Design and scale-up of polycondensation reactors ... · DESIGN AND SCALE-UP OF POLYCONDENSATION REACTORS Hydrodynamics in horizontal stirred tanks and pervaporation membrane modules

Design and scale-up of polycondensation reactors :hydrodynamics in horizontal stirred tanks andpervaporation membrane modulesvan der Gulik, G.J.S.

DOI:10.6100/IR559776

Published: 01/01/2002

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Citation for published version (APA):Gulik, van der, G. J. S. (2002). Design and scale-up of polycondensation reactors : hydrodynamics in horizontalstirred tanks and pervaporation membrane modules Eindhoven: Technische Universiteit Eindhoven DOI:10.6100/IR559776

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DESIGN AND SCALE-UP OFPOLYCONDENSATION REACTORS

Hydrodynamics in horizontal stirred tanks andpervaporation membrane modules

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven op gezag van de

Rector Magnificus, prof.dr. R.A. van Santen, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop dinsdag 10 december 2002 om 14.00 uur

door

Gerardus Johannes Stefanus van der Gulik

geboren te Uithuizen

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. J.T.F. Keurentjesenprof.dr.ir. W.P.M. van Swaaij

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Gulik, Gerardus J.S. van der

Design and scale-up of polycondensation reactors : hydrodynamics in horizontal stirred tanks andpervaporation membrane modules / by Gerardus J.S. van der Gulik. – Eindhoven : TechnischeUniversiteit Eindhoven, 2002.

Proefschrift. – ISBN 90-386-2704-1

NUR 913

Trefwoorden: chemische reactoren ; opschaling / hydrodynamica / membraantechnologie ;pervaporatie / menging / warmteconvectie / numerieke stromingsleer ; CFD / ultrasonecomputertomografie

Subject headings: chemical reactors ; scale-up / hydrodynamics / membrane technology ;pervaporation / mixing / convective heat transfer / computational fluid dynamics ; CFD /ultrasonic computer tomography

© Copyright 2002, G.J.S. van der GulikOmslagontwerp: Paul VerspagetDruk: Universiteitsdrukkerij TU/e

Voor mijn ouders

Het in dit onderzoek beschreven onderzoek werd financieel gesteund door het StanAckermans Instituut, Akzo Nobel, Tejin Twaron en de NOVEM.

SUMMARY

Condensation polymers are an important class of polymers, and have alreadybeen produced for a long time. For new processes, reactor design is often a timeconsuming process of trial and error, resulting in a reactor, for which little is knownabout the occurring physical and chemical conditions. Consequently, the conditionsare difficult to reproduce at a different scale, making the quality of the producedpolymer sensitive to the scale of operation. A proper insight into the hydrodynamicswould enlarge the operating window and provides a route to a more straightforwardscale-up. This includes the Residence Time Distribution, flow patterns, fluidvelocities, and the mixing in the reactor under all flow conditions. The removal ofwater during polycondensations is also strongly influenced by the hydrodynamics.In this thesis, hydrodynamic aspects are studied for the design and scale-up of twotypes of reactor for the production of condensation polymers. The first type ofreactor is a horizontal stirred tank reactor for which it is not straightforward to keepmixing conditions constant upon scale-up. In the second type of reactor, pervapo-ration membranes are used for the removal of water to favor product formation.Controlling hydrodynamics is important for increasing the heat and mass transferbetween the membrane surface and the bulk liquid.

Hydrodynamic aspects of the horizontal stirred tank reactor have been studiedexperimentally and numerically. The experimental characterisation has beenperformed using Planar Laser Induced Fluorescence, Pulse Response Measure-ments, Power measurements and Laser Doppler Anemometry. These techniqueslead to information on overall circulation, poorly mixed zones, macro-mixing times,power consumption, velocities and turbulent quantities. Under laminar conditionsthe mixing in this reactor appears to be chaotic, which identifies the reactor as agood laminar mixer. Under turbulent conditions, however, the fluid mainly rotateslike a solid body, which classifies the reactor as a moderate turbulent mixer. Macro-mixing times have been measured as a function of geometrical parameters andoperation conditions. By correlation, equations are provided for scale-up of thereactor, while keeping the macro-mixing times constant.

In more detail, the hydrodynamics in the horizontal reactor under turbulentconditions has been studied numerically using Computational Fluid Dynamics. Forthe turbulence modelling, the isotropic k-ε model and the anisotropic DifferentialStress Model have been applied. A comparison of fluid velocities and turbulentquantities has been made with data from Laser Doppler Anemometry. The tur-bulence models proved to be able to describe the flow properties equally good.Passive scalar mixing could only be described properly using the anisotropicDifferential Stress Model. Using these numerical tools, new routes come availablefor designing an improved reactor.

Computational Fluid Dynamics has also been used for the design of apervaporation membrane reactor. In this type of reactor it is important to reduceconcentration and temperature polarization to obtain high water fluxes duringoperation. Polarization can effectively be reduced with secondary flow as inducedby density differences. The secondary flow is found to be most effective in

horizontal set-ups increasing water fluxes up to 50%. The temperature segregationin the system, as induced by the secondary flow, has been measured usingUltrasonic Computer Tomography. With this experimental technique the pro-pagation time of sound waves between several transducers is measured. Theaverage temperature between the transducers can be calculated as the propagationtime depends on temperature. With Computer Tomography, a 2D-distribution canbe constructed from a number of average temperatures. The constructed 2D-temperature distribution shows temperature segregation, which is a result of thesecondary flow.

This study is a compilation of numerical and experimental work performed ontwo reactors for polycondensation processes. From the work presented in this thesisit can be concluded that hydrodynamics has a major impact on the design anddevelopment of reactors for polycondensation reactions. CFD has been used as anumerical tool for studying hydrodynamics and designing reactors. The majorchallenge for the future will be to combine the insights in hydrodynamics withkinetic reaction schemes using CFD. This should lead to new reactor designs andmodes of operation, providing reliable and sustainable processes for the future.

SAMENVATTING

Condensatiepolymeren vormen een belangrijke klasse van polymeren die reedsgedurende lange tijd geproduceerd wordt. Bij het vergroten van de produc-tieprocessen, moeten de reactoren veelal worden opgeschaald. Het reactorontwerpis dan echter vaak een tijdrovend proces van trial and error, resulterend inreactoren waarvan weinig bekend is over de heersende fysische en chemischecondities. Bij schaalvergroting zijn de condities derhalve moeilijk te reproduceren,wat leidt tot ongewenste variaties in productkwaliteit. Goed inzicht in dehydrodynamica zou het werkgebied van de reactor kunnen vergroten en zou hetopschalen kunnen vergemakkelijken. Onder de hydrodynamica wordt onder andereverstaan de verblijftijdspreiding, de mengpatronen, lokale vloeistofsnelheden en demenging in de reactor bij verschillende stromingsregimes.

In dit proefschrift worden de resultaten beschreven van het onderzoek naarhydrodynamische aspecten van twee type reactoren waarin polycondensatiesworden uitgevoerd. Het doel is de verkregen inzichten te gebruiken voor hetopschalen en ontwerpen van de reactoren. Het eerste type reactor is een horizontalegeroerde tankreactor waarbij de nadruk ligt op het voorspellen en controleren vande menging op verschillende schaalgroten. In het tweede type worden pervaporatiemembranen toegepast voor het verwijderen van water uit het reactiemengselwaardoor productvorming wordt bevorderd. Het kunnen voorspellen en controlerenvan de hydrodynamica in dit type reactoren is belangrijk omdat daarmee dewarmte- en stofoverdracht kunnen worden verbeterd tussen het membraanoppervlaken de vloeistofbulk.

De hydrodynamische aspecten van de horizontaal geroerde reactor zijn zowelexperimenteel als numeriek onderzocht. Experimenteel onderzoek is uitgevoerd metbehulp de experimentele methoden Planar Laser Induced Fluorescence, puls-responsie-metingen, Laser Doppler Anemometrie en vermogensmetingen. Dezemethoden leveren inzicht in stromingspatronen, slecht gemengde zones,macromengtijden, vermogensverbruik, lokale vloeistofsnelheden en turbulentegrootheden. Onder laminaire condities heeft de menging in de reactor een chaotischkarakter waardoor de menging relatief goed en snel verloopt. Onder turbulenteomstandigheden draait de vloeistof onder invloed van de centrifugaalwerkingvoornamelijk rond zijn as als een star lichaam. De menging is dan niet erg effectief.Macromengijden zijn voor verscheidene condities bepaald als functie van geome-trische variaties. Door correlatie van de data zijn empirische relaties afgeleid diekunnen worden toegepast om bij schaalvergroting de macromengtijden te kunnenvoorspellen.

Met behulp van de numerieke techniek Computational Fluid Dynamics, is dehydrodynamica in meer detail bestudeerd. Voor de turbulente modellering zijn hetisotrope k-ε-model en het anisotrope Differential Stress Model gebruikt. Berekendevloeistofsnelheden en turbulente grootheden zijn vergeleken met data uit de LaserDoppler Anenometrie metingen. Beide turbulentiemodellen voorspellen de hoofd-stromingen even goed. Opmenging van passieve scalairen, hetgeen wordt gebruiktom de mengsnelheid te kunnen kwantificeren, wordt alleen goed voorspeld met

behulp van het Differential Stress Model. Uitgebreidere toepassing van Computa-tional Fluid Dynamics technieken zou het ontwerpen van dit type reactoren kunnenvergemakkelijken en versnellen.

Compuational Fluid Dynamics technieken zijn ook toegepast bij het ontwerpenvan het tweede type reactor; een pervaporatie membraanreactor. In dit typereactoren is het van belang de concentratie- en temperatuurpolarisatie te reducerenteneinde hoge waterfluxen te verkrijgen tijdens het productieproces. Polarisatie kaneffectief worden gereduceerd door secundaire stroming die onstaat onder invloedverschillen in dichtheid in de vloeistof. Het is gebleken dat deze secundairestroming het meest effectief is in horizontale opstellingen. Een toenamen van 50%in de waterflux wordt voorspeld.

Aan temperatuurssegregatie in een horizontale modelopstelling zijn metingenverricht met behulp van Ultrasone Computer Tomografie. Met deze experimenteletechniek wordt de voortplantingssnelheid gemeten van geluidsgolven tussenverschillende transducers. De gemiddelde temperatuur tussen twee transducers kanhiermee indirect worden vastgesteld omdat de voortplantingssnelheid afhankelijk isvan de temperatuur. Met Computer Tomografie kan van een groot aantalgemiddelde temperaturen een 2-dimensionale temperatuursverdeling gereconstru-eerd worden. Deze verdeling laat zien dat in de onderzochte horizontalemembraanmodules de voorspelde temperatuurssegregatie inderdaad optreedt.

Dit onderzoek is een compilatie van numeriek en experimenteel werk aan tweetypen reactoren voor de productie van condensatiepolymeren. De hydrodynamica isvan eminent belang voor het goed functioneren en kunnen ontwerpen van beidetype reactoren. Grote vooruitgang kan worden geboekt door uitgebreideretoepassing van Computational Fluid Dynamics technieken. De grote uitdaging is tevinden in het combineren van de verkregen inzichten in de hydrodynamica enkinetische reactieschema’s in Computational Fluid Dynamics. Dit biedt perspectiefvoor nieuwe revolutionaire reactor ontwerpen en operationele condities, resulterendin betrouwbare en duurzame productieprocessen voor de toekomst.

CONTENTS

1. THE NEED FOR CONTROLLING HYDRODYNAMICSIN POLYCONDENSATION REACTORS 1

2. HYDRODYNAMICS IN A HORIZONTAL STIRREDTANK REACTOR 15

3. HYDRODYNAMICS AND SCALE-UP OF HORIZONTALSTIRRED REACTORS 37

4. FLUID FLOW AND MIXING IN AN UNBAFFLEDHORIZONTAL STIRRED TANK 59

5. HYDRODYNAMICS IN A CERAMIC PERVAPORATIONMEMBRANE REACTOR FOR RESIN PRODUCTION 87

6. MEASUREMENT OF 2D-TEMPERATURE DISTRIBUTIONSIN A PERVAPORATION MEMBRANE MODULE USINGULTRASONIC COMPUTER TOMOGRAPHY 107

7. FUTURE PERSPECTIVES FOR PROCESS ENGINEERINGOF POLYCONDENSATION REACTIONS 131

DANKWOORDCURRICULUM VITAE

THE NEED FOR CONTROLLINGHYDRO

POLYCONDE

1
DYNAMICS IN
NSATION REACTORS

Chapter 12

→←

1.1 Condensation polymers

Condensation polymers are an important class of macromolecular products andinclude synthetic materials used as high-strength and/or high-toughness plastics andfibers (e.g., polyamides, polyesters and polycarbonates) as well as almost all hardresins (e.g., unsaturated polyesters, epoxy resins, urea-, melamine-, and phenolformaldehyde resins), thus covering a wide range of applications. Additionally,members of the same product family are used in relatively small but importantsectors as sealants, elastomers, foams, and adhesive coatings (e.g., silicones, alkydresins, and polyimides, Parodi et al., 1989).

Condensation polymerization, also called step-growth polymerization,involves one or more reactants (monomers) possessing at least two reactivefunctional groups (Manaresi et al., 1989). In general, a polycondensation reactioncan be described schematically in the following way:

(n+1) A R1 A + n B R2 B A R1 ( C R2 C R1 )n A + 2 n Q (R.1)

End groups A and B react, forming a group C that becomes part of the polymerchain and a small molecule Q. The monomer molecules will disappear rapidly, andconsequently, after a while only chains of monomers are coupling with otherchains. In principle, the degree of polymerization is determined by a small excess ofone of the monomers. When the relative excess equals 1/n, the average chain willcontain n units1.

1 The excess origins from the ratio of reactants in reaction R.1: (n+1)/n = 1 + 1/n

10

100

1000

0.9 0.92 0.94 0.96 0.98 1

Xc [-]

Pn [

-]

n/(n+1)1.0

0.995

0.99

0.9750.950.9250.9

Figure 1.1: Average degree of polymerization as a function ofconversion Xc at different initial stoichiometric ratios n/(n+1).

The need for controlling hydrodynamics in polycondensation reactors 3

The conversion, Xc, can be expressed as the degree of conversion of endgroups, either based on A or B. In Figure 1.1, the number-averaged degree ofpolymerization n has been plotted against the degree of conversion Xc for severalinitial stoichiometric ratios n/(n+1). This figure shows that to obtain 100 units perchain with an initial stoichiometric ratio of 0.99, Xc has to be 0.995. Thus, to obtainlong polymer chains both the stoichiometric ratio and Xc have to be close to unity.Often such long polymer chains are required to achieve the desired product quality.

Chapter 14

1.2 Reaction engineering of polycondensations

The very high conversions that are required often lead to major technicalproblems (Thoenes, 1994; Vollbracht, 1989). For example, the small molecule Q(often water) may have to be removed to force the reaction to completion(Carothers, 1936; Sawada, 1976). This can be achieved by evaporation or chemicalbonding. The removal of the small molecule from a viscous molten polycondensateis difficult as the surface tension and the viscosity of the liquid hinder thenucleation of vapor bubbles. Even when bubbles are formed, they have such a smallvolume that their rising velocity is extremely low.

To obtain high conversions, the Residence Time Distribution (RTD) has to bevery narrow (Thoenes, 1994; Westerterp et al., 1984; Biesenberger et al., 1983).Therefore, most often a batch reactor is used. A disadvantage of a batch reactor isthat the viscosity of the reactor content increases dramatically during the process.This usually induces flow conditions to change from turbulent to laminar. A seconddisadvantage of a batch reactor is that at the beginning of the polycondensationprocess the reaction rate is high as the concentration of end groups is highest.Consequently, most heat of reaction has to be removed at the beginning of theprocess. It is difficult to remove all the heat within a short period of time from alarge batch reactor because the surface to volume ratio is relatively low. Therefore,external cooling has to be applied or the scale of production has to be reduced. Anumber of reactors in series would sometimes be highly desirable, because differenttypes of impeller can be used during different stages of the process and the heatproduction can be distributed over several reactors. However, a series of reactorscan only be used when the RTD shows exact plug flow behavior or can be keptextremely narrow.

For a polycondensation process it is difficult to find a suitable reactor, becausethe various requirements are often contradictory. The route to a successfulpolycondensation reactor is a time-consuming process of trial and error. From theresulting reactor, frequently little is known about the physical and chemicalconditions. Consequently, these conditions are difficult to reproduce at a differentscale, making the product quality sensitive to the scale of operation. However,successful operation upon scale-up can sometimes be maintained when severalreactor-engineering aspects are known. Such aspects, for which generally thecollective term ‘hydrodynamics’ is used, comprise the RTD of a reactor system, theflow patterns, the mixing under various flow conditions, and the mixing power. Inthis thesis, hydrodynamic aspects are studied for the design and scale-up of twotypes of reactor for the production of condensation polymers. The first type ofreactor is a horizontal stirred tank reactor for which it is not straightforward to keepthe mixing conditions constant upon scale-up. In the second type of reactorpervaporation membranes are used for the removal of the byproduct water.


1.3 Tools for studying hydrodynamics

For the optimization of polymerization reactors, characterization of theoccurring hydrodynamics is of prime importance. Two characterization methods areavailable: Experimental and Computational Fluid Dynamics (known as EFD andCFD, respectively). EFD is a collective term for several experimental techniquesused to characterize the hydrodynamics. In this thesis the following experimentaltechniques have been used:

• Planar Laser Induced Fluorescence (PLIF) – With PLIF a fluorescentdye can be traced in a 2-dimensional laser sheet. Using a high-speedcamera, a digital film can be made of the mixing pattern of the tracer in thereactor (Chapter 2; Miller, 1981; Owen, 1976).

• Pulse Response Measurements – With Pulse Response Measurements aninert tracer is injected in a batch reactor while at a different position theconcentration of the tracer is monitored as a function of time. From thesemeasurements macro-mixing times can be deduced (Chapter 3; Westerterpet al., 1984).

• Laser Doppler Anemometry (LDA) – LDA is an optical method for fluidflow research based on a combination of interference and Doppler effects.LDA allows the measurement of the local, instantaneous velocities ofparticles suspended in the flow. LDA has a high resolution power in timeand is non-invasive (Chapter 4; Durst et al., 1981).

• Ultrasonic Computer Tomography (U-CT) – U-CT is based on thedependence of the propagation velocity of ultrasound on the temperature ofa medium. Over a line between a speaker and a microphone the velocity ofsound is measured and converted into an average line temperature. Bymeasuring a large number of lines in a plane and using ComputerTomography, a 2-dimensional temperature distribution can be constructed.The technique is non-invasive (Chapter 6; Norton et al., 1984; Peyrin et al.,1983).

• Magnetic Resonance Imaging (MRI) – With MRI the proton density in amedium can be measured over a line. By measuring a large number of linesin a plane and using advanced computational techniques, a 2-dimensionalproton density distribution can be constructed. This distribution canprovide information on the location of different chemicals. The techniqueis non-invasive. Additionally, MRI can be used to measure local velocitiesby applying pulse field gradients (Chapter 7; Hornak).

CFD is a numerical tool for studying hydrodynamics. Fluid flow, mixing, heatand mass transfer in a prescribed geometry can be calculated on a computer. Forthis, several commercial packages are available, including CFX (AEA Technology,Harwell, UK), Fluent (Fluent Inc., Lebanon, USA), Star-CD (ComputationalDynamics Ltd., London, UK). Only CFX has been used, in which both laminar and

Chapter 16

turbulent conditions can be handled. For turbulent flows, the Reynolds-AveragedNavier-Stokes equations can be solved with the use of appropriate turbulencemodels.

Figure 1.2: Flow state reactants in the Twaron

TDC

H O2

PPD

and initial location of the® polymerization process.


1.4 Scale-up of a Horizontal Stirred Tank Reactor.

The horizontal stirred tank reactor studied in this thesis is of the Drais typeand is further referred to as the Drais reactor. The aromatic polyamide Twaron isproduced in this type of reactor (Vollbracht, 1989; Banneberg-Wiggers et al., 1998).During the reaction, the flow regime changes from turbulent to laminar, due to atremendous increase in viscosity. In both regimes mixing has to be sufficient as ithas a large influence on the final product quality. The Twaron polyamide PPTA(Para-Phenylene TerphthalAmide) is produced via the condensation reaction ofPPD (Para-Phenylene Diamide) with TDC (Terephthaloyl DiChloride). The fastpropagation step is given as reaction (R.2):

(R.2)

(PPD) (TDC) (PPTA or Twaron® polymer)

The degree of polymerization is controlled by adding a small amount of waterthat can terminate a reactive acylchloride group via the slow reaction (R.3):

(R.3)

Water is added at the start of the process because at the end of the process theviscosity is too high for mixing the water sufficiently with the reactor content. Theinitial presence of water, however, complicates the process, as it is able to terminategrowing chains too early when mixing is insufficient, leading to short chains and abroad molecular weight distribution (MWD). The set of reactions (R.2 and R.3) isusually described as competitive-parallel.

In production, the reactor is partially filled with the diamide component(containing the required amount of water), to which the diacylchloride componentis added semi-batchwise. To allow for an exact stoichiometric ratio of the twomonomers, only one injection point is used, as schematically depicted in Figure 1.2.The use of a single injection point implies that a good overall circulation is neededto allow the acylchloride molecules to react with all the amide-containing moleculesthroughout the reactor before termination with water occurs. Thus, a proper insightinto the flow pattern, the mixing, and the mixing times are mandatory to guaranteeconstant product quality upon scale-up and to allow for quality improvements inexisting equipment. These hydrodynamic aspects will be treated in the chapters 2 to4 of this thesis.

R3 C

O

Cl + H 2O + H ClR3 C

O

OH

C ClO

CO

ClNH2H2N(n+1) + n NH

N CH O

CO

H N NH2

H

n

+ n HCl

Chapter 18

1.5 Design of the membrane reactor

Alkyd resins are low molecular weight polyesters, formed by the equilibriumreaction between a di-alcolhol and a di-acid, as given in (R.4):

(R.4)

Reaction conditions require elevated temperatures, typically above 200 °C. At thesetemperatures, the mixtures are moderately viscous (µ = 5-25 mPa·s). To obtain highyields, the reaction equilibrium has to be shifted to the right. This can be achievedby using a large excess of the alcohol (usually as solvent) and by removing thewater. In current processes, water is removed afterwards or during the reaction bydistillation (Figure 1.3A), which is not very efficient in the case of azeotropeformation. Additionally, due to the large reflux ratios required, energy consumptioncan be significant (Keurentjes et al., 1994). In this operation the energy input can be10 times larger than the energy required for the removal of water.

n R 1 O H + n R 2 C O O H R e s i n + H 2 O

Figure 1.3A-C: A) Current batch process for the production of resins with a distillation unitfor the removal of water. B) Improved batch process in which the distillation unit has beenreplaced with a pervaporation membrane module. C) Continuous process in which thereactor and the distillation unit have been replaced with one pervaporation membranereactor, running in a once-through continuous mode.

2Pervaporationmembrane reactor

Resin

Reactants

H O2

Pervaporationmembranemodule

Reactor

ReactantsResin

H O2

Distillation unit

Reactor

ReactantsResin A

B

C H2O


With ceramic pervaporation membranes, water can be removed selectivelythrough the membrane by evaporation (Ho et al., 1992; Nunes et al., 2001;Rautenbach et al., 1984, 1985; Bakker et al., 1998; Koukou et al., 1999; Verkerk etal., 2001; Jafar et al., 2002). These membranes are selective to water as they areextremely hydrophilic and have pores that are about the kinetic diameter of water.Additionally, they are resistant to high temperatures. Use of these membranesallows for two alternative process schemes. In Figure 1.3B, a pervaporationmembrane module is used instead of the evaporator. This bath-wise concept inFigure 1.3B is relatively easy to implement in current production processes, becauseonly one unit operation has to be replaced. In Figure 1.3C, both the reactor andevaporator have been replaced by a membrane reactor. This concept is less easy toimplement but has advantages, as it can be operated in continuous mode, whichmakes it easier to control the product quality. In both concepts, the reduction inenergy consumption should be significant as, in principle, the only energy requiredis used for the evaporation of water.

It can be anticipated that concentration and temperature polarization near themembrane represent a major problem. The polarization effects are schematicallydepicted in Figure 1.4. As water is transported through the membrane, the waterconcentration near the membrane surface will be low. As a result the resinconcentration will be high, as the reaction will locally be in equilibrium. Also, thetemperature will be low as water evaporates at the membrane surface adiabatically.Consequently, viscosity near the membrane will be relatively high. As a result, arelatively thick stagnant layer can be formed, severely reducing water transportfrom the bulk to the membrane surface.

Optimizing the flow conditions near the membrane can reduce the occurringpolarization effects. For the two process options given, different flow conditions arepossible. For the process in Figure 1.3B, polarization can be reduced by applying

Water concentration

Water

Resin concentration

Temperature

Arbitrary units

Viscosity

δ T δ C

Membrane

Figure 1.4: Concentrations, temperature and viscosity inarbitrary units near the membrane surface.

Chapter 110

high turbulence levels. For example, a powerful pump can recycle the liquid withhigh velocities over the membrane. However, this will reduce the energy savings.Using a powerful pump is not appropriate in the process depicted in Figure 1.3C,because velocities have to be low in order to limit equipment dimensions.Additionally, in Figure 1.3C nearly plug flow characteristics are required to obtain anarrow molecular weight distribution (MWD).

It will be difficult to design a membrane reactor according to the process inFigure 1.3C with a flow pattern that reduces the polarization effects effectively,while fulfilling the requirements. In this thesis, the application of buoyancy forcesis considered for obtaining a flow pattern that will effectively reduce polarizationeffects. Also, a relatively new experimental technique has been implemented forstudying accompanying temperature effects.


1.6 Outline of this thesis

This thesis describes the hydrodynamic aspects occurring in the two types ofpolycondensation reactor using both EFD and CFD. For the horizontal stirred tankreactor, fluid flow and mixing will be described both quantitatively andqualitatively. Insight into these hydrodynamic aspects will lead to efficient scale-upof the reactor in such a way that the product quality can be kept constant uponscale-up. For the membrane reactor the characterization of the flow and polarizationeffects should lead to the design of a once-through continuous reactor in which flowconditions are laminar.

In Chapter 2, flow patterns of inert tracers under turbulent and laminarconditions are described. PLIF is used as the experimental technique. Experimentshave been performed in three small-scale models of the Drais reactor. Mixing timeshave been determined, leading to guidelines for scale-up.

In Chapter 3, the power input upon agitation has been measured in order toclassify the configuration relative to other standard configurations. The energy inputalso provides the energy dissipation rate ε, which is crucial for defining well-mixedand poorly mixed regions. Mixing times have been measured using Pulse ResponseMeasurements. Combined with the power measurements this leads to insight intothe mixing efficiency as a function of scale.

Chapter 4 is the final chapter on the Drais reactor describing LDAmeasurements under turbulent conditions. Mean and fluctuating velocities provide ageneral insight into the velocity scales and mixing processes occurring in the Draisreactor. LDA and PLIF measurements are compared with CFD calculations. Toincorporate the turbulent properties, several turbulence models are available inCFD. Comparison shows that for a correct prediction of the PLIF experiments theselection of the appropriate turbulence models is the crucial parameter in CFD.

Chapters 5 and 6 concern the pervaporation reactor. Chapter 5 contains aCFD study in which the effect of the orientation of a single membrane tube withrespect to the gravity field has been studied. For these cases the relevance ofbuoyancy effects at laminar conditions has been examined. As an additionalparameter the superficial velocity has been varied. Heat and mass transfer ratesprovide the essential parameters to distinguish the most optimal configuration.

Chapter 6 describes the successful implementation of U-CT (UltrasonicComputer Tomography) in a pervaporation membrane module. With this relativelynew experimental technique, 2-dimensional temperature distributions can bemeasured. The measured temperature distributions are compared with CFDcalculations.

Chapter 112

A review will be given in Chapter 7. Additionally, the implications for scale-up and future designs will be discussed.

The set-up of this thesis is such that each chapter can be read separately.Consequently, some information will be repeated at more than one location. Thisapproach has been chosen deliberately in order to enable the reader to go onlythrough specific chapters of interest and to avoid a long list of references to otherparts of this thesis.


References

Bakker, W.J.W.; Bos, I.A.A.C.M.; Rutten, W.L.P.; Keurentjes, J.T.F.; Wessling, M.;“Application of ceramic pervaporation membranes in polycondensation reactions”, Int.Conf. Inorganic Membranes, Nagano, Japan, 1998, 448-451.

Bannenberg-Wiggers, A.E.M.; Van Omme, J.A.; Surquin, J.M.; “Process for the batchwisepreparation of poly-p-terephtalamide”, U.S. Pat., 5,726,275, 1998.

Biesenberger, J.A.; Sebastian, D.H.; “Principles of Polymerization engineering”, John Wileyand Sons, New York, 1983.

Carothers, W.H.; Trans. Faraday. Soc., 1936, 32, 39.Durst, F.; Melling, A.; Whitelaw, J.H.; “Principles and Practice of Laser Doppler Anemo-

metry”, Academic Press, London, UK, 1981.Ho, W.S.W.; Sirkar, K.K. (Eds.); “Membrane Handbook”, Chapman & Hall, New York,

1992.Hornak, J.P.; “The basics of MRI”, http://www.cis.rit.edu/class/schp730/bmri/bmri.htm.Jafar, J.J.; Budd, P.M.; Hughes, R.; “Enhancement of esterification reaction yield using

zeolite A vapour permeation membrane”, J. Membr. Sci., 2002, 199 (1-2), 117-123.Keurentjes, J.T.F.; Janssen, G.H.R.; Gorissen, J.J.; “The esterification of tartaric acid with

ethanol: kinetics and shifting the equilibrium by means of pervaporation”, Chem. Eng.Sci., 1994, 49, 4681-4689.

Koukou, M.K.; Papayannakos, H.; Markatos, N.C.; Bracht, M.; Van Veen, H.M.; Roskam,A.; “Performance of ceramic membranes at elevated pressure and temperature: effect ofnon-ideal flow conditions in a pilot scale membrane separator”, J. Membr. Sci., 1999,155, 241-259.

Manaresi, P.; Munari, A.; “Factors affecting rate of polymerization”, ComprehensivePolymer Science, Step Polymerization, 1989, 5, 35.

Miller, J.N.; “Standard in fluorescence spectrometry”, Chapman & Hall, London, UK, 1981.Norton, S.J.; Testardi, L.R.; Wadley, H.N.G.; “Reconstruction internal temperature distri-

butions from ultrasonic time-of-flight tomography and dimensional resonancemeasurements”, J. Res. Natl. Bur. Stand., 1984, 89(1), 65-74.

Nunes, S.P.; Peinemann, K.-V. (Eds.); “Membrane Technology in the Chemical Industry”,Wiley-VCH, Weinheim, 2001.

Owen, F.R.; “Simultaneously laser measurements of instantaneous velocity and concentra-tion in turbulent mixing flows”, AGARD-CP193, Paper No. 27, 1976.

Parodi, F.; Russo, S.; “Polycondensation and Related Reactions”, Comprehensive PolymerScience, Step Polymerization, 1989, 5, 1.

Peyrin, F.; Odet, C.; Fleischmann, P.; Perdrix, M.; “Mapping of internal material tempera-ture with ultrasonic computed tomography”, Ultrason. Imag., Conference Proceeding,July 1983, Halifax, 31-36.

Rautenbach, R.; Albrecht, R.; “On the behavior of asymmetric membranes inpervaporation”, J. Membr. Sci., 1984, 19, 1-22.

Rautenbach, R.; Albrecht, R.; “The separation potential of pervaporation. Part 2. Processdesign and economics”, J. Membr. Sci., 1985, 25, 25-54.

Sawada, H.; “Thermodynamics in polymerization”, Chapter 6, Marcel Dekker, New York,1976.

Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers, Dordrecht,The Netherlands, 1994.

Verkerk, A.W.; Van Male, P.; Vorstman, M.A.G.; Keurentjes, J.T.F.; “Description ofdehydration performance of amorphous silica pervaporation membranes”, J. Membr.Sci., 2001, 193(2), 227-238.

http://www.cis.rit.edu/class/schp730/bmri/bmri.htm

Chapter 114

Vollbracht, L.; “Aromatic Polyamides”, Comprehensive Polymer Science, Step Polymeriza-tion, 1989, 5, 374.

Westerterp, K.R.; Van Swaaij, W.P.M.; Beenackers, A.A.C.M.; “Chemical Reactor Designand Operation”, John Wiley and Sons, New York, 1984.

HYDRODYNAMSTIRRED

In this study, the hreactor is investigated. Thpolycondensation processeand macro-mixing timesturbulent (Re > 105) and laser induced fluorescenceoverall circulation is codiameter ratio is varied. Ube chaotic. The poorly milife span at different lengtmixing times under tuparameters variations andlength-to-diameter ratio, conditions, macro-mixinunambiguously, but they aconditions.

This chapter is a slightly modified Van der Gulik, G.J.S.; Wijers, J.GStirred Tank Reactor”, Ind. Eng. C

2
ICS IN A HORIZONTAL TANK REACTOR
Abstract

ydrodynamics in a horizontal stirred tankis type of reactor is used in industry for fasts. Overall circulation, poorly mixed zones

are determined in scale models underlaminar (Re < 300) conditions using planar. At both sets of conditions, the observed

mplex and changes when the length-to-nder laminar conditions, the flow appears toxed zones change in location, number, andh-to-diameter ratios. Dimensionless macro-rbulent conditions are correlated with show nonlinear relationships in fill ratio,and Reynolds number. Under laminar

g times could not be determinedre only 2.5 times larger than under turbulent

version of the publication:.; Keurentjes, J.T.F.; “Hydrodynamics in a Horizontal

hem. Res., 2001, 40(3), 785-794.

Chapter 216

2.1 Introduction

The design of a reactor for fast polycondensations is a major challenge forchemical engineers, as often several conflicting needs have to be fulfilled.Generally, two types of agitation are needed as the flow regime changes fromturbulent to laminar, because of a tremendous increase in viscosity. In both regimesthe mixing has to be sufficient as it has a large influence on the final product quality(Thoenes, 1994; Manaresi et al., 1989). This influence can easily be understood byconsidering the production process of Twaron, an aromatic polyamide producedvia the polycondensation reaction of a diamide with a diacyl chloride (Gaymans etal., 1989; Vollbracht, 1989).The fast propagation step is given as reaction R.5:

(R.5)

Theoretically, a small excess of the diamide determines the degree ofpolymerization. When the relative excess equals 1/n, the average chain will containn+1 units. In practice, the degree of polymerization is controlled by adding waterthat can terminate a reactive acyl chloride group via the slow reaction R.6:

(R.6)

Water is added at the start of the process because, at the end of the process, theviscosity is too high to mix water sufficiently to molecular scale. The initialpresence of water complicates the process, as water is able to terminate chains tooearly when mixing is insufficient, leading to short chains and a broad molecularweight distribution (MWD).

Table 2.1: Order of magnitude of reaction time for reactions R.5 and R.6 (Jeurissen et al.;Borkent, 1976).

Viscosity [Pa·s] Reaction time for propagation R.5 [s] Reaction time for termination R.6 [s]10-3 10-4 102

10 10 102

In Table 2.1, the orders of magnitude of the reaction half-life times for bothreactions are given for low and high viscosity levels. Going from low to highviscosity, the propagation rate slows by 5 orders of magnitude, while thetermination rate remains unchanged. The propagation reaction is slowed down fortwo reasons. Firstly, RNH3

+Cl--groups are formed, which are less reactive than

R3 C

O

Cl + H 2O + H ClR3 C

O

OH

Cl C

O

R2 C

O

ClH2N R1 NH2(n+1) + n

OOH

CR2CNR1

H

NH N R1 N H

HH

+ n H Cl

n

Hydrodynamics in a Horizontal Stirred Tank Reactor 17

NH2-groups (Gaymans et al., 1989). Secondly, the reactive end groups are moresterically hindered upon an increase in molecular weight. The chain stopper (water)will not be hindered as much, because it is a relatively small molecule.

In production, the reactor is filled with the diamide-component (containing therequired amount of water), to which the diacyl-component is added semi-batchwise.To allow for an exact stoichiometric ratio of the two monomers, only one singleinjection point is used. This implies that good overall circulation is needed to allowthe acyl molecules to react with all the amide-containing molecules throughout thereactor before termination occurs. This implies that a fundamental insight into thehydrodynamic behavior of this type of reactor is mandatory to guarantee constantproduct quality upon scale-up and to allow for quality improvements in existingequipment.

The polymerization described above is performed in a horizontal stirred tankreactor of the Drais type (Vollbracht, 1989). This multifunctional reactor, asdepicted in Figures 2.1 and 2.2, can be used for powder mixing, turbulent fluidmixing and kneading at high viscosities with an energy dissipation up to 200 W/kg.Literature on the hydrodynamics in horizontal mixing vessels is very limitedcompared to the literature on vertical vessels. There is some literature on turbulentmixing, but literature on laminar mixing is absent. Ando et al. (1971a) studiedpower consumption and flow behavior under turbulent conditions in an unbaffledhorizontal vessel with Rushton turbine impellers with Di/D = 0.9. They distinguishtwo flow states A and B. State A is obtained at a relatively low stirrer speed. The

C

D

L

Hatch

Top

FrontSide

Injection

h

w α

β

Figure 2.1: The Drais reactor, given in front, top and side views with length L, diameterD, blade angles α and β, blade width w, blade height h, and a hatch. The arrows pointout the direction of rotation during operation.

Chapter 218

liquid is then pushed up by the impellers and sprayed, leading to the formation offine liquid droplets and fine air bubbles. State B, the so-called hollow state, isobtained at a higher stirrer speed, providing a ring of fluid. As their research mainlyfocused on applications to gas-liquid absorption, mainly state A in baffled vesselswas investigated (Ando et al., 1971b, 1974, 1981; Fukuda, 1990). This is due to alarger gas-liquid interface in the A state compared to state B. Ando et al. (1990)also studied turbulent mixing in an idealized horizontal vessel with baffles andmultiple impellers. Macro-mixing times were measured, and a model was proposedfor predicting them. It was established that the dimensionless macro-mixing timeN⋅t is proportional to L/D. The information available in the literature on turbulentmixing in horizontal stirred tank reactors is not directly applicable topolycondensations in the Drais reactor for two reasons. First, the impeller geometryis completely different. Second, the fluid in the polycondensation process is in thehollow state (or the B state according to Ando et al., 1971a) because of a high stirrerspeed.

Therefore, we conducted an experimental study on the hydrodynamics in thistype of reactor. For this purpose, mixing patterns, the life span of poorly mixedzones, and the macro-mixing time have been established experimentally. This isdone for turbulent as well as laminar conditions for different reactor fill ratios.Subsequently, scaling rules will be defined based on these macro-mixing times.


Figure 2.2: Stirrer geometry for reactor-15 (top) and reactor-20 (bottom). The whiteblades transport fluid to the right during rotation, the black stirrers transport fluid tothe left. Thus, both have a pumping action toward the center of the reactor. Theblades are evenly distributed over the shaft and are not drawn in perspective.

Chapter 220

2.2 Experimental Section

The reactor used in this study was a horizontal stirred tank of the Drais type(Turbulent Schnellmischer, Drais Ltd, Mannheim, Germany), as depicted in Figures2.1 and 2.2. Typical for the unbaffled cylindrical reactor is its horizontal positionand the heavily designed impeller. This impeller can provide a high mixing power,needed to achieve sufficient mixing under highly viscous conditions. The reactor ischaracterized by length L and diameter D. Typical for the reactor is the clearance C,the distance between the blades and the reactor wall. For a small clearance, theblades perform a scraping action that keeps the reactor walls free from polymermaterial and also provides good heat exchange with the cooled walls. The bladeshave a pumping action towards the reactor center, providing an easy way to emptythe reactor through the opened hatch.

To determine macro-mixing times and the life span of poorly mixed zones,planar laser induced fluorescence (PLIF) was used. With PLIF, it is possible tomake a digital film of the mixing of a tracer in a 2-dimensional plane in which thepoorly mixed zones can easily be located. In contrast with other studies (Distelhoffet al., 1997; Kersting et al., 1995; Mayr et al., 1994; Moo-Young et al., 1972;Perona et al., 1998). PLIF is unobtrusive, has a small measurement volume, and isflexible in changing the monitoring point, as only the position of the laser sheet hasto be changed. PLIF experiments were performed in three small-scale models of theDrais reactor. These reactors, as depicted in Figures 2.1 and 2.2, were all 0.18 m indiameter but differed in length, being 0.20, 0.27, and 0.36 m, providing L/D ratiosof 1.1, 1.5, and 2.0, respectively. From this point on, these scale-models are referredto as reactor-11, reactor-15, and reactor-20, respectively. The blade width w wasequal to 0.1 m, the blade height h was equal to 0.015 m, and the shaft had adiameter of 0.03 m. The blades were evenly distributed over the shaft for eachreactor. The mutual angle of the blades was 180, 120, and 135° for reactor-11,reactor-15, and reactor-20, respectively. These angles correspond to industrialconfigurations and proved to provide the most stable fluid ring in partially filledreactors. The sidewalls, the shaft, and the impeller blades were made of stainlesssteel, and the cylindrical wall of glass. Under turbulent conditions, tap water wasused as the reactor content for the three fill ratios 40, 60, and 100%. The rotationalspeed varied between 3.8 and 11.6 Hz, resulting in turbulent flow with Reynoldsnumbers ranging from 123,000 to 375,000, as defined by ρNDi

2/µ. When thereactor was partially filled, the rotational speeds always resulted in the hollow stateor the B state according to Ando et al. (1971a). For laminar conditions, glycerin(Heybroek, Amsterdam; purity >99.9%) was used at Reynolds numbers rangingfrom 90 to 270. Glycerin limits the use of PLIF to the examination of completelyfilled reactors because air bubbles lead to an untransparent fluid in partially filledreactors.

Figure 2.3 shows the experimental arrangement for the PLIF experiments,which is comparable to the set-up used by Schoenmakers et al. (1997). The laserbeam was generated by a 2-W Ar/Kr-laser (model Stabilite 2017-005, Spectra


Physics) and had a wavelength of 488 nm. The beam was converted to a laser sheetwith a thickness of 0.5 mm by a cylindrical lens (Dantec 9080XO.21). The positionof the laser sheet in the vessel geometry was always parallel to the shaft. Therefore,the observed mixing process was always the mixing in the axial and radialdirections. This is secondary mixing, superimposed on the mixing in the tangentialdirection.

Disodium fluorescein (C20H10O5Na2) was used as the fluorescent dye (Merck,Darmstadt; purity >98wt%). This dye emits light with an intensity depending on thepower of the laser light, the concentration, and the pH of the solvent. The power ofthe laser light was kept constant at 0.4 W. During every experiment, the final dyeconcentration was around 10-7 M. Therefore, the amount of injected solution, withdye concentration of 2·10-3 M, varied between 0.3 and 0.5 mL, depending on thereactor volume and fill ratio. The pH of both the injected solution and the reactorcontent was kept constant at a value of 10, as the intensity of the emitted light isindependent of the pH at pH > 8.

A high-speed camera (JAI CV-M30), connected to a PC, with an EISAcompliant frame grabber (Magic) recorded the light that was emitted by thefluorescein molecules in the laser light plane. The commercial software packageDMA-MAGIC was used for data acquisition. The value of the recorded gray scalesranged from 0 to 255, providing a resolution of 256 values. For complete mixing, agray scale of around 150 was obtained. In the range from 0 to 255 the gray scalecorresponds linearly with concentration. In Figure 2.3, the evaluated region isdepicted as a dotted rectangle. This region was always set to the left part of thereactor. It was sufficient to monitor mixing in one half of the reactor because weobserved that mixing was symmetrical with respect to the reactor center. Thenumber of recorded images per second ranged between 30 and 120 and dependedon impeller speed and expected mixing time. The number of pixels per image in theaxial direction, i.e., from injection point to sidewall, ranged from 90 to 150 pixels

Laser .

Camera

Lasersheet

PC

δ(t)

Lens

Region ofinterest

x

Figure 2.3: Sketch of the experimental set-up.

Chapter 222

for reactor-11 and reactor-20, respectively. In the radial direction, i.e., from shaft tocylindrical wall, the number was 68 for every image. This results in a spatialresolution of about 1×1×0.5 mm per pixel.


2.3 Results and Discussion

This section starts with a brief description of the mixing pattern and theestablishment of poorly mixed zones or islands, as observed in the PLIF images.Then, the technique for measuring concentrations and mixing times is shown.Finally, the mixing times are correlated with process parameters in order toformulate empirical correlations for scale-up.

2.3.1 Mixing Patterns and Chaotic Mixing

The mixing of the injected dye in the axial and radial direction in reactor-11 isshown in Figures 2.4 and 2.5 for turbulent and laminar conditions, respectively.Although not the same, these mixing patterns show a resemblance. In Figures 2.4band 2.5a, it can be seen that the dye is mainly transported in the axial direction as itflows from the central injection point towards the sidewall. Figures 2.4d and 2.5bshow that the dye is subsequently transported in the radial direction along thesidewall, followed by transport to the bulk along the shaft in the axial direction. Theoverall circulation in Figures 2.4 and 2.5 is therefore turning counterclockwise. Thisis the opposite of what was expected, based on the center-oriented pumping actionof the stirrer blades.

2.4a: 0.35 s. 2.4b: 0.55 s. 2.4c: 0.93 s.

2.4d: 1.65 s. 2.4e: 2.75 s. 2.4f: 5.6 s.

Figure 2.4: Mixing pattern in water in reactor-11 at N = 6 Hz.

Chapter 224

2.5a: 0.67 s. 2.5b: 1.0 s. 2.5c: 1.5 s.

2.5d: 2.37 s. 2.5e: 4.0 s. 2.5f: 7.4 s.

Figure 2.5: Mixing pattern in glycerin in reactor-11 at N = 8 Hz.

2.7a: 0.7 s. 2.7b: 1.5 s.

2.7d: 3.1 s. 2.7e: 7.7 s. 2.7f: 8.0 s.

2.7c: 2.2 s.

Figure 2.7: Mixing pattern in glycerin in reactor-20 at N = 6 Hz.

2.6a: 0.48 s.

2.6d: 2.50 s.

2.6c: 1.50 s.

2.6f: 9 s.

2.6b: 0.79 s.

2.6e: 4.01 s.Figure 2.6: Mixing pattern in water in reactor-20 at N = 6 Hz.


Figures 2.6 and 2.7 show how the dye is mixed in reactor-20. The PLIF imagesshow that two regions are present that differ in overall circulation: one at the left-hand side of the PLIF image and one at the right-hand side. The circulation at theleft-hand side is counterclockwise and thus shows resemblance with the circulationin reactor-11. The circulation at the right-hand side is clockwise. This circulation isvisible in Panels a and b of Figures 2.7 in which the injected dye is transported firstin the radial direction towards the shaft and subsequently in the axial directionalong the shaft. By way of illustration, the generalized overall circulation in theradial and axial directions is depicted in Figure 2.8 for reactor-11, reactor-15 andreactor-20.

The flow in Figures 2.5 and 2.7 can be considered as a 3-dimensionaldiscontinuous periodic flow in which the impeller blades provide the discontinuousmovement with a period equal to 1/N. This allows a comparison with the two-dimensional flow in a cavity as studied by Leong and Ottino (1990) and Ottino(1991). Clearly, the laminar flow has a chaotic nature as it is capable of stretchingand folding a region of fluid and returning it - stretched and folded - to its initiallocation after one period, i.e., one impeller revolution. Furthermore, the formedstriations are reoriented when the impeller blades cross them. This event is visiblein Figure 2.5d, in which the vertical impeller arm, as present in the right-hand sideof the laser sheet, plows through the horizontal striations. These reorientationsfurther enhance chaotic mixing.

Figure 2.8: Observed overall circulation in reactor-11, reactor-15, and reactor-20.

Chapter 226

A quantitative indication for chaotic behavior is the Liapunov exponent σ inthe formula PE=PE0·exp(σ·t). This formula (Leong and Ottino, 1989; Ottino, 1991)represents the stretching rate of the flow as it describes the perimeter ofintermaterial area between the dye and the clear fluid as a function of time. Apositive exponent σ implies exponential generation of intermaterial interface and,hence, implies chaotic flow. The inset in Figure 2.9 gives the perimeter against forreactor-20 at N=11 Hz. The initial exponential increase yields the Liapunovexponent by a fit procedure for which all the perimeter values after the maximumhas been reached are ignored. The perimeter decreases after the maximum has beenreached because the striations are lost when they become smaller than the pixelsize. For all experiments under laminar conditions, the Liapunov exponent is givenin Figure 2.9 as a function of impeller speed. The positive exponents increaselinearly with impeller speed and appear to be independent of L/D. This indicatesthat chaotic mixing in all three reactors occurs in a similar fashion.

2.3.2 Poorly Mixed Zones and Islands

Under turbulent conditions, poorly mixed zones are visible in Figures 2.4 and2.6 as the areas in the reactors that remain dark the longest. These areas are visiblein Figures 2.4e and 2.6e (reactor-11 and reactor-20), near the shaft between the twoouter impeller blades (the same holds for reactor-15, although not shown here).Apparently, under turbulent conditions the location does not depend on L/D.Nevertheless, the poorly mixed zones are not very stagnant, as they disappearwithin seconds through turbulent dispersion.

σ = 0.22N

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12N [Hz]

Lia

puno

v-ex

pone

nt σ

[s-1

]

Reactor-11

Reactor-15

Reactor-20

0

20

40

60

80

100

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2t [s]

Per

imet

er [

cm] Reactor-20

N = 11 Hz

Figure 2.9: Liapunov exponents σ as a function of impeller frequency N. The inset providesthe perimeter against time for reactor-20 at 11 Hz. The curve represents PE=PE0·exp(σ·t).


Under laminar conditions, the islands (as a poorly mixed zone is usuallyreferred to under laminar conditions) in reactor-15 and reactor-11 (Figure 2.5e) arelocated below the injection point. These islands are unstable and consequentlydisappear within seconds (compare e.g., panels e and f of Figure 2.5). Thisobservation leads to the conclusion that reactor-11 and reactor-15 are globallychaotic. Leong and Ottino (1989) indicate that the existence of multiple folds alongthe island boundary is indicative for the instability of islands. The presence of a‘rough’ island boundary in Figure 2.5e can be regarded as an indication for this.

In reactor-20 (Figure 2.7e and f) four islands are visible throughout the reactor.These islands are very stable in the range of impeller frequencies applied. Thisleads to the conclusion that reactor-20 is not globally chaotic. The four islands, infact, form four segregated torii, fluid elements that have often been observed inmixing vessels (Dong et al., 1994; Hoogendoorn et al., 1967; Lamberto et al., 1996;Lamberto et al., 1999; Nomura et al., 1997). These fluid elements act as barriers tomixing and are therefore highly undesirable in the polycondensation process as theirexistence allows early termination, hence leading to an undesired broad MWD.Probably, the segregated torii can be terminated by changing the mutual anglebetween the blades or by periodically changing the impeller speed (Harvey III et al.,1997; Lamberto et al., 1996; Leong et al., 1989; Unger et al., 1999).

2.3.3 Macro-Mixing Times under Turbulent Conditions

For turbulent conditions, mixing is quantified by determining macro-mixing

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 1 2 3 4 5 6 7 8

Time (in s)

G/G

n [-

]

6 123.5 24 18

Figure 2.10: Normalized gray scale, plotted versus time, in reactor-11 at 6 Hz with a fill ratioof 40%. The dots represent the raw data, and the line represents the data after FFT filtering.The frequency plot from the Fourier transformation is depicted in the inset.

Chapter 228

times from response curves. These curves are created by plotting gray scales at acertain position in the PLIF images against the elapsed time. The chosen position islocated as depicted in Figure 2.3 by an ‘x’ and is exactly two pixels from thesidewall and two pixels from the cylindrical wall. This position is selected to berepresentative for the mixing in the whole reactor as it is covered with fluid at everyfill ratio.

An example of a response curve is depicted in Figure 2.10 for the mixing inreactor-11, 40% filled with water. The dots represent the raw data afternormalization on final gray scale. The maximum in gray scale is often obtainedbecause of the appearance of the blades and air bubbles in the measurement point asa consequence of the presence of 60% air in the reactor. Because the stirrer had aconstant speed, the disturbance of the blades could be removed by using fastFourier transformation filtering (FFT filtering) from the commercial softwarepackage TablecurveTM by Jandel Scientific. A standard 40% smoothing level isused to zero 80% of the higher frequency components and all stirrer-relatedfrequencies, resulting in the line in Figure 2.10. The accompanying frequency plotis also given in Figure 2.10 and is discussed in the Appendix.

Figure 2.11 shows normalized response curves after FFT filtering at 6 Hz in allthree reactors that are completely filled. The profiles for reactor-11 and reactor-15exceed unity, meaning that the dye is preferentially transported in the axialdirection, toward the position where the gray scales are recorded. For reactor-20 theprofile gradually rises to unity, without exceeding this limit. From Figure 2.11 it isconcluded that when L/D is increased, it will take longer to mix the dye to the finalconcentration.

Macro-mixing times are determined from the response curves after FFTfiltering, like in Figure 2.11. As a representative value, the time that the normalizedconcentration differed by less than 10% from the final concentration was chosen. InFigure 2.12, these mixing times, hereafter referred to as t10, are represented as afunction of the stirrer speed. The mixing times decrease with increasing impeller

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Time [s]

G/G

n [-

]

1.1

1.5

2

Figure 2.11: Normalized gray scale plotted versus time at 6 Hz in reactor-11,reactor-15, and reactor-20 at a fill ratio of 100%.


speed, whereas the macro-mixing times increase with increasing L/D. This is due tothe fact that the distance between injection point and measuring position is largerwith higher L/D. A fill ratio of 60% results in shorter mixing times than those for100% or 40%. Mixing at a fill ratio of 60% can be shorter than mixing at 100%,because the slowest mixed zone near the shaft is absent at 60%. The difference fromthe mixing times at 40% can be a result of a lower overall circulation at 40%. Fromthis it is suggested that, for a good circulation, a minimum amount of fluid isneeded. These results also reveal that the mixing time is not linearly dependent onthe fill ratio.

It is supposed that the macro-mixing time will depend on the process andreactor variables as follows:

(2.1)

Then, using dimensional analysis, the functional relationship can be arranged as

(2.2)

which applies to mixing vessels in general. The polycondensation process isoperated in the hollow state in which the Froude-number N2·D/g is irrelevant. Thisleaves L/D as the only relevant geometrical dimension. When the obtained mixingtimes are correlated with the parameters varied by applying the relevant

t t N D g geometrical dimensions of the systemm m= ( , , , , , )ρ µ

Figure 2.12: Mixing time versus impeller frequency for the three reactors at three differentfill ratios for turbulent conditions.

0

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9 10 11 12

N [Hz]

T10

[s]

Reactor-11, x=0.4

Reactor-11, x=0.6

Reactor-11, x=1.0

Reactor-15, x=1.0

Reactor-20, x=0.4

Reactor-20, x=0.6

Reactor-20, x=1.0

N t t N D N D g

geometrica l dimensions as ratios m m ⋅ = ⋅ ⋅ ⋅ ( ) ρ

µ

2 2 , ,

Chapter 230

dimensionless numbers as given in equation 2.2, the following empirical correlationholds for D = 0.18m:

(2.3)

with

(2.4)

A power series was chosen to describe the dependence of the mixing time onthe fill ratio. The dimensionless mixing time N·t10 increases with the Reynoldsnumber, although the contribution is less than 15%. The influence is small, as themixing is turbulent over the entire range of applied impeller speeds. However, thepositive power indicates that an increase in impeller frequency results in an increasein dimensionless mixing time. This suggests that mixing is less efficient at highstirrer speeds as the fluid tends to more solid-body rotation. The power of 1.2 inL/D indicates that the mixing mechanism is a combination of convection anddispersion, as 1 would indicate full convective flow and 2 full dispersive flow.

As mentioned in the Introduction, Ando et al. (1971a) studied mixing inhorizontal vessels with baffles in order to prevent the tendency to solid-bodyrotation. Their dimensionless mixing time correlated with L/D, indicating a largerconvective contribution due to the baffles. Mixing in the reactor investigated here ismore dispersive because of the absence of baffles, resulting in a dependency of(L/D)1.2 and, therefore, larger mixing times. Applying baffles can decrease mixingtimes under turbulent conditions. For high viscosity levels, however, baffles are notrequired, as viscous shear will damp out behind these baffles.

2.3.4 Macro-Mixing Times for Laminar Conditions

Using the same black-box approach as used for turbulent conditions, macro-mixing times could be determined for laminar conditions. In the range of theparameters varied and with D = 0.18m, the empirical correlation N·t10 = 60·(L/D)can be obtained. This result is in good agreement with the correlations provided byHoogendoorn and Hartog (1967) and Novak and Rieger (1975) for helical ribbonimpellers in vertical vessels. According to this equation, mixing under laminarconditions is only 2.5 times slower than under turbulent conditions. However,establishing the mixing time in this manner is of course troublesome as thesegregated torii in reactor-20 do not disappear within the determined macro-mixingtimes.

A second method for quantifying mixing times is setting the macro-mixingtime equal to the life span of the islands in reactor-11 and reactor-15. In thepolycondensation process it is important to minimize this life span as the presenceof islands implies a concentration ratio deviating from unity. This deviation resultsin an increased possibility of early termination, thereby leading to an undesired

f x x( ) . ( . )= + −0 22 0 70 2

N t 16 f x Re L

D 0.11

1.21 ⋅ = ⋅ ⋅ ⋅ ( ) 10 ( )


broad MWD. In Figure 2.13, the life spans are plotted against impeller frequency.From Figure 2.13, it follows that the life span decreases with increasing impellerspeed. It also follows that, at low impeller frequency, the life span in reactor-15 islarger than in reactor-11, whereas the two are comparable at high frequencies.

2.3.5 Mixing at Intermediate Viscosities

In this study, we use only water and glycerin, both with Newtonian behavior.For the polycondensation process, one can imagine that viscosity and rheologicbehavior will change continuously, and mixing behavior will go through a widerange of scenarios. Leong and Ottino (1990) investigated chaotic mixing inviscoelastic 2-dimensional flows at various viscosities. Upon an increase inviscosity, islands grow, and chaotic regions shrink. From this observation, itappears plausible to conclude, that, between our extreme cases (water and glycerin),nothing dramatic will occur. However, additional experiments by Leong and Ottinoin which the shear rate was increased show that, in viscoelastic fluids, the numberof islands increases, whereas in Newtonian fluids, this number is constant.Translating this observation to our practical situation indicates that, at higherimpeller speeds (higher shear rate) more islands are formed. These observationsshow that studying hydrodynamics in the Drais reactor with viscoelastic fluids ismandatory for obtaining a complete picture of the mixing process.

Figure 2.13: Life span of islands versus N, in reactor-11and reactor-15 for laminar conditions.

0

5

10

15

20

25

30

35

40

4 5 6 7 8 9 10 11 12

N [Hz]

Mix

ing

time

[s]

L/D=1.5

L/D=1.1tNm ∝

12 6.

tNm ∝11 3.

Chapter 232

2.4 Conclusions

This study provides information about flow patterns, the presence of poorlymixed zones, and macro-mixing times in three industrial configurations of the Draisreactor. Under turbulent conditions, the flow pattern shows flow circulation whichis opposite to the pumping action of the impeller blades. The location of poorlymixed zones is the same in all three reactors. The dimensionless macro-mixing timeN·t10 is correlated with L/D, stirrer frequency N, and fill ratio x. The obtainedempirical correlation shows the following three relationships:

• N·t10 is at a minimum value at fill ratios around 60%.• N·t10 increases more than linearly with increasing L/D ratio.• The incorporated Reynolds number has a positive power, indicating that

mixing at high stirrer speeds becomes less efficient.

Under laminar conditions, the flow patterns found indicate that the mixing ischaotic. Reactor-11 and reactor-15 are globally chaotic, whereas reactor-20 appearsto have elements of order. This behavior shows that mixing in the Drais reactor iscomplex and complicates effective scale-up. The location, number, and life span ofthe islands, as well as the overall flow pattern in the Drais reactor, change whenL/D is enlarged. Macro-mixing times could not be determined unambiguously, asthe islands do not disappear within the measurement time. However, the macromixing time seems to be only 2.5 times larger than under turbulent conditions.


Nomenclature

C clearancem

Di impeller diameter mD vessel diameter mG gravitational constant m/s2

h blade height mL vessel length mN number of revolutions HzPE perimeter mPE0 perimeter at t = 0 mRe Reynolds number -t time stm mixing time st10 time at which concentration only differs 10% from final concentration sV reactor volume m3

w blade width mx fill ratio -

Greek

α angle between impeller blade and shaft °β angle between impeller blades in the tangential direction °µ dynamic viscosity kg/(m·s)ρ liquid density kg/m3

σ Liapunov exponent 1/s

References

Ando, K.; Hara, H.; Endoh, K.; “Flow behavior and power consumption in horizontal stirredvessels”, Int. Chem. Eng., 1971a, 11, 735.

Ando, K.; Hara, H.; Endoh, K.; “On mixing time in horizontal stirred vessel”, KagakuKogaku, 1971b, 35, 806.

Ando, K.; Fukuda, T.; Endoh, K.; “On mixing characteristics of horizontal stirred vesselwith baffle plates”, Kagaku Kogaku, 1974, 38, 460.

Ando, K.; Shirahige, M.; Fukuda, T.; Endoh, K.; “Effects of perforated partition plate onmixing characteristics of horizontal stirred vessel”, AIChE J., 1981, 27(4), 599.

Ando, K.; Obata, E.; Ikeda, K.; Fukuda, T.; “Mixing time of liquid in horizontal stirredvessels with multiple impellers”, Can. J. Chem. Eng., 1990, 68, 278.

Borkent, G.; Tijssen, P.A.T.; Roos, J.P.; Van Aartsen, J.J.; “Kinetics of the reactions ofaromatic amines and acid chlorides in hexamethylphosphoric triamide”, Recl. Trav.Chim. Pays-Bas, 1976, 95, 84.

Distelhoff, M.F.W.; Marquis, A.J.; Nouri, J.M.; Whitelaw, J.H.; “Scalar mixing measure-ments in batch operated stirred tanks”, Can. J. Chem. Eng., 1997, 75, 641.

Dong, L.; Johanson, S.T.; Engh, T.H.; “Flow induced by an impeller in an unbaffled tank”,Chem. Eng. Sci., 1994, 42, 549.

Chapter 234

Fukuda, T.; Idogawa, K.; Ikeda, K.; Ando, K.; Endoh, K.; “Volumetric Gas-phase masstransfer coefficient in baffled horizontal stirred vessel”, J. Chem. Eng. Jpn., 1990, 13(4),298.

Gaymans, R.J.; Sikkema, D.J.; “Aliphatic polyamides”, Comprehensive Polymer Science,Step Polymerization, 1989, 5, 357.

Harvey III, A.D.; Wood, S.P.; Leng, D.E.; “Experimental and computational study ofmultiple impeller flows”, Chem. Eng. Sci., 1997, 52(9), 1479.

Holmes, D.B.; Voncken, R.M.; Dekker, J.A.; “Fluid flow in turbine-stirred, baffled tanks - I-Circulation time”, Chem. Eng. Sci., 1964, 19, 201.

Hoogendoorn, C.J.; den Hartog, A.P.; “Model studies on mixers in the viscous flow region”,Chem. Eng. Sci., 1967, 22, 1689.

Jeurissen, F.T.H.; Surquin, J.; Private communication.Kersting, Ch.; Prüss, J.; Warnecke, H.J.; “Residence time distribution of a screw-loop

reactor: experiments and modelling”, Chem. Eng. Sci., 1995, 50, 299.Lamberto, D.J.; Muzzio, F.J.; Swanson, P.D.; Tonkovich, A.l.; “Using time-dependent RPM

to enhance mixing in stirred vessels”, Chem. Eng. Sci., 1996, 51(5), 733.Lamberto, D.J.; Alvarez, M.M.; Muzzio, F.J.; “Experimental and computational investiga-

tion of the laminar flow structure in a stirred tank”, Chem. Eng. Sci., 1999, 54, 919.Leong, C.W.; Ottino, J.M.; “Experiments on Mixing due to Chaotic Advection in a Cavity”,

J. Fluid Mech., 1989, 209, 463-499.Leong, C.W.; Ottino, J.M.; “Increase in regularity by polymer addition during chaotic

mixing in two dimensional flows”, Phys. Rev. Lett., 1990, 64(8), 874.Manaresi, P.; Munari, A.; “Factors affecting rate of polymerization”, Comprehensive

Polymer Science, Step Polymerization, 1989, 5, 35.Mayr, B.; Nagy, E.; Horvat, P.; Moser, A.; “Scale-up on basis of structured mixing models:

A new concept”, Biotechnol. Bioeng., 1994, 43, 195.Moo-young, M.; Tichar, K.; Takahashi, A.L.; “The blending efficiencies of some impellers

in batch mixing”, AIChE. J., 1972, 18, 178.Nomura, T.; Uchida, T.; Takahashi, K.; “Enhancement of mixing by unsteady agitation of an

impeller in an agitated vessel”, J. Chem. Eng. Jpn., 1997, 30(5), 875.Novák, V.; Rieger, F.; “Homogenization efficiency of helical ribbon and anchor agitators”,

Chem. Eng. J., 1975, 9, 63.Ottino, J.M.; “Unity and diversity in mixing: Stretching, diffusion, breakup and aggregation

in chaotic flows”, Physics of Fluids A3, 1991, 3(5), 1417.Perona, J.J.; Hylton, T.D.; Youngblood, E.L.; Cummins, R.L.; “Jet mixing of liquids in long

horizontal cylindrical tanks”, Ind. Eng. Chem. Res., 1998, 37, 1478.Schoenmakers, J.H.A.; Wijers, J.G.; Thoenes, D.; “Determination of feed stream mixing

rates in agitated vessels”, Proc. 8th European Mix. Conf. (Paris): Récents Progrès enGénie des Procédés, ed. Lavoisier, 1997, 11(52), 185.

Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers: Dordrecht,1994.

Unger, D.R.; Muzzio, F.J.; “Laser-induced fluorescence technique for the quantification ofmixing in impinging jets”, AIChE J., 1999, 45(12), 2477.

Vollbracht, L.; Comprehensive Polymer Science, Step Polymerization, 1989, 5, 374.


Appendix

In this appendix some remarks are made on the frequency plot in Figure 2.10.The plot shows spikes at frequencies that are characteristic for raw data like theimpeller speed, i.e., 6 Hz and accompanying higher frequencies such as 12, 18 and24 Hz. The peak around 3.5 Hz originates from differences in dye concentration inthe tangential direction and is responsible for the large fluctuations in the responsecurve. In the response curve of reactor-11 in Figure 2.11, the fluctuations reappearwith a time period of 1/3.5 s, the so-called circulation time tc (Holmes et al., 1964).As 10 periods can be distinguished, one can estimate a mixing time in the tangentialdirection of 3 s. Because the macro-mixing time t10 for this experiment was found tobe 4.0 s, distributive mixing in the tangential direction is faster than in the axial andradial direction.

The mixing in the tangential direction was also faster than that in the axial andradial direction in reactor-15 and reactor-20 because no large fluctuations wereobserved in the accompanying response curves in Figure 2.11. In these reactors, thedye is homogeneously distributed in the tangential direction before it has reachedthe position where gray scales are read. It is suggested that the clockwise circulationin the larger reactors, as depicted in Figure 2.8, enhances the tangential mixing.

According to the frequency plot in Figure 2.10, the concentration in thetangential direction in reactor-11 fluctuates with a specific frequency. However, thisfrequency is not constantthroughout the reactor. Figure2.14 shows the ratio of thefrequency and impellerfrequency in one-fourth ofreactor-11. The Figure showsthat, near the wall, the fluid isretained more than in the bulkas the ratio is smaller. Thelargest ratio is 0.9, positionedbetween the two impellerblades. The fluid in that arearotates almost as a solid-bodyand will therefore not be well-mixed, as is confirmed by thepresence of the poorly mixedzone in Figure 2.4e at thesame position.

-0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 -0.00

Axial distance [m]

0.03

0.06

0.09

0.12

0.15

0.18

Rad

ius

[m]

Injection

Figure 2.14: Ratio of fluid velocity to stirrerspeed in reactor-11.

HYDRODYNAMHORIZONTAL

In this chapter, thereactors are investigated. mixing times have been di.e., 'slosh' and 'ring', can btwo states that shows hystby measuring the tempenumber appears to be covertical vessels. A variatiodissipates uniformly throuUnder turbulent conditionvessel wall. Using pulse-rhave been determined. Tinput and liquid volume, large and small scales. Apower shows that, at smmixing is most energy-effi

This chapter is a slightly modifiedVan der Gulik, G.J.S.; Wijers, J.Horizontal Stirred Reactors”, Ind.

3
ICS AND SCALE-UP OF STIRRED REACTORS
Abstract

hydrodynamics in horizontal stirred-tankThe flow state, agitation power, and macro-etermined experimentally. Two flow states,e distinguished, with transition between theeresis. The agitation power was determinedrature increase upon mixing. The powermparable to power numbers in unbaffledn in fill ratio indicates that agitation energyghout the reactor under laminar conditions.s, however, most energy is dissipated at theesponse measurements, macro-mixing timeshe mixing times correlate with momentumthus indicating different hydrodynamics at

combination of mixing times and agitationall scale and intermediate fill ratios, thecient.

version of the publication:G.; Keurentjes, J.T.F.; “Hydrodynamics and Scale-Up ofEng. Chem. Res., 2001, 40(22), 4731-4740.

Chapter 338

3.1 Introduction

Horizontal stirred-tank reactors are widely used in industry. A commercialexample is the unbaffled Drais reactor that can be used for multiple purposes suchas powder mixing in catalyst preparation, liquid mixing and kneading during CMC-production, and polycondensation processes (Vollbracht, 1989; Bannenberg-Wiggers et al., 1998). In all applications, the reactor is only partially filled. Aschematic representation of the Drais reactor is given in Figure 2.1. Typical for theunbaffled cylindrical reactor is its horizontal position and the heavily designedimpeller. The impeller makes possible the application of high mixing power, whichis needed to achieve sufficient mixing in viscous fluid processes. The reactor ischaracterized by its length L, diameter D, and clearance c, which is the distancebetween the blades and the reactor wall. Because the clearance is small, the bladesperform a scraping action that keeps the reactor wall free from sticking material.The clearance also provides a region with high shear rates and good heat exchangewith the cooled walls. The blades have a pumping action towards the reactor center,providing an easy way to discharge the reactor through the open hatch asrepresented in Figure 2.1.

Despite the wide application of Drais reactors in liquid mixing, little is knownabout the hydrodynamics, mixing performance, and scale-up of such reactors. Aliterature survey shows that the literature on hydrodynamics in horizontal vessels israther limited, in contrast with that on vertical stirred vessels. Some literature existson the hydrodynamics under turbulent conditions. Ganz (1957) describes powermeasurements in horizontal stirred gas absorbers. Ando et al. (1971a) measuredpower input upon stirring in partially filled horizontal vessels in relation to flowbehavior. In the vessel, two flow states could be distinguished. The 'slosh'-state isobtained at low stirring speed. The liquid is then pushed upward by the impeller andsprayed, which is ideal for use in gas absorption processes (Ganz, 1957; Ando et al.,1971b; 1974; 1981; Fukuda et al., 1990). The 'ring'-state, which is obtained at highstirrer speeds, results in a cylindrical liquid layer on the inside wall. Ando et al.(1990) also studied turbulent mixing in a horizontal vessel with baffles and multipleimpellers. Macro-mixing times were measured, and a model was proposed forscale-up purposes. It has been established that the dimensionless macro-mixing timeNtm is proportional to L/D.

Because literature on mixing in horizontal vessels under laminar conditions isvirtually absent, we have to rely on mixing studies in vertical unbaffled tanks, forwhich many studies are available. The usual purpose of these studies has been tofind geometries that provide good mixing performance. Data on mixing vary fromauthor to author as a result of differences in geometry, definitions, experimentaltechniques, and fluid properties. However, a general consensus emerges concerningimpeller designs. Judging from power input and mixing time experiments (Ando etal., 1990; Hoogendoorn et al., 1967; Novák et al., 1967), it can be concluded thatthe flow pattern of a good laminar mixer should include 1) axial flow, 2) allstreamlines passing through the impeller region, 3) no closed streamlines occurring

Hydrodynamics and Scale-Up of Horizontal Stirred Reactors 39

outside the impeller region, and 4) frequent disruption of fluid along the wall.Figure 2.1 shows that conditions 2 and 4 will probably be achieved in the Draisreactor, because the impeller blades pass through the entire reactor volume. InChapter 2, the flow patterns in a Drais reactor have been described, as investigatedusing planar laser induced fluorescence. It was found that axial flow in the reactor israther effective. Also, closed streamlines can be present, acting as toroidal vortices.

We have studied the macro-mixing in a Drais reactor filled with low- andhigh-viscosity liquids with the aim of obtaining a better understanding of themixing during a polycondensation process in which the viscosity strongly increasesas a result of the formation of polyaramid molecules (Bannenberg-Wiggers et al.,1998). To obtain a polymer product with the required quality in terms of MW(molecular weight) and MWD (molecular weight distribution), it is important toexpose the polymerizing liquid to high shear rates (Agarwal et al., 1992). The highshear rates increase reaction rates through molecular orientation and rotationaldiffusion of the rods. This has been shown in an experimental study by Agarwal andKhakhar (1992; 1993), using two reactors in series in which the first reactor is avertical reactor with a high-speed stirrer ensuring good overall mixing. The secondreactor provides Couette-flow hydrodynamics, with nearly homogeneous high shearflow over the entire reactor but with little overall mixing. The authors were able toimprove product quality considerably by increasing the shear rate in the secondreactor. The Drais reactor investigated here, combines the important features ofboth reactors (Vollbracht, 1989; Bannenberg-Wiggers et al., 1998): a large impellerprovides overall mixing while at the same time high shear rates occur in the smallclearance between impeller and vessel wall.

As the clearance of the Drais reactor is small, its volume is small compared tothe total liquid volume: in completely filled reactors, this volume ratio is 2⋅10-4.Therefore, macro-mixing in the reactor has to be optimized so that all liquid in thebulk will pass the high-shear region in the clearance frequently. A second reasonthat emphasizes the importance of short macro-mixing times is that thepolycondensation is performed in a semi-batch manner: one reactant is fed to theother and has to be mixed quickly throughout the reactor to prevent the occurrenceof a premature termination reaction (Bannenberg-Wiggers et al., 1998; Chapter 2).Judging from the available literature, it is unclear what the macro-mixing time willbe and how it will evolve in scale-up. Therefore, we have conducted anexperimental study at different scales in which we established macro-mixing timesby means of pulse-response measurements. The applied agitation power was alsomeasured to link the mixing performance with power consumption. Allmeasurements were performed in the 'ring'-state as the polycondensation process isperformed at high stirrer frequencies, thus forcing the liquid into the 'ring'-state. Asit is known that application of the 'ring'-state is required for a high MW to beobtained (Bannenberg-Wiggers et al., 1998), we determined the impeller frequencyat which the 'ring'-state forms or disappears during operation. Also, flow behaviorwas studied as a function of vessel and stirrer geometry, impeller frequency, andfluid viscosity.

Chapter 340

3.2 Scale-up theory

For scale-up of the polycondensation process, it is important to know how themixing time evolves upon reaction. From the literature it is well-known that themixing time will be a function of the process conditions and reactor configuration:

tm = ƒ(ρ,µ,N,DI,g,geometrical dimensions of the system) [s] (3.1)

Using dimensional analysis and omitting the Froude number (Fr), the functionalrelationship can be rearranged to:

Ntm = ƒ(Re, geometrical dimensions as ratios) [-] (3.2)

Ntm is the dimensionless mixing time and is expected to be independent of theReynolds number (Re) under both turbulent and laminar conditions (Harnby et al.,1992). Under intermediate conditions, Ntm will be a power law in Re. In ourspecific case, Fr can be omitted as Fr is only important under conditions at whichtransition occurs between 'slosh'- and 'ring'-state. We are only interested in macro-mixing times in the 'ring'-state as the polycondensation process is performed in thisstate.

To maintain tm constant during scale-up, a constant impeller frequency N isrequired according to equation 3.2. This will also result in a constant average shearrate aγ& , which is related to N by

Nkdy

dva 1=−=γ& [1/s] (3.3)

with k1 close to unity (Metzner et al., 1957; Thoenes, 1994). The highest shear rate,

maxγ& occurs in the clearance and can be estimated using

Nk/D/D

ND

c

ND

dy

dv

i

i

cmax 222

0∝

−=

−≈−=

ππγ& [1/s] (3.4)

Thus, maxγ& is constant when N is kept constant, provided that the clearance is kept

in a constant ratio with the vessel diameter. Keeping N constant in scale-up,however, is strongly reflected in the power requirement. Under turbulent conditions,the required power is given by

53iP DNNP ρ= [W] (3.5)


and the average energy dissipation rate per unit of mass is

x

DNfN

V

P igP

l

23

==ρ

ε [m2/s3] (3.6)

where ρ is the liquid density [kg/m3], NP is the power number [-], N is the agitationrate [1/s], Di is the impeller diameter [m], x is the fill ratio [-], Vl is the liquidvolume [m3], and fg is the geometrical factor, which is 1.05 [-].

For geometrically similar systems, the power number NP can be rewritten as afunctional relationship of dimensionless groups. Under turbulent conditions, onlyRe is relevant for the Drais reactor; thus

ap RekN 3= [-] (3.7)

Under laminar conditions, the following relation applies

324 iDNkP µ= [J/s] (3.8)

and consequently

25 Nk

V

P

l

νρ

ε == [J/(kg⋅ s)] (3.9)

with k4 and k5 as constants and ν the kinematic viscosity. For these equations, it isassumed that Np depends on Re-1 only, which is a good approximation when Re <100.

From equations 3.5 and 3.8, it can be seen that, for constant N, P increaseswith Di

5 and Di3, respectively, which are both highly impracticable. Therefore, on

larger scales, lower specific power input has to be applied which is usually obtainedby reducing the impeller speed. This results in an increase in tm, which isundesirable in the polycondensation process. With this experimental study, weidentify the limitations that can be faced during scale-up.

Chapter 342

3.3 Experimental Section

3.3.1 The Drais reactor

To investigate the mixing process at different scales, four scale models of theDrais reactor were available. Typical geometrical data are given in Table 3.1. Thesmall models are named reactor11, reactor15 and reactor20, where the numbers 11,15, and 20 refer to the L/D-ratio. The large-scale model is reactor11-60, whichrefers to the L/D-ratio and the diameter. Reactor11 and reactor11-60 aregeometrically similar. Reactor15 and reactor20 differ from reactor11 in angle βbetween blades (as shown in Figure 2.1) and in length. A detailed description hasbeen given in Chapter 2.

Table 3.1: Geometric data regarding the four reactors.

Small-scale Large-scaleParameter Reactor11 Reactor15 Reactor20 Reactor11-60

Reactor length L [m] 0.198 0.27 0.36 0.66Reactor diameter D [m] 0.18 0.6Reactor volume Vr [L] 5.0 6.7 8.9 185Impeller diameter Di [-] D⋅(29/30) D⋅(29/30)

Blade width w [-] D/2 D/2Blade height h [-] D/12 D/12Blade thickness [-] D/90 D/54Shaft diameter [-] D/6 D/10

β (°) 180 135 120 180Materials

Shaft Stainless steel Stainless steelCylindrical wall Glass Perspex

Side walls Stainless steel Perspex

3.3.2 Transition in flow state

The flow state was determined for different reactor geometries and fluidviscosities by observing the impeller frequency at which a fluid ring was formed orcollapsed. The formation was complete when the inner gas/liquid surface was flat.Both transitions were clearly visible.

3.3.3 Power measurements

Power measurements were performed in reactor11, reactor20, and reactor11-60. Depending on the desired value of Re, tap water, glycerin (purity > 99.9%Heybroek, Amsterdam), or a mixture of the two was used as the working fluid. The


power input to the liquid by stirring was measured by determining the temperatureincrease of the liquid with time. The experimental set-up is depicted in Figure 3.1.Two Pt100-elements, denoted T1 and T2, measured the temperature in the reactorsduring agitation. There was no difference observed between T1 and T2 underturbulent conditions. Under laminar conditions, the difference never exceeded 0.2°C. Pt100-element T3 measured the ambient temperature, Ta, during theexperiments. All temperatures were measured with an accuracy of 0.1 °C. Thepower input followed from the energy balance over the mixing vessel, as given inequation 3.10.

)( alp TThPdt

dTVC −−=ρ [J/s] (3.10)

where ρ is the liquid density [kg/m3]; Cp is the heat capacity [J/(kg⋅K)]; Vl is theliquid volume [m3]; T is the liquid temperature [K]; Ta is the ambient temperature[K]; P is the dissipated stirring energy [J/s]; and h is overall heat transfer coefficient[J/(K⋅s)].

The term on the left-hand side in equation 3.10 represents the accumulationterm, P represents the dissipated stirring energy, and the last term represents thelumped losses to the environment. Assuming density, heat capacity, overall heattransfer coefficient and ambient temperature to be constant and temperature-independent, differential equation 3.10 can be solved by considering the initialcondition T(0) = T0, resulting in

−−+

−−+=

lplpaa VC

ht

h

P

VC

htTTTtT

ρρexp1exp)()( 0 [K] (3.11)

Figure 3.1: Schematic representation of the experimentalset-up for the temperature measurements.

Chapter 344

3.3.4 Pulse-response experiments

The experimental set-up for the pulse-response experiments is depicted inFigure 3.2. It was decided to perform measurements outside the vessel, because theclearance was too small for a probe to be placed inside. Therefore, the reactorcontents were circulated through a spectrophotometer (A) placed in an externalloop. The liquid was withdrawn from an outlet, placed at the same height as theclearance and fed back at the top in the reactor center. The flow rate in the loop wasmonitored by a flow meter (F). The measurement started when a small amount of aconcentrated aqueous methylene blue solution was injected as a pulse, δ(t), in thereactor center. The concentration was measured by the spectrophotometer, whichwas connected to a PC for automatic monitoring.

For reactor11-60, the volume of the piping was 450 mL, and the flow rate was50 mL/s. Because the liquid volume was between 72 and 180 L, higher ordermixing effects can be ignored. The time needed for the tracer to leave the reactorand reach the spectrophotometer was 0.9 s. This delay time was measured byinjecting a small amount of tracer at the reactor outlet. Macro-mixing times werecorrected for this delay. For the smaller reactors, the volume of the piping was 250mL, the flow rate was 25 mL/s, the total liquid volume varied between 2 and 10 L,and the delay was 2.1 s. The total amount of injected solution was 3 mL. Using ahigh-speed camera, as described in chapter 2, the injection time was determined tobe 0.20 ± 0.03 s. This time was negligible compared to the mixing time.

An example of a tracer concentration response curve with time is given inFigure 3.3. The plotted response on the left y-axis was normalized using:

)0(

)0()()(

CC

CtCtCn −

−=

∞

[-] (3.12)

A

t

PCA

δ

F

Figure 3.2: Schematic representation of the experimental set-upfor the pulse-response measurements.


On the right y-axis of Figure 3.3 is plotted the variance of concentration, σ2, around

the equilibrium value, as defined by

22 ))(1()( tCt n−=σ [-] (3.13)

The macro-mixing time was defined to be the time at which the variance was below10-3. This resulted in tm = 12.3 s for the experiment in Figure 3.3. Every presentedmacro-mixing time (shown in Figures 3.11-14) is the average of at least threemeasurements.

Table 3.2 provides the impeller frequencies used in the pulse-responseexperiments. Under turbulent conditions, the chosen frequency ensured that theliquid was in the 'ring'-state. Consequently, the values of Fr at large and small scalewere similar. The values of Re were not similar, which was less important as itfollowed from literature that Ntm was independent of Re at the high applied valuesof Re (Coulson and Richardson, 1996). Using glycerin in reactor11-60 provides Revalues up to 1730, which does not really justify the laminar classification. However,for convenience we have grouped all measurements using glycerin.

Table 3.2: Range of impeller frequency and the corresponding shear rate γ& , tip speed, Re and

Fr values as applied in the pulse-response measurements.

Kinematic DynamicReactor id. D [m] Regime N [Hz] γ& [1/s]

Vtip[m/s] Re [-] Fr [-]

Laminar 3.0-11.2 560-2100 1.7-6.3 100-360 0.17-2.3Reactor11,

-15, and -200.18 Turbu-

lent5.5-9.1 1040-1715 3.1-5.1

1.77⋅105-

2.95⋅105 0.55-1.5

Laminar 0.6-4.8 110-900 1.1-9.0 210-1730 0.02-1.4

Reactor11-60 0.6 Turbu-lent

1.8-4.8 340-900 3.4-9.06.48⋅105-1.73⋅105 0.20-1.4

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25Time [s]

Cn

[-]

0.00001

0.0001

0.001

0.01

0.1

1

10

σ2 [

-]

tm = 12.3

Figure 3.3: Typical response curve with the corresponding variancefor D = 0.6 m, L = 0.66 m, x = 100 %, and N = 4.8 Hz.

Chapter 346

3.4 Results and discussion

3.4.1 Flow state in water

For the reactor filled with water, the Fr values at which fluid rings form orcollapse, are plotted in Figure 3.4 against the fill ratio for reactor11, reactor20, andreactor11-60. The solid symbols mark the Fr value at which the fluid ring formswith increasing impeller speed, and the open symbols mark the Fr value at whichthe ring collapses with decreasing impeller speed. From Figure 3.4, it follows that,at all fill ratios hysteresis occurs between the values of Fr for ring formation andcollapse. The values of Fr at which the transition occurs coincide for the ringcollapse, however, they do not coincide for ring formation. At low as well as at highfill ratios, flow changes occur at higher values of Fr than is the case at intermediatefill ratios. This implies that, at intermediate fill ratios, weaker inertial forces, i.e.,less power, are required to form a fluid ring.

From Figure 3.4, the value of Fr can be determined at which a fluid ring ispresent. For the large-scale reactor, ring formation occurs around Fr = 1, which canpractically be used as a criterion for ring formation. This information is important,as the fluid in the pulse-response measurements and power measurements has to bein the 'ring'-state for the polycondensation process (Bannenberg-Wiggers et al.,1998). Because of the hysteresis effect, however, it is possible to maintain the 'ring'-state at stirrer frequencies lower than the frequency needed for ring formation.

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

Fill ratio [-]

Fr

[-]

Ring formation: Reactor11-60 Reactor11 Reactor20

Ring collapse: Reactor11-60 Reactor11 Reactor20

Figure 3.4: Flow state transition, expressed in Fr as function of the fill ratiofor reactor11-60, reactor11, and reactor20.


3.4.2 Flow state in glycerin

In reactors partially filled with glycerin, no fluid ring is formed at high stirrerspeeds. For illustration, an image of the flow state is recorded with a high speedcamera and is presented in Figure 3.5. Viscous forces are too high to obtain a fluidring as in water. The gas phase is finely dispersed in the liquid, providing a milkyfluid with small gas bubbles. Figure 3.5 also shows the presence of large holes inthe fluid in the wake behind the blades. The holes do not completely extent to thecylindrical wall, indicating that the wall is entirely covered by a liquid film. Thepresence of the liquid film was proven to exist as small gas bubbles were found,using a stroboscope, that are transported tangentially by the impeller with a lowerspeed than the impeller speed. The presence of the liquid film probably isimportant, as this is the region with the highest shear rate.

3.4.3 Power measurements

In Figure 3.6, the temperature rise is plotted as a function of time for fivedifferent situations. The time for measurement ranged from 5 minutes in reactor11with glycerin to over 2 hours in reactor11-60 with water. By fitting the temperatureprofiles with equation 3.11, the applied agitation power P was obtained. The overallheat transfer coefficient h also follows from the fitting procedure. However, as theinterpretation of these results appeared not to be very straightforward, these datahave not been included.

In Figure 3.7, the power number NP, as defined by P/ρN3D5, is given as afunction of Re. The solid line represents measurements in a completely filledreactor11 and reactor11-60, which are geometrically similar but differ only in size.The measurements are in close agreement with data for a propeller mixer in anunbaffled vertical vessel as reported by Rushton et al. (1950). NP becomesindependent of Re at high values of Re, i.e., under turbulent conditions.Accordingly, the power 'a' of Re in equation 3.7 approaches 0. The applied power isplotted against Re in Figure 3.8. From the slope of the lines at high values of Re, itfollows that the applied power is proportional to the cubic root in impeller speed,which is in accordance with equation 3.5. Under laminar conditions, NP becomesinversely proportional to Re (the exponent 'a' in equation 3.7 approaches -1). FromFigure 3.8, it follows that

Figure 3.5: Photograph of glycerin in reactor20 at a fill ratio of 40 % at 8 Hz.

Chapter 348

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1000 2000 3000 4000 5000 6000 7000 8000

Time [s]

T-T

0 [°

C]

N = 12 Hzx = 40 %P = 154 W

N = 8 Hzx = 80 %P = 87.4 W

N = 12 Hzx = 100 %P = 48.6 W

N = 8 Hzx = 100 %P = 17.5 W

N = 2.4 Hzx = 100 %P = 202 W

Glycerin, small scale model

Water, large scale model

Water, small scale model

Figure 3.6: Temperature rise versus elapsed time for the small- (D = 0.18 m) and large-scale(D = 0.6 m) models, filled with glycerin or water.

100 1000 10000 100000 1000000

0.1

1

Np(0.4,1.1) Np(0.6,1.1) Np(0.8,1.1) Np(1.0,1.1)

Np(0.4,2.0) Np(0.6,2.0) Np(0.8,2.0) Np(1.0,2.0)

Np [-]

Re [-]Figure 3.7: Power number as a function of Re with the fill ratio and L/D ratio as parameters.

1

10

100

1000

10 100 1000 10000 100000 1E+06 1E+07

Re [-]

P [W

]

P(0.001)

P(0.003)

P(0.014)

P(0.099)

P(0.297)

P(0.527)

P(1.4)

P(0.001)

P~N2.1

P~N3.0

D=0.18

D=0.60

P~N2.6

Figure 3.8: Power as a function of Re with viscosity and diameter as parameters.


the applied power at low values of Re is proportional to the square root of theimpeller speed, which is in accordance with equation 3.8.

The power number NP is given as a function of Fr in Figure 3.9. Figure 3.9shows that NP is virtually independent of Fr, apart from the case in which thereactor is completely filled with glycerin. For a horizontal reactor, Ando et al.(1971a) also found that NP is independent of Fr.

In Figures 3.10A-D, the power P and dissipated energy per unit mass ε aregiven as functions of fill ratio for water and glycerin. With glycerin, the appliedpower was around 5 times higher than it was with water. From Figures 3.10A and3.10B, it follows that, for water and glycerin, the applied power increases withincreasing fill ratio. For water, the increase is less than proportional, as followsfrom Figure 3.10C in whichε decreases with increasing fill ratio. In water, mostenergy is dissipated at the impeller tips and at the cylindrical wall. At low fill ratios,this region represents a relatively larger volume than at high fill ratios. Therefore,ε will be higher at low fill ratios.

Figure 3.10D shows that, in glycerin, ε is independent of the fill ratio,suggesting that energy is dissipated more uniformly throughout the fluid than inwater. This also follows from Figure 3.10F, in which ε is plotted against Re. The

values forε at all fill ratios coincide and follow a power law of 2.1. As ε ishomogeneous, the shear rate is also homogeneous. Consequently, the contributionof the high shear rate in the clearance toε is limited, which can be explained by thesmall clearance volume.

0.01

0.1

1

10

0 0.5 1 1.5 2 2.5 3 3.5 4

Fr [-]

Np

[-]

Water

Np (0.4)

Np (0.6)

Np (0.8)

Np (1.0)

Glycerin

Np (0.4)

Np (0.6)

Np (0.8)

Np (1.0)

Figure 3.9: Power number as a function of Fr with fill ratio and fluid type as parameters.

Chapter 350

Using multivariable analysis, the relationship in equation 3.14 between NP andthe varied parameters can be obtained for laminar conditions as

89.0

85.0

92.0

Re93

=

D

LxN P [-] (3.14)

and for turbulent conditions as

Figure 3.10: Results of the power measurements: (A) power in water against fill ratio,(B)power in glycerin against fill ratio, (C)ε in water against fill ratio, (D) ε in glycerin against

fill ratio, (E) ε in water against Re, (F) ε in glycerin against Re. Parts (A)-(D) have N andL/D as parameters, and Parts (E) and (F) have fill ratio and L/D as parameters.

0

5

10

15

20

25

0.4 0.5 0.6 0.7 0.8 0.9 1

Fill ratio [-]

ε [m

2 /s3 ]

8 , 1.1

10 , 1.1

12 , 1.1

14 , 1.1

8 , 2.0

12 , 2.0

C) Water

N L/D

0

100

200

300

400

500

600

0.4 0.5 0.6 0.7 0.8 0.9 1

P [

W]

B) Glycerin

0

10

20

30

40

50

60

70

80

90

0.4 0.5 0.6 0.7 0.8 0.9 1

Fill ratio [-]

ε [m

2 /s3 ]

8 , 1.1

10 , 1.1

12 , 1.1

14 , 1.1

8 , 2.0

D) Glycerin

N L/D

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500Re [-]

ε [m

2 /s3 ]

0.4 , 1.1

0.6 , 1.1

0.8 , 1.1

1.0 , 1.1

F) Glycerin

x L/D

0

10

20

30

40

50

60

70

80

0.4 0.5 0.6 0.7 0.8 0.9 1

P [

W]

A) Water

0

5

10

15

20

25

100000 200000 300000 400000 500000Re [-]

ε [m

2 /s3 ]

0.4 , 1.1

0.6 , 1.1

0.8 , 1.1

1.0 , 1.1

0.4 , 2.0

0.6 , 2.0

0.8 , 2.0

1.0 , 2.0

E) Water x L/D


44.036.015.0

=

D

LxN P [-] (3.15)

The standard errors of the exponents are given in Table 3.3. For laminarconditions, NP proves to be nearly linear with liquid volume as the exponent for thefill ratio comes close to unity. Exponent 'a' approaches –1, which is in agreementwith equation 3.7.

Summarizing, it follows that, the value of NP in this case is similar to that of apropeller in an unbaffled tank. In water,ε is a function of the fill ratio and is

highest near the vessel wall. In glycerin,ε is independent of the fill ratio and ishomogeneous over the reactor.

Table 3.3: Standard errors of the constants in the correlations for NP.

Conditions Pre-exp. factor Exponent in x Exponent in (L/D) Exponent in Re

Lam. (Re < 400) 93 ± 9.6 0.92 ± 0.066 0.89 ± 0.064 -0.85 ± 0.02

Turb. (Re > 105) 0.15 ± 0.0055 0.36 ± 0.064 0.44 ± 0.072 -

3.4.4 Macro-mixing times in water under turbulent conditions

The dimensionless mixing time Ntm in the large reactor11-60 is depicted inFigure 3.11A as a function of the fill ratio. It shows that, at every fill ratio, Ntm isindependent of Re. Also, Ntm increases with increasing fill ratio, indicating thatmore impeller revolutions are required for a given degree of mixing with increasingliquid volume.

From Figure 3.11B-D, in which Ntm is given for reactor11, reactor15, andreactor20, respectively, it follows that the lowest Ntm is found at a fill ratio of 0.7,which is in agreement with the results in chapter 2. This suggests that at low fillratios (x < 0.4), the total amount of fluid is too low to provide good overallcirculation in the fluid ring, whereas at high fill ratios (x ≈ 1), the poorly mixedzone near the shaft reduces the advantages of the better overall circulation.

A comparison of Ntm for reactor11, reactor15, and reactor20 shows that Ntm

increases with increasing L. Also, Ntm shows to be independent of Re, although Ntmincreases slightly with increasing Re in reactor20. This increase indicates that thefluid tends toward solid-body rotation at higher stirrer speeds. Solid-body rotationcan occur more easily in reactor20 than in reactor11 and reactor15, because of therelatively small effect of the sidewalls in reactor20.

The above observations show that reactor11 and reactor11-60 differsignificantly in hydrodynamic behavior despite their geometric similarity. Inreactor11-60, the shortest mixing time is found at a fill ratio of 0.4, which is thelowest fill ratio applied, whereas in reactor11, it is found at a fill ratio of 0.7. TheNtm values at the two scales are also different. Apparently, turbulent and convective

Chapter 352

mixing on both scales are different. An important difference between reactor11 andreactor11-60 is the wall-surface/liquid-volume ratio. This ratio is higher at smallscale in reactor11 (22.2 m2/m3) than at large scale in reactor11-60 (6.66 m2/m3). Asa result, the fluid tends more toward solid-body rotation at large scale and,therefore, shows increased mixing times.

The observed trends indicate that the mixing time depends on scale and liquidvolume in contrast to the conclusion of Harnby et al. (1992), who state that, forgeometrically similar systems, Ntm is constant. Fox and Gex (1956), Middleton(1979), and Mersmann et al. (1976) have observed that, in vertical vessels, mixingtimes depend on liquid volume. Fox and Gex (1956) have correlated mixing timeswith liquid volume and momentum input according to:

( ) 42042

50

.

.l

mDN

Vt ∝ [s] (3.16)

Application of this approach has the advantage that the fill ratio can easily beimplemented. When we use all presented mixing data, the following relationship isobtained:

0

1020

30

4050

60

7080

90

500000 1000000 1500000 2000000

Nt m

[-]

2.6 5.2 7.8 10.4

vtip [m/s]

A) Reactor11-60

No stable fluid ring

0

1020

30

4050

60

7080

90

150000 200000 250000 300000

2.6 3.0 3.4 3.8 4.2 4.6 5.0

vtip [m/s]

10.70.60.50.4

B) Reactor11 x [-]

0

10

20

30

40

50

60

70

80

90

150000 200000 250000 300000Re [-]

Nt m

[-]

2.6 3.0 3.4 3.8 4.2 4.6 5.0

C) Reactor15

0

10

20

30

40

50

60

70

80

90

150000 200000 250000 300000Re [-]

2.6 3.0 3.4 3.8 4.2 4.6 5.0

D) Reactor20

Figure 3.11: Dimensionless macro-mixing time against Re and impeller speed using water for(A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20.


( )( ) 42042

6802466

.

.

mDN

LDxt

π

⋅= [s] (3.17)

Obviously, this is in good agreement with equation 3.16. This empiricalrelationship can be used for scale-up of the Drais reactor for processes underturbulent conditions (Re > 105).

3.4.5 Macro-mixing times in glycerin under laminar conditions

When the reactor is filled with glycerin at fill ratios below 1, the gas phaseis finely dispersed in the liquid. In the milky liquid thus obtained, as shown in thephotograph in Figure 3.6, it is impossible to perform spectrophotometric pulse-response measurements with the current set-up. Therefore, experiments have onlybeen performed using completely filled reactors.

In Figure 3.12A, Ntm is plotted against Re. It follows that the mixing timein glycerin is approximately 2.5-5 times longer than in water for the small and largereactors. Ntm is hardly dependent on Re on the different scales. Thus, in accordancewith the observation of Harnby et al. (1992), Ntm is independent of Re for bothlaminar and turbulent conditions. However, this is only valid for a given scale, asthe Ntm values for reactor11 and reactor11-60 differ despite their similargeometries. This emphasizes our previous observation that the hydrodynamics inthe two geometrically similar systems are significantly different.

Fox and Gex (1956) also provided a correlation for mixing times underlaminar conditions:

( ) 25.142

5.0

DN

Vt l

m ∝ [s] (3.18)

100 100050

60708090

100100

200

300

400

500

Reactor11-60

Reactor20

Reactor15

Reactor11

Nt m

[-]

Re [-]

0.1 1 10

10

100

Reactor11-60

Reactor20

Reactor15

Reactor11

t m [

s]

ε [m2/s

3]ε

A

Figure 3.12: (A) Dimensionless macro-mixing time against Re and impeller speed, using

glycerin. (B) tm as a function ofε using glycerin.

B

Chapter 3 54

Using the same approach, we obtain

� �� 44.042

98.024530

DN

LDtm

�

�� [s] (3.19)

The standard errors are given in Table 3.4. The differences between equations 3.18 and 3.19 can be a result of different geometries. Also, the results of Fox and Gex (1956) might be biased, as their experiments were performed in one single vessel in which only the liquid height was varied. Table 3.4: Standard errors of the constants in the correlations for t m

Conditions Pre-exp. factor Exponent in �/4 D2L Exponent. in N2 D4

Laminar (Re < 1730) 519 ± 36.4 0.98 ± 0.038 -0.44 ± 0.014 Turbulent (Re > 105) 66 ± 3.1 0.68 ± 0.022 -0.45 ± 0.022

3.4.6 Macro-mixing efficiencies

The term Ntm is an efficiency parameter that represents the number of revolutions required to obtain the desired mixing at time tm. According to this term, the highest mixing efficiency is found in reactor11 at a fill ratio of 0.7. Less than 10 impeller revolutions are required to obtain the required mixing which is 5 times more efficient than in reactor11-60 or reactor20.

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

10.70.60.50.4

x [-]

B) Reactor11

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

t m [s

]

A) Reactor11-60

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12[m2/s3]

t m [s

]

C) Reactor15

�

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12[m2/s3]

D) Reactor20

�

Figure 3.13: tm as a function of � for (A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20.


In Figure 3.12B and 3.13A-D, the mixing times are plotted against the averageenergy dissipation ε for laminar and turbulent conditions, respectively. ε iscalculated from equation 3.6, from which NP is calculated using the exponents inTable 3.3. From these figures, it can be concluded that, even though ε is the same,on average, tm in reactor11-60 is 5 times longer than in reactor11, although thelength and diameter are only 3 times greater. In reactor11 at a fill ratio of 0.7, 1m2/s3 is required to obtain mixing times of 1 s, whereas in reactor20, the same εvalue provides a mixing time of 9 s. Therefore, an important conclusion is thatscale-up of the Drais reactor based on energy consumption only, will lead to longermixing times.

3.5 Concluding remarks

In this study, we have examined the hydrodynamics in the Drais reactor atdifferent scales. It has been concluded that the specific power consumption as afunction of Reynolds number is the same on small and large scales. Over a widerange of Re values, the power number Np is comparable to Np of vertical unbaffledreactors with propellers. However, it has been shown that scale-up with constantpower consumption will lead to longer mixing times at larger scale. This is reflectedin Ntm, which depends linearly on the fill ratio at large scale and with the squareroot at small scale (Chapter 2). Thus, an important conclusion is that thehydrodynamics at different scale are different, making scale-up of the Drais reactora difficult task. For scale-up purposes, however, the macro-mixing times have beencorrelated, using an approach as suggested by Fox and Gex (1956) that includesliquid volume and flux of momentum. Especially for turbulent conditions, theagreement in the obtained correlation is remarkable.

Two flow states occur in this reactor, i.e., a 'slosh'-state at low vales of Fr anda 'ring'-state at high values of Fr. The transition between the two states as a result ofchanging impeller speed shows hysteresis. Using the criterion Fr=1 in scale up, onecan determine the number of revolutions required to obtain the 'ring'-state. The'ring'-state only applies for low-viscosity liquids. Using high-viscosity liquids, weobserve a liquid phase with a dispersed gas phase. Nevertheless, still a liquid filmexists on the cylindrical wall. The existence of this thin liquid film is important,because the highest shear rates are obtained in this film, which is crucial forobtaining products with high molecular weights in polymerization reactions(Agarwal et al., 1992).

Mixing times under laminar conditions are approximately only 2.5 timeshigher than under turbulent conditions. In a previous paper, we showed that mixingunder laminar conditions is very efficient, because of the chaotic nature. Striationswere made visible using planar laser-induced fluorescence and shown to stretch,fold, and reorient throughout the whole reactor. The deformation process of thestriations appears to occur homogeneously, which corresponds to the observedhomogeneous energy dissipation as described in this paper. Because of thishomogeneous dissipation, it can be concluded that the Drais reactor is a veryefficient mixer, especially under laminar conditions.

Chapter 356

Nomenclature

a constant -b constant -c clearance mC arbitrary concentration -Cn normalized concentration -C∞ final concentration -Cp heat capacity J/(kg⋅K)D vessel diameter mDi impeller diameter mfg geometrical factor -Fr Froude number -g acceleration of gravity m/sh overall heat transfer coefficient J/(K⋅s)h blade height mk1-k5 constants -L length of the vessel mNP power number -N agitation rate 1/sP dissipated stirring energy J/sRe Reynolds number -t time stm macro-mixing time sT liquid temperature KTa ambient temperature KT0 temperature at t = 0 KVl liquid volume m3

Vr reactor volume m3

vtip impeller speed m/sw blade width mx fill ratio -y distance m

Greek

α blade angle °,degreesβ mutual blade angle °,degrees

γ& shear rate 1/s

ε average power input per unit of mass m2/m3

µ dynamic liquid viscosity kg/(m⋅s)ν kinematic viscosity m2/sρ liquid density kg/m3

σ2 variance -


References

Agarwal, U.S.; Khakhar, D.V.; “Enhancement of polymerization rates for rigid rod-likemolecules by shearing”, Nature, 1992, 360, 53.

Agarwal, U.S.; Khakhar, D.V.; “Shear flow induced orientation development duringhomogeneous solution polymerization of rigid rodlike molecules”, Macromolecules,1993, 26, 3960.

Ando, K.; Hara, H.; Endoh, K.; “Flow behavior and power consumption in horizontal stirredvessels”, Int. J. Chem. Eng., 1971a, 11, 735.

Ando, K.; Hara, H.; Endoh, K.; “On mixing time in horizontal stirred vessel”, KagakuKogaku, 1971b, 35, 806.

Ando, K.; Fukuda, T.; Endoh, K.; “On mixing characteristics of horizontal stirred vesselwith baffle plates”, Kagaku Kogaku, 1974, 38, 460.

Ando, K.; Shirahige, M.; Fukuda, T.; Endoh, K.; “Effects of perforated partition plate onmixing characteristics of horizontal stirred vessel”, AIChE J., 1981, 27(4), 599.

Ando, K.; Obata, E.; Ikeda, K.; Fukuda, T.; “Mixing time of liquid in horizontal stirredvessels with multiple impellers”, Can. J. Chem. Eng., 1990, 68, 278.


Coulson, J.M.; Richardson, J.F.; Backhurst, J.R.; Harker, J.H.; “Coulson and Richardson’sChemical Engineering”, Vol. 1, 5th ed., Fluid flow, heat transfer and mass transfer;Butterworth-Heinemann Ltd: Oxford, England, 1996.

Fox, E.A.; Gex, V.E.; “Single-phase blending of liquids”, AIChE J., 1956, 2(4), 539.Fukuda, T.; Idogawa, K.; Ikeda, K.; Ando, K.; Endoh, K.; “Volumetric Gas-phase mass

transfer coefficient in baffled horizontal stirred vessel”, J. Chem. Eng. Jpn., 1990, 13(4),298.

Ganz, S.N.; Zh. Prikl.Khin., 1957, 30, 1311.Harnby, N.; Edwards, M.F.; Nienow, A.W. (Eds); “Mixing in the Process Industries”, 2nd

ed., Butterworth-Heinemann Ltd: London, England, 1992.Hoogendoorn, C.J.; den Hartog, A. P.; “Model studies on mixers in the viscous flow region”,

Chem. Eng. Sci., 1967, 22, 1689.Mersmann, A.; Einenkel, W.D.; Kappel, M.; “Design and scale up of mixing equipment”,

Int. Chem. Eng., 1976, 16, 590.Metzner, A.B., Otto, R.E.; “Agitation of Non-Newtonian Fluids”, AIChE J., 1957, 3(1), 3.Middleton, J.C; “Measurement of circulation within large mixing vessels”, Proc. 3rd Eur.

Conf. On Mixing, University of York, BRHA Fluid Eng. Cranfield, England, 1979, A2,15.

Novák, V.; Rieger, F.; “Homogenization efficiency of helical ribbon and anchor agitators”,Chem. Eng. J., 1975, 9, 63.

Rushton, J.H.; Costisch, E.W.; Everett, H.J.; “Power characteristics of mixing impellers,Parts I and II”, Chem. Eng. Prog., 1950, 46, 395 & 467.

Tatterson, G.B.; “Fluid mixing and gas dispersion in agitated tanks”, McGraw-Hill Inc: NewYork, United States of America, 1991.

Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers: Dordrecht,The Netherlands, 1994.

Vollbracht, L.; “Aromatic Polyamides”, Compr. Polym. Sci., Step Polym., 1989, 5, 374.

FLUID FLOW AND MIXING INAN UNBAFF

STI

The turbulent flow fbeen examined using LaseFluid Dynamics. The LDreactor the mean velocitiorder of magnitude highedirections the turbulent amean velocities. Therefodetermined by turbulent di

In the CFD calculanisotropic Differential Stthe turbulent properties oreasonably well predicunderestimate the fluctuaare simulated properly onModel is used. This impmodel (isotropic k-ε modis based on flow propertiereacting scalars might anisotropic turbulence is e

This chapter is a slightly modifiedVan der Gulik, G.J.S.; Wijers, JUnbaffled Horizontal Stirred Tank

4
LED HORIZONTAL
RRED TANK

Abstract

ield in a horizontal stirred tank reactor hasr Doppler Anemometry and ComputationalA experiments show that in the unbaffledes in tangential direction are at least oner than in axial and radial direction. In thesend periodic fluctuations dominate over there, macro mixing in these directions isspersion.ations the isotropic k-ε model and theress Model have been applied to incorporatef the flow. The mean flow properties areted with both models. Both modelsting properties. Scalar mixing experimentsly when the anisotropic Differential Stresslies that when the choice for a turbulenceel or anisotropic Differential Stress Model),s only, mixing simulations with passive andfail completely for those cases wherexpected.

version of:.G.; Keurentjes, J.T.F.; “Fluid Flow and Mixing in an”, Submitted for publication in AIChE. J.

Chapter 460

4.1 Introduction

Horizontal stirred tank reactors are widely used in industry. A commercialexample is the unbaffled horizontal reactor of the Drais type (TurbulentSchnellmischer, Drais Ltd, Mannheim, Germany) as depicted in Figure 2.1. Thisreactor can be used for multiple purposes like powder mixing in catalystpreparation, liquid mixing and kneading during CMC production orpolycondensation processes (Vollbracht, 1989; Bannenberg-Wiggers et al., 1998).In all applications the reactor is only partially filled (40% < x < 75%). The reactoris characterized by length L, diameter D and clearance B, which is the distancebetween the blades and the reactor wall. Since the clearance is small, the bladesperform a scraping action that keeps the reactor wall free from sticking material.The small clearance also implies the presence of a region with high shear rates andgood heat exchange. The blades have a pumping action towards the reactor centerplane at xL = 0, providing an easy way to discharge the content of the reactorthrough the opened hatch as depicted in Figure 2.1.

We have previously studied the mixing in the horizontal reactor (Van derGulik et al., 2001a and 2001b) within the framework of the application inpolycondensation processes in which viscosity increases several orders ofmagnitude during reaction. The choice for the Drais reactor has been made based onthe good mixing performance when the reactor content is highly viscous. However,at the start of the polycondensation process, when the components are added andmixed, the reactor content has a low viscosity, resulting in turbulent conditions.Consequently, the mixing performance has also to be sufficient at turbulentconditions as this determines the local monomer ratio. We have characterized themixing in the horizontal reactor at both low and high viscous conditions bydetermining power consumption (Van der Gulik et al., 2001a), mixing times (Vander Gulik et al., 2001a and 2001b) and the flow pattern (Van der Gulik et al.,2001b). Because the reactor is unbaffled and the impeller speed is usually high, ithas been found that the fluid mainly flows in tangential direction. The flow in axialand radial direction can be interpreted as secondary flow superpositioned on thetangential flow. The secondary flow pattern has been studied by using Planar LaserInduced Fluorescence from which it follows that flow in axial direction dominatesover the flow in radial direction (Van der Gulik et al., 2001b). Pulse responsemeasurements have confirmed this observation (Van der Gulik et al., 2001a). For aquantitative description of the internal flow field computational fluid dynamics(CFD) calculations are becoming increasingly popular. CFD requires models thatdescribe the turbulent properties of the flow. The k-ε model, as proposed byLaunder and Spalding (1974), is based on an eddy viscosity hypothesis. One of themain assumptions is that the turbulence is isotropic. Although turbulence is ingeneral anisotropic in mixing vessels (Kresta, 1998), the k-ε model is often usedbecause of simplicity and computational convenience (Schoenmakers, 1998;Montante et al., 2001; Brucato et al., 2000; Togatorop et al., 1994; Read et al.,1997; Rousseaux et al., 2001; Brucato et al., 1998). Reynolds Stress Models (RSM)

Fluid flow and mixing in an unbaffled horizontal stirred tank 61

are computationally much more demanding (5 additional equations compared to thek-ε model) but are able to incorporate the anisotropic nature of turbulence. CFDstudies on unbaffled vessels that compare the viability of both eddy viscositymodels and RSM show different results. For example, Armenante et al. (1997) andCiofalo et al. (1996) could only reproduce their Laser Doppler Anenometry data(LDA data) properly using RSM. Montante et al. (2001) used the k-ε model and aRSM to study the effect of the impeller clearance on the flow pattern in a verticalvessel. The k-ε model sufficed for all cases, except for the smallest clearance.

In this paper we have examined CFD for describing the hydrodynamics in theDrais reactor. As a start we have checked the viability of the k-ε model and theDifferential Stress Model (DSM), which is a RSM in differential form (Launder etal., 1975). For validation purposes, we have performed LDA measurementsproviding local velocities and turbulent quantities. Insight into these parametersshould allow for effective scaling up and should allow definition of routes forimprovement on the current reactor. Using CFD we have also performed scalarmixing simulations in the obtained flow fields for examining the mixingperformance. The results of these simulations have been compared with previouslyreported PLIF data (Planar Laser Induced Fluorescence, Van der Gulik et al.,2001b) providing an additional validation on the turbulence modeling.

+U

-r

- Ua +

0.0 x 0.17 0.33L

0.05

0.17 = w⋅sin(30)

Figure 4.1: Quarter of the reactor in which the axial coordinate xL has beendefined: xL = 0.0 in the reactor center and xL = 0.33 at the side wall. Thedirection of the axial velocity Ūa and the radial velocity Ūr have beenexemplified.

Chapter 462

4.2 Experimental and numerical set-up

4.2.1 The horizontal vessel

The vessel as depicted in Figure 2.1 was used in this study. Detailedgeometrical data were given previously (Van der Gulik et al., 2001b). The diameterand the length were equal to 0.60 and 0.66 m, respectively, providing an L/D-ratioof 1.1. The clearance B was equal to 0.01 m. The impeller blades make an angle of30° with the shaft. More detailed dimensions of the blades are given in Figure 4.1.The total blade length ‘w’, as defined in Figure 2.1, was 0.3 m. In Figure 4.1, thelength, projected on the x-y-plane, was equal to 0.17 m. The blade height ‘h’ had amaximum of 0.05 m in the middle of the blade and was equal to 0 at the far ends ofthe blade.

4.2.2 LDA measurements

LDA is an optical method for fluid flow research, based on a combination ofinterference and Doppler effects. LDA allows the measurement of the local,instantaneous velocities of particles suspended in the flow. LDA has a highresolution power in time and is non-invasive. A detailed description of the methodhas been given by Durst et al. (1981). Theoretical aspects of the turbulent flowoccurring in the Drais reactor and how the turbulent flow has been investigatedusing LDA, are given in Appendix 4A.

The LDA system used here has been described by Schoenmakers (1998) andSchoenmakers et al. (1997). A 2W Argon laser (Spectra Physics, Model Stabilite2017-055) was used to provide two colored beams with wavelengths of 514.5 and488 nm, respectively. Using a lens with a focal length of 400 mm both laser beamsconverged in a elliptical control volume of approximately 0.15 × 0.15 × 3.3 mm. Anautomated traverse system with the laser probe and the receiving optics was utilizedin Backscatter acquisition mode. Two photo multipliers and two Burst SpectrumAnalysers (Dantec, Denmark) were used for measuring two velocity componentssimultaneously.

The LDA measurements were performed in a perspex model of the Draisreactor that was placed in a square perspex container. The impeller frequency wasset to 3 Hz. All measurements were performed using water. The square perspexcontainer was also filled with water to minimize lensing effects. Positioning of themeasurement volume was realized using the procedures of Kehoe and Prateen(1987). Nevertheless, it was not possible to measure the tangential and radialcomponents close to the cylindrical wall and the side wall. The measurement pointsare summarized in Table 1. To obtain the three velocity components, two separatemeasurements were taken for the same (r, Z) point: one in the plane at 0°, yieldingthe tangential and axial components, and one in the plane at -90°, yielding the axialand radial components. In axial direction steps were made of 0.01 m. Only half thereactor was examined since flow proved to be symmetrical over an axial plane at x


= 0. In each location 30,000 measurements were performed with an average datarate of 0.3 kHz. For seeding we used an aqueous dispersion of polystyrene particleswith an average diameter of 4 µm.

Table 4.1: Radial locations for performed measurements between axial positions 0 < xL< 0.29.

r r/R Tangential (0°) Axial (0° and –90°) Radial (-90°)0.295 0.983 √0.29 0.967 √0.285 0.95 √ √0.28 0.933 √ √0.27 0.9 √ √ √0.25 0.833 √ √ √0.22 0.733 √ √ √0.29 0.633 √ √ √0.16 0.533 √ √ √0.13 0.433 √ √ √

The experimental error in mean velocity, periodic and turbulent fluctuationswas less than 10%. The mean and fluctuating axial velocities in the 0° and -90°plane appeared to be the same within 10%, indicating that the average flowcharacteristics are rotation symmetric. Therefore, all experimental data will bepresented as if measured in a single plane as presented in Figure 4.1. The directionof the axial and radial velocities has also been exemplified in Figure 4.1. A negativeaxial velocity is directed towards the reactor center whereas a positive velocity isdirected towards the reactor side wall. A negative radial velocity is directed towardsthe shaft and a positive radial velocity is directed towards the cylindrical wall.

4.2.3 Computational Fluid Dynamics

For the CFD calculations the Finite Volume Package CFX-4.2 was used (AEATechnology), installed on a Silicon Graphics Origin 200 workstation. The grid usedis depicted in Figure 4.2. The grid was built up in Cartesian coordinates andcomprised of 402 blocks and 374,820 cells. Because the angle between the impellerblades and the shaft is 30°, the grid strongly deviates from a computationally idealorthogonal grid. The Block-Stone solver was required to solve the differentialequations.

Chapter 464

Figure 4.2: The grid as used in the CFD Calculations, in front and side view.

The SIMPLE algorithm was adopted to couple the continuity and Navier–Stokesequations. Two turbulence models were used: the DSM and the standard k-ε model.Mathematical details are given in Appendix 4B. Because of the distorted grid, itwas necessary to adopt the fully deferred correction for the ε equation in bothturbulence models. On walls the conventional linear logarithmic “wall functions”were used (Launder and Spalding, 1974). The y+-values ranged from an averagevalue of approximately of 300 on both the cylindrical wall and the impeller blade toover 1000 at the shaft. The latter region was thought to be less important. The 1st-order accurate Hybrid differencing scheme was used for all advection terms.Additional calculations were performed using the 2nd-order CCCT-scheme(Curvature Compensated Convective Transport) for all equations. This scheme is amodification of the 3rd-order Quick-scheme in that it is bounded.

Simulations were only carried out at the impeller speed at which the LDAmeasurements were carried out (i.e., 3 Hz). The rotating action of the impeller wasimplemented using the “rotating coordinates” approach. In this approach thecoordinates of the complete grid rotated with 3 Hz while the vessel wall rotatedback with 3 Hz. Consequently, the net velocity of the vessel wall equals 0 m/s. TheCoriolis forces are implemented automatically in the “rotating coordinates”approach.

The simulations were conducted as a transient event. The simulations with thek-ε model were started from still fluid conditions. Then 80 steps with a time stepcorresponding to 90° impeller rotation were applied using 25 iterations per timestep. This corresponds to 20 complete impeller rotations of 6.66 seconds real time at3 Hz. These large time steps were applied to obtain fluid motion in the tank. Theattained solution was subsequently refined by allowing 12 rotations of the impellerwith time steps covering an angular extent of 360/56 = 6.43°. The refinedsimulation thus consisted of 672 time steps and 25 iterations per time step.


For the simulations with DSM, the final solution obtained with the k-ε modelwas used as an initial guess. Subsequently, the geometry was allowed to rotate withtime steps covering an angular extent of again 6.43° with a total of 672 time steps,i.e., 4 seconds of real time. Here also 25 iterations were used per time step.

It was found that using the above-described time-marching procedure, apseudo steady state was obtained in the vessel. This implies that at the end of thetransient calculation the solution only differed from the solution in the previoustime step in that the complete field was rotated with the required angular extent,indicating that sufficient convergence was obtained by the procedure adopted.

4.2.4 Scalar Mixing

In the obtained velocity field, the mixing of an inert tracer was studied. In acell next to the inlet (which is depicted in Figure 2.1) an initial concentration equalto 1 was defined. The progress of mixing was monitored at the position that ismarked in Figure 2.3 with an ‘×’ (xL = 0.32, r = 0.295). The tracer was mixed in thepreviously mentioned pseudo steady state. Time steps in the tracer mixingsimulations again covered an angular extent of 6.43°.

Chapter 466

4.3 Results and discussion LDA

4.3.1 Mean flow characteristics

The time average tangential velocity Ūt proves to be almost independent of theaxial position. Therefore, all the measurements performed at one radial coordinatehave been averaged. These averages are presented as a function of the radius r inFigure 4.3 with a solid line. The location of the shaft, impeller arm, impeller bladeand clearance are depicted on the x-axis. The impeller speed Uim as a function of theradius r is also depicted with a dotted line. The tip speed is equal to Uim(r = 0.29) =2⋅π⋅r⋅N = 2⋅π⋅0.29⋅3 = 5.47 m/s.

According to Figure 4.3, there appears to be a bulk region (0 < r < 0.22) wherethe tangential velocity is almost equal to the local impeller speed. The flowbehavior in this bulk region can be qualified as ‘solid body rotation’. A secondregion can be distinguished near the cylindrical wall (0.22 < r < 0.3) where thetangential velocity is fairly constant (Ūt ≈ 3.8 m/s). In this region viscous forcesreduce the tangential velocity. As the velocity of the cylindrical wall is equal to zerothere will be a steep fall in tangential velocity very close to the wall. This could notbe quantified as no measurements could be performed sufficiently close to the wall.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

Ūt(r

)/U

im(0

.29)

[-]

0.00

0.55

1.09

1.64

2.19

2.73

3.28

3.83

4.37

4.92

5.47

Ūt(r

) [m

/s]

Ūim(r)

Uim(0.29) = 5.47 [m/s]

Clearance .Shaft Blade arm Blade .

Figure 4.3: Tangential velocity Ūt as a function of the radial coordinate. On the left y-

axis Ūt has been made dimensionless through division by the impeller tip speed. On theright y-axis the absolute velocity is given. On the x-axis, the location of the shaft, theblade arm, the blade and the clearance have been exemplified.


The time averaged axial velocities Ūa are depicted in Figure 4.4A as a functionof the axial coordinate. The maximum absolute velocity is 0.2 m/s at (xL,r) =(0.1,0.29). At this position Ūt is equal to 3.8 m/s. Based on these measurements itcan be concluded that the axial velocities are much smaller than the local tangentialvelocities. The axial velocity appears to be a function of the radius. There areregions with positive velocities (e.g., at r = 0.22) and region with negative velocities(e.g., at r = 0.28). As these regions cannot be determined easily from Figure 4.4A,the xL-averaged axial velocities are given as a function of the radius in Figure 4.4B.Near the cylindrical wall and near the impeller blade (0.3 < r < 0.27) the velocitiesare negative, meaning that these velocities are directed towards the reactor center.For 0.26 < r < 0.2 the axial velocities are a positive, thus directed towards the sidewall.

In Figure 4.5, the radial velocities are given. On average the velocity isnegative, which violates with continuity. However, their order of magnitude issimilar to the axial velocities. The highest velocities are measured near the side wallat high xL-values.

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

0.275

0.3

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1Ua [m/s]

r [m

]

0.00

0.03

0.05

0.08

0.10

0.13

0.15

0.18

0.20

0.23

0.25

0.28

0.30

Clearance

Impeller blade

Impeller arm

Shaft

r = R = 0.3 m

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2 0.25 0.3

xL [m]

Ua

[m/s

]

0.29

0.28

0.27

0.25

0.22

0.19

r [m]

A B

|

_

Figure 4.4: A) Time averaged axial velocity Ūa at several radii r, as a function of theaxial coordinate xL. B) Averages of the velocities as presented in Figure 4.4A.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.05 0.1 0.15 0.2 0.25 0.3

xL [m]

Ur [

m/s

]

0.27

0.25

0.22

0.19

0.16

|

r [m]

Figure 4.5: Time averaged radial velocity Ūr at several radii r,as a function of the axial coordinate xL.

Chapter 468

4.3.2 Turbulent quantities

In Figure 4.6A, the time averaged turbulent fluctuations are given at a radialcoordinate r = 0.27 m. The turbulent fluctuations for the three directions are of thesame size, indicating that the turbulence shows isotropic behavior. The fluctuationsin the vicinity of the side wall are higher than in the reactor center. Similar isotropicbehavior in unbaffled vessels has also been observed by Montante et al. (2001).

The periodic fluctuations at a radial coordinate r = 0.27 are presented in Figure4.6B and show to be of the same order of magnitude as the turbulent fluctuations.Anisotropic behavior is observed near the side wall where the tangentialfluctuations show a strong increase. In this region, the side wall scraper probably isresponsible for the stronger periodic fluctuations.

In Figure 4.6C and 4.6D, the xL-averaged values for the turbulent and periodicfluctuations, respectively, are plotted against the radial coordinate. The turbulentfluctuating velocities in Figure 4.6C indicate isotropy as the fluctuations in all threedirections are of the same order of magnitude over the complete radius. Theperiodic fluctuations in Figure 4.6D show differences for the three directions,indicating the presence of an anisotropic turbulent mesostructure.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3xL [m]

u T [

m/s

]

Axial

Tangential

Radial

Turbulent fluctuations on r = 0.27 m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3xL [m]

u P [

m/s

]

Periodic fluctuations on r = 0.27 m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

uT [

m/s

]

Average turbulent fluctuations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

uP [

m/s

]

Average periodic fluctuations

A B

C D

Figure 4.6: A) Time averaged turbulent fluctuation on r = 0.27 for axial, radial andtangential direction. B) Time averaged periodic fluctuation on r = 0.27 for axial, radial andtangential direction. C) xL- averages of the fluctuations presented in Figure 4.6A, but for allradii. D) xL-averages of the fluctuations presented in Figure 4.6B, but for all radii.


In Figure 4.7, the energy dissipation rate ε is given in a 2-dimensional plot.This area represents 80% of the total reactor volume. Only the turbulent fluctuatingpart is included as the periodic fluctuations are supposed to represent pseudoturbulence. In the measured region, ε ranges from 2.5 m2/s3 near the cylindrical wallto 0.01 near the shaft. The maximum of 2.5 m2/s3 is located in the region where theimpeller tips overlap. Integration of ε and multiplication by 2 to take the other halfof the reactor into account, provides a dissipated energy of 45 W. On average, εequals 0.31 m2/s3. Previously reported results (Van der Gulik et al., 2001a) provideda value of 375 W for the complete reactor so that almost all energy dissipates veryclose to the wall.

4.3.3 Overall remarks

The detailed LDA information provides guidelines that can be useful in the useand scale-up of Drais reactors. For example, Figure 4.6B shows that the periodicfluctuations in the reactor center are lower than near the side wall. Consequently,mixing will be more efficient near the side wall. From this observation one canexpect that the L/D-ratio should not be made too large, as the fluctuations in thevicinity of the reactor center lower upon an increase in L/D, thus decreasing themixing performance. This is in agreement with results presented in a previous paperin which we have shown that mixing times increase more than linearly upon anincrease in L/D (Van der Gulik et al., 2001a and 2001b).

In Table 2 the average velocity and average turbulent components are givenfor r = 0.27 which are representative for the reactor. Note that the turbulent andperiodic fluctuations in axial and radial direction are much higher than the averaged

0 0.05 0.1 0.15 0.2 0.25 0.3

xL [m]

0

0.05

0.1

0.15

0.2

0.25

0.3

r [m

]

ε [m2/s3]

Figure 4.7: Distribution of the energy dissipation rate ε for the measurable region.

Chapter 470

velocities. In axial direction, the difference is one order of magnitude. Therefore,the flow in axial and radial direction is exclusively determined by turbulent andperiodic fluctuations. Consequently, macro mixing in these directions will bedetermined by turbulent dispersion. In tangential direction the convective flow isdetermining.

The reactor can be considered as a moderate turbulent mixer as the liquid ismainly rotated in tangential direction and overall mixing is slow. Moreover, at radiibelow r = 0.22 the liquid rotates almost as a solid body. Figures 4.6C and 4.6Dshow that the fluctuations reduce at decreasing radii, meaning that the mixingperformance reduces. Therefore, the reactor is usually used at fill ratios below 50%,so that at high impeller speeds a ring of fluid exists above r = 0.2 where at leastsome mixing occurs. This limitation in fill ratio up to 50% can be seen as adisadvantage of the Drais reactor.

Table 4.2: RMS values in m/s of velocity and fluctuations scales at r = 0.27.

r = 0.27 Tangential Axial RadialMean velocity 3.73 0.035 0.14

Turbulent fluctuations 0.32 0.32 0.31Periodic fluctuations 0.2 0.18 0.16


4.4 Results and Discussion CFD

4.4.1 Mean flow characteristics

The calculated tangential velocities Ūt only depend on the radius which hasalso been observed in the LDA experiments. Therefore, all tangential velocities arexL-averaged analogous to the LDA velocities as represented in Figure 4.8. Overall,DSM combined with the Hybrid differencing scheme provides the best results asthe data are the closest to the LDA data. Especially around the impeller blade at r =0.27, the comparison is relatively good. The k-ε model with the Hybrid differencingscheme underestimates the tangential velocity around r = 0.27 whereas the DSMwith the CCCT-differencing scheme underestimates the tangential velocity. Thedissipative nature of the Hybrid differencing scheme seems to be effective inpredicting the measured velocity. In the LDA data the highest tangential velocity ismeasured at r = 0.22m. All CFD solutions underestimate the velocity at this radius.As will be shown, axial velocities at this radius are also underestimated.

In Figure 4.9 simulated mean axial and radial velocities at r = 0.27 arecompared with LDA data. The axial velocities in Figures 4.9A are all of the sameorder of magnitude and follow the same trend. In detail the velocities differ butregarding the quality of the numerical grid and the large periodic fluctuations thathave been observed in the LDA measurements, the comparison is as good asexpected. The simulated radial velocities in Figure 4.9B differ substantially from

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

Ūt(r

)/U

im(0

.29)

[-]

0.00

0.55

1.09

1.64

2.19

2.73

3.28

3.83

4.37

4.92

5.47Ūt

(r)

[m/s

]

LDA-data

k-e/hybrid

dsm/hybrid

dsm/ccct

Uim(r)

Uim(0.29) = 5.47 m/s

Tangential velocities .

Figure 4.8: xL-averaged tangential velocity Ūt as a function of the radial coordinatefor the LDA measurements and the CFD calculations.

Chapter 472

the measured velocities. Near the side wall a relatively strong radial flow (0.3 m/s)towards the shaft has been obtained. The effect of this flow phenomenon will bediscussed in section 4.5. The strong radial velocity could not be confirmed usingLDA data as no measurements could be performed near the side wall.

In Figure 4.10A the xL-averaged axial velocities are compared with LDAmeasurements. The k-ε model and the DSM are well matched despite some minordifferences near the shaft. The profile of the simulated velocities has the samenature as the LDA data, meaning that alternating negative (towards reactor center)and positive (towards side wall) velocities occur as a function of the radius r. Theorigin of the alternating direction can be explained from the vector plot in Figure4.10B which has been obtained using the DSM with the Hybrid scheme. In thisvector plot, the axial and radial components of the total velocity vector are

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 0.05 0.1 0.15 0.2 0.25 0.3xL [m]

Ūa [

m/s

]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3xL [m]

Ūr [

m/s

]

ldadsmk-e

r = 0.27

Figure 4.9: A) Comparison of the axial velocity Ūa, and B) the radial velocity Ūr. Both

time averaged measured with LDA and calculated using the k-ε model and the DSM.

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

0.275

0.3

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

Ua [m/s]

r [m

]

LDA-data

k-e/hybrid

dsm/hybrid

dsm/ccct

Axial velocities

DSM/Hybrid

Plane with axial andradial velocity vectors

A B

C

_

Figure 4.10: A) xL-averaged velocities Ūa as a functionof the radius r. B) Vector plot representing the axialand the radial components. C) Exemplification of thepoint of view in the reactor for obtaining Figure 4.10B.


projected on a plane that crosses the shaft and the impeller blade near the side wall.The point of view is along the impeller blade as depicted in Figure 4.10C. Theimpeller blade (0.24 < r < 0.29) transports the fluid towards the reactor center whilethe flow reverses between the impeller blade and the shaft. Between impeller tipand cylindrical wall the flow appears to be directed towards the side wall which isnot observed in the LDA measurements.

4.4.2 Turbulent quantities

The periodic velocity fluctuations are not directly available from CFD data,but can be calculated using equation 4.1:

( ) ( ) ( )[ ]∑ −+−+−=0

0

N

1

233

222

211N

121

P UUUUUUk [m2/s2] (4.1)

in which Ui are instantaneous local velocities and Ūi the time averaged velocities(Montante et al., 2001). N0 is the number of cells in the tangential direction and kP

represents the energy content of the periodic fluctuations. The results are presentedin Figure 4.11A. The data presented are the averages of all kP in tangential and axialdirection. kP as calculated with both the k-ε model and DSM are of the same size asthe measured kP. However, they differ in trend: the simulated kP remains more orless constant over the radius, the measured kP decreases with decreasing radius. Theoverestimated kP at low radii is probably due to the poor mesh quality near theshaft. Near the cylindrical wall, the differences in kP are approximately one order ofmagnitude.

The turbulent fluctuations in all three directions are represented by theturbulent kinetic energy kT, which has been calculated using equation 4A.5 inAppendix 4A. Although the use of kT for DSM masks anisotropic properties, itmakes comparison with kT from the k-ε model straightforward. The presented kT

values in Figure 4.11B are xL- and time averaged. Clearly the k-ε model predicts thekT values better than DSM. However, both simulated kT are lower than themeasured kT. Similar differences have been observed before and have beenrecognized as one of the major of discrepancies between LDA measurements andCFD calculations (Montante et al., 2001; Brucato et al., 1998; Armenante et al.,1997; Ng et al., 1998).

Figure 4.11C shows the (xL-averaged and time-averaged) energy dissipationrate for the LDA measurements and the simulations. The simulated ε is significantlylower than the measured one. The simulated ε shows a strong increase near thecylindrical wall. The total dissipated energy using both turbulence models is givenin Table 4.3. The predicted values appear to be about 30% of the measured value,for both the complete reactor and the part of the reactor in which LDAmeasurements could be performed (Van der Gulik et al., 2001a). The ratio betweenthe values for the complete and for part of the reactor are the same for LDA andCFD data, meaning the distribution in dissipated energy appears to be similar.

Chapter 474

Table 4.3: Dissipated energy [W] in the complete reactor and in a part of the reactor usingdifferent experimental techniques and numerical models.

Experimental CFDRegion Temperature

measurements LDA k-εHybrid

DSMHybrid

Complete reactor 375 (100%) - 101 (100%) 93 (100%)Part of the reactor

-0.29<xL<0.29 0<r<0.285- 45 (12%) 13.9 (14.8%) 7.7 (8.2%)

Concluding, based on the main flow characteristics the DSM is preferredabove the k-ε model. According to the turbulent properties, the k-ε model ispreferred although the differences with measured quantities are significant for bothturbulence models.

0.0001

0.001

0.01

0.1

1

10

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

ε [m

2 /s3 ]

dsm

k-e

LDA

0.0001

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2 0.25 0.3r [m]

k t [

m2 /s

2 ]

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2 0.25 0.3

r [m]

k P [

m2 /s

2 ]

A B

C

Figure 4.11: A) Periodic fluctuations, represented by the periodic kinetic energy kP,A) Turbulent fluctuations, represented by the turbulent kinetic energy kT, and C) theenergy dissipation rate ε. All are time and xL-averaged.


4.5 Scalar mixing

In the flow fields obtained using the k-ε model and the DSM, mixing of anadded scalar has been monitored. This is especially relevant for simulations withchemical reactions in which the components are represented by reactive scalars.

In Figure 4.12, the scalar concentrations for the DSM and the k-ε model aregiven against real time. The concentrations are monitored in the position that ismarked with an ‘x’ in Figure 2.3. Scalar concentrations are normalized to their finalconcentration. It can be seen that is takes about 2 seconds for the scalar to reach themonitoring point. The concentration initially fluctuates strongly for bothsimulations because concentration differs in tangential direction. The maindifference between the k-ε model and DSM is that the curve for the k-ε modelgradually rises towards the final value of 1.0 while the curve for the DSM exceedsthe final concentration and then gradually lowers to the final value.

In Figure 4.12 also a curve has been given that results from a pulse-responseexperiment with a colored tracer as described before (Van der Gulik et al., 2001a).The curve does not show the strong fluctuations because the monitoring probe wasplaced outside the reactor. The shape of the response curves shows strongresemblance with the curve for the DSM. Clearly the DSM represents the mixingexperiment closer than the k-ε model does.

In Figures 4.13A and B the calculated scalar concentration distribution is givenfor the DSM and the k-ε model, respectively, after a mixing time of 4 seconds. Thered color represents the region with a higher concentration than the finalconcentration. Figure 4.13B emphasizes that using the k-ε model, the mixing inradial direction is faster than in axial direction as the red region extends until theshaft. This can be explained by the isotropic turbulence making the mixing speed inall directions equal, but in radial direction apparently the fastest because of thereduced volume towards the shaft. Using the DSM, the mixing in radial and axial

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 5 10 15 20 25 30 35 40Time [s]

Nor

mal

ized

sca

lar

conc

. [-

]

Mixing time measurement [3]

dsm

k-ε

Figure 4.12: Passive scalar concentration plotted against time for a pulse response experiment(Van der Gulik et al., 2001a) and for CFD calculations using the k-ε model and the DSM.

Chapter 476

direction apparently proceeds at the same speed. This difference in mixing behavioralso clarifies the shape of the response curves in Figure 4.12. The monitoringposition differs only in axial coordinate from the inlet position, so that using theDSM the scalar will be traced before it will be traced using the k-ε model.

Previously we have reported on PLIF experiments (Planar Laser InducedFluorescence, Van der Gulik et al., 2001b) in a Drais reactor with a diameter of 0.18m at an impeller speed of 6 Hz. This reactor is geometrically similar to the reactorcurrently investigated. The mixing in the complete reactor using PLIFmeasurements is presented in Figure 4.13C. The measured mixing pattern in thisfigure resembles the mixing pattern in Figure 4.13A as calculated using the DSM.This resemblance and the resemblance of the response curve shows that forsimulations of scalar mixing in this reactor the DSM performs much better than thek-ε model. An important conclusion from this work is that the difference in scalarmixing behavior should make CFD users cautious, as based on mean flow andfluctuating properties no clear preference could be made between the k-ε model andDSM.

A B

C

Figure 4.13: Distribution of passive scalars, modeled using A) the DSM, B) the k-ε model,and C) determined experimentally using PLIF (Van der Gulik et al., 2001b).


Nomenclature

A constant in eq. A.7. -B clearance mCµ,C1,C2 parameters in k-ε modelD vessel diameter mE one dimensional power spectrum s/m2

G production of turbulent kinetic energy kg/(m⋅s3)h blade height mk turbulent kinetic energy m2/s2

L vessel length mLI integral length scale mp pressure PaRi time autocorrelation function -TI integral time scale su` velocity fluctuations m/sU instantaneous velocity m/sŪ time averaged velocity m/s

w blade length mxL axial coordinate m

Greek

α angle between blade and shaft 60°β angle between blades 180°δij Kronecker delta -ε energy dissipation rate m2/s3

µ dynamic viscosity kg/(m⋅s)ρ density kg/m3

σk, σz parameters in k-ε model

Subscripts

a axiali one of three directionsim impellerP periodicr radialrms root mean squaret tangentialT turbulent

Chapter 478

References

Adrian, R.J.; Yao, C.S.; “Power spectra of fluid velocities measured by laser Dopplervelocimetry”, Exp. in Fluids, 1987, 5, 17.

Armenante, P.M.; Luo, C.; Chou, C.; Fort, I.; Medek, J.; “Velocity Profiles in a Closed,Unbaffled Vessel: Comparison Between Experimental LDV Data and Numerical CFDPredictions”, Chem. Eng. Sci., 1997, 52(20), 3483.

Bannenberg-Wiggers, A.E.M.; Van Omme, J.A.; Surquin, J.M.; “Process for the BatchwisePreparation of Poly-p-terephtalamide”, U.S. Pat., 5,726,275, 1998.

Bird, R.B.; Stewart W.E.; Lightfooth, E.N.; “Transport phenomena”, John Wiley & Sons, US1960.

Brucato, A.; Ciofalo, M.; Grisafi, F.; Micale, G.; “Numerical Prediction of Flow Fields inBaffled Stirred Vessels: A Comparison of Alternative Modeling Approaches”, Chem.Eng. Sci., 1998, 53(21), 3653.

Brucato, A.; Ciafalo, M.; Grisafi, F.; Tocco, R.; “On the Simulation of Stirred Tank Reactorsvia Computational Fluid Dynamics”, Chem. Eng. Sci., 2000, 55, 291.

Ciofalo, M.; Brucato, A.; Grisafi, F.; Torraca, N.; “Turbulent Flow in Closed and Free-Surface Unbaffled Tanks Stirred by Radial Impellers”, Chem. Eng. Sci., 1996, 51(14),3557.

Clarke, D.S.; Wilkes, N.S.; “The calculation of turbulent flows in complex geometries using adifferential stress model”, UKAEA Report AERE-R 13428, Harwell, UK.

Cutter, L.A.; “Flow and turbulence in a stirred tank”, AIChE J., 1966, 12, 35.Durst, F.; Melling, A.; Whitelaw, J.H.; “Principles and Practice of Laser Doppler

Anemometry”, Academic Press, London, UK, 1981.Kehoe, A.B.; Prateen, V.D.; “Compensation for Refractive-Index Variations in Laser

Doppler Anemometry”, Applied Optics, 1987, 26(13), 2582.Kresta, S.; “Turbulence in Stirred Tanks: Anisotropic, Approximate, and Applied”, Can. J.

Chem. Eng., 1998, 75, 563.Kusters, K.A.; “The influence of turbulence on aggregation of small particles in agitated

vessels”, Ph.D. Thesis, Eindhoven University of Technology, The Netherlands, 1991.Launder, B.E.; Reece, G.J.; Rodi, W.; “Progress in the Development of a Reynolds Stress

Turbulence Closure”, J. of Fluid Mechanics, 1975, 68, 537.Launder, B.E.; Spalding, D.B.; “The Numerical Computation of Turbulent Flows”, Comp.

Meth. Appl. Mech. Eng., 1974, 3, 269.Montante G.; Lee, K.C.; Brucato, A.; Yianneskis, M.; “Numerical Simulation of the

Dependency of Flow Pattern on Impeller Clearance in Stirred Vessels”, Chem. Eng. Sci.,2001, 56, 3751.

Mujumdar, A.R.; Huang, B.; Wolf D.; Weber, M.E.; Douglas, W.J.M.; “Turbulenceparameters in a strirred tank”, Can. J. Chem. Eng., 1970, 48, 475.

Ng, K.; Fentiman, N.J.; Lee, K.C.; Yianneskis, M.; “Assessment of Sliding Mesh CFDPredictions and LDA Measurements of the Flow in a Stirred by a Rushton Impeller”,Trans. Inst. Chem. Eng. Res. Des., 1998, 76, 737.

Nieuwstad, F.T.M.; “Turbulentie, inleiding in de theorie en toepassingen van turbulentestromingen (In Dutch)”, Epsilon uitgaven, The Netherlands, 1998.

Read, N.K.; Zhang, S.X.; Ray, W.H.; “Simulations of a LDPE Reactor using ComputationalFluid Dynamics”, AIChE J., 1997, 43(1), 104.

Rogers, M.M.; Mansour, N.N.; Reynolds, W.C.; “An algebraic model for the turbulent fluxof a passive scalar”, J. Fluid Mech., 1989, 203, 77.

Rousseaux, J.M.; Vial, C.; Muhr, H.; Plasari, E.; “CFD Simulation of Precipitation in theSliding-Surface Mixing Device”, Chem. Eng. Sci., 2001, 56, 1677.


Schoenmakers, J.H.A.; “Turbulent Feed Stream Mixing in Agitated Vessels”, Ph.D. Thesis,Eindhoven University of Technology, The Netherlands, 1998.

Schoenmakers, J.H.A.; Wijers, J.G.; Thoenes, D.; “Non Steady-State Behavior of the Flowin Agitated Vessels”, Proc. 4th World Conf. on Experimental Heat Transfer, FluidMechanics and Thermodynamics, ExHFT 4, Brussels, Ed. Giot, Mayinger & Celata,1997, Vol. 1, 477-481.

Taylor, G.I.; “Statistical Theory of Turbulence Parts I-IV”, Proc. Roy. Soc., 1935, A151,421.

Tennekes H.; Lumley, J.L.A.; “First course in turbulence”, MIT Press, USA, 1972.Togatorop, A.; Mann, R.; Schofield, D.F.; “An Application of CFD to Inert and Reactive

Tracer Mixing in a Batch Stirred Vessel”, AIChE Symp. Series, 1994, 299, 19.Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics and Scale-Up of

Horizontal Stirred Reactors”, Ind. Eng. Chem. Res., 2001a, 40(22), 4731.Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics in a Horizontal

Stirred Tank Reactor”, Ind. Eng. Chem. Res., 2001b, 40(3), 785.Van der Molen, K.; Van Maanen, H.R.E.; “Laser-Doppler measurements of the turbulent

flow in stirred vessels to establish scaling rules”, Chem. Eng. Sci., 1978, 33, 1191.Van ‘t Riet, K.; Smith, J.H.; “The trailing vortex system produced by a Rushton turbine

agitators”, Chem. Eng. Sci., 1975, 30, 1093.Vollbracht, L.; “Aromatic Polyamides”, Compr. Polym. Sci., Step Polym., 1989, 5, 374.

Chapter 480

Appendix 4A: Theoretical Aspects of LDA Measurements

Turbulent velocities

At turbulent conditions, instantaneous fluid velocities (U1, U2, U3) can berepresented by a mean and a fluctuating component, )uU,uU,uU( '

33'22

'11 +++ ,

which is the so called Reynolds decomposition for three dimensions. The meanflow in direction i, iU , is time averaged and is represented by:

∆∞→∆= ∫

∆+ tt

t

ii dtUt

1

t

limU [m/s] (4A.1)

The absolute averaged velocity is obtained using a vector summation:

23

22

21 UUUU ++= [m/s] (4A.2)

Often this is referred to as convective velocity or speed in CFD. The root-mean-square (rms) value of the turbulent fluctuations is given by:

)u+u+u(=u2'

3

2'

2

2'

121

rms [m/s] (4A.3)

The intensity of the turbulent fluctuations, I, is equal to:

U

uI rms= [-] (4A.4)

Turbulent fluctuations

The velocity fluctuations represent a significant amount of turbulent kineticenergy. The contribution per unit of mass is equal to:

)u+u+(u=k2'

T3,

2'T2,

2'T1,2

1T [m2/s2] (4A.5)

in which 'T,iu is the turbulent part of the velocity fluctuations that can be

distinguished from periodic fluctuations as will be discussed later. The energy kT isinitially present in large eddies generated by the impeller. The large eddies break-upinto smaller eddies transferring the energy to smaller scales until the Kolmogorovscale is reached where the kinetic energy is dissipated into heat by viscosity. Therate of energy transfer from large to small scale is thought to be proportional to kT


divided by the time scale of the largest eddies i.e., L/u. The rate of energy transfer isusually referred to as the energy dissipation rate ε:

L

u~

3rmsε [m2/s3] (4A.6)

This can be interpreted as the amount of energy u2rms that a turbulent eddy

dissipates in a timescale T~urms/L. Using the ‘inviscid estimate’ as postulated byTennekes and Lumley (1972), ε can be rewritten as:

I

T

L

kA

23

=ε [m2/s3] (4A.7)

‘A’ has been found to be equal to 1 ± 0.2 which is to be expected based on theenergy balance (Cutter, 1966; Kusters, 1998; Nieuwstadt, 1998).

Integral length and time scales

LI represents the integral length scale of the eddies. Using Taylor’s hypothesis(1935), LI can be calculated from experimentally determined local time-averagedvelocity Ū and the integral time scale TI. This hypothesis assumes that temporal

velocity fluctuations measured in an Eulerian frame can be interpreted as spatialturbulence along a line parallel to the average convective velocity Ū. Hence, itfollows that:

II TUL ⋅= [m] (4A.8)

Another formulation of Taylor’s hypothesis is that the average convective velocityŪ transports the eddies with such a high speed through a measurement point that theeddies do not change during the measurement. The turbulent fluctuations thenappear to be frozen.

The integral time scale TI is equal to the sum of the time scales for the threedirections:

∑=

=3

1iiI TT [s] (4A.9)

The time scale for each direction i can be obtained from zero frequency intercept ofthe one-dimensional power spectrum Ei(f) as proposed by Mujumdar et al. (1970):

)f(E0f

lim

u4

1T i2'

i

i →= [s] (4A.10)

Chapter 482

Ei(f) is the Fourier cosine transform of the time autocorrelation function Ri(τ):

ττ= ∫∞

τπ de)(Ru4)f(E0

if2i

2'ii [m2/s] (4A.11)

with

2i

iii

u

)t(u)t(u)(R

τ+⋅=τ [-] (4A.12)

τ is the time over which the correlation is being determined. The length scale of thelargest eddies are expected to be equal to the height of the impeller blades.

An equidistant time signal is required to calculate spectra or auto correlationfunctions of a signal, which is not obtained from LDA measurements due to thePoison-like shape of the inter arrival time distribution. The interpolation procedureused to obtain an equidistant velocity signal, is the Sample-Hold techniqueproposed by Adrian and Yao (1987). The first moment of the velocity signal provedto deviate only a few percent from the variance of the velocity signal from spectralanalysis using the Sample-Hold as a interpolation procedure.

Periodic fluctuations

The flow in the Drais reactor is mechanically driven by the impeller. In thistype of flow the velocity fluctuations can be split up in a turbulent and periodic part.The turbulent part is random. The periodic part results from the trailing vorticescoming from behind the rotating stirring device (Van ‘t Riet and Smith, 1975). Theperiodic part has a frequency equal to the impeller frequency multiplied by thenumber of revolutions. Also sub- and higher harmonics occur. Van der Molen andVan Maanen (1978) showed that the turbulent and periodic parts are not correlatedand can be separated using:

2'P,i

2'T,i

2'i uuu += [m2/s2] (4A.13)

Periodic fluctuations are visible as strong peaks in the power spectrum whichcan be constructed using equation 4A.11. The contribution of the periodicfluctuations can be obtained by integrating the corresponding peaks. The remainingfrequencies are assigned to the turbulent fluctuations which are used to obtain theturbulent kinetic energy in equation 4A.5.


Appendix 4B: Computational Fluid Dynamics

The equations that are solved in Computational Fluid Dynamics for a turbulentflow are the Reynolds-averaged continuity and momentum equations. With constantdensity and viscosity, the continuity equation has the following form for a Cartesiangrid:

0z

U

y

U

x

U)(div

zyx =

∂∂

+∂∂

+∂∂

ρ=ρ iU [kg/(m3⋅s)] (4B.14)

The momentum equation for the x-component has the following form:

∂∂

+∂

∂+

∂∂

ρ−∇µ+∂∂

−=

∂

∂+

∂∂

+∂

∂+

∂∂

ρz

uu

y

uu

x

uuU

x

p

z

)UU(

y

)UU(

x

)UU(

t

U 'x

'z

'x

'y

'x

'x

x2zxyxxxx

[kg/(m2⋅s2)] (4B.13)

or more condensed:

( ) )(divUx

pUdiv

t

Ux

2x

xxiU τ−∇µ+

∂∂

−=

+

∂∂

ρ [kg/(m2⋅s2)] (4B.14)

For the y and z-direction similar equations can be derived. τx, τy, and τz are the rowvectors comprising the traceless part of the turbulent stress tensor τij. These vectorsare the components that represent the Reynolds stresses acting on surfaces ofnormal x, y, and z, respectively, with the normal components depleted by p. For theclosure of these equations, turbulence models are used.

The k-ε turbulence model

The first model used is the well-known k-ε model which is based on an eddyviscosity hypothesis (Launder and Spalding, 1974). The Reynolds stresses aremodeled as stress terms with a viscosity µT that depends on flow properties. In mostgeneral form, the stresses can be written as:

iji

j

j

iT

'j

'i k

3

2

x

U

x

Uuu δρ−

∂∂

+∂∂

µ=ρ− [kg/(m⋅s2)] (4B.15)

in which i and j refer to either x, y or z. The term 2/3ρkδij, in which δij is theKronecker delta (Bird et al., 1960), is added so that continuity and the definition ofturbulent kinetic energy are satisfied. The turbulent viscosity is described as:

Chapter 484

ερ=µ µ

2

Tk

c [kg/(m⋅s)] (4B.16)

k and ε are the turbulent kinetic energy and energy dissipation rate, respectively, forwhich the following equations apply:

ρε−+

∂∂

σµ

∂∂

=∂∂

ρ+∂∂

ρ Gx

k

xx

kU

t

k

jk

T

jjj [kg/(m⋅s3)] (4B.17)

k)CGC(

xxxU

t 21j

T

jjj

ερε−+

∂ε∂

σµ

∂∂

=∂ε∂

ρ+∂ε∂

ρε

[kg/(m⋅s4)] (4B.18)

In these equations G is the production of turbulent kinetic energy by deformationand is equal to:

j

i'j

'i x

UuuG

∂∂

ρ−= [kg/(m⋅s3)] (4B.19)

The standard values of the constants in the model were used: Cµ=0.09, C1=1.44,C2=1.92, σk=1.0, and σε =1.3.

The DSM turbulence model

The second model used, was the DSM (Differential Stress Model) by Launderet al. (1975). In this model the turbulent stresses in the stress tensor τij werecomputed by solving the six independent transport equations, that have thefollowing general form:

ijijijijij F)(divG)(div +ε+φ++=τρ iji dU [kg/(m⋅s3)] (4B.20)

In this equation the terms on the right-hand side represent generation, diffusion,pressure strain correlation, dissipation and generation by fluctuating body forces,respectively. In the DSM a further transport equation for the dissipation ε is alsosolved, while the turbulence energy kT can be computed as the half sum of thediagonal terms τkk. The DSM is especially suitable for problems in whichturbulence is expected to be strongly anisotropic in time and space. Thedisadvantage of the DSM is the numerical cost of solving 10 strongly coupledpartial differential equations. The details of the model formulation are cumbersomeand will not be given here. The implementation of the DSM in the CFX-4 code ispresented by Clarke and Wilkes.


Scalar mixing

The transport of a passive scalar in a turbulent flow field can be described bythe following equation (after Reynolds decomposition and time averaging):

i

'j

'i

2i

2

ii

x

cu

x

CD

x

CU

t

C

∂

∂−

∂

∂=

∂∂

+∂∂

[1/s] (4B.21)

In this expression the turbulent transport of the scalar 'j

'i cu is closed by invoking the

gradient diffusion assumption:

jT

'j

'i x

Ccu

∂∂

Γ−= [m/s] (4B.22)

ΓT can be interpreted as an effective turbulent diffusion coefficient. Using the k-εmodel ΓT has the following form:

c

TT ρσ

µ=Γ [m2/s] (4B.23)

in which µT is the previously mentioned turbulent viscosity and σc the turbulentSchmidt number which is equal to 0.7. The turbulent diffusion coefficient in the k-εmodel implies isotropic behavior in that ΓT is the same in all directions. Using theDSM, ΓT is treated as a tensor for which the details are given by Rogers et al.(1989). This tensor invokes the non-isotropic mixing of the passive scalar.

HYDRODYNAMICS IN A CERAMICPERVAPORATION MEMBRANE

REACTOR FOR RESIN PRODUCTION

Abstract

The hydrodynamics in a pervaporation membrane reactor for resinproduction has been investigated. In this type of reactor it is importantto reduce concentration and temperature polarization to obtain highwater fluxes during operation. The influence of secondary flow onpolarization, as induced by small density differences, is studied usingComputational Fluid Dynamics in a model system. This model isoperated in three parallel flow situations: horizontal, vertical opposed,and vertical adding flow. Density-induced convection is found to bemost effective in the horizontal situation, increasing water fluxes up to50%. Water fluxes were also determined experimentally using thehorizontal set-up. The influence of density-induced convection wasobserved experimentally.

This chapter is a slightly modified version of the publication:Van der Gulik, G.J.S.; Janssen, R.E.G.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics ina Ceramic Pervaporation Membrane Reactor for Resin Production”, Chem. Eng. Sci., 2001,56, 371-379.

5

Chapter 588

5.1 Introduction

Resins are polycondensation products, formed by the following equilibriumreaction:

(R.7)

Reaction conditions require elevated temperatures, typically above 200 °C. Themixtures are moderately viscous at these temperatures (µ = 5-25 mPa·s). To obtainhigh yields, the reaction equilibrium has to be shifted to the right. This can beachieved using a large excess of one of the starting reagents (usually the alcohol,which is also used as the solvent). However, this results in a relatively inefficientuse of reactor space, and requires an efficient separation afterwards. The reactioncan also be forced to completion by removing the water. In current processes wateris removed by distillation, which is not very efficient in the case of azeotropeformation. Additionally, due to the large reflux ratios required, energy consumptioncan be significant (Keurentjes et al., 1994).

Alternatively, pervaporation membranes can be used for selective waterremoval. This has shown to be effective for low molecular weight esterifications(Keurentjes et al., 1994; David et al., 1991; Okamoto et al., 1993). For this purpose,polymeric membranes have been used, mainly based on polyvinyl-alcohol.However, in resin production only ceramic membranes are appropriate, as they areresistant to the high reaction temperatures (Bakker et al., 1998; Koukou et al., 1999;Verkerk et al., 2001). These membranes are selective to water, as they areextremely hydrophilic, and have pores that are about the kinetic diameter of water(Van Veen et al., 1999).

It can be anticipated that concentration and temperature polarization will be amajor problem in the application of ceramic membranes for the production ofresins. The polarization effects are schematically depicted in Figure 5.1. As water istransported through the membrane, the water concentration near the membranesurface will be low. Resin concentration will be high, as the reaction will locally bein equilibrium. Also, the temperature will be low, as water evaporates at themembrane surface adiabatically. Consequently, the viscosity near the membranewill be relatively high. As a result, a relatively thick stagnant layer will be formed,severely reducing water transport from bulk to the membrane surface.

To reduce these polarization effects, the hydrodynamics near the membranehas to be optimized. Applying high turbulence levels in the main flow field caneffectively reduce polarization effects. In laminar flows, Dean vortices can beinduced by curved polymer (Moulin et al., 1999; Winzeler et al., 1993) or ceramicmembranes (Broussous et al., 1997). Also, vibrating devices have been proposed(Vane et al., 1999).

n R1 OH + n R2 COOH Resin + H2O

Hydrodynamics in a pervaporation membrane module 89

The application of high turbulence levels is not appropriate in the resinproduction process, because we aim to develop a membrane reactor that is operatedin a once-through continuous mode. Therefore, flows have to be low in order tolimit equipment dimensions. Additionally, plug flow characteristics are required toobtain a narrow molecular weight distribution (MWD).

The objective of this study is to design a membrane module, in whichhydrodynamics is optimized for maximum water removal. For this purpose, thehydrodynamics in an annular membrane module has been investigated. Modelcalculations have been performed using computational fluid dynamics (CFD).Additionally, flux measurements have been performed in an experimental set-up.

Figure 5.1: Concentrations, temperature and viscosity inarbitrary units near the membrane surface.

Water concentration

Water

Resin concentration

Temperature

Arbitrary units

Viscosity

δ T δ C

Membrane

Figure 5.2: Schematic representation of the experimental set-up.

80°C1 bar

Recycle

v(x,r)

v(x,r)

Module

Inlet pipe Membrane Vacuum pumpOutlet pipe

T T T

Heater= Temperature sensorTTank

(10 l.)

P Mixture

Chapter 590

5.2 Experimental

5.2.1 The model system

The model system, used in this study, is depicted in Figure 5.2. The horizontalmodule consisted of a glass pipe with a length of 1650 mm, outer and innerdiameter of 50 and 40 mm, respectively. The hydrophilic membrane (supplied byECN, The Netherlands, (Van Veen et al., 1999) consists of a support tube, made ofα-Al2O3, on which on the outside an amorphous SiO2 top layer is applied with anaverage pore diameter smaller than 0.5 nm and a thickness δ of 200 nm. Porosity isestimated to be between 50 and 80%. The length of the membrane was 900 mm,with an outer and inner diameter of 14 and 8 mm, respectively. An inlet pipe, with alength of 350 mm, was positioned in front of the membrane to ensure a fullydeveloped flow. An outlet pipe (250 mm) was positioned after the membrane tominimize disturbances due to outflow effects.

Aqueous solutions of glycerin or 1,4-butanediol (Heybroek, Amsterdam,Purity > 99.7%) were fed under laminar conditions. The flow direction is parallel tothe membrane. Weight percentages of glycerin and 1,4-butanediol up to 75% wereused, in order to provide moderately viscous conditions. Reynolds numbers, asdefined by ρvsupdh/µ, were below 300.

5.2.2 Flux measurements

Flux measurements were performed at several superficial velocities andtemperatures. To evaluate changes in membrane properties, a pure water flux wasmeasured before and after changes in fluid composition. The permeate wascondensed in a cold trap with liquid N2. Permeate pressure ranged from <1 mbar to50 mbar, depending on composition and temperature.

The water flux φw through the membrane was found to depend linearly on thedriving force over the membrane (Verkerk et al., 2001):

)PP(C *

p,w*

r,ww −⋅=δ

φ [kg/(m2⋅h)] (5.1)

in which C is a constant, and δ the thickness of the top layer of themembrane. *

r,wP and *p,wP represent the equilibrium vapor pressure of water in the

retentate and permeate (vapor system), respectively. *r,wP is equal to:

°⋅⋅= www*

r,w PxP γ [Pa] (5.2)


in which γw is the activity coefficient of water in the mixture, xw is the mole fractionof water, and °

wP is the vapor pressure of the pure water. γw is calculated with the van

Laar correlation for activity coefficients. *p,wP is equaled to the absolute pressure in

the vacuum system when the separation selectivity is high. This selectivity isdefined by:

)w(w

)w(wS

r,wr,w

p,wp,w

−

−=

1

1[-] (5.3)

with ww,p and ww,r as water fraction in the permeate and retentate, respectively. Inorder to determine separation selectivity, the permeate composition was measuredwith HPLC for glycerin/water and refractometry for 1,4-butanediol/water mixtures.Karl-Fisher titration was used to determine retentate compositions.

Chapter 592

5.3 CFD simulations

Flow simulations on the model system were performed using CFX 4.2, a finitevolume fluid dynamics package (AEA Technology, Harwell, UK) which wasinstalled on a Silicon Graphics Origin 200 workstation (IRIX 6.4). To study thecontribution of natural convection in the horizontal set-up, 3D simulations havebeen undertaken on a half-tube. The grid used (90×30×25), is depicted in Figure5.3. A symmetry plane is applied in the x-z-plane, as the problem is consideredsymmetrical, thus significantly reducing computation time. In the verticalconfiguration of the model system, 2D calculations were sufficient, as the systembecomes rotation symmetrical. Therefore, the z-component was omitted in thesecases. To demonstrate the independence of the solution from the grid, thesimulations were repeated with 1.5 times the number of cells without a noticeablechange in solution.

The CFD simulations were only performed on 60/40 w% glycerin/watermixtures at 80°C. The water weight fraction (ww) is simulated as a scalar that doesnot represent any volume. All fluid properties depend on this scalar and ontemperature as given in the appendix. The membrane is simulated as a wall patch,on which a sink term is defined for the scalar. The water flux is equal to:

φw = C⋅P*w,m [kg/(m2⋅h)] (5.4)

with C as a constant, and P*w,m as the equilibrium feed-phase vapor pressure of

water at the membrane calculated using the Van Laar correlation for activity

Figure 5.3: Representation of the grid (90×30×25) of the model system. Geometricprogression of the grid was 1.1 in all directions. Distances are given in mm.

z x y

40 Outlet pipe (250)

Membrane (900)

Inlet pipe (350)

14


coefficients. Required data were obtained from Gmehling et al. (1980). C is chosensuch that φw is equal to 1.82 kg/m2⋅h based on inlet conditions. The temperature atthe membrane decreases adiabatically and is accounted for in a sink term asfollows:

φh = φw ⋅∆Hv [W] (5.5)

with ∆Hv as the heat of vaporization of water.Special attention is given to obtain a converged solution. Incorporating natural

convection results in rather unstable CFD simulations. Underrelaxation factorsdown to 0.0001 for temperature and v-velocity (parallel to gravity vector) areneeded. In addition, 5 inner iterations were performed on temperature beforechanges were implemented on the body force. Up to 40,000 iterations was neededto obtain a converged solution, requiring over three day’s simulation time for thefull 3D simulation. The upwind scheme was used in all calculations.

0

2

4

6

8

10

12

0 50 100 150 200 250 300P°

w,r-P°w,p [mbar]

φw [kg/m

2 .h]

Water (5 mm/s) Water (8.3 mm/s)

66% 1,4-Butanediol (5 mm/s) 66% 1,4-Butanediol (8.3 mm/s)

60% Glycerin (5 mm/s) 60% Glycerin (8.3 mm/s)

Figure 5.4: Water flux as a function of the difference in retentate andpermeate water vapor pressure.

Chapter 594

5.4 Results


In Figure 5.4, the flux is given as a function of the difference between theequilibrium vapor pressure of water in the retentate and permeate. In this figure, thewater flux is presented for two superficial velocities for pure water, glycerin, and1,4-butanediol mixtures. The influence of the superficial velocity on the fluxappears to be small, indicating that hydrodynamics is mainly determined by naturalconvection. The observed selectivity appeared to be close to 100%. In the cold trapneither glycerin nor 1,4-butanediol could be detected. However, selectivity wasthought to be around 99.5% as small condensed droplets were formed in the outletpipe, containing glycerin or 1,4-butanediol.

5.4.2 CFD simulations

CFD simulation in which natural convection is omitted

Initially, CFD simulations have been performed, in which natural convection isomitted, creating a 2-dimensional flow problem that is fully determined by forcedconvection. The results in Figures 5.5a-f show the relevant variables in a slicethrough the module at x = 0.2 m. The profile of the axial velocity over the annulusin Figure 5.5a is in agreement with the profile according to laminar conditions. Thewater fraction ww, temperature T, density ρ, and viscosity µ are given in Figures5.5b-e, respectively. From Figures 5.5b and 5.5c it can be observed that the waterfraction (ww) profile over the annulus is less developed than the temperature profile.Comparison of density and viscosity profiles with the water fraction profile showsthat density and viscosity are mainly determined by the water fraction, as theprofiles show a strong resemblance. Additionally, it can be seen that relativelystrong concentration and temperature polarization occurs.

Horizontal set-up with natural convection.

When natural convection is incorporated, gravity force induces complexsecondary flow as a result of density variations. In Figure 5.6, the induced flowpattern is depicted in a slice through the module at x = 0.2 m. Near the membranethe flow direction is downwards, while flow is upwards in the bulk, creating acircular flow next to the membrane. Two circular flow regions are present belowand above the membrane, resulting in a total of three circulation regions. Hattori etal. (1979) also observed three circulation zones. The highest downward velocityoccurs below the membrane tube.


Figure 5.7a-f: Axial velocity u, z-velocity v, fraction water ww, temperature T, density ρ, andviscosity µ in the simulation with natural convection (vsup = 5⋅10-3 m/s, x = 0.2m).

Figure 5.5a-e: Axial velocity u, fraction water ww, temperature T, density ρ, and viscosity µin the simulation without natural convection (vsup = 5⋅10-3 m/s, x = 0.2 m).

a: u [m/s] b: ww [-] c: T [K] d: ρ [kg/m3] e: µ [kg/(m⋅s)]

a: u [m/s] b: v [m/s] c: ww [-] d: T [K] e: ρ [kg/m3] f: µ [kg/(m⋅s)]

Chapter 596

Figure 5.8a-f: Axial velocity u, weight fraction water ww, and temperature T for thesimulations with opposing natural convection at Vsup = 0.005 and 0.1 m/s. The horizontal linesindicate the location of the membrane.

vsup = 0.005 m/s vsup = 0.1 m/s a: u [m/s] b: ww [-] c: T [K] d: u [m/s] e: ww [-] f: T [K]

Figure 5.9a-f: Axial velocity u, weight fraction water ww, and temperature T for thesimulations with adding natural convection for Vsup = 0.005 and 0.1 m/s. The horizontal linesindicate the location of the membrane.

v = 0.005 m/s v = 0.1 m/s a: u [m/s] b: ww [-] c: T [K] d: u [m/s] e: ww [-] f: T [K]

Flo

w d

irec

tion

Flo

w d

irec

tion


Figure 5.7a-f shows all relevant parameters inthe same slice as in Figure 5.5 (at x = 0.2 m).Figures 5.7c-f show that the stagnant layers, asobserved in Figure 5.5, are largely reduced. Asa result, concentration and temperaturepolarization are effectively reduced. Fluid withlow temperature and low water fraction isconcentrated at the bottom of the annulus.Consequently, this region will have a higherviscosity, slowing down the forced convectionlocally. This behavior is observed in Figure5.7a.

Vertical set-up with opposing naturalconvection

In figures 5.8a-f, the v-velocity, weightfraction water, and the temperature are given forthe vertical set-up with upward flow atsuperficial velocities vsup = 0.005 (5.8a-c) and0.1 m/s (5.8d-f), respectively. In this situation,the natural convection acts opposed to theforced convection. In figure 5.8a, the position ofthe membrane is depicted at the left. It can beseen that at low superficial velocities, a strongdownward flow is induced at the membranesurface. Flow behavior is fully determined by natural convection. As a result, thecooled mixture subsides towards the entrance. At higher superficial velocities, nodownward flow occurs. The cooled mixture is then pushed upwards to the outlet.

Vertical set-up with adding natural convection

In Figures 5.9a-f, the v-velocity, weight fraction water, and the temperature aregiven for the vertical set-up with downward flow at superficial velocities vsup =0.005 m/s (5.9a-c) and 0.1 m/s (5.9d-f). In these situations, the natural convection iscalled adding. Figure 5.9a shows that at low superficial velocities, naturalconvection accelerates the downward flow near the membrane. At higher velocities,no acceleration is observed, resulting in flow conditions that are fully determined byforced convection.

Nu and Sh numbers

For all simulations, the heat and mass transfer along the membrane can beexpressed in the dimensionless heat and mass transfer numbers Nu and Sh. Thesewere calculated using the following equations:

Figure 5.6: Secondary flow in ahorizontal positioned annulus,induced by free convection withvsup = 5⋅10-3 m/s at x = 0.2 m.

Chapter 598

[-] (5.6)

[-] (5.7)

with φw as the water flux, dh the hydraulic diameter, Dw the diffusion coefficient ofwater, ρin the inlet density, in

ww the weight fraction water at the inlet, mww the weight

fraction water at the membrane, ∆Hv the heat of vaporization, λin the inletconductivity, Tin the inlet temperature, and Tm the temperature at the membrane.

In Figures 5.10 and 5.11, the Nu and Sh numbers are plotted for four flowsituations described above: without natural convection, horizontal flow, verticalopposing, and vertical adding flow. In Figure 5.10a and 5.11a, the situations withand without natural convection are compared for the horizontal situation. Nu and Shdecline very rapidly for both situations. However, with natural convection, Nu andSh stay on a level that is 4 times higher than without natural convection. Theenhanced heat and mass transfer for this situation is evident.

The Sh and Nu numbers for the vertical situation with adding naturalconvection, in Figures 5.10b and 5.11b, resemble the situation without naturalconvection. At high superficial velocities, the numbers are comparable as forcedconvection dominates. At low superficial velocities, the numbers are considerablyhigher than in the situation without natural convection.

Table 5.1: Dimensionless groups for the three different flow situations (Pr=19.4, Sc=735).Horizontal Vertical, opposing flow Vertical, adding flow

vsup

[m/s]Re Gr

2Re

Gr

L

GrdPr h Gr2Re

Gr

L

GrdPr h Gr2Re

Gr

L

GrdPr h

(·10-5) (·10-5) (·10-6) (·10-5) (·10-6) (·10-5)

0.001 15.7 - - - 1.20 4854 6.72 1.26 5085 7.040.002 31.5 - - - 1.17 1177 6.52 1.10 1108 6.130.005 78.7 7.93 128 4.43 1.13 182 6.30 1.08 175 6.060.01 157 7.70 31.1 4.31 1.11 45.0 6.22 1.06 42.8 5.930.02 315 7.58 7.55 4.18 1.12 11.3 6.26 1.03 10.4 5.770.05 786 7.31 1.18 4.09 1.39 2.25 7.79 9.51 1.54 5.310.1 1571 7.25 0.29 4.05 1.05 0.42 5.86 8.46 0.34 4.73

For the vertical opposing situation, the Sh and Nu curves in Figures 5.10c and5.11c are similar to the adding flow situation at high velocities. For both situations,flow behavior is determined by forced convection. However, at low velocities, allSh and Nu numbers coincide as flow behavior is fully determined by naturalconvection. At vsup = 0.05 m/s, a transition takes place from natural convection toforced convection, determining the flow behavior.

The ratio Gr/Re2 is often used to estimate which flow behavior prevails. Gr isthe Grashof number representing the ratio of buoyancy and viscous forces:

))x(TT(

3600dH)x()x(Nu

))x(ww(D

3600d)x()x(Sh

minin

hvw

mw

inwinin

hw

−⋅λ⋅⋅∆⋅φ

=

−⋅ρ⋅⋅⋅φ

=


23 µρ∆ρ ⋅⋅⋅= hdgGr [-] (5.8)

Re is defined as:

µρ ⋅⋅⋅= hdvRe [-] (5.9)

In the vertical opposing flow situation, the transition takes place at vsup = 0.05m/s which corresponds to Gr/Re2 = 2.25, as given in Table 5.1. In the other twosituations, this transition is not obvious from the results presented here.Nevertheless, it can be expected that it will be at about the same number.

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x [m]

Nu(

x) [

-]

0.005

0.01

0.02

0.05

0.1

With natural convection

Without natural convection

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x [m]

Sh(

x) [

-] 0.005

0.01

0.02

0.05

0.1


With natural convection

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x [m]

Nu(

x) [

-]

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x [m]

Sh(

x) [

-]

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x [m]

Nu(

x) [

-]

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x [m]

Sh(

x) [

-]

0.001

0.002

0.005

0.01

0.02

0.05

0.1

A: Horizontal A: Horizontal

B: Vertical, adding B: Vertical, adding

C: Vertical, opposing C: Vertical, opposing

Figure 5.10a-c: Nu-numbers along themembrane for horizontal, vertical adding,and vertical opposing flow.

Figure 5.11a-c: Sh-numbers along themembrane for horizontal, vertical adding,and vertical opposing flow.

Chapter 5100

In a situation with only natural convection, the Rayleigh number Ra, asdefined by Pr·Gr·d/L, determines the transition from laminar to turbulent flow. Raexceeds 109 under turbulent conditions, which is not obtained in our configurations,as can be seen in Table 5.1. In a situation with only forced convection, turbulencecan also be obtained by applying high superficial velocities. This transition occursat Re = 2300, which is also not reached in our situation. In mixed convectionsituations, turbulent conditions can already be obtained at Re numbers around 900at Rayleigh numbers of 105, for both horizontal and vertical configurations (Metaisand Eckert, 1964). The occurrence strongly depends on the presence of secondaryflow (Hallman, 1961). Some of our simulations have been carried out at Re valuesaround 900 and higher, while a strong secondary flow is observed. Therefore, ourpresumed laminar flow may not be valid, which will be subject for further research.

Simulated fluxes

In Figure 5.12 the obtained water fluxes are plotted against superficialvelocity. It is observed that the horizontal set-up provides the highest fluxes.Natural convection increases the fluxes up to 50% at low superficial velocities.Calculations were performed down to 5 mm/s for the horizontal set-up. Below thisvelocity, the flow problem is fully determined by natural convection. Numerics thenbecome unstable, indicating that the solution will be time dependent. Therefore,these results are not included. At high superficial velocities all flow situationsbecome determined by forced convection. As a result, all fluxes tend to coincide.When the velocity will be increased further to fully turbulent conditions, with nopolarization, the maximum flux of 1.82 kg/m2⋅h will be obtained (based on inletconditions). Note that the horizontal set-up is only 23 % below this maximum. Thisemphasizes the significant contribution of the natural convection to the performanceof the membrane module.

Figure 5.12: Flux versus superficial velocity for simulations with and without natural

0.6

0.8

1.0

1.2

1.4

1.6

0.001 0.010 0.100Superficial velocity [m/s]

Flux

[kg/

m2.

h]

Horizontal


Vertical, adding

Vertical, opposing


5.5 Conclusions

This report presents the occurring hydrodynamics in a model system of amembrane reactor for resin production. Flux measurements have been performedexperimentally at different superficial velocities, temperatures, and fluidcompositions, showing that hydrodynamics is determined by forced and naturalconvection simultaneously.

CFD calculations have also been performed on the model system Thesecalculations show that the hydrodynamics is indeed determined by forced andnatural convection, simultaneously. It was shown that natural convection iseffective in reducing concentration polarization. Natural convection has beenstudied in three different configurations: horizontal, vertical adding, and verticalopposing flow. The horizontal configuration showed an increase in water flux up to50% compared to a simulation in which natural convection is omitted. Thedimensionless heat and mass transfer coefficients Nu and Sh are up to 4 timeshigher.

Nomenclature

A12,A21 binary parameters in Van Laar correlation for activity coefficients -Cp heat capacity J/(kg⋅K)dh hydraulic diameter mD diffusivity m2/sGr Grashof number -∆HV heat of vaporization J/kg∆HV0 heat of vaporization at T0 J/kgL membrane length mM molecular weight g/molNu Nusselt NumberPr Prandtl number -P° vapor pressure of pure component PaP* equilibrium vapor pressure PaRa Rayleigh number -Re Reynolds number -S separation selectivity -Sc Schmidt number -Sh Sherwood number -T temperature KTc critical Temperature Ku axial velocity m/sv velocity in z-direction m/sV molar volume m3/kmolvsup superficial velocity m/sw weight fraction -x mole fraction -

Chapter 5102

Greek

δ thickness of the top layer mφw flux of water kg/(m2⋅h)φh heat flux J/(m2⋅h)γ activity coefficient -λ heat conductivity J/(m⋅s⋅K)µ dynamic viscosity kg/(m⋅s)ρ density kg/m3

Super and subscripts

g glycerinin inlet conditionsm at the membrane surfacemix mixture of water and glycerinp permeater retentatew water

References

Bakker, W.J.W.; Bos, I.A.A.C.M.; Rutten, W.L.P.; Keurentjes, J.T.F.; Wessling, M.;“Application of ceramic pervaporation membranes in polycondensation reactions”, Int.Conf. Inorganic Membranes, Nagano, Japan, 1998, 448-451.

Broussous, L.; Ruiz, J.C.; Larbot, A.; Cot, L.; “Stamped ceramic porous tubes for tangentialfiltration”, Proc. Euromembrane '97, Twente, The Netherlands, 1997, 394-396.

David, M.-O.; Gref, R.; Nguyen, T.Q.; Néel, J.; “Pervaporation-esterification coupling: PartI, Basic kinetic model”, Trans. Inst. Chem. Eng., 1991, 69A, 335-340.

Gmehling, J.; Oncken, U.; Rary-Nies, J.R.; “Vapor-liquid equilibrium data collection,Aqueous systems”, Suppl. 2, Dechema Chemistry data series, I-1b, 1980, 182-188.

Hallman, T.M.; “Combined forced and free-laminar heat transfer in a vertical tube withuniform internal heat generation”, Trans. ASME, 1956, 78, 1831-1841.

Hattori, N.; “Combined free and forced-convection heat-transfer for fully developed laminarflow in horizontal concentric annuli (numerical analysis)”, Heat transfer Jpn. Res., 1979,8, 27.

Keurentjes, J.T.F.; Janssen, G.H.R.; Gorissen, J.J.; “The esterification of tartaric acid withethanol: kinetics and shifting the equilibrium by means of pervaporation”, Chem. Eng.Sci., 1994, 49, 4681-4689.

Koukou, M.K.; Papayannakos, H.; Markatos, N.C.; Bracht, M.; Van Veen, H.M.; Roskam,A.; “Performance of ceramic membranes at elevated pressure and temperature: effect ofnon-ideal flow conditions in a pilot scale membrane separator”, J. Membr. Sci., 1999,155, 241-259.

Metais, B.; Eckert, E.R.G.; “Forced mixed and free convection regimes”, J. Heat Transfer,1964, 86C, 295-296.

Moulin, P.; Manno, P.; Rouch, J.C.; Serra, C.; Clifton, M.J.; Aptel, P.; “Flux improvementby Dean vortices: ultrafiltration of colloidal suspensions and macromolecular solutions”,J. Membr. Sci., 1999, 156, 109-130.

Newman, A.A.; “Glycerol”, Morgan-Gampian, London, 1968.


Okamoto, K.; Yamamoto, M.; Otoshi, Y.; Semoto, T.; Yano, M.; Tanaka, K.; Kita, H.;“Pervaporation-aided esterification of oleic acid”, J. Chem. Eng. Japan, 1993, 26, 475-481.

Reid, R.C.; Prausnitz, J.M.; Poling, B.E.; “Properties of gases and liquids”, 4th ed., McGraw-Hill Book Co, 1988.

Van Veen, H.M.; Van Delft, Y.C.; Engelen, C.W.R.; Pex, P.P.A.C.; “Dewatering of organicsby pervaporation with silica membranes”, Proc. Euromembrane '99, Leuven, Belgium,1999, 209.

Vane, L.M.; Alvarez, F.R.; Giroux, E.L.; “Reduction of concentration polarization inpervaporation using a vibrating membrane module”, J. Membr. Sci., 1999, 153, 233-241.

Vargaftik, N.B.; Vinogradov, Y.K.; Yargin, U.S.; “Handbook of physical properties ofliquids and gases: pure substances and mixtures”, 3rd augm. and rev. ed., Begell Mouse,New York, 1996.

Verkerk, A.W.; Van Male, P.; Vorstman, M.A.G.; Keurentjes, J.T.F.; “Description ofdehydration performance of amorphous silica pervaporation membranes”, J. Membr.Sci., 2001, 193(2), 227-238.

Vignes, A.; “Diffusion in binary solutions”, Ind. Eng. Chem. Fundam., 1966, 5, 189.Weast, P.C.; Astle, M.J.A.; “Handbook of chemistry and physics”, 63rd edition, CRC Press,

1982.Wilke, C.R.; Chang, P.; “Correlation of diffusion coefficients in dilute solutions”, AIChE J.,

1955, 1, 264.Winzeler, H.B.; Belfort, G.; “Enhanced performance for pressure-driven membrane

processes: the argument for fluid instabilities”, J. Membr. Sci., 1993, 80, 35-47.

Chapter 5104

Appendix

Heat of vaporization of water (Reid et al., 1988).

38.0

00 /1

/1

−−

⋅∆=∆c

cVV TT

TTHH

with: ∆Hv0 = 2.26⋅106 J/kg; T0 = 298 K; Tc = 647.3 K.

Density and viscosity (Empirical correlations, based on experimental data fromNewman, 1968).

)T/lTkTji(w)w1(

)T/hTgTfe()w1()T/dTcTba(wX2

ww

2w

2w

+⋅+⋅+⋅⋅−

++⋅+⋅+⋅−++⋅+⋅+⋅=

X = ρmix

(273<T<373, 0<xw<1)X = ln(µmix⋅103)

(333<T<363, 0.09<xw<0.5)a 2114 -1.534⋅104

b -2.288 44.34c 6.75⋅10-4 -4.274⋅10-2

d -1.478⋅105 1.769⋅106

e 1767 -1504f -1.262 4.284g 2.654⋅10-4 -4.096⋅10-3

h -46056 1.784⋅105

i -581.7 3.274⋅104

j 0.2971 -94.51k 9.638⋅10-4 9.093⋅10-2

l 1.166⋅105 -3.783⋅106

Heat capacity, 273<T<373 K (Weast et al., 1982; Vargaftik et al., 1996).

gwwwmix

3426g

34275w

Cp)w1(CpwCp

T10680.1T2628.0T3.158T/10712.338368Cp

T10504.9T261.1T0.629T/10165.110355.1Cp

⋅−+⋅=

⋅⋅+⋅−⋅+⋅+−=

⋅⋅+⋅−⋅+⋅+⋅−=−

−


Heat conductivity, 273<T<623 K (Reid et al., 1988).

)()w1(w72.0)w1(w

)726/T(

)726/T1(235.0

T1037.6T1025.5384.0(

gwwwgwwwmix

61

38.0

g

263w

λ−λ⋅−⋅⋅−λ⋅−+λ⋅=λ

−⋅=λ

⋅⋅−⋅⋅+−=λ −−

Diffusion coefficients of water in mixture (Wilke et al., 1955; Vignes, 1966).

2

6.0

16

6.0

16

)1(2112

)1(2121exp

)ln/ln/(

5.110173.1

26.210173.1

00

−⋅+⋅

−⋅⋅=

⋅=

⋅

⋅⋅⋅⋅=

⋅

⋅⋅⋅⋅=

−

−

ww

ww

wwx

wg

x

gww

wg

g

wg

gw

wgw

xAxA

xAA

xddDDD

V

MTD

V

MTD

gw

o

o

γ

γ

µ

µ

with: Mw = 18 g/mol, Mg = 92.1 g/mol, Vg = 0.096 m3/kmol; Vw = 0.0756m3/kmol, A12 = 0.2305, A21 = -0.6978.

MEASUREMENT OF 2D-TEMPERATUREDISTRIBUTIONS IN A PERVAPORATION

MEMBRANE MODULE USINGULTRASONIC COMPUTER TOMOGRAPHY

Abstract

Temperature polarization effects can considerably limit theoverall performance of pervaporation modules. This study shows themeasurement of temperatures in a pervaporation membrane moduleusing Ultrasonic Computer Tomography. With this technique,complete 2D temperature distributions that occur in the module can bemeasured within seconds. Temperature distributions were alsodetermined using Computational Fluid Dynamics. Both temperaturedistributions are comparable, and show that in the tubularpervaporation system studied, temperature segregation occurs. Thissegregation is a result of the occurring hydrodynamics in the systemthat is determined by mixed convection. Also, the measured waterfluxes through the membrane show to be dependent of forced andnatural convection.

This chapter is a slightly modified version of the publication:Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Measurement of 2D-TemperatureDistributions in a Pervaporation Membrane Module using Ultrasonic ComputerTomography and Comparison with Computational Fluid Dynamics Calculations”, J. Membr.Sci., 2002, 204, 111-124.

6

Chapter 6108

6.1 Introduction

During the past decades, pervaporation has emerged as a promising separationtechnique. Using hydrophilic membranes, water can be removed from organicmixtures, whereas using hydrophobic membranes, organic molecules can beremoved from aqueous solutions (Ho et al., 1992; Nunes et al., 2001). In both cases,concentration and temperature polarization occur, which can limit the feasibility ofthe separation.

Temperature polarization occurs as a result of the energy required forevaporation of the compound removed. As the phase change occurs in themembrane, the temperature of the membrane will be lower than the bulk liquidtemperature. In a heat transfer study, Rautenbach and Albrecht (1984, 1985) havereported on these temperature effects during pervaporation by measuringtemperatures of feed, retentate, and permeate. More recent, heat transfer has beenstudied by Karlsson and Trägårdh (1996), indicating that in some cases temperaturepolarization is even rate limiting for the pervaporation process.

Using CFD (Computational Fluid Dynamics), we have recently (Van der Guliket al., 2001) investigated temperature and concentration polarization effects in aceramic pervaporation membrane reactor for resin production. In this process,polarization effects are thought to limit the feasibility of pervaporation severely, astemperature polarization will lead to strong viscosity gradients, thus hamperingmass transfer. We have observed that under laminar conditions, natural convectioninduces secondary flow, which reduces the thickness of the hydrodynamic layers.As a result, the polarization effects are effectively reduced, leading to a fluxincrease up to 50% compared to cases without natural convection. Other secondaryflows are Dean vortices, that can be induced by curved membranes, which hasshown to be feasible for polymer (Winzeler et al., 1993; Moulin et al., 1999) andceramic mem-branes (Broussous et al., 1997).

In this paper, we report on experimentally determined temperaturedistributions occurring during pervaporation. We have implemented theexperimental technique U-CT (Ultrasonic Computer Tomography) in thepervaporation system, as previously used to study hydrodynamics under laminarconditions (Van der Gulik et al., 2001). U-CT is based on the dependence of thepropagation velocity of ultrasound on the temperature of the medium. Themeasurements provide line average temperatures that can be reconstructed to a 2D-temperature distribution by using computerized tomography. The technique hasbeen used to measure temperature distributions in solids (Norton et al., 1984; Peyrinet al., 1983), gases (Mizutani et al., 1997, 1999; Funakoshi, et al., 2000; Wright etal., 1998), and liquids (Beckford, 1998; Basarab-Horwath et al., 1994). The U-CTtechnique is non-invasive, and can be used for real time temperature measurementsin 2D, with fast computers nowadays. Background on the technique is given in theexperimental section.

Measurement of 2D-Temperature distributions using U-CT 109

This paper will demonstrate the successful implementation of the U-CTtechnique for the measurement of temperature profiles during pervaporation.Additionally, the measurements are used to validate CFD calculations.

Chapter 6110

6.2 Experimental

6.2.1 The pervaporation system

The pervaporation system, as depicted in Figure 6.1A, was an annulusconfiguration. The outer glass tube had a length of 1650 mm, and an inner diameterof 40 mm. The wall thickness was 5 mm. The inner tube consisted of three parts: aglass inlet pipe, the membrane, and a glass outlet pipe. The hydrophilic membranewas supplied by ECN, The Netherlands, and consisted of a support tube, made of α-Al2O3 on which an amorphous SiO2 top layer was applied, with an average porediameter smaller than 0.5 nm, and a thickness δ of 200 nm (Van Veen et al., 1999).Porosity was estimated to be between 50 and 80%. The length of the membrane was900 mm, with an outer and inner diameter of 14 and 8 mm, respectively. The inletand outlet pipes had lengths of 110 and 680 mm, respectively. The outlet pipe washollow, and provided the connection between the vacuum pump and the inside ofthe hollow membrane. The complete inner tube could be translated in the axialdirection between two extreme positions as depicted in Figures 6.1A and 6.1B.

In this study, demineralized water was used as the working fluid, which was indirect contact with the membrane. The water temperature in the storage tank was setto 313 K. The inlet and outlet temperature of the liquid depended on operatingconditions. The ambient temperature TA was monitored continuously. U-CTmeasurements were performed with and without a vacuum applied. In case ofvacuum, the permeate was condensed in a cold trap with liquid N2. The permeatepressure was around 1 mbar, and the flux was determined gravimetrically.

MembraneVacuum pumpOutlet pipe

Liquid recycleT T

Heater = Temperature sensorTTank(10 l.)

P900 mm855 mm

Module

900 mm390 mm

Block withtransducers

A

B

Inlet pipe

Figure 6.1: Set-up of the pervaporation system during a U-CT-measurement at the end ofthe membrane (Case A, with the block at x = 855 mm) and during a U-CT-measurementhalfway the membrane (Case B, with the block at x = 390 mm).


In Figure 6.2, the membrane module is represented schematically, in which Frepresents volume flows [m3/s], T temperatures [K], and Q energy flows [J/s].Without vacuum, there is no evaporation, so that Fvap = 0 and Qvap = 0. Then, thefollowing energy balance over the system can be set up:

0=++ lossoutin QQQ [J/s] (6.1)

As the in- and outlet temperatures are known, the energy losses to the environment,Qloss, can be calculated:

( )inoutinploss TTFcQ −= ρ [J/s] (6.2)

with ρ the density [kg/m3] and specific heat capacity cp [J/(kg⋅K)] of water. Whenapplying vacuum, the energy balance becomes:

0=+++ vaplossoutin QQQQ [J/s] (6.3)

Qloss can be estimated, by assuming that the mass flow at the inlet is equal to themass flow at the outlet (i.e., |ρF|in≈|ρF|out>>|ρF|vap) and using the known inlet andoutlet temperatures:

( ) vapvapinoutinploss FHTTFcQ ∆ρρ −−= [J/s] (6.4)

with ∆Hvap the heat of vaporization for water [J/kg]. The energy flows Qloss and Qvap

have been calculated for each experiment, and are tabulated in Table 6.2. Thesedata are also required as input for the CFD calculations.

Qloss

Fin Fout

Tin membrane Tout

Qin Qout

Fvap Qvap

Figure 6.2: Schematic representation of the pervaporation system indicating the energyflows and the mass flows.

Chapter 6112

2.2 U-CT Method

U-CT measurements are based on the measurement of time-of-flight (TOF) ofsound between a speaker and a microphone. The TOFL over a distance L is givenby:

dl)(v

TOFL

L ∫= ρ1

[s] (6.5)

in which v(ρ) is the sound velocity, that depends on ρ. The velocity depends ondensity, being a function of temperature and composition (and pressure when gasesare applied, which is currently not the case). As only water is used, the soundvelocity is only a function of temperature T (Suurmond, 1998):

( ) ( )59-46-

3-42

T103.0449T101.45262-

T103.31636T0.0579506-T5.033581402.336Tvv

⋅⋅+⋅⋅

⋅⋅+⋅⋅+==ρ[m/s] (6.6)

with T in °C. From equation 6.6 the average line temperature can be derived. InFigure 6.3, v(T) and dv/dT are shown as a function of temperature, in which dv/dTdetermines the accuracy of the measurements. From the figure, it can be seen thatv(T) has a maximum at 346K (73°C). Obviously no measurements can beperformed when dv/dT = 0.

For speaker and microphone, in-house made transducers were applied in whichpiezo-electric elements were mounted. The transducers were placed in an aluminumblock. The block was placed between the two glass tubes that together made up themembrane module. The block could rotate independently from the glass tubes. The

1400

1420

1440

1460

1480

1500

1520

1540

1560

273 293 313 333 353 373

T [K]

v [m

/s]

-2

-1

0

1

2

3

4

5

6

0 20 40 60 80 100

T [°C]

dv/d

T [

m/s

K]

Figure 6.3: Sound velocity and its derivative, plotted against temperature.


fluid under study flowed through the block from one glass tube to the other, whilethe transducers were in direct contact with the fluid. The complete construction isshown in Figure 6.1A.

The peripheral equipment, needed for the U-CT measurements, is depicted inFigure 6.4. The data acquisition system controlled and synchronized all actions. Thepulse generator (Synthesized Function Generator Yokogawa FG120 / 2 MHz)generated a sound pulse for the speaker with a frequency of 1.2 MHz. The signal, ascollected by the microphone, was depicted on an oscilloscope (LeCroy 9314MQuad 300MHz, 100 Ms/s, 50kbps/Channel), and was transferred to the PC thatprocessed the signal. A typical incoming signal is given in Figure 6.5A. Post-processing started with a 40 degree low (1.1 MHz) and a high (1.3 MHz)Cheybyshev bandpass filter, thus focussing on the speaker frequency of 1.2 MHz.This resulted in the signal as depicted in Figure 6.5B. Then, a Hilbert transform wasapplied that enveloped the absolute signal. The TOFL was chosen to be the time thatthe signal level was equal to 20% of the maximum of the enveloped signal, asdepicted in Figure 6.5C. The obtained time was not the real TOFL, as a constanterror was introduced, which was removed by calibration. Wakai et al. (1990) have

M

embrane

Transducers

Sound waves

ModuleOscilloscope

Data acquisitionsystem

Pulse generator

Figure 6.4: Required peripheral equipment for the U-CT-measurements.

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -tp

10 µs

0 time

A B C

I [-]

0 time 0 time

I /Imax

Figure 6.5: Signal processing in three steps. A) The incoming signal, obtained from theoscilloscope. B) The signal after the Cheybyshev filter. C) The enveloped signal with thededicated time at 20% of the maximum in absolute signal.

Chapter 6114

studied different methods for determining the TOFL, which showed that, whenusing the enveloped signal, the most accurate results could be obtained. Refractioneffects introduced an error that could not easily be removed. Estimating calculationsshowed that the error was at least one order smaller than the errors generated bynoise.

In Figure 6.6A, the location of the transducers in the block, relative to themembrane, is given in more detail. A 3D representation of the x, y, and z directionsis given in Figure 6.8. The transducers had a diameter of 5mm. In z-direction, 8transducers were evenly distributed over 7 mm, whereby the top of transducer 8matched the underside of the membrane. In x-direction, the transducers were evenlydistributed over 28 mm. Although the transducers differed in x-coordinate, forreconstruction of the 2D temperature distribution it was presumed that thetransducers coincide at x = 14 mm. A 3D representation is given in the photographin Figure 6.6B.

Only the rays that are depicted inFigure 6.4 were used. It was also possibleto use oblique rays up to two transducersto the left or to the right, but these rayswere not used, as the density of the givenrays was sufficiently high for recon-struction purposes. By rotating the blockindependently from the glass tubes andthe membrane, the straight rays coveredthe complete section. It was chosen toperform measurements after each rotationover 30°. So, for completion of the total360°, 12 rotations were needed, providinga ray distribution as depicted in Figure6.7.

Calibration of the system wasrequired, as the exact distance between

A B

5

Figure 6.6: A) Location of 8 transducers relative to the membrane with distances in mm. B)A photograph representing the location of the 16 transducers in three dimensions.

Figure 6.7: Ray distribution after rotationof the block over 360° in 12 steps.


transducers was unknown. Therefore, TOFL measurements at a constant andhomogeneous temperature were performed. To obtain this situation, the flow andheating were turned off, and the apparatus was maintained at rest for 24 hours.During calibration, the temperature of the liquid and the environment weremonitored continuously within ± 0.1 K. Using TOFL and the measured temperature,the exact distances between the transducers could be calculated using equations 6.5and 6.6.

The complete inside tube (glass tube – membrane – glass tube) could bemoved in axial position, relative to the U-CT recorder and the outside tube, thusallowing to perform U-CT measurements at several positions along the membrane.The U-CT measurements, as presented in Figure 6.1, were performed at the end ofthe membrane (x = 855 mm, Figure 6.1A) and approximately halfway themembrane (x = 390 mm, Figure 6.1B). The experimental set-up was used with andwithout vacuum. Applying two flow rates, 300 and 500 mL/min, 6 situations werestudied as summarized in Table 6.1. For each TOFL measurement, the average of 10consecutive measurements was taken in order to reduce noise contribution.Standard deviations will be presented as a result. A TOFL measurement then tookabout 10 seconds. With 8 rays, the total measurement time of one angle took 80seconds. After rotation of the block over 30°, the system set-up was given 300seconds to stabilize before the next TOFL measurement was performed.Consequently, the measurement of a complete section took 360°/30°∗(300+80) =4560 seconds.

Table 6.1: Conditions of experiments 1 to 6.

Flow [mL/min] Condition Fig. 1 x-coordinate [mm] Experiment no.No vacuum A 855 1

A 855 2300Vacuum

B 390 3No vacuum A 855 4

A 855 5500Vacuum

B 390 6

For reconstruction of the 2D-temperature distribution it was assumed that allmeasurements were performed simultaneously. This implies, that we assumed thecomplete system to be at the equilibrium state during the complete U-CTmeasurement. Based on the inlet, outlet and ambient temperatures this assumptioncould hold as these remained constant within 0.1K.

2.3. Reconstruction

The experiments provided 96 rays, that form the basis for reconstruction of thetemperature distribution. The experimental set-up was symmetrical around thevertical centerline, which allowed for averaging of symmetrical measurements.

The reconstruction method used, was filtered back-projection. The temperaturedistribution that had to be reconstructed, is infinitely-dimensional, and there were

Chapter 6116

only 96 measurements, indicating that the problem was severely underdetermined.Therefore, regularization techniques were needed to obtain numerical stability. Asuitable method was a collocation method in which the Moore-Penrose generalizedinverse of a large matrix was computed (Suurmond, 1998; Golub et al., 1983).

A temperature distribution with circular shape had to be constructed. Thisshape was built up in a rectangular grid with 7854 cells. The temperatures in thesecells were restricted by using linear independent basis pictures pi. In this case, thenumber of basis pictures was equal to the number of measurements, i.e., 96. Theanticipated temperature distribution was built up by these 96 pictures. The basispictures were designed to take into account the fact that the speed of sound couldnot vary too much in space. One choice of basis pictures that was in agreement withthis, were smooth ‘hills’, as modeled by negative exponential functions:

( )2||xx||exp)x(p ii −⋅−= λ (6.7)

where the xi were positioned in a regular grid, that covered the reconstruction regionaround position x. i is an index, ranging from 1 to the number of basis pictures (=96). λ is the damping factor of the basis functions. The use of these basis pictures pi

provided much better results than using the original ART methods (Herman, 1980).Norton et al. (1984) showed that Bessel-functions are also good candidates forobtaining smooth distributions.

From the anticipated temperature distribution, as built up by the basis pictures,a matrix R could be constructed, consisting of the back-projection of discrete raytransforms of the basis pictures:

( )( )ijiij ,xpR θΡ= (6.8)

in which P was a linear operator, representing the back projection pj at position(xi,θi). j Is an index ranging from 1 to the number of rays (= 96). Now the problemcould be represented as a matrix equation: given measured data y, find a vector ξsuch that Rξ best approximates the measured data y:

Rξ = y (6.9)

In other words, find the solution vector ξ by minimizing:

||yR|| −ξ (6.10)

for which SVD (Singular Value Decomposition) was used, including the Moore-Penrose generalized inverse matrix (Suurmond, 1998; Golub et al., 1983)..

The solution ξ could be spoiled, because of small singular values, that arose inthe system matrix of the tomography problem. As an efficient approach toovercome this problem, a filter was used that modified the singular values, such thatthe resulting matrix did not have any near-zero singular values. A soft Tichonov


regularization filter was applied with filter parameter η. Using this parameter, thesolution ξ was computed by minimizing the following function:

22 ||||||yR|| ξηξ +− (6.11)

In this way, a compromise was obtained between accuracy and smoothness:large values of η would result in smoother reconstructions, while smaller values ofη would cause Rξ to follow the data y more closely. In this paper a value of 0.001was used. The choice was based on the maximum values obtained in the legend ofFigure 6.13A, i.e., 311.46 and 304.16K, respectively. These values had to match theextremes in vector y. This matching introduced an error, as the ‘real’ maximum wasalways higher or equal to the maximum ray-averaged temperature. Smaller valuesfor η could compensate for this, but it was found that smaller values causedspurious high-frequency oscillations in the obtained temperature distribution.

The temperature distributions were reconstructed for a complete circle, assurrounded by the glass wall. The circle included the inner area that is covered bythe membrane, although no U-CT measurements could be performed in that area.Reconstruction was performed here, because it was impossible to implement aboundary condition in the reconstruction software that describes the temperature onthe membrane. Also, no values were available beforehand to set up a boundarycondition.

2.4. Computational Fluid Dynamics

Flow simulations of thepervaporation system wereperformed to obtain a refer-ence for the U-CT experi-ments. CFX 4.2 was used forthis purpose, which is a finitevolume fluid dynamics pack-age (AEA Technology, Har-well, UK). The software wasinstalled on a Silicon Graph-ics Origin 200 workstation(IRIX 6.4). To study thecontribution of natural con-vection in the horizontal set-up, 3D simulations were per-formed on a half-tube. TheCartesian grid (74×30 ×35),is depicted in Figure 6.8. Asymmetry plane was appliedin the x-z-plane, as theproblem was considered

z

x

y

Inlet pipe(110 mm)

Membrane (890 mm)

Outlet pipe (680 mm)

14 mm

40 mm

Figure 6.8: Representation of the grid (74×30×35) ofthe pervaporation system in position A, as described inFigure 6.1. From the inner tube, only the surface gridof the membrane is depicted.

Chapter 6118

symmetrical, thus significantly reducing computation time. To demonstrate theindependence of the solution from the grid, the simulations were repeated with 1.5times the number of cells, without a noticeable change in solution.

The membrane was simulated as a wall patch, on which a sink term wasdefined that represented the water flux at arbitrary numeric cell i (φw,i) as follows:

∗⋅= i,wi,w PCφ [kg/(m2⋅s)] (6.12)

with C as a constant and P*w,i as the feed-phase vapor pressure of water at the

membrane. Required data for the equation of P*w,i were obtained from Reid et al.

(1988). C was chosen such, that the sum of φw,i was equal to the experimentallyobserved flux φw, thus:

mw

n

iii,w AA φφ 3600

1

=∑=

[kg/s] (6.13)

with Ai the membrane area of numeric cell i, and Am the total membrane area. Theamount of energy required for evaporation was accounted for in a sink term asfollows:

vapi,wi,h H∆φφ ⋅= [J/(m2⋅s)] (6.14)

with ∆Hvap in J/kg. Viscosity, heat capacity, heat of vaporization, and conductivityof water were assumed to be temperature-independent.

Heat losses to the environment were also taken into account, becauseexperiments showed that losses were significant. On the outside wall a combinedDirichlet-Neumann boundary condition was defined as follows:

( )iAii TThAQ −= [J/s] (6.15)

in which Qi was the heat loss in a single numeric cell, h the lumped heat transfercoefficient, Ai the cell area, TA the ambient temperature, and Ti the cell temperature.In a converged solution, the experimentally determined total heat loss Qloss, was setequal to the sum of the heat losses in all outer cells:

( ) loss

n

iiAi

n

ii QTThAQ =−=∑∑

== 11

[J/s] (6.16)

h was kept constant for the complete tube. The ambient temperature TA was knownfrom measurements.


6.3 Results and discussion


The relevant experimental settings and the measured and calculated quantitiesduring the flux measurements are given in Table 6.2. The calculated Reynoldsnumber indicates that conditions are laminar. The measured inlet temperature at 300mL/min is 0.4 K lower than at 500 mL/min, as losses from the feed tube towardsthe environment are larger at lower flow speed. The outlet temperature is lower thanthe inlet temperature, also as a result of energy losses towards the environment. Theenergy loss over the membrane module, Qloss, can be calculated from the differencein inlet and outlet temperature, and Fin, using equation 6.2.

Table 6.2: Settings, measured quantities, and calculated energy flows for the 6 experimentsperformed.

Set Measured Calculated

ExpNo.

Fin

[mL/min]vsup

[mm/s]Re[-]

VacuumTin

[K]Pvac

[mbar]φw

[kg/m2⋅h]Tout

[K]Qin-Qout

[W]Qvap

[W]-Qloss

[W]

1 No - - 309.2 50.4 - 50.4

2&3300 4.53 118

Yes311.6

1 2.22 307.0 96.6 53.7 42.9

4 No - - 310.4 57.0 - 57.0

5&6500 7.56 196

Yes312.0

2 2.77 308.6 119 63.0 56.0

When the vacuum is applied, water is evaporated at the membrane. Theobtained water flux, φw, appears hardly to be affected by the superficial velocityvsup, indicating that hydrodynamics are mainly determined by natural convection.The fluxes are in good agreement with results reported previously (Van der Gulik etal., 2001). The permeate pressure at 500 mL/min was found to be twice as high ascompared to 300 mL/min. Based on the results presented by Rautenbach andAlbrecht (1980), the permeate pressure should be less than 25% of the vaporpressure to eliminate the influence of the permeate pressure on the water flux.Meanwhile, the vapor pressure is determined by the temperature of the membrane,which is unknown. However, assuming a membrane temperature equal to 293 K,the vapor pressure is calculated to be 23 mbar. Consequently, no effect of thepermeate pressure is expected.

The losses towards the environment with the vacuum applied, appear to belower than the losses without vacuum applied. This is because the averagetemperature in the module is lower, resulting in a lower driving force for heattransfer towards the environment.

Chapter 6120

6.3.2 CFD simulations

CFD simulations have been performed on the 6 cases, that are summarized inTable 6.1. Using the measured water and energy fluxes in the boundary conditionsas described in equations 6.12-6.16, secondary flow patterns as given in Figure 6.9are obtained. In Figure 6.9A, the secondary flow pattern is given for 300 mL/min atx = 855 mm without vacuum. Due to heat losses towards the environment, theoutside wall temperature is lower than the bulk temperature. Consequently, thedensity is higher, inducing a downward velocity at the wall, and an upward velocityin the bulk near the membrane, creating one large cell at the top of the annulus, asmarked with ‘1’. The temperature distribution at the same x-coordinate (x = 855mm) is given in Figure 6.10A. The figures show that fluid with low temperature isexpected to be collected at the bottom of the annulus.

When vacuum is applied, the temperature at the membrane also drops.Although the energy flux as a result of the evaporation is comparable to the lossesto environment (compare Qvap and Qloss in Table 6.2), the temperature drop at the

1

2

1

z

y

g

z

y

g

A) Experiment 1 B) Experiment 2

Figure 6.9: Secondary flow pattern predicted by CFD for experiment 1 (no vacuum, Fin =300 mL/min, x = 855mm) and experiment 2 (vacuum, Fin = 300 mL/min, x = 855mm).


membrane is larger, because the membrane area is substantially smaller than theoutside wall. As a result, the maximum temperature difference over the annulus is 4K without vacuum (Figure 6.10A), while a maximum temperature difference of 9 Kis found when the vacuum is applied (Figure 6.10C). Because the temperaturedifferences are larger, the density differences are also larger. Consequently, theinduced secondary flow around the membrane is stronger than at the outside wall.As both induced secondary flows are pointed downward, they suppress each otherin the annulus. Therefore, the cell at the outside wall in Figure 6.9A is reduced inFigure 6.9B (marked with ‘1’), but a second cell is formed at the bottom (markedwith ‘2’). Both cells provide local mixing at the top and the bottom, respectively,but overall mixing is poor. As a result, a large temperature difference is obtainedbetween the top and the bottom of the annulus as can be seen in Figures 10C and10D.

6.3.3 U-CT measurements

In order to determine the calculated temperature profiles experimentally,Ultrasonic Computer Tomography measurements have been performed. Firstly, theaccuracy of the U-CT measurements will be discussed. Secondly, the performanceof the reconstruction software will be demonstrated, by using average raytemperatures extracted from a CFD calculation. Finally, the experimental U-CTmeasurements are reconstructed into a 2D-temperature distribution, demonstratingthe capabilities of the technique.

To study the noise level of our equipment, standard deviations in temperaturehave been determined. These deviations follow from 10 consecutive calibrationmeasurements, in which the distance between the 8 speaker-microphonecombinations has been determined at constant temperature. The determineddeviations in distance can be converted into deviations in temperature usingequations 6.5 and 6.6. Using equation 6.6, the deviations are extrapolated to thehigher temperatures that are relevant in our experiments. The deviations are givenin Figure 6.11 for all 8 speaker-microphone combinations. At 313 K, the deviationsare between 0.3 and 0.7 K, and at 298 K between 0.2 and 0.4 K, respectively.Therefore, using water, lower noise levels can be obtained at low temperatures. Thedeviations do not correlate with the distance between speaker and microphone andare thought to be introduced by the transducers themselves. The deviations definethe lower limit in measurable temperature differences.

To test the performance of the reconstruction procedure and the value for theparameter η of the Tichonov regularization filter, a reconstruction has been made ofthe temperature distribution in Figure 6.10C that have been calculated using CFD.This CFD calculation corresponds to experiment 2, at 300 mL/min, with thevacuum applied (see Table 6.1 and 2 for details). For this purpose, the averagetemperatures over imaginary rays in the temperature distribution in Figure 6.10Chave been determined. These average temperatures are presented in Figure 6.12A asa function off the rotation angle. At 0°, the rays cover the bottom of the module,while at 180° the rays cover the top of the module. These ray temperatures arereconstructed to a temperature distribution again that is given in Figure 6.13A. The

Chapter 6122

reconstruction performs according to the correspondence with Figure 6.10, althoughsome finer structures in the temperature distributions near the membrane are misseddue to the absence of rays near the membrane.


A B C DT

empe

ratu

re [

K]

Figure 6.10: Temperature distributions predicted by CFD for A) Experiment 1 (no vacuum,Fin = 300 mL/min, x = 855mm), B) Experiment 4 (no vacuum, Fin = 500 mL/min, x =855mm), C) Experiment 2 (vacuum, Fin = 300 mL/min, x = 855mm), and D) Experiment 5(vacuum, Fin = 500 mL/min, x = 855mm).

0.2

0.3

0.4

0.5

0.6

0.7

295 300 305 310 315T [K]

σT [

K]

1 2

3 4

5 6

7 8

Figure 6.11: Standard deviation in the temperature measurements at 25 °C for the 8transducer combinations and extrapolated towards higher temperatures. The numbers fromone to eight refer to the transducer numbers, that are given in Figure 6.6.

Chapter 6124

311.46311.15

310.15

309.15

308.15

307.15

306.15

305.15

304.16

311.00

a 309.15

307.15

305.15

303.15

301.15

299.15

297.15296.95

A T [K] B T [K]

Figure 6.13: A) Reconstruction of the ray temperatures in Figure 6.12A, derived from theCFD-calculation in Figure 6.10C. B) Reconstruction of measured ray temperatures in Figure6.12B. Both figures correspond with experiment 2 (vacuum, Fin = 300 mL/min, x = 855mm).

309.13

a 307.15

305.15

303.15

301.15

299.15

297.15296.94

309.06

308.15

307.15

306.15

305.15

304.15

303.15

302.15

301.15

300.15

299.15

298.48

A T [K] B T [K]

Figure 6.14: Reconstructed images of the ray temperatures at x = 390 mm in A) experiment 3(vacuum, Fin = 300 mL/min) and B) experiment 6 (vacuum, Fin = 500 mL/min).

300

302

304

306

308

310

312

0 90 180 270 360Position [°]

Tem

pera

ture

[K

]

1

2

3

4

5

6

7

8300

302

304

306

308

310

312

0 90 180 270 360Position [°]

Tem

pera

ture

[K

]

1

2

3

4

5

6

7

8

Figure 6.12: A) Ray temperatures for a CFD calculation, set-up according to experiment 2.B) Measured ray temperatures for experiment 2 (vacuum, Fin = 300 mL/min, x = 855mm).

A B


The highest and the lowest values in the legends of Figures 10C and 13A are usedto check the previously determined parameter η. Using η = 0.001 the match appearsto be good, although the highest and the lowest values in the legend of thereconstructed image in Figure 6.13A are higher than in the original image in Figure6.10C. This can be attributed to the pronounced extremes at the bottom, and at thetop of the annulus in the CFD calculation.

In Figure 6.12B, the averaged temperatures of U-CT measurements are givenfor experiment 2 (300 ml/min, vacuum, x = 855 mm). The experimental conditionsare equal to the CFD calculation in Figure 6.12A. At 0°, the rays measure a lowertemperature than at 180°, which corresponds to the CFD calculations. Overall, themeasured temperatures are lower than in the CFD calculation. The values of theCFD distribution are thought to represent the experiments better, as the calculationswere fitted on the macroscopic values that are given in Table 6.2.

Figure 6.13B shows the reconstruction of the temperatures in Figure 6.12B.Figure 6.13B clearly shows a region with a low temperature below the membrane,and a region with a high temperature above the membrane. This segregationcorresponds to the CFD calculations in Figure 6.11C and 13A. The temperatures atthe top of the annulus (311 K) and at the bottom glass wall (304 K), correspondvery well. The main difference between the reconstructed CFD calculation, andreconstructed U-CT measurement is the temperature just below the membrane. TheCFD calculation in Figure 6.10C predicts 302 K, while the reconstruction in Figure6.13B yields 297 K. We assume that reconstruction shows a too low temperature.The temperature is the outcome of a balance between the ray temperatures in thereconstruction, and the ray temperature of the measurement according to equation6.11. Probably the value η = 0.001 for the Tichonov-filter is too strict, leading to alocal unbalance. On the other hand, the value of η = 0.001 seems adequate, as thetoo high temperatures of ray 7 in Figure 6.12B do not return in the reconstruction asthey are distributed over the image by the Tichonov regularization filter.

The reconstructions of U-CT experiments 3 and 6 are given in Figures 6.14Aand 6.14B (x = 390 mm, vacuum, 300, and 500 mL/min, respectively). As expected,the temperature distributions are similar, but overall, the temperatures at 300mL/min are lower than at 500 mL/min. Compared to the CFD calculation, thetemperatures are about 2 K lower. At 300 mL/min, the temperatures at the bottomof the module are lower than at 500 mL/min, which is as expected, since at 300mL/min natural convection determines hydrodynamics to a larger extent than at 500mL/min (Van der Gulik et al., 2001). Overall, the temperature distributions areadequate. The distributions, as well as the maximum temperatures, correspond withthe CFD calculations. Also the occurrence of natural convection, as predicted by theCFD calculation (Van der Gulik et al., 2001), has been shown to exist.

Reconstruction of the ray temperatures of the experiments without vacuum(experiments 1 and 4), did not result in realistic temperature distributions. Theoccurring temperature differences in the module are of the same order of magnitudeas the standard deviations that are given in Figure 6.8. Thus, the noise level hasbeen too high. Consequently, to be able to perform U-CT measurements without thevacuum, the technique has to be optimized further, with emphasis on noisereduction and the determination of TOFL. Wakai et al. (1990) have concluded that

Chapter 6126

the enveloped signal after Hilbert transformation provides the best results fordetermining the TOFL. Analogously, we have chosen the TOFL to be the time thatthe signal is equal to 20% of the maximum of the enveloped signal. However, adetailed study of the signals has shown that declination can occur when themaximum decreases. Therefore, future developments should include an improveddetermination method, probably using a sweep signal that shifts the speakerfrequency from 1.2 to 1.4 MHz. After applying the Cheybyshev-filter on themicrophone signal, the speaker frequency shift will be visible. Then by a uniqueoverlap of both signals, the TOFL is expected to be determined straightforward.


6.4 Concluding remarks

This study illustrates the implementation of U-CT measurements for themeasurement of temperature distributions in pervaporation modules. The measuredtemperature distributions agree well with the CFD calculations. The temperaturedistributions prove the occurrence of natural convection in the pervaporationmembrane module, as previously predicted by CFD calculations.

Improving the experimental technique by reducing the noise level can broadenthe application to larger and more complex systems, revealing temperaturedistributions inside pervaporation modules with multiple tubes. Also, because inprinciple the U-CT technique is based on density differences, it should be possibleto measure composition differences. Moreover, the measurements can be performedin various types of equipment, provided density differences are present over thesampling plane. Reducing measurement times can extent the application to dynamicprocesses, even with prospects in control systems.

Nomenclature

A area m2

C constant -cp specific heat J/(kg⋅K)F volume flow m3/sh overall heat transfer coefficient W/(m2⋅K)n number of numeric cells -P pressure PaP* vapor pressure Papi basis picture -R matrix of back-projected ray transformsRe Reynolds number -Q energy flow J/sT temperature KTOFL time of flight over length L sv velocity m/sx vector position of the basis picturexi vector position of points around x

Greek

δ thickness of the top layer on the membrane m∆Hvap heat of vaporization J/kgφw water flux kg/(m2⋅h)φh energy flux J/(m2⋅h)η Tichonov Regularization filter parameter -λ damping factor of the basis pictureξ solution vectorρ density kg/m3

Chapter 6128

Subscripts

A ambienth heatI ith cellin inletloss losses to environmentout outletsup superficial velocityvap vaporizationvac vacuumw water

References

Basarab-Horwath, I.; Dorozhevets, M.M.; “Measurement of the temperature distribution influids using ultrasonic tomography”, IEEE Ultrasonics Symposium, Cannes, 1994, 1891-1894.

Beckford, P.; Höfelmann, G.; Luck, H.O.; Franken, D.; “Temperature and velocity flowfields measurements using ultrasonic computer tomography”, Heat Mass Transfer, 1998,33, 395-403.

Broussous, L.; Ruiz, J.C.; Larbot, A.; Cot, L.; “Stamped ceramic porous tubes for tangentialfiltration”, Proc. Euromembrane '97, Twente, The Netherlands, 1997, 394-396.

Funakoshi, A.; Mizutani, K.; Nagai, K.; Harakawa, K.; Yokoyama, T.; “Temperaturedistribution in circular space reconstructed from sampling data at unequal intervals insmall numbers using acoustic computerized tomography”, Jpn. J. Appl. Phys., 2000, 39,3107-3111.

Golub, G.H.; Van Loan, C.F.; “Matrix computations”, North Oxford Academic, 1983.Ho, W.S.W.; Sirkar, K.K. (Eds.); “Membrane Handbook”, Chapman & Hall, New York,

1992.Herman, G.T.; “Image reconstruction from projections. The fundamentals of computerized

tomography”, Academic Press, Inc., 1980.Karlsson, H.O.E.; Trägårdh, G.; “Heat transfer in pervaporation”, J. Membrane Sci., 1996,

119, 295-306.Mizutani, K.; Nishizaki, K.; Nagai, K.; Harakawa, K.; “Measurement of temperature

distribution in space using ultrasound computerized tomography”, Jpn. J. Appl. Phys.,1997, 36, 3176-3177.

Mizutani, K.; Funakoshi, A.; Nagai, K.; Harakawa, K.; “Acoustic measurement oftemperature distribution in a room using a small number of transducers”, Jpn. J. Appl.Phys., 1999, 38, 3131-3134.

Moulin, P.; Manno, P.; Rouch, J.C.; Serra, C.; Clifton, M.J.; Aptel, P.; ”Flux improvementby Dean vortices: ultrafiltration of colloidal suspensions and macromolecular solutions”,J. Membrane Sci., 1999, 156, 109-130.

Norton, S.J.; Testardi, L.R.; Wadley, H.N.G.; “Reconstruction internal temperaturedistributions from ultrasonic time-of-flight tomography and dimensional resonancemeasurements”, J. Res. Natl. Bur. Stand., 1984, 89(1), 65-74.

Nunes, S.P.; Peinemann, K.-V. (Eds.); “Membrane Technology in the Chemical Industry”,Wiley-VCH, Weinheim, 2001.


Peyrin, F.; Odet, C.; Fleischmann, P.; Perdrix, M.; “Mapping of internal materialtemperature with ultrasonic computed tomography”, Ultrason. Imag., ConferenceProceeding, Halifax, 1983, 31-36.

Rautenbach, R.; Albrecht, R.; “Separation of organic binary mixtures by pervaporation”, J.Membrane Sci., 1980, 7, 203.

Rautenbach, R.; Albrecht, R.; “On the behavior of asymmetric membranes in pervapora-tion”, J. Membrane Sci., 1984, 19, 1-22.

Rautenbach, R.; Albrecht, R.; “The separation potential of pervaporation. Part 2. Processdesign and economics”, J. Membrane Sci., 1985, 25, 25-54.

Reid, R.C.; Prausnitz, J.M.; Poling, B.E.; “Properties of gases and liquids”, 4th ed. Mc Graw-Hill Book Co., 1988.

Suurmond, R.T.; “Tomographic reconstruction of temperature distributions from acousticmeasurements”, Postgraduate report, Stan Ackermans Institute, Eindhoven, 1998, ISBN90-5282-873-3.

Van der Gulik, G.J.S.; Janssen, R.E.G.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics ina ceramic pervaporation membrane reactor for resin production”, Chem. Eng. Sci., 2001,56, 371-379.

Van Veen, H.M.; Van Delft, Y.C.; Engelen, C.W.R.; Pex, P.P.A.C.; “Dewatering of organicsby pervaporation with silica membranes”, Proc. Euromembrane '99, Leuven, Belgium,1999, 209.

Wakai, K.; Shimizu, S.; Ishikawa, S.; “Instantaneous measurement of two-dimensionaldistribution of temperature of water by means of ultrasonic-CT”, Proc. of 2nd KSME-JSME Fluids Engineering Conf., Seoul, 1990, Vol.1, 299-304.

Winzeler, H.B.; Belfort, G.; “Enhanced performance for pressure-driven membraneprocesses: the argument for fluid instabilities”, J. Membrane Sci., 1993, 80, 35-47.

Wright, W.M.D.; Schindel, D.W.; Hutchins, D.A.; Carpenter P.W.; Jansen, D.P.; “Ultrasonictomographic imaging of temperature and flow fields in gases using air-coupledcapacitance transducers”, J. Acoust. Soc. Am., 1998 104(6), 3446-3455.

FUTURE PERSPECTIVES FORPROCESS ENGINEERING OF

POLYCONDENSATION REACTIONS

Abstract

In this chapter the most important results described in theprevious chapters will be summarized. Some additional aspects will bediscussed for describing the hydrodynamics in the two types of reactorinvestigated here. Additionally, some issues will be discussed for thedesign and development of processes for polycondensation reactionsthat have not been described in the previous chapters.

7

Chapter 7132

7.1 Hydrodynamics in polycondensation reactors

7.1.1 Drais horizontal stirred tank reactor

A large number of flow parameters of the Drais reactor have been establishedexperimentally. This has been done for turbulent as well as laminar conditions,which are the occurring flow regimes during the polymerization process. The flowparameters include the fluid velocities, turbulence parameters, energy dissipationrates, mixing patterns, mixing times, and power numbers. The results have beendescribed in Chapters 2 and 3. From these results it can be concluded that mixing inthe Drais reactor is relatively good under laminar conditions due to the occurrenceof chaotic patterns upon mixing, resulting in short macro-mixing times. Thedimensionless macro-mixing Ntm appears to increase upon liquid volume. It wasshown that the hydrodynamics on large and small scale are different. For thepolycondensation reaction investigated, the Drais reactor volume cannot be madetoo large, as the macro-mixing time tm will increase to undesirable values.Increasing the impeller frequency can keep tm constant upon scale-up but this maylead to unrealistic power requirements.

To increase the understanding of the flow phenomena in the Drais reactor, thehydrodynamics have been studied numerically using computational fluid dynamics(CFD) as described in Chapter 4. In these CFD calculations a distinction has beenmade between fluid flow and mixing. The performance of two turbulence modelshas been tested for describing the turbulence parameters occurring in the Draisreactor. The simple (and computationally friendly) k-ε model and the complexDifferential Stress Model (DSM) predict fluid flow in the Drais reactor equallywell. However, mixing can only be predicted correctly using the DSM. This impliesthat in CFD the choice of an appropriate turbulence model is of key importance forthe correct description of scalar mixing. Moreover, if the choice for a turbulencemodel is only based on the comparison of mean flow parameters, the description ofscalar mixing may fail completely.

7.1.2 Membrane reactor

As a result of a combination of temperature polarization and concentrationpolarization, the latter leading to an increased reaction rate near the membranesurface, a strong viscosity polarization has initially been expected to be present inthe pervaporation membrane reactor. With the CFD simulations described inChapters 5 and 6, buoyancy forces have shown to be responsible for the occurrenceof a secondary flow, which refreshes the membrane surface and leads to increasedwater fluxes through the membrane. Horizontal configurations showed higherfluxes than vertical configurations. In horizontal configurations, a water fluxincrease of 50% was observed due to the secondary flow.

In order to be able to measure temperature distributions, an UltrasonicComputer Tomography (U-CT) methodology has been developed successfully. The

Future perspectives for process engineering of polycondensation reactions 133

method has been implemented in the horizontal membrane module as described inChapter 6. Using U-CT, it has been shown that the temperature distributionspredicted by means of CFD were indeed present in the module.

3

3,1

3,2

3,3

3,4

3,5

3,6

3,7

3,8

3,9

4

0 0,2 0,4 0,6 0,8 1x [-]

Ut [

m/s

]

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Ua

[m/s

]

Ut(r=0.28)

Ut(r=0.25)

Ua(r=0.29)

Ua(r=0.28)

Ua(r=0.25)

xL = 0.1 m

0,001

0,01

0,1

1

0 0,2 0,4 0,6 0,8 1x [-]

ut2 [

m2 /s

2 ] ut2(r=0.28)

ut2(r=0.25)

ua2(r=0.29)

ua2(r=0.28)

ua2(r=0.25)

xL = 0.1 m

Turbulent f luctuations222

p,it,ii uuu = +

Figure 7.1A-B: A) Tangential and axial velocities as a function of fill ratio.B) Axial and tangential fluctuations as a function of fill ratio.

1.05 eq HCl1.05 eq ECA

1 eq NaOH

Figure 7.2: The equivalencies in ECA, HCl and NaOH for chemicalexperiments to study the mixing performance in the Drais reactor.

Chapter 7134

7.2 Design and scale-up of the Drais reactor

In the previous chapters several aspects of the hydrodynamics in the Draisreactor have been described. Empirical formulas are provided that can be used topredict macro-mixing times and power consumption on various scales. Also,detailed data on fluid velocities and turbulence quantities have been determinedusing LDA and CFD. In this section, additional LDA data are provided that extendthe applicability of the findings from the LDA measurements to fill ratios below100%. To allow for a verification of the future implementation of chemicalreactions in CFD, chemical reactions have been performed experimentally.

7.2.1 LDA measurements at intermediate fill ratios

The LDA measurements as described in Chapter 4 have been performed incompletely filled reactors. However, in practice the reactor is used at fill ratios of40 to 50%. It is interesting to know to what extent the presented data are applicableto lower fill ratios. Therefore, additional LDA measurements have been performedat intermediate fill ratios, providing axial and tangential velocities and turbulenceproperties. The experimental conditions are similar to the conditions described inChapter 4. Measurements have only been performed at the axial coordinate xL =0.1, and radii r = 0.29, 0.28, and 0.25, respectively.

The results are summarized in Figures 7.1A and 7.1B in which the averagedvelocities and the turbulent fluctuations, respectively, are given. Both figures showthat upon decreasing the fill ratio (x = 1 → 0), both the local velocities and theirfluctuating components remain approximately constant and only deviate below x =0.5. It is possible that the measurements at fill ratios below x = 0.4 are somewhatinaccurate because the interface between the liquid ring and gaseous core mightinterfere with the measurement volume. Nevertheless, it can reasonably be assumedthat the data as presented in Chapter 4 are applicable down to fill ratios of 40%.Apparently, for the liquid near the cylindrical wall it does not matter whether agaseous core or a liquid that rotates as a solid body is present.

7.2.2 Combining reaction kinetics and mixing

The polymerization reaction that takes place in the Drais reactor has beenclassified as a parallel-competitive reaction system (R.1, Chapter 1). A similarmodel reaction system can been used to relate reaction selectivity and micro mixing(Bourne et al., 1994; Schoenmakers, 1998; Verschuren, 2001). The reaction systemis usually referred to as the 3rd Bourne reaction and comprises the following tworeactions:

NaOH + HCl → H2O + NaCl (R.1)NaOH + ClCH2COOCH2CH3 → C2H5OH + ClCH2COO-Na+ (R.2)


Reaction R.1 is an instantaneous reaction with a reaction coefficient of k1 =1.3⋅1011 L/(mol⋅s). Reaction R.2 is relatively slow with k2 = 30.4 L/(mol⋅s). Inpractice, NaOH is added to premixed HCl and ECA (ClCH2COOCH2CH3).Depending on the availability on a molecular level, NaOH will react with eitherHCl or ECA. When mixing is ideal, there will always be HCl present to react withNaOH and no conversion of ECA will be obtained. The selectivity for reaction R.1

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16Impeller speed N [Hz]

Am

ount

NaO

H u

sed

for

R.2

[%

]

0

1

2

3

4

5

6

Mac

ro m

ixin

g tim

e [s

]

Chemical experiments

tm PLIF (Chapter 2)

tm RTD (Chapter 3)

L/D = 1.1x = 40%

Figure 7.3: Percentage NaOH used for Reaction 2 and macro-mixingtime tm against impeller frequency in reactor11 at a fill ratio of 40%.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Fill ratio [-]

Am

ount

NaO

H u

sed

for

R.2

[%

]

0

1

2

3

4

5

6

Mac

ro m

ixin

g tim

e [s

]

Chemical experimentstm PLIF (Chapter 2)tm RTD (Chapter 3)

L/D = 1.1N = 10

Figure 7.4: Percentage NaOH used for Reaction 2 and macromixing time tm against fill ratio in reactor11 at N = 10 Hz.

Chapter 7136

is then equal to unity. When mixing is moderate, locally all the HCl will beconsumed and the remaining NaOH can react with ECA, resulting in a selectivitysmaller than unity. Thus, the consumed amount of ECA is an indication for thequality of the mixing process.

The characterization of the macro-mixing performance using this reactionsystem has been obtained with the experimental set-up as depicted in Figure 7.2. Anexactly known amount of NaOH is added to a premixed aqueous solution of ECAand HCl. In Figure 7.3 the percentage of NaOH used for reaction R.2 is given as afunction of impeller speed for reactor11 at a fill ratio of 40%. The percentage ofNaOH used for reaction R.2 increases with decreasing impeller speed. Macromixing times as previously presented in Chapters 2 and 3 are also depicted. Thereappears to be a strong correlation between both profiles, indicating that macromixing determines the selectivity.

Figure 7.4 shows the percentage of NaOH used for reaction R.2 as a functionof fill ratio in reactor11 at an impeller speed of 10 Hz (Re = 3.2⋅105). Also themacro-mixing times from Chapter 2 and 3 are depicted. The highest ECAconversion occurs in completely filled reactors, analogous to the macro-mixingtimes. The shortest macro-mixing times are obtained at fill ratios of 60-70%. Thesedo not correspond with the lowest ECA conversion, which has been obtained at afill ratio of 40%. Therefore, there seems to be no direct correlation. However, theinjection time of NaOH varied between 0.3 and 1 second over these experiments.This injection time is relatively large compared to the macro-mixing times.Therefore, masking effects introduced by the variance in injection time cannot beruled out.

7.2.3 Scale-up

In a patent, several reactor designs have been described that provide Twaron®

polymer with the required product quality (Bannenberg-Wiggers et al., 1998). Twoof these designs are selected to show how the production process can be scaled-upto 2.5 m3 while keeping the macro-mixing times constant. Data are provided inTable 7.1.

The reaction in example 4A of this patent takes 7 minutes in a liquid volumeof 1.75 m3. With the provided dimensions of the reactor, the macro-mixing timestm,t and tm,l for turbulent and laminar conditions can be calculated using equations4.7 and 4.9, respectively. When this process is scaled up to a liquid volume of 2.5m3 according to route 1 in Table 7.1, while keeping x, N and L/D constant, thisleads to a larger diameter (1.55 m) but to comparable mixing times. However, therequired power inputs would double to 24 and 124 kW for turbulent and laminarconditions, respectively. The reactor can also be scaled-up to a liquid volume of 2.5m3 by keeping the diameter D constant, aiming at a reduced power input becausePt~D5. Then the reactor length has to increase to 4 m, and consequently the impellerspeed to 3.2 Hz in order to keep the mixing times constant. The resulting powerinputs of 28 and 136 kW for turbulent and laminar conditions, respectively, areeven larger than for the first route.


The reaction in example 2C of the patent takes 30 minutes in a liquid volumeof 1.3 m3. Chemical conditions differ from example 4A for which the reader isreferred to Bannenberg-Wiggers et al. (1998). Scale-up of this process to 2.5 m3

while maintaining x and L/D = 2 requires a diameter of 1.72 m and a slight increasein impeller speed to 2 Hz. The required power has more than quadrupled to 15 and124 kW, respectively. Based on these findings it can be concluded that thebottleneck for scale-up will probably be the enormous power input under laminarconditions, required to keep mixing times constant.

Table 7.1: Typical numbers for three different strategies for scale-up of the Twaron® Process to2.5m3.

Vl D L L/D N tm,t tm,l(1) εt εl Pt Pl

(2)

Conditions[m3] [m] [m] [-] [Hz] [s] [s] [m2/s3] [kW]

Original 1.75 1.37 2.73 2 2.5 27 240 6.4 37 11 65

Scale-up 1 2.5 1.55 3 2 2.5 27 270 8.5 50 21 124

Exa

mp

le 4

A t

r = 7

min

. [

H2O

] =

675

ppm

[P

PT

] =

11.

2 w

t% x

= 0

.437

Scale-up 2

2.5 1.37 4 3 3.2 28 280 11 53 28 136

Original 1.3 1.38 1.66 1.1 1.7 30 250 2.3 18 4 24

Exa

mp

le 2

C t

r = 3

0 m

in.

[H

2O]

= 4

00 p

pm [

PP

T]

= 1

2.4

wt%

x =

0.5

2

Scale-up 3

2.5 1.72 2.07 1.1 2 28 270 6.2 50 15 124

1 The equation for the mixing time is only valid for x = 1.0;2 The power input at laminar conditions is based on equation 4.8 at Re=100 with k4 = x⋅200.

Chapter 7138

7.3 Design of a pervaporation membrane reactor forpolycondensation processes

The secondary flow pattern, which has been calculated using CFD and hasbeen demonstrated using U-CT, provides an efficient route to reduce concentrationpolarization in the model system studied. Although it might be somewhat prematureto design a membrane reactor based on the work described in this thesis, there aresome thoughts on reactor designs in which advantage is taken from the secondaryflows as observed in the membrane module. The next paragraph will elaborate onthese reactor designs. It is expected that it will be difficult to control the residencetime distribution in membrane reactors because of the possible presence ofshortcuts. An experimental technique that can be used to study these shortcuts isMRI (Magnetic Resonance Imaging). The shortcuts can be determined bymeasuring velocities using Pulse Field Gradients techniques in MRI (Neling et al.,1997; Gibss et al., 1997). They can also be studied by measuring proton densities(Hornak, 2002). This has been done as will be discussed in paragraph 7.3.2. Somepreliminary results will be presented.

7.3.1 Module design

First, a process has to be defined for which a membrane module is required.The object process scheme has been given in Figure 1.3c: a once-throughcontinuous process in which a low molecular weight alkyd resin (Mw = 1,000) isproduced with an average polymerization degree of 10. From a production capacityof 10,000 ton/year, it can be calculated that around 1,600 ton/year of water has to be

Products

External loops

Reactants Heat exchangers

Figure 7.5: Proposed geometry (1) for a membrane reactorfor low molecular resins operated at laminar conditions.


removed. Assuming an average water flux of 4kg/(m2⋅h), a membrane surface area of 50 m2 isrequired. Using a packing density equivalent totube-and-shell heat exchangers of 100 m2/m3, thisleads to a reactor volume of approximately 0.5 m3.

An example of a possible reactor is given inFigure 7.5, consisting of a horizontal bundle ofmembrane tubes in a larger manifold. Along thesemembranes the reactants are fed in parallel. Due toevaporation of the water at the membrane surface,temperature segregation occurs. The cold and thusheavy liquid flows to the bottom of the modulewhere it is removed from the module. Afterheating, it is fed back to the top of the reactor. Theliquid is removed and heated at several axialpositions along the tube to compensate the energyloss caused by evaporation.

The most difficult part in the design will be tomaintain plug flow behavior, which is mandatoryto obtain a narrow MWD. Using the relativelysmall buoyancy forces to force the liquid in theright direction might result in unsteady flowbehavior, and shortcuts for reactants directly to theoutlet are likely to occur. Particle tracking in CFDcalculations or MRI (see below) can give insightinto these unwanted events, thus providing meansto prevent shortcuts.

A possibility to control the average degree ofpolymerization is by removing cold fluid at thebottom where the polycondensation is in equili-brium. Then, just enough energy has to be added inthe recycle for the evaporation of water that isreleased by a next propagation step. Theoretically,a reactor with 10 recycles can then provide anaverage degree of polymerization of 10. For such aprocess a set-up of several reactors in series ispresented in Figure 7.6.

i = 1

Pn = 1

∆H

i = 2

Pn = 2

i = n

Pn = n

Figure 7.6: Proposed geometry (2)for a set-up of several membrane re-actors in series for the production oflow molecular resins operated atlaminar conditions.

Chapter 7140

7.3.2 Magnetic Resonance Imaging (MRI)

A possible experimental technique for studying shortcuts is MagneticResonance Imaging (MRI). In a nuclear magnetic resonance experiment themagnetic moments of the hydrogen nuclei are manipulated by suitably chosenalternating radio-frequency fields, resulting in a so called spin-echo signal. Theamplitude of the spin-echo signal is proportional to the amount of nuclei excited bythe radio-frequency field. The resonance condition for the nuclei is given by:

(7.1)

Here f is the frequency of the alternating radio-frequency field, γ is thegyromagnetic ratio (γ /2π = 42.58 MHz/T for 1H), and Bo is the externally appliedstatic magnetic field. Because of the resonance condition, the method can be madesensitive to hydrogen only. When also a known magnetic field gradient is applied,the resonance condition and hence the NMR signal will depend on the position ofthe nuclei. By applying the magnetic field gradients along a large number ofdifferent directions and using back-projecting techniques, analogous to the methodsdescribed in Chapter 6, the proton density can be made visible in a 2-dimensionalplot (Kak, 1984).

The spin-echo signal also gives information about the rate at which theexcitation of the magnetic moments decays. The system will return to equilibriumby two mechanisms: interactions between the nuclei themselves, causing the so-called spin-spin relaxation, and interactions between the nuclei and their environ-

oBfπγ2

=

Figure 7.7A-B: A) Relative NMR signal strength as a function of repetition time. Thepercentage in the legend refers to the amount of 1,4-butanediol. T1 is the spin latticerelaxation time determined from fits of an exponential function to the data. B) T1 as afunction of the 1,4-butanediol mass fraction (Van der Sande, 2000).


ment, causing the so-called spin-latticerelaxation.

Assuming that both mechanismsgive rise to a single exponentialrelaxation and that the spin-lattice relax-ation is much slower than the spin-spinrelaxation, the magnitude of the NMRspin-echo signal is given by(Vlaardingerbroek et al., 1996):

(7.2)

In this expression, ρ is the density of thehydrogen nuclei, T1 the spin-lattice orlongitudinal relaxation time, TR therepetition time of the spin-echo experi-ments, T2 the spin-spin or transverserelaxation time, and TE the so-calledspin-echo time. T1 and T2 depend on thenuclei surrounding the hydrogen nucleiat resonance, and hence the T1 and T2 ofwater and 1,4-butanediol will bedifferent. In this study, it was chosen torelate T1 of the mixture to the 1,4-butanediol mass fraction. If TE << T2, the T1 can be determined by changing the TRof the spin-echo experiment. In Figure 7.7A the signal is given as a function of TRfor various 1,4-butanediol mass fractions. As can be seen the T1 of the mixtureclearly depends on the 1,4-butanediol mass fraction. In Figure 7.7B the calibrationof the T1 of the mixture as a function of the 1,4-butanediol mass fraction is given.

To illustrate the power of MRI, a lot of effort has been made to experimentallyshow the presence of concentration polarization in the membrane module. For sucha MRI experiment, a 67/33 wt% mixture of 1,4-butanediol and water has beenchosen. This mixture combines a moderate viscosity (0.015 Pa⋅s) and moderatewater fluxes: at an inlet temperature of 70-80°C and 300 mL/min the flux isbetween 1 and 1.5 kg/m2⋅h (Chapter 5 and 6; Van der Gulik et al., 2001). Adedicated CFD calculation has been performed for these conditions to show thepolarization effects that could be expected. The calculated concentrationdistribution in mass fractions at the end of the membrane (at x = 0.85 m of amembrane 0.9 m long) is plotted in Figure 7.8. Water and 1,4-butanediol haveapproximately the same density as a function of temperature (Sun et al., 1992,Hawrylak et al., 1998). Therefore, the buoyancy effects are not profound, resultingin a strong concentration polarization, which is helpful in MRI as a largeconcentration difference is easier to measure.

A special MRI set-up has been built for measuring concentration polarizationin the membrane module, which is depicted in Figure 7.9 (Blok, 1999; Van derSande, 2000). This MRI set-up uses an electromagnet generating a magnetic field of

)1( 12 // TTRTTE eeS −− −= ρ

w% H2O

Figure 7.8: Mass fraction distribution ofwater in a 67/33 w% mixture of 1,4-butanediol and water along a pervapo-ration membrane at x = 0.85m.

Chapter 7142

0.4 Tesla. Unfortunately, with this equipment no concentration polarization couldbe demonstrated, since the noise levels were too high and the spatial resolution ofthe MRI equipment was too low. The mediocre spatial resolution was demonstratedby analyzing the signal strength over a line from the liquid bulk to the center of themembrane. A very sharp transition in proton density exists across the membranesurface going from unity in the bulk liquid (high proton density) to zero in themembrane tube (low proton density). In a 2-dimensional representation of themeasured proton density, 4 mm is required to span the decrease from unity to zero.This length is too much to visualize the 0.75-mm thick boundary layer in Figure 7.8obtained with CFD calculations. The lack of spatial resolution was mainly a resultof inhomogeneity of the main magnetic field. The best result obtained is depicted inFigure 7.10 in which a measured concentration distribution is given at conditionsthat apply for the distribution given in Figure 7.8. The off-circle shape is a result ofinhomogeneity of the main magnetic field. A slightly higher 1,4-butanediolconcentration (+3 w%) can be observed in red below the membrane.

Measuring concentration polarization requires MRI equipment with morerestrictive characteristics. Currently, a new 1.5 Tesla MRI machine has beeninstalled, providing a much more homogeneous magnetic field, in whichconcentration polarization might be measurable. However, with the current set-upshortcuts in membrane modules might still be measurable because in that case thebulk of the liquid will be monitored, which requires less spatial resolution.

Main magnet

Membranemodule

Heating unitfor liquid

Transport unit

MRI Electronics

Figure 7.9: Setup of the MRI equipment.


Figure 7.10: Measured concentration distribution around membrane tube in the horizontalmembrane module. Green is the bulk liquid, orange indicates an increased concentration.

Bulk liquid

Membrane tube

Increasedconcentration

Chapter 7144

7.4 Outlook to the future

From the work presented in this thesis it can be concluded that hydrodynamicshave a major impact on the design and development of reactors forpolycondensation reactions. A major challenge for the future will be to combine theinsights in hydrodynamics with kinetic reaction schemes. This should allow for theincorporation of all hydrodynamic characteristics and evaluation of their effect onfinal product properties like the molecular weight distribution. Obviously, CFDpackages should eventually be able to handle this type of complex question. Thisshould allow for a proper comparison between different reactor geometries andmodes of operation, leading to the development of reliable and sustainableprocesses for the future.

References

Blok, R.; “Design and validation of a 3d MRI setup to study chemical separation processes”,M.Sc. Thesis, FIK/FTI99-05, Eindhoven University of Technology, 1999.

Bourne, J.R.; Yu, S.; “Investigation of micromixing in stirred tank reactors using parallelreactions”, Ind. Eng. Chem. Res., 1994, 33(1), 41-55.


Gibbs, S.J.; Haycock, D.E.; Frith, W.J.; Ablett, S.; Hall, L.D.; “Strategies for rapid NMRRheometry, by magnetic resonance imaging velocimetry”, J. Magn. Reson., 1997, 125,43-51.

Hawrylak, B.; Gracie, K.; Palepu, R.; “Thermodynamic properties of binary mixtures ofbutanediols with water”, J. Solution Chem., 1998, 27(1), 17-31.

Hornak, J.P.; “The basics of MRI”, http://www.cis.rit.edu/class/schp730/bmri/bmri.htm.Kak, A.C.; “Image reconstruction from projections”, Digital image processing techniques,

Academic Press Inc., ISBN 0-12-236760-X, 1984.Neling, B.; Gibbs, S.J.; Derbyshire, J.A.; Xing, D.; Hall, L.D.; Haycock, D.E.; Firth, W.J.;

Ablett, S.; “Comparisons of magnetic imaging velocimetry with computational fluiddynamics”, J. Fluids Eng., 1997, 119, 103-109.

Schoenmakers, J.H.A.; “Turbulent feed stream mixing in agitated vessels”, Ph.D. Thesis,Eindhoven University of Technology, 1998.

Sun, T.; DIGuillo, M.; Teja, A.S.; “Densities and viscosities of four butanediols between 293and 463 K”, J. Chem. Eng. Data, 1992, 37, 246-248.

Van der Gulik, G.J.S.; Janssen, R.E.G.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodyna-mics ina ceramic pervaporation membrane reactor for resin production”, Chem. Eng. Sci., 2001,56, 371-379.

Van der Sande, J.J.; “Design and validation of a NMR-measurement system”, M.Sc. Thesis,NF/FIK 2000-07, Eindhoven University of Technology, 2000.

Verschuren, I.L.M.; “Feed stream mixing in stirred tank reactors”, Ph.D. Thesis, EindhovenUniversity of Technology, 2001.

Vlaardingerbroek, M.T.; Den Boer, J.A.; “Magnetic Resonance Imaging”, Springer-Verlag,Germany, ISBN 3-540-60080-0, 1996.

DANKWOORD

Hoewel dit dankwoord voor velen waarschijnlijk het meest begrijpelijke deelvan dit proefschrift is, was het misschien wel het moeilijkste deel om te schrijven.Is het bij het schrijven van een wetenschappelijk getinte tekst vaak al moeilijk dejuiste afbakening te vinden, bij een dankwoord is dit welhaast onmogelijk. Vandaardat ik allereerst wil beginnen met iedereen, maar dan ook íedereen, te bedanken, dieop welke wijze dan ook geholpen heeft bij de totstandkoming van dit boekwerk.Toch wil ik een aantal mensen bij naam bedanken.

Te beginnen met mijn promotor Jos Keurentjes en mijn coach Johan Wijers.Jos, ik bewonder je onuitputtelijke enthousiasme dat ook nog eens aanstekelijkwerkt. Daarmee heb je mij over de streep getrokken om te gaan promoveren. Ik wilje bedanken voor het vertrouwen, de begeleiding en de bijzonder vlotte afhandelingvan de stapels papier. Johan wil ik graag bedanken voor zijn grote betrokkenheid,de open deur en zijn humoristische doch kritische blik op de alledaagse dingen.Tenslotte is het mij opgevallen dat jullie opmerkingen t.a.v. een wetenschappelijketekst volledig complementair zijn. Soms leek het afgesproken werk.

De sponsoren van het verrichtte werk wil ik bedanken voor de financiële steun.Hoewel vaak op grote afstand, wil ik Jan Surquin speciaal bedanken, ook gezienzijn voortdurende interesse in het onderzoek. Ook is het prettig te constateren datsponsoren bereid zijn het vervolgonderzoek van 8 manjaar mee te willenfinancieren.

Dit proefschrift staat bol van het experimentele werk. Voor het vervaardigenvan de opstellingen heb ik waarschijnlijk iedereen van de Technische Dienst weleens ingeschakeld. Zonder de illusie te hebben volledig te zijn, wil ik in elk gevalde volgende mensen bedanken: Peter Brinkgreve, Anton Bombeeck, ErwinDekkers, Piet van Eeten, Henk Hermans, Rinus Janssen, Frans Kuijpers, TheoMaas, Jovita Moerel en Hans Wijtvliet.

Baukje Osinga wil ik bedanken voor het uitvoeren van een groot aantalpervaporatie-experimenten en het veelvuldig verslepen van de opstelling van deFaculteit Scheikundige Technologie naar Technische Natuurkunde v.v. Daarwerden we altijd hartelijk ontvangen door Leo Pel, Roland Blok en/of Joris van derSande van de groep Fysische informatica en klinische fysica (FIK). Een speciaalwoord van dank gaat uit naar hen. Met grote inzet en doorzettingsvermogen hebbenzij een complete MRI-opstelling gebouwd om aan concentratiepolarisatie in demembraanmodule te kunnen meten. Het is nauwelijks voor te stellen dat uit zoveelnoeste arbeid, zo weinig bevredigend resultaat kan voortkomen. Gelukkig kunnenjullie de opstelling ook nog voor andere doeleinden gebruiken.

Het bedrijf Innovation Handling verdient zeker een bedankje voor deassistentie bij het berekenen van temperatuurverdelingen zoals die ontstaan tijdensde pervaporatie-experimenten. Ook al is onze samenwerking niet gelopen zoalsgepland, jullie reconstructiesoftware is uit de kunst!

Ik heb het genoegen gehad een vijftal afstudeerders te mogen begeleiden. Zijhebben veel werk verricht waarvan het grootste gedeelte is opgenomen in ditproefschrift. Achtereenvolgens waren dit: René van der Zande (de man met debekertjes glycerol), Eric Rossou (voor de mooiste PLIF-plaatjes), Roger Janssen

(“Het personeel”), Hanny van Enschot (chaotisch mengen) en Dirk van Asseldonk(de duizendpoot). Allen bedankt voor jullie inzet, de goede samenwerking en degezelligheid, ook na vijven.

Alle leden van de capaciteitsgroep Procesontwikkeling bedank ik voor deplezierige werksfeer, in het bijzonder alle kamergenoten die ik in de loop der tijd alspromovendus heb versleten, te weten Aico de Volder, Geurt Swanenberg, MaartjeKemmere, Jaco Boelhouwer en Marc Jacobs.

Twee heren zullen 10 december nog een beetje op (me) moeten letten, Marc enPeter, de beide paranimfen. Bedankt dat jullie deze belangrijke taak willenuitvoeren. Kunnen jullie meteen een beetje wennen aan het podium…

Tenslotte wil ik alle vrienden en familie bedanken voor de steun en degetoonde interesse aangaande de voortgang van het boekje, in het bijzondernatuurlijk Pa, Ma, Pieter en Floor. Uiteraard wil ik Arina en de kinderen bedanken.Arina, het zal je nog tot vervelens toe gezegd worden dat dit niet mogelijk waszonder jouw liefde, steun en ruimte. Maar dat klopt dan ook als een bus. Joanne,Huub en Mieke, jullie wil ik bedanken voor het opeisen van tijd voor ontspanning.Jullie geven mij het inzicht om alles op de juiste waarde in te schatten.

Ommen, Oktober 2002

CURRICULUM VITAE

Gert-Jan van der Gulik werd op 3 juni 1969 geboren te Uithuizen. In 1986behaalde hij het HAVO diploma aan het Ludgercollege te Doetinchem. Aansluitendstudeerde hij Chemische Technologie aan de Hogeschool Enschede. Na een jaaronder de wapenen te hebben gestaan, werd de studie Chemische Technologievervolgd aan de Universiteit Twente. In 1995 studeerde hij af onder leiding vanprof.dr.ir. J.A. Lercher in de Vakgroep Katalytische Processen en Materialen. In1996 begon hij aan de 2de-fase opleiding Proces- en productontwerp aan deTechnische Universiteit Eindhoven. De eindopdracht omvatte het opschalen van deDrais reactor waarin het polymeer voor de Twaron-vezel wordt geproduceerd. Dezeopdracht is vervolgens 2,5 jaar voortgezet en uitgebreid met onderzoek naar dehydrodynamica in pervaporatie membraanmodules. De resultaten van beide zijnbeschreven in dit proefschrift. Het onderzoek is uitgevoerd binnen decapaciteitsgroep Procesontwikkeling onder supervisie van prof.dr.ir. J.T.F.Keurentjes.

Sinds 1 november 2000 is Gert-Jan van der Gulik werkzaam als CFD-consultant bij BuNova Development BV te Zwolle.

Design and scale-up of polycondensation reactors ... · DESIGN AND SCALE-UP OF POLYCONDENSATION REACTORS Hydrodynamics in horizontal stirred tanks and pervaporation membrane modules

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