfrom the region where stable operation can be obtained;dotted lines separate the regions where conversion is larger(top part) or lower than the specified value; dashed linesseparate the regions where maximum solid temperature ishigher (top part) or lower than the specified value.


concentration does not require any control action as the systemremains in the optimal working region. In the example consid-ered in Figure 4 (top graph) the switching time in increased toface against the catalyst overheating resulting from the increaseof the surface velocity (bottom graph). It is important to stressthat this control system based on the model of the process isable to avoid any control action when this is useless and it isable to fulfill any constraints on process operation, being suf-ficient to indicate them in the working map.

The performance of the control system was also experi-mentally tested. The requirements considered were 1 ppmVfor the semicycle averaged outlet concentration, and 650 °Cfor the maximum temperature, which was experimentallydemonstrated to be the maximum temperature that the cat-alyst can endure without deactivation. The changes of theinlet concentrations, the times at which they were intro-

duced, and responses of the control system are included inTable 2. The negative perturbations of the inlet concentra-tion endanger the stability of the reactor and can lead to anincrease of the outlet concentration and, eventually, to theextinction. By contrast, the risk of positive perturbations is

Figure 4. Bottom graph: maximum solid temperatureand outlet methane conversion in a RFR whenthe inlet surface velocity is varied according tothe top graph.Variation of the switching time is considered to maintain thesystem in the optimal region. The behavior with (solid line)and without (dashed line) control action is compared. Distur-bance and control action are evidenced on the stability map(top graph).

Table 2. Perturbations, Time at Which They WereIntroduced, and Responses of the Control System

Perturbation Perturbation

yG,0 106 initial 3500 3500yG,0 106 final 3050 3950t, perturbation* 9900s 4400stc initial 990s 120stc final 150s 990s

*Time elapsed from the beginning of the experiment to the perturbation.

Figure 3. Bottom graph: maximum solid temperatureand outlet methane conversion in a RFR whenthe inlet concentration is varied according tothe top graph.Variation of the switching time is considered to maintain thesystem in the optimal region. The behavior with (solid line)and without (dashed line) control action is compared. Distur-bance and control action are evidenced on the stability map(top graph).


the overheating. This is why the interesting variable tofollow in the former case is the outlet concentration,whereas in the latter one is the maximum temperature.

Four experimental runs, with and without applying the con-trol system for a negative and a positive perturbation, werecarried out, and the control system performed as theoreticallyexpected for both types of disturbances in all cases. Figure 5shows the concordance with the corresponding simulated re-sults. Moreover, in this figure it can be observed that the outletconcentration in the uncontrolled response increases with time,whereas in the controlled one tends to zero. In this way, thediminution of the switching time is in practice demonstrated tobe an effective measure to avoid the outlet concentration in-crease that takes place as a consequence of a negative pertur-bation.

The results for a positive perturbation are shown in Figure 6,pointing out that, although instant maxima appear in the con-trolled response in the first minutes after the perturbation, thecontrol system eventually dampens the temperature increase,otherwise occurring in the uncontrolled response.

Conclusions

A control system based on a detailed model was designed fora catalytic afterburner. A detailed one-dimensional model wasused to simulate the operation in a bench-scale reactor with aninnovative system for compensation of thermal losses. Thecontrol strategy, based on the appropriate choice of the switch-ing time according to the operating conditions (inlet gas ve-locity, concentration), was demonstrated to be effective inpreventing both catalyst overheating and reaction extinction(and thus reactant emissions), avoiding any control actionwhen this is useless, even if the inlet flow rate and concentra-tion changes, but remaining in the optimal working region.When acting on the switching time is not sufficient, maximumtemperature is decreased by dilution of the feed, which hasproved to be the simplest and more effective control action toachieve the desired result and unitary conversion is guaranteedby injection of auxiliary fuel. The pumping costs associatedwith the air dilution and those attributed to the auxiliary fuel,which also generates additional carbon dioxide emissions, arereduced because this measure will be adopted just when largeperturbations occur.

To apply this control strategy we need to know both the inletflow rate (which is easy to do and inexpensive) and the inletconcentration. Because on-line measurements may be difficult

Figure 5. Experimental test of the performance of thecontrol system for a negative inlet concentra-tion perturbation (see Table 2).Comparison between experimental value of the outlet meth-ane concentration (symbols) and simulated values (lines)when no control action is undertaken (top graph) and whenthe control system is working (bottom graph).

Figure 6. Comparison between the simulated (topgraph) and the experimental values (bottomgraph) of the maximum temperature (amongthe seven axial positions in which the temper-ature is measured) of the reactor to a positiveinlet concentration perturbation (see Table 2)with and without control system.


and expensive, a soft sensor based on high-gain techniques canbe used to estimate the inlet concentration from some temper-ature measurements.

AcknowledgmentsThis work was financially supported by the European Community (Con-

tract ENV4-CT97-0599). The financial support of Italian and SpanishMinistries of University and Research (Project “Integrated Actions Italy–Spain”) is also gratefully acknowledged.

Notation

av � external particle surface area per unit volume of reactor, m�1

c � molar concentration, mol m�3

cp � specific heat at constant pressure, J kg�1 K�1

Deff � effective mass dispersion coefficient, m2 s�1

dP � pellet diameter, mDR � reactor diameter, mEa � activation energy, J kmol�1

�Hf � molar enthalpy of formation, J mol�1

h � gas-solid heat transfer coefficient, J m�2 K�1 s�1

k � preexponential factor, s�1

k� � frequency factor, mol kg�1 s�1 Pa�1

keff � effective heat dispersion coefficient, J m�1 K�1 s�1

kG � gas–solid mass transfer coefficient, mol m�2 s�1

kr � kinetic constant, s�1

L � total reactor length, mNR � number of reactionsnr � number of components in the mixturePr � Prandt numberR � ideal gas constant, J K�1 mol�1

R� � reaction rate, s�1

ReP � particle Reynolds numberSc � Schmidt numberT � temperature, Kt � time, s

tc � switching time, su � surface velocity, m s�1

v � interstitial velocity, m s�1

x � axial reactor coordinate, my � molar fractionz � nondimensional axial reactor coordinate, z � x/L

Greek letters

� � bed void fraction� � effectiveness factor � thermal conductivity, J m�1 K�1 s�1

st � thermal conductivity of the stagnant gas, J m�1 K�1 s�1

� � stoichiometric coefficient� � density (or apparent density for the solid), kg m�3

Subscripts and superscripts

0 � inlet conditionse � external value

G � gas phasei � internal value

max � maximum valueS � solid phase or solid surface

W � reactor wall

Abbreviations

RFR � reverse-flow reactorPSS � periodic steady state

Literature Cited1. Matros YS, Bunimovich GA. Reverse-flow operation in catalytic re-

actors. Catal Rev Sci Eng. 1996;38:1-68.2. Van de Beld L, Westerterp KR. Operation of a catalytic reverse flow

reactor for the purification of air contaminated with volatile organiccompounds. Can J Chem Eng. 1997;75:975-983.

3. Nieken U, Kolios G, Eigenberger GA. Fixed-bed reactors with peri-odic flow reversal: Experimental results for catalytic combustion.Catal Today. 1994;20:335-350.

4. Nieken U, Kolios G, Eigenberger GA. Control of the ignited steadystate in autothermal fixed-bed reactors for catalytic combustion. ChemEng Sci. 1994;49:5507-5518.

5. Cittadini M, Vanni M, Barresi AA, Baldi G. Development and designof a forced unsteady-state reactor through numerical simulation. In:Proceedings of the 10th European Symposium on Computer AidedProcess Engineering (Computer-Aided Chemical Engineering Series).Vol. 8. Amsterdam: Elsevier; 2000:697-702.

6. Cunill F, Van de Beld L, Westerterp KR. Catalytic combustion of verylean mixtures in a reverse flow reactor using an internal electricalheater. Ind Eng Chem Res. 1997;36:4198-4206.

7. Barresi AA, Vanni M. Control of catalytic combustors with periodicalflow reversal. AIChE J. 2002;48:648-652.

8. Mancusi E, Russo L, di Bernardo M, Crescitelli S. Zeno trajectories ina non-smooth model of a controlled reverse flow reactors. Proc ofEuropean Symposium on Computer-Aided Chemical Engineering(ESCAPE-14), Lisbon, Portugal, May 19–23; 2004.

9. Eigenberger G, Nieken U. Catalytic combustion with periodical flowreversal. Chem Eng Sci. 1988;43:2109-2115.

10. Grozev GG, Sapundzhiev CG. Modelling of reversed flow fixed bedreactor for catalytic decontamination of waste gases. Chem Eng Tech-nol. 1997;20:378-383.

11. Sapundzhiev C, Chaouki J, Guy C, Klvana D. Catalytic combustion ofnatural gas in a fixed bed reactor with flow reversal. Chem EngCommun. 1993;125:171-186.

12. Purwono S, Budman H, Hudgins RR, Silveston PL, Matros YS.Runaway in packed bed reactors operating with periodic flow reversal.Chem Eng Sci. 1994;49:5473-5487.

13. Budman H, Kzyonsek M, Silveston P. Control of nonadiabatic packedbed reactor under periodic flow reversal. Can J Chem Eng. 1996;74:751-759.

14. Edouard D, Schweich D, Hammouri H. Observer design for reverse-flow reactor. AIChE J. 2004;50:2155-2166.

15. Van de Beld L. Air Purification by Catalytic Oxidation in an AdiabaticPacked Bed Reactor with Periodic Flow Reversal. PhD Thesis. En-schede, The Netherlands: University of Twente; 1995.

16. Fissore D, Barresi AA, Baldi G, Hevia MAG, Ordonez S, Dıez F.Design and testing of small scale unsteady-state afterburners andreactors. AIChE J. 2005;51:1654-1664.

17. Hindmarsh AC. ODEPACK, A Systematized Collection of ODE Solv-ers. Stepleman RS, et al., eds. (Vol. 1 of IMACS Transactions onScientific Computation). Amsterdam: North-Holland; 1983.

18. Van de Beld L, Borman RA, Deckx OR, Van Woezik BAA, Wester-terp KR. Removal of volatile organic compounds from polluted air ina reverse flow reactor: An experimental study. Ind Eng Chem Res.1994;33:2946-2956.

19. Ben-Tullilah M, Alajem E, Sheintuch M. Flow-rate effects in flowreversal reactors: Experiments, simulations, approximations. ChemEng Sci. 2003;58:1135-1146.

20. Dıez F, Vega A, Ordonez S, Hevia MAG, Fissore D, Cittadini M,Vanni M, Barresi AA, Baldi G. Dispositivo para el control de flujo decalor a traves de la pared en equipos pequenos. Spanish PatentApplication No. P200400625; 2004.

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Manuscript received Sep. 9, 2004, revision received Feb. 18, 2005, and final revisionreceived Apr. 26, 2005.


Stationary Transversal Hot Zones in AdiabaticPacked-Bed Reactors

G. Viswanathan and D. LussDept. of Chemical Engineering, University of Houston, Houston, TX 77204

A. Bindal and J. KhinastDept. of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ 08854


A pseudo-homogeneous model predicts that a stationary, stable, transversal nonuni-form temperature pattern cannot form in any cross section of an adiabatic packed-bedreactor used to conduct a single reaction, if its dynamics depends only on the concen-tration of a limiting reactant and temperature. This is due to the fact that the transversalheat dispersion in the reactor is larger than that of the limiting reactant. This conclusionis proved for a shallow packed-bed reactor described by a pseudo-homogeneous model.Extensive numerical simulations showed that a two-phase model of an adiabatic packed-bed reactor exhibits the same behavioral features. A stable, stationary hot zone can formin the cross section of the reactor only under the unrealistic assumption that thetransversal species dispersion exceeded that of the temperature. © 2005 American Instituteof Chemical Engineers AIChE J, 51: 3028–3038, 2005Keywords: packed-bed reactor, hot spot, neutral stability, temperature pattern, nonuni-form perturbations

Introduction

Localized hot zones have been reported to exist in packed-bed reactors. They have in general a deleterious impact on theperformance of the reactor, that is, decreasing the yield of thedesired products. Moreover, they may initiate undesired re-sults, leading to runaway reactions. Puszynski and Hlavacek1

observed a periodic generation of a downstream traveling hotregion during CO oxidation in a packed-bed reactor. Wicke andOnken2,3 observed a similar periodic, traveling temperaturewave during the oxidation of either CO or ethylene in apacked-bed reactor. They pointed out that these periodic tem-perature waves form, when the catalytic reaction rate in theupstream section of the reactor is oscillatory. Rovinski andMenzinger4 observed a periodic sequence of traveling concen-tration pulses when they conducted a Belousov–Zhabotinski

reaction in a catalytic packed-bed reactor under excitable con-ditions. Sheintuch research group5-7 conducted extensive theo-retical analyses of the formation of these traveling hot zones.

A uniform temperature is expected to exist at any crosssection of adiabatic packed-bed reactors. There have beenseveral attempts to predict the conditions that may lead toevolution of a stable, nonuniform transversal temperature fieldin such a reactor. Matros8 reported formation of hot zones in apacked-bed reactor in which the catalyst was not uniformlypacked. This type of hot zone formation may be circumventedby improving the packing of the bed and will not be addressedhere. Barkelew and Gambhir9 reported the formation of smallclumps of molten catalyst (clinkers) during hydrodesulfuriza-tion in trickle-bed reactors. Wicke and Onken2,3 noted that thetemperature at two locations in the same cross section of thereactor was different, that is, transversal (normal to the flowdirection) temperature patterns existed. When local hot regionsexist next to the reactor walls, they may decrease its strength,leading to severe safety problems. A thorough understandingand ability to predict when such local hot zones form are

Correspondence concerning this article should be addressed to D. Luss [email protected].


REACTORS, KINETICS, AND CATALYSIS


paramount for a rational reactor design and operation thatcircumvent this undesired behavior.

Several studies attempted to predict the conditions leading toformation of the transversal hot zones. Viljoen et al.10 pointedout that an exothermic chemical reaction may generate variousconvection and temperature patterns in a porous media. Strohand Balakotaiah,11 Nguyen and Balakotaiah,12,13 and Subrama-nian and Balakotaiah14 all showed that spatiotemporal flow andtemperature patterns may evolve in packed-bed reactors, whenthe feed flow rate is rather low. Schmitz and Tsotsis15 foundthat stationary patterns can form in a chain of interactingcatalysts (cell model) only when the rate of species exchangeexceeded that of heat exchange. Balakotaiah et al.16 showedthat the uniform stable state in an adiabatic packed-bed reactormay become unstable, leading to evolution of a nonuniformstate. They showed that the larger the diameter of the reactor,the larger the number of possible stationary nonuniform statesthat may form. Yakhnin and Menzinger17 pointed out that theanalysis of Balakotaiah et al.16 was for a case in which thetransversal effective heat dispersion was lower than that of thespecies dispersion. This condition is similar to that enablingformation of stationary patterns in two variable reaction–dif-fusion systems, if the diffusion of the inhibitor is larger thanthat of the activator.18,19 However, in practice the transversaldispersion of heat in a packed-bed reactor is always larger thanthat of the species dispersion.

The goal of this study is to determine under what conditionsand reaction rates stable, stationary, nonuniform temperaturestates exist in packed-bed reactors, and to analyze the qualita-tive features and stability characteristics of the branches of thenonuniform states.

Mathematical Model

We use a heterogeneous or two-phase model to describe anadiabatic packed-bed reactor. The corresponding energy andlimiting species balances for both the gas and the solid phasesare as follows

��Cp�f��Tf

�t� ��Cp�fv

�Tf

�z� ��f,a

�2Tf

�z2

� ��f,��2 Tf � hav�Ts � Tf� (1)

��Cf

�t� �v

�Cf

�z� �Df,a

�2Cf

�z2 � �Df,��2 Cf � kav�Cf � Cs�

(2)

��Cp�s�1 � ��Ts

�t� �1 � ��s,��

2 Ts � �1 � ��s,a

�2Ts

� z2

� hav�Ts � Tf� � ��H�r��Cs, Ts� (3)

dCs

dt� �kav�Cf � Cs� � r��Cs, Ts� (4)

where Tf and Ts are the fluid- and solid-phase temperatures, Cf

and Cs are the fluid- and solid-phase concentrations, v is thesuperficial gas velocity, r�(Cs, Ts) is the reaction rate, and

��2 � �1

r

�

�r �r�

�r� �1

r2

�2

��2� (5)

The corresponding boundary conditions are

��Df,a

�Cf

�z� v�Cin � Cf�

��f,a

�Tf

�z� v��Cp�f�Tin � Tf�

�Ts

�z� 0

� z � 0 (6)

�Cf

� z� 0

�Tf

� z� 0

�Ts

� z� 0 z � L (7)

�Cf

�r� 0

�Tf

�r� 0

�Ts

�r� 0 r � R (8)

Because the characteristic reaction time is very small relativeto the others in the system the solid reactant concentration maybe described by a pseudo-steady-state equation (that is, ignor-ing the time derivative in Eq. 4). For certain kinetics thisenables one to obtain an explicit solution for Cs that can besubstituted into the rate expression. For example, for a first-order reaction

Cs �Cf

1 � � k�

kav�exp��E

R� Ts� (9)

This substitution reduces the model to three equations for Tf,Cf, and Ts. By introducing the following dimensionless vari-ables

� �T � Tin�/Tin x � �Cin � C�/Cin

�E

R� Tin

� ��H�Cin

��Cp�fTin

Stm � vL/�kav� Sth � vL/�hav�

Pef,ah �

vL

��f,a/��Cp�fPef,a

m � vL/�Df,a

Pes,ah �

vL

�1 � ��s,a/��Cp�fPef,�

h �vR2

L��f,�/��Cp�f

Pef,�m � vR2/�LDf,�� Pes,�

h �vR2

L��s,�/��Cp�f

Le � � � �1 � ��Cp�s

��Cp�f(10)

the model becomes

�f

��

1

� � 1

Pef,�h ��

2 f �1

Pef,ah

�2f

� 2 ��f

� � Sth�s � f�� (11)


� xf

��

1

� � 1

Pef,�m ��

2 xf �1

Pef,am

�2xf

� 2 ��xf

� � ��s, xf�� (12)

�s

��

1

�Le � �� 1

Pes,�h ��

2 s �1

Pes,ah

�2s

� 2 � Sth�s � f�

� ��s, xf�� (13)

where, for a first-order reaction,

��s, xf� � Da� exp� s

s � 1�1 �

Da

Stmexp� s

s � 1��1 � xf�

Da �Lk�e�

v(14)

The corresponding boundary conditions are

1

Pef,ah

�f

� � f

1

Pef,am

�xf

� � xf

�s

� � 0 � 0 (15)

�f

� � 0

� xf

� � 0

�s

� � 0 � 1 (16)

�f

�� 0

� xf

�� 0

�s

�� 0 � � 1 (17)

When the transport resistances between the solid and fluidphase are small, the reactor may be described by the simplerpseudo-homogeneous (��) model

�

��

1

Le�� 1

Pe��,�h ��

2 �1

Pe��,ah

�2

� 2 ��

� � ��, x��

(18)

� x

�� 1

Pe��,�m ��

2 x �1

Pe��,am

�2x

� 2 ��x

� � ��, x�� (19)

subject to boundary conditions

1

Pe��,ah

�

� �

1

Pe��,am

�x

� � x � 0 (20)

�

� � 0

� x

� � 0 � 1 (21)

�

�� 0

� x

�� 0 � � 1 (22)

where, for a first-order irreversible reaction,

��, x� � Da exp�

� 1��1 � x� (23)

The dimensionless quantities in the pseudo-homogeneousmodel are

Le�� Cp��/��Cp�f� Pe��,ah �

vL

��,a/��Cp�f

Pe��,am �

vL

D��,a

Pe��,�h �

vR2

L��,�/��Cp�fPe��,�

m �vR2

LD��,�(24)

Vortmeyer and Schaeffer20 derived a relation for the effectivethermal conductivity in this model. Recent studies by Dommetiet al.15 and Balakotaiah and Dommeti21 indicate that this rela-tion is not always valid.

Steady-state numerical solutions were obtained by discretiz-ing the spatial derivatives by finite differences in all threedirections (102 grid points in the axial direction and 60–100grid points in the radial and azimuthal directions). In somecases, global orthogonal collocation was used to discretize theradial and azimuthal directions (10–25 points in each direc-tion). The steady-state solutions were computed either by asparse inexact Newton–Krylov subspace method using ILUpreconditioned BiCGSTAB22 or by a Newton–Raphson itera-tion procedure using a nonlinear equation solver (NLEQ),direct (LAPACK), and sparse (MA28) solver.23 The bifurca-tion diagrams were computed using pseudo-arc length contin-uation.24 The branches of the 2-D and 3-D states and of thecodimension-1 were calculated using the Library of Continu-ation Algorithms (LOCA) software.25,26

Dynamic simulations were conducted using the Linear Im-plicit Extrapolator (LIMEX).27,28 The 1-D calculations werecomputed using the LAPACK option. The 2-D and 3-D calcu-lations were performed using either the direct linear solver(MA28) or the sparse iterative linear solver (GMRES/BiCG-STAB). The sparse iterative solver in the linear step usuallyconverged in two to three iterations.

Both the steady-state and dynamic simulations of the fullmodel are rather demanding. Thus, we first studied a simplifiedversion of the full model, that of a shallow reactor, to gaininsight into the structure of the bifurcation diagrams of thestates with the nonuniform transversal temperature, and theirstability features and transitions. A similar approach was usedby Balakotaiah et al.29 in their study of pattern formation in ashallow monolith reactor. This shallow reactor model wasobtained by a Liapunov–Schmidt reduction30 of the full model.Details of the reduction procedure used in this case are reportedby Viswanathan.31 We denote by f, xf, and s the spatiallyaveraged quantities in the shallow reactor model, that is

f � �0

1

fd xf � �0

1

xfd s � �0

1

sd (25)

The shallow-reactor two-phase model is


df

d��

1

� � 1

Pef,�h ��

2 f � f � Sth�s � f�� 0 (26)

dxf

d��

1

� � 1

Pef,�m ��

2 xf � xf � ��s, xf�� 0 (27)

ds

d��

1

�Le � �� 1

Pes,�h ��

2 s � Sth�s � f� � ��s, xf�� 0

(28)

with �(s, xf) defined by Eq. 14. The corresponding boundaryconditions are

�f

�� 0

� xf

�� 0

�s

�� 0 � � 1 (29)

The pseudo-homogeneous model can be reduced in a similarway.

The reduced model was discretized using a second-ordercentral difference scheme. To circumvent the singularity at thecenter, the grid points were placed at the radial positions �j (2j � 1)/(2N � 1) @ j 1, N.32 After discretization, thereduced model is similar to a cell model in a circular crosssection, with the diffusion terms (in the continuous model)mimicked by exchange coefficients among the cells.

Stationary nonuniform temperature in a shallow reactor

We consider here the evolution and dynamics of stationarytemperature patterns in an adiabatic shallow packed-bed reac-tor (��) using a pseudo-homogeneous model. This limitingmodel provides important insight about the dynamic features ofthe more complex models, the analysis and simulations ofwhich are much more demanding. The mathematical model ofthis two variables system is

d

d��

1

Le�� 1

Pe��,�h ��

2 � � ��, x�� 0 (30)

dx

d�� 1

Pe��,�m ��

2 x � x � ��, x�� 0 (31)

with �(, x) defined by Eq. 23. The corresponding boundaryconditions are

�

�� 0

� x

�� 0 �j � 1 (32)

Linear stability analysis predicts that a uniform steady stateis stable with respect to uniform perturbations, if the twoeigenvalues of

�1 � ��1

Le��

��

Le��

��

�

�, x�ss

�

Le��

��

�x

�, x�ss

��

�

�, x�ss

�1 ��

�x

�, x�ss

� (33)

have a negative real part. In a packed-bed reactor, Le 1 andit can be proven that the stability condition

det��1� � 0 (34)

implies that in this system tr(�1) � 0. Thus, condition 34 isnecessary and sufficient for stability and no Hopf bifurcationcan occur from a state satisfying Eq. 34. Next we consider asmall nonuniform spatial perturbation of the form

�mn��, �� 1Jm��mn��eim�

�2Jm��mn��eim�� (35)

where m and n are the azimuthal and radial mode numbers andeim� and Jm(�mn�) are the corresponding eigenfunctions. Be-cause of the no-flux boundary condition, �mn satisfies therelation

dJm��mn��

d�

�1

� mJm��mn� � �mnJm�1��mn� � 0 (36)

The first nine eigenvalues �mn of the spatial perturbations arereported in Table 1. Although all the perturbations with m 0have no azimuthal dependency, all the others depend on both �and �. Figure 1 describes the first six eigenmodes. Linearstability analysis of a uniform solution of the steady state of thepseudo-homogeneous model following a small perturbation of�mn, defined by Eq. 35, shows that it is stable to the inhomo-geneous perturbations, if both eigenvalues of

Table 1. First Nine Eigenvalues Satisfying Eq. 36

No. m n �mn

1 1 1 1.84122 2 1 3.05423 0 1 3.83174 3 1 4.20125 4 1 5.31766 1 2 5.33147 5 1 6.41568 2 2 6.70619 0 2 7.0155

Figure 1. The first six transversal eigenmodes.


�� 1 � �mn

2 �1� (37)

have a negative real part, where

�1 � �1

Le��Pe��,�h 0

01

Pe��,�m

� (38)

The neutral stability curve, at which a transition from auniform to a nonuniform state occurs, is obtained by a simul-taneous solution of the uniform steady-state solutions of Eqs.30–32 and

det�� 0 (39)

Condition 39 requires that

��

�mn2

Pe��,�h ��

��

��, x�ss

� 1� ��mn

2

Pe��,�h �

�Le�� det��1�� mn

2

Pe��,�h �1 �

��

�x�, x�ss

�� (40)

where

�� Pe��,�

m

Pe��,�h �

��,�

D��,��Cp��

(41)

The above condition may be rewritten as

�1 ��mn

2

Pe��,�h �Le�� det��1��

�mn2

Pe��,�h � � �mn

2

Pe��,�h �2

�

�1 � ��1 ��mn

2

Pe��,�h �Le�� det��1��

�mn2

Pe��,�h �

��

��, x�ss

�(42)

The effective heat diffusion in a pseudo-homogeneousmodel of a packed-bed reactor is always larger than that of thespecies dispersion, that is, �� 1. The lefthand side of Eq.42 is positive when det(�1) 0. Thus, the condition for atransition from a uniform to a nonuniform state of det(��

��) 0 is not satisfied if �� 1. This leads to the importantconclusion that for reactions described by the above model abifurcation to a stationary nonuniform state from a uniformstate, stable to uniform perturbations, cannot occur for therealistic cases of �� 1. However, when �� 1, anunstable bifurcation to a nonuniform state may occur from theunstable branch of uniform states for which det(�1) � 0.Extensive numerical calculations showed that all the solutionson a branch of nonuniform states are unstable whenever thebranch emanates and terminates on the branch of unstableuniform states. Thus, stable nonuniform states do not exist forthe realistic case of �� 1. A stable bifurcation to anonuniform state may occur from the stable branch only for�� 1.

The two-phase shallow reactor model (��) consists ofthree state variables. Its stability analysis is similar, butmore intricate. Assuming a small spatial perturbation of theform

��1Jm��mn��eim�

�2Jm��mn��eim�

�3Jm��mn��eim�� (43)

the uniform steady-state solution is stable with respect to thisperturbation if all eigenvalues of

�� 2 � �mn

2 �2� (44)

have a negative real part, where (the subscript �� denotestwo-phase)

�2 � �1

��1 � Sth� 0

Sth

�

01

� ��1 ��

�xf

�f, xf,s�ss

� 1

�

��

�s

�f, xf,s�ss

Sth

�Le � ��

�

�Le � ��

��

�xf

�f, xf,s�ss

1

�Le � �� Sth � ��

�s

�f, xf,s�ss

�� (45)

and


�2 � �1

�

1

Pef,�h 0 0

01

�

1

Pef,�m 0

0 01

�Le � ��

1

Pes,�h

� (46)

The neutral stability curve is obtained by a simultaneous solu-tion of the uniform steady-state solutions of Eqs. 26–29 and

det�� 0 (47)

Unfortunately, we were unable to derive in this case a boundon

�� Pef,�

m

Pef,�h �

�f,�

Df,��Cp�f(48)

above which a stable uniform state cannot bifurcate to a non-uniform state. Extensive calculations indicated that for thekinetic model used here a uniform steady state may bifurcateonly to a stationary nonuniform state. However, we are unableto prove this.

Before proceeding to the regular reactor case, we shallpresent some numerical results describing the impact of ��

on the evolution of the nonuniform states for the two-phase��. In all the numerical simulations, unless otherwise stated,we used the following set of parameters

� � 0.36 Le � 1416

� � 0.4 � 15

Sth � 300L

dpStm � 200

L

dp

Pef,ah �

L

dpPef,a

m � 5.0L

dp

Pes,ah � 0.25

L

dpPef,�

h � 2.0�R/dp�

2

L/dp

Pes,�h � 1.0

�R/dp�2

L/dp(49)

We used L dp in all the shallow reactor simulations andL 30dp in all the regular reactor simulations. Figure 2 showsthe neutral stability curve of the bifurcation to the first mode(�mn 1.8412) for the shallow reactor. Figure 2a shows thatfor �� 1 the stability curve is bounded between the Davalues corresponding to the ignition (Dai) and extinction (Dae).However, this graph does not show from which branch ofuniform states the bifurcation occurs. Thus, from now on wereport the stability curves in the plane of R/dp vs. f,exit, theeffluent temperature, because these plots show from whichbranch of uniform solutions the bifurcation occurs. Figure 2bshows that for all �� 1, the bifurcation occurs from thebranch of unstable uniform states because the effluent temper-ature is bounded between the ignition and extinction tempera-

tures (i and e). However, for �� 0.165 and R/dp 7.5one bifurcation to nonuniform temperature state occurs from astable, uniform high-temperature state ( e), and a secondone from an unstable, uniform temperature state (e � � i).For �� 0.075 and R/dp 6.5 one bifurcation is from astable, uniform high-temperature state and the second from astable, uniform low-temperature state.

Figure 3 is a bifurcation diagram of the uniform and non-uniform states of this reactor. The nonuniform states wereobtained by perturbing the uniform ones close to the bifurca-tion points by the first mode and numerical continuation tocompute the branch. The transition from uniform to nonuni-form states occurred at a pitchfork bifurcation, at which a pairof nonuniform branches emerged. The nonuniform states bi-furcating from a uniform unstable branch are unstable. In allthe cases we studied, the pitchfork bifurcation from the branchof uniform stable states is subcritical, that is, the bifurcatingnonuniform states are unstable. (In all the bifurcation diagramsshown here, except for Figure 10, dashed lines represent un-stable transversal nonuniform states and solid segments onthese branches represent stable transversal nonuniform states.)In the case shown in Figure 3 for �� 1.5, all the states onthe two nonuniform branches are unstable. In contrast, some of

Figure 2. Neutral stability curves corresponding to thefirst transversal mode (�mn � 1.8412) for atwo-phase model of a shallow packed-bed re-actor (L � dp) for various �� in (a) the R/dp vs.Da and (b) R/dp vs. �f,exit plane.


the nonuniform states for �� 0.165 are stable. The non-uniform states are rather similar in shape to the first mode, thatis, half of the cross section is hot and half is cold. The neutralstability curve in Figure 2 indicates that for �� 0.075 andR/dp 6.5 the branch of the nonuniform state emanates froma stable, uniform ignited state and terminates at a stable,uniform extinguished state. In the special case of mode 1, thetwo states (on either branches of the nonuniform states) aremirror images of each other and have the same effluent tem-perature. Thus, the two states are represented by a singlebranch in Figure 3.

Figure 4 shows the neutral stability corresponding to thethird mode (�mn 3.8317), which has no azimuthal depen-dency and has the shape of a target. Again, bifurcation fromstable uniform states do not occur for values of �� 1.However, bifurcations from stable uniform states occur forsufficiently small values of ��.

Figure 5 is a bifurcation diagram of the uniform steadystates and the stationary nonuniform ones corresponding tothe third mode. The states corresponding to �� 1.5emanate from unstable, uniform states and all the states onthis branch are unstable. For �� 0.165 the branch ofnonuniform state bifurcates (subcritically) from a stable,uniform ignited state and terminates at an unstable, uniformstate. Some of the nonuniform states on this branch arestable (solid line in Figure 5), whereas the others are unsta-ble. The nonuniform states were rather similar in shape tothe third mode, that is, had the shape of a target. Twobranches of nonuniform states exist in this case. This occursbecause the pitchfork bifurcation leads to formation oftwo nonuniform states that have different effluent tempera-tures.

Stationary nonuniform temperature in packed-bedreactors

We consider here the formation of nonuniform transversaltemperature patterns in an adiabatic packed-bed reactor using atwo-phase model. At the neutral stability curve a transitionfrom a uniform to a nonuniform state occurs following anonuniform perturbation. It is obtained by solving simulta-neously the transversally uniform steady-state form of equa-tions (Eqs. 11–17) and the eigenvalue problem

Figure 3. Dependency of the uniform and nonuniformstates corresponding to the first mode on Dafor a two-phase model of a shallow packed-bed reactor (L � dp) for �� 0.165 and�� 1.5.Dotted line denotes unstable uniform state; dashed line de-notes unstable nonuniform state; solid line denotes stablestates.

Figure 4. Neutral stability curves corresponding to thethird transversal mode (�mn � 3.8317) for atwo-phase model of a shallow packed-bed re-actor (L � dp) for various ��.

Figure 5. Dependency of the uniform and nonuniformstates corresponding to the third mode on Dafor a two-phase model of a shallow packed-bed reactor (L � dp) for �� 1.5 and �� 0.165.Dotted line denotes unstable uniform state; dashed line de-notes unstable nonuniform state; solid line denotes stablestates.


1

� � 1

Pef,ah

�2�1

� 2 ��1

� �

�mn2 �1

Pef,�h � Sth��3 � �1�� 1 (50)

1

� � 1

Pef,am

�2�2

� 2 ��2

� �

�mn2 �2

Pef,�m �

��

�xf

�f, xf,s�ss

�2 ��

�s

�f, xf,s�ss

�3� � ��2 (51)

1

Le � � � 1

Pes,ah

�2�3

� 2 ��mn

2 �3

Pes,�h � Sth��3 � �1�

� ��

�xf

�f, xf,s�ss

�2 � ��

�s

�f, xf,s�ss

�3� � ��3 (52)

subject to the boundary conditions

1

Pef,ah

��1

� � �1

1

Pef,am

��2

� � �2

��3

� � 0 � 0 (53)

��1

� �

��2

� �

��3

� � 0 � 1 (54)

��1

��

��2

��

��3

�� 0 � � 1 (55)

Linear stability predicts the nature of the bifurcating non-uniform solutions. When a single negative eigenvalue � be-comes positive upon crossing the neutral stability curve, astationary nonuniform state evolves, whereas if a pair of com-plex eigenvalues crosses the imaginary axis an oscillatorynonuniform state evolves. Extensive numerical simulationssuggest that this three-variable model leads only to bifurcationsto a stationary nonuniform state, although we could not provethis. Similarly, we were unable to derive a criterion predictingthe impact of the ratio between the effective transversal dis-persion of heat and mass on the stability of the uniform steadystate from which the nonuniform states can bifurcate.

Figure 6 shows the neutral stability curve of the bifurcationto the first mode (�mn 1.8412). Figure 6a shows that for all�� 1 the stability curve is bounded between the Da valuescorresponding to the ignition (Dai) and extinction (Dae). Figure6b shows that for all �� 1 the nonuniform states bifurcatefrom the branch of unstable uniform states because their efflu-ent temperature is between the ignition and extinction temper-atures (i and e). This indicates that, as predicted by theshallow reactor model, no bifurcation from a stable uniformsteady-state solution can occur under the practical case of ��

� 1. However, for �� values of 0.1 and 0.075 bifurcations tononuniform temperature state occur from the stable, uniformhigh-temperature states ( e). The qualitative behavior ofthis model is similar to that of the shallow reactor shown inFigure 2. However, a nonuniform state bifurcated from stableuniform states for �� 0.165 for the shallow reactor. Nosuch bifurcation occurred in the regular packed-bed reactormodel. Numerical simulations indicated that when the neutral

stability curve has a fold in the (R/dp vs. Da) plane, such as for�� 0.075 in Figure 6a, then a bifurcation can occur froma stable uniform state.

The neutral stability curves for the higher modes have ashape similar to that for mode 1. For example, Figure 7 showsthe neutral stability for nonuniform states corresponding to thethird mode (�mn 3.8317) in Figure 1. Again, bifurcationfrom stable uniform states do not occur for values of �� 1.However, bifurcations from stable uniform state occur forsufficiently small values of ��. Comparison of Figures 6band 7 shows that the neutral stability curve for the higher modeare shifted to higher values of R/dp. The fact that the neutralstability curves for higher modes are shifted to higher values ofR/dp implies that the larger the reactor diameter, the larger thenumber of possible stationary nonuniform states.

The bifurcation diagram of the uniform steady state and ofthe nonuniform ones corresponding to the third mode areshown in Figure 8. The states corresponding to �� 1.5emanate from unstable, uniform states and all the states on thisbranch are unstable. This occurs for all the states satisfying therealistic condition that �� 1. For �� 0.075 the branchof nonuniform state bifurcates (subcritically) from a stable,uniform ignited state and terminates at an unstable, uniformstate. Some of the nonuniform states on that branch are stable

Figure 6. Neutral stability curves corresponding to thefirst transversal mode (�mn � 1.8412) for atwo-phase model of a packed-bed reactor (L �30dp) for various �� in (a) the R/dp vs. Da and(b) R/dp vs. �f,exit plane.


(solid line in Figure 8), whereas the others are unstable. Thenonuniform states are rather similar in shape to the third mode.Two branches of nonuniform states exist in this case. Thisoccurs because the pitchfork bifurcation leads to formation oftwo nonuniform states with different effluent temperatures.

The solutions corresponding to mode 3 have no azimuthaldependency and have the shape of a target at any cross sectionof the reactor. Figure 9 describes the temperature profileswithin such a reactor of such a stable nonuniform state. It wascomputed for �� 0.075. Thus, it corresponds to the con-

dition that the effective heat diffusivity is smaller than that ofthe limiting reactant.

The branches of the nonuniform states corresponding to thesecond mode (�mn 3.0542) are shown in Figure 10 for ��

1.5, that is, for a realistic situation, in which the heatdispersion exceeds that of the limiting reactants. In this case thetwo branches of the nonuniform states emanate from an unsta-

Figure 7. Neutral stability curves corresponding to thethird transversal mode (�mn � 3.8317) for atwo-phase model of a packed-bed reactor (L �30dp) for various ��.

Figure 8. Dependency of the uniform and nonuniformstates corresponding to the third mode on Dafor a two-phase model of a packed-bed reac-tor (L � 30dp) for �� 1.5 and �� 0.075.Dotted line denotes unstable uniform state; dashed line de-notes unstable nonuniform state; solid line denotes stablestates.

Figure 9. Reactor temperature of stable nonuniformstates in a packed-bed reactor predicted by atwo-phase model and corresponding to thethird mode for Da � 134 and �� 0.075 when(a) ��f,exit� � 0.237 and (b) ��f,exit� � 0.325.

Figure 10. Dependency of the uniform and nonuniformstates corresponding to the second mode onDa for a two-phase model of a packed-bedreactor (L � 30dp) for �� 1.5.Dotted line denotes unstable uniform state; dashed linedenotes unstable nonuniform state; solid line denotes stablestates.


ble uniform steady state, located on the intermediate branch ofsolutions. Detailed linear stability calculations showed that allthe states on the two branches were unstable and there was nochange in stability at the limit points on the branches of thesestates.

Conclusions and Remarks

A pseudo-homogeneous model predicts that a stable, station-ary nonuniform temperature pattern cannot form in any crosssection of an adiabatic packed-bed reactor used to carry out asingle reaction, having a conventional rate expression thataccounts only for reactant surface adsorption, desorption, andtemperature. This is explained by the fact that the transversalheat dispersion in the reactor is larger than that of the limitingreactant. This conclusion is true for all the possible modes. Thisconclusion was proved for a shallow packed-bed reactor de-scribed by a pseudo-homogeneous model. Extensive numericalsimulations showed that both a regular and shallow packed-bedreactor described by a two-phase model exhibited the samequalitative behavioral features. A stable, stationary pattern mayform in the cross section of the reactor only under the assump-tion that the transversal species dispersion exceeds that of thetemperature. As indicated by Schmitz and Tsotsis14 and Yakh-nin and Menzinger,17 this condition is not satisfied in packed-bed reactors and is the same as that generating a Turing patternin reaction–diffusion systems.

This study shows that the analysis of pattern formation in ashallow packed-bed reactor provides useful insight and guid-ance about pattern evolution and stability in packed-bed reac-tors. Moreover, it drastically decreases the numerical effortneeded to conduct such studies. This finding should be used infuture studies of pattern formation involving different classesof reactions.

Our study is the first to present the branches of stationarytransversal pattern states in a 3D packed-bed reactor. We foundthat all the stationary patterned states on the branch wereunstable, when the branch bifurcated from two states that wereunstable to uniform disturbances. Stable stationary states wereobtained only when the branch bifurcated from a uniform statestable to uniform disturbances. Only some of the states on thebranch of nonuniform states were stable.

The analysis of the pattern formation in an adiabatic packed-bed reactor exploits the fact that the concentration and temper-ature satisfy the same (no-flux) boundary condition at thereactor walls. Consequently, the radial perturbations of all thevariables can be expressed by the same Bessel functions. Amuch more intricate analysis is required when heat transferoccurs at the reactor walls. In this case different Bessel func-tions describe the concentration and temperature disturbancesand the concentration and temperature patterns are no longersimilar. It is important to conduct future analysis of this case togain insight into the impact of the heat losses on the patternevolution and stability.

This study, however, does not provide an answer to theimportant question: Which reaction mechanism and operatingcondition can lead to transversal pattern formation in adiabaticpacked-bed reactors? Several industrial reports and laboratoryexperiments mentioned in the introduction reported formationof hot zones in packed-bed reactors. This study suggests that abifurcation to a stable nonuniform state may occur only if the

reaction rate depends—in addition to the temperature and sur-face adsorption and desorption of reactants—on other rateprocesses, such as a periodic variation of the surface catalyticactivity33or impact of subsurface adsorption–desorption of re-actants.34,35 The pattern formation for these more complexkinetic mechanisms will be addressed in a future publication.Our analysis is of a bed that is uniformly packed with nointernal flow obstructions. Clearly, internal obstruction of theflow36 or nonuniform packing of the bed8 may also generate hotzones.

AcknowledgmentsWe gratefully acknowledge support of this research by grants from the

National Science Foundation, the Welch Foundation and the San DiegoSupercomputing Center. We thank Dr. Andrew Salinger for helpful dis-cussions and advice.

Notation

av specific surface area, m2/m3

C concentration, mol/m3

Cp specific heat capacity, J kg�1 k�1

dp diameter of particle, mD species diffusion coefficient, m2/s

Da Damkohler numberE activation energy, J/molh interfacial heat transfer coefficient, W K�1 m�2

J Bessel function of first kindk interfacial mass transfer coefficient, m/s

k� intrinsic reaction rate constant, 1/sL reactor length, m� first Frechet derivative

Le Lewis number, defined by Eq. 10Le�� Lewis number, defined by Eq. 24

� ratio of transversal heat to mass dispersionsN radial grid points� transversal perturbation matrix

Pe Peclet number, defined by Eq. 10Pe�� Peclet number, defined by Eq. 24

r radial coordinate, mR reactor radius, mr� reaction rate, mol m�3 s�1

� dimensionless reaction rateR� universal gas constant, J mol�1 K�1

Stm Stanton number for mass, defined by Eq. 10Sth Stanton number for heat, defined by Eq. 10

t time, sT temperature, Kv linear velocity, m/sx conversion, defined by Eq. 10z axial coordinate, m

Greek letters

� adiabatic temperature rise, defined by Eq. 10 dimensionless activation energy, defined by Eq. 10� bed voidage dimensionless axial coordinate dimensionless temperature, defined by Eq. 10� thermal conductivity, W/(m.K)� transversal eigenmode number� dimensionless radial coordinate� eigenvalue� dimensionless time, defined by Eq. 10� azimuthal coordinate�i ith component of nonuniform perturbation

��H heat of the reaction, J/mol

Other

� � � averaged quantity


Subscripts

a axialf fluid phase

in inletm azimuthal mode numbern radial mode number

�� pseudo-homogeneouss solid phase

ss transversally uniform steady state�� two-phase

� transversal

Superscripts

e extinctionh massi ignition

m mass�� shallow reactor

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Manuscript received Dec. 23, 2004, and revision received Mar. 7, 2005.


CFD Simulation of UV Photocatalytic Reactors forAir Treatment

Fariborz Taghipour and Madjid MohseniDept. of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada


A photocatalytic reactor was simulated through computational fluid dynamics (CFD)with surface reaction for trichloroethylene (TCE) oxidation at various pollutant concen-trations, flow rates, and reactor lengths. The results were compared with those fromexperiments. The experimental work involved using a differential photoreactor for kineticsstudies and an annular flow photoreactor for overall removal investigations under variousconditions. The modeling predictions agreed closely with the experimental data within therange in which results were examined. The modeling results indicated significant radialTCE concentration gradient and nonuniform flow distributions in the annular photore-actor. CFD was applied to predict the performance of a number of UV photocatalyticreactor design concepts, to study the impacts of some design parameters on the reactorefficiency. The modeling results demonstrated that under similar flow rate conditions, thethickness of the contaminated air layer flowing over the photocatalyst surface couldsubstantially influence the reactor performance. Thinner contaminated air layers providedmore uniform radial concentration distribution of TCE and improved the reactor perfor-mance. © 2005 American Institute of Chemical Engineers AIChE J, 51: 3039–3047, 2005Keywords: CFD, photocatalysis, trichloroethylene (TCE), UV, air treatment

Introduction

Heterogeneous photocatalysis, using semiconductor parti-cles such as TiO2 in combination with ultraviolet (UV) irradi-ation, has proved to be very effective for the abatement ofcontaminated air streams.1-3 This effectiveness along with theincreasingly stringent environmental regulations has intensifiedresearch toward the development of efficient photoreactors forlarge-scale applications. To that end many researchers haveinvestigated different photoreactor configurations for air treat-ment,1,4-6 and have attempted to understand the fundamentalsof photoreactor design and development using mathematicalmodeling.2,5 More recently, computational fluid dynamics(CFD) has been used as a tool to simulate photoreactor perfor-mance.7,8

CFD, which uses the conservation of mass and momentumequations to model fluid flow, is a valuable tool for simulating

reacting flows with high precision. It is often superior tosimplified hydrodynamic models, such as plug flow or com-pletely mixed, which do not usually reflect the real flow patterninside the reactors. By incorporating detailed description offlow in complex reactor geometries using CFD, the use ofplug-flow or completely mixed assumptions and correlationsfor mass transfer can be avoided. In simulating reactive flows,in addition to mass and momentum equations, the mixing,transport, and production/consumption of chemical speciesshould also be considered through species transport equations,which represent the conservation of each species in the system.The system of equations is solved in the computational domainusing one of the numerical techniques (such as finite-volume).

CFD has been applied to modeling fluid flow and overallperformance of single-phase and multiphase chemical reac-tors,9-12 as well as photoreactors. In photoreactors with volu-metric reactions, such as UV disinfection or oxidation reac-tions, reaction kinetics is usually a function of radiation fluencerate. As a result, radiation models describing the fluence ratedistribution are required for considering the radiation effect onreactor performance. In modeling photoreactors, the reaction

Correspondence concerning this article should be addressed to F. Taghipour [email protected].



kinetics and radiation distribution are integrated with the reac-tor hydrodynamics to predict reactor performance. Kucuk Un-luturk et al.13 performed CFD modeling of the UV dose dis-tribution in a thin-film UV reactor for processing of applecider, and observed reasonable agreement between simulationand experimental UV dose values. Pareek et al.7 conductedCFD modeling of a three-phase photocatalytic reactor using anEulerian–Eulerian approach and reported on the model ade-quacy for predicting the experimental results and reactor per-formance. Models for simulating the performance of ozone-UVreactors14 and UV disinfection reactors15 have been developed,taking into account parameters such as kinetics, hydrodynam-ics, and fluence rate within the reactor. The results indicatedthat decomposition of contaminants is influenced by both re-actor hydrodynamics and fluence rate distribution.

CFD has also been applied to simulate surface reactions incatalytic reactors. Endo et al.16 presented a CFD model for the rateof carbon nanotubes production by catalytic decomposition ofxylene in a chemical vapor deposition reactor. Using gas-phaseand surface reactions, the authors reported good agreement of themodel with the experimental data. Seo et al.17 carried out CFDsimulation of the surface reactions of a catalytic heat exchangerand found reasonable agreement between the modeling and ex-perimental results. Heitsch18 performed CFD analysis of catalyticsurface reactions in a catalytic recombiner and compared theresults with the reported test data. Finally, in a study of photocat-alytic destruction of gas-phase vinyl chloride (VC), Mohseni andTaghipour8 demonstrated that both CFD modeling and experi-mental approaches showed similar trends with respect to VCremoval as a function of VC loading rate.

The primary objectives of this study were: to evaluate CFD’scapability in modeling the performance of UV photocatalyticreactors for use in air treatment; to use CFD to provide a betterunderstanding of the flow characteristics and their impacts onthe overall reactor performance; and to explore CFD applica-tion as a tool for improving photoreactor design. The perfor-mance of a UV photocatalytic reactor was predicted by CFDand the results were compared with those from experimentsunder various pollutant concentrations, gas flow rates, andreactor lengths. CFD was also applied to predict the perfor-mance of a few UV photocatalytic reactor design concepts tostudy the impacts of some design parameters on the reactorefficiency.

CFD Modeling

The transport and chemical reactions were modeled by solv-ing mass, momentum, and species conservation equations, us-ing Fluent 6.0 CFD software.19 The simulation of the systemwas performed with a three-dimensional, steady-state, laminar-flow model including the photocatalytic surface reaction.19 Thegeneral forms of the governing equations for modeling thesystem are as follows:

Conservation of Mass

��

�t� � � �� 0 (1)

Conservation of Momentum

��

�t� � � �� P � � � �� g (2)

Conservation of Species Yi

��Yi�

�t� � � ��Yi� � �� J�i � Ri (3)

In Eqs. 1–3, � is density, v� is velocity, P is pressure, �� is stresstensor, g is gravitational force, and J� i is diffusion flux ofspecies i. In the species equation (Eq. 3), in addition to “con-vective” and “diffusive” terms, the “reactive” term also ap-pears. In this equation, Ri is the rate of production or depletionof the compounds by chemical reaction.

Reactions may occur at the photoreactor wall surface and/orin the homogeneous phases. For surface reaction, the concen-tration of the species on reacting surface is based on a balancebetween the convection/diffusion of each species to/from thesurface and the consumption/production rate at the surface:

J� i � n� � Ris (4)

where n� is a unit vector normal to the surface and Ris is the rate

of the production/depletion of species i arising from surfacereaction.

Ris � Mw,i �

r�1

NR

Ri,rs (5)

where Mw,i is the molar weight of the species i and Ri,rs is the

rate of production/depletion of species i in surface reaction r.Ri

s is computed as the sum of the reaction sources over NR

reactions in which the species i is involved.The general form of the rth surface reaction can be consid-

ered as

�i�1

N

��i,rMiO¡kf,r �

i�1

N

� �i,rMi (6)

where N is the total number of species in the system, v�i,r is thestoichiometric coefficient for reactant i in reaction r, v �i,r is thestoichiometric coefficient for product i in reaction r, Mi is thesymbol denoting species i, and kf,r is the forward rate constantfor reaction r.

The molar rate of production/depletion of species i in reac-tion r, Ri,r

s , is given by

Ri,rs � ��i,r � � �i,r��kf,r �

j�1

Nr

[Cj,r]��j,r� (7)

where Nr is the number of chemical species in reaction r, Cj,r

is the molecular concentration of each reactant and productspecies j in reaction r, and ��j,r is the forward rate exponent foreach reactant and product species j in reaction r.

In the vapor phase, trichloroethylene (TCE) absorbs UV


slightly (�1000 M�1 m�1) at a wavelength of 254 nm,20 andthus the homogeneous photolytic reaction in the photoreactorsirradiated by 254-nm lamps was insignificant. This was furthersubstantiated in our research, where photolytic experimentsconducted with no photocatalyst present in the photoreactor,showed that UV photolysis provided 5% TCE removal fromthe contaminated air. Given this insignificant contribution,heterogeneous photocatalysis on the wall surface (Ri

s) was theprimary reaction that took place in the system. Thus, thefollowing one-step reaction on the surface of the photocatalystwas considered

C2HCl3 �gas phase� 3 C2HCl3 �surface� (8)

C2HCl3 �surface� � •OH 3 products�CO2H2OHClintermediates�

(9)

The overall degradation of TCE, involving adsorption to thephotocatalyst surface followed by the oxidation reaction, wasassumed to follow a first-order kinetics. This assumption wassupported by the experimental results obtained from the dif-ferential glass plate photoreactor (refer to Experimental sec-tion). Thus, the molar rate of TCE degradation in the system(Ri,r

s in Eq. 7) is given by

Ri,rs � k�C2HCl3� (10)

where [C2HCl3] is the molar concentration of TCE.The “laminar finite rate” model, in which the transport

equations are solved for species mass fractions with the pre-defined chemical reaction mechanism, was chosen for reactionmodeling. The reaction rates appear as reactive terms in thespecies conservation equations. Models of this type are suitablefor simulating a broad range of reacting systems. The Reynolds

number for the reactor at the highest air flow rate of 3 10�4

m3 s�1 was about 500, which is within the laminar regime.The three-dimensional physical domain of the photoreactor

was discretized with approximately 42,000 hexahedral andtetrahedral cells. Increasing the number of cells to 100,000through solution-adaptive mesh refinement changed the con-centration results by 3%, indicating a reasonable sensitivityof the solution to cell size. Physical and chemical properties ofthe species (that is, TCE and air as contaminant-carrying me-dium) were specified for calculating the coefficients of thegoverning equations. The no-slip boundary condition was se-lected at the surfaces of the lamp and the reactor. Constant fluidvelocity with TCE mass fractions and pressure were specifiedas boundary conditions for the inlet and outlet flows, respec-tively. A segregated implicit solution algorithm was used tonumerically solve the governing equations. The relative errorbetween two successive iterations was specified with a conver-gence criterion of 10�4 for each scaled residual component ofmass, velocity, and TCE concentration.

Experimental

Two types of photoreactor setups were used for the photo-catalytic experiments involved in this research (complete de-scriptions are presented elsewhere8). Figure 1 shows the sche-matic of the glass-plate differential photoreactor used todetermine the rate of TCE photocatalytic oxidation. It consistedof a 25-mm-wide aluminum reactor designed to allow thecontaminated air to flow over the TiO2 photocatalyst (DegussaP-25, Degussa AG, Dusseldorf, Germany), that was coated onglass plates and placed in the reactor. The reactor was coveredby quartz window; gaskets between the quartz window and thealuminum block created a flow passage of 25 mm (width) 3mm (height) above the photocatalyst glass plates. The amountof TiO2 coated onto glass slides was about 16 g m�2 and the

Figure 1. Glass plate differential photoreactor.


total length of TiO2-coated slides was 22.5 mm. Two low-pressure monochromatic germicidal lamps (G10T51/2/L, LightSources Inc., Orange, CT) with 254-nm output, provided UVillumination to the photocatalyst. The UV irradiance was reg-ulated at 30 W m�2 by adjusting the distance between thelamps and the catalyst surface. UV irradiance was measured bya radiometer (IL1700, SED240 detector, International LightInc., Newburyport, MA).

The annular flow photoreactor setup (Figure 2), on which theCFD analysis was conducted, consisted of a glass annularreactor, a mass flow controller, and a syringe pump for deliv-ering TCE. The annular reactor (custom-made, Trojan Tech-nologies Inc., London, Ontario, Canada) was equipped with aUV lamp and a quartz sleeve, which was made of diffusedquartz to allow UV to penetrate through and irradiate/activatethe TiO2 photocatalyst (Degussa P-25, Degussa AG) coated onthe outside wall of the reactor. Dimensions for the annularreactor were 0.035 m OD, 0.024 m ID (also the OD of thequartz sleeve) and 0.29 m height, with the effective volume of9 10�5 m3. The UV irradiance in the annular reactor and atthe surface of the photocatalyst was constant at 73 W m�2 forall experimental runs.

TCE-contaminated air entering the photoreactor was gener-ated using the syringe pump that allowed for the introduction ofpure TCE (Fisher certified ACS) into a stream of clean air(pressurized building air) controlled by the mass flow control-ler (Figures 1 and 2). The retention times and inlet concentra-tions were adjusted using the mass flow controller for air andthe injection rate of the syringe pump, respectively. Coveringdifferent portions of the UV lamp in the annular photoreactorallowed one to vary the effective photocatalyst surface area andphotoreactor length. This provided an opportunity to run theexperiments with different photoreactor lengths while main-taining the same inlet concentrations.

Gas samples were taken from the inlet and outlet streams andwere analyzed for the primary contaminant (that is, TCE).Stainless steel tubing (1.6 mm diameter) was used for onlinesamplings from the inlet and outlet streams. Gas analysis was

conducted using a gas chromatograph equipped with massspectrometer detector (GC/MS, Saturn 2200, Varian Associ-ates, Palo Alto, CA) and a megabore capillary column (CPSil-8CP5860). The injector temperature was 200°C. The initialcolumn temperature was 50°C, held constant for 120 s, afterwhich it was increased to 110°C at a rate of 0.67°C/s. Gassamples were injected into the GC/MS using a six-port sam-pling/injection valve (Valco Instruments Co., Inc., Houston,TX). A minimum number of three replicate samples wereinjected.

Results and DiscussionKinetics of TCE photocatalytic oxidation

The TCE photocatalytic oxidation rate was directly calcu-lated from the change in TCE concentration and gas retentiontime in the differential glass-plate photoreactor.8 The effect ofmass transfer was investigated by studying the oxidation rate atvarious air flow rates. At flow rates above 1.4 10�4 m3 s�1,the reaction rate remained constant, indicating that the processwas not mass transfer limited. This was in agreement with theresults obtained by Keshmiri et al.21 using the same experi-mental setup and conditions. To ensure gas-phase mass transferdid not influence the process in further experiments, the oxi-dation rate was determined at a flow rate of 1.7 10�4 m3 s�1.

Differential mode operation (that is, very low conversion) ofthe glass-plate photoreactor was confirmed by monitoring thefractional changes of pollutant concentrations that were 10%for all the experiments. Also, control experiments were con-ducted without a photocatalyst and under direct photolyticconditions. The results showed no significant oxidation of TCEduring the photolysis experiments, indicating that the removalsobtained during the photocatalytic oxidation tests were primar-ily attributed to the presence of TiO2 and reactions involvingOH radicals generated on the surface of the photocatalyst.

Figure 3 shows that the photocatalytic oxidation of TCE onTiO2 photocatalyst followed first-order kinetics with respect toTCE concentrations up to 1.6 g m�3. These findings agree with

Figure 2. Setup of the annular photoreactor for experimental analysis of TCE photocatalysis.Annular photoreactor was used for the CFD analysis.


those of other researchers2,22 who reported first-order oxidationkinetics for relatively low concentrations of TCE. Because theconcentration gradients along the reactor and in reactor crosssection are negligible throughout the differential reactor, therate constant of the surface reaction could be obtained directlyfrom the TCE concentration in the reactor inlet. The kineticinformation obtained from the differential reactor was thencorrected for the irradiance before being used for the CFDmodeling of the annular reactor. This was because the irradi-ance at the catalyst surface was different for the two reactors asa result of different reactor geometries and configurations. Thecorrection for irradiance from the differential to the annularphotoreactor was made based on separate experiments thatshowed a relatively linear relationship between the reactionrate and irradiance. When the reaction rate constant was cor-rected for the UV-irradiance difference in the annular anddifferential reactors, the surface rate constant for the annularphotoreactor was 1.2 10�2 m3 m�2 s�1.

The first-order dependency of the reaction rate on UV irra-diance obtained in this work and used for CFD modeling of theannular photoreactor is in agreement with the literature. Up-adhya and Ollis22 reported that for systems with relatively highphoto-efficiency, the rate of TCE photocatalysis follows afirst-order irradiance dependency, which is believed to be theresult of fairly high gas-phase TCE concentration (between 0.1and 1.6 g m�3) and high photo-efficiency in the photoreactor.

Comparison between the modeling and experimentalresults

CFD results were compared to those obtained experimen-tally for various reactor lengths (0–0.25 m), TCE concentra-tions (0.08–0.67 g m�3), and flow rates (3 10�5–3 10�4

m3 s�1). Figure 4 compares the CFD-simulated and experimen-tal results for TCE concentrations along the length of theannular photoreactor during photocatalytic oxidation (flow rate1.7 10�4 m3 s�1, inlet concentration 0.5 g m�3). As ex-pected, the TCE concentration decreases along the photoreac-tor. The results show a higher initial oxidation rate near theinlet of the photoreactor, which then gradually decreases as theconcentration of the pollutant decreases along the length of the

reactor. The higher rate near the inlet was accurately predictedby CFD, and is the result of both the higher concentration ofTCE in these areas and the first-order kinetics behavior forTCE photocatalysis.

Figure 5 shows the effect of flow rate on TCE degradation atan inlet concentration of approximately 0.5 g m�3. The exper-imental data indicated that increasing the flow rate (or decreas-ing the retention time) reduces the percentage of TCE removed,which was predicted by CFD simulation. The relationshipbetween TCE removal and flow rate was to some extent dif-ferent from that predicted by the first-order oxidation kinetics.Although first-order kinetics behavior predicts an exponentialdrop in TCE removal with increasing flow rate, a nearly linearrelationship was observed at low flow rates. This behavior,observed experimentally and predicted by CFD simulation,could be the result of mass-transfer limitations. At low flow

Figure 3. Rate of TCE photocatalytic oxidation as afunction of concentration.Error bars represent the standard deviation for a minimumreplication of three samples.

Figure 4. Modeling and experimental results of the con-centrations of TCE along the length of the pho-toreactor (flow rate 1.7 � 10�4 m3 s�1; inletconcentration 0.5 g m�3).(�) Represents experimental data; solid line represents CFDsimulation. Error bars represent the standard deviation for aminimum replication of three samples.

Figure 5. Modeling and experimental results of TCE re-moval at various flow rates (inlet concentration� 0.5 g m�3).(�) Represents experimental data; solid line represents CFDsimulation. Error bars represent the standard deviation for aminimum replication of three samples.


rates, there would be a limited degree of radial mixing alongthe photoreactor that, in turn, prevents TCE molecules that arefarther away from the surface boundary layer from reaching thesurface of the photocatalyst for adsorption and subsequentdegradation. At relatively high gas velocity, increasing the flowrate will induce some degree of turbulence and mixing withincertain regions of the photoreactor. Such a phenomenon wouldincrease the diffusion rate of the chemicals to and from thephotocatalyst surface. Thus, with reduced mass-transfer limi-tation at higher flow rates, the percentage removal and reactorbehavior approach an exponential shape and the removal rate isless dependent on flow rate.

The removal of TCE at various concentrations is illustratedin Figure 6. Both experimental data and CFD modeling indi-cate the percentage removal of TCE remains constant at rela-tively high concentrations (�0.2 g m�3). Such behavior can beexplained based on the first-order kinetics for the oxidation ofTCE. For first-order kinetics, the rate of oxidation increaseswith concentration, resulting in higher amounts of TCE re-moved from the system (Figure 6). This, in turn, corresponds toa relatively constant percentage removal of TCE throughout therange of concentrations between 0.2 and 0.7 g m�3. At con-centrations 0.2 g m�3, however, there is a discrepancybetween the CFD modeling and experimental results; themodel does not predict the high degree of removal observedexperimentally. This difference was likely the result of degra-dation caused by photolysis that is generally insignificant,especially at higher concentrations. However, when the con-centration of TCE decreases and the photocatalytically initiatedsurface reactions become less pronounced, the small amount ofremoval obtained by photolysis could become more significant.This hypothesis was further examined using separate photolyticexperiments with the glass-plate photoreactor at TCE inletconcentrations of 0.1 g m�3. About 4–5% TCE removal wasobserved, indicating the potential, albeit small, impact of pho-tolysis to the overall removal of TCE. Given that CFD simu-

lation did not account for homogeneous reactions such as thoseinitiated by photolysis, it showed some deviations from theexperimental data at low TCE concentrations (Figure 6).

Overall, Figures 4 through 6 provide an evaluation of theCFD simulation of the photoreactor under various operatingconditions. The results demonstrate close agreement betweenthe modeling prediction and the experimental data within therange of reactor lengths, contaminant concentrations, and flowrates that were examined. Such close agreements between CFDpredictions and experimental results indicate that CFD couldbecome a valuable tool to understand the system and enhancethe design of the photoreactors for optimum photocatalystactivities. Preliminary investigations in this area are presentedbelow.

CFD modeling of different reactor designs

A number of attempts were made to examine the potentialfor using CFD simulation to improve the photoreactor perfor-mance. Different photoreactor configurations and designs wereevaluated by studying their effects on reactor hydrodynamicsand overall TCE removal efficiency. These results were thencompared to the performance of the prototype photoreactorevaluated experimentally in this research.

Figure 7, Design A, shows the CFD modeling of TCEconcentration along the length of the annular prototype photo-reactor and the TCE deposition rate on the photocatalyst sur-face. Clearly, the concentration is higher on the left-hand sideof the UV lamp, opposite the inlet. This could be explained bythe nonuniform flow distribution in the reactor. On the oppositeside of the inlet, the flow rate is considerably higher becausethe inlet position leads the flow to one side of the reactor. Thisresults in a higher gas velocity, which shortens residence timeand produces a higher TCE concentration in these regions. Thedeposition rate on the surface of the photocatalyst is also higheron the left-hand side, where the gas flow rate is higher. In anattempt to influence these reactor hydrodynamics, the reactorinlet position was changed from perpendicular (Design A) toparallel (Design B) to the reactor length. As shown in Figure 7,Design B produces significant changes in the concentrationdistribution along the reactor. Also, the deposition rate at thesurface of the photocatalyst (that is, along the reactor wall)became more uniform (Figure 7, Design B). The overall reactorperformance, however, was not significantly improved, in-creasing only slightly from 67 to 69% (Table 1). This could bethe result of a balance between the two factors associated withDesign B. On the one hand, less mixing and lower diffusion ofTCE to the surface of photocatalysts have a negative impact onthe overall degradation of TCE (disadvantages of configurationB to A). On the other hand, more uniform flow distribution andoptimum use of the photocatalyst in all regions of the photo-reactor bring about a positive impact on TCE degradation(advantages of configuration B to A). Despite their similarperformances, Design B has some advantages for the long-termoperation of the photoreactor. This design results in a moreuniform shear stress on the photocatalyst surface, which couldbe important if photocatalyst stability is an issue, such as insituations where the coated photocatalyst is prone to shearstress and abrasion. Furthermore, Design B provides a moreuniform deposition/adsorption, which could be of consider-ation during the treatment of some organic compounds (such as

Figure 6. Modeling and experimental results of TCE re-moval at various concentrations (flow rate1.7 � 10�4 m3 s�1).(�) Represents experimental data of percentage removal; (f)represents experimental data of amount removed; solid linerepresents CFD simulation of percentage removal; dashed linerepresents CFD simulation of amount removed. Error barsrepresent the standard deviation for a minimum replication ofthree samples.


aromatics) that form polymeric intermediates that inactivate thephotocatalyst. Finally, uniform flow through the reactor resultsin a more consistent use of the photocatalyst, which could alsobe regenerated more effectively/consistently upon inactivation.

In Figure 8, Design A depicts CFD modeling of mass frac-tion of TCE at the prototype photoreactor cross section, located0.085 m from the reactor inlet (flow rate 1.33 10�4 m3 s�1,inlet concentration 0.54 g m�3). The results show that TCEconcentration is not symmetric in the reactor cross section andthat the overall concentration is higher at the opposite side ofthe inlet (the left-hand side), where there is a higher gas flowrate and shorter local residence time. Design A also shows aradial concentration gradient between the outer and inner sur-faces of the reactor with lower concentration at the outersurface of the reactor, where the photocatalyst is present in theform of coating on the reactor wall. This concentration gradientis anticipated because TCE degradation takes place only on thesurface of the TiO2 photocatalyst. Therefore, changing thecontaminated air layer flowing over the photocatalyst surface,by altering the distance between the lamp sleeve and thephotocatalyst surface, could provide more uniform radial con-centration distribution and potentially improve reactor perfor-mance.

Two other designs were simulated by changing the diameterof the photoreactor sleeve (from 23 to 27.5 and 14 mm) to

increase/decrease the air layer by a factor of 2 (from 4.5 mm inDesign A, to 9.0 mm in Design C, and to 2.25 mm in DesignD). In all the design alternatives, the inlet flow rate and TCEconcentration remained the same (flow rate 1.33 10�4 m3

s�1, inlet concentration 0.54 g m�3). Also, the photoreactordiameter and, in turn, the photocatalyst surface area, remainedunchanged. Figure 8 illustrates that a thicker air layer (DesignC) creates a nonuniform concentration distribution, but with athinner air layer (Design D), the concentration distribution ismore uniform in the photoreactor cross section. In addition, asthe air layer in the photoreactor decreases (Designs C, A, andD, respectively), the radial concentration gradient becomesmore uniform. This results in an increase in the diffusion rateof the chemicals to the photocatalyst surface and, thus, en-hancement in the efficiency of the catalyst for eliminating thecontaminant of interest. The CFD modeling results (Table 1)predict an improvement in the photoreactor performance (from67 to 75% degradation) as a result of reducing the air layer bya factor of 2 (from 4.5 mm in Design A to 2.25 mm in DesignD). This improvement, predicted by CFD, was substantiated bypreliminary experiments that showed the rate of TCE removalincreased by almost 150% when the air layer was decreased bya factor of 3 in a separator flat-plate reactor configuration.Although this experiment was performed in a different reactorconfiguration, the preliminary test proved that smaller air lay-

Table 1. Reactor Performance at Various Conceptual Designs*

Design Description TCE Mass Fraction Outlet/Inlet TCE Removal (%)

A Inlet cross flow, air layer 4.5 mm 1.5 10�4/4.4 10�4 67B Inlet parallel flow, air layer 4.5 mm 1.4 10�4/4.4 10�4 69C Inlet cross flow, air layer 9.0 mm 1.9 10�4/4.4 10�4 56D Inlet cross flow, air layer 2.25 mm 1.1 10�4/4.4 10�4 75

*The performance is reported in terms of the outlet/inlet TCE mass fraction and the TCE removal (flow rate 1.33 10�4 m3 s�1).

Figure 7. CFD modeling of contours of mass fraction and surface deposition rate (kg m�2 s�1) of TCE for thecross-flow inlet (A) and parallel-flow inlet (B) photoreactors (flow rate 1.33 � 10�4 m3 s�1; inlet concentra-tion 0.54 g m�3).Inlet is on top of the photoreactor (Z direction). There is no photocatalyst in the areas where the deposition rate is zero (at the top and bottomof the photoreactors).


ers can increase the removal rate. Although narrowing the airlayer will increase the pressure drop, it is anticipated that thegreater reactor performance will compensate for this disadvan-tage. Reaching an optimal air layer might be achieved byconsidering a balance between the highest possible removalrates at an appropriate pressure drop.

Conclusions

Photocatalytic oxidation of TCE followed first-order oxida-tion kinetics for the range of TCE concentrations between 0.1and 1.6 g m�1. CFD modeling of a photocatalytic reactor withsurface reaction for TCE oxidation at various reactor lengths,pollutant concentrations, and flow rates predicted the experi-mental data well. CFD modeling of the annular photocatalyticreactor provided detailed information on concentration gradi-ents of the contaminant in the reactor and provided accurateanalysis and explanation of the experimental results. Improve-ment in photoreactor performance could be achieved by usingthe information provided by modeling. CFD modeling demon-strated that under similar flow rate conditions, the thickness ofcontaminated air flow (the contaminated air layer flowing overthe photocatalyst surface) influences the reactor performance.A thinner contaminated air layer provided more uniform radialconcentration distribution of TCE and improved the reactorperformance, within the range of air layers (2.25–9.0 mm) andflow rates (1.33 10�4 m3 s�1) examined.

AcknowledgmentsThe authors acknowledge financial support from the Natural Sciences

and Engineering Research Council of Canada (NSERC), Canada Founda-tion for Innovation (CFI) New Opportunity Fund, and British ColumbiaKnowledge Development Fund (BCKDF).

Notation

� � density, kg m�3

�� stress tensor, N m�2

v�i,r � stoichiometric coefficient for reactant i in reaction rv �i,r � stoichiometric coefficient for product i in reaction r

�j,r � forward rate exponent for each reactant and product species jin reaction r

Cj,r � molecular concentration of each reactant and product speciesj in reaction r, kg m�2 s�1

g � gravitational force, m s�2

J� i � diffusion flux of species i, kg m�2 s�1

k � first-order surface reaction rate constant, m3 m�2 s�1

Mi � symbol denoting species iMw,i � molar weight of the species i, kg kmol�1

N � total number of species in the systemP � pressure, N m�2

r � photocatalytic reaction rate, g m�2 min�1

Ri � rate of production or depletion of species i arising fromvolumetric reactions, kg m�3 s�1

Ris � rate of surface production or depletion of species i, kg m�2

s�1

Ri,rs � rate of molar production or depletion of species i, kmol m�2

s�1

v� � velocity, m s�1

Yi � species i

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22. Upadhya S, Ollis DF. A simple kinetic model for the simultaneousconcentration and intensity dependence of TCE photocatalyzed de-struction. J Adv Oxidat Technol. 1998;3:199-202.

Manuscript received Sep. 12, 2004, and revision received Mar. 1, 2005.


Characterization and Drainage Kinetics ofColloidal Gas Aphrons

Divesh Bhatia, Gaurav Goel, Sidhartha K. Bhimania, and Ashok N. BhaskarwarDept. of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi, 110 016, India


The variation of air holdup and stability of colloidal gas aphrons (CGAs) with stirringtime, surfactant concentration [6, 8.1, 10, and 12 mM sodium lauryl sulfate (SLS)], andstirring speed (3500–7000 rpm) was investigated, and an empirical correlation betweenthe air holdup and stirring time was obtained. A first-order model for the drainage ofCGAs was proposed and it correlated very well with the experimental data. A populationbalance model was proposed and correlated to the first-order drainage model to obtainan expression for the size distribution of CGA bubbles. A narrower size distribution atlarge stirring times was observed, which was correlated to the stability of CGAs. Thestabilization of CGA dispersion with attainment of equilibrium size distribution at longerstirring times, analogous to homogenization of emulsions, was observed and an empiricalexpression for this time was reported. Mixing of two oppositely charged CGAs, generatedusing SLS (anionic) and cetyl trimethyl ammonium bromide (CTAB) (cationic), showed noeffect on their stability, air holdup, and drainage kinetics. © 2005 American Institute ofChemical Engineers AIChE J, 51: 3048–3058, 2005Keywords: colloidal gas aphrons, drainage kinetics, size distribution, oppositely chargedCGAs, stability; air hold-up

Introduction

Sebba1 first described colloidal gas aphrons (CGAs) as mi-crofoam with colloidal properties. In a later study, Sebba2

defined CGAs as microbubbles created by intense stirring(5000–10,000 rpm) of a surfactant solution. Characterizationof CGAs—that is, the study of effects of various parameters,such as surfactant concentration, salt concentration, stirringtime, pH, and temperature on air holdup and stability ofCGAs—was done by Jauregi et al.3 A review of the literatureon CGAs was presented by Jauregi and Varley.4 An updatedsummary of the literature on stability and drainage of CGAs isgiven in Table 1.

TheoryStructure and properties of CGAs

The term “colloidal” was used because of the small size ofthe bubbles (10–100 microns in diameter). Sebba2 proposedthat CGAs consisted of a gaseous inner core surrounded by athin aqueous surfactant film composed of two surfactant layersand an outer electrical double layer that stabilized the structure,as shown in Figure 1. However, there is no real conclusiveevidence available for this structure. CGAs are gas micro-bubbles and thus dynamic systems, continuously undergoingchanges resulting from creaming, bubble breakage, coales-cence, and disproportionation. Therefore, they possess a lim-ited stability.

Some of the salient features of CGAs can be summarized asfollows:

(1) Larger specific surface area and higher stability thanthose of normal bubbles because of their small size, likecharges, and an electrical bilayer.

(2) Flow properties similar to those of water and a high gas

Correspondence concerning this article should be addressed to A. N. Bhaskarwar [email protected].


MATERIALS, INTERFACES, AND ELECTROCHEMICAL PHENOMENA


content (55–65%), making them one of the lightest compress-ible liquids at ordinary temperature and pressure conditions.

(3) Hydrophobicity of the encapsulating surfactant shells,enabling them to collect oil globules, protein molecules, and soforth.These features of CGAs provide applications such as the re-moval of naphthalene from a contaminated soil matrix,9 sepa-ration of fine fibers from a lean slurry of cellulosic pulp in aflotation column,10 and extraction of copper from aqueousacidic solution in LIX 622lkerosene (mixture of 5-dodecylsali-cyal-dioxime and tridecanol in kerosene),11 to cite but a few.

Characterization of CGAs

According to Sebba,2 the formation of CGAs occurs asfollows. The surface waves, generated as the result of intensestirring, strike against the baffles. They entrap a thin film ofatmospheric air between the liquid and the surface of baffles,and reenter the solution. This film of air breaks into micro-

bubbles or CGAs. Characterization of CGAs should involveconsiderations of the stability of the dispersion of microbubblesand the air holdup.

Stability. The stability of CGAs, measured in terms ofhalf-life (tdh), is defined as the time taken by “the CGAsdispersion–bulk liquid interface” to reach half its final height.The aphrons phase separates easily from the bulk-liquid phasebecause of its buoyancy. Figure 1 schematically shows theformation and drainage of CGAs as well as the state at half-lifeof CGAs.

Air Holdup. Air holdup is defined as the volume percent-age of entrapped air in the CGA dispersion, expressed as

� ��VTo � x��

VTo� 100 (1)

where VTo is the initial volume of CGA dispersion (m3) and x�is the initial volume of surfactant solution (m3).

Table 1. Summary of the Open Literature on Stability and Drainage of CGAs

Author(s) Surfactant* Parameters StudiedDrainageVelocity

BubbleSize

Stabilityand AirHoldup Conclusions

Jarudilokkulet al.5

Tween 20(n)

Initial protein concentration,pH, protein/CGA volumeratio, surfactantconcentration, rpm,stirring time, NaClconcentration

Yes Stability increases with increasing surfactantconcentration and stirring time.

Separation decreases with increasing surfactantconcentration.

pH should be such that electrostatic interactionsfavor separation.

Jauregi etal.6

AOT (a) pH, surfactantconcentration, saltconcentration,temperature

Yes Yes Yes Aphron diameters could be predicted usingmodels proposed for bubble breakage inagitated tanks and liquid drainage in foamsand CGA dispersions.

Drainage rate was predicted using foam drainagemodel modified for CGAs.

Electron microscopy and X-ray diffractiontechniques indicate the existence of aninterface of many layers.

Jauregi andVarley4

AOT (a) Stirring time, ionic strength,initial mass of protein,pH, protein/CGA volumeratio, magnetic stirrerspeed

Yes Speed of stirring and pH had little effect onseparation parameters.

Concentration of protein and surfactant in theinitial mixture had an important effect onseparation. The optimum protein recovery was95% without significant loss of lysozymeactivity.

Jauregi etal.3

AOT (a) AOT concentration,temperature, stirring time,salt concentration, pH

Yes Surfactant and salt concentration had maximumeffect. Empirical equation for air holdup andhalf-life determined.

Power required for generating CGAs decreaseswith increase in gas holdup.

Chaphalkaret al.7

HTAB (a)SDBS (c)Tergitol15-S-12 (n)

Drainage time, type ofsurfactant, surfactantconcentration, ionicstrength

Yes Yes Bubble size in the range 30–300 �m.Mean diameter for Tergitol was reported to be

less than that for SDBS and HTAB.Increase in surfactant concentration reduced the

mean diameter for all three surfactants.Increase of ionic strength reduced mean diameter

for ionic surfactants, but it had no effect onthe nonionic surfactants.

Amiri et al.8 TTAB (c) Yes Yes Drainage of CGAs was shown to be consistentwith hindered rising of 35 �m sphericalbubbles with 0.75 �m shells in downwardflowing solution.

The assumption of no coalescence may not holdgood at later intervals. Ascertaining CGAdispersion—foam interface provides anotherlimitation.

*n, nonionic surfactant; c, cationic surfactant; a, anionic surfactant.


Figure 1. CGA diagrams.


Models for liquid drainage from CGAs

Model 1. The rate of drainage of liquid from CGAs isdefined as

r � �dVa

dtd(2)

where td is drainage time (s), and Va is the volume of CGAdispersion at time td (m3).

A first-order model is proposed for liquid drainage whereinthe rate of drainage at a particular time is proportional to thevolume of CGA dispersion still present at that time.

Thus, the first-order drainage equation is

�dVa

dtd� mVa (3)

where m is the rate constant for drainage (s�1).From the volume balance on CGA dispersion–bulk liquid

system, assuming no air escapes from the system, we obtain

VTo � VL � Va (4)

where VL is the volume of surfactant solution in the bulk-liquidphase collected at the bottom at time td, (m3).

By combining Eqs. 3 and 4, we obtain

d�VTo � VL�

dtd� �m�VTo � VL� (5)

The initial condition (a) for Eq. 5 is VL � 0 at td � 0, and thesolution of Eq. 5 subject to initial condition (a) is

ln�VTo � VL

VTo� � �mtd (6)

Model 2. Consistent with actual observations, a foam layerof constant thickness is assumed to be present at the top of theCGA dispersion. Because of buoyancy, CGAs rise to theaphron–foam interface, where they release most of the associ-ated surfactant solution that drains down to enter the bulk-solution pool at the bottom. The following assumptions havebeen made while performing a population balance on theaphrons in the CGA dispersion:

(1) Aphrons are spherical and they are numerous enough toapproximate their size distribution by a continuous function.

(2) Each aphron has an identical trajectory in the aphron-phase space.

(3) There is no breakage of aphrons.(4) No new aphrons are formed in the system.(5) The size of a particular aphron does not change with

time.A population balance for aphrons in a fixed subregion R1 of

the phase space is

�R1

��n/�td � � � �vxn� � � � �vin� � D � B�dR � 0

where n is the number of aphrons per unit length; vx is thevelocity of an aphron in the vertical direction; vi � dL/dtd,where L is the diameter of the aphron; D is the number ofaphrons collapsing in unit time per unit length; and B is thenumber of aphrons forming in unit time per unit length.

Because the subregion R1 is arbitrary, the integrand mustvanish identically. D, B, and vi are all equal to zero for oursystem because of the assumptions (3), (4), and (5), respec-tively. Thus, the population balance becomes

�n

�td�

��vxn�

� x� 0 (7)

where n � n(x, td, L).Suppose the aphrons are moving at their terminal velocity

and Stokes law is applicable for individual aphrons of differentsizes, even if in the presence of other sized aphrons. Thevelocity of the aphrons is

vx�L� �gL2�

18�� cL2 (8)

where � is the difference between the densities of the aphronsand the surfactant solution and c (�g�/18�) is a constant forthe system.

Substituting the value of vx(L) from Eq. 8 into Eq. 7, weobtain

�n

�td� c

��L2n�

� x� 0

It has been assumed that the diameter L does not vary withrespect to time, and thus for rising aphrons with x. Thus,

�n

�td� cL2

�n

� x� 0 (9)

Boundary conditions (b) and (c) for Eq. 9 are, respectively, n(td� 0) � n0 and n(x � 0) � 0.

Solving Eq. 9 using Laplace transformation with boundaryconditions (b) and (c), we obtain

n � n0 0 td x

cL2

n � 0 td x

cL2

Let, at a particular time td, a bubble of diameter L� reach aheight x. Equation 8 would then yield


L� � � x

ctd(10)

and the above solution becomes

n � n0 0 L L� (11)

n � 0 L L� (12)

Consider an equivalent cylinder of diameter Le and height Lhaving the same volume as that of a spherical aphron ofdiameter L. Thus,

�L3

6�

�Le2

4L (13)

That is, Le � (2/6)/L.The volume of aphrons of sizes between diameter L and L �

dL leaving the aphron–foam interface in time dtd is given by

dVGL � �dtd�vx�L��Le

2

4n�H, td, L�dL�

where H is the height of the aphron–foam interface.Thus, the total volume of aphrons leaving the aphron–foam

interface is

dVG � ��0

� vx�L��Le2

4n�H, td, L�dL�dtd

That is,

dVG

dtd� ��

0

� vx�L��Le2

4n�H, td, L�dL

By inserting the values of vx(L) and Le from Eqs. 8 and 13,respectively, we obtain

dVG

dtd� �

�g�

108� �0

�

L4n�H, td, L�dL

Recalling the values of n from Eqs. 11 and 12, we obtain

dVG

dtd� �

�g�

108� ��0

L�

L4n0�L�dL ��L�

�

L4n0�L�dL��

�g�

108� �0

L�

L4n0�L�dL (14)

The volume of liquid drained is

VL � VTo�1 � �0� �1 � �

�VG (15)

where �0 is the initial air holdup and VG is the volume of gasin the CGA dispersion.

Equations 14 and 15, with the assumption of constant gasholdup with drainage time, yield

dVL

dtd�

�1 � �0��g�

108��0�

0

L�

L4n0�L�dL (16)

For monodisperse aphrons in CGA dispersion (that is, n0(L) �n0 � a constant), Eq. 14 leads to

dVG

dtd� �

�g�n0

108� �0

L�

L4dL

and further, in view of Eqs. 8 and 10, we obtain

dVG

dtd� �

n0�

30 �H

td�5/ 2�18�

g��3/ 2

Using Eq. 15 and the definition of c, we obtain

dVL

dtd�

�1 � �0�n0�

30�0�H

td� 5/ 2

c�3/ 2 (17)

Thus, Eq. 17 represents the liquid-drainage rate for a mono-disperse CGA dispersion.

To obtain n0 as a function of the aphron diameter, we equatethe value of dVL/dtd from Eq. 6 to that from Eq. 16:

mVToe�mtd �

�1 � �0��g�

108��0�

0

L�

L4n0�L�dL

that is,

�0

�H/ctd

L4n0�L�dL �108�mVTo�0

�1 � �0��g�e�mtd

It is assumed that L4n0(L) is a slowly varying function oftime. Taking the derivative with respect to td, and substitutingthe value of td from Eq. 10, we obtain

n0�L� �3888�2m2VTo�0H

�1 � �0��g2��2 �e�18m�H/g�L2

L7 � (18)

Thus,

n0�L� �a

L7 exp��b

L2 � (19)


where a and b are constants for a given system given by

a �3888�2m2VTo�0H

�1 � �0��g2��2 (20)

b �18m�H

g�(21)

By inserting the aphron-size distribution function n0(L) fromEq. 19 into Eq. 16, we obtain

dVL

dtd�

�1 � �0�

�0

�ac

12be��bctd/H�

which leads to

VL ��1 � �0�

�0

�aH

12b2 �1 � e��bctd/H� (22)

We therefore deduce the following equation:

dVL

dtd� �bc

H �� 1 � �0��aH

12b2�0� VL� (23)

Thus, Eq. 23 predicts a first-order liquid-drainage rate for theCGAs with the aphron size distribution represented by Eq. 19.

Experimental Approach

The experimental setup consists of a 5-L beaker with twovertical inverted L-shaped baffles as shown in Figure 1. Thesurfactant solution is stirred in the beaker with a spinning discat a constant stirring speed ( 3000 rpm). The initial surfactant-solution level was 2 cm above the spinning disc.

To study the effects of parameters—surfactant concentration,stirring speed, and time of stirring—numerous sets of experimentswere performed using surfactant concentrations below the criticalmicelle concentration (CMC) of SLS (6 mM), equal to the CMC(8.1 mM), and above the CMC (10 and 12 mM). One preliminaryexploratory experiment was run for a long enough time such thatthere was no further change in the air holdup, that is, the steadystate was achieved. Experiments were performed for various stir-ring speeds, ranging between 3000 and 7000 rpm. For the abovesets of experiments performed, samples (�100 mL) of the stirredCGAs dispersions were taken out into measuring cylinders atdifferent stirring-time intervals, and the rise of the (CGA-disper-sion)–(bulk-liquid) interface with drainage time was noted.

In a separate set of experiments, CGAs were prepared froman anionic surfactant (8.1 mM SLS) and a cationic surfactant (2mM CTAB) simultaneously, in two separate CGA generators.Methyl red dye was added to the SLS-based CGA generator,which provided a yellow color to the CGAs. Equal volumes (50mL each) of the two oppositely charged CGAs were thenstirred together on a magnetic stirrer for 30 s, and then pouredinto a 100-mL measuring cylinder. Simultaneously, 100-mLsamples of SLS- and CTAB-based CGAs were separatelycharged to similar measuring cylinders, and the rise of the(CGA-dispersion)–(bulk-liquid) interface with drainage timewas recorded for the three cases.

Figure 2. Half-life vs. stirring time for various SLS concentrations and stirring speeds.

Figure 3. Air holdup vs. stirring time for various SLS concentrations and stirring speeds.


Results and DiscussionStability

Figure 2 shows the plots of half-life times vs. stirring timefor different surfactant concentrations (sodium lauryl sulfate,6–10 mM), each at three to four stirring speeds ranging from3600 to 6500 rpm. For short stirring times, the (CGA-disper-sion)–(bulk-liquid) interface was not clear, indicating a broadsize distribution of aphrons. Data for such short times aretherefore not being reported here. Figure 2 implies that the sizedistribution of aphrons improves with time, surfactant concen-tration, and stirring speed, seen in a general increase in thehalf-life time. The concentration range used by Jauregi et al.,3

who reported an increase in half-life with concentration, wasvery large (0.1–34 mM for AOT). In contrast, the concentrationrange in the current experiments was much smaller (6–10 mMfor SLS). As a result, there was only a limited increase in thehalf-life, that is, about 50–60 s. The increase in the stability ofCGAs with surfactant concentration perhaps results from theincreased repulsive forces between aphrons at higher concen-trations of surfactant either in the surfactant shells or in thebulk-liquid phase.13 Some of these experiments were run in

duplicates and the average experimental error in measurementswas estimated to be within �7%.

Air holdup

Figure 3 shows the semilog plots of percentage air holdupvs. stirring time (nondimensionalized with unit time) for fourdifferent surfactant (sodium lauryl sulfate) concentrations (6,8.1, 10, and 12 mM), and each at three to four stirring speeds(3800–6500 rpm). Figure 3 shows that as the time of stirringincreases, the percentage air holdup increases irrespective ofthe surfactant concentration and stirring speed. This is becausemore air is entrapped with time into the system until thesteady-state dispersion is reached. At low stirring speeds(�3000 rpm), there was no observable CGA formation. Withan increase in stirring speed beyond 3000 rpm, for a fixed timeof stirring and surfactant concentration, the air holdup in-creased, which can be attributed to the fact that with an in-crease in the stirring speed, the number of surface wavesstriking the baffles per unit time increased, thereby entrappingmore air microbubbles in the CGA dispersion during the sametime period.

Empirical correlation for air holdup

Figure 3 shows that for a given surfactant concentration, thesemilog plots of air holdup vs. stirring time (nondimensional-ized with unit time) are linear with almost equal slopes, andwith increasing intercepts at higher stirring speeds. In an effortto bring all experimental data points onto a single curve, airholdup was scaled with the maximum air holdup (�max) for aparticular stirring speed, and stirring time (t) was scaled withthe time required to attain steady state (ts). The data for fourdifferent surfactant concentrations (that is, 6, 8.1, 10, 12 mMSLS) mapped onto a single line of the form:

�/�max � k ln�t/ts� � p (24)

Figure 4 shows that the final air holdup correlation for theSLS–water system is

Figure 4. Best-fit line for air holdup fraction vs. ln(stir-ring time fraction).

Figure 5. Variation of �max and ts with stirring speed.


�/�max � 0.2407 ln�t/ts� � 0.9669 (25)

with a correlation coefficient of R2 � 0.9275.This correlation has an implicit dependency of �max and ts on

the surfactant concentration and the stirring speed. Two out ofthe four surfactant concentrations used were above the CMC ofSLS (8.1 mM), and one concentration was below the CMC.Figure 5 shows the variation of �max and ts, respectively, withthe stirring speed for all four surfactant concentrations. Thecorresponding R2 values of 0.6784 and 0.6959 imply that thesurfactant concentration, within the range used, does not havea significant effect on the air holdup. This observation issupported by Matsushita et al.,14 who have reported no increasein air content for concentrations of CTMAB 0.5 g L�1.Interestingly, however, we find from our experiments for SLSthat a concentration 25% lower than the CMC also leads toonly a slight decrease in air holdup. The empirical dependencyof �max and ts on the stirring speed, as obtained from Figure 5,is

�max � 0.004s � 44.017 (26)

and

ln ts � �2.0858 ln s � 20.74 (27)

where s is stirring speed (rpm).By combining Eqs. 25–27, we obtain the empirical depen-

dency of air holdup on the stirring time and stirring speed as

� � �0.004s � 44.017��0.2407 lnt

s�2.0858 � 4.025�(28)

Figure 6 shows a comparison of the experimental values of airholdup with the calculated values from Eq. 28. An excellent

Figure 6. Comparison between experimental air holdupvalues and the values obtained from empiricalcorrelation.

Figure 7. Plots of ln[(VTo � VL)/VTo] vs. td at 6 mM SLSand 5700 rpm.

Table 2. Variation of Rate Constant with Stirring Speed and Stirring Time

6 mM 8.1 mM 10 mM

StirringSpeed(rpm)

StirringTime(min)

Negativeof Slope

(Eq. 6), mCorrelationCoefficient

StirringSpeed(rpm)

StirringTime(min)

Negative ofSlope (Eq.

6), mCorrelationCoefficient

StirringSpeed(rpm)

StirringTime(min)

Negative ofSlope (Eq.

6), mCorrelationCoefficient

5700 6 0.0021 0.72 3600 12 0.0032 0.99 4800 8 0.0026 0.4810 0.0011 0.91 16 0.0032 0.95 12 0.0012 0.6314 0.0007 0.95 20 0.0026 0.97 16 0.0009 0.6820 0.0007 0.92 25 0.0016 0.99 21 0.001 0.9224 0.0007 0.98 30 0.0015 0.96 26 0.0007 0.9828 0.0008 0.99 36 0.0012 1 29 0.0006 0.95

6000 8 0.0012 0.8 5230 10 0.002 1 5500 6 0.0015 0.5412 0.0007 0.94 14 0.0013 0.97 10 0.001 0.8716 0.0007 0.99 18 0.0012 0.99 14 0.0008 0.9320 0.0006 0.98 23 0.0009 0.99 18 0.0007 0.8325 0.0007 0.99 29 0.0008 0.99 22 0.0007 0.86

35 0.0008 0.99 26 0.0007 0.985750 6 0.0014 0.98 30 0.0007 0.99

10 0.0007 1 6000 6 0.0013 0.814 0.0006 0.99 10 0.0007 118 0.0007 1 14 0.0006 1

6500 6 0.0015 0.98 18 0.0007 0.978 0.0011 0.97 22 0.0006 0.99

12 0.0008 1 26 0.0006 116 0.0013 0.54


agreement between the two sets of values (R2 � 0.8352)implies that Eq. 28 can express the experimental results verywell within the surfactant (SLS) concentration range explored.

Liquid drainage from CGAs

There was a rise in the (CGA-dispersion)–(bulk-liquid) in-terface with time as a result of the creaming, coalescence ofCGAs, and the accompanying liquid drainage. Figure 7 showsthe plots of ln[(VTo � VL)/VTo] vs. td, for 6 mM SLS and fordifferent stirring times at a speed of 5700 rpm. The plots forvarious other stirring speeds and surfactant concentrations weresimilar and are summarized in Table 2 in terms of the slopesand correlation coefficients. The high correlation-coefficientvalues show that the experimental data correlated well with thefirst-order drainage kinetics represented by Eq. 6. The values ofthe drainage-rate constant varied from 0.0032 to 0.0006, de-creasing with increasing stirring speeds and stirring times, asshown in Table 2. This indicates an increase in the stability ofCGA dispersion, which can be attributed to the attainment of anarrower size distribution of CGAs in the dispersion. Theconvergence of m values to a constant value of 0.0007 withincreasing stirring times indicates that an equilibrium sizedistribution will be attained for all stirring speeds above thecritical stirring speed for CGA generation. However, the timerequired to attain this equilibrium size distribution, te, will belarger for lower stirring speeds. It decreases exponentially with

stirring speed as shown in Figure 8. The stirring time wasscaled with te, and the drainage-rate constant m was plottedagainst this nondimensional stirring time. A fourth-degreepolynomial fit was used for these data as shown in Figure 9.

The best-fit values of n0 were obtained by fitting the exper-imental liquid-drainage rates to Eq. 17 for various values ofsurfactant concentrations, stirring times, and stirring speeds.Figure 10 shows a comparison of the experimental values ofdrainage rates with the predicted values from Eq. 17. Figure 10also shows that the model predicts a faster decrease in drainagerate with time in contrast to the experimental data. The com-parisons for other stirring speeds, surfactant concentrations,and stirring times yielded similar trends.

The experimentally measured surfactant-solution volumewas next used in Eq. 22 to find the best-fit values of a and b forvarious values of surfactant concentrations, stirring times, andstirring speeds. These were further used to calculate n0(L) fromEq. 19. Figure 11 shows the variation of n0 with the aphrondiameter for 6 mM SLS and different stirring times at a speedof 5700 rpm. Figure 11 also shows that the size distribution ofaphrons becomes narrower with increasing stirring time, fur-ther consolidating the analogous notion of homogenization.The variation of n0(L) for various other stirring speeds andsurfactant concentrations yielded similar results. The increaseof stability with stirring time can thus be explained by anarrower size distribution of aphrons at longer stirring times.At short stirring times, the size distribution is broad, leading to

Figure 8. Equilibrium time vs. stirring speed.

Figure 9. Plot of m vs. nondimensional stirring time forvarious stirring speeds.

Figure 10. Comparison of experimental and model-pre-dicted liquid drainage rate.

Figure 11. Aphron size distribution for 6 mM, 5700 rpm,and various stirring times.


a hazy interface between dispersion of CGAs and the bulksurfactant solution.

The values of a and b obtained by fitting the experimentaldata to Eq. 22 were compared with the values calculated usingEqs. 20 and 21. Figure 12 shows that the values of a and bobtained using Eqs. 20 and 21 are in agreement with thoseobtained by fitting the experimental data to Eq. 22 to within anorder of magnitude. The size distribution of aphrons, repre-sented by Eq. 19, can therefore broadly explain the first-orderdrainage kinetics of CGAs represented by Eq. 23.

Mixing of oppositely charged CGAs

Visual Observation. A 50-mL solution of SLS-basedCGAs, colored yellow with methyl red dye, was charged to a100-mL measuring cylinder. A CTAB-based solution of CGAs(50 mL) was then slowly added to the cylinder. It was observedthat the whole dispersion turned lighter yellow, as a result ofdilution, indicating the mixing of the oppositely chargedCGAs.

Drainage-Rate Measurements. Figure 13 shows ln[(VTo �VL)/VTo] vs. td plots for the drainage data obtained from exper-iments similar to those described earlier.

The following observations may be made:(1) The drainage curves overlap, indicating practically no

change in the drainage kinetics on mixing of the oppositely

charged CGAs, in contrast to oppositely charged foams that areknown to quickly destroy each other.2

(2) Upon mixing of oppositely charged CGAs, no apprecia-ble changes in stability and air holdup are observed, as seen inTable 3.

(3) The foam formed above the mixed (oppositely charged)CGAs, however, collapsed much faster than that above thedispersions of individual CGAs, indicating the instability offoam resulting from the oppositely charged surfactants. Thisfeature may be useful in quickly destroying the creamed foamformed in applications of CGAs, such as for separation of oils,pigments/dyes, and proteins, from aqueous emulsions, disper-sions, or solutions.

Conclusions

Characterization of CGAs has been done, and an empiricalcorrelation for the variation of air holdup with stirring time andstirring speed has been developed. First-order kinetics fordrainage of CGAs has been proposed and the resultant expres-sion was validated experimentally. A new population balancemodel has been developed, which for the first-order drainagekinetics yielded an expression for the size distribution ofaphrons. This size distribution has been used to explain theeffect of stirring time on the stability of CGAs. Convergence ofthe drainage-rate constant values indicates that an equilibriumsize distribution of aphrons for the dispersion is attained at afinite time te for all stirring speeds above the critical speed forformation of CGAs. An empirical correlation for te was foundto be an exponentially decreasing function of the stirring speed.For the first time, the behavior upon mixing of two oppositelycharged CGAs has been studied and the results surprisinglyshow no effect on drainage and stability, except in the tail-endfoam stage.

Figure 12. Comparison between parameters a and b obtained by curve fitting and model.

Figure 13. Plot of ln[(VTo � VL)/VTo] vs. drainage time forSLS, CTAB, and their mixture.

Table 3. Effect of Mixing of Oppositely Charged CGAs onStability and Air Holdup

Air Holdup(%)

Half-life(s)

SLS 59.25 194CTAB 65.9 229Mixture 64.23 223


Notation

B � number of aphrons forming in unit time per unit length, s�1 m�1

dVG � volume of aphrons leaving the aphron–foam interface in time dtd,m3

dVGL � volume of aphrons between diameter L and L � dL leaving the

aphron–foam interface in time dtd, m3

D � number of aphrons collapsing in unit time per unit length, s�1

m�1

H � height of the aphron–foam interface, mL � diameter of the aphron, m

Le � diameter of cylinder of height L and having the same volume asa sphere of diameter L, m

m � rate-constant for drainage, s�1

n � number of aphrons per unit length, m�1

n0 � number of aphrons per unit length at td � 0, m�1

r � rate of drainage of liquid from CGAs, m3 s�1

s � stirring speed, rpmt � stirring time, min

td � drainage time, stdh � half-life of CGAs, ste � time required for attaining equilibrium size distribution, mints � time required for attaining steady state for a given stirring speed,

mint� � nondimensionalized stirring time [�t/(1 min)], dimensionlessvx � velocity of an aphron in the vertical direction, ms�1

Va � volume of CGA dispersion at time td, m3

VG � volume of gas in the CGA dispersion at time td, m3

VL � volume of surfactant solution in bulk-liquid phase collected at thebottom at time td, m3

VTo � total initial volume of CGA dispersion, m3

x � vertical coordinate, mx� � initial volume of surfactant solution, m3

� � percentage air holdup of CGAs, dimensionless�max � maximum air holdup for a given stirring speed, dimensionless

�0 � percentage air holdup of CGAs at td � 0, dimensionless� � difference in densities of the aphrons and the surfactant solution,

kg/m3

Literature Cited1. Sebba F. Microfoams—An unexploited colloid system. J Colloid

Interface Sci. 1971;35:643-646.2. Sebba F. Foams and Biliquid Foams: Aphrons. 1st Edition. Chichester,

UK: Wiley; 1987.3. Jauregi P, Gilmour S, Varley J. Characterisation of colloidal gas

aphrons for subsequent use for protein recovery. Chem Eng J. 1997;65:1-11.

4. Jauregi P, Varley J. Colloidal gas aphrons: A novel approach to proteinrecovery. Biotechnol Bioeng. 1998;59:471-481.

5. Jarudilokkul S, Rungphetcharat K, Boonamnuayvitaya V. Protein sep-aration by colloidal gas aphrons using nonionic surfactant. Sep PurifTechnol. 2004;35:23-29.

6. Jauregi P, Mitchell GR, Varley J. Colloidal gas aphrons (CGA):Dispersion and structural features. AIChE J. 2000;46:24-36.

7. Chaphalkar PG, Valsaraj KT, Roy D. A study of the size distributionand stability of colloidal gas aphrons using a particle size analyzer. SepSci Technol. 1993;28:1287-1302.

8. Amiri MC, Woodburn ET. A method for the characterization ofcolloidal gas aphron dispersions. Trans IChemE. 1990;68A:154-160.

9. Roy D, Kongara S, Valsaraj KT. Applications of surfactant solutionsand colloidal gas aphron suspensions in flushing naphthalene from acontaminated soil matrix. J Hazard Mater. 1995;42:247-263.

10. Hashim MA, Gupta BS. The application of colloidal gas aphrons in therecovery of fine cellulose fibres from paper mill wastewater. Biore-source Technol. 1998;64:199-204.

11. Save SV, Pangarkar VG, Kumar SV. Liquid–liquid extraction usingaphrons. Sep Technol. 1994;4:104-111.

12. Jauregi P, Varley J. Colloidal gas aphrons: Potential applications inbiotechnology. Trends Biotechnol. 1999;17:389-395.

13. Noble M, Brown A, Jauregi P, Kaul A, Varley J. Protein recoveryusing gas–liquid dispersions. J Chromatogr B Biomed Sci Appl. 1998;711:31-43.

14. Matsushita K, Mollah AH, Stuckey DC, Cerro CD, Bailey AI. Predis-persed solvent extraction of dilute products using colloidal gas aphronsand colloidal liquid aphrons: Aphron preparation, stability and size.Colloids Surf. 1992;69:65-72.

Manuscript received Apr. 6, 2004, and revision received Mar. 4, 2005.


Numerical Study on the Underground CoalGasification for Inclined Seams

Lanhe YangCollege of Resources and Geosciences, China University of Mining and Technology, Xuzhou,

Jiangsu Province, 221008, China


According to the characteristics for combustion and gasification reactions occurring inthe gasification gallery, the mathematical functional relationship between the chemicalreaction rate and every influencing factor is studied. The dynamic nonlinear couplingmathematical models on underground coal gasification of inclined seams are established.The determination methods of major model parameters are introduced. Additionally, thecontrol volume method is adopted to find the numerical solution to the mathematicalmodels. The patterns of development and variation for temperature field, concentrationfield and pressure field in gasification panel are studied. On the basis of the model test,calculation results are analyzed. From the distribution of temperature field, its calculationvalue is a little higher than the experimental one, with the relative error of everymeasuring point virtually within 17%. Research shows that, the experiment value of gasheat value and calculated value take on a good conformity; due to the influence oftemperature, in the high temperature zone, the change gradient of the experiment valuefor concentration field of gas compositions is greater than that of the calculation value.The simulated results indicate that the relative error of the pressure field calculation is4.13%–12.69% and 8.25%–17.47%, respectively, 7 h and 45 h after the ignition. The droprate for the fluid pressure is 6.01% and 10.91%, respectively. Research shows that thesimulated values conform with experimental values comparatively well, which demon-strates that the numerical simulation on the “three fields” in underground coal gasifica-tion is correct. © 2005 American Institute of Chemical Engineers AIChE J, 51: 3059–3071, 2005Keywords: underground coal gasification, mathematical models, temperature field, con-centration field, pressure field

Introduction

Field test and laboratory model test verify that the ways ofthe underground coal gasification is closely related to theoccurrences of coal seams. It is easy for the channel gasifica-tion to be formed in the gently-inclined coal seams during theunderground gasification, while in the steep coal seams, thepercolation-patterned gasification is in a primary position.1

Research shows that in the underground gasification of inclined

coal seams, the heat institution and fluid mechanics institutionof gasification process are ever-changing. Its thermal condi-tions and dynamics conditions fall between those of gently-inclined coal seams gasification and those of steep coal seamsgasification.1,2 With the enlargement of the channel diameter,the podzolization of the wall plane for the channel producesgradually, which, in turn, affects the temperature field neededfor keeping the normal gas production, leads to the beginningof the combustion–gasification effect weakening for coalseams, and worsens the gasification conditions gradually. How-ever, as the process of gasification occurs, the effect of hightemperature and gravitation cause the coal seams on the roof ofthe channel to continually inbreak. This leads to the gradual

Correspondence concerning this article should be addressed to L. Yang [email protected].


ENVIRONMENTAL AND ENERGY ENGINEERING


change from free gasification channel to percolation-patternedporous loose channel. Thus, the specific surface area for thereaction between gas and solid carbon increases, improving thegas production conditions.1–3

For the development and exploration of the undergroundcoal gasification technique, experiment certainly is an impor-tant means, but, recently, people have tended to adopt themethod of the combination of experiment and mathematicalmodels. Because the underground coal gasification process israther complicated, it is comparatively difficult to establish andsolve its mathematical models. In spite of those, a number ofstudies have been made by many scholars home andabroad.4–13 At one time, a large amount of experimental andtheoretical research on the “three fields (temperature field,concentration field and pressure field)” was carried out incountries such as the Former Soviet Union, the United States,China, Germany and France, resulting in great progress.14–20

While their research results are confined to the general analysismethod, experimental method or comparatively simple 1-D,2-D steady or unsteady-numerical simulation. The dynamiccoupling mathematical models on the underground gasificationof the inclined coal seams are established in this paper. What’smore, the numerical analysis is made. The calculation resultsare checked against the model test.

The Chemical Reactions in the Underground CoalGasification Process

During the underground coal gasification, the followingseven kinds of chemical reactions mainly take place in thegasification channel

�1� C � O2O¡R1

CO2 �H1 � �393.8 MJ/kmol

�2� C � CO2O¡R2

2CO �H2 � �162.4 MJ/kmol

�3� C � H2OO¡R3

CO � H2 �H3 � �131.4 MJ/kmol

�4� C � 2H2O¡R4

CH4 �H4 � �74.9 MJ/kmol

�5� CO �1

2O2O¡

R5

CO2 �H5 � �285.1 MJ/kmol

�6� H2 �1

2O2O¡

R6

H2O �H6 � �242.0 MJ/kmol

�7� CO � H2OO¡R7

CO2 � H2 �H7 � �41.0 MJ/kmol

The above reactions are the main basis for the conservationof mass equation of the chemical reactions. (1), (2), (3) and (4)reactions take place on the wall plane of the coal seams, while(5), (6) and (7) reactions occur at the gaseous stage. During theunderground gasification, due to the high temperature, the coalseams crack. The combustion reactions must cause the O2 to

diffuse to the surface of the carbon where reaction has not yetoccurred. Theory proves that, under the underground combus-tion and gasification condition, the diffusion mass transfer ofthe gas passing through the surface grieshoch is the majorfactor controlling the carbon reaction rate. The gasificationreaction is controlled by the diffusion process. The diffusionrate of the mole number for gas compositions equals thedynamic reaction rate of coal surface.21 On the basis of theabove, we can obtain every chemical reaction rate Rj. Theinfluence of temperature on the reaction rate is mainly reflectedon the constant of reaction rate, the value of K.

According to literature,21 the Arrhenius formula is

K � Acexp��E/RT� (1)

According to the Eq. 1 and the research findings22 of V.Fredersdorff and M. A. Elliott, we can obtain

R1 � Acexp��E1/RT�PO2 (2)

R2 � Acexp�a/T � bT�PCO2 (3)

R3 � Acexp��E3/RT��PH2O �1

KfPCOPH2� (4)

where

Kf � exp��G/RT� (5)

where

R4 � Acexp��E4/RT��PH2

2 �1

KfPCH4� (6)

R5 � Acexp��E5/RT�PCO (7)

R6 � Acexp��E6/RT�PH2 (8)

R7 �

� K1

K�1� 1/ 2

PH2

1/ 2�K2K3PH2PCO2 � K�2K�3PCOPH2O�

K�2PCO � K3PH2

(9)

where

K1 � a1T�b1exp��E1/RT� (10)

K2 � a2T�b2exp��E2/RT� (11)

K�2 � a�2exp��E2/RT� (12)

K�3 � a�3exp��E3/RT� (13)

K3 � a3exp��E3/RT� (14)

K�1 �a�1

T(15)


where Kj is the equilibrium constant of chemical reaction j, Kf

is the equilibrium constant expressed in terms of the partialfugacity of every composition; �Ej is the activation energy ofchemical reaction j, R is universal gas constant; �G is the freeenthalpy of standard formation when the temperature stands atT, Pi is the partial pressure of composition i in the mixed gas.

Coupled Mathematical ModelsAssumed conditions

In the course of the combustion and gasification for the coalseams, various complex physical and chemical reactions occurin the gasifier. Meanwhile, there are energy and mass transferbetween gas phase and solid phase. In this paper, in order tosimplify the calculation, the following assumptions are made:

(1) The gasifier itself is in a stable working state. Majorphysical and thermodynamic parameters, such as coefficient ofheat conductivity, specific heat and coefficient of heat ex-change do not vary with time,23

(2) Thermal diffusion and pressure diffusion will be ig-nored,24

(3) The effect of thermal resistance will be ignored;(4) The change in the mass of gas current in the oxidation

zone due to the chemical reactions will be ignored;25

(5) For the inclined coal seams, since its thickness is muchsmaller than its slant height and strike length, temperatureconduction and the movement of the mixed gas in the coallayer can be simplified into a 2-D problem (Figure 1).

In the multicomponent gas, take a control body with anencompassed area of F, in which every component occupiesthe same area. Tracing the changes of the control body with themovement gives the changes of various parameters with thetime. If a certain characteristic variable of the control body is�, the total of the � in the control body will be �. Then, in thecylinder-coordinate system, we have

��t� � ��F

�� x, r, t�dF (16)

Conservation equation of the compositions

In the process of underground coal gasification, there aremainly seven kinds of compositions in the product gas, in thepaper, only the balance of these seven kinds of compositions istaken into consideration, which are, in turn, O2 (1), CO2 (2),CO (3), H2O (4), H2 (5), CH4 (6) and N2 (7). According to thedocument,26 considering axial flow and axial and radial diffu-

sion, we can obtain the conservation equation of the composi-tions

� yi

�t�

� yi

� x�uyi� �

�

� x �D� yi

� x� ��

�r �D� yi

�r � � D1

r

� yi

�r

� Si i � 1, 2, . . . , 7 (17)

where yi is the mole fraction of the composition i (O2, CO2,CO, H2O, H2, CH4, N2); u is only taking the gas flowing speedalong the axial direction into consideration; D is regarding thediffusion rate of every composition along the axial and radialdirection the same, therefore, Deff,i � D; Si is the generatingrate of the composition i in the chemical reaction.

From the earlier seven kinds of chemical reaction equations,we can obtain

S1 � �R5 � R6 (18)

S2 � 2 R5 (19)

S3 � �2R5 (20)

S4 � 2 R6 (21)

S5 � �2R6 (22)

S6 � S7 � 0 (23)

Initial conditions and boundary conditions:(1) Boundary ConditionsMeasuring and analyzing the gas compositions at the inlet of

the gasification gallery, we can obtain the concentration ofgasification agent at the inlet, that is

At x � 0, yi � yi0, 0 � r � r1, t � 0 (24)

where yi0 is the mole fraction of gasification agent at the inletof gallery.

Because there is little change in the gradient of concentrationfor every gas composition at the outlet of gasification gallery,the gradient of concentration for the gas composition i at theoutlet can be regarded as zero, that is

At x � L,�yi

�x� 0, 0 � r � r1, t � 0 (25)

where L is the length of gallery, the subscript “0” of thevariable denotes the known value of parameter.

At r � r1, that is, at the boundary of the wall plane for thecoal seams, the boundary conditions are

D� y1

�r�

r1

2R1 � 0 (26)

D� y2

�r�

r1

2��R1 � R2 � R7� � 0 (27)

Figure 1. Model gasifier.


D� y3

�r�

r1

2��2R2 � R3 � R7� � 0 (28)

D� y4

�r�

r1

2�R3 � R7� � 0 (29)

D� y5

�r�

r1

2�R4 � R3 � R7� � 0 (30)

D� y6

�r�

r1

2 �R4

2 � � 0 (31)

D� y7

�r� 0 (32)

(2) Initial ConditionsBefore the gasification, the content for the composition i in

the gasification agent is known, that is

t � 0, yi � yi0, 0 � r � r1, 0 � x � L (33)

Conservation equations of energy solid phase

Considering the interior heat conduction gives the tempera-ture field equation of coal seams26,27

SCS

�TS

�t�

�

� x �S

�TS

� x � ��

�r �S

�TS

�r � �1

rS

�TS

�r� QS

(34)

where TS is the temperature of solid phase, CS is the specificheat of solid phase, QS is the heat losses of solid phase, S isthe coefficient of heat conductivity for the solid phase, S is thedensity of solid phase.

Equation 34 must meet the following boundary conditionsand initial conditions: According to the measurement resultsfor the temperature field in the gasifier, the temperature of theouter coal seam with a radius of r0, and temperature at the inletof gasification gallery can be known. Because there is littlechange in the gradient of temperature for coal seams at theoutlet of gasification gallery, it can be regarded as zero, that is

At r � r0, TS � TS�, 0 � x � L, t � 0 (35)

At x � L,�TS

�x� 0, r1 � r � r0, t � 0 (36)

At x � 0, TS � T0, r1 � r � r0, t � 0 (37)

where TS� is the temperature of the outer coal seam with aradius of r0, assumed to be a constant.

At the gas-solid interface where r � r1, because there is heatconvection between gas and solid and heat and mass transfer,the following conditions should be met

S

�TS

�t� ��TS � Tg� � �

i, j�1

7

mi�Hj � RHT (38)

where mi is the mass flowing quantity of the composition i,�Hj is the reaction heat of the chemical reaction j, � is thecoefficient of heat convection, Tg is the temperature of gasphase, RHT is the quantity of heat exchange by radiation,26 thatis

RHT � F1 �q�TS4 � Tg

4� (39)

where is Stefan-Baltzmann constant, �q is the coefficient ofheat radiation, F1 is the area of heat radiation,26

F1 � 2� �0

L

r� x, t�dx (40)

Initial conditions

According to the measured results for the temperature ofcoal seams before ignition, the temperature for the solid phasecan be known when

t � 0, TS � TS0, 0 � z � L, 0 � r � r1

where TS0 is the known temperature of the solid phase.

Gas phase

According to the documents,28,29 we can obtain the conser-vation equation of energy for gas phase

� � � �i, j�1

7

�jHi � �mi� � � � �g � �Tg� � SH � Qg

� gCgi

�Tg

�t� ugCgi

�Tg

� x(41)

where �j is the coefficient of weighing and measuring for thechemical reaction j, Hi is the enthalpy of formation for the gascomposition i, SH is the heat of formation for the gas phase,Cgi is the specific heat of the composition i for the gas phase,g is the coefficient of heat conductivity for the gas phase, g

is the density of the mixed gas, Qg is the heat losses of the gasphase where �j � gD.21

Simplifying Equation 41 gives

�i, j�1

7 �gmi�Tg�Cgi�Tg��Tg

�t� Cgi�Tg��Tg�j�mi� � �g�Tg

� �i�1

7

Himi � Qg � R5�H5 � R6�H6 (42)


Boundary conditions

Measuring the gas temperature in the gasification gallery, thegas temperature at the inlet and temperature gradient for the gasat the outlet can be known, that is

At x � 0, Tg � Tgb, 0 � r � r1, t � 0 (43)

At x � L,�Tg

�x� 0, 0 � r � r1, t � 0 (44)

Wall plane boundary

In the proximity of wall plane, there is similarly heat con-vection and heat radiation between heat current and wall plane,whose boundary conditions can be shown as

At r � r1, g

�Tg

�r� ��TS � Tg� � RHT, 0 � x � L, t � 0

(45)

At r � 0, Tg � Tgp, 0 � x � L, t � 0 (46)

Initial conditions

According to the measured results for the gas temperature inthe gasification gallery before ignition, the initial temperatureof the gas can be known, that is, when

t � 0, Tg � Tg0, 0 � x � L, 0 � r � r1 (47)

where Tg0 is the initial temperature of the gas.

Equation of the gas phase movement

As stated earlier, in the process of the underground coalgasification, the flow of the mixed fluid in the gallery can beregarded as the seepage movement of the gas in the porousmedia.1 Thus, in so doing, the equation of the gas movementcan be obtained

�Pg

�t� ac�� 1

a0 � b0u�Pg� � Wg� (48)

where Pg is the fluid pressure; ac is the coefficient of conduc-tion pressure; Wg is the source-sink item, depending on thegenerating rate of the gas phase; parameters a0 and b0 aredetermined by the following two formulas, respectively

a0 � �/ge�rg, b0 � �d/gn� e�rg

where � is the coefficient of movement viscosity, g is thegravitational acceleration, e� is the permeability, rg is the weightdegree, � is the coefficient of geometrical shape for the mediaparticles, d is the average compromising particle size, n� is theporosity.

Gas state equation

In the process of underground coal gasification, the gas stateequation can be shown as follows

Pg � CRTg (49)

where C is overall mole concentration of the gas.

Unified form of the conservation and movementequations

The earlier conservation and movement equations can bewritten into a unified form, that is

��

�t� div�u� � �grad �� S� (50)

where � is the dependent variable; � and S� indicate thecoefficient of exchange and source corresponding to �, respec-tively. The equation can be regarded as being made up of fouritems, which are transient item, convection item, diffusion itemand source item, in turn.

Numerical Calculation

In view of the nonlinearness for the earlier control equationsand the strong coupling among equations, so it is difficult tosolve them through analysis method, making us have no choicebut to employ the numerical solution. In this article, the controlvolume method26,30 is adopted, which is a discretizationmethod belonging to the finite difference method in the form,but being not fundamentally different from finite elementmethod in methodology. The method aims at the integralequilibrium in the control volume, uses knot to represent con-trol volume. The discretization of the domain to be solvedusually includes two kinds, namely, even grid and uneven grid.On the basis of the characteristics of the problem to be ad-dressed in the paper, every variable is the function of time forits spatial distribution. Hence, in the process of solution, thegrid will be divided according to the even grid.

The discretization of Eq. 50 depends on the following twobasic assumptions:

(1) The function value is evenly distributed in each grid,that is, the function value on the node of the grid represents itsvalue everywhere within the grid, or we can say, the size of thegrid determines the spatial resolution ratio of the function.

(2) The function value is evenly distributed on each inter-face for each grid, that is, the function value at an arbitrarypoint of the boundary of the grid can represent the functionvalue at that boundary. Usually, the crossover point betweenthe ligature of the two neighboring points on both sides of theboundary for a certain grid and the boundary is selected as therepresentative point of that boundary.

Integrate Eq. 50 on the grid centered around point P, we canobtain the commonly-used form of the difference equation

�

�t��F��P � �

e,wn,s

�u� � �grad �� A� � �S�F��P (51)


where F� indicates the area of the grid, varying with time,under this circumstance, u takes the velocity of the fluidrelative to the moving boundary of the grid. The absolute valueof A indicates the length of a certain interface for the grid,whose direction is that of the outer normal for the interface, ¥indicates taking a sum of the four boundaries for the grid.

Discretization of the transient item

The expression of the discretization for the transient item is

�t��t

t �

�t��F��Pdt � ��F��P � ��F��P0 (52)

where superscript “0” denotes the state of the point P at theprevious time, t � �t; �t is time step size.

Discretization of the source item

Assuming adopting the fully implicit form when making theintegration of the source item in Eq. 51 relative to time, that is

�t��t

t

�S�F��Pdt � �S�F��P�t (53)

If the source item is a constant one, its discretized expressionwill be the form of Eq. 53.

If the source item is the function of the variable �, generallythe linearization method will be adopted to further discretizethe source item, that is, write the source item S� in the generalform of the differential Eq. 50 personally as

S� � SC � SP� (54)

where SC and SP are constants relative to �, actually they caneither be a real constant or the function of the previous iterativevalue �* for the variable �.

Discretization of the total flux item

The item in the ¥ of the Eq. 51 combines the influence of theconvective flux, and the diffusion flux, known as the total fluxitem. Write the total flux as J

J � �u� � �grad �� (55)

In view of the similarity for the principles and treatmentmethod in which the total flux item is discretized at eachinterface, now we will take the calculation of the total flux foran arbitrary grid interface (for example, e plane) as an example(Figure 2), so as to illustrate the discretization method of thetotal flux item.

At the e plane of the grid P, the general flux item is

Je � J � Ae/Ae � ��u��e � e�d�

dx�e

� (56)

For the convenience of the writing, the discretized expres-sion of the general flux item at the e plane can be written as

AeJe � Ce�e � De��E � �P� (57)

where

Ce � �uA�e (58)

De � eAe/��x�e (59)

Ce indicates the convection (or flowing) intensity at the eplane, whose sign is determined by the direction of u, De

indicates the diffusion intensity at the e plane, always to bepositive.

Normally, the exchange coefficient e at the interface takesthe arithmetic average of the exchange coefficient for twoneighboring grid points. In the even grid system

e �1

2�P � E� (60)

Discretized equation

On the basis of the numerical model established earlierfor the 2-D problem, the author developed the generally-used computer program used to solve for the mathematicalmodels on the 2-D nonlinear dynamic reaction flow in theunderground coal gasification, and realized the calculationof the models in this article. The general form and itscorresponding coefficients for the discretized (finite differ-ence) equations of the control differential, Eq. 50, in 2-Dunsteady problem, are as follows

aP�P � aE�E � aW�W � aN�N � aS�S � bC (61)

Figure 2. Grid points and grid interface.


where

aE � DeA��Pe�� Ce, 0��

aW � DwA��Pw�� Cw, 0��

aN � DnA��Pn�� Cn, 0��

aS � DsA��Ps�� Cs, 0��aP0 � P0F�/�tbC � SCF� � aP0�P0

aP � aE � aW � aN � aS � aP0 � SPF�

� (62)

In the coefficient expressions, aI indicate coefficients ofdifference equation corresponding to different grid points (I �E, W, N, S); bC is constant term, P0 is the fluid density of thepoint P at the previous time, Ck and Dk are the convectionintensity and diffusion intensity of the k interface, respectively,Pk is the ratio of the two; their definition expressions are

Ck � �uA�k

Dk �kAk

�k

Pk �Ck

Dk

� (63)

k � e, w, n, s indicates the grid interface between two gridpoints, Ak is the length of the k interface, �k is the distancebetween two grid points on the vertical k interface.

The distribution of the variable in the control volume takesthe power function, so, for different circumstances, the formfunction A(�P� �) may be selected various forms, selecting powerlaw form in the article, that is

A��P� �� 0, �1 � 0.1�P� ��5�� (64)

where P� is Peclet number, indicating the relative intensity ofconvection and diffusion; the operator [[ ]] denotes selectingthe maximum in the square brackets.

According to the difference equations established, we canobtain the simultaneous algebraic equation set of the controlequations. On the basis of this, it is comparatively easier toadopt the separated iterative solution.

Model ParametersSpecific heat of gas phase, Cgi

The specific heat of the gas phase mixture normally dependson its thermodynamics state. The relational form for the spe-cific heat used in this article is as follows

Cgi � A� i � B� iTn1 � C� iT

n2 (65)

where A� i, B� i, C� i, n1 and n2 are coefficients of the expression,which vary with the kinds of substances and are irrelevant totemperature, whose specific values can be referred to in refer-ence 26.

Specific heat of solid phase, CS

The experimental research shows that the specific heat forsolid substances containing carbon is linked to its characteristiccomposition and temperature, whose relationship is shown inFigure 3. In the process of inversion calculation, CS will bedetermined according to the Figure 3.

Coefficient of heat convection, �

Gas-solid phase reactions mainly occur in the pores of solidphase, the heat of the heterogeneous reaction is applied to thesolid phase. When the oxidation reaction, the solid-phase trans-fers heat to gas phase, while reduction, dry and distillation, thegas phase transfers heat to solid phase.

There are heat-transfer by convection and heat-transfer byradiation in the gas-solid heat exchange. When the temperatureis high, heat-transfer by radiation will play a major role. Ad-ditionally, when the diameter of the particle is comparativelybig, the heat conduction of the particle itself has to be takeninto consideration. In the gas-solid heat transfer, the tempera-ture of every point is shown in Figure 4. The quantity of heattransfer is calculated as follows31

Q � �FS�Tg � TS� (66)

where

� �1

1

�L � ��S�

1

�K � �S

(67)

�L �2

� 1

Rr�

1

RK�RK

2

(68)

��S � �S��TK3 � TK

2 TK1 � TKTK12 � TK1

3 � (69)

�K �Nug

DK(70)

�S � �0�0�TK13 � TK1

2 Tg � TK1Tg2 � Tg

3� (71)

Figure 3. Relation between the specific heat and tem-perature of coal.


�0 �1

1

�S�

1

�g� 1

, �S � 0.7 � 0.8 (72)

In the earlier formula, �L and ��S are the coefficient of heatconvection, and that of heat-exchange by radiation through ashdreg layers, respectively; �K and �S are the coefficient of heatconvection and that of heat-exchange by radiation throughreduced film, respectively; �S and �g is the radiation rate ofsolid phase and gas phase, respectively; Nu is Nulet dimen-sionless number; � and �0 are the thickness of the ash dreglayer and that of the reduced film, respectively; FS is thespecific surface area of the solid particle; DK is the diameter ofthe particle; Rr is the radius of the reaction core, RK is theradius of the particle; TK is the temperature of reaction core;and TK1 is the temperature of the solid particle.

Permeability, e�

Experimental research shows that, under the nonisothermalcondition, the permeability of the porous media is a function oftemperature.1,2,29 With the movement of the flame workingface, in the process of combustion and gasification, since thereare great temperature differences for the coal layers at varioussections, great changes will take place in the permeability.Under the nonisothermal condition, the measured results of thepermeability for the coal layers media used in the experimentalmodel are shown in Figure 5. The input of this parameter ismainly based on the experimental data shown in Figure 5.

Coefficient of heat conductivity, �S

Under the condition of the combustion and gasification forthe coal layers, the total coefficient of the heat conductivity forthe media also consists of heat conduction and heat radiation,whose expression is as follows1

S � �1�T� � �2�T� (73)

where �1(T) is the function of the coefficient of heat conduc-tivity for coal mass; �2(T) is the function of the coefficient ofheat-exchange by radiation among coal chunks.

Other parameters

Other parameters in the mathematical model, such as thecoefficient of diffusion, the enthalpy of formation and thecoefficient of viscosity, are all functions of temperature underthe nonisothermal condition, whose specific calculationmethod can be referred to in the relevant documents.32–36

Because temperature has an important effect on model pa-rameters, in the process of inversion calculation, the values ofparameters shall be initially determined, based on the relation-ship between each parameter and temperature and temperaturevalues at different nodes. Then through positive calculation, wewill judge the reasonability of the model parameters selected. Ifthe errors between calculated values and experimented valuesare within tolerance, the parameters selected are right, other-wise, further adjustments or correction should be made onevery parameter till comparatively ideal calculated results havebeen obtained.

Model Test

The overall size of the model gasifier body is 7.35 m long by1.45 m high by 1.0 m thickness, consisting of base and lid, asshown in Figure 6. Under the bottom of the gasifier wasinstalled a line of hydraulic jacks, which makes the gasifier restat any angle. The hearth is so spacious that it can be filled withother materials, which are used to simulate coal seams withdifferent thickness. On the sides of the gasifier there are anumber of gas inlets, outlets, and slip casting holes, used tosimulate various gasifiers and study different modes of airpumping and gasification channel with different lengths. Acircular air (steam) injection pipeline is equipped on the plat-form in hope for supplying the air (steam) in the same orcounter direction and with moving pumping points. The gas-ifier adopts hydraulic sealing technology. On the upside of the

Figure 4. Temperature distribution for gas-solid heattransfer.

Figure 5. Changing curve of permeability and tempera-ture.

Figure 6. Cutaway view of the gasifier model.


gasifier is installed a piece of board pushed by hydraulic pushrod, which imposes a certain pressure on the coal seamsthrough the board in the course of gasification, in order tosimulate the pressure on the top of the coal seams.

The structure of the gasifier is made up of three layers. Theouter one is sealant made of thick steel plate, with the middleone thermal insulating made of vermiculite, the inner one flameretardant coating. The size of simulated coal seams in thegasifier is 6.7 m 1.3 m 0.45 m, with a dip angle of about45°. The type of coal is gas-fat coal. The data of the proximateanalysis can be consulted in the document.1 In the process ofcoal injection, use as much lump coal as possible, so we cankeep the natural state of the coal seams at best. The intersticeswill be filled with small pieces of coal. Finally, smear theinterstices with the mixture of cement and coal powder. Thegasification panel is 6.0 m 1.1 m 0.42 m. The initialequivalent diameter of the gasification gallery is 80 mm.

The system of the model test is shown in Figure 7. Thepipeline system is designed as an armillary circuit. Through thereversal valve, we can pump the air or steam negatively andpositively. Coal gas cleaner is mainly to remove the tar fromthe gas and lower the temperature of the resulting gas. In orderto measure the temperature distribution in the gasifier accu-rately, we deploy the temperature-measuring points inten-sively. In the gasification panel, 17 rows of temperature-mea-suring points are buried, with 7 in each row. The number totals

119. The temperature-measuring elements adopt the strictly-standardized NiCr-NiSi thermal couples. The automatic dataacquisition will be made regularly, and the data will be dis-played on the screen of the computer and recorded. The result-ing gas will be analyzed with gas-phase chromatographer,which can tell the contents of different compositions. Everybore hole deployed at the sides of gasifier along the direction ofgasification gallery also has the monitoring function. In theexperiment, the real-time monitor has been made against thepressure and gas compositions in the gasifier, so as to learn thestate of changes for the pressure field and concentration field atdifferent places and various times within the gasification gal-lery.

The operating conditions: the gasification agent, air; thepressure at the inlet, 2550–2600 Pa; the flowing capacity at theinlet, 25–30 m3/h. The results of the experiment are shown inFigures 8–15.

Analysis

The calculation results of numerical simulation are shown inFigures 8–15. From Figure 8, it can be shown that the calcu-

Figure 8. Experiment value and calculation value of thetemperature field at various times of gasifica-tion.

Figure 7. System of model experiment.

Figure 9. Experiment value and calculation value of theconcentration of O2 at various times of gasifi-cation.

Figure 10. Experiment value and calculation value of theconcentration of CO2 at various times of gas-ification.


lated values of temperature field virtually conform with themeasured ones. Except the measuring points in the combustionzone, where the relative error between the calculated valuesand measured ones of temperature is rather high (certain points,over 20%), the relative error of other points is no more than17%, majority of which is within 14%, completely meeting theprecision requirement of numerical simulation on the temper-ature field. Judging from the distribution of temperature field,the calculated values are a bit bigger than the measured ones.The reasons are the following: first of all, the term of heatlosses in the mathematical model is determined on such con-ditions as the calculation formula for the coefficient of heattransfer with the composite structure and fixed setting (temper-ature, wind velocity). In the process of experiment, naturalventilation or forced ventilation makes the coefficient of heatlosses increase, which contributes to the slight drop in themeasured temperature. Then, during calculation, the coefficientof heat conductivity for coal seams is not that of body coal butthe equivalent coefficient of heat conductivity for coal seamstaking the influence of convection and heat radiation intoconsideration. The coal seams in the model test are piled upselecting various sizes of coal chunks. Though the intersticesare filled with pulverized coal, its pores are far bigger than

those of the natural ones. Therefore, heat transfer is no longera single form of heat conduction. With the effect of convectionintensifying, the coefficient of apparent heat conduction for thecoal seams heightens, which influences the calculation resultsprecision.

Because the reaction rate heightens rapidly with the rise intemperature, which leads to the increase in the reaction con-version rate, as a result, the influence of the temperature causesthe calculated values of the concentration for various compo-sitions in gas to be a little bigger than the experimental ones;the change gradient of the composition concentration for ex-perimental value in high-temperature zone is bigger than that ofthe calculated value (Figures 9–13).

According to Figure 14, with the prolonging of gasificationtime and the increase in the gasification channel length, theheating value of gas increases gradually, and there is virtuallyno fluctuation for it. Therefore, the experimental values and thecalculated ones take on a good conformity. However, beyondthe reduction zone, the extent of increase for the heating valuedecreases. The influence of temperature field on the heatingvalue of gas is remarkable. 7 h after the ignition, in spite of the

Figure 12. Experiment value and calculation value of theconcentration of H2 at various times of gas-ification.

Figure 14. Experiment value and calculation value of thegas heat value at various times of gasifica-tion.

Figure 11. Experiment value and calculation value of theconcentration of CO at various times of gas-ification.

Figure 13. Experiment value and calculation value of theconcentration of CH4 at various times of gas-ification.


comparatively long gasification channel, due to the short periodof time and the low temperature in the oven, the heating valueof gas is comparatively low; 45 h after the ignition, the tem-perature of the oven increases. Although the length of thegasification channel is relatively short, the heating value of theoutlet gas is higher than that of previous one. Therefore,maintaining an ideal temperature field and comparativelylonger gasification channel is conducive to the stability andimprovement of the heating value for the gas.

Figure 15 demonstrates that the simulated calculation valuesof the gas pressure basically conform with the measured ones.The relative error at the second time between calculated valuesand measured ones is 4.13%–12.69%, with the average droprate of fluid pressure 6.01%. From Figure 15, we can knowthat, 45 h after the commencement of gasification, the error ofthe simulated calculation for pressure is 8.25%–17.47%, withthe average drop rate of the gas pressure along the gasificationchannel 10.91%. It can be concluded that, with the prolongingof gasification time, the extent of drop for the fluid pressurerises and the calculation error increases. The major reasons arethe following: at the very beginning of the experiment, the gasmoves along the free gasification channel with little resistanceon the fluid. The fluctuation for the curve of the experiment issmall, so the drop rate and the calculation error are low, but thecalculation value is comparatively high. With the developmentof the gasification process, the coal layers over the gasificationchannel, due to the effect of high temperature, expands, splin-ters and falls onto the gasification channel because of dead-weight, which fills the bottom of the gasification channel withloose coal chunks. Thus, the free gasification channel trans-forms into a percolation-patterned one. The seepage movementof the gas will go on in the porous media. The resistance on thegas movement increases by a large margin, so does the extentof the pressure drop, which also tends to be stable. The curvebetween the experiment and calculation takes on a good con-formity. Considering the changes in movement conditions, inthe preliminary determination of major model parameters, suchas the coefficient of conduction pressure, permeability, theinvolvement of personal or empirical factors in the consider-ation of relevant calculation coefficients results in a certainerror in the initial parameters calculation, which causes oscil-latory occurrence in the value of initial numerical simulation.

In short, the simulated results indicate the calculated valuescan conform with the measured values, which shows that theestablishment of mathematical models on heat and mass trans-fer in the process of underground coal gasification, the deter-mination of parameters, the analysis and treatment of boundaryconditions and the solution method are correct. This will pro-vide necessary theoretical evidence and scientific guidance forthe further comprehensive quantitative study and productionpractice of underground coal gasification.

Conclusion

(1) According to the features of gasification process, on thebasis of the model test, the nonlinear coupled mathematicalmodels on the underground coal gasification for inclined coallayers are established. The simulated results demonstrate thecalculated values can conform with the measured ones com-paratively well, which proves that the numerical simulation onthe temperature field, concentration field and pressure field isreasonable in the underground coal gasification under the ex-perimental condition.

(2) The numerical simulation results show that, in hightemperature zone, the calculated value of the temperature fieldis a bit bigger than the measured one; the change gradient of themeasured value of the concentration for various compositionsin the gas is bigger than that of the calculated value; temper-ature field has a great influence on the heating value of the gas.The heating value of the gas increases gradually with theprolonging of gasification time. The calculated value of heatingvalue is basically the same with the experimental one. Appar-ently, the ideal temperature field is conducive to the improve-ment on the gas quality and the stability of the gasificationprocess.

(3) According to the calculation results, the relative errorbetween the calculation value and measurement one of the fluidpressure and its drop rate increases gradually with the processof gasification. The change of the flow condition in gasificationchannel is mainly responsible for the change of the calculationerror.

(4) The numerical calculation results basically reflect thereal patterns of variation for the temperature field, concentra-tion field and pressure field in the underground coal gasificationfor the inclined coal seams, which will provide necessaryscientific theoretical evidence for the further quantitative studyon the underground coal gasification process and the predic-tions of its variation patterns.

AcknowledgmentsThis work was supported by the National Natural Science Foundation of

China (Ratification No.: 59906014), and by the National Natural ScienceFoundation of China (Ratification No.: 50276066, 20207014). The techni-cal contributions of Professor Yu Li and Dr. Wang Zaiquan are gratefullyacknowledged by the author. The author also gratefully thanks ProfessorXiang Youqian, Wang Jialian, and Gao Wenjun, for helpful discussion.Support received for this research from the Engineering Research Centre(ERC) for Clean Coal Technology of the China University of Mining andTechnology (CUMT) is gratefully acknowledged.

Notation

a � coefficient in Eq. 3am � coefficients in Eqs. 10–15 (m � 1, 2, �2, �3, 3, �1)aI � coefficients of difference equation (61) (I � E, W, N, S)

Figure 15. Pressure field in the gasification tunnel 7 hand 45 h after the ignition.


a0 � parameter in Eq. 48, determined by a0 � �/ge�rg

ac � the coefficient of conduction pressure, m2/sAc � coefficient in Eqs. 1–4 and Eqs. 6–8A � length vector of a certain interface for the grid, m

Ae � length of the e plane, mAe � length vector of the e plane, mAk � length of the k interface (k � e, w, n, s), mAi � coefficient of the expression (65), which varies with the kinds of

substancesA(�P� �) � form functionA(�Pk�)� form function of the k interface (k � e, w, n, s)

b � coefficient in Eq. 3bh � coefficients in Eqs. 10 and 11 (h � 1, 2)bC � constant term of difference Eq. 61b0 � parameter in Eq. 48, determined by b0 � �d/gnerg

B� i � coefficient of the expression (Eq. 65), which varies with thekinds of substances

C � overall mole concentration of gas, mol/m3

Ce � convection (or flowing) intensity at the e plane in Figure 2Ck � convection intensityC� i � coefficient of the expression (Eq. 65), which varies with the

kinds of substancesCgi � specific heat of the composition i for the gas phase, J/kg KCS � the specific heat of solid phase, J/kg K

d � average compromising particle size, mD � diffusion rate of every composition along the axial and radial

direction, m2/sDe � diffusion intensity at the e plane in Figure 2

Deff,i � effective diffusion coefficient of the composition i, m2/sDk � diffusion intensityDK � diameter of the particle, m

e � interface between two grid pointse� � permeability, �m2

F � encompassed area of a control body, m2

F1 � area of heat radiation, m2

FS � specific surface area of the solid particle, m2/m3

F� � area of the grid, m2

g � gravitational acceleration, m/s2

Hi � enthalpy of formation for the gas composition i, J/molJ � vector of total flux item

Je � vector of general flux item at the e plane in Figure 2K � rate constantKj � equilibrium constant of chemical reaction jKf � the equilibrium constant expressed in terms of the partial fugac-

ity of every compositionL � length of gallery, m

mi � mass flowing quantity of the composition i, kg/m3

n � interface between two grid pointsn1 � coefficient of the expression (Eq. 65)n2 � coefficient of the expression (Eq. 65)

n� � porosityNu � Nulet dimensionless numberP� � Peclet numberPi � partial pressure of composition i in the mixed gas, PaPg � fluid pressure, PaPk � ratio of the Ck and Dk

Q � quantity of heat transfer, WQg � heat losses of the gas phase, J/m3 sQS � heat losses of solid phase, J/kg s

r � r-coordinater0 � distance between boundary of the outer coal seam and x-coor-

dinate axis, mr1 � radius of gallery, mrg � weight degree, N/m3

R � universal gas constant, J/kmol KRHT � quantity of heat exchange by radiation, W

Rj � chemical reaction rate, mol/m3 sRK � radius of the particle, mRr � radius of the reaction core, m

s � interface between two grid pointsSC � constant relative to � in Eq. 54Si � generating rate of the composition i in the chemical reaction,

mol/m3 sSH � heat of formation for the gas phase, J

SP � constant relative to � in Eq. 54S� � source item

t � time, sT � temperature, K

T0 � temperature of solid phase at the inlet of gallery, KTg � temperature of gas phase, K

Tgb � temperature of the gas at the inlet of the gallery, KTgp � temperature of the gas at wall plane boundary, KTg0 � initial temperature of the gas at the inlet of the gallery, KTK � temperature of reaction core, K

TK1 � temperature of the solid particle, KTS � temperature of solid phase, K

TS0 � known temperature of the solid phase, KTS� � temperature of the outer coal seam, K

u � velocity of the fluid, m/sw � interface between two grid points

Wg � source-sink itemx � x-coordinateyi � mole fraction of the composition i

yi0 � mole fraction of gasification agent at the inlet of gallery

Greek letters

� � coefficient of heat convection, W/m2 K�L � coefficient of heat convection through ash dreg layers, W/m2 K�K � coefficient of heat convection through reduced film, W/m2 K�S � coefficient of heat-exchange by radiation through reduced film,

W/m2 K��S � coefficient of heat-exchange by radiation through ash dreg lay-

ers, W/m2 K�j � coefficient of weighing and measuring for the chemical reaction

j�S � radiation rate of solid phase, W/m3 K4

�g � radiation rate of gas phase, W/m3 K4

� � thickness of the ash dreg layer, m�0 � thickness of the reduced film, m�k � distance between two grid points on the vertical k interface, m�x � distance between two grid points, m

�1(T) � function of the coefficient of heat conductivity, W/m K�2(T) � function of the coefficient of heat-exchange by radiation, W/m K

� � coefficient of movement viscosity, m2/s� � coefficient of geometrical shape for the media particles� � dependent variable

e � exchange coefficient at the e plane in Figure 2, determined byEq. 60

E � exchange coefficient at the grid point E in Figure 2k � exchange coefficient at the k interface (k � e, w, n, s)P � exchange coefficient at the grid point P in Figure 2� � coefficient of exchange corresponding to the variable �

� � characteristic variable of the control body�I � variables of difference Eq. 61 (I � E, W, N, S)�* � previous iterative value for the variable �

�Ej � activation energy of chemical reaction j, J/mol�G � free enthalpy of standard formation, J/mol�Hj � reaction heat of the chemical reaction j, J/mol

�t � time step size, s � Stefan-Baltzmann constant, W/m2 K4

�q � coefficient of heat radiationg � coefficient of heat conductivity for the gas phase, W/m KS � coefficient of heat conductivity for the solid phase, W/m K

� fluid density, kg/m3

g � density of the mixed gas, kg/m3

P0 � fluid density of the point P at the previous time, kg/m3

S � density of solid phase, kg/m3

[[ ]] � selecting the maximum in the square brackets� � Hamilton operator

Subscripts

b � boundaryc � coefficientC � constant

eff � effectivef � partial fugacity


g � gasi � composition ij � chemical reaction jk � k interfacep � wall planeP � grid point P

P0 � state of the point P at the previous timeq � heat radiationr � radiusS � solid� � variable� � outer coal seam0 � known value

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Measurement of Flow Field in Biofilm Reactorsby 3-D Magnetic Resonance Imaging

Kevin P. Nott, Frank P. Heese, and Laurance D. HallHerchel Smith Laboratory for Medicinal Chemistry, University of Cambridge School of Clinical Medicine,

University Forvie Site, Robinson Way, Cambridge, CB2 2PZ, U.K.

Lynne E. Macaskie and Marion Paterson-BeedleSchool of Biosciences, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K.


3-D Magnetic resonance imaging (MRI) was used to measure the flow field of water in apacked-bed column containing Serratia sp. biofilm supported on polyurethane foam, andsubsequently to follow a reaction which precipitates lanthanum phosphate on the biofilm.Sensitizing the MR image contrast to the fluid flow along the axis of the bioreactor providedbetter image-contrast between the foam and fluid compared to that based on MR signalintensity alone. After reaction, that same “velocity contrast” effectively defined the differencebetween blocked and unblocked regions by distinguishing between regions of flow and no flow.Data acquired during progressive blockage of reactors challenged at two different flow ratesaccord with reactor theory; thus, the faster flow rate replenished the reactants uniformly,whereas at the slower flow rate the reactants were concentration limited. MRI velocimetry wasused to generate data that can be used to model reactors where the efficiency is progressivelycompromised by blockage due to precipitation. © 2005 American Institute of Chemical EngineersAIChE J, 51: 3072–3079, 2005Keywords: MRI, serratia, bioreactor, flow, blockage

Introduction

With increasing environmental legislation concerning therelease of industrial wastewaters, there is a growing need forefficient and economical remediation methods to treat suchfluids. It is encouraging, therefore, that there is an increasingbody of evidence to suggest that microorganisms can be usedfor the removal of heavy metals from industrial effluents whichcan be detrimental to health.1,2 Thus, an effective method hasbeen developed that can typically remove 85–95% of heavymetal ions by passage through a bioreactor containing immo-bilised Citrobacter biofilm;3 a cell-bound phosphatase liberatesinorganic phosphate from an organic “donor” molecule, whichthen promotes the formation of cell-bound metal phosphate.4 Asimple model has been derived from the Michaelis-Menten

equation to relate the efficiency of removal of metal ions5,6

from a uranium mine wastewater semiquantitatively to theinput flow rate, total enzyme loading and bioreactor activity.6

However, that model assumes ideal plug-flow behavior and islimited to bulk parameters that describe the overall reactorperformance, and cannot account for the spatial and temporalchanges in flow field induced by blockage formation. Althoughassumption of ideal plug-flow behavior is generally fundamen-tal to fixed-bed bioreactor modeling, that assumption is diffi-cult to prove experimentally within the 3-D dynamic system ofan operational flow-through reactor. Use of controlled block-age deposition does provide a tool whereby these factors can beinvestigated quantitatively, but that “black box” approachbased on analysis of input- and exit-solutions, provides nospatio-temporal information, and, hence, is of little use inprocess optimization.

It is known that MRI is a powerful tool that can measure,noninvasively, the velocity of opaque fluids in three dimen-

Correspondence concerning this article should be addressed to K. P. Nott now [email protected].


sions, and, hence, it is invaluable for many process engineeringapplications.7–9 Increasingly, MRI is also being used for envi-ronmental engineering and biotechnology applications.10,11 Ofspecific relevance to this study, MRI has previously been usedto visualise the formation of biofilm as well as the flow velocityof fluid in a bioreactor cell;12–16 it has also been used to followthe absorption of heavy metals in alginate, immobilised cellsand algal biosorbents.17–20

As part of a broad ranging study using noninvasive imagingtechniques to investigate the function of a biofilm bioreactor, ithas already been demonstrated that MRI of the aqueous me-dium can visualize biofilm on polyurethane reticulated foam, aswell as the removal of Cu2�/La3� ions by biosorption andinorganic precipitation.21 This is possible because the MRproperties of water protons are dependent on the concentrationand molecular environment of the water molecules. Thus,heavy metal ions (La3�) induce a concentration-dependentrelaxivity, or in the case of paramagnetic ions (Cu2�), a sus-ceptibility effect which alters the MR properties of vicinalwater molecules, and thereby provides image contrast with thebulk of the aqueous medium. Extensive phosphatase-mediatedbioaccumulation of LaPO4 was observed, but no evidence fordeposition of Cu3(PO4)2 was obtained by X-ray diffraction andproton-induced X-ray emission analyses;21 those observationswere confirmed by MRI. Since Cu2� was not taken up into thebiofilm it was concluded that Cu2� could be added as an MRIcontrast agent to increase signal-to-noise from the perfusateduring experiments to monitor the biodeposition of LaPO4,which shows as regions of low signal intensity because of thelack of water. On that basis, it was decided that this is a goodmodel system for a noninvasive study of the function of abiofilm reactor by MRI velocimetry; in turn, that prompted thepresent study, which seeks to answer the following questions:

(1) Can MRI quantitate in three dimensions the entire 3Dflow velocity field of fluid flowing through the reactor?

(2) If so, are such data sensitive to the progressive deposi-tion of metal as it is removed from the wastewater stream?

(3) What are the effects of the air bubbles entrained in thesupport matrix on the fluid flow, and is it possible to distinguishthem from the metal precipitate?

(4) Can MRI be used to generate data to refine the existingenzyme kinetic-based model, and hence help the design of amore efficient reactor?

Flow Imaging

The theory of MRI and its application to the study of abiofilm reactor has been described previously.22 MRI of wateris based on the spatial localization of the nuclear magneticresonance (NMR) signal in a homogeneous magnetic field viathe application of linear magnetic field gradients. If, in additionto those spatial encoding gradients, a pair of gradient pulseswith strength g, both with duration � and separated by � areapplied in a particular direction, the phase of the MRI signal issensitised to flow in that direction;23–25 a typical pulse sequenceas used in this study is shown in Figure 1. The phase �, of thesignal at each pixel of the resulting pulsed field gradient (PFG)velocity-encoded MR image is related to velocity v, by theequation

� � ��g�v � �

where � is the magnetogyric ratio of the 1H nucleus and � isthe phase offset caused by any magnetic field inhomogeneity.Therefore, a series of images can be acquired with differentgradient strengths to obtain a value of flow velocity for eachpixel. Velocity images can be acquired sequentially for allthree orthogonal ( x, y, z) directions, where required, to pro-vide the complete 3-D flow field.

ExperimentalPreparation of Reactor Packed-Bed Systems for MetalRemoval by Serratia Biofilm

Citrobacter sp. NCIMB 40259, now reassigned to the genusSerratia by modern molecular and biochemical methods,26 wasused under licence from Isis Innovation (Oxford, U.K.). Theconditions for controlled growth of Serratia sp. biofilm havebeen established.4 The biofilm was grown on polyurethanereticulated foam (Filtren TM 30 cubes, 1 cm3) supplied byRecticel (Belgium) in an air-lift fermenter in continuous cul-ture, as described previously.4 The reticulated foam cubes,coated with biofilm of average thickness of 26 �m27 werewithdrawn after six days of continuous culture. The phospha-tase specific activity of the cells from the fermenter was deter-mined by the release of p-nitrophenol (PNP) from p-nitrophe-nyl phosphate (PNPP); one unit is defined as nmol PNPliberated per min per mg bacterial protein.4 In all experimentsresting cells were used since no further growth medium wasprovided; effectively the biofilm behaves as an immobilisingmatrix for the phosphatase enzyme.

All measurements were made using cylindrical glass col-umns (16 mm dia., 70 mm length) packed, without compres-sion, with 88, 0.125 cm3 cubes of polyurethane foam with

Figure 1. Gradient echo imaging sequence with inter-leaved Pulsed Field Gradient (PFG) pair.Each radio frequency (RF) pulse tips the magnetisation of thewater protons through a flip angle �°, and its magnitude andphase are recorded at the end of the “echo time” (TE). TE isset to reflect the spin-spin relaxation time (T2) of the protons;the shorter the T2, the shorter must be TE. Successive RFpulses are separated by the “repetition time” (TR), which islong enough for the magnetisation to relax fully before thenext RF pulse is applied. 3-D spatial encoding is achieved byusing a fixed z gradient whilst incrementing the x and ygradients; thus, the total scan time is Nx � Ny � TR, whereNx and Ny are the number of pixels in the x and y directionsrespectively. Velocity encoding is achieved by application ofa bipolar gradient pair, in this example in the z directionparallel to the bulk flow.


immobilised Serratia sp. biofilm21 (made by dividing 1 cm3

cubes obtained from the fermenter into eight).

MRI Velocity Measurements

All 1H MRI images were acquired using a Bruker MSL-100imaging console (Bruker Analytische Messtechnik GmbH,Karlsruhe, Germany) operating under TOMIKON software(Bruker Medizintechnik GmBH., Karlsruhe, Germany) con-nected to an Oxford Instruments 2 Tesla, 31 cm horizontal boresuper-conducting magnet (Oxford, U.K.). The 11 cm internaldia. gradient set and radio-frequency (RF) probe (internal dia.5.5 cm) were built “in house.” Each axis of the gradient set waspowered by two Techron� amplifiers (Models 7560 and 7570,Elkhart, IN, U.S.A.) giving gradient strengths up to 195 mT/m.The data were transferred to a Linux workstation and processedinto images using CaMReS software written by Dr. N. J.Herrod.

3-D dimensional flow encoded images (256 � 64 � 64pixels) were acquired using a PFG gradient echo sequence(TR/TE/� 100 ms/7.5 ms/90°) with 312.5 �m isometric spatialresolution in a scan time of 6 min 50 sec; a schematic is givenin Figure 1. A total of ten images was acquired, with fivedifferent velocity encoding gradient strengths (�117, �19.5, 0,�19.5, �117 mT/m), with and without flow, giving a total scantime of 1.1 h for each velocity direction either transverse ( xand y) or parallel ( z) to the bore of the magnet. The fluid flowwas provided by an electromagnetically driven gear pump(Integral Series 120 pump, Micropump Inc., Vancouver, WA)which was used to replenish water in a constant head gravita-tional flow vessel held at a given height in order to provide thereactor with steady-state flow. The velocity encoded imagesacquired without flow were used to correct for errors caused byimperfections in the motion encoding gradients. Initially x, y,and z velocity measurements were carried out on a bioreactorcontaining foam cubes with biofilm, using a 20 mM CuSO4

solution to enhance the signal-to-noise of the experiment andconsequently the accuracy of the velocity measurement. Con-centrations normally toxic to the biofilm could be used since nobiomass growth was required and the probe solution was re-cycled without any consequence to the remaining experiment.The velocity-encoded images were processed using a Bayesianstatistical method for analysis in order to maximize accuracyover a wider range of velocities using fewer gradient steps.28

Bioreactor Challenge Experiments

Metal accumulation was achieved using two “fresh” col-umns side-by-side inside the magnet as shown in Figure 2, andchallenged with a fluid containing La (NO3)3 � 6H2O (1 mM;for lanthanum phosphate precipitation onto the biomass),CuSO4 � 5H2O (1 mM; not precipitated appreciably onto thebiomass21), and sodium glycerol-2-phosphate (5 mM; phos-phate donor) in sodium citrate buffer (2 mM; to prevent metalhydrolysis and maintain metal solubility in the input solution),pH 6.0. Two different flow rates (11 and 128 ml/hr, hereaftertermed “slow” and “fast” flow) were generated by an externalperistaltic pump (Watson-Marlow, 505U, Wilmington, MA);these conditions were chosen, using data from outflow analysesin preliminary tests, to give contrasting results; in the absenceof obstructions those would have Reynolds numbers (NRe) of

0.25 and 2.87, respectively. The Peclet numbers for the reactorsat the slow and fast flow rates are ca. 22 and 261, respectively,both calculated from the equation Pe � UL/Da where: U is theaverage velocity; L the pore size of the foam; and, Da is thedispersion coefficient.29 Initially the columns were challengedfor two hours to allow La3� and Cu2� ions to be background-absorbed onto the biofilm. Subsequently, MRI velocity mea-surements were made under constant head gravitational flow; 1mM CuSO4 solution was used in order to maximise the signal-to-noise and consequently the accuracy of the measurement,without being toxic to the phosphatase enzyme (necessary forthe next metal accumulation phase); the effluent was not recy-cled to prevent recirculation of any biofilm cells which hadsloughed off. The flow rate used was over 100 times faster thanthat for the La3� supplemented solution, in order to producevelocities which were measurable by MRI, but also too high forthe metal deposition reactions to occur during the flow resi-dence time. The flow rate was measured gravimetrically at theoutlet pipe, and was maintained at approximately 3120 ml/h inboth reactors for all the flow measurements; this corresponds toa Reynolds number (NRe) of 70 without any obstructions.Importantly, since the flow data are acquired over many hoursit was important to ensure that the flow is steady and nonturbulent to enable measurement of velocity.

Subsequently, the reactors were challenged for three longerperiods (each ca. 20 h) each followed by MRI velocity mea-surements to spatially probe the progressive blockage (256 �128 � 64 pixels for both reactors, giving a total acquisitiontime 2.3 h). The time-dependent precipitation of lanthanumphosphate was also monitored by analysis of bioreactor out-flows using spectrophotometry (see later).

Figure 2. Continuous experiment carried out in situ inthe magnet.Once the scanner had been tuned with the reactors inside theRF probe, the experimental conditions could be controlledsequentially. Initially, the challenge solution was pumpedthrough two reactors at different flow rates (11 and 128 ml/h,termed “slow” and “fast”) for a 2 h period; that solution wasthen displaced by 1 mM aqueous CuSO4 for measurement ofthe flow velocity over the next 2.3 hr. Subsequently, this fluidwas displaced using a reservoir of challenge solution and the“slow” and “fast” flows of challenge solution restored for afurther defined period (ca. 20 h) before a second set ofvelocity measurements based on the CuSO4 solution. Thiscycle was repeated two more times to effectively create a 4-Ddata set of the time dependence of the 3-D spatial effects.


Spectrophotometric Analyses of La3�, Cu2� andInorganic Phosphate

The La3�, Cu2� and phosphate contents of the outflows ofthe reactors were measured as described by Yong and Ma-caskie,30 Nott et al.,21 and Yong and Macaskie,31 respectively.

Environmental Scanning Electron Microscopy

Samples of Serratia sp. biofilm grown on polyurethane re-ticulated foam with and without lanthanum phosphate depositswere visualized using environmental scanning electron micros-copy (ESEM) (FEI Philips FEG ESEM XL30) in their hydratedstate. The samples containing lanthanum phosphate were takenfrom a “slow” reactor.

Magnitude Data Analysis

Whereas the phase of the MRI signal was sensitized tovelocity, the magnitude of the MRI signal acquired simulta-neously, is primarily dependent on water proton density.Hence, the blockage could be quantitated using the magnitudeof the MRI image acquired without either flow or flow encod-ing, by segmenting high signal intensity from the challengesolution from that of any obstructions, such as the foam,biofilm, precipitate and/or air bubbles (low signal intensity).The cut off intensity was chosen to be 25% of the maximumsignal intensity from the challenge solution, and each data setwas referenced against the first, which had no appreciableblockage despite the two hours exposure to the La3� supple-mented solution.

Flow Data Analysis

In general, 15% of the length of the MR images at either endof the reactor was unusable for quantitative analysis of the flowdata because the magnetic field homogeneity in those regionswas poor. In addition, it is apparent from a preliminary studyusing sagitally orientated slices (Figure 3) that the availablegradient linearity (magnetic field strength vs. position) waspoor. Thus, although the transverse image dimensions werecircular across the bioreactor, the MRI sagittal image along itslength was distorted towards the inlet and outlet. Consequently,the cross-sectional area, required for calculation of the averagez velocity (including zeros), had to be determined separatelyfrom each slice using the following procedure programmedusing Matlab (Mathworks Inc., Natick, U.S.A.):

(1) The centers of the circular transverse images at each endof the usable region (central 70% of the length) were definedmanually.

(2) An imaginary straight line was drawn between them todefine the center of the intervening slices.

(3) For each transverse slice, starting from the estimatedcenter, the size of a circle was increased until it had the largestradius (that is, the maximum distance) which included a nonzero velocity pixel.

(4) For each of the resultant circular regions-of-interest(ROI’s) including areas of zero velocity, the average z velocityVavg was calculated; separately the areas of zero velocity wereexcluded for estimation of the non stagnant average z velocityVns.

(5) The blockage fraction (BF) was calculated using theequation, 1 � Vavg/Vns.

Results

Figure 3 shows sagittal and transverse slice images takenfrom the 3-D data sets of signal magnitude (related to waterdistribution), plus the x, y, z vectorial velocity taken from threeseparate phase encoded 3-D maps of fluid flow through areactor before treatment. The heavy Cu2� doping (20 mM)dominates the relaxation behavior of water protons in both thebiofilm and surrounding fluid, hence, there is little contrast inthe signal magnitude image compared to a previous study;21 infact the upper image resembles that of reactor with no biofilmand visualizes the small and large voids associated with thefoam and air bubbles, respectively. Although the x and yvelocities were too low to be measured accurately, and thefoam structure is not clearly discernible than for z, the zvelocity image is very sensitive to the foam structure. Thehighest z velocities are found in the gaps between the foamcubes which disperse the high velocity jet as it enters thebioreactor; the fluid then follows the path of least resistanceand channels between the foam cubes and down the sides.Since neither the x nor y components provided additionalinformation, all subsequent velocity measurements were car-ried out only in the z-direction, parallel to the direction of flow;this reduced the total scan time for the flow measurements from6.8 to 2.3 h.

Figure 4 shows specimen data of the concentrations oflanthanum, copper and phosphate ions and the pH duringchallenge by an aqueous solution of lanthanum nitrate (1 mM),copper sulphate (1 mM) and sodium glycerol-2-phosphate (5mM) in a sodium citrate buffer (2 mM), pH 6.0 at the twodifferent flow rates; samples were taken from the outflowbefore each imaging experiment. As was found previously,21

under both conditions the concentration of copper ions in theoutflow was 1 mM indicating little or no copper removal. Incontrast, the majority of the La3� was removed by both the“fast” and “slow” flow reactors. The pH in both reactorsincreased in the latter half of the experiment. Thus, the “slow”

Figure 3. Sagittal and transverse MRI images.Sagittal and transverse images (taken at the position of thedotted line) of water in a reactor containing biofilm immobil-ised on foam cubes. The pair of upper traces was taken froma 3-D data set of signal magnitude and shows the spatialdistribution of water. The image intensity in the lower threetraces has been sensitised to the flow velocity in the x-, y- andz-directions.


reactor gave 1.65 mM phosphate in the outflow (at 57.1 h) anda higher pH upshift (6.00 to 6.88) whereas the “fast” reactorgave 0.69 mM phosphate in the outflow (at 57.1 h) and, asmaller pH upshift (6.00 to 6.45). This can be explained by thereaction of the PO4

3� ion generated by the system, that is,cleaved from the disodium glycerol 2-phosphate, with waterinitially to form HPO4

2� and then H2PO4�, as follows

PO43� � H2O7 HPO4

2� � OH� (1)

HPO42� � H2O7 H2PO4

� � OH� (2)

The basic dissociation constants for Eqs. 1 and 2 are Kb1 �2.22 � 10�2 and Kb2 � 1.58 � 10�7, respectively.32 In thiscase Kb1 �� Kb2, therefore, essentially all the OH� arises fromthe first reaction. The fractions (�) of HPO4

2� and H2PO4� at pH

6.45 and 6.88 were determined using the plot of distribution ofvarious phosphate species (�) as a function of pH.33 Theestimated concentrations of the phosphate species ([phosphate-containing species] � � � total phosphate concentration) inthe outflow of the “slow” reactor were [HPO4

2�] � 0.54 mMand [H2PO4

�] � 1.11 mM, and in the “fast” reactor were[HPO4

2�] � 0.11 mM and [H2PO4�] � 0.58 mM. Thus, the

“slow” reactor, with a higher pH in the outflow, contains ahigher concentration of HPO4

2� species in solution than doesthe “fast” reactor.

During metal accumulation, all the “slow” reactor columnsshowed a heavy deposition near the inlet, whereas with the

“fast” reactors this was more evenly dispersed along the lengthof the column. This is to be expected since the bulk of thesubstrate cleavage, and hence metal precipitation, would occurnear the reactor inlet at slow flow rates; in contrast, at fasterflows more unreacted substrate can pass to the column’s distalareas, thereby giving a more even distribution of precipitateand hence less local blockage. Figure 5 shows ESEM micro-graphs of Serratia sp. biofilm containing lanthanum phosphatedeposits taken from a “slow” reactor. The blockage fractionaccumulated during the experiment was measured at threedifferent times by thresholding the magnitude MR-images ofboth the “fast” and “slow” flow reactors. The results in Figure6 show that even in the early stages precipitation was moreuniform in the “fast” flow reactor (Figure 6a), whereas the“slow” flow reactor showed preferential blocking at the inlet

Figure 5. ESEM micrographs.(a) Serratia sp. biofilm on reticulated polyurethane foam(2136 magnification), and samples of Serratia sp. biofilm onreticulated foam containing lanthanum phosphate depositstaken from: (b) the middle (2000 magnification), and (c) thebottom (1,600 magnification) of a “slow” reactor.

Figure 4. Phosphate-release and metal-removal by re-actors.Challenged at (a) 128 ml/h (fast), and (b) 11 ml/h (slow) flowwith aqueous La3� (1 mM) and Cu2� (1 mM) in the presenceof glycerol-2-phosphate (5 mM) and citrate buffer (2 mM),pH 6.0. F � Residual Cu2�. ■ � Residual La3�. ‚ �Phosphate released. E � pH of outflow solution. The arrowindicates the time (57.1 h) at which the concentration of freephosphate species were estimated.


region, which was particularly pronounced after 57 h (Figure6b).

The “fast” flow rate was chosen, by prior experiment, as thatwhich gave a small amount of La3� in the outflow solution(that is, the threshold value at which not all of the La3� wasremoved within the flow residence time). Thus, La3� wasremoved to �95% at the slow flow rate (Figure 4b), but onlyca. 90% at the fast flow rate (Figure 4a); the correspondingphosphate concentrations found in the flow were ca. 1.7 mMand 0.7 mM; at the slow flow rate more glycerol 2-phosphatecleavage occurred within the flow residence time. Figure 7shows plots of the values of the non stagnant average z velocity(Vns), average z velocity (Vavg), and blockage fraction (BF) forthe central section of the “fast” flow reactor; there was littlevariation in the same region of the “slow” reactor for furtheranalysis (blockage was largely confined to the post-inlet re-gion). In theory, Vavg should be the same for each transverseslice across the reactor; however, in practice this was not thecase, probably due to the presence of the air bubbles whichdistorted the image, especially the phase data used to calculatethe flow velocity. However, since Vavg and Vns are both calcu-lated using the same region of interest, each acts as an internalreference for the other, and, hence, these fluctuations cancel outbecause the air bubbles are stable throughout each 2.3 h ex-periment. The data show clearly the development of blockageswith time in the 7 mm and 28 mm regions (Figure 7c).

These data are further quantified in Figure 8, which showsthe correlation between blockage fraction and non stagnantaverage z velocity for the data given in Figure 7, with pooleddata from two replicate experiments. The curve was fitted tothe equation BF � 1 � Vavg/Vns; hence, the intercept on theVns axis when BF � 0 equates to Vavg and the value of 0.47cm/s is the same as that obtained by volumetric analysis. Thewide spread of data is because the signal-to-noise of MRI is

poor on a pixel-to-pixel basis; however, using the pixels col-lectively increases the signal-to-noise and, as a consequence,the confidence of the measurements as a whole.

Discussion

These experiments demonstrate the potential utility of MRInot only in this specific application, but for many other pro-cesses which take place in columnar reactors, for which theresultant efficiency is changed by blockage; for example, fil-tration. The image data can provide significant informationregarding the spatial distribution of blockage caused a by

Figure 7. Sample data from a representative MRI veloci-metry experiment for the “fast” reactor.Plots of: (a) Nonstagnant average z velocity, Vns; (b) averagez velocity, Vavg, and (c) blockage fraction, BF (�1 � Vavg/Vns), calculated using each slice, transverse to the direction ofthe flow, along the length of the reactor taken from the 3-Ddata set. Although in theory, Vavg in (b) should be the same foreach slice along the reactor, in practice this was not the case.However, Vns which was calculated in (a) using the sameregion of interest, showed similar fluctuations which cancelout for the calculation of BF shown in (c).

Figure 6. Blockage fraction within the reactor as a func-tion of time.Each of the 2-D slice images was taken from a 3-D magnitudeimage after 57.1 h of flow either at (a) 128 ml/h (fast), and (b)11 ml/h (slow). The blockage fractions were calculated bythresholding the signal intensities for scans measured at 17.6,37.1 and 57.1 h after initiation. All of the data are in goodaccord with reactor theory: thus, the “fast” reactor is moreefficient with the reactants distributed uniformly, whereas the“slow” reactor is concentration limited.


progressive reaction, which could not otherwise be gatheredfrom analysis of the bulk outflow (the “black box” approach) orbe visualized by eye through the opaque precipitate. In the caseof an enzymatic reaction the local flow residence “time” iscritical since the reaction is time dependent; static flow instagnant areas “wastes” capacity, whereas rapid flow in otherswould allow insufficient time to retain Vmax (maximum reac-tion velocity) locally. The overall bioreactor activity is definedto relate to the concentration of substrate and available bio-mass,5,6 but only at a constant flow rate which maintains thelocal substrate concentration high enough to maintain Vmax

throughout.Analysis of the concentration of La3�, Cu2�, and phosphate

ions as well as the pH of the outflow solution from the “fast”and “slow” flow reactors showed some differences (Figure 4);however, one cannot predict from those data how efficient thereactors will be in the longer term. Even after the first day, thesignal magnitude of the MR image reveals that the “fast”reactor is more efficient overall since the reactants, and, hence,as a consequence the precipitate, are both more uniformlydistributed; however, in contrast the “slow” reactor is concen-tration limited since it cannot replenish the reactants fastenough along the length, which leads to blockage at the inlet.That this combination of results is in accord with reactortheory, is well illustrated in Figure 6. The MR images alsoreveal the presence of air bubbles which cannot be observeddirectly by eye, yet are clearly distinguished as zero signalintensity regions compared to the grey scale appearance of theprecipitate.

In contrast to “thresholding” of the magnitude images whichcan be subjective, MRI quantitation of flow through the reac-tors not only defines the difference between blocked and un-blocked regions by distinguishing regions of flow and no-flow,but the resultant “velocity contrast” is also sensitive to smallerobstructions, such as the foam. Areas of apparent zero velocityinclude features, such as air bubbles; however, it is importantto note that since those images may also include areas wherethe signal-to-noise is not sufficient to support the Bayesiananalysis of the velocity encoded images (that is, below thesignal-to-noise threshold), they will tend to overestimate extent

of the blockage. In the present study, comparison of the block-age fraction across the “fast” flow reactor calculated from thevelocity data (Figure 7c) with that from the equivalent regionof the MR magnitude image (between the dotted lines in Figure6a) shows that although the precise values are different, thepeaks and troughs are similar.

If the overall bulk flow rate through the reactor vessel isconstant throughout the duration of the experiment, then theaverage velocity Vavg, should remain the same for each trans-verse slice across the reactor. However, local blockage withinany one region will cause the flow to be constricted to smallerareas causing the non stagnant average z velocity, Vns, toincrease. This is clearly demonstrated in Figure 8 which showsa progressive increase in flow velocity and a quantitative rela-tionship with blockage.

Conclusions

This study clearly demonstrates that MRI can quantitate theflow field in a biofilm reactor, distinguishing the lower velocityflow within the polyurethane reticulated foam cubes from thehigher velocity flow between them, and along the inner edgesof the reactor tube. Flow imaging is also sensitive to thebuildup of metal precipitate, and, hence, enables quantitation ofthe blockage fraction; however, it is not possible to distinguishair bubbles from the precipitate by flow data alone since bothregions are characterised by zero flow. The experimental dataalready available from this study are reminiscent of “text book”theory, and, hence, in principle, could be used to model andsubsequently to optimize the efficiency of this biofilm reactorfor scaleup. Further data could be acquired to refine the currentmodel, developed from applied enzyme kinetics, which as-sumes ideal plug flow, or that the deviations from ideal behav-ior are constant throughout, and that the value of the Km ofenzymes in immobilized bacteria (Km apparent) is a constant.5,6,34

That the former is not the case is clearly demonstrated by theradial profiles taken along the biofilm reactor with increasedblockage; the adjacent local flow rate is increased, and, hence,the amount of substrate which flows past the enzyme is avariable parameter, the value of which depends on the extent oflocal blockage and the exact position of each enzyme moleculein the reactor. Future models need to take this into account andfurther refinements of the MRI technique and subsequent dataanalysis may be required for future studies of such systems.

AcknowledgmentsThis work was supported by the U.K. Biotechnology and Biological

Sciences Research Council (Grant Nos. EO9214 and 6/E1464), and by theHerchel Smith Endowment. The authors thank Recticel (Belgium) for thebiofilm support. KPN, FPH and LDH would like to thank Dr. N. J. Herrodand Dr. D. Xing for the computer software and facilities, respectively; andRichard Smith, Simon Smith and Cyril Harbird for supply and maintenanceof the MR, and other hardware.

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Manuscript received May 28, 2004, and revision received Feb. 25, 2005.


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