CFD modelling of gas-sparged ultrafiltration in tubular membranes

Journal of Membrane Science 210 (2002) 13–27

CFD modelling of gas-sparged ultrafiltrationin tubular membranes

Taha Taha, Z.F. Cui∗Department of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ, UK

Received 16 August 2001; received in revised form 15 July 2002; accepted 22 July 2002

Abstract

In ultrafiltration processes, injecting gas to create a gas–liquid two-phase crossflow operation can significantly increasepermeate flux and, moreover, can improve the membrane rejection characteristics. It has been shown that controlled pulseinjection to generate slug flow is more advantageous than uncontrolled gas sparging, especially when the gas flow rate islow. The slug size and frequency affect the performance of ultrafiltration, and there exits an optimal slug size and frequencyto achieve high permeate flux. Previous studies have been based on the analysis of the experimental data and mass-transfercorrelations. In this work, an attempt is made to model the slug flow ultrafiltration process using the volume of fluid (VOF)method with the aim of understanding and quantifying the details of the permeate flux enhancement resulting from gassparging. For this numerical study, the commercial CFD package, FLUENT, is used. The first part of the model uses theVOF method to calculate the shape and velocity of the slug, the velocity distribution and the distribution of local wall shearstress in the membrane tube (neglecting the wall permeation effect). The second part links the local wall shear stress to thelocal mass-transfer coefficient that is then used to predict the permeate flux. In order to validate the model, experimental datareported in the literature over a wide range of gas and liquid velocities, slug frequencies, and transmembrane pressures arecompared with the CFD predictions. Good agreement is obtained between theory and experiment.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Ultrafiltration; Gas sparging; Enhancement; CFD; Hydrodynamics

1. Introduction

Ultrafiltration has become an established unit op-eration with a great potential in various applicationsin the dairy, water, chemical and pharmaceutical in-dustries. However, its practical use has been limitedby the high process cost, including both capital andoperational costs. A relatively high energy cost isassociated with the high crossflow velocities that arenecessary to control concentration polarisation and

∗ Corresponding author. Tel.:+44-1865-273-118/017;fax: +44-1865-283-273.E-mail address:[email protected] (Z.F. Cui).

membrane fouling, thereby maintaining an acceptablyhigh permeate flux. The need for frequent cleaningwith chemicals and detergents also contributes signifi-cantly to the operational cost of membrane processes.Gas sparging, i.e. injecting air bubbles into the liquidfeed to generate a two-phase flow stream, has provedto be an effective, simple and low-cost technique forenhancing ultrafiltration processes[1,2].

Gas and liquid flowing together in a pipe distributein an annular flow pattern when gas rate is high. Atlow gas rates in vertical flow, the pattern observedis bubbly. Over a wide range of flow rates betweenthese two limits, theslugflow pattern exists. Such flowpattern is characterised by a quasi-periodic passage

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0376-7388(02)00360-5

14 T. Taha, Z.F. Cui / Journal of Membrane Science 210 (2002) 13–27

Nomenclature

C concentration of solute (kg/m3)Cb bulk concentration of solute (kg/m3)Cw wall concentration of solute (kg/m3)d diameter of the tubular membrane (m)dh equivalent hydraulic diameter (m)D diffusion coefficient (m2/s)�F external body forces (N)g acceleration due to gravity (m/s2)Jv permeate flux (kg/(m2 h))k mass-transfer coefficient (m/s)L length of the tubular membrane (m)n normal vector to the bubble surfacep static pressure (Pa)ps surface tension induced pressure

difference (Pa)�P transmembrane pressure (Pa)QL liquid flow rate (l/min)QG gas flow rate (l/min)Rc cake resistance (Pa s/m)Rm membrane resistance (Pa s/m)Re Reynolds numbert time (s)Uinlet inlet velocity (m/s)UL liquid velocity (m/s)UTB Taylor bubble velocity (m/s)Uwall wall velocity (m/s)v velocity vector (m/s)x axial tube coordinate (m)y perpendicular tube coordinate (m)

Greek lettersαG volume fraction of the gas phase

in the computational cellαL volume fraction of the liquid phase

in the computational cell�π osmotic pressure difference (Pa)γ shear rate (l/s)κ free surface curvatureµ molecular viscosity (kg/(m s))ρG gas density (kg/m3)ρL liquid density (kg/m3)σ surface tension (N/m)

of long round-nosed bubbles—usually referred to as‘Taylor bubbles’ or ‘slugs’—separated by liquid plugs.

Slug flow, which has proved advantageous overbubbly flow in enhancing membrane filtration[3], canbe achieved even at low gas flow rates by injectingthe gas in a controlled manner with a timer and asolenoid valve in order to give the desired slug sizeand frequency. In particular, for ultrafiltration, it wasfound, firstly, that the gas flow rate required to effectsubstantial improvements in permeate flux is verysmall. Secondly, that the liquid crossflow velocity haslittle effect on the permeate flux in gas-sparged ul-trafiltration. The combination of these two particularaspects of this technique provides the possibility of asignificant saving on energy costs[1,2].

Yet, published literature in this field so far hasmainly dealt with performance assessment of exper-imental methods. Suggestions for flux enhancementhave failed to be quantitative and, at times, have beenmerely speculative. Ghosh and Cui[4] used approxi-mated hydrodynamic models[5,6] to calculate theaverage velocities in the film and the wake regions.Adopting the Dittus and Boelter correlation[7], theycalculated the mass-transfer coefficient from the veloc-ity values to predict the permeate flux. Otherwise, sev-eral mechanisms involved in flux enhancement havebeen identified and qualitatively described. For tubularmembranes, it was postulated that the two-phase flowgenerated complex hydrodynamic conditions insidethe filtration module that limited the accumulationof particles or molecules by creating local velocityand pressure fluctuations related to intermittence[8].Cui and Wright[1,2] speculated that the mixing zonein the bubble wake induced secondary flows that areresponsible for enhancing the permeate flux. Bellaraet al. [9] hypothesised that physical displacementof the mass-transfer boundary layer is responsiblefor the enhancement in the hollow fibre membranesystems. High shear stresses were thought to be themain reasons for the observed flux improvement[3].

In order to optimise the process efficiency, it isessential to understand and quantify the details ofslug flow dynamics and to identify their effect onultrafiltration performance. In this paper, an attemptis made to explain permeate flux enhancement dueto gas sparging by examining the hydrodynamicsof gas–liquid two-phase flow and the increase in

T. Taha, Z.F. Cui / Journal of Membrane Science 210 (2002) 13–27 15

mass-transfer, in the special case of upward slug flowin a tubular membrane module. All the publishednumerical methods to model slug flow in verticaltube assume either the shape of the bubble or a func-tional form for the shape[10,11]. These assumptionsconstrain the nature of the solution while the ap-proach adopted here (VOF method) lays no such apriori foundations. The solution domain in the presentmodel not only includes the field around the bubble,as in the study of Mao and Duckler[12], but alsoextends behind the bubble, allowing field informationto be obtained in the wake region.

2. Formulation of the problem and thesolution strategy

The first part of the proposed model uses the VOFmethod to calculate the Taylor bubble shape and ve-locity, the velocity and the pressure fields and thewall shear rate around the slug unit in a verticalclosed-wall pipe. The second part uses a polarisationand osmotic model to predict the permeate flux usingthe output data from the first part, namely the wallshear rate, to evaluate the mass-transfer coefficientwith a standard correlation[13]. Wang et al.[14]proved that the existence of realistic wall fluxes doesnot alter the bulk flow fields. This justifies the use ofthe previous consecutive steps.

The CFD software FLUENT (Release 5.4.8, 1998)was used to simulate the motion of a single Taylorbubble rising in a flowing liquid through a tube ofa circular cross-section. In FLUENT, the controlvolume method—sometimes referred to as the finitevolume method—is used to discretize the transportequations. The movement of the gas–liquid interfaceis tracked based on the distribution ofαG, the volumefraction of gas in a computational cell, whereαG = 0in the liquid phase andαG = 1 in the gas phase[15].Therefore, the gas–liquid interface exists in the cellwhereαG lies between 0 and 1. The geometric re-construction scheme that is based on the piece linearinterface calculation (PLIC) method[16] is applied toreconstruct the bubble-free surface. The surface ten-sion is approximated by the continuum surface forcemodel of Brackbill et al.[17]. Turbulence is intro-duced by the Renormalization Group basedk-epsilonzonal model.

2.1. Governing equations

2.1.1. The continuity equation∂

∂t(ρ) + ∇ · (ρ�v) = 0 (1)

2.1.2. The momentum equationA single momentum equation is solved throughout

the domain, and the resulting velocity field is sharedamong the phases. The momentum equation, shownlater, is dependent on the volume fractions of all phasesthrough the propertiesρ andµ.

∂

∂t(ρ�v) + ∇ · (ρ�v�v)

= −∇p + ∇ · [µ(∇�v + ∇�vT)] + ρ �g + �F (2)

2.1.3. The volume fraction equationThe tracking of the interface between the gas and

liquid is accomplished by the solution of a continuityequation for the volume fraction of gas[15].∂

∂t(αG) + �v · ∇αG = 0 (3)

The volume fraction equation will not be solved forthe liquid; the liquid volume fraction will be computedbased on the following constraint:

αG + αL = 1 (4)

2.1.4. Surface tensionThe surface tension model in FLUENT is the

continuum surface force (CSF) model proposed byBrackbill et al. [17]. With this model, the additionof surface tension to the VOF calculation results ina source term in the momentum equation. In thegas–liquid free surfaces, the stress boundary conditionfollows the Laplace–Young equation as

ps = σκ (5)

where ps is the surface tension induced pressuredifference,σ the surface tension, andκ is the freesurface curvature defined in terms of the divergenceof the unit normal,̂n as[17]

κ = ∇ · n̂ = 1

|n|[(

n

|n| · ∇)

|n| − (∇ · n)

](6)

where

n̂ = n

|n| , n = ∇αG (7)


2.2. Differencing schemes

The solution of the momentum equation is ap-proximated by the second order up-wind differencingscheme in order to minimise numerical diffusion.The pressure-implicit with splitting of operators(PISO) pressure–velocity coupling scheme, part ofthe SIMPLE family of algorithms, is used for thepressure–velocity coupling scheme[18]. Using PISOallows for a rapid rate of convergence without anysignificant loss of accuracy. As large body forces(namely, gravity and surface tension forces) exist inmultiphase flows, the body force and pressure gra-dient terms in the momentum equation are almostin equilibrium, with the contributions of convectiveand viscous terms small in comparison. Segregatedalgorithms converge poorly unless partial equilibriumof pressure gradient and body forces is taken into ac-count. FLUENT provides an optional “implicit bodyforce” treatment that can account for this effect, mak-ing the solution more robust[19]. Eq. (3) is solvedusing an explicit time-marching scheme and the maxi-mum allowed Courant number is set to 0.25. A typicalvalue of 10−3 was used as the time step. A total timerun of 1.0 s was used for each run of the simulations.

2.3. Phyiscal properties

The properties of liquid or gas are used in thetransport equations when the computational cell is inthe liquid or the gas phase, respectively. When it isin the interface between the gas and liquid phases,the mixture properties of the gas and liquid phases onthe volume fraction weighted average are used. If thevolume fraction of the gas being tracked, the densityand viscosity in each cell are given by

ρ = αGρG + (1 − αG)ρL (8)

µ = αGµG + (1 − αG)µL (9)

2.4. Interface tracking

To overcome the problem of diffusion which moststandard differencing schemes suffer, the geometricreconstruction scheme is used[16]. It assumes that theinterface between two fluids has a linear slope withineach cell, and uses this linear shape for calculation ofthe advection of fluid through the cell faces.

The first step in this reconstruction scheme iscalculating the position of the linear interface rela-tive to the centre of each partially filled cell, basedon information about the volume fraction and itsderivatives in the cell. The second step is calculatingthe adverting amount of fluid through each face us-ing the computed linear interface representation andinformation about the normal and tangential velocitydistribution on the face. The third step is calculatingthe volume fraction in each cell using the balance offluxes calculated during the previous step[19].

2.5. Model geometry

A two-dimensional coordinate system assuming ax-ial symmetry about the centreline of the pipe was used.The grid used to generate the numerical results wasuniform and contained 26× 280 quadrilateral controlvolume. Thus, the length of the domain is 11d, whered is the pipe diameter. The grid was refined until theshape and terminal velocity of the bubble no longervaried with additional grid refinement. The grid wasrefined near to the wall with the intention of resolvingthe laminar sublayer. The simulation was initialisedwith an arbitrarily shaped bubble and allowed to rununtil a steady bubble shape was established.

In Fig. 1, the boundary conditions used in the sim-ulation are displayed. The no-slip wall condition isapplied to the walls. The fluid mass flux at the inletis specified using a profile for a fully developed flowthrough a pipe. The previous equations are solved fora domain surrounding a Taylor bubble in a frame ofreference attached to the rising Taylor bubble (Fig. 1).With these coordinates, the bubble becomes stationaryand the pipe wall moves with a velocityUwall, equalto that of the Taylor bubble rise velocity,UTB. Theliquid is fed at the inlet with a velocityUinlet, whichis equal toUTB − UL.

2.6. Permeate flux evaluation

In this work, the local shear stress was evaluatedby the CFD simulation and its absolute value wasaveraged over the length of the membrane module.The average mass-transfer coefficient can then beestimated as follows:

k = 1.62

(dγD

dhL

)0.33

(10)


Fig. 1. Taylor bubble rising in a vertical pipe in a moving coor-dinate.

It should be pointed out that the previous equationwas developed under steady shear rates[14]. In thiscalculation, the absolute values of the wall share rateare averaged over the length of the membrane. Thetransient behaviour of the wall shear rate is to be consi-dered in our future investigation. For the special caseof total solute rejection, the permeate flux is calculatedby using the concentration polarisation equation for to-tal rejection, together with the osmotic pressure model:

Jv = k ln

(Cw

Cb

)(11)

Jv = (�P − �π)

(Rm + Rc)(12)

The previous correlations are based on steady-stateultrafiltration. Rc is significant only when gel layerformation takes place. In ultrafiltration of a macro-molecule such as dextran, the value ofRc is expectedto be negligible in comparison toRm and hence maybe neglected. The experimental results reported by Liet al. [20] and Sur et al.[21] are simulated here inorder to test the proposed model. They used dextran167 and 283 kDa, respectively. The osmotic pres-sure for dextran is calculated using the correlations[22]

logπ(150 kDa) = 0.248+ 0.2731C0.35 (13)

logπ(283 kDa) = 0.1872+ 3.343C0.3048 (14)

3. Results and discussion

3.1. Hydrodynamics and mass-transfer

In vertical pipes, Taylor bubbles are axisymmetricand have round noses, while the tail is generally as-sumed to be nearly flat (Fig. 2). The Taylor bubbleoccupies most of the cross-sectional area of the tube.When the bubble rises through a moving liquid, theliquid that is flowing ahead of the nose of the bubble ispicked up and displaced as a liquid film—it begins toflow downwards in the annular space between the tubewall and the bubble surface. Alongside the bubble,the liquid film accelerates until it reaches its terminalvelocity under the condition of a long enough bubble.At the rear of that bubble the liquid film plunges intothe liquid plug behind the bubble as a circular walljet and produces a highly agitated mixing zone in thebubble wake. This mixing zone is generally believedto have the shape of a toroidal vortex[23]. This wakeregion is believed to be responsible for mass- andheat-transfer enhancement.

Fig. 3shows the wall shear stress around a slug unit(Taylor bubble+ liquid plug) together with the liquidfilm thickness. The wall shear stress sign changestwice in a slug unit. The first change takes place nearthe nose of the Taylor bubble and the second nearthe top of the liquid plug. The negative shear stress,indicating upflow, exists over the liquid plug ahead of


Fig. 2. Calculated velocity field around a Taylor bubble witha frame of reference moving with the bubble: membranelength= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l;QL = 1.0 l/min; Vb = 8.3 ml; TMP = 1.0 bar.

the bubble and persists beyond the nose of the Taylorbubble, before becoming positive as the downflow isestablished in the liquid film around the bubble. Theinverse transition from the downward film to an up-ward one in the liquid plug is of a burst-like type. Thebrief fluctuations of the wall shear stress in the filmregion correspond to the wavy nature of the bubblesurface. Near the slug tail, the wall shear stress startsto fluctuate. The previous features were observed ex-perimentally[24]. The predicted permeate flux of theearlier case study was found to be 13.31 kg/(m2 h)comparing to 14.47 kg/(m2 h), the value reported inliterature with 8.0% error.

3.2. Effect of TMP

The variation of permeate flux with the transmem-brane pressure (TMP) is shown inFig. 4. Experimentswere performed for ultrafiltration of industrial gradedextran having average molecular weight of 283 kDausing a tubular PVDF membrane having a molecu-lar cut-off of 100 kDa[21]. The length of the tubularmembrane was 1.18 m and the diameter was 12.7 mm.The slug frequency was controlled using a solenoidvalve and set to 1.0 Hz. It can be seen that at a fixedliquid flow rate the permeate flux increases with TMP.The CFD predicted values clearly capture the sametrend. The predicted values underestimate the exper-imental ones due to the fact that the model does notconsider the transient nature of the shear stress.

3.3. Effect of liquid flow rate

The response of the permeate flux to increasingliquid velocity, observed experimentally[21] andcalculated theoretically is shown inFigs. 5 and 6,respectively. The permeate flux decreases with an in-creased liquid flow rate from 1.5 l/min (Re= 2494) to4.0 l/min (Re = 6651) but increases when the liquidflow rates increases to 6 l/min (Re= 9978). This be-haviour is repeated for all the TMP values examined.The theoretically calculated values follow the sametrend with reasonable accuracy. Cui and Wright[1]also observed the same trend experimentally whenusing uncontrolled gas sparging. Explanations to thisphenomenon can be found in a closer examinationof the hydrodynamics of a rising slug in a flowingliquid.


Fig. 3. Wall shear stress distribution around a slug unit and the liquid film thickness: membrane length= 1.18 m;D = 12.7 mm; dextran–airsystem;Cb = 10 g/l dextran (100 kDaMW); QL = 1.0 l/min; Vb = 8.3 ml; TMP = 1.0 bar.

Fig. 4. Effect of TMP on permeate flux: membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran (283 kDaMW); QL = 0.6 l/min; QG = 0.6 l/min; slug frequency= 1.0 l/s.


Fig. 5. Effect of liquid flow rate on permeate flux[21]: membrane length= 1.18 m;D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran(283 kDaMW); QG = 0.6 l/min; slug frequency= 1.0 l/s.

Fig. 6. Effect of liquid flow rate on permeate flux (Theory): membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/ldextran (283 kDaMW); QG = 0.6 l/min; slug frequency= 1.0 l/s.


Fig. 7. Wall shear stress distribution around a slug unit: (1)QL = 1.5 l/min; (2) QL = 4.0 l/min; (3) QL = 6.0 l/min; membranelength = 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran (283 kDaMW); QG = 0.6 l/min; slug frequency= 1.0 l/s;TMP = 1.0 bar.

Figs. 7 and 8illustrate the calculated wall shearstress around a slug unit and the liquid film axialvelocity around the gas slug respectively for threedifferent liquid rates. The character of the wall shearstress distribution is similar for all cases. The wallshear stress rapidly decreases to zero and attains themaximum positive value near the slug bottom (Fig. 7).In the liquid plug, the wall shear stress recovers. Asthe liquid flow increases, the portion of downwardflow becomes shorter (Fig. 8). The previous trend wasalso reported in the literature[24].

Fig. 9 shows the calculated mass-transfer coeffi-cient for three different mass-transfer zones: the filmregion, surrounding the bubble; the wake zone, ahighly agitated region behind the bubble tail; and theliquid plug zone, the region separating two bubbles.In the film zone, increasing the liquid flow rate causesa decrease in the mass-transfer coefficient which isattributed to the decrease in the average shear rate.The mass-transfer coefficient in the wake region showthe same response of the permeate flux (Fig. 5). In-tuitively, since the film velocity just before plunginginto the liquid behind the bubble actually decreasesby increasing the flow rate (Fig. 8), one expects the

mass-transfer coefficient in the wake region to increasewith the liquid flow rate.Fig. 10shows the turbulenceintensity in the wake, defined as ratio of the magni-tude of the rms of turbulent fluctuations to the meanvelocity. The contribution from the turbulent intensityin the wake region is more significant for higher liquidflow rates. It can be deduced that the turbulent inten-sity in the wake depends on the relative velocities ofthe circular jet of liquid film and the flowing liquidbehind the bubble[20]. As far as the optimisation ofgas-sparged ultrafiltration processes is concerned, oneshould consider the contribution from the three distinctzones of the slug unit in permeate flux enhancement.

3.4. Effect of the gas flow rate and slug frequency

Li et al. [20] used a solenoid valve to generateslug flow with defined frequency and bubble size.They carried out their experiments for ultrafiltrationof industrial grade dextran having average molecularweight of 167 kDa using a tubular PVDF membranehaving a molecular cut-off of 100 kDa. The length ofthe tubular membrane was 1.18 m and the diameterwas 12.7 mm. Their experimental results are modelled


Fig. 8. Axial liquid film velocity: (1)QL = 1.5 l/min; (2) QL = 4.0 l/min; (3) QL = 6.0 l/min; membrane length= 1.18 m; D = 12.7 mm;dextran–air system;Cb = 10 g/l dextran (283 kDaMW); QG = 0.6 l/min; slug frequency= 1.0 l/s; TMP= 1.0 bar.

here.Fig. 11 illustrates the bubble shapes calculatedfor the parameters in their experiments. It can beseen that the shape of the nose of the bubble doesnot change with a change in the bubble length which

Fig. 9. Mass-transfer coefficient in the different zone: membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran(283 kDaMW); QG = 0.6 l/min; slug frequency= 1.0 l/s; TMP= 1.0 bar.

agrees with observations in the literature[25–28].The bubble interface becomes wavy in nature near tothe tail when the bubble is long. This phenomenonwas observed by Nakoryacov et al.[24].


Fig. 10. Turbulence intensity: (1)QL = 1.5 l/min; (2) QL = 4.0 l/min; (3) QL = 6.0 l/min; membrane length= 1.18 m; D = 12.7 mm;dextran–air system;Cb = 10 g/l dextran (283 kDaMW); QG = 0.6 l/min; TMP = 1.0 bar.

The calculated wall shear stress is presented inFig. 12. There is a smooth transition from the upwardto the downward flow, occurring in the film zone.The reverse transition, however, from the downwardto the upward flow is rather sudden. For the longerbubble, fluctuations in the wall shear stress near to

Fig. 11. Liquid film thickness: (�) QG = 0.66 l/min; (�) QG = 1.0 l/min; (+) QG = 1.5 l/min; (−) QG = 2.5 l/min; QL = 1.0 l/min;membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran (100 kDaMW); TMP = 1.0 bar.

the bubble tail—corresponding to the wavy bubblesurface (Fig. 11)—and in the wake can be clearlyseen inFig. 12.

Fig. 13 shows experimental and theoretically cal-culated values for permeate flux as a function of gasflow rate for a fixed sparging frequency. The response


Fig. 12. Wall shear stress distribution around a slug unit: (1)QG = 66.7 ml/min; (2) QG = 100 ml/min; (3) QG = 150 ml/min;4-QG = 250 ml/min; QL = 1.0 l/min; membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran (100 kDaMW);TMP = 1.0 bar.

Fig. 13. Effect of gas flow rate on permeate flux: membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran(100 kDaMW); QL = 1.0 l/min; slug frequency= 0.5 l/s; TMP= 1.0 bar.


Fig. 14. Effect of slug frequency on permeate flux: membrane length= 1.18 m; D = 12.7 mm; dextran–air system;Cb = 10 g/l dextran(100 kDaMW); QL = 1.0 l/min; Vb = 8.3 ml; TMP = 1.0 bar.

of the increase in the permeate flux, which is its re-sponse to an increase in sparging frequency with afixed gas flow rate, is shown inFig. 14. The calcu-lated values by Ghosh and Cui[4] are also included.

Fig. 15. Parity plot of permeate flux.

A good agreement between the experimental and pre-dicted data is obtained. As has been already noted,the model generally underestimates the experimentalvalues. The experimental results obtained by Sur et al.


Table 1Measured permeate flux (Jv, kg/(m2 h)) data[21]

QL (l/min) TMP (bar)

0.5 1.0 1.5 2.0

1.5 10.17 11.51 13.33 13.912.0 9.93 11.14 12.23 13.043.0 8.90 11.16 11.62 12.104.0 9.62 11.31 12.23 13.126.0 12.23 14.71 16.42 17.35

Gas flow rate: 0.6 l/min; dextran–air system,Cb = 10 g/l (283 kDaMW); slug frequency: 1.0 l/s.

[21] and modelled in this work are summarised inTable 1. Fig. 15 shows a parity plot of the permeateflux measured by Li et al.[20] and Sur et al.[21] andthe calculated permeate flux. The agreement betweenexperiment and theory is quite encouraging and themodel may be improved by considering the transientbehaviour of the wall shear stress.

4. Conclusion

The CFD software FLUENT with the method ofvolume of fluid (VOF) was adopted to model the mo-tion of a Taylor bubble rising in a flowing liquid insidea tubular membrane module. The shape of the Taylorbubble can be predicted by the VOF method with rea-sonable accuracy. The shear stress behaviour in a slugunit was calculated and agreed qualitatively with thepublished experimental findings.

The permeate flux enhancement due to gas-spargedultrafiltration with tubular membranes can be pre-dicted with reasonable accuracy. The agreement be-tween the values calculated by the previous model andexperimental data reported in literature is quite en-couraging. The enhancement can be explained by theincrease in the mass-transfer coefficient. The turbu-lence just behind the air bubble, caused by the annularfilm flowing downward, is of significant intensity; andit plays a pivotal role in permeate flux enhancement intubular membranes. Contribution from the film regionand the liquid plug region has also been discussed.Based on the present model, guidance for the design,operation, and control of gas-sparged ultrafiltrationcan be prepared for achieving optimal design, as wellas safe and economical operation.

Acknowledgements

T. Taha is grateful to the Karim Rida Said Founda-tion for financial support.

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ttp://http://mecko.nichd.nih.gov/Lpsb/docs/OsmoticStress.html

ttp://http://mecko.nichd.nih.gov/Lpsb/docs/OsmoticStress.html

CFD modelling of gas-sparged ultrafiltration in tubular membranes

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