fluid mechanics and mass transport in centrifugal membrane

Journal of Membrane Science 176 (2000) 277–289

Fluid mechanics and mass transport in centrifugal membrane separation

J.G. Pharoah, N. Djilali∗, G.W. VickersDepartment of Mechanical Engineering, University of Victoria, Victoria, BC, Canada V8W 3P6

Received 20 October 1999; received in revised form 12 April 2000; accepted 24 April 2000

Abstract

Centrifugal membrane separation (CMS) is a novel technology proposed for the treatment of industrial process streamsand waste waters. This membrane separation process benefits from inherent energy recovery and from the favorable effectsof centrifugal and Coriolis acceleration in alleviating concentration polarization and membrane fouling. A numerical studyof both conventional membrane separation and CMS is presented and used to quantify and analyze the effects of centrifugaland Coriolis accelerations.

The numerical model consists of a 3-D flow channel with a permeable membrane surface. The membrane is modeled usinga boundary condition representing the preferential removal of one component of a solution. The Navier–Stokes equations,coupled with a scalar transport equation which accounts for dissolved species, are solved for both stationary and rotatingmembranes. The model is validated against measurements obtained in a parallel investigation.

In the case of CMS, secondary flow structures are identified and found to enhance the mixing of the feed solution and toincrease the permeate flux over the non-rotating case. Modeled surface salt concentrations increase up to 28% above the feedconcentration for non-rotating separations, while with CMS it is possible to keep the surface concentration within 4% of thefeed. The relative effects of centrifugal and Coriolis accelerations are investigated for various membrane orientations, and itis shown that the alleviation of concentration polarization and the resulting increase in permeate production are largely dueto Coriolis acceleration. © 2000 Elsevier Science B.V. All rights reserved.

Keywords:Fluid mechanics; Reverse osmosis; Concentration polarization; Coriolis acceleration; Water treatment

1. Introduction

1.1. Centrifugal membrane separation

Centrifugal membrane separation (CMS) is a noveltechnology proposed to enhance and extend the appli-cation of membrane separation processes. CMS ben-efits from the inherent energy recovery of thecentrifugal reverse osmosis (CRO) process and from

∗ Corresponding author.E-mail addresses:[email protected] (J.G. Pharoah), [email protected] (N.Djilali), [email protected] (G.W. Vickers)

the potentially favorable effects of centrifugal andCoriolis acceleration in alleviating concentration po-larization and membrane fouling. In this process,a pressurized feed stream flowing across the mem-brane is separated into two effluents, as illustratedin Fig. 1, the ‘permeate’ which passes through themembrane, and the ‘concentrate’ which retains dis-solved substance or suspended particles rejected bythe membrane.

In conventional separation processes such as reverseosmosis, process pressures are achieved using highpressure pumps, and, typically, a turbine is requireddownstream to recover energy from the high pressure

0376-7388/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0376-7388(00)00457-9

278 J.G. Pharoah et al. / Journal of Membrane Science 176 (2000) 277–289

Fig. 1. Cross-flow filtration.

exhaust stream. Significant energy efficiency gainshave been demonstrated with CRO, in which processpressure is developed within a spinning centrifuge.The feed stream in CRO enters the axis at low pres-sure and is pressurized as it flows radially outwardsto the membrane. After exiting the membrane, the re-tentate stream is depressurized as it returns to the axisand exits the rotor. Thus, the feed stream in CROen-ters at low pressureand the exhaust streamleaves atlow pressure, allowing inherent energy recovery with-out the addition of an auxiliary turbine. Reduction inspecific energy consumption of more than 35% overnon-rotating RO have been reported by Wild et al. [1]for a prototype producing 7.5 m3 per day of fresh wa-ter. It was also shown that the theoretical energy effi-ciency of CRO increases with system capacity, and upto 70% reduction in specific energy consumption waspredicted for units producing over 1000 m3 per day.

Two of the major problems associated with mem-brane separation processes are fouling and concentra-tion polarization. Fouling occurs when the membraneis physically obstructed, either by a buildup of

Fig. 2. Membrane stack.

particulates on the surface or by membrane com-paction, whereas concentration polarization refersto the formation of a high concentration boundarylayer adjacent to the membrane which results in alocal increase in the osmotic pressure and a reduc-tion of the permeate flux. To capitalize further onthe rotational effects present in CRO, a combinedexperimental/numerical investigation was undertakento develop membrane configurations which reducefouling and concentration polarization due to thesecondary flows induced by centrifugal and Coriolisaccelerations. This work complements the current ex-perimental work which is aimed at applying the CMSprocess in reverse osmosis [2] and in ultrafiltration ofprotein solutions and colloidal mixtures [3] where thepotential benefits are greater still.

Fig. 2 depicts a custom membrane stack comprisedof a series of lexan discs with membrane materialglued (around the periphery) to one side and a rectan-gular channel milled out of the other side. These stacksare inserted into the CMS test centrifuge, shown inFig. 3, in order to obtain the desired orientation withrespect to the rotational axis. This orientation, whichaffects both Coriolis and centrifugal accelerations, isdefined by assigning a reference configuration, andthree successive rotations, (pitch, roll, yaw) about thez, x andy axes, respectively. Fig. 4 depicts the refer-ence (0, 0, 0) position and three sample orientations.

The objectives of this work are to develop a nu-merical model of membrane separation and to use this

J.G. Pharoah et al. / Journal of Membrane Science 176 (2000) 277–289 279

Fig. 3. CMS experimental apparatus.

model to investigate the effect of system rotation onreverse osmosis membrane separation. Of specific in-terest is an in-depth understanding of the parametersinfluencing the permeate flux, and the membrane sur-face concentrations in the experimentally difficult en-vironment of a CMS device. The model will also beused to determine optimal membrane configurationsand orientations.

1.2. Fluid mechanical background

Flows in partly porous stationary and rotating ductsare also found in heat exchangers, solar energy col-lectors, porous walled flow reactors, pulp and paperprocessing, and fuel cell stacks. Laminar flow in a 2-Dchannel with porous walls was considered by Berman[4] who obtained a first-order perturbation solutionoften used for validation. An experimental study oflaminar flow and heat transfer in square ducts witha single porous wall [5] has shown that significantchanges occur with respect to pressure drop, Nusseltnumber and entrance lengths depending on the wallvelocity. Numerical modeling of membrane flows to1994 was reviewed by Bouchard et al. [6], whereinall models used simple analytic solutions, such as thatof Berman, for the fluid flow and numerically solveda convection diffusion equation to account for solutetransport. The most comprehensive numerical modelrelated to membrane flows is that of Pellerin et al.[7], in which the steady 2-D Navier–Stokes solutions

were solved in conjunction with a transport equationfor the dissolved phase and Darcy’s law to determinethe transpiration velocity. Turbulence was modeledusing the density-weighted ensemble-averaged equa-tions (Favre averaging) together with thek–ε model.The model did not properly account for low Reynoldsnumber turbulence, but nonetheless highlighted thesignificant differences arising from turbulence.

The potential benefits of fluid dynamical instabili-ties in alleviating concentration polarization and foul-ing have been recognized for some time. Belfort [8]discussed for instance the effect of unsteadiness due tooscillating pressure gradients, ‘furrowed’ flow chan-nels and centrifugal acceleration. In a series of subse-quent papers, Belfort and co-workers investigated theeffectiveness of Dean vortices in flux enhancement andfouling reduction (see, e.g. [9]). The reduction in con-centration polarization due to centrifugal accelerationwas also investigated by Andeen [10] using commer-cial hollow-fiber membranes. The same system wassubsequently used to demonstrate reduced particulatefouling as well [11]. Pulsatile flow is another methodthat has been proposed for controlling and minimizingconcentration polarization and fouling [12,13]. In thepresent study the focus is the combined effect of cen-trifugal and Coriolis acceleration. To our knowledge,no work has been reported on the latter in the contextof membrane separation.

A general review of the fluid mechanics of centrifu-gal separation of mixtures in rotating vessels is givenin [14], with an emphasis on the importance of Corio-lis forces under various conditions and a discussion ofvarious approaches for modeling monodisperse sus-pensions and their limitations. 3-D numerical inves-tigations of flow and heat transfer in rotating squareducts, together with reviews of earlier work, are pre-sented in [15] for laminar flow and in [16] for turbu-lent flow. Investigations of flow in rotating ducts witha porous wall are scarcer, but a recent 3-D numericalstudy of the mixed convection problem is presentedin [17]. A vorticity-velocity formulation was used andthe equations were parabolized to allow a marchingintegration in the axial direction.

In this paper, we present 3-D computations, usingthe full Navier–Stokes equations, of the low Reynoldsnumber laminar flow of a salt-water solution in a longchannel with a reverse-osmosis filtration membraneon one side. Both stationary and rotating cases are


Fig. 4. Membrane orientation with respect to the axis of rotation. Angles represent successive rotations in (pitch, roll, yaw). The darkshaded sections represent reference planes on which visualizations will be presented. Note: in the physical device the rotation axis is heldconstant and the membrane orientation is altered.

considered and the relative effects of centrifugal andCoriolis accelerations are investigated.

2. Computational model

2.1. Governing equations

The flow in a CMS device is governed by the con-servation of mass, the Navier–Stokes equations, and,when a dissolved second phase is present, a scalartransport equation. The Reynolds number of the flowin the membrane channel is of the order of 102 andthe flow may therefore be assumed to be laminar. The

governing equations in a rotating frame of referencetake the following conservation form:

∂ρ

∂t+ ∂

∂xj

(ρuj ) = 0 (1)

∂

∂t(ρui) + ∂

∂xj

(ρujui) + ρRi = − ∂P

∂xi

+ ρgi

+ ∂

∂xj

{µ

(∂ui

∂xj

+ ∂uj

∂xi

)−2

3µ

∂ul

∂xl

δij

}(2)

∂

∂t(ρφ) + ∂

∂xj

(ρujφ) = ∂

∂xj

(0

∂φ

∂xj

)(3)


where φ is the mass fraction of dissolved species,gi the gravity vector andRi is the acceleration termwhich can be used to solve the equations in rotatingframes of reference,

Ri = εilmωlεmnkωnxk + 2εilmωlum (4)

or, alternatively in vector notation, as

ER = E� × ( E� × Er) + 2E� × EU (5)

where the first term corresponds to centrifugal accel-eration, the second term corresponds to Coriolis ac-celeration,ωl( E�) is the rotation vector,xk(Er) is theposition vector with respect to the rotational axis andεijk is the alternating tensor, which takes a value of 1for cyclic permutations ofijk, −1 for acyclic permu-tations and 0 otherwise.

In addition to the standard boundary conditions (in-let, outlet, no-slip), a selective membrane boundarycondition must be specified for the CMS problem.

2.2. Membrane boundary condition

The specification of a membrane boundary condi-tion is essential to modeling the flow in a CMS device.This boundary condition must represent the selectivityof the membrane and can be derived using irreversiblethermodynamics [18]. Assuming that dynamic re-versibility is valid for all cases considered, and thatthe membrane is a perfect rejector with respect to con-vection (see Appendix A for details), then the flux ofsolution and solvent are, respectively, given by [19]:

jv = Lp(1P − 15) (6)

Js = jv(1 − R)

R(cf − cp) (7)

whereLp is the membrane permeability,R the rejec-tion of the membrane,1P and15 the hydrostatic andosmotic pressure differences across the membrane,cfandcp are the concentrations of the feed and perme-ate respectively. In the above equations, the solutionflux is presented as a volume flux and the solute fluxis presented as a mass flux.

2.3. Implementation

The present model of the CMS process has beenimplemented into the commercial CFD package

Fig. 5. Flux element on membrane surface.

TASCflow3d [20]. TASCflow3d is a Navier–Stokescode which solves the Cartesian form of these equa-tions over collocated body fitted grids using a finitevolume based finite element method. Steady solutionsare arrived at by stepping through time using first or-der backwards differencing. While the code providesseveral options for the discretization of the convectiveterms, all of the simulations presented here employeda flux limited higher order upwind scheme for theseterms.

Fig. 5 depicts a flux element on the membrane sur-face which is made up of partial faces from the fournodes bounding the flux element. Properties and vari-ables are constant over a flux element and are deter-mined as numerical averages of the four surroundingnodal values. The solute flux is specified by setting thevalue ofφout based on the experimentally determinedrejection. This value ofφout is then used in calculatingall fluxes through the membrane surface. All the otherterms are based on the flux element values so that theviscous terms and the fluxes through the other facesare still based on the higher concentration present onthe feed side of the membrane.

The velocity components parallel to the membranesurface are set to zero in accordance with the no-slipcondition, and the normal component of velocity is setaccording to Eq. (6).

3. Results and discussion

3.1. Model validation

3.1.1. Berman solutionThe present model, designated as the porous wall

model (PWM), was initially validated against the


Fig. 6. Geometry used for initial validation against Berman solu-tion.

Berman solution [4] consisting of symmetrical flowbetween two parallel membrane plates as in Fig. 6.The channel dimensions used for these simulationswere those of the actual CMS channel, and the feedsolution was taken to be 22,000 ppm salt at 26◦C(densityρ = 1012.12 kg/m3 and kinematic viscosityν = 9.0344× 10−4 kg m/s). Figs. 7 and 8 presentcomparisons of the calculated velocity componentsat various distances along the channel for a channelReynolds number of Reh = 2hU/ν = 250, and a wallReynolds number Reh = hvw/ν = 0.1. Two differentgrid sizes were used, and excellent agreement wasobtained on both grids.

3.1.2. Conventional (non-rotating) membraneseparation

Next, the model was further validated consideringconventional (non-rotating) reverse osmosis desalina-tion. Experimental data was generated on a static testbench, designed to house the custom CMS membranestacks described above, and employed nine parallel

Fig. 7. Profiles of longitudinal velocity at various distances alongthe channel. Reh = 250 and Rew = 0.1.

Fig. 8. Profiles of normal velocity at various distances along thechannel. Reh = 250 and Rew = 0.1.

flow channels, four of which had an active membraneon the one side of the channel. A full description ofthe experimental program is given in [2]. The feedflow rate was 2 l/min, which results in an averagechannel Reynolds number of Reh = 72.5, whereh isthe channel half height. Fig. 9 compares the predictedpermeate fluxes with the measurements. The numeri-cal results were obtained using the parameters outlinedin the validation column of Table 1 for three inlet con-centrations and, with one exception, assumed the flowwas two dimensional. Additional predictions wereobtained for feed concentrations of 35,000 ppm NaCl

Fig. 9. Comparison of experimental permeate mass flux data andCFD calculations for conventional membrane separation. Concen-trations are in ppm NaCl.


Table 1Parameters used for the numerical simulations

Properties Validation CMS

Temperature (◦C) 26 25Permeability,Lp × 1012 (m/(s Pa)) 1.83 2.24Rejection,R (%) 98.5 98.0

using feed rates of 4 l/min (Reh = 145) and for a 3-Dcase (500 psi) which accounts for side wall effects.

Two trends are immediately apparent in examiningFig. 9; the model correctly predicts the trends shownin the concentration polarization experiments, butover-predicts the experimentally determined perme-ate fluxes. Also, as expected, increasing the feed flowrate increases the permeate flux and accounting forside walls decreases the permeate flux. The absoluteerror between the simulations and the experiments isof the order of 4× 10−4 kg/(m2 s), giving a relativeerror of 6% at the highest flux. The relative differencebetween the slopes ranges from less than 1%–4.8%.These differences are not unreasonable consideringthe difficulty in controlling experimental parameterssuch as membrane permeability, feed concentrationand applied pressure.

3.2. The effect of system rotation

Rotation can be expected to induce asymmetry andspanwise motion and hence a breakdown of the 2-Dfeature. In order to investigate the relative effects ofchannel orientation, 3-D simulations were carried outin a channel of half heighth = 0.381 mm, of length220h, and aspect ratio 8. A constant inlet velocity, cor-responding to a Reynolds number of 145, was speci-fied with an NaCl concentration of 22,000 ppm (φ =0.022). The membrane boundary condition is appliedon they = −h face betweenx/h=7.4 and 213 withthe remainder of they = −h surface set as an im-permeable wall. A fully developed outlet conditionwas used on thex = 220h face, and while this isnot strictly correct with wall permeation, it is antici-pated that this will affect the simulation only locallynear the exit since membrane fluxes are small com-pared to the bulk flow. The remaining boundary faces(y = h, z = ±8h) are specified as impermeable walls.The axis of rotation is located 1650h (0.6286 m) away

Table 2Simulations performed to investigate the effect of system rotationfor various membrane orientationsa

Orientation Properties Coriolis Centrifugalb

Static Variable 0 0(0, 0, 0) Variable +y +y

(0, 0, 90) Variable 0 +y

(0, 180, 180) Variable +y −y

(90, 90, 0) Variable −z +x

(90, 90, 270) Variable −z −z

(90, 90, 0) Constant −z 0(90, 90, 270) Constant −z 0

a The direction in which the additional acceleration terms actare given for each case.

b Centrifugal accelerations are listed as 0 when no densitygradients exist.

from the center of the membrane. A uniform grid of(120, 20, 153) was used for all simulations.

The various simulations performed are summarizedin Table 2, which also includes the sense of the Cori-olis and centrifugal accelerations. The simulation pa-rameters are listed in the CMS column of Table 1.

3.2.1. Secondary flow patterns and their effect onsalt concentration

Fig. 10 depicts the distribution of salt concentration(in terms of mass fraction) on the membrane sur-face. All simulations involving additional accelerationterms show a greatly reduced surface concentrationand accordingly, a significant increase in permeateproduction over the conventional process. Each case,however, features a distinct concentration pattern,which can be understood by investigating the flowpatterns inside the feed channels. Fig. 11 shows thesecondary, or in-plane, velocity vectors for varioussections along the channel for the (0, 0, 0) case. Ini-tially, there are only wall-bound vortices which growtowards the center of the channel as the distance fromthe inlet is increased. These vortices account for thereduced concentration adjacent to the side walls forthis case. As these vortices meet, additional vortexpairs are formed, first at the center of the channel, andfinally filling the entire channel. Again, the effect ofthe formation of these additional vortex pairs is clearlyevident on the surface concentration distribution.

Figs. 12 and 13 show the corresponding in-planevelocity vectors and salt concentration contours atx/h = 200 for the four cases presented above. In


Fig. 10. Surface salt concentration along the channel for variousorientations. Dark shading indicates high concentrations, as percontour legend at top.

the static case, there is no secondary motion and alarge build up of concentration occurs. Orientation(0, 0, 0) features a number of vortices, or roll cells,which form adjacent to the membrane and mix thefluid locally. Orientation (90, 90, 0) features two elon-gated cells which cover the entire span causing acomplete breakdown in the symmetry of the velocityprofile and concentration distribution. Finally, orien-tation (0, 0, 90), which has no Coriolis accelerationon the bulk flow, features very weak convection cells(<10% of the strength of the previous cases) whichare considerably smaller and more numerous thanthe Coriolis induced structures evident in the (0, 0, 0)case. These cells first form next to the channel sidewalls, and then spread inwards to fill the channel.

The relative strength of these secondary motions isdemonstrated in Fig. 14 which shows relief surfacesof the streamwise velocity at the same position in thechannel. The static case is typical of unperturbed lam-inar flow with a maximum velocity at the center of

Fig. 11. Secondary velocity vectors for orientation (0, 0, 0).

Fig. 12. In-plane velocity vectors atx/h = 200 for various sim-ulations. Velocity vectors are magnified 10 times in the (0, 0, 90)case.


Fig. 13. In-plane salt concentration contours atx/h = 200 forvarious simulations. Fifteen evenly spaced contour lines are plottedbetweenφ = 0.0222 and 0.026.

the channel, whereas the effects of the secondary mo-tions are readily apparent in the cases (0, 0, 0) and(90, 90, 0). In orientation (0, 0, 0), the entire profile isshifted towards the membrane surface (y/h = −1)and the influence of the roll cells is clearly evident. Inorientation (90, 90, 0) the secondary structures carryfluid towards the lagging side wall(z = 8h) and signif-icantly broaden the profile perpendicular to the mem-brane. This broadening of the profile, which is evidentto a lesser degree in the (0, 0, 0) orientation, will serveto increase the shear stress on the membrane surface.There is no noticeable effect of the secondary motionsin the case (0, 0, 90), which has no Coriolis accelera-tion on the streamwise velocity component, as mighthave been anticipated from the weakness of the sec-ondary motions.

3.2.2. Spanwise averaged evolution of membranesurface salt concentration

In order to assess the overall effect of system rota-tion on surface concentration, the values presented inFig. 10 were averaged in the spanwise direction(z) ateach position(x) along the membrane surface. Thesespanwise averaged salt concentrations are presented inFig. 15 and provide a good overview of the effect ofrotation. Both orientations using a pitch and a roll of90◦ (Coriolis in the spanwise direction) feature a flatconcentration distribution which is only marginallyhigher (≈ 4%) than the feed concentration, offeringthe potential for both significantly enhanced permeateproduction and for the use of longer flow channels.The remaining rotating cases are similar in the sense

that the surface concentration builds until it reaches amaximum and then rapidly decreases to a seeminglyconstant level. Simulations (0, 0, 0) and (0, 180, 180),which both have Coriolis accelerations directed awayfrom the membrane surface, have the same initial con-centration growth rate; but in the former case, wherethe centrifugal acceleration is aligned with the Cori-olis acceleration, the drop-off occurs byx/h ≈ 60.In the (0, 180, 180) case, the Coriolis and centrifugalaccelerations are opposed, and the concentration dropoff does not occur untilx/h ≈ 125. The static caseand (0, 0, 90), neither of which have Coriolis accelera-tion on the bulk flow, have the same initial concentra-tion growth rate, with the latter beginning to decreaseby x/h ≈ 60, but remaining higher than the case withCoriolis accelerations on the bulk flow.

Several of these results are well correlated withthe parallel experimental program presented in theaccompanying paper [2]. In particular, a decrease inperformance of the (0, 0, 90) and the (0, 180, 180)orientations with respect to the (0, 0, 0) orientation isfound. The overall agreement of the computationalmodel with observations is very satisfactory, particu-larly when noting that experimental variations due tochanges in permeability from membrane to membraneare not adjusted for in the numerical model.

3.3. Further analysis of system rotation

3.3.1. The effect of density variationFig. 16 compares the (90, 90, 0) and (90, 90, 270)

orientation with and without density variations. In thelatter case the centrifugal instabilities are removed byswitching off the terms corresponding to variation ofdensity with concentration in the numerical solution.The figure shows that the relative increase in surfaceconcentration when density variation is used to ac-count for centrifugal accelerations is less than 0.5%for both cases. At this level of refinement, a small dis-tinction can be made between the (90, 90, 0) case andthe (90, 90, 270) case: in the latter the Coriolis and cen-trifugal accelerations are aligned and the performanceis marginally improved. Another effect of density isrevealed by referring back to Fig. 15. The only differ-ence between the (0, 0, 0) case, and the (0, 180, 180)case is that the sense of the centrifugal accelerationterm has changed with respect to the density gradient


Fig. 14. Relief plot of streamwise velocity component for conventional (static) membrane separation.

created at the membrane surface (the sense of theCoriolis acceleration is the same in both cases). In thecase of (0, 0, 180), the feed solution is stably stratifiedwith respect to centrifugal acceleration, and the devel-opment of the roll cells in the interior of the channelis delayed by approximately 60 channel half-heights.This shows that while centrifugal instabilities on theirown produce weak secondary motions there are clearinteractions between the two instabilities. The over-all significance of this interaction, however, woulddiminish greatly with increasing channel length.

3.3.2. Shear on the membraneAll membranes considered in this work have a

surface normal that is parallel to they-coordinatedirection. For this configuration, we can derive two

components of the shear on the membrane surface,

τxy = µ

(∂u

∂y+ ∂v

∂x

)(8)

τzy = µ

(∂w

∂y+ ∂v

∂z

)(9)

and the magnitude of the shear stress,

τ =√

τ2xy + τ2

zy (10)

Since, in the present work, the viscosity varies as afunction of the local concentration, the above shearstresses are calculated using the local viscosity, and ashear rate is defined using the feed viscosity. Fig. 17presents the spanwise distribution of membrane sur-face concentration and shear rate,S = (τ/µfeed), at


Fig. 15. Evolution along the channel of the spanwise averaged saltconcentrations.

x/h = 200. Each of the four cases presented aremarkedly differently, with, in general, a maximum inshear rate corresponding to a minimum in concen-tration. The static case features relatively flat profilesand a high concentration, while all other cases havegradients in the spanwise direction. The (0, 0, 0) casefeatures a large amplitude, low wavelength, variation,

Fig. 16. Spanwise averaged increase in surface concentration withthe addition of centrifugal forces through variable density.

Fig. 17. Distribution of membrane surface concentration and shearrate across the channel atx/h = 200.

while the (0, 0, 90) is characterized by a small ampli-tude short wavelength variation. Clearly, in compar-ing the static and (0, 0, 90) cases, the magnitude of theshear rate is not the determining factor in the reduc-tion of membrane surface concentration.

4. Concluding remarks

A numerical model applicable to membrane sepa-ration processes has been developed and implementedinto the TASCflow CFD code. The model is able tocorrectly model the flux versus pressure behavior in


conventional membrane separation. This model hasalso been used to demonstrate the potential for signifi-cantly improved permeate production rates in the caseof CMS and to identify three distinct flow regimesdepending on channel orientation. This improvementis made possible largely by the Coriolis acceleration,with centrifugal accelerations providing a further mod-est gain. When considering flows susceptible to con-centration polarization, there is no direct correlationbetween high shear rate and reduced surface concen-tration — small secondary motions are adequate tosignificantly reduce flux decline associated with con-centration polarization.

All CMS simulations presented above were car-ried out at a rotational speed of 50 rad/s while theactual CMS experimental apparatus runs at 158 rad/sto develop this pressure. Physically, the simulationscorrespond to a combination of a centrifuge and ahigh pressure pump to develop the process pressure.Higher rates of rotation are currently under investiga-tion and in the case of orientation (0, 0, 0), it appearsthat the flows are unsteady at a rotation rate of 75 rad/swith the previously discussed roll cells moving sideto side, merging with the wall bound vortices and re-forming near the channel center. One simulation wascarried out at a rotation rate of 158 rad/s, the shapeof the surface concentration distribution was verysimilar to the curve generated at 50 rad/s. This wouldindicate that while the flows do become unsteady, thisdoes not drastically change the physics, at least in anaverage sense.

Acknowledgements

Financial support for this work was provided by theNatural Sciences and Engineering Research Council(NSERC) and the BC Information, Science and Tech-nology Agency (First Jobs in Science and Technol-ogy). The authors are indebted to Dr. Tom Fyles forhis invaluable suggestions and discussions throughoutthe course of this work.

Appendix A. Flux equations for flow throughsemi-permeable membranes

The flux equations for the transport of both soluteand solvent can be derived using irreversible thermo-

dynamics [18]. This approach states that the entropyincrease due to reversible processes may be related tothe sum of all fluxes and their driving forces, and thatnear equilibrium the forces and the fluxes are linearlyrelated. In the membrane processes with which we areconcerned, there are two driving forces, the chemicalpotential gradient of the solvent (1µ1) and the solute(1µ2), and two mass fluxes, the solvent flux (J1) andthe solute flux (J2)

J1 = −L111µ1 − L121µ2 (A.1)

J2 = −L211µ1 − L221µ2 (A.2)

whereLij are the coupling coefficients.The chemical potential can be separated into a hy-

drodynamic pressure difference,1P , and a concen-tration gradient represented by the osmotic pressuredifference, 15, where an ideal solution has beenassumed in relating activities to concentrations. Fol-lowing [18], Eqs. (A.1) and (A.2) can be cast in termsof a total volume flux,jv and a solute mass flux,Js.

jv = Lp(1P − σ 15) (A.3)

Js = (1 − σ ′)c̄jv + ω 15 (A.4)

whereLp is the hydraulic permeability of the mem-brane,ω the solute permeability,̄c the average con-centration difference across the membrane, andσ andσ ′ are transport coefficients referred to as reflectioncoefficients. It is common at this point to invoke theprinciple of microscopic dynamic reversibility, orOnsager’s principle [21], to equateσ andσ ′.

In the literature, a perfectly rejecting membrane issaid to have a reflection coefficient of 1, a non-selectivemembrane is said to have a reflection coefficient of0, and a selective membrane to have a reflection co-efficient between 0 and 1. This term, in this context,relates only to the convective flux of solution, as it isapparent from Eqs. (A.3) and (A.4) that with

σ = σ ′ = 1.0 (A.5)

there is still a diffusive flux of solute which is propor-tional to the solute permeability. The rejection,R, is atrue measure of the solute rejection, and is defined as

R = 1 − ρpφp

ρf φf(A.6)

where ρφ is the concentration,c, the subscript ‘p’refers to the permeate, and the subscript ‘f’ refers to


the feed. If we assume that the membrane is a perfectrejector with respect to convection, given by Eq. (A.5),then the flux equations reduce to

jv = Lp(1P − 15) (A.7)

Js = ω 15 (A.8)

Since the osmotic pressure difference is representativeof the concentration gradient across the membrane,Eq. (A.8) may be rewritten as

Js = ω′ 1c (A.9)

and noting that the concentration of the permeate is

cp = Js

jv(A.10)

Eqs. (A.6), (A.9) and (A.10) can be used to solve forthe modified solute permeability

ω′ = jv(1 − R)

R(A.11)

giving a solute mass flux of

Js = jv(1 − R)

R(cf − cp) (A.12)

This means that if dynamic reversibility is assumedand if the membrane is a perfect rejector with respectto convection, the flux equations can be completelyspecified with only two experimentally determinedconstants, as was done in this paper. The hydraulicpermeability,Lp can be determined from Eq. (A.7) bymeasuring the slope of the flux versus pressure curvefor pure solvent since the15 term goes to zero inthe limit of pure solvent. The solute permeability canbe set according to Eq. (A.12) in order to achieve anexperimentally observed rejection.

References

[1] P.M. Wild, G.W. Vickers, N. Djilali, The fundamentalprinciples and design considerations for the implementationof centrifugal reverse osmosis desalination, J. ProcessMech. Eng. (Proc. Inst. Mech. Eng. E) 211 (1997) 67–81.

[2] A. Bergen, T.M. Fyles, D.S. Lycon, G.W. Vickers, Fluxenhancement in reverse osmosis using centrifugal membraneseparation, J. Membr. Sci. 176 (2000) 257–266.

[3] D.S. Lycon, Flux enhancement and fouling reduction in acentrifugal membrane process, Ph.D. Thesis, University ofVictoria, 1999.

[4] A.S. Berman, Laminar flow in channels with porous walls,J. Appl. Phys. 14 (9) (1953).

[5] Y.C. Cheng, G.J. Hwang, Experimental studies of laminarflow and heat transfer in a one-porous-wall square duct withheat transfer, Int. J. Heat Mass Transfer 38 (18) (1995) 3475–3484.

[6] C.R. Bouchard, P.J. Carreau, T. Matsuura, S. Sourirajan,Modeling of ultrafiltration: prediction of concentrationpolarization effects, J. Membr. Sci. 97 (1994) 215–229.

[7] E. Pellerin, E. Michelitsch, K. Darcovich, S. Lin, C.M. Tam,Turbulent transport in membrane modules by CFD simulationin two dimensions, J. Membr. Sci. 100 (1995) 139–153.

[8] G. Belfort, Fluid mechanics in membrane filtration: recentdevelopments, J. Membr. Sci. 40 (1989) 123–147.

[9] H. Mallubhotla, G. Belfort, Flux enhancement during Deanvortex microfiltration. 8. Further diagnostics, J. Membr. Sci.125 (1997) 75–91.

[10] G.B. Andeen, Effects of acceleration on reverse osmosisdesalination, Desalination 36 (1981) 265–275.

[11] J.C. Eid, G.B. Andeen, Effects of acceleration on particulatefouling in reverse osmosis, Desalination 47 (1983) 191–199.

[12] S. Ilias, R. Govind, Potential applications of pulsed flowfor minimizing concentration polarization in ultrafiltration,Sep. Sci. Technol. 25 (1990) 1307–1324.

[13] C.D. Bertram, H. Li, D.E. Wiley, Mechanisms by whichpulsatile flow affects cross-flow filtration, AIChE J. 44 (9)(1990) 1950–1961.

[14] U. Schaflinger, Centrifugal separation of a mixture, FluidDyn. Res. 6 (1990) 213–249.

[15] T.C Jen, A.S. Lavine, G.J. Hwang, Simultaneously developinglaminar convection in rotating isothermal square channels,Int. J. Heat Mass Transfer 35 (1) (1992) 239–254.

[16] S. Dutta, M.J. Andrews, J.C. Han, Prediction of turbulent heattransfer in rotating smooth square ducts, Int. J. Heat MassTransfer 39 (12) (1996) 2505–2514.

[17] Wei-Mon Yan, Effects of wall transpiration on mixedconvection in a radial outward flow inside rotating ducts, Int.J. Heat Mass Transfer 38 (13) (1995) 2333–2342.

[18] M. Mulder, Basic Principles of Membrane Technology,Kluwer Academic Publishers, Dordrecht, 1996.

[19] J.G. Pharoah, Computational modeling of centrifugalmembrane and density separation, Master’s Thesis, Universityof Victoria, 1997.

[20] Advanced Scientific Computing Ltd., TASCflow3d V2.4Theory Documentation, ASC Ltd., 1995.

[21] L. Onsager, Reciprocal relations in irreversible processes I,Phys. Rev. 37 (1931) 405–426.

fluid mechanics and mass transport in centrifugal membrane

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