Memb_2009_Kostoglou_Mathematical Analysis of Fluid Flow and Mass Transfer in a Cross Flow Tubular

8/12/2019 Memb_2009_Kostoglou_Mathematical Analysis of Fluid Flow and Mass Transfer in a Cross Flow Tubular

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Mathematical Analysis of Fluid Flow and Mass Transfer in a Cross Flow TubularMembrane

M. Kostoglou* , and A. J. Karabelas

DiVision of Chemical Technology, Department of Chemistry, Aristotle Uni Versity, Uni V. Box 116, 541 24Thessaloniki, Greece and Chemical Process Engineering Research Institute, Center for Research and Technology s Hellas, P.O. Box 60361, GR 570 01, Thermi-Thessaloniki, Greece

A mathematical analysis is presented for a simplied model of cross-ow in tubular membranes, for whicha numerical treatment was recently reported. It is shown, step by step, that by using several asymptotic andanalytical techniques the number of dimensionless numbers governing the problem can be reduced. Resultsare derived in terms of algebraic or integral equations, with accuracy comparable to those obtained from thecomplicated numerical solution. The analysis performed here permits an improved insight into the structureof the problem, compared to the numerical technique, and facilitates explanation of the numerically obtainedresults. The present approach can, in principle, be generalized for application to more complicated models of the particular process.

1. Introduction

Cross ow tubular membranes are extensively used in manypractical applications aimed at the removal of species of verydiverse size from salt ions (reverse osmosis) to colloidal particles(microltration). Mathematical modeling of the operation of such membranes is crucial in order to understand the detailedstructure of the ow and concentration elds, which areinaccessible experimentally, and to optimize the design and theoperation of membrane modules. In the case of ion removal,the major feature of the process is the accumulation of the soluteclose to the porous wall (concentration polarization layer) whichreduces the ux of permeate due to the osmotic pressure builtup. However, in the case of colloidal particles, the inevitablefouling layers can simultaneously exhibit characteristics of concentration polarization and of porous deposit mechanisms.Modeling of colloidal particle cross-ow ltration is verycomplicated and can be performed using continuous models, 1

discrete models, 2 or their combination. 3 Modeling of soluteremoval is relatively easier, especially in the case of completerejection of the solute. 4

There are many studies concerning the hydrodynamics intubes with porous walls. It appears that the rst attempt wasmade by Berman, 5 which was followed by several otherapproximate and analytical studies. 68 A relevant review up to1989 can be found elsewhere. 9 More recently detailed modelsbased on numerical techniques are employed; i.e., models basedon nite elements 10 and nite volumes. 11 Whereas detailedmodels are important for process design and optimization,simplied ones can be used for understanding the basic featuresof the process, for order of magnitude estimates as well as foreducational purposes. A rather simple model for the develop-ment of the concentration polarization layer in a tubularmembrane presented in ref 12 will be examined in detail here.In this model, the coupling between the ow and the polarizationlayer is only through the wall ow resistance, i.e., no concentra-

tion dependence of viscosity or diffusivity is considered. Thissimplied model is solved 12 using fairly complicated numerical

techniques requiring a ne discretization. In addition, very fewresults are given; specically, only ve curves are provided forthe polarization layer thickness along the ow. However,according to Aris, 13 simplied models are really of interest if they admit simplied solutions. Here, special approximate andasymptotic techniques are developed for the simplied model, 12

a new parametric dependence is revealed and several resultsare obtained leading to a fairly complete understanding of thenature of the problem. The proposed solution techniques canalso be used for more realistic and complicated models.

The structure of this paper is as follows: First, the mathemati-cal model is formulated in terms of the Navier - Stokesequations, the mass conservation equation, and the pressure drop

at the wall model. Then an analytical velocity prole is obtainedand validated against a numerically computed one. The masstransfer equation is derived and it is transformed into auniversal form for the case of a thin concentration polarizationlayer with constant permeation velocity. Approximate solutiontechniques are developed for the mass transfer equation permit-ting insight into the problem and applied to the general case of non-constant permeation velocity. Finally, several results arepresented and discussed.

2. Problem Formulation

The problem under consideration is as follows. The uidenters the porous tube (tubular membrane) having an elevatedpressure, P e, whereas the uid pressure outside the tube is P o.The pressure difference leads to a wall velocity V w with outwarddirection and to a two-dimensional velocity prole in the tube.The outward directed ow leads to a reduction of the ow ratealong the tube and since the solute cannot penetrate the porouswall, its concentration increases, especially at the membrane -liquid interface. This phenomenon is usually referred to asconcentration polarization. The concentration eld of the soluteis described by a convection - diffusion equation. The masstransfer problem exhibits a two way coupling with the uiddynamics problem through the increase of the resistance to theoutward ow caused by the accumulation of solute near thewall leading to reduced wall velocities. This reduction of

* To whom correspondence should be addressed. Tel: + 30-2310997767. Fax: + 30-2310997759. E-mail: [email protected].

Division of Chemical Technology, Department of Chemistry,Aristotle University.

Chemical Process Engineering Research Institute, Center forResearch and Technology s Hellas.

Ind. Eng. Chem. Res. 2009, 48, 58855893 5885

10.1021/ie900056c CCC: $40.75 2009 American Chemical SocietyPublished on Web 05/13/2009


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the wall velocity (trans-membrane ux) is the basic feature of the concentration polarization. Regarding the other parametersof the problem, L is the length of the tube, R its internal radius,U o the average inlet uid velocity, and C o the inlet soluteconcentration.

A rigorous derivation of the mathematical problem wouldrequire consideration of the effect of the solute concentrationon the local properties (density, viscosity, diffusion coef-cient) of the uid. 14,15 A simpler alternative, albeit ap-

proximate, approach (usually employed in the literature12

)is to assume constant uid properties and to account for theconcentration polarization through an increase of the effectiveow resistance of the porous wall. This approach, permittingsignicant analytical insight into the problem, will befollowed here. The mathematical problem to be solved ispresented in the following. To avoid repetition of the systemequations, they are given directly in terms of dimensionlessvariables. In particular, all velocities are normalized withrespect to U o (including V w), all lengths with respect to R,the concentration is normalized with respect to C o and thepressure with respect to FU o2. The axial and radial coordinatesin the tube are denoted as x and r , respectively.

Continuity equation,

U x

+ V

r ) 0 (1)

Axial momentum equation,

U U x

+ V U r

) - p x

+ 2 N R(

2U x 2

+ 1r

r

U r ) (2)

Radial momentum equation,

U V x

+ V V r

) - p r

+ 2 N R(

2V x 2

+ 1r

r

V r ) (3)

In the above equations, U and V are the axial and radial

velocities, respectively, and N R is the main ow Reynoldsnumber dened as N R ) 2U o RF / where F is the constant densityand is the constant viscosity of the solution.

The solute transport equation is given as follows:

U c x

+ V c r

) 2ScN R(

2c x 2

+ 1r

r

c r ) (4)

where Sc is the Schmidt number for the solution.The boundary conditions for the above system of equations

are as follows: Inlet boundary (x ) 0) A fully developed ow prole is considered.

U (0, r ) ) 2(1 - r 2

) (5a)V (0, r ) ) 0 (5b)

c(0, r ) ) 1 (5c)

Outlet boundary (x ) L)Fully developed ow and concentration proles are consid-

ered i.e.:

U x

) V

x )

c x

) 0 (6)

It is noted that this is a weak boundary condition, i.e.,provided that L is long enough to avoid inlet transient, therange of inuence of the outlet boundary condition is lessthan one radius of the pipe. For example, the solutions for L) 100 and L ) 200 are exactly the same up to x ) 99.

Axial symmetry conditions (r ) 0)

U r

) c r

) V ) 0 (7)

Porous wall surface (r ) 1)

U ) 0 (8a)

V ) V w (8b)

V wc ) 2Sc N R c r

(8c)

Finally, a relation between the local wall ux and the solutedistribution in the concentration polarization layer is needed. Itis assumed (as is usual in practical situations) that the axialpressure drop in the pipe (although it is the main design variablefor tubular membranes) is insignicant in comparison with thedifference ( P e - P o); thus, the axial evolution of V w is due onlyto the respective variation of the resistance to uid permeation.A fairly general expression for the axial variation of permeationvelocity V w( x ) based on the resistances in series model 16 is asfollows:

V w( x ) )V wo

1 + R (c( x , r )) (9)

The symbol R is not a function but a generalized functionalacting on the local (with respect to x direction) radial distributionof the solute concentration. It denotes the ratio of concentrationpolarization layer resistance to the pure wall resistance. Thevelocity V wo is the wall velocity in the absence of solute (i.e.,R (0) ) 0).

3. Problem Solution

3.1. Fluid Dynamics Problem. The uid dynamics problemwill be dealt with rst to determine the ow eld for an arbitrary

wall velocity distribution V w( x ). A numerical solution can beobtained using standard techniques to treat the Navier - Stokesequations but in any case over 10 5 degrees of freedom (unknownvalues of discretized variables) are required to achieve highaccuracy. Fortunately, under the conditions usually prevailingin practice, a drastic simplication can be safely made, leadingto explicit expressions for the ow eld.

The approach here is similar to the one employed in the well-known lubrication approximation based on the slow variationof variables along the x -direction. 17 The main idea is thedecomposition of the ow eld to a x -dependent and anr -dependent component; i.e., a self-similar ow eld with respectto the r -prole. According to this approach, the pressure isuniform in each cross-section, the U velocity is obtained fromeq 2, the V velocity from the continuity eq 1, whereas eq 3 isignored as its terms are much smaller than those of eq 2. Theexpressions for the two components of the velocity which satisfythe continuity eq 1, are given as follows:

U ( x , r ) ) U ave( x )d f (r 2)dr 2

(10)

V ( x , r ) ) V w( x ) f (r 2)

r (11)

where the dimensionless cross-sectionally averaged axial veloc-ity can be determined from a total mass balance as follows:

U ave( x ) ) (1 - 2 0 x

V w( y) d y) (12)

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As long as V w( x )/ U ave( x ) , 1 and V w( x ) varies smoothly along x , the radial function f can be obtained from the solution of anonlinear one-dimensional boundary value problem having asparameter the local value of N Rw( x ), which is a very importantparameter called wall-Reynolds number and dened as N Rw) V w( x )U o2 RF / . This number accounts for the inertia contribu-tion to the radial ow. The rst order, with respect to N Rw,velocity elds are given in ref 5 for planar geometry and in ref 18 for cylindrical geometry. In the limit of N Rw f 0, thecorresponding nonlinear term can be ignored, and the equationfor f is solved analytically, leading to the following velocityproles:

U ( x , r ) ) 2(1 - 2

0 x

V w( y) d y)(1 - r 2) (13)

V ( x , r ) ) V w( x )(2r - r 3) (14)

In general, the velocity proles are a weak function of N Rw,and the above expressions can be used for N Rw as large as 0.5.To conrm the validity of the analytical solution for the radialvelocity, the hydrodynamic problem was solved using the niteelement method. The results for the particular case with N R )250, N Rw ) 0.1 and L ) 200, using a discretization grid of 66000 elements, are presented in Figure 1. In particular, the radialvelocity proles for x ) 1, x ) 10, and x ) 50 are shown andcompared with the analytical prole eq 14. It is noted that asthe uid enters the porous tube, the radial velocity prole adjusts

itself to the fully developed asymptotic state very quickly (i.e.,at x much smaller than 1). At x ) 1, a prole which isqualitatively (but no quantitatively) similar to the asymptoticone is already established. The prole converges slowly to theasymptotic one which is reached at x ) 50. It is obvious thatthe asymptotic prole is actually the one given by eq 14. Thewiggles in the numerical proles are due to numerical errors,even though a ne discretization grid is employed. It is stressedthat ignoring the ow eld evolution at the entrance is notimportant since for the present application only the region closeto the wall (where the concentration polarization layer isdeveloped) is of interest. In Figure 1, one observes that in theregion r > 0.9, the velocity prole is very close to the asymptoticeven for x ) 1.

3.2. Mass Transfer Problem. The solution for the masstransfer will be presented step by step in order to make clear

the nature of the problem and the approximation technique, aswell as to derive several asymptotic results. As discussed above,for cases of interest, the ow eld is not dependent on the twoReynolds numbers; thus the inuence of the N R, N Rw, and Scon mass transfer is through the two Peclet numbers Pe ) N RwSc/2 and Pe x ) N RSc. However, in any case Pe x /Pe . 1(affected by the ratio U o / V w) and, because in practice Pe isgenerally large, the axial diffusion term can be safely ignored.Here it is noted that the uniform wall velocity problem has been

solved19

for both planar and cylindrical geometry using expan-sion in eigenfunctions and the Frobenius method for solvingthe corresponding eigenvalue problems. In addition, approximateexpressions for the wall concentration were derived. 19 Thesetechniques and expressions cannot be used for the case of variable wall velocity which is of interest in the present work.

3.2.1. Uniform Wall Velocity. First, the problem of constantpermeation velocity V w along the tube is examined. The masstransfer equation in this case takes the form:

2(1 - r 2)U o(1 - 2V w x ) c x

+ U oV w(2r - r 3)

c r

) D R

1r

r r

c r

(15)

Zero Wall Curvature. According to eq 15, a concentrationpolarization layer will be developed close to the porous wall.The thickness of this layer will increase in the main owdirection. In case of , R, it is sufcient to solve eq 15 onlyin a thin region close to the wall, so the coordinate r can bewritten as r ) 1 - s. Substituting the new coordinate s in eq15, expanding its terms with respect to s and retaining only theleading order terms (since s , 1) leads to:

4U os(1 - 2V w x ) c x

- U oV w c s

) D R

2c s2

(16)

Employing the new independent variable z ) - ln(1 - 2V w x )in place of x and substituting in eq 16 one obtains 8 s( c)/( z)

- ( c)/( s) ) (1)/(Pe)( 2

c)/( s2

). The nal step is the introduc-tion of the normalized variables s* ) sPe and z* ) zPe2 /8. Themass balance takes the form:

s* c z*

- c s*

) 2c s*2

(17)

with the boundary conditions,

c ) - c s*

at s* ) 0 (18a)

c ) 1 at s* f (18b)

c ) 1 at z* ) 0 (18c)

The second condition 18b can be set to innity due to theassumption of small .Equation 17, which also holds for the case of two parallel

porous plates is a universal one, i.e., it contains no explicitdependence on the parameters of the present problem whichare absorbed in the independent variables. As this equationcannot be solved analytically, a numerical solution is neededto determine the concentration eld c( z*, s*). The nite elementmethod with a ne grid is used. The s* direction is truncatedand a zero normal ux boundary condition is used at the positionof the truncation. The truncation distance must be far enoughfrom the wall so that it does not inuence the concentrationproles. The normalized concentration proles [ c( z*,s*) -1]/[c( z*,0) - 1] in the s* direction for several values of z* areshown in Figure 2. These proles can be at least qualitativelydescribed by an exponential function; such a t is shown for

Figure 1. Numerical and analytical normalized radial velocity proles atseveral positions in a tubular membrane ( N R ) 250, N Rw ) 0.1).

Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009 5887


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one of the curves. It is noted that the tting must be done notusing a general exponential but an exponential with value 1 ats* ) 0. The nearly exponential prole is compatible with theconvection - diffusion (in s* direction) terms of eq 17. The thirdterm of eq 17 leads to a slower variation in the z* directionand can be obtained approximately in explicit form. Thus, letus consider that

c ) 1 + e- ms* (19)From boundary condition 18a, one obtains that m ) ( +

1)/ .Integrating eq 17, with respect to s* from zero to innity

leads to the following:

dd z* 0

s*c ds* - (c( z*, ) - c( z*, 0)) ) ( c s*)s*) - ( c s*)s*) 0

(20)

Using boundary conditions 18a the above equation istransformed to the following:

dd z* 0

s*c ds* ) 1 0

s*c ds* ) z* (21)

The constant term in the expression for concentration c, eq 19,is independent of z; thus, all the z-dependence is attributed to theexponential term. Substituting the relation c ) e- (1 + )/( )s* in eq21 and performing the integration leads to

3(1 + )2

) z* (22)

By numerically solving this equation, the function ( z*) canbe determined. The value of is related to the wall concentrationas follows:

) c( z*, 0) - 1 (23)In Figure 3 the value of z* versus cw - 1, computed by the

approximate eqs 22, 23, and by the numerical solution of 17, isshown and compared with the numerically obtained wallconcentration. It is obvious that the theory leads to qualitativelycorrect results. From eqs 22 and 23, one obtains that theasymptotic slope of the approximate z* versus cw - 1 curve is1. The same behavior was observed in the numerically computedcurve. Thus, there is a constant difference between the ap-

proximate and exact z* values. This means that as z* increases,the relative error decreases. It is noted that in order to achievethe universal solution some terms of order s are disregardedin the expansion of the velocity eld with respect to s. Thisresults in a nonconservative ow eld in eq 16 and a nal errorof order V w x . Consequently, according to these arguments theuniversal solution is exact in the limit V w x , 1, V w x Pe2 . 1.Furthermore, it can be shown, that in this limit, the approximateexpression for the wall concentration in tubular membranes,given in ref 19, can be expanded in a form similar to the oneresulting from the universal equation; i.e., cw ) V w x Pe2 /4.

After the validation of the approximation approach for thecase of the universal mass transfer equation, it is applied for

the variables s, z. To demonstrate the dependence of the problemon the typical dimensionless numbers, the ratio N Rw / N R will beused in place of V w in the rest of the work. The only differenceis that the equation for the evolution of is derived byperforming a total mass balance up to the position z; i.e., thesolute excess must match the amount of the permeated solvent.The nal result is as follows:

3

(1 + )2 )

Pe2

4 N Rw N R

x (24)

c ) 1 + e- + 1

Pes (25)

After an initial development, the function goes from 0 to

a value much larger than unity and then the following asymptoticsolution is valid.

c ) Pe2

4 N Rw N R

x e- Pes (26)

Finite Wall Curvature. As in the case of zero curvature(boundary layer very thin with respect to the pipe radius), theasymptotic concentration prole is the solution of the convectiondiffusion equation in radial direction. To determine the shapeof the asymptotic prole in the case of nite curvature (nitewith respect to tube radius boundary layer thickness) thecorresponding radial convection diffusion equation mustbe solved:

(2r - r 3) c r

) 1Pe

1r

r

r c r

(27)

Figure 2. Normalized concentration proles ( c - 1)/(cw - 1) from thenumerical solution of the universal thin layer equation at three positionsalong the main ow direction. The exponential tting for the x ) 10 curveis also shown. Figure 3. Comparison between numerically and approximately computed

relation between distance along the ow z* and wall concentration(cw - 1), for the universal thin layer equation.



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It is noted that this equation is not compatible with the

boundary condition at r ) 1 and, since it gives only the radialprole of the concentration, it is normalized to give c ) 1 at r ) 1. The result is as follows:

c )

r 1r

ePe(r 2- r 4 /4) dr

1 1

r ePe(r

2- r 4 /4) dr (28)

A small value is used instead of zero because the integralsdiverge at zero (a consequence of the cylindrical geometry andthe approximate character of the solution). The important issueis that eq 28 converges to a particular prole for a range of values of . One of these values must be chosen for thecomputation of the integrals. It is proposed to use the valuewhich minimizes the integrands in eq 28. Setting the derivativeof the integrand equal to zero leads to the expression:

) (1 - (1 - 1/Pe)1/2)1/2 (29)which for Pe . 1 is simplied to ) (2Pe)- 0.5.

At this point, it is important to examine the proles of c forseveral values of Pe. As shown in Figure 4, if the independentvariable scales as Pe(1 - r ) the concentration proles coincidefor Pe > 20. Only the prole for Pe ) 10 is somewhat different.In addition, all of the proles can be tted very well by a simpleexponential. Actually, the t is much better than the corre-sponding t for the zero-curvature case. The above asymptoticprole is necessary to derive the shape of the trial function forthe solution of the mass transfer problem in the tube. Theanalysis shows that despite the cylindrical geometry and thenonuniform radial velocity proles the simple exponentialfunction is the best choice.

The procedure is the same as in the case of zero curvature.An exponential trial function is chosen and the application of the wall boundary condition leads to the following:

c ) 1 + e- + 1

Pe(1 - r ) (30)

The difference here is in the mass balance where thecylindrical geometry must be accounted for:

01

4(1 - r 2)r e- + 1

Pe(1 - r ) dr )

2 N Rw x

N R (1 -

2 N Rw x

N R )- 1

(31)

Performing the integration and considering that the termexp(Pe(1 + )/ ) is vanishingly small, the evolution equationfor ( x ) is obtained,

8 (( (1 + )Pe)2

- 3( (1 + )Pe)3

+ 3( (1 + )Pe)4) )

2 N Rw x N R (1 -

2 N Rw x N R )

- 1

(32)

The far eld asymptotic result ( . 1) is as follows:

c ) N Rw x Pe

2

4 N R (1 -2 N Rw x

N R )- 1

e- Pe(1 - r ) (33)

The concentration boundary layer thickness c is dened asthe distance from the wall where the difference between thelocal and the inlet concentration is smaller than 0.1% (as in ref 12). Then according to the above analysis,

c )

(1 + )Peln(10 3 ) (34)

with the far eld asymptotic result,

c ) 1Pe ln[103 N

Rw x Pe2

4 N R (1 -2 N

Rw x

N R )- 1

] (35)3.2.2. Non-Uniform Permeation Velocity. In reality, the

wall velocity varies in the main ow direction and this variationis associated with the concentration prole in the porous tube.The mass transfer equation which must be solved in this casetakes the form:

2(1 - r 2)U o(1 - 2 N Rw N R x ) c x

+V wo

1 + R (c)(2r - r 3)

c r

)

D1r

r r

c r

(36)

with the wall boundary condition,V wo

1 + R (c)c )

2Sc N R

c r

(37)

The effect of the variation of wall velocity in the main owdirection, causes a variation of the local Pe number, andcorrespondingly, a variation of the concentration boundary layerstructure. However, the axial ow variation is very slow in thescale of the transverse phenomena; thus, the approximatetechnique developed in the previous section, based on thedecomposition of the problem, to fast transverse phenomenaand slow axial phenomena, can still be used. Assuming againan exponential prole, substituting to the wall boundary

condition and performing a solute mass balance between theentrance of the tube and the location x, leads to the followingsystem of equations (where Pe and N Rw are dened using theinitial wall velocity value V wo):

c( x , r ) ) 1 + e- + 1

Pe

1+ R (c)(1- r ) (38)

8 (1 + R (c))2

Pe2 ((

(1 + ))2

- 3( (1 + ))31 + R (c)

Pe +

3( (1 + )Pe)4(1 + R (c)Pe )

2) )2 N Rw N R

0 x 1

1 + R (c) d x (1 - 2 N Rw N R 0 x 11 + R (c) d x )

- 1

(39)

For a particular functional, R , the system of the aboveequations can be solved for the evolution of the function ( x )

Figure 4. The function c versus normalized radial distance for several valuesof Pe.



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along the tube. Here, two specic forms will be tested but it isstraightforward to generalize the analysis for any form of R (c).The rst form of the function (Model I) corresponds to theassumptions made that the resistance of the polarization layeris just the product of the concentration polarization layerthickness and a specic resistance which is assumed to be aconstant (the Carman - Kozeny equation is invoked for this inref 12). This model is quite unrealistic since it is implied thatregions with concentration 0.2% larger than the inlet concentra-

tion contribute equally to regions with innitely large concentra-tion. Despite its inadequacy, this model is used here to comparethe analytical with the numerical results of ref 12. Thecorresponding form of the functional R is as follows:

R (c) ) a c (40)

It is noted that this model is extremely susceptible toapproximation errors either using numerical discretizationtechniques or approximation techniques as the present one forthe solution of the mass balance equation. Even a small localerror in the value of concentration (which is unavoidable dueto the discretization) leads to a large error in the estimation of polarization layer thickness c and this is transformed im-

mediately to a global error through eq 40. Even if the overallproblem is stable, its convergence rate is very small and a veryne grid is needed to obtain convergent results.

The second model used here (Model II) is based on the properapplication of the Carman - Kozeny relation to the particularproblem. In the limit of large porosity, taking also into accountthe variation of the porosity in the polarization layer, one obtains,

R (c) ) a 0

(c - 1)2 d y (41)

It is noted that the excess concentration c - 1 is used insteadof c to denote that only the local accumulation of the soluteleads to increase of the pressure drop across the wall.

Equation 41 does not relate global to local quantities as eq

40; thus, the convergence difculties of Model I are notencountered here.

Case I. Applying the equation dening c (i.e., from c( x , c)) 0.001) and using eqs 38 and 40 leads to the followingexpression for c:

c )

(1 + )Peln(10 3 )(1 - a (1 + )Pe ln(10 3 ))

- 1

(42)

Substituting in eq 39, the following expression is derived

R (c) ) a (1 + )Pe

ln(103 )(1 - a (1 + )Pe ln(10 3 ))- 1

(43)

which is a nonlinear integral equation for ( x ). This equationis solved numerically to determine the function ( x ) andsubsequently c from 42 and V w from V w ) V wo /(1 + a c).

Case II. First, the concentration distribution eq 38 issubstituted in eq 41, leading to the following:

R (c) ) a(1 +R (c)) 3

2(1 + )Pe R (c) )

a 3

2(1 + )Pe(1 - a 3

2(1 + )Pe)- 1

(44)

Again substitution of eq 44 in 39 leads to a nonlinear integralequation which must be solved numerically for the function ( x ).Then the evolution of c and V w is computed in a straightforwardmanner through the relations:

c )

(1 + )Peln(103 )[1 + a 32(1 + )Pe(1 - a

3

2(1 + )Pe)- 1](45)

V w ) [1 + a 32(1 + )Pe(1 - a 3

2(1 + )Pe)- 1]- 1 (46)

It is noted that a very similar problem to the one discussedhere was studied recently in ref 20. The focus in that work ison planar geometry with a relation for the permeation velocitytaking into consideration the osmotic pressure increase alongthe ow (instead of the ow resistance in the present work).The mathematical problem is solved analytically in ref 20. Theresulting concentration proles in the transverse to the mainow direction are rather complicated including Airy functions.These proles are substituted in the integral mass balance toget a much more complex nal computational procedure forthe variation along the ow than the simple integral equationof the present work. The results in ref 20 are not comparedwith a numerical solution of the complete problem, whereashere the results are compared with those of ref 12.

4. Results

In what follows, the effect of the operating conditions on theevolution of the thickness of the concentration polarization layeralong the tubular membrane is examined through the presenta-tion of typical results. The case of constant wall Velocity isconsidered rst. Evidently, three dimensionless numbers (Sc, N R, N Rw) affect the problem. The evolution of the concentrationpolarization layer thickness along the membrane is shown inFigure 5 for N Rw ) 0.1 and two values of N R and Sc numbers(in the range of parameters of practical relevance used by 12 ).The evolution curves consist of two periods. In the rst period,the boundary layer is developed very quickly (the penetrationperiod). During this period, most of the accumulating solutecontributes to the expansion of the boundary layer and not to

the increase of its average concentration. At some axialdistance, a pseudoequilibrium is established, and the con-centration prole attains a constant shape; then, the onlyincrease of the concentration boundary layer thickness is dueto the increase of the wall value of the solute concentration.

Figure 5. The evolution of concentration polarization layer thickness alongthe main ow direction for N Rw ) 0.1 and several values of Sc and N R.



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This evolution is rather articial, and it is brought about bythe particular denition of the boundary layer thickness; thekey point is that it is dened with respect to the inletconcentration (0.1% excess). If it were dened with respectto the boundary concentration at each cross section, then thevalues in the second period would be constant. The evolutionof c in the second period can be described by a logarithmicfunction ( c ) c1 + c2ln( x )), which under certain conditionsresembles the linear growth rate referred to in ref 12, e.g.,see the two curves for N R ) 1000 in Figure 5.

From Figure 5, it is obvious that c is reduced with increasing N R and Sc, as expected. However, as it has been already shown,the ow eld (provided N Rw < 0.5) does not depend on the twoReynolds numbers. These Reynolds numbers have an effect onlythrough the total mass balance and enter only through theparameter ( N Rw / N R) x . In addition, the thickness depends directlyon the Pe number which is related the product of Sc and N Rw.Thus, using the new length parameter ( N Rw / N R) x and the Penumber, the four parameters Sc, N Rw, N R, x used in ref 12 canbe reduced to two, namely c ) c(Pe, [ N Rw / N R] x ). Thepolarization layer thickness normalized by 1/Pe for properscaling is shown in Figure 6, using the new length coordinatefor several values of Pe. Actually, this gure includes all of the

information that can be derived for the constant wall velocityproblem.The above reduction of the problem parameters holds even

in the case of adding a pressure drop model for the evolutionof V w. For the two models I and II employed in the presentwork (section 3.2.2) the only additional parameter (the thirdone) is the coefcient a of the pressure drop model. In Figure7, the evolution of the thicknes c for the particular case Pe )100 is shown for several values of the parameter a and thepressure drop Model I. With increasing a, the wall velocity tendsto decrease along the ow leading to a reduction of the localPeclet number. But a reduced Peclet number implies an increaseof the effect of diffusion and thickening of the concentrationboundary layer. This thickening further reduces the wall velocityand so on. From Figure 7, one can deduce that the localthickness has qualitatively a nearly linear dependence on the

parameter a . The reduction of the wall velocity along the owis shown in Figure 8 (in case of a ) 0, V w / V wo ) 1 identically).A very fast initial reduction is followed by a slow smoothdecrease of the velocity.

The evolution of thickness c for the pressure drop Model IIis shown in Figure 9 for several values of a. In this case, thelocal thickness exhibits a kind of logarithmic dependence onparameter a. In particular, the evolution of the wall velocityalong the tube is very different from that of Model I presentedin Figure 7. The velocity decrease is zero at the beginning, asshown in Figure 10, and then after passing through an inectionpoint tends to increase. The difference between the two modelsis due to the fact that the resistance to the wall ow increasesmuch slower than the increase of the average concentration in

Figure 6. The evolution of concentration polarization layer thickness alongthe main ow direction, using normalized variables for N Rw ) 0.1 andseveral values of Pe.

Figure 7. The evolution of polarization layer thickness along the main ow

direction, for pressure drop model I, Pe ) 100 and several values of parameter R.

Figure 8. The evolution of permeation velocity along the main owdirection, for pressure drop model I, Pe ) 100 and several values of parameter R.



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the pseudoequilibrium period and much faster in the initialpenetration period.

It is noted that the approximate semianalytical results obtainedhere for the evolution of the concentration polarization layerare quite comparable (regarding accuracy) to the results obtainedfrom a numerical solution with discretization of the equationsusing a uniform grid (e.g., in ref 12). This is probably due tothe fact that the concentration boundary layer is very thin andmay be inadequately represented by a rather limited number of discretization points if a uniform grid is used. However, theapproximate method presented here is free from restrictions onthe thickness of the polarization layer.

Another important point is that the approximation methodbased on the decomposition to fast radial and slow axialdynamics, derived here and applied to the simplest case of constant properties, can be extended to the more realistic caseof concentration-dependent ow properties (e.g., ref 15).Another possible extension may be to the case of a non-uniformbase ow (static or dynamic) which can be generated byturbulent promoters and inserts (e.g., refs 21 and 22). Thesedirections will be taken in the future.

5. Conclusions

A rather simple model for the development of the concentra-tion polarization layer along a tubular membrane, previouslystudied in the literature using numerical tools, 12 is extended,reformulated, and studied here using approximate analyticalmeans. It is shown that the complete ow eld can be describedby closed form expressions. The mass transfer equation in thelimit of a very thin polarization layer can be written in universalform, i.e., all of the parameters are absorbed into the problemvariables. Additionally, it is shown that the parameters thatdetermine the problem are the wall Peclet number, the properlynormalized distance in the main ow direction, and a wallpressure drop model parameter. It is recognized that the masstransfer equation, with the exception of an initial period of fastpolarization layer growth, can be decomposed into fast termsacting in the radial direction and slow terms acting in theaxial direction. Approximate solution techniques based on thisdecomposition are developed, leading to rather simple equationsfor the evolution of the boundary layer thickness and the wallvelocity, permitting the elucidation of the problem and a betterinterpretation of the numerically obtained results.

Nomenclature

a parameter of the law for the permeation velocity (seeeqs 40, 41)

c solute concentration (dimensionless by C o)C o inlet solute concentrationcw wall concentration D solute diffusivity L length of the membrane tube (dimensionless by R) N R main ow Reynolds number ( ) 2 RFU o / ) N Rw wall ow Reynolds number ( ) 2 RFU oV w / )

P e pressure inside the tubePe wall Peclet number ( ) V wU o R / D)Pe x axial Peclet number ( ) U o2 R / D)P o pressure outside the tube R radius of the membrane tubeR (c) dimensionless functional of the radial concentration

prole associated with the ow resistance (eq 9)s distance from the wall (dimensionless by R)Sc Schmidt number ( ) /(FD))U axial velocity (dimensionless by U o)U ave cross-sectional average velocity in the tube (dimen-

sionless by U o)

U o cross sectional average inlet velocityV w wall velocity (dimensionless by U o)

Figure 9. The evolution of polarization layer thickness along the main ow

direction, for pressure drop model II, Pe ) 100 and several values of parameter R.

Figure 10. The evolution of permeation velocity along the main owdirection, for pressure drop model II, Pe ) 100 and several values of

parameter R.



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x ,r axial and radial coordinates in the tube (dimension-less by R)

Greek Letters

parameter of the r direction concentration prolevarying along the ow

concentration boundary layer thickness c concentration boundary layer thickness (dimension-

less by R) uid viscosityF uid density

Literature Cited

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(2) Yoon, S.-H.; Lee, C.-H.; Kim, K.-J.; Fane, A. G. Three dimensionalsimulation of the deposition of multi-dispersed charged particles andprediction of resulting ux during cross ow microltration. J. Membr. Sci.1999 , 161 , 720.

(3) Chen, J. C.; Kim, A. S. Monte Carlo simulation of colloidalmembrane ltration: Principal issues for modeling. Ad V. Colloid InterfaceSci. 2006 , 119 , 3553.

(4) Sekino, M. Precise analytical model of hollow ber reverse osmosismodules. J. Membr. Sci. 1993 , 85 , 241252.

(5) Berman, A. S. Laminar ow in channels with porous walls. J. Appl.Phys. 1953 , 24 , 12321235.

(6) White, F. M. Viscous Flow ; McGraw Hill: New York, 1974.(7) Weissberg, H. Laminar ow in the entrance region of a porous pipe.

Phys. Fluids 1959 , 2 , 510516.(8) Kinney, R. Fully developed frictional and heat transfer characteristics

of laminar ow in porous tubes. Int. J. Heat Mass Transfer 1968 , 11 , 13931401.

(9) Belfort, G. Fluid mechanics in membrane ltration: Recent develop-ments. J. Membr. Sci. 1989 , 40 , 123147.

(10) Nassehi, V. Modeling of combined Navier - Stokes and Darcy owsin crossow membrane ltration. Chem. Eng. Sci. 1998 , 53, 12531265.

(11) Damak, K.; Ayadi, A.; Zeghmati, B.; Schmitz, P. A new Navier -Stokes and Darcys law combined model for uid ow in crossow ltrationtubular membranes. Desalination 2004 , 161 , 6777.

(12) Pak, A.; Mohammadi, T.; Hosseinalipour, S. M.; Allahdini, V. CFDmodeling of porous membranes. Desalination 2008 , 222 , 482488.

(13) Aris, R. Mathematical Modeling Techniques ; Dover: New York,1994.

(14) Wiley, D. E.; Fletcher, D. F. Techniques for computational uiddynamics modeling of ow in membrane channels. J. Membr. Sci. 2003 ,

211 , 127137.(15) Bessiere, Y.; Fletcher, D. F.; Bacchin, P. Numerical simulation of colloid dead-end ltration: Effect of membrane characteristics and operatingconditions on matter accumulation. J. Membr. Sci. 2008 , 313 , 5259.

(16) Chapter 12: Membrane Filtration in Water Treatment Principlesand Design ; Wiley: New York; 2005.

(17) Kostoglou, M.; Karabelas, A. J. On the structure of the single-phase ow eld in hollow ber membrane modules during ltration. J. Membr. Sci. 2008 , 322 , 128138.

(18) Yuan, S. W.; Finkelstein, A. B. Laminar ow with injection andsuction through porous wall. Trans. ASME 1956 , 78 , 719724.

(19) Sherwood, T. K.; Brian, P. L. T.; Fisher, R. E.; Dresner, L. Saltconcentration at phase boundaries in desalination by reverse osmosis. Ind. Eng. Chem. Fund. 1965 , 4 , 113118.

(20) Kim, A. S. Permeate ux injection due to concentration polarizationin crossow membrane ltration: A novel analytic approach. Eur. Phys. J. E. 2007 , 24 , 331341.

(21) Koutsou, C. P.; Yiantsios, S. G.; Karabelas, A. J. Numericalsimulation of the ow in a plane channel containing a periodic array of cylindrical turbulence promoters. J. Membr. Sci. 2004 , 231 , 8190.

(22) Koutsou, C. P.; Yiantsios, S. G.; Karabelas, A. J. A numerical andexperimental study of mass transfer in spacer-lled channels: Effects of spacer geometrical characteristics and Schmidt number. J. Membr. Sci. 2009 ,326 , 234251.

Recei Ved for re View January 14, 2009 ReVised manuscript recei Ved March 31, 2009

Accepted April 6, 2009

IE900056C


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