Journal of Membrane Science, 28 (1986) 191-208 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

191

FLUID FLOW IN AN IDEALIZED SPIRAL WOUND MEMBRANE MODULE

SIDDHARTH G. CHATTERJEE and GEORGES BELFORT

Department of Chemical Engineering and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 (U.S.A.)

(Received November 15, 1985; accepted in revised form March 4, 1986)

Summary

Analytical and numerical solutions for steady laminar incompressible flow in an idealized spiral wound membrane (annular) duct are developed and computed. Axial and radial velocity profiles and axial pressure distributions for injection and suction are presented. The analytical method uses a perturbation technique in wall Reynolds num- ber (Re,) to solve the Navier-Stokes problem. The usual assumptions of uniform injection or suction and similar velocity profiles with axial distance are invoked. For most membrane processes where Re, < 1, the analytical solution is sufficient and the assumptions are valid after the developing entrance region. For Re, > 1, numerical calculations reveal that similarity in the velocity profiles (assumed for the analytical development) disappear and that the radial velocity profile exhibits inflections. These results could be used for estimating pumping requirements and for effecting design changes in spiral wound modules.

Introduction

With the development about 25 years ago of asymmetric high-flux syn- thetic membranes, a series of new pressure-driven membrane processes such as hyperfiltration (also called reverse osmosis), ultrafiltration and microfiltration have recently gained acceptance for industrial scale separ- ations [l] . The membranes have been packaged in pressure modules of various design. In a majority of the designs the feed solution moves across the membrane in tangential flow (often called cross-flow), allowing the permeation to be removed by suction through the duct or channel walls. Common cross-sectional geometries for these ducts include tubular, slit or rectangular with one or two porous faces, and spiral wound with both walls porous. It has been known for a while that the performance of pressure- driven membrane processes is related to the tangential fluid mechanics across the membrane face and that concentration polarization of dissolved solutes (ions, macromolecules) can be predicted and controlled at the membrane-solution interface through an understanding of the fluid mech- anics and mass transfer [2-6].

TA

BL

E

1

Flow

in

por

ous

du

cts:

A

lit

erat

ure

su

mm

ary

Geo

met

ry

Wal

l fl

ow

Ass

um

pti

ons*

S

olu

tion

m

eth

od

Con

clu

sion

s R

efer

ence

Sli

t S

uct

ion

(2

por

ous

wal

ls)

Su

ctio

n/i

nje

ctio

n

(2 p

orou

s w

alls

)

Su

ctio

n/i

nje

ctio

n

2D,

SS,

IC,

L, P

, U

S/I

, (2

por

ous

wal

ls);

NF,

ig

nor

e (V

*Vu

) al

so f

or t

ub

e in

erti

al t

erm

s

Su

ctio

n/i

nje

ctio

n

(2 p

orou

s w

alls

) 2D

, IC

, L,

US

/I,

NF

Su

ctio

n

(2 p

orou

s w

alls

)

Su

ctio

n

(On

e p

orou

s w

all)

2D,

SS,

IC,

L, P

, U

S,

NF

2D,

SS,

IC,

L, P

, U

S/I

, N

F

2D,

IC,

L, P

, U

S,

NF

2D,

IC,

L,

P, U

S,

NF

Per

turb

atio

n

for

Re,

<

1

Per

turb

atio

n

and

pow

er

seri

es R

e,

, sm

all,

larg

e

Fou

rier

an

alys

is f

or

Re,

<O

Fin

ite

dif

fere

nce

(P

atan

kar

an

d S

pal

din

g m

eth

od)

Fin

ite

dif

fere

nce

fo

r d

oub

le

par

abol

ic

entr

ance

fl

ow

Per

tub

atio

n

for

Re,

<

1

Vel

ocit

y,

pre

ssu

re

exp

ress

ion

s

Vel

ocit

y f’

( y)

an

d w

all

skin

fu

nct

ion

(-

f'

(y))

. Yu

an’s

w

ork

only

co

rrec

t fo

r in

ject

ion

R

e,

<

0

Lim

itin

g an

d a

rbit

rary

se

epag

e ra

te

Fric

tion

an

d v

eloc

ity

equ

atio

n

in d

evel

opin

g en

tran

ce

regi

on

Ob

tain

sk

ewed

Rai

thb

y p

rofi

les

Vel

ocit

y,

pre

ssu

re

exp

ress

ion

s’

Ber

man

, 19

53

1131

Ter

rill

, 19

64

[I41

Koz

insk

i et

al.,

19

70

[15]

Dou

ghty

an

d

Per

kin

s,

1970

iI

61

Dou

ghty

, 19

74

iI71

Gre

en,

1980

1181

Tab

le

1 (c

onti

nu

ed)

Geo

met

ry

Wal

l fl

ow

Ass

um

pti

onsa

S

olu

tion

m

eth

od

Con

clu

sion

s R

efer

ence

Tu

be

Su

ctio

n/i

nje

ctio

n

Su

ctio

n/i

nje

ctio

n

Su

ctio

n/i

nje

ctio

n

Su

ctio

n

Su

ctio

n/i

nje

ctio

n

Su

ctio

n

Su

ctio

n

Sam

e as

ab

ove

bu

t w

ith

cl

ose-

end

tu

be

AS

, SS

, IC

, L,

P,

US

, N

F,

for

suct

ion

eq

ual

s to

tal

flow

, cl

osed

-en

d

tub

e

Su

ctio

n/i

nje

ctio

n

AS

, SS

, IC

, L,

P,

NF

AS

, SS

, IC

, L,

P,

US

/I,

NF

AS

, SS

, IC

, L,

P,

US

/I,

NF

AS

, SS

, IC

, L,

P,

US

/I,

NF

AS

, SS

, IC

, L,

P,

US

, N

F,

for

suct

ion

eq

ual

s to

tal

inle

t fl

ow

AS

, SS

, IC

, L,

P,

US

, N

F,

wal

l ve

loci

ty

not

ind

epen

den

tly

set

bu

t d

epen

den

t on

Dar

cy’s

L

aw

Per

turb

atio

n

for

Re,

<

1

Vel

ocit

y,

pre

ssu

re,

wal

l sk

in

Yu

an

and

fr

icti

on

exp

ress

ion

s Fi

nk

elst

ein

, 19

56

[19]

Pow

er

seri

es

Pow

er

seri

es

Fin

ite

dif

fere

nce

Kar

man

-Poh

lhau

sen

in

tegr

al

mom

entu

m

and

4t

h

ord

er

Ru

nge

-Ku

tta

Sam

e as

ab

ove

Per

turb

atio

n

and

pow

er

seri

es

An

alyt

ical

, p

oten

tial

fl

ow s

up

erim

pos

ed

on

Poi

seu

ille

fl

ow

Inst

abil

itie

s ap

pea

r w

ith

wal

l W

hit

e,

1962

su

ctio

n

dou

ble

sol

uti

ons

for

[201

0

<

Re,

<

2.

3 (b

ack

flow

); d

oub

le s

olu

tion

s fo

r 9.

1 <

Re,

<

*

(bac

kfl

ow)

Dou

ble

sol

uti

ons

for

all

valu

es

Ter

rill

an

d

of R

e,

for

inje

ctio

n

and

T

hom

as,

1969

su

ctio

n

exce

pt

2.3

<

Re,

<

[2

1 I

9.1

wh

ere

no s

olu

tion

s ex

ist

Su

btl

e d

iffe

ren

ces

from

B

erm

an’s

so

luti

on

Frie

dm

an

and

G

illi

s, 1

967

[221

V

eloc

ity,

p

ress

ure

, sh

ear

and

fri

ctio

n

fact

or

Gal

owin

an

d

DeS

anti

s,

1971

[2

3]

Sam

e as

ab

ove

Gal

owin

et

al.,

1974

[2

4]

Con

firm

s T

erri

ll a

nd

Th

omas

Q

uai

le a

nd

[ 2

11

and

th

eory

m

atch

es

Lev

y,

1975

[ 2

51

exp

erim

ents

R

e,

<

3

Var

iab

le v

eloc

ity

suct

ion

T

erri

ll,

1983

p

rofi

le,

axia

l p

ress

ure

d

rop

[=

I + ._

Tab

le 1

(co

ntin

ued)

Geo

met

ry

Wal

l fl

ow

Ass

um

pti

on8

Sol

uti

on

met

hod

C

oncl

usi

ons

5

Ref

eren

ce

An

nu

lus

Su

ctio

n

and

2D

, SS

, L,

US

/I,

IC,

NF

-- P

ertu

rbat

ion

V

eloc

ity

and

pre

ssu

re

Ber

man

, 19

58

inje

ctio

n

at e

ach

p

rofi

les

1271

fa

ce

Su

ctio

n/i

nje

ctio

n

AS

, Z

D,

SS,

L, U

S/I

, IC

, P

ertu

rbat

ion

an

d p

ower

V

eloc

ity

for

vari

ous

case

s T

erri

ll,

1966

N

F se

ries

12

81

Su

ctio

n/i

nje

ctio

n

AS

, SS

, IC

, L,

US

, N

F,

Per

turb

atio

n

and

pow

er

Inje

ctio

n

pre

ssu

re d

rop

dat

a G

up

ta a

nd

fo

r su

ctio

n

only

th

rou

gh

seri

es

agre

e w

ith

th

eory

(s

imu

l-

Lev

y,

1974

in

ner

wal

l w

ith

clo

sed

- ta

neo

usl

y)

bu

t su

ctio

n

dat

a [2

91

end

du

ct

only

ag

ree

for

low

Re,

<

1

.O

aTh

e fo

llow

ing

cod

e is

use

d:

AS

-

axis

ymm

etri

c;

2D -

Z

-dim

ensi

onal

; S

S -

st

ead

y st

ate;

L

-

lam

inar

; P

-

par

abol

ic;

US

/I -

u

nif

orm

su

ctio

n/i

nje

ctio

n;

IC -

in

com

pre

ssib

le;

NF

- no

ext

ern

al

forc

es.

Re,

m

ay b

e d

efin

ed d

iffe

ren

tly

in v

ario

us

pap

ers.

195

Recent efforts in our laboratory have focused on membrane fouling due to dilute colloid suspensions in which the fluid flow in a permeable slit with one porous wall [7] or a porous tube [S] is coupled with an inertial analysis to obtain the particle trajectory as it traverses the channel [9, lo]. Current efforts are to include multiple particles moving in Poiseuille flow in a two- dimensional slit in the creeping flow regime [ll] and then hopefully for larger Reynolds number flows [12] .

Most mass transfer models for describing concentration polarization and fouling in porous membrane ducts require some knowledge of the fluid mechanics. Mass transfer correlations for porous membrane systems are usually borrowed from studies of flows in non-porous ducts [ 61.

Flow in porous ducts During the past 30 years many analytical studies of fluid flow through

different geometry ducts with mass transfer (suction of injection) at the walls have been conducted. Many practical applications depend on an understanding of the fluid mechanics in such systems. Examples include gaseous diffusion technology, transpiration cooling, control of fluids in nuclear reactors, boundary layer stabilization, and more recently synthetic membrane technology.

Most of the previous analyses have been for slits, tubes and annuli as shown in Table 1. All three are common geometries for membrane systems. Two categories of problems have been solved: fully developed flow, where the shape of the nondimensional velocity profiles is considered similar or invariant with axial distance, and developing flow at the duct entrance, where the shape of the nondimensional velocity profile is changing with axial distance. To our knowledge, all previous workers, besides Galowin and co-workers [23, 241, and Terrill [26], have a priori assumed constant mass transfer at the permeable walls with axial distance. This crucial assump- tion implies that a variable pressure exists outside the duct, or even less likely, that the walls are of variable permeability and/or thickness with axial distance. It is, of course, possible to cast synthetic membranes with variable axial permeability, but this has generally not been done commer- cially. Implicit in the work of Galowin and co-workers [23, 241 and Terrill [ 261 is the similarity assumption.

Numerical solutions assuming constant wall suction and similarity of velocity profiles are summarized by Berman [30] and White 1311. Two solutions exist for 0 < Re, < 2.3, one without and one with backflow. No solutions exist for 2.3 < Re, < 9.1, while double and possibly even more solutions exist for Re, > 9.1. White’s conclusion from this is that similarity probably does not hold in a porous pipe for Re, > 2.3, while Weisberg [32] confirmed this for a Poiseuille flow entering a porous tube with Re, = 3.

Very few experimental studies have ,been conducted for flow in porous ducts. Most of the studies have been conducted for the turbulent flow of air [33-371 and for laminar flow of air [38] in porous tubes. Even less has

196

been measured with liquids. Quaile and Levy [25] studied the laminar flow of a silicone fluid in a dead-ended porous tube (100% recovery of inlet flow through the walls).

For relatively short path lengths and for systems in which the velocity of fluid removed through the walls is small compared with the bulk axial flow velocity, both similarity and constant wall flux with axial distance are reasonable approximations. The purpose of this paper is to compare an analytical solution obtained by regular perturbation theory with a numerical solution of the NavierStokes equations, to determine the range of validity of the analytical solution. Uniform suction and injection and similarity of the velocity profile are both assumed for the analytical development.

Theory

Although Berman’s [13] and Yuan and Finkelstein’s [ 191 solutions have been used for describing flow in membrane slits [9] and tubes [2, lo], respectively, little attention has been given to the fluid mechanics in a spiral wound geometry. This is in spite of the fact that the spiral wound geometry is widely used in commercial membrane plants (hyperfiltration and ultra- filtration).

Background Berman [27] using a perturbation method, was the first to solve the

steady-state lammar flow of an incompressible fluid in an annulus with porous walls. He considered fluid withdrawal and fluid injection simul- taneously occurring at each wall, Using a similar method, Terrill [28] developed equations for either suction or injection flow at both walls of an annulus. Only a theoretical analysis was presented without any analysis of the results. Later, Gupta and Levy [ 291, using a perturbation and power series method, solved for flow in a closed-ended annulus assuming suction at one wall and similarity.

Idealized spiral wound geometry In this paper, similarity solutions using Berman’s perturbation method

and numerical solutions are presented for flow in an annulus with two porous walls. Our analytical approach is similar to that of Terrill [28] ; however, we present a detailed analysis with special reference to fluid flow in an idealized spiral wound module. To simplify the geometry, a spiral annulus will be idealized by an annulus formed from concentric tubes as shown in Fig. 1. The fluid flows into the end of the membrane spiral or annuli traversing along the two porous membrane faces. A cross- section of one annulus is shown in Fig. 2 and used to define the coordinate system.

Assuming a logarithmic spiral from the center outwards where R0 is the initial radius (for w = 0) and a is a constant, we note that as w varies from

197

F RMEATE FLOW

PERMEATE FLOW

IDEALIZATION

c3

POROUS WALLS

FLOW PASSAGE FLOW PASSAGES

SPIRAL-WOUND MEMBRANE SANDWICH CONCENTRIC CYLINDERS

END-VIEW END-VIEW

Fig. 1. Idealization of spiral-wound geometry by a series of concentric cylinders.

“W

CENTERLINE

I%__ P_L_ CENTERLINE OF CONCENfRlC CYLINDERS

Fig. 2. Fluid flow through annulus with wall suction.

circle of radius R, to that of a 0 to Zn, the ratio of the circumference of a spiral is given by

Ccircle 27rR, 27ra -- Cspiral = R,J,fn exp (ao)do = exp (2na) - 1 (1)

For a < 1, exp (2na) may be expanded in a Taylor series, and, neglecting terms of 0 (a2) and higher, eqn. (1) shows that the ratio approaches 1. For a < 0.01, one obtains

Ccircle ___ > 0.97. Cspiral

Under these circumstances we expect the annular wound geometry to be a very good approximation,

idealization of the spiral

198

Assumptions The two-dimensional coordinate system is chosen as shown in Fig. 2,

where x and r are the axial and radial coordinates, respectively, and ri, r, are the inner and outer radii of the annulus, respectively. Assumptions include: steady-state conditions, the fluid is incompressible with constant density and viscosity, no external forces act on the fluid, laminar flow, uniform suction or injection at both walls independent of axial distance, and axisymmetric entrance flow.

Governing equations The NavierStokes equations for the system shown in Fig. 2 are as follows:

Ww + am4 = o - __

ax ar

u*+v+ = -1 ap P ah ax at-

_-+_ - +G!f+~ P ax i p ar2 1

ua?l+v&! =-lap P ah ax ar

-_+_ P ar (

2 +$ +$$_2 P ax r2 1

(2)

(3)

(4)

where u and v are the x- and r-components of velocity at any point, res- pectively, p is the pressure, and p and ~1 are the fluid density and viscosity, respectively.

The associated boundary conditions are:

at r = ri, u = 0, v = - v, (suction)

atr = rO, u = 0, (5)

v = v, (suction)

The set of governing equations (eqns. 2-4) can be reduced to a simpler form as is shown below [19, 201.

The continuity equation (2) is satisfied identically by the axisymmetric stream function $(x, r) as follows:

u(x r) = LawA 1 aax, 9 3

r ar ’ v(x,r) = -;

ax (6)

From the law of conservation of mass, the stream function can be defined as

4 $/(x,X) = y ii(O) - 2% t f(N [ 1

where rn = rO - ri is the hydraulic radius, h = (r/m )2, U(0) is the average axial velocity at the inlet position x = 0 and f(h) is a function to be deter- mined. Using eqns. (6) and (7), the velocity components become

u(x, h) = [G(O) - 2v,x/rH I f’(A)

‘UN, N = v,f(N/G = v(V

(8)

(9)

199

The equations of motion are now transformed to those containing the un- known function f(h). Equations (3) and (4) then become

aP -= 2X

ap -= ax

where

U(0) - wvx

rH

Xf"' + f" + R!! (f'2 -ff") 2 1 (10)

(11)

Re, = v, prH /p is the wall Reynolds number and primes denote differentiations with respect to A.

In eqn. (ll), the right-hand side is a function of X only and therefore a’p/axaA = 0, implying that the pressure gradient is not constant. Then eqn. (10) gives

Xf”” + Zf”’ + R!? (f'f" - ff"') = 0

The boundary conditions from eqn. (5) become

f’(hi) = 0, f(hi) = -~

f’(L) = 0, f(L) = 4c

(12)

(13)

where Ai = (ri/rn)2, h, = (r, /ru )2. Thus the original set of partial differ- ential equations are reduced to eqn. (12), which is a homogeneous ordinary differential equation.

Perturbation solution In membrane processes, the wall Reynolds number Re, is typically much

less than 1. Thus, treating small values of Re, as a perturbation parameter, the solution to eqn. (12) may be expressed by a power series developed near Re, = 0 as follows

f = f,(A) + Re,f,0) + Re;fz(X) + . . . + Rekf,(N (14)

Substituting eqn. (14) into eqn. (12) and equating the coefficients of the various powers of Re, to zero, leads to a consecutive set of differential equations:

Zeroth order: Af;j” + 2fl’ = 0 (15)

First order: Xfl”’ + Zf;” + l(fAfcY - f,fF) = 0 (16)

fAChi) = O, f&L) = 0 (17)

fnCxi) = fnCxo) = fiCxi) = fL(ho) = O; n>l

200

._

f,

2 I

2 0

201

+

202

Consider a first-order perturbation solution, i.e. f = f. + Re, fl , where terms of higher order are negligible due to small wall Reynolds numbers We, < 1). The expression for f. may be found by successive integration of eqn. (15). Then fl is determined from eqn. (16). The results obtained are

f*(X) = Ah(l-ln h) l kB$ +Ch+D (18) and

f,(h) = fX2 5lnX---(lnX)‘-~

Yh3 +3 lnh_i _!?f_ ( i ah(ln h)’ Fh2

2 -EA(ln h-l) + 2

+Gh+H (19) where expressions for A to H and (Y to 8 are given in Table 2.

The calculation procedure is as follows. The constants A to D are deter- mined from eqns. (20)-(23) in Table 2. Then, o to 8 are found from eqns. (28)-(33) and used to calculate K, to K4 from eqns. (34)-(37). Finally E to H are determined from eqns. (24)-( 27).

The first-order solution for f(h) becomes:

f(A) = f00) + Re,fl (A) (38)

with f. and fl given by eqns. (18) and (19). The normalized radial and axial velocities are obtained from eqns. (9) and (8), respectively, as

v = v/vu, = f(Wfi (39) and

u = u/ii(O) = [I-gy)f’(A) (40)

where Re = U( O)pr, /p is the entry Reynolds number and f’(X) is obtained by differentiating eqn. (38) and substituting the differentiated forms of eqns. (18) and (19) with

f;(h) = -Alnh+BX+C (41)

and

f; (h) = - cpX(ln h - 2)2 + BX(ln A - 1) + 6X2 - o In h 1 + !n$ ( 1

--oh3 -ElnX+FX+G

(42)

203

The normalized axial pressure drop is readily obtained from eqn. (10) as

P(O, N -p(x, A) :m2w

- = &(Z -1) (xf”‘+f” ++(f’2 -ff’.)) (43)

with

f”(h) = -A/X + I3 + Re, [2plnhjl-?)+26h+flInX

+ 2yX(ln X - 1.5) - 36X2 - cr(1 + In A)

x -!+F 1 r

f”‘(A) = $ + Re, I

2~(1 - In X) h +26+:+27(111X-0.5)-613X

arlnX E + 12 +jp

1

(44)

(45)

From eqn. (43) it is seen that the axial pressure drop is independent of A. This is because the term within second braces, Xf “’ + f” + (Re, /2) ( f ‘2 - ff”), is constant, as may be readily seen by integrating eqn. (12) once. The radial pressure drop can be obtained from eqn. (11).

For injection, the wall Reynolds number Re, is given a negative sign in the equations above.

Results

The first-order solutions expressed by eqn. (40) for the dimensionless axial velocities are plotted in Fig. 3 against the dimensionless CrOSS-SeCtiOn of the annulus for different values of Rew, axial distance, 3t/rH, and for injection and suction. The entrance axial Reynolds number Re = 1000 and the outer and inner annulus dimensions r, /rn = 5 and ri/rH = 4, respectively, were chosen. These values si.mulate the geometry and flow in a commercial membrane module. The axial velocity profiles are asymmetrical toward the inner wall about the annulus centerline, and show increased and decreased maxima with path length for uniform injection and suction, respectively. The steeper velocity gradients at the porous walls for injection as com- pared to suction illustrate their increased wall shear rate.

The first-order solution expressed by eqn. (39) for the dimensionless radial velocities is given in Fig. 4. The uniform (with axial distance) radial velocity profile is similarly skewed toward the inner wall and is zero at the same radial position where the axial velocity reaches a maximum.

For uniform suction, the fluid is eventually all lost through the walls. The region of validity of the solution is then found from eqn. (7) as

0 < x/rH < O.EiReJRe, (46)

204

2.4

2.0

/-

/_

/

/_

/

/_

/

If- / d- /

1’

0.f

0.C

0.C .4.c I 4.2 4.4 4.6 4.8 5.0

r/r,

CURVE Re, x/r,, BOTH WALLS ---

I -0.8 250 INJECTION

2 -0.8 50 INJECTION

3 0.8 50 SUCTION

4 0.8 250 SUCTION

Re = 1000

~:0=r~/r~=5i~i=ri/rH=4

Fig. 3. Axial velocity variation with radial distance. 1.0 r Re = IOOUSJ

MEMBRANE+

i

Re, - 0.8 0.8

./I, q ra /rH = 5

o.6 J& = ri/rn= 4

4.6 4.8 5.

Fig. 4. Lateral velocity variation with radial distance,

205

The aid pressure drop in the annulus is plotted against axial Position

in Fig. 5 for small and large injection and suction. AS expected, the axial pressure drop increases sharply with axial path distance and is significantly higher for injection than suction.

To check the range of validity of the analytical solution presented above,

eqns. (2)-( 4) with boundary conditions eqn. (5) and initial condition of u = C(O) at x = 0 for all r, were solved numerically using the SIMPLE (semi-implicit method for pressure-linked equations) algorithm of Patankar [39]. A comparison of the numerical and analytical solutions showed excel- lent agreement for axial path distances x/rH > 20. This is when fully de- veloped flow has been established. For example, at x/rH = 50, the analytical solution shown in Fig. 3 for the axial velocities is indistinguishable from the numerical solution for the scale of Fig. 3. This is also true for the radial velocity profile shown in Fig. 4, where the similarity assumption on which the analytical solution is based is confirmed. For tie, > 1 the analytical and numerical solutions diverge. This is shown in Fig. 6 where V calculated

Fig. 5. Pressure drop versu.s axial distance.

6-

C

4-

12 -

IO -

8-

6-

4-

2-

Od 0

-

IJRVE Rer BOTH WALLS --

l -0.8 INJECTlON

2 -0.1 INJECTION

3 -0.01 INJECTION

4 0. I SUCTION

5 0.8 SUCTION

Re = 1000

~:o’ro/rH=5;~i=li/r~‘4

100 150

x/r”

250

206

IO

0.8

0.6

0.4

-06

-0.8

tte = 1000.0

ReVI =20 Jx,= r&=5

&-= r,,/r, =4

.O 4.2 4.4d 4.6 4.8 5

r/r"

Fig. 6. Non-similar radial velocity profiles for large Re,.

numerically is dependent on x/rH . Clearly, the similarity assumption cannot hold for these conditions.

List of symbols

A, B, C, D, E, F, G, H f K,,K,,KJ,& P r

rH

ri

k

Re, u

U(O) u

V

Constants defined by eqns. (20)-( 27) Function of X Constants defined by eqns. (34)-( 37) Pressure Radial coordinate Hydraulic radius, r, - ri Inner radius of annulus Outer radius of annulus Entry Reynolds number, rn zZ(O)p/p Wall Reynolds number, rn v, p/p Axial velocity of fluid Mean axial velocity of fluid at inlet Normalized axial velocity, u/U (0) Radial velocity

207

Vu, V X

Wall suction velocity Dimensionless radial velocity, P/P,

Axial coordinate

Greek letters

QJ, P, Y, 6, P, 0 Constants defined by eqns. (28)-( 33)

h Dimensionless radial coordinate, ( r/rH )2

P Fluid density

M Fluid viscosity

4 Stream function 0 Angle

References

6

7

8

9

:10

11

12

13

14

0. Belfort (Ed.), Synthetic Membrane Processes, Theory and Water Applications, Academic Press, New York, NY, 1984. P.L.T. Brian, Mass Transfer in reverse osmosis, in: U. Merleu (Ed.), Desalination by Reverse Osmosis, M.I.T., 1966, Chap. 5. W.N. Gill, L.J. Derzansky and M.R. Doshi, in: E. Matijevic (Ed.), Surface and Colloid Science, vol. 4, Wiley-Interscience, New York, NY, 1971, pp. 261-360. R.F. Probstein, Desalination: Some fluid mechanical properties, Trans. ASME, (June 1972) 266-313. C. Kleinstreuer and G. Belfort, Mathematical modelling of fluid flow and solute distribution in pressure-driven membrane modules, in: G. Belfort (Ed.), Synthetic Membrane Processes, Theory and Water Applications, Academic Press, New York, NY, 1984, Chap. 5, pp. 131-190. G. Belfort and N. Nagata, Fluid mechanics and cross flow filtration: Some thoughts, Desalination, 53 (1985) 57-79. J.R. Otis, F.W. Altena, J.J. Mahar and G. Belfort, Measurement of single spherical particle trajectories with lateral migration in a slit with one porous wall under laminar flow conditions, Exp. Fluids, 4 (1986) l-10. G. Belfort, R.J. Weigand and J.J. Mahar, Particulate membrane fouling and recent developments in fluid mechanics of dilute suspensions, in: S. Sourirajan and T. Matsuuka (Eds.), Reverse Osmosis and Ultrafiltration, ACS Symposium Series No. 781, American Chemical Society, Washington, DC, 1985. F.W. Altena and G. Belfort, Lateral migration of spherical particles in porous flow channels: Application to membrane filtration, Chem. Eng. Sci., 39 (1984) 343- 355. R.J. Weigand, F.W. Altena and G. Belfort, Lateral migration of spherical particles in laminar porous tube flows: Application to membrane filtration, Physiochem. Hydro- dyn., 6 (1985) 393-413. J.A. Schonberg, D.A. Drew and G. Belfort, Viscous interactions of many neutrally buoyant spheres in Poiseuille flows, J. Fluid Mech., (1986) in press. J.A. Schonberg, D.A. Drew and G. Belfort, Transient interactions for two neutrally buoyant spheres in Poiseuille flow between two plates. Chem. Eng. Sci., (1986) sub- mitted. AS. Berman, Laminar flow in channels with porous walls, J. Appl. Phys., 24 (1953) 1232-1235. R.M. Terrill, Laminar Flow in a uniformly porous channel, Aeronaut. Q., 15 (1964) 299-310.

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