- Email: [email protected]
Performance optimization of hollow fiber reverse osmosis membranes, part I. development of theory
journal of MEMBRANE SCIENCE E LSEVI E R
Journal of Membrane Science 103 (1995) 257-270
Performance optimization of hollow fiber reverse osmosis membranes, Part I. Development of theory Victor M. Starov u, Jim Smart b, Douglas R. Lloyd b,. a Moscow Academy of Food Industry, Moscow 125080, Russian Federation b Department of Chemical Engineering, Separations Research Program, The University of Texas at Austin, Austin TX 78712-1062, USA
Received 16 June 1994;accepted in revised form 19 December 1994
Abstract The overall performance of hollow fiber membranes can be engineered based upon the interplay of fiber productivity and fiber selectivity. Fiber length and diameter can be optimized to render the desired balance of fiber performance. Fundamental analytical models are developed for feed outside (concurrent and countercurrent flow configurations) and feed inside. Part II of this series of papers will evaluate the sensitivity of fiber performance to such operating conditions as pressure, packing density, and fiber diameter. Keywords: Reverseosmosis; Modules; Modeling
I. Introduction Hollow fiber membranes for desalination offer the advantage of withstanding large pressure drops and provide the economic advantage of enormous surface area to volume ratios. In addition, when the feed is directed through the interior of the tube bundle, the high fluid velocities helps reduce surface fouling [ 1 ]. Membrane manufacturers sell modules in a few standard diameters and lengths with a few fiber dimensions (radii, wall thickness, and length) based on in-house experience and the resulting empirical correlations. The objective of the research reported here was to develop a model capable of optimizing hollow fiber dimensions and module operating parameters in terms of productivity and selectivity. * Corresponding author. Tel (512) 471-4985, Fax (512) 4717060. 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD10376-7388(95)00008-9
Productivity or recovery ( q~, defined as the fraction of feed recovered as permeate) and selectivity or rejection coefficient can be studied in terms of a dimensionless metric parameter o" defined below as a combination of fiber length, inner and outer fiber radii, permeability coefficient, and solution viscosity. As membrane productivity is increased, selectivity suffers and vice versa; consequently, a balance must be struck in membrane design, module design, and operating conditions to satisfy specific application requirements. Consider the case of feed on tlae outside of a single hollow fiber of fixed inner radius, fixed fiber thickness, and short length. If the fiber length is extended, thereby increasing o', more surface area is available and increased productivity results. The permeate volume produced by the fiber continues to increase with additional fiber length until a "critical length" is approached. This critical length is defined to be the point at which ocorresponds to q~= 1.0. This situation is illustrated in
V.M. Starov et al. / Journal of Membrane Sc&nce 103 (1995) 257-270
258
1.0
1.0
o.o ................. T.................:.................i ................i...................................................................i................ i................o., 0.8
0.8
0.7
0.7
0,6
0.6
:~ 0.5
0.5
|
i
~
i
0.4
0,4
0.3
- 0.3
0.2
- 0.2
0.1
- 0.1
0.0
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
dlrnemdonleu length permnet~, 0
Fig. 1. Typical hollow fiber productivity q) and selectivity R as a function of dimensionless length parameter o'. Fig. 1. If fiber productivity tends to 1.0, then selectivity decreases to zero. The objective of this paper is to demonstrate the generation of these curves. Notice in Fig. 1 that permeate quality (that is, selectivity) diminishes with increased fiber length. This phenomenon is a result of two effects: ( 1 ) As fiber length is increased, the solute concentration on the feed side increases along the length of the fiber. Consequently, the local driving force for solute transport across the membrane increases with increased fiber length. (2) Rejection or selectivity decreases with decreasing pressure drop across the membrane wall, as illustrated in Fig. 2. In the case of increasing the length of a hollow fiber, the pressure drop across the membrane wall falls as a result of the additional length [2]. The end result is less rejected salt, with consequent higher salt concentration of the permeate. Early attempts to model hollow fiber reverse osmosis performance in the 1970s produced analytic expressions for productivity in parallel and radial flow patterns [3,4]. However, the system was oversimplified by assuming complete solute rejection across the membrane wall and an arbitrary concentration profile on the shell side of the module. Subsequent attempts to
develop predictive models for parallel and radial flow reverse osmosis assumed constant rejection; that is, rejection independent of fiber length [ 5-7 ]. Replacing this constant rejection model with a solution-diffusion transport model [8], and later a solution-diffusionimperfection model [9], failed to overcome significant deviations between predicted and observed rejection for regions of high productivity and high feed concentrations. In all of these previous studies, hollow fiber productivity was modelled and studied, but fiber selectivity was essentially ignored. Optimum fiber length was discussed by Gill, et al. [ 10], who noted permeation flow (vol/time) could be increased if a single fiber open at one end (current design of Du Pont B-9 permeator module [ 11 ] ) could be opened at both ends. This amounts to shortening the fiber length by one-half and doubling the number of fibers. Starov et al. [ 12,13 ] recognized that fiber length can be adjusted to optimize the balance between productivity and selectivity. The research reported here builds upon these previous studies by: • Developing mathematical models to describe and predict hollow fiber productivity and selectivity as a
1
R~ .......................................
i
i
r
j
5
10
15
20
25
pressure drop across membrane(kPa)
Fig. 2. Dependency of selectivity R of a flat reverse osmosis membrane on the applied pressure across the membrane. Rm~ is a maximum selectivity.
V.M. Starov et aL / Journal of Membrane Science 103 (1995) 257-270
259
Q, feed pressure at the entrance of the module and pressure at the exit are assumed to be fixed. The following conditions are assumed: t ( 1 ) Equivalent annulus or free surface model [ 1416]. P-O (2) Steady laminar flow, ~X~Re <~1, parallel to fibers. 3...~ ~ s.......~ ..................................... I Transverse flow is not considered. (3) Flux through the membrane wall is relatively small. This assumption has been verified by previous Fig. 3. Concurrent flow (case AI). Feed solution supplied from investigators [ 3,9]. outside the fiber: 1, feedsolution;2, bilayermembrane(s, skinlayer; (4) Newtonian fluid with constant physical properb, supportinglayer); 3, permeate. ties. (5) Constant membrane permeability coefficient. function of fiber length, inside fiber radius, and outside (6) System symmetry; no angular velocity compofiber radius. nents; no velocity slip at the wall. • Developing these models for three cases: feed to (7) Temperature constant; no free convection the outside of the fiber in a concurrent flow (case A1; effects. see Fig. 3) and countercurrent flow (case A2; see Fig. (8) Dilute feed; no osmotic pressure effects; no pre5), and feed through the inside of the lumen of the fiber cipitation of dissolved salts. Osmotic pressure will be (case B; see Fig. 6). an important consideration when using more concen• Optimizing fiber performance, which is defined in trated feed solutions. terms of productivity and selectivity. (9) Small diameter fibers such that ~Pec < 1. While the results reported here deal only with reverse These assumptions were selected as a basis for the osmosis, the productivity equations are equally applidevelopment of a simple, fundamental model. Imporcable to ultrafiltration. Ultrafiltration is primarily a sizetant effects such as osmotic pressure considerations will exclusion separation process and the selectivity be included in Part II. of this series. equations presented in this study may have limited application in cases of ultrafiltration. Inside the fiber lumen (0 < r < rl , 0 < x < L) In this section, relationships between fluid pressures, Since r i/L << 1, a Navier-Stokes equation and equafluid velocities, and fiber geometries are developed tion of continuity can be written within three regions: (1) inside the fiber lumen, (2) within the membrane, and (3) outside the fiber. These 1 d P 3 1 0 Ou3 (1) relationships will be used to assemble an expression tx dx = r O r W Or for productivity (qb) and selectivity (R). .......................................
--
.
O_ _
--
--
F L O W
I1
. . . .
3
I
o~(rv3) q-o-~(ru3)=0 2. M a t h e m a t i c a l model: case A1. Feed exterior to fiber; c o n c u r r e n t flow
2.1. Productivity
Feed is introduced outside the hollow fiber under an imposed pressure difference: P l o - P l f = A P > O. See Fig. 3 for definitions of terminology. The assumptions made and development of productivity theory presented here are similar to those of Gill and Bansal [ 6 ] ; however, they developed a model based upon a fixed flow rate, Q. In the following model, instead of fixed
(2)
The boundary conditions are symmetry in the center of the fiber, non-slip condition at the walls, impermeability of the sealed end of the fiber, and atmospheric pressure at the fibers open end Ou3 Or r = O = t ) 3 l r = O = 0
(3)
u31r=~=0
(4)
dP3
~lx=o=O
(5)
P3=0, atx=L
(6)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
260
Here, P3 is the pressure, while u3, and v3 are pressure and velocity components along x and r axes. From these equations and boundary conditions,
Hv2
0X
=
L2
(7)
Volumetric flow rate from Eq. (7) can be expressed as
J
2
ro\ H ] Or
Hence, from Eq. (12) 10 or( rV2) = 0 or rv2 = constant.
t'i
2,trf ru3dr = - ,m.4 d p 3
H
iO(rvj i(rov2]=(~)<
1 dP3
u3: 4;
Q3
OU2
(8)
r
From this result, it is concluded
0
From the equation of continuity, Eq. (2), a conservation law is deduced ri
fru3dr+rv3, o
(13)
roVlw= riV3w
Since rv 2 is constant, this result can be expressed as k* v2 = - - -
(14)
r
0 or
d -T'Q3 + ri2~V3w = 0 Ox
(9)
Combining Eqs. (8) and (9), the expression for pressure distribution inside the fiber lumen becomes d2p3 16~ dx2 ~ "V3w
(10)
In the membrane (ri < r < ro) According to Darcy's law 02 ~---
Or
and
U2 =
--
Ox
(11)
where H = r o - r~ is the total membrane thickness. From the equation of continuity inside the fiber wall Ou2. 1 O W-f-7~rtrV2)
=0
k*
.
OP 2
k*
(15)
where Ks and Kb are permeabilities of the skin layer and the support layer, respectively. Upon rearrangement, pressure within the membrane can be expressed as k* r PE=~ln~o+P1, ( r o - h ) < r < r o
(16)
p E = k * l n r + p3, r i < r < ( r o - h )
(17)
(12)
where h is the thickness of the skin layer. The following boundary conditions were used P2 = P1 (x) at r = ro and PE=P3(x) at r=ri. Pressure inside the membrane is continuous everywhere including the point r = r o - h . Using this condition, the unknown constant k* is found to be equal to k*=
(18)
(P1-P3)
in(fro h ) (o--
the magnitude of the ratio is evaluated as
.. OP 2
7=t~S.~r,,-7-=t~bOr
Kbrl
where K is the membrane permeability coefficient. Using approximations offered by the derivative and forming appropriate ratios, Darcy's law suggests u2 L L =-->>1 u2 t o - ri H
where k* is an undetermined constant. The asymmetric membrane can be further subdivided into a skin layer (s) and a supporting layer (b) as illustrated in Fig. 3. Using Darcy's equation and Eq. (14),
Ks
in( ro - h) k
ri
Kb
V.M. Starov et al. /Journal of Membrane Science 103 (1995) 257-270
Upon substituting this expression back into Eq. (14), using the equality of Eq. (13), and the condition (h < < ro), the final equation for the velocity on the surface of the membrane becomes riU3w = --
Ks
ro" h + Ks(rolnrO_ h) Kb ri
261
From the equation of continuity, the following conservation law is deduced d -7--Q1 - 2~rrovlw= 0 (IX
(26)
Combining Eqs. (26) and (9) results in (19)
d(Ql +
Q3) -4- 27r(riV3w - -
F0Olw ) =
0
(27)
(P1 - P 3 ) = - r o K ( P l - P3)
where K is a permeability of the membrane wall as defined by Eq. (19). Usually roKJKb < < h, and therefore, K = Ks/ h.
Outside the fiber (ro < r < rex0 Using Navier-Stokes equations and the appropriate boundary conditions, as in Eq. (1), (3), and (4), it is possible to write the following expression I OPI Iz
1 O.
(20)
with boundary conditions: ( 1 ) no shear stress and no permeation through the free surface boundary [6-8] Oul
- - = v l =0, at r=rext Or
(22)
Applying the equation of continuity, O/Ox(rul) + O~ 0r(rvl) = 0, yields an expression for velocity outside the fiber dP1
/~extdP1 r In2/x dx ro
(23)
Combining the definition for volumetric flow,
01 = 27rlruldr and Eq. (23) gives rO (24)
3 4 A2 1 f,(A) = - ~ A 4 2 8
Z
dx
=Q
(29)
Introducing the following dimension less expressions, y = 8 (~i0)4fl(A) and q__
Q
(30)
\8gLJ
dPl= dx
l dP3
y dx
q Plo y L
(31)
Upon integration of Eq. (31 ) PI(X) =
P3(x) y
qPlzx+a*-%
(32)
(33)
= Plf
dP31 _ dx x=O-0 and P3(L) = 0
(34)
Using boundary conditions (33) and (34), the unknown constant a* can be determined from Eq. (32) atx=Z
where
h4
/z
P1 (0) = Plo and Pl (L)
7rrgfl (A) dP1 /z dx
+-=-In A, where A=
8/~ dx
where a* is an unknown constant of integration. The following boundary conditions apply
rext
Q1 =
7rr4A(A) dP1
Eq. (29) can be rewritten as
ul =0, at r=ro
. . . 4/x dx
(28)
where Q is an integration constant to be determined below. From Eqs. (8), (24), and (28),
(21)
(2) non-slip condition at the wall
(r2--~)
d -7"(01-t-03) = 0 or Q1 + Q 3 = Q = constant (IX
7rr4 dP3
Oul.
Ox =r ~ r ' ~ r )
ul .
Recall ffomEq. (13), qv3w- ~vlw =0. Therefore, it can be concluded
rext> 1 ro
(25)
qPlo +a* or a* Y
Plf . . . .
=Plf+--
qPlo
Y
(35)
V.M. Starov et al./ Journal of Membrane Science 103 (1995) 257-270
262
Substituting Eq. (35) into Eq. (32), gives the final expression for pressure distribution outside the fiber
Using dimensionless parameters, this can be expressed
Pl(x) = P3(x__....~)+plf+qPlo(1 x 3' 3" - Z )
p(0) = 3'(~- 1) +q (36)
as
Combining Eqs. (43) and (45), results in 3'(~- 1) + q =
Assembling the expression for fiber productivity Combining Eqs. (10), (19), and (36), the following equation for pressure distribution inside the fiber can be produced d2p3 16/~roK dx2 r4
(37)
(45)
q tanh/3
~3'
/3(1+3')
(l+3')cosh/3 + q + - -~y l+y l+y
Rearrangement produces an expression for the dimensionless flow rate,
[ ' 3 ( 1 +1) --Vlf- ~-~( 1 - L) ] Introducing the following dimensionless variables and quantities, P3
P=P-£o
(38)
q=Y
~=~o
'|
(46)
/ where dimensionless flow rate q must be positive. Productivity is defined as • --
Plf
" _ tTnh~
QI(0) - QI(L)
(39)
Q,(0)
= 1
QI(L) -
-
(47)
-
QI(0)
Using Eqs. (24) and (31), productivity @can be recast X
Y=L
(40)
as
dp(l) dy q
- -
/32_
16/zroKL2( 3' + 1) Y
r¢
• =1
(41)
(42)
q sinh[/3(1 - y ) ] /3( 1 + y) cosh/3 s¢3' cosh/3y_~ q(1-y_____))~_ s¢3' 1+3' coshfl 1+3" 1+3"
(48)
ly = 1 -- (
1 + 3') cosh/3
#3"/3sinh/3 q + - (1+ y)cosh/3 (1+3,)
(49)
Substituting Eq. (49) into (48), yields the final expression for productivity, 1
(43)
1 - - - -
cosh/3
1+3'
Now, an expression for q can be deduced from the rest of the boundary conditions. At x= 0, Eq. (36) can be rewritten as P3(O) = (Plf - Plo) 3' + qPlo
1 dp(1) q dy
q
with boundary conditions dp/dyly =o and p( 1) = 0. Solution of Eq. (42) yields the result
p(y) =
-
From Eqs. (43) and (46)
Eq. (37) can be written as d2P 2 q ~----y2=/3 [p - 1--~---y(1 - y) - 1--~+~y]
+ q
(44)
{
, tanh/3]
~fl tanh fl ~3"t - - - - ~ j +
(50)
(1 + 3 ' ) [ 1 + 3 ' - ' ( 3 ' + c o ~ hfl)]
V.M. Starovet al. / Journal of MembraneScience 103 (1995) 257-270
1.0-
!
Concentration distribution inside the fiber, outside the fiber, and in the membrane are used to derive an equation for rejection coefficient.
- 1.0
.s !
0.9 .....
- 0.9
0.8-=
- 0.8
0.7 ......
- 0.7
0.6 .....
- 0.6
263
Inside the fiber lumen (0 < r < ri , 0 < x < L) The equation of continuity for dissolved molecules can be expressed as
O(ru3C3 )
|
(52)
=c
¢ o
-2--0 = O__(D Or_~2C3t q-or (rv3C3) Or\ Or]
0.5
0.5 o
;/
0.4
8
~->
!/
where D is a diffusion coefficient of dissolved molecules in the bulk solution. Integration with boundary condition OC3/Orl r=o = v3 Ir=o = 0 yields
0.4
0.3
- 0.3
diru3C3dr=-riJ3w
0 - 0.2 0.2 f - 0.1
0.1
/7/I
:~
0.0 0.0
...........
....
'"*"
I - ~ - - ~ - L ' ~ ' ~
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
II
-0.0
0.9
1.0
dlmenslonlen length parameter, t~
Fig. 4. Examples of calculated dependencies of productivity and selectivity on dimensionless length parameter o-: --, case A 1; -.-, case A2; . . . . , case B. Note, productivity q~= ~ ( 3,,/3,sc) is a function of only three independent dimensionless parameters. See Fig. 4 for an example of a plot of productivity versus dimensionless length parameter generated from Eq. (50). Since the assumption of dilute feed was used in the development of Eq. (50), productivities near 1.0 (concentrated salt solutions) in Fig. 4 do not strictly apply. The curves are included here for illustrative purposes. The abscissa of Fig. 4 is the dimensionless length parameter, 0-, where 0-2= (16txKL2)/~ fl and 0- are related by/~2 = 0 . 2 X 2 , where X 2 = [ ( 3' + 1 ) / 3'] (ro/ri). This transformation is done to provide for consistency between all cases of flow configurations.
2.2. Selectivity
- C3(L)
C1 (0)
~ D~ u3C3< < ] / ~0C3, -,/~~ I o r
or
v3ri/D< < 1
ri
At v3 ~ 1 0 - 4 c m / s , r i ~ 10 -4 cm and D ~ 10 -5 cm2/s. Therefore, the ratio v3ri/D, which is the Peclet number (INPec), is ~ 10 -3 < < 1. Thus, it is possible to assume that concentration depends only on the axial coordinate. Hence, the above equations indicate d
r
"~[ C3fru3dr]
=
-- riJ3w
o
(53) ri
Upon substitution of Q3 = 2"rrjru3 dr into Eq. (53), the final expression becomes d ~ ( C 3 Q 3 ) = - 2~rriJ3w
(54)
Inside the membrane skin [(ro - h) < r < ro, h < < ro]
Selectivity is defined in terms of the rejection coefficient:
R -Cn(O)
where J3w is a flux of dissolved molecules through the inside fiber walls. It is assumed below that concentration depends on the x coordinate only; that is, C3 = C3 (x). This assumption means that radial diffusion makes the concentration profile uniform in the radial direction; that is,
1 - (L-------~) C3
Cl (0)
(51)
Molar flux moving from ro, in the normal direction toward ri can be expressed as [ 12,17,18] : dC Js = a s V s C - Ds--:oz
(55)
264
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
where z = r o - r , Ds is the diffusion coefficient in the skin layer, and a s is a constant that corrects for the deviation of the dissolved molecules convective velocity from the convective velocity of the solution [ 12, 171. In the membrane supporting layer, concentration is assumed to be the same as inside the hollow fiber; that is, C3 = C3(x). This assumption is valid under the same conditions as above; that is, small Peclet number. Velocity in the skin layer vs is introduced using Eqs. (13) and (19):
d
~(C303)
Combining Eqs. (8), (60), and (61) yields the final equation for the concentration within the hollow fiber: - ~-~
dp3
c
= 27rroK(P1 - P 3 ) .
rasK(P1-P3)h] "~"/-
~exp/ .
(57)
eVsC(h) = C3(x)
(58)
These boundary conditions were introduced in [ 18] and are briefly explained as follows: chemical potentials of the dissolved molecules are assumed to be equal at z = 0 and z = h to the chemical potentials of corresponding adjacent bulk solutions. These conditions result in the following expressions (see [19] for details) In C~ = I n C(0) +Us and In C3=ln
(59)
Solution of Eq. (56) with boundary conditions (57) and (58) yields,
Os
J
) - 1
It is desirable to decouple C3 and Ca to evaluate selectivity. Moving into the environment outside of the fiber will allow development of expressions to accomplish this decoupling.
Outside the fiber (ro < r < rext) Once again, using the condition of continuity,
_~x(rU,Cl) +_~r(rvlCl ) "=-~r(Dr-~-r 0 OCI)
(63)
Upon integration of this equation,
d f r u l C l d r = - roJlw
) - C3
(64)
ro
Ignoring concentration polarization effects along the membrane wall, it is possible to show as before that C~ = Ca(x). Eq. (64) can then be rewritten as d
-~(C1Q1) =
- 27rroJs
(65)
Now, from Eqs. (61) and (65) it is possible to conclude, d
~ ( C I Q 1 + C3Q3) = 0 or
[asvsh~
CIQ l + C3Q 3 = CoQ
where Co = C1 (0). Upon rearrangement:
Js = vs(oS][ exp( asvsh]
(60)
tT3 t-El-l]
C1 =
roars
As before, rJ(r) = - roJs, or J ( r ) = . Therefore, at r = ri :J3w = -roars Similar to Eq. (54f, rl
c~
J
rext
where Us is the potential of a specific interaction of the dissolved solute molecules with the membrane expressed in kT units (where k is the Boltzmann constant and Tis the temperature in K) [ 19]. Solving for C~ and C3 from the latter equalities result in boundary conditions (57) and (58). Let Os represents the distribution coefficient, defined as [ 19]
Clexp~'-~
Ds .
(62)
C(h) +Us
Os = eUs
.
[
The following boundary conditions on the skin layer interfaces, z = 0 and z = h, are eVsC(0) = C a ( x )
L .
loser lasK(P~- - - P a ) h | -exp
(56)
Us = -- Vlw = K ( P 1 - e 3 )
(61)
= 27rroJs
CoO - C303 al
(66)
This is the necessary connection between concentrations inside and outside the fiber.
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
Assembling the expression for fiber selectivity Substituting the expression for C 1 from Eq. (66) back into Eq. (62) and using a rearranged version of Eqs. (36) and (42), a dimensionless equation is formulated:
-~y[p'(y)c(y)]
O__~/[l_exp(
%KhPlo ~, ,~l -DsyCOshJtY)JJ
where q cosh/3(l-y) (l+y) cosh/3
C3(L) R = l-c(1) = 1---
Co
~7/3 sinh/3y ( 1 + 7) cosh/3
(1+3,)
is deduced from Eq. (43) and
f(y) -flsinh[/3(1 - y ) ] + ~cycosh fly Now, using the compact notation,
b
C3
(68)
Co
asKhPlo
=~--Q-Q3
q
andQ3=
q+P'
Q3
_
Q1 Q-t-Q3
p' q+p'
/32
~y[p'(y)c(y)l = - (Y+ 1)coshJ(Y)
q+~ f q
See Fig. 4 for an example of a plot of selectivity versus dimensionless length parameter. This typical curve was generated from Eq. (70) with boundary condition (71 ) using the Runge-Kutta method. Again, a transformation was performed to express the abscissa in terms of the length parameter, o-. See details below Eq. (50). Since the assumption of dilute feed was used in the development of Eq. (70), selectivities near 0.0 to 0.3 (concentrated salt solutions) in Fig. 4 do not strictly apply. The curves are included here for illustrative purposes.
3. Mathematical model: case A2. Feed exterior to the fiber, countercurrent flow
and substituting all of these expressions into Eq. (67) yields, d
(72)
(69)
DsTcosh/3
and the following ratios
Q1
(71)
and this value of c(0) is used below as a boundary condition for Eq. (70). Notice that Eq. (71) coincides with the dimensionless concentration in the permeate in the case of flat membranes [ 19]. Solving for the concentrations and making the appropriate substitutions, the selectivity can be evaluated since R is defined as:
Q3 ~ll_C(y)/q [ ~ , ~+exp [ D~--~'~'~tY)JJa~KhP' :, ,-]]-°
c=
1
c(0) -
\ O~s
- Q
p, d p = dy
of the concentration with respect to y is finite at y = 0. After some calculations it is found from Eq. (70) that c(0) must be as follows
1 + (J?s_ 1)[1-e - ~:(°>]
- 16tzroKL2 , r4ycosh/3 JtY)
,]k
265
As in Case AI, feed solution is introduced outside the hollow fiber under an applied pressure difference: P 1 f - P l o = zIP > 0. Notice that in the case under consideration direction of the feed solution flow is opposite to the flow inside the fiber. See Fig. 5.
(70)
c(y)[ q+p'(y) -I-e-by(y) P'(Y) ]t a~[1-e -by(y)]
It is possible to show that y = 0 is a singular point for Eq. (70). To make its solution regular at this particular point it is necessary to require that the first derivative
3.1. Productivity Derivation of expressions for flow rates inside the fiber, Q1, and outside the fiber, Q3,are exactly the same as in the previous case. Hence, expressions for Q1 and Q3coincide with Eqs. (24) and (8), respectively. As above, Eq. (28), which represents the conservation of water, can be deduced. Though the form of Eq. (28)
266
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
is now exactly the same as before, the physical meaning is different. Q~ is now negative and actually Q is a difference between flow rates in the permeate and the feed solution; hence, Q must be negative. Eq. (29) is still valid, but some modifications are in order. In the current case a dimensionless flux rate q is introduced in a slightly different way. q = Q~ (Trr4p,f/8tzL)
+ (q)PH-(1 - y ) 3'
Plf-P3(x)
Y
p(y) = _
+q(1-y) +y (l+y)
(74)
q=
(75)
Again, the differences between the definitions of countercurrent flow and concurrent flow [Eqs. (38) and (39) ] are noted. In the same way as above, the equation for p(y) dependence is q(1 - y ) ( 1 + y)
(l+y)(1-O-(1-cosh/3) tanh/3 y+--
3' ] ( 1 + y)
(76)
with boundary conditions p(1) = 0
(77)
(80)
/3 Productivity is now defined as qo=QI(L)-QI(O)
~ = e l o / e l f < 1 and p = P 3 / P l f
(79)
Now the unknown quantity q can be determined with the help of Eqs. (74) and (79) a t y = 0 ,
=
Q,(L)
The following dimensionless quantities are introduced
-d2p - = /32[p dyE
(q/fl)sinh[fl( 1 - y ) ] + y cosh/3y ( 1 + y)cosh/3
(73)
Introducing q according to Eq. (73) differs from the corresponding value q used in Eq. (30). In the case of countercurrent flow, q is negative, and the maximum pressure is P lf. This pressure is used in the denominator instead of the previously used Pro. Integration of Eq. (28) yields PI(x)
where p' indicates the first derivative o f p with respect toy. Solution of the Eqs. (76)-(78) results in an equation for pressure distribution inside the fiber
p'(1)
p'(1) +q
which results in the following expressions for the productivity of the hollow fiber for the countercurrent case q~=
(cosh f l - 1) [ 1 + ~(1 + y) ] + y(/]sinh fl-cosh fl+ 1) y(/Jsinh f l - cosh ~ + 1) +cosh ,8- 1 + ( - ~ (1 + "y)(),cosh fl + 1)
(81)
3.2. Selectivity Eq. (62) and the following conservation law are still valid for the countercurrent case: d(C1al + C303) =0 dx which after integration becomes
and p'(O) = 0
(78) FLOW
",o .............
"- .............
.. o
T ,, 3
'--I
t
H/MMM/HMMHHMM//MHMMMMMHMMMMMMM/ I ZE* = C, (L)Q, (L) + C3(L) Q3(L)
.L~___
".,~
FLo,
(82)
E* is an integration constant, which can be determined from a boundary condition at x = L
P.f I -'- .....
C1QI + C3Q3 = E*
D --x--3 p-o
Fig. 5. Countercurrent flow (case A2). Feed solution supplied from outside the fiber: 1, feed solution; 2, bilayer membrane (s, skin layer; b, supporting layer); 3, permeate.
(83)
In this case the constant E" includes the unknown concentration C3(L) to be determined below. Eq. (82) provides a necessary connection between concentrations inside and outside the fiber E* Cl(x) = - Ql(x)
C3(x)Qa(x ) Ql(x)
After simple manipulations Eq. (84) yields
(84)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
Q3(x) =
Q,(x)
267
4. Mathematical model: case B. Feed interior to fiber
P'(Y) q+p'(y)
E* = C q + p ' ( 1 ) R Ql(x) o q+p,(y)
(85)
where Co - C1 (L) is a given feed concentration and
C3(L) R= 1 - - Co
(86)
R is a rejection coefficient in the countercurrent case. Introducing as before dimensionless concentration
Schematic presentation of the case under consideration is presented in Fig. 6. In this case, a feed solution with an initial concentration Co is supplied at x = 0 in the lumen of the hollow fiber and flows under an imposed pressure difference AP=P3o-P3f> O. Outside the hollow fiber pressure is equal to atmospheric pressure, which is considered as a reference pressure. The concentration in the permeate is denoted as Cp. This concentration is to be determined.
4.1. Productivity
C3(x)
c(y) = ~ Co
Combining Eqs. (1) and (2) with boundary conditions (3) and (4) results in Eq. (8) for the flow rate inside the fiber. Using the same arguments used for the derivation of Eq. (19), the following equation is obtained
and 17
g(y) =~sinh[fl( 1 - y ) ] + T cosh fly
V3w(X)
=KP3(x)
(89)
Eq. (62) yields the required equation for the concentration profile inside the hollow fiber
where the membrane permeability is
d --~y[p'(y)c(y)] ffi -
K=
fl2g(y)
(87)
( 1+ T)cosh
Ks
g(y)
q+p'(1)R p'(y) ~-c(y)[ ~ - e x p ( q+p'(y) q+p'ty)
- b=-:'~") ] ¢
O.[a,1- exp(-/~ff)) ]
As in the case of concurrent flow, it can be shown that y = 0 is a singular point in Eq. (87). As in the previous case, dc/dy at y = 0 must be finite, which results in the following boundary condition
Because of the cylindrical geometry, permeability is slightly different from that determined in Eq. (19). But when the condition riKs/Kb<
l+p'(1)R c(0) =
q
&
1+--[ ~s
1 -
bg(O) ~1 expl,T / J
~ ] PERMEATE
(88)
Eq. (87) with boundary condition (88) determines the profile of concentration of dissolved molecules inside the fiber. Both equation and boundary condition include unknown rejection coefficient R = l - c ( 1 ) . Eqs. (87) and (88) can be solved using the RungeKutta method with iterations. A typical result of such a solution is presented in Fig. 4.
q %
"°
~ ....
_',_t___'_.L
. . . . . . .
s
2s--~
J ~,o.,
~.
x-L %
....
2b
Fig. 6. Feed solution inside (case B): 1, permeate; 2, bilayer membrane (s, skin layer; b, supporting layer); 3, feed solution; Cp, concentration in the permeate.
268
V.M. Starov et al. /Journal of Membrane Science 103 (1995) 257-270
dP3_(161XK]p 3 dx I d f
dC
(90)
or introducing dimensionless quantities p (y) = P3 (x) / P30, ~--P3f/P30 < 1, 0 -2= 16#KL2/~, and y = x/ L, Eq. (90) becomes
,,_d2p p = ~----y2= 0-2p
(91)
with the following boundary conditions p(O) = 1 a n d p ( 1 ) = ~
(92)
Solution of Eq. (91) with boundary conditions (92) results in the following equation for pressure distribution inside the hollow fiber
p(y) =
# sinh 0-y + sinh [ 0- ( 1 - y) ]
(93)
sinh 0-
Productivity in the case of feed inside the hollow fiber should be determined as c19=[Q3(O)-Q3(L)]/ Q3(0), or using Eqs. (8) and (93) p'(1) qb= 1 - - - p'(0)
(~+l)(coshG-1)
(94)
cosh 0 - - #
J3w = asvsC-Ds"~z, 0 < z < h
(95)
dC
J3w=vsC-Db-'-i --, h < z < H - h oz
(96)
with boundary conditions that express equality of chemical potentials of dissolved molecules at the skin layer interfaces z = 0 a n d z = h C(0)/2 s = C3(x)
(97)
C(h- )12s=C(h+ )
(98)
C(H) = Co
(99)
where concentration Cp in the permeate outside the hollow fiber is to be determined below. Eqs. (95) and (96) with boundary conditions (97) to (99) are solved and the solution leads to the unknown flux J3w of dissolved molecules J3w =
exp[
o,
ob
J
7
+exd_ ~.vsh][1_exp( t's(.Do~ h).)]] J "~, D~ ]L
-
IJ
(100)
Previously, the dimensionless length parameter owas used and it is used for all three cases: A1, A2, and B of Fig. 4.
In the same way as above the conservation law for dissolved molecules, Eq. (54), can be applied, where Q3(x) is given by the following equation
4.2. Selectivity
O3(x) =
Local variable z = r - ri is introduced with the origin on the inner surface of the membrane. See Fig. 7. Flux J3w of dissolved molecules consists as before of two parts: convective transport coupled with water flow and diffusive flux
percmeate
"trr4p3o dp(y) 8/zL dy
(101)
and the dependencep(y) is calculated according to Eq. (93); the dependence ofJ3w(X ) is given by Eq. (100). The concentration in the permeate Cp is now determined in terms of fluxes of water and dissolved molecules. According to [ 13] this concentration is equal to the ratio Cp= {total amount of dissolved molecules permeated through the hollow fiber walls}/{total amount of water permeated through the hollow fiber walls } or L
f J3w(x)dx
~ / / / / / / / / / / / / / / / / / / / A
°1
ca(x)
s
"~
feed
CpX
Fig. 7. Local coordinate with origin on the inner surface of the membrane wall.
o L
f V3w(x)dx 0
(102)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
269
The integrals in this equation can be calculated using Eqs. (9) and (54)
with a boundary condition in the same dimensionless units
L
c(O) = 1
L
fV3w(x)dx=
1 (dQ3dx 2~-Tri,J "-~ O
0
1 --
L
-[Q3(L) 27rr i
- Q3(0)
]
(103)
]
(104)
L
f J3w(x)dx = - --
2~'r~J dx O
o
(109)
In the case under consideration, Eq. (108) does not have a singular point at y = 0 and can be easily solved by the Runge-Kutta method using iterations. Concentration in the permeate Cp is expressed using concentration c( 1 ) at the hollow fiber end, which can be found in the process of iteration. Results of calculations according to Eqs. (108) to (109) is shown in Fig. 4.
1 -- - - - [ C 3 ( L ) Q 3 ( L
27rri
) - C3(0)Q3(0)
5. Conclusions
Substituting Eqs. (103) and (104) into definition (102) the following expression for concentration in the permeate is obtained
C3(L)Q3(L) -
COO3(0) Q3(L) - Q3(0)
Cp =
(105)
Introducing dimensionless concentrations c---C3/Co and % = Cp/Co and using Eq. (101), Eq. (105) gives
dp
C(1)~yyly=
1--~yly=O
co= ~y,>,=,-~y,,,=o
(lO6)
Dependence o f p ( y ) is given by Eq. (93). After some rearrangement of Eq. (106) Cp= ( F + 1) - c( 1)F
(107)
where F - ( 1 - ~cosh o-) / [ ( ~c+ 1 ) (cosh o" - 1) ]. It is important to notice that concentration in the permeate is determined by an unknown concentration c(1) at the end of the hollow fiber, which can be calculated in the process of solution only. Applying Eq. (54) to the case under consideration, let us introduce the following dimensionless groups M= KP3o(H- h)/O b and w = ashDb/(H- h)Ds. Using these notations, Eq. (54) can be recast as d -~y[C(y)p' (y) ] =
An analytical model that provides a method to optimize performance of hollow fiber membranes is developed from fundamental principles. Fiber performance is defined in terms of a balance between fiber productivity and fiber selectivity. Fig. 1 illustrates the dynamic interplay of these coupled phenomena. A dimensionless length parameter is designated as the independent variable to gage how fiber performance changes as a function of fiber length. Other independent variables such as operating pressure, fiber packing density, and fiber diameter are studied in Part II of this series of papers. The sensitivity of fiber performance to these variables will be illustrated with appropriate plots.
6. List of symbols
b c C D F
c(y) - ceexp ( - p ( y ) M ( w + 1) ) ~l~[ _ e x p ( - p ( y ) w M ) ] + e x p ( - p ( y ) M w ) [ 1 - e x p ( - p ( y ) M ) ] as
M see definition after Eq. (107) ~ dimensionless number p dimensionless pressure diffusion coefficient P pressure ( 1 - ~ c o s h ~r)/ q dimensionless flow [ (~+ 1) (cosh rate
,r-l)]
(108)
fl o'2p(y)
see definition Eq. (69) dimensionless concentration concentration
f(y)
see definition Eq. (25) see definition after Eq. (67)
Q flow rate r radial coordinate
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
270
g ( y ) see definition after h H J K
Eq. (86) skin layer thickness thickness of hollow fiber wall dissolved molecule flux permeability
L
fiber length
y Z
x/L r--ri
R rejection coefficient or selectivity u axial coordinate U potential of interaction v radial velocity w see definition after Eq. ( 1 0 7 ) x axial coordinate
Greek symbols a
departure o f convective velocity of dissolved molecules from convective velocity o f water. see definition Eq. ( 4 1 )
7 8(ro/rl)'f~(X) A re~to> 1 ratio of imposed pressure tr dimensionless length parameter; see definition after Eq. ( 5 0 ) /~ viscosity productivity
X { [ ( T + l ) / T ] / ( r o / r i ) } °'s distribution coefficient
Subscripts 1 2 3 i o ext w s b 0 f p Re Pec
exterior region (outside the fiber) in the m e m b r a n e (fiber wall) interior ( l u m e n ) region inner outer outer b o u n d a r y (free surface) at the outside fiber wall skin layer support layer at the origin at the axial coordinate L in the permeate Reynolds number Peclet n u m b e r
Superscript * integration constant
Acknowledgements The authors are grateful to the State of Texas A d v a n c e d Technology Program and The University of Texas at Austin Separations Research Program for the financial support of this project.
References [ 1] G. Belfort, Membrane modules: Comparison of different configurations using fluid mechanics, J. Membrane Sci., 35 (1988) 245-270. [21 M. Soltanieh and W.N. Gill, A note on the effect of fiber length on the productivity of hollow fiber modules, Chem. Eng. Commun., 22 (1983) 109-113. [3 ] J.J. Hermans, Hydrodynamics of hollow fiber reverse osmosis modules, Membrane Dig., 1(3) (1972) 45. [4] J.K. Lawson, J.D. Bashaw, and T.A. Orofino. Hollow fiber technology for advanced waste treatment, ASME J., April (1973) 1-13. [5] B. Bansal, A Theoretical and Experimental Study of Hollow Fiber Reverse Analysis, Clarkson College of Technology, Ph.D. Dissertation, 1973. [6] W.N. Gill and B. Bansal, Hollow fiber reverse osmosis systems analysisand design. AIChE J., 19 (1973) 823-831. [7] B. Bansal and W.N. Gill, Theoretical experimental study of radial flow hollow fiber reverse osmosis, AIChE Symp. Ser., 70 (1974) 136-149. [8] M.S. Dandavati, M.R. Doshi, and W.N. Gill, Hollow fiber reverse osmosis: experiments and analysis of radial flow systems, Chem. Eng. Sci., 30 (1975) 877-886. [9] M. Soltanieh, Experimental Study of Hollow Fiber Modules in Desalination Processes, State University of New York at Buffalo, Ph.D. Dissertation, 1979. [ 10] M.R. Doshi, W.N. Gill, and V.N. Kabadi, Optimal design of hollow fiber modules, AIChE L, 23 (1977) 765-768. [ I 1] PerraasepEngineering Manual, E.1.DuPont de Nemours, 1982. [ 12] N.V. Churaev and V.M. Starov, The theory of reverse osmosis separation of solutions on hollow fiber membranes, L Colloid Interface Sci., 89 (1982) 77-85. [ 13] VM. Starov and A.M. Torkunov, Theory of reverse osmosis separations of solutions of hollow fibers in a continuous flow, Khimiya i Tekhnologiya Vody, 13 (1990) 195-201. [ 14] J. Happel, Viscous flow relative to array of cylinders, AIChE J., 5 (1959) 175. [15] E.M. Sparrow and A. Locffler, Longitudinal laminar flow between cylinders arranged in regular array, AIChE J., 5 (1959) 325. [ 16] E.M. Sparrow, A.L. Loeffler, and H. Hubbard, Heat transfer to longitudinal laminar flow between cylinders, ASME J. Heat Trans., 415 (1961).
Journal of Membrane Science 103 (1995) 257-270
Performance optimization of hollow fiber reverse osmosis membranes, Part I. Development of theory Victor M. Starov u, Jim Smart b, Douglas R. Lloyd b,. a Moscow Academy of Food Industry, Moscow 125080, Russian Federation b Department of Chemical Engineering, Separations Research Program, The University of Texas at Austin, Austin TX 78712-1062, USA
Received 16 June 1994;accepted in revised form 19 December 1994
Abstract The overall performance of hollow fiber membranes can be engineered based upon the interplay of fiber productivity and fiber selectivity. Fiber length and diameter can be optimized to render the desired balance of fiber performance. Fundamental analytical models are developed for feed outside (concurrent and countercurrent flow configurations) and feed inside. Part II of this series of papers will evaluate the sensitivity of fiber performance to such operating conditions as pressure, packing density, and fiber diameter. Keywords: Reverseosmosis; Modules; Modeling
I. Introduction Hollow fiber membranes for desalination offer the advantage of withstanding large pressure drops and provide the economic advantage of enormous surface area to volume ratios. In addition, when the feed is directed through the interior of the tube bundle, the high fluid velocities helps reduce surface fouling [ 1 ]. Membrane manufacturers sell modules in a few standard diameters and lengths with a few fiber dimensions (radii, wall thickness, and length) based on in-house experience and the resulting empirical correlations. The objective of the research reported here was to develop a model capable of optimizing hollow fiber dimensions and module operating parameters in terms of productivity and selectivity. * Corresponding author. Tel (512) 471-4985, Fax (512) 4717060. 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD10376-7388(95)00008-9
Productivity or recovery ( q~, defined as the fraction of feed recovered as permeate) and selectivity or rejection coefficient can be studied in terms of a dimensionless metric parameter o" defined below as a combination of fiber length, inner and outer fiber radii, permeability coefficient, and solution viscosity. As membrane productivity is increased, selectivity suffers and vice versa; consequently, a balance must be struck in membrane design, module design, and operating conditions to satisfy specific application requirements. Consider the case of feed on tlae outside of a single hollow fiber of fixed inner radius, fixed fiber thickness, and short length. If the fiber length is extended, thereby increasing o', more surface area is available and increased productivity results. The permeate volume produced by the fiber continues to increase with additional fiber length until a "critical length" is approached. This critical length is defined to be the point at which ocorresponds to q~= 1.0. This situation is illustrated in
V.M. Starov et al. / Journal of Membrane Sc&nce 103 (1995) 257-270
258
1.0
1.0
o.o ................. T.................:.................i ................i...................................................................i................ i................o., 0.8
0.8
0.7
0.7
0,6
0.6
:~ 0.5
0.5
|
i
~
i
0.4
0,4
0.3
- 0.3
0.2
- 0.2
0.1
- 0.1
0.0
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
dlrnemdonleu length permnet~, 0
Fig. 1. Typical hollow fiber productivity q) and selectivity R as a function of dimensionless length parameter o'. Fig. 1. If fiber productivity tends to 1.0, then selectivity decreases to zero. The objective of this paper is to demonstrate the generation of these curves. Notice in Fig. 1 that permeate quality (that is, selectivity) diminishes with increased fiber length. This phenomenon is a result of two effects: ( 1 ) As fiber length is increased, the solute concentration on the feed side increases along the length of the fiber. Consequently, the local driving force for solute transport across the membrane increases with increased fiber length. (2) Rejection or selectivity decreases with decreasing pressure drop across the membrane wall, as illustrated in Fig. 2. In the case of increasing the length of a hollow fiber, the pressure drop across the membrane wall falls as a result of the additional length [2]. The end result is less rejected salt, with consequent higher salt concentration of the permeate. Early attempts to model hollow fiber reverse osmosis performance in the 1970s produced analytic expressions for productivity in parallel and radial flow patterns [3,4]. However, the system was oversimplified by assuming complete solute rejection across the membrane wall and an arbitrary concentration profile on the shell side of the module. Subsequent attempts to
develop predictive models for parallel and radial flow reverse osmosis assumed constant rejection; that is, rejection independent of fiber length [ 5-7 ]. Replacing this constant rejection model with a solution-diffusion transport model [8], and later a solution-diffusionimperfection model [9], failed to overcome significant deviations between predicted and observed rejection for regions of high productivity and high feed concentrations. In all of these previous studies, hollow fiber productivity was modelled and studied, but fiber selectivity was essentially ignored. Optimum fiber length was discussed by Gill, et al. [ 10], who noted permeation flow (vol/time) could be increased if a single fiber open at one end (current design of Du Pont B-9 permeator module [ 11 ] ) could be opened at both ends. This amounts to shortening the fiber length by one-half and doubling the number of fibers. Starov et al. [ 12,13 ] recognized that fiber length can be adjusted to optimize the balance between productivity and selectivity. The research reported here builds upon these previous studies by: • Developing mathematical models to describe and predict hollow fiber productivity and selectivity as a
1
R~ .......................................
i
i
r
j
5
10
15
20
25
pressure drop across membrane(kPa)
Fig. 2. Dependency of selectivity R of a flat reverse osmosis membrane on the applied pressure across the membrane. Rm~ is a maximum selectivity.
V.M. Starov et aL / Journal of Membrane Science 103 (1995) 257-270
259
Q, feed pressure at the entrance of the module and pressure at the exit are assumed to be fixed. The following conditions are assumed: t ( 1 ) Equivalent annulus or free surface model [ 1416]. P-O (2) Steady laminar flow, ~X~Re <~1, parallel to fibers. 3...~ ~ s.......~ ..................................... I Transverse flow is not considered. (3) Flux through the membrane wall is relatively small. This assumption has been verified by previous Fig. 3. Concurrent flow (case AI). Feed solution supplied from investigators [ 3,9]. outside the fiber: 1, feedsolution;2, bilayermembrane(s, skinlayer; (4) Newtonian fluid with constant physical properb, supportinglayer); 3, permeate. ties. (5) Constant membrane permeability coefficient. function of fiber length, inside fiber radius, and outside (6) System symmetry; no angular velocity compofiber radius. nents; no velocity slip at the wall. • Developing these models for three cases: feed to (7) Temperature constant; no free convection the outside of the fiber in a concurrent flow (case A1; effects. see Fig. 3) and countercurrent flow (case A2; see Fig. (8) Dilute feed; no osmotic pressure effects; no pre5), and feed through the inside of the lumen of the fiber cipitation of dissolved salts. Osmotic pressure will be (case B; see Fig. 6). an important consideration when using more concen• Optimizing fiber performance, which is defined in trated feed solutions. terms of productivity and selectivity. (9) Small diameter fibers such that ~Pec < 1. While the results reported here deal only with reverse These assumptions were selected as a basis for the osmosis, the productivity equations are equally applidevelopment of a simple, fundamental model. Imporcable to ultrafiltration. Ultrafiltration is primarily a sizetant effects such as osmotic pressure considerations will exclusion separation process and the selectivity be included in Part II. of this series. equations presented in this study may have limited application in cases of ultrafiltration. Inside the fiber lumen (0 < r < rl , 0 < x < L) In this section, relationships between fluid pressures, Since r i/L << 1, a Navier-Stokes equation and equafluid velocities, and fiber geometries are developed tion of continuity can be written within three regions: (1) inside the fiber lumen, (2) within the membrane, and (3) outside the fiber. These 1 d P 3 1 0 Ou3 (1) relationships will be used to assemble an expression tx dx = r O r W Or for productivity (qb) and selectivity (R). .......................................
--
.
O_ _
--
--
F L O W
I1
. . . .
3
I
o~(rv3) q-o-~(ru3)=0 2. M a t h e m a t i c a l model: case A1. Feed exterior to fiber; c o n c u r r e n t flow
2.1. Productivity
Feed is introduced outside the hollow fiber under an imposed pressure difference: P l o - P l f = A P > O. See Fig. 3 for definitions of terminology. The assumptions made and development of productivity theory presented here are similar to those of Gill and Bansal [ 6 ] ; however, they developed a model based upon a fixed flow rate, Q. In the following model, instead of fixed
(2)
The boundary conditions are symmetry in the center of the fiber, non-slip condition at the walls, impermeability of the sealed end of the fiber, and atmospheric pressure at the fibers open end Ou3 Or r = O = t ) 3 l r = O = 0
(3)
u31r=~=0
(4)
dP3
~lx=o=O
(5)
P3=0, atx=L
(6)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
260
Here, P3 is the pressure, while u3, and v3 are pressure and velocity components along x and r axes. From these equations and boundary conditions,
Hv2
0X
=
L2
(7)
Volumetric flow rate from Eq. (7) can be expressed as
J
2
ro\ H ] Or
Hence, from Eq. (12) 10 or( rV2) = 0 or rv2 = constant.
t'i
2,trf ru3dr = - ,m.4 d p 3
H
iO(rvj i(rov2]=(~)<
1 dP3
u3: 4;
Q3
OU2
(8)
r
From this result, it is concluded
0
From the equation of continuity, Eq. (2), a conservation law is deduced ri
fru3dr+rv3, o
(13)
roVlw= riV3w
Since rv 2 is constant, this result can be expressed as k* v2 = - - -
(14)
r
0 or
d -T'Q3 + ri2~V3w = 0 Ox
(9)
Combining Eqs. (8) and (9), the expression for pressure distribution inside the fiber lumen becomes d2p3 16~ dx2 ~ "V3w
(10)
In the membrane (ri < r < ro) According to Darcy's law 02 ~---
Or
and
U2 =
--
Ox
(11)
where H = r o - r~ is the total membrane thickness. From the equation of continuity inside the fiber wall Ou2. 1 O W-f-7~rtrV2)
=0
k*
.
OP 2
k*
(15)
where Ks and Kb are permeabilities of the skin layer and the support layer, respectively. Upon rearrangement, pressure within the membrane can be expressed as k* r PE=~ln~o+P1, ( r o - h ) < r < r o
(16)
p E = k * l n r + p3, r i < r < ( r o - h )
(17)
(12)
where h is the thickness of the skin layer. The following boundary conditions were used P2 = P1 (x) at r = ro and PE=P3(x) at r=ri. Pressure inside the membrane is continuous everywhere including the point r = r o - h . Using this condition, the unknown constant k* is found to be equal to k*=
(18)
(P1-P3)
in(fro h ) (o--
the magnitude of the ratio is evaluated as
.. OP 2
7=t~S.~r,,-7-=t~bOr
Kbrl
where K is the membrane permeability coefficient. Using approximations offered by the derivative and forming appropriate ratios, Darcy's law suggests u2 L L =-->>1 u2 t o - ri H
where k* is an undetermined constant. The asymmetric membrane can be further subdivided into a skin layer (s) and a supporting layer (b) as illustrated in Fig. 3. Using Darcy's equation and Eq. (14),
Ks
in( ro - h) k
ri
Kb
V.M. Starov et al. /Journal of Membrane Science 103 (1995) 257-270
Upon substituting this expression back into Eq. (14), using the equality of Eq. (13), and the condition (h < < ro), the final equation for the velocity on the surface of the membrane becomes riU3w = --
Ks
ro" h + Ks(rolnrO_ h) Kb ri
261
From the equation of continuity, the following conservation law is deduced d -7--Q1 - 2~rrovlw= 0 (IX
(26)
Combining Eqs. (26) and (9) results in (19)
d(Ql +
Q3) -4- 27r(riV3w - -
F0Olw ) =
0
(27)
(P1 - P 3 ) = - r o K ( P l - P3)
where K is a permeability of the membrane wall as defined by Eq. (19). Usually roKJKb < < h, and therefore, K = Ks/ h.
Outside the fiber (ro < r < rex0 Using Navier-Stokes equations and the appropriate boundary conditions, as in Eq. (1), (3), and (4), it is possible to write the following expression I OPI Iz
1 O.
(20)
with boundary conditions: ( 1 ) no shear stress and no permeation through the free surface boundary [6-8] Oul
- - = v l =0, at r=rext Or
(22)
Applying the equation of continuity, O/Ox(rul) + O~ 0r(rvl) = 0, yields an expression for velocity outside the fiber dP1
/~extdP1 r In2/x dx ro
(23)
Combining the definition for volumetric flow,
01 = 27rlruldr and Eq. (23) gives rO (24)
3 4 A2 1 f,(A) = - ~ A 4 2 8
Z
dx
=Q
(29)
Introducing the following dimension less expressions, y = 8 (~i0)4fl(A) and q__
Q
(30)
\8gLJ
dPl= dx
l dP3
y dx
q Plo y L
(31)
Upon integration of Eq. (31 ) PI(X) =
P3(x) y
qPlzx+a*-%
(32)
(33)
= Plf
dP31 _ dx x=O-0 and P3(L) = 0
(34)
Using boundary conditions (33) and (34), the unknown constant a* can be determined from Eq. (32) atx=Z
where
h4
/z
P1 (0) = Plo and Pl (L)
7rrgfl (A) dP1 /z dx
+-=-In A, where A=
8/~ dx
where a* is an unknown constant of integration. The following boundary conditions apply
rext
Q1 =
7rr4A(A) dP1
Eq. (29) can be rewritten as
ul =0, at r=ro
. . . 4/x dx
(28)
where Q is an integration constant to be determined below. From Eqs. (8), (24), and (28),
(21)
(2) non-slip condition at the wall
(r2--~)
d -7"(01-t-03) = 0 or Q1 + Q 3 = Q = constant (IX
7rr4 dP3
Oul.
Ox =r ~ r ' ~ r )
ul .
Recall ffomEq. (13), qv3w- ~vlw =0. Therefore, it can be concluded
rext> 1 ro
(25)
qPlo +a* or a* Y
Plf . . . .
=Plf+--
qPlo
Y
(35)
V.M. Starov et al./ Journal of Membrane Science 103 (1995) 257-270
262
Substituting Eq. (35) into Eq. (32), gives the final expression for pressure distribution outside the fiber
Using dimensionless parameters, this can be expressed
Pl(x) = P3(x__....~)+plf+qPlo(1 x 3' 3" - Z )
p(0) = 3'(~- 1) +q (36)
as
Combining Eqs. (43) and (45), results in 3'(~- 1) + q =
Assembling the expression for fiber productivity Combining Eqs. (10), (19), and (36), the following equation for pressure distribution inside the fiber can be produced d2p3 16/~roK dx2 r4
(37)
(45)
q tanh/3
~3'
/3(1+3')
(l+3')cosh/3 + q + - -~y l+y l+y
Rearrangement produces an expression for the dimensionless flow rate,
[ ' 3 ( 1 +1) --Vlf- ~-~( 1 - L) ] Introducing the following dimensionless variables and quantities, P3
P=P-£o
(38)
q=Y
~=~o
'|
(46)
/ where dimensionless flow rate q must be positive. Productivity is defined as • --
Plf
" _ tTnh~
QI(0) - QI(L)
(39)
Q,(0)
= 1
QI(L) -
-
(47)
-
QI(0)
Using Eqs. (24) and (31), productivity @can be recast X
Y=L
(40)
as
dp(l) dy q
- -
/32_
16/zroKL2( 3' + 1) Y
r¢
• =1
(41)
(42)
q sinh[/3(1 - y ) ] /3( 1 + y) cosh/3 s¢3' cosh/3y_~ q(1-y_____))~_ s¢3' 1+3' coshfl 1+3" 1+3"
(48)
ly = 1 -- (
1 + 3') cosh/3
#3"/3sinh/3 q + - (1+ y)cosh/3 (1+3,)
(49)
Substituting Eq. (49) into (48), yields the final expression for productivity, 1
(43)
1 - - - -
cosh/3
1+3'
Now, an expression for q can be deduced from the rest of the boundary conditions. At x= 0, Eq. (36) can be rewritten as P3(O) = (Plf - Plo) 3' + qPlo
1 dp(1) q dy
q
with boundary conditions dp/dyly =o and p( 1) = 0. Solution of Eq. (42) yields the result
p(y) =
-
From Eqs. (43) and (46)
Eq. (37) can be written as d2P 2 q ~----y2=/3 [p - 1--~---y(1 - y) - 1--~+~y]
+ q
(44)
{
, tanh/3]
~fl tanh fl ~3"t - - - - ~ j +
(50)
(1 + 3 ' ) [ 1 + 3 ' - ' ( 3 ' + c o ~ hfl)]
V.M. Starovet al. / Journal of MembraneScience 103 (1995) 257-270
1.0-
!
Concentration distribution inside the fiber, outside the fiber, and in the membrane are used to derive an equation for rejection coefficient.
- 1.0
.s !
0.9 .....
- 0.9
0.8-=
- 0.8
0.7 ......
- 0.7
0.6 .....
- 0.6
263
Inside the fiber lumen (0 < r < ri , 0 < x < L) The equation of continuity for dissolved molecules can be expressed as
O(ru3C3 )
|
(52)
=c
¢ o
-2--0 = O__(D Or_~2C3t q-or (rv3C3) Or\ Or]
0.5
0.5 o
;/
0.4
8
~->
!/
where D is a diffusion coefficient of dissolved molecules in the bulk solution. Integration with boundary condition OC3/Orl r=o = v3 Ir=o = 0 yields
0.4
0.3
- 0.3
diru3C3dr=-riJ3w
0 - 0.2 0.2 f - 0.1
0.1
/7/I
:~
0.0 0.0
...........
....
'"*"
I - ~ - - ~ - L ' ~ ' ~
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
II
-0.0
0.9
1.0
dlmenslonlen length parameter, t~
Fig. 4. Examples of calculated dependencies of productivity and selectivity on dimensionless length parameter o-: --, case A 1; -.-, case A2; . . . . , case B. Note, productivity q~= ~ ( 3,,/3,sc) is a function of only three independent dimensionless parameters. See Fig. 4 for an example of a plot of productivity versus dimensionless length parameter generated from Eq. (50). Since the assumption of dilute feed was used in the development of Eq. (50), productivities near 1.0 (concentrated salt solutions) in Fig. 4 do not strictly apply. The curves are included here for illustrative purposes. The abscissa of Fig. 4 is the dimensionless length parameter, 0-, where 0-2= (16txKL2)/~ fl and 0- are related by/~2 = 0 . 2 X 2 , where X 2 = [ ( 3' + 1 ) / 3'] (ro/ri). This transformation is done to provide for consistency between all cases of flow configurations.
2.2. Selectivity
- C3(L)
C1 (0)
~ D~ u3C3< < ] / ~0C3, -,/~~ I o r
or
v3ri/D< < 1
ri
At v3 ~ 1 0 - 4 c m / s , r i ~ 10 -4 cm and D ~ 10 -5 cm2/s. Therefore, the ratio v3ri/D, which is the Peclet number (INPec), is ~ 10 -3 < < 1. Thus, it is possible to assume that concentration depends only on the axial coordinate. Hence, the above equations indicate d
r
"~[ C3fru3dr]
=
-- riJ3w
o
(53) ri
Upon substitution of Q3 = 2"rrjru3 dr into Eq. (53), the final expression becomes d ~ ( C 3 Q 3 ) = - 2~rriJ3w
(54)
Inside the membrane skin [(ro - h) < r < ro, h < < ro]
Selectivity is defined in terms of the rejection coefficient:
R -Cn(O)
where J3w is a flux of dissolved molecules through the inside fiber walls. It is assumed below that concentration depends on the x coordinate only; that is, C3 = C3 (x). This assumption means that radial diffusion makes the concentration profile uniform in the radial direction; that is,
1 - (L-------~) C3
Cl (0)
(51)
Molar flux moving from ro, in the normal direction toward ri can be expressed as [ 12,17,18] : dC Js = a s V s C - Ds--:oz
(55)
264
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
where z = r o - r , Ds is the diffusion coefficient in the skin layer, and a s is a constant that corrects for the deviation of the dissolved molecules convective velocity from the convective velocity of the solution [ 12, 171. In the membrane supporting layer, concentration is assumed to be the same as inside the hollow fiber; that is, C3 = C3(x). This assumption is valid under the same conditions as above; that is, small Peclet number. Velocity in the skin layer vs is introduced using Eqs. (13) and (19):
d
~(C303)
Combining Eqs. (8), (60), and (61) yields the final equation for the concentration within the hollow fiber: - ~-~
dp3
c
= 27rroK(P1 - P 3 ) .
rasK(P1-P3)h] "~"/-
~exp/ .
(57)
eVsC(h) = C3(x)
(58)
These boundary conditions were introduced in [ 18] and are briefly explained as follows: chemical potentials of the dissolved molecules are assumed to be equal at z = 0 and z = h to the chemical potentials of corresponding adjacent bulk solutions. These conditions result in the following expressions (see [19] for details) In C~ = I n C(0) +Us and In C3=ln
(59)
Solution of Eq. (56) with boundary conditions (57) and (58) yields,
Os
J
) - 1
It is desirable to decouple C3 and Ca to evaluate selectivity. Moving into the environment outside of the fiber will allow development of expressions to accomplish this decoupling.
Outside the fiber (ro < r < rext) Once again, using the condition of continuity,
_~x(rU,Cl) +_~r(rvlCl ) "=-~r(Dr-~-r 0 OCI)
(63)
Upon integration of this equation,
d f r u l C l d r = - roJlw
) - C3
(64)
ro
Ignoring concentration polarization effects along the membrane wall, it is possible to show as before that C~ = Ca(x). Eq. (64) can then be rewritten as d
-~(C1Q1) =
- 27rroJs
(65)
Now, from Eqs. (61) and (65) it is possible to conclude, d
~ ( C I Q 1 + C3Q3) = 0 or
[asvsh~
CIQ l + C3Q 3 = CoQ
where Co = C1 (0). Upon rearrangement:
Js = vs(oS][ exp( asvsh]
(60)
tT3 t-El-l]
C1 =
roars
As before, rJ(r) = - roJs, or J ( r ) = . Therefore, at r = ri :J3w = -roars Similar to Eq. (54f, rl
c~
J
rext
where Us is the potential of a specific interaction of the dissolved solute molecules with the membrane expressed in kT units (where k is the Boltzmann constant and Tis the temperature in K) [ 19]. Solving for C~ and C3 from the latter equalities result in boundary conditions (57) and (58). Let Os represents the distribution coefficient, defined as [ 19]
Clexp~'-~
Ds .
(62)
C(h) +Us
Os = eUs
.
[
The following boundary conditions on the skin layer interfaces, z = 0 and z = h, are eVsC(0) = C a ( x )
L .
loser lasK(P~- - - P a ) h | -exp
(56)
Us = -- Vlw = K ( P 1 - e 3 )
(61)
= 27rroJs
CoO - C303 al
(66)
This is the necessary connection between concentrations inside and outside the fiber.
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
Assembling the expression for fiber selectivity Substituting the expression for C 1 from Eq. (66) back into Eq. (62) and using a rearranged version of Eqs. (36) and (42), a dimensionless equation is formulated:
-~y[p'(y)c(y)]
O__~/[l_exp(
%KhPlo ~, ,~l -DsyCOshJtY)JJ
where q cosh/3(l-y) (l+y) cosh/3
C3(L) R = l-c(1) = 1---
Co
~7/3 sinh/3y ( 1 + 7) cosh/3
(1+3,)
is deduced from Eq. (43) and
f(y) -flsinh[/3(1 - y ) ] + ~cycosh fly Now, using the compact notation,
b
C3
(68)
Co
asKhPlo
=~--Q-Q3
q
andQ3=
q+P'
Q3
_
Q1 Q-t-Q3
p' q+p'
/32
~y[p'(y)c(y)l = - (Y+ 1)coshJ(Y)
q+~ f q
See Fig. 4 for an example of a plot of selectivity versus dimensionless length parameter. This typical curve was generated from Eq. (70) with boundary condition (71 ) using the Runge-Kutta method. Again, a transformation was performed to express the abscissa in terms of the length parameter, o-. See details below Eq. (50). Since the assumption of dilute feed was used in the development of Eq. (70), selectivities near 0.0 to 0.3 (concentrated salt solutions) in Fig. 4 do not strictly apply. The curves are included here for illustrative purposes.
3. Mathematical model: case A2. Feed exterior to the fiber, countercurrent flow
and substituting all of these expressions into Eq. (67) yields, d
(72)
(69)
DsTcosh/3
and the following ratios
Q1
(71)
and this value of c(0) is used below as a boundary condition for Eq. (70). Notice that Eq. (71) coincides with the dimensionless concentration in the permeate in the case of flat membranes [ 19]. Solving for the concentrations and making the appropriate substitutions, the selectivity can be evaluated since R is defined as:
Q3 ~ll_C(y)/q [ ~ , ~+exp [ D~--~'~'~tY)JJa~KhP' :, ,-]]-°
c=
1
c(0) -
\ O~s
- Q
p, d p = dy
of the concentration with respect to y is finite at y = 0. After some calculations it is found from Eq. (70) that c(0) must be as follows
1 + (J?s_ 1)[1-e - ~:(°>]
- 16tzroKL2 , r4ycosh/3 JtY)
,]k
265
As in Case AI, feed solution is introduced outside the hollow fiber under an applied pressure difference: P 1 f - P l o = zIP > 0. Notice that in the case under consideration direction of the feed solution flow is opposite to the flow inside the fiber. See Fig. 5.
(70)
c(y)[ q+p'(y) -I-e-by(y) P'(Y) ]t a~[1-e -by(y)]
It is possible to show that y = 0 is a singular point for Eq. (70). To make its solution regular at this particular point it is necessary to require that the first derivative
3.1. Productivity Derivation of expressions for flow rates inside the fiber, Q1, and outside the fiber, Q3,are exactly the same as in the previous case. Hence, expressions for Q1 and Q3coincide with Eqs. (24) and (8), respectively. As above, Eq. (28), which represents the conservation of water, can be deduced. Though the form of Eq. (28)
266
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
is now exactly the same as before, the physical meaning is different. Q~ is now negative and actually Q is a difference between flow rates in the permeate and the feed solution; hence, Q must be negative. Eq. (29) is still valid, but some modifications are in order. In the current case a dimensionless flux rate q is introduced in a slightly different way. q = Q~ (Trr4p,f/8tzL)
+ (q)PH-(1 - y ) 3'
Plf-P3(x)
Y
p(y) = _
+q(1-y) +y (l+y)
(74)
q=
(75)
Again, the differences between the definitions of countercurrent flow and concurrent flow [Eqs. (38) and (39) ] are noted. In the same way as above, the equation for p(y) dependence is q(1 - y ) ( 1 + y)
(l+y)(1-O-(1-cosh/3) tanh/3 y+--
3' ] ( 1 + y)
(76)
with boundary conditions p(1) = 0
(77)
(80)
/3 Productivity is now defined as qo=QI(L)-QI(O)
~ = e l o / e l f < 1 and p = P 3 / P l f
(79)
Now the unknown quantity q can be determined with the help of Eqs. (74) and (79) a t y = 0 ,
=
Q,(L)
The following dimensionless quantities are introduced
-d2p - = /32[p dyE
(q/fl)sinh[fl( 1 - y ) ] + y cosh/3y ( 1 + y)cosh/3
(73)
Introducing q according to Eq. (73) differs from the corresponding value q used in Eq. (30). In the case of countercurrent flow, q is negative, and the maximum pressure is P lf. This pressure is used in the denominator instead of the previously used Pro. Integration of Eq. (28) yields PI(x)
where p' indicates the first derivative o f p with respect toy. Solution of the Eqs. (76)-(78) results in an equation for pressure distribution inside the fiber
p'(1)
p'(1) +q
which results in the following expressions for the productivity of the hollow fiber for the countercurrent case q~=
(cosh f l - 1) [ 1 + ~(1 + y) ] + y(/]sinh fl-cosh fl+ 1) y(/Jsinh f l - cosh ~ + 1) +cosh ,8- 1 + ( - ~ (1 + "y)(),cosh fl + 1)
(81)
3.2. Selectivity Eq. (62) and the following conservation law are still valid for the countercurrent case: d(C1al + C303) =0 dx which after integration becomes
and p'(O) = 0
(78) FLOW
",o .............
"- .............
.. o
T ,, 3
'--I
t
H/MMM/HMMHHMM//MHMMMMMHMMMMMMM/ I ZE* = C, (L)Q, (L) + C3(L) Q3(L)
.L~___
".,~
FLo,
(82)
E* is an integration constant, which can be determined from a boundary condition at x = L
P.f I -'- .....
C1QI + C3Q3 = E*
D --x--3 p-o
Fig. 5. Countercurrent flow (case A2). Feed solution supplied from outside the fiber: 1, feed solution; 2, bilayer membrane (s, skin layer; b, supporting layer); 3, permeate.
(83)
In this case the constant E" includes the unknown concentration C3(L) to be determined below. Eq. (82) provides a necessary connection between concentrations inside and outside the fiber E* Cl(x) = - Ql(x)
C3(x)Qa(x ) Ql(x)
After simple manipulations Eq. (84) yields
(84)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
Q3(x) =
Q,(x)
267
4. Mathematical model: case B. Feed interior to fiber
P'(Y) q+p'(y)
E* = C q + p ' ( 1 ) R Ql(x) o q+p,(y)
(85)
where Co - C1 (L) is a given feed concentration and
C3(L) R= 1 - - Co
(86)
R is a rejection coefficient in the countercurrent case. Introducing as before dimensionless concentration
Schematic presentation of the case under consideration is presented in Fig. 6. In this case, a feed solution with an initial concentration Co is supplied at x = 0 in the lumen of the hollow fiber and flows under an imposed pressure difference AP=P3o-P3f> O. Outside the hollow fiber pressure is equal to atmospheric pressure, which is considered as a reference pressure. The concentration in the permeate is denoted as Cp. This concentration is to be determined.
4.1. Productivity
C3(x)
c(y) = ~ Co
Combining Eqs. (1) and (2) with boundary conditions (3) and (4) results in Eq. (8) for the flow rate inside the fiber. Using the same arguments used for the derivation of Eq. (19), the following equation is obtained
and 17
g(y) =~sinh[fl( 1 - y ) ] + T cosh fly
V3w(X)
=KP3(x)
(89)
Eq. (62) yields the required equation for the concentration profile inside the hollow fiber
where the membrane permeability is
d --~y[p'(y)c(y)] ffi -
K=
fl2g(y)
(87)
( 1+ T)cosh
Ks
g(y)
q+p'(1)R p'(y) ~-c(y)[ ~ - e x p ( q+p'(y) q+p'ty)
- b=-:'~") ] ¢
O.[a,1- exp(-/~ff)) ]
As in the case of concurrent flow, it can be shown that y = 0 is a singular point in Eq. (87). As in the previous case, dc/dy at y = 0 must be finite, which results in the following boundary condition
Because of the cylindrical geometry, permeability is slightly different from that determined in Eq. (19). But when the condition riKs/Kb<
l+p'(1)R c(0) =
q
&
1+--[ ~s
1 -
bg(O) ~1 expl,T / J
~ ] PERMEATE
(88)
Eq. (87) with boundary condition (88) determines the profile of concentration of dissolved molecules inside the fiber. Both equation and boundary condition include unknown rejection coefficient R = l - c ( 1 ) . Eqs. (87) and (88) can be solved using the RungeKutta method with iterations. A typical result of such a solution is presented in Fig. 4.
q %
"°
~ ....
_',_t___'_.L
. . . . . . .
s
2s--~
J ~,o.,
~.
x-L %
....
2b
Fig. 6. Feed solution inside (case B): 1, permeate; 2, bilayer membrane (s, skin layer; b, supporting layer); 3, feed solution; Cp, concentration in the permeate.
268
V.M. Starov et al. /Journal of Membrane Science 103 (1995) 257-270
dP3_(161XK]p 3 dx I d f
dC
(90)
or introducing dimensionless quantities p (y) = P3 (x) / P30, ~--P3f/P30 < 1, 0 -2= 16#KL2/~, and y = x/ L, Eq. (90) becomes
,,_d2p p = ~----y2= 0-2p
(91)
with the following boundary conditions p(O) = 1 a n d p ( 1 ) = ~
(92)
Solution of Eq. (91) with boundary conditions (92) results in the following equation for pressure distribution inside the hollow fiber
p(y) =
# sinh 0-y + sinh [ 0- ( 1 - y) ]
(93)
sinh 0-
Productivity in the case of feed inside the hollow fiber should be determined as c19=[Q3(O)-Q3(L)]/ Q3(0), or using Eqs. (8) and (93) p'(1) qb= 1 - - - p'(0)
(~+l)(coshG-1)
(94)
cosh 0 - - #
J3w = asvsC-Ds"~z, 0 < z < h
(95)
dC
J3w=vsC-Db-'-i --, h < z < H - h oz
(96)
with boundary conditions that express equality of chemical potentials of dissolved molecules at the skin layer interfaces z = 0 a n d z = h C(0)/2 s = C3(x)
(97)
C(h- )12s=C(h+ )
(98)
C(H) = Co
(99)
where concentration Cp in the permeate outside the hollow fiber is to be determined below. Eqs. (95) and (96) with boundary conditions (97) to (99) are solved and the solution leads to the unknown flux J3w of dissolved molecules J3w =
exp[
o,
ob
J
7
+exd_ ~.vsh][1_exp( t's(.Do~ h).)]] J "~, D~ ]L
-
IJ
(100)
Previously, the dimensionless length parameter owas used and it is used for all three cases: A1, A2, and B of Fig. 4.
In the same way as above the conservation law for dissolved molecules, Eq. (54), can be applied, where Q3(x) is given by the following equation
4.2. Selectivity
O3(x) =
Local variable z = r - ri is introduced with the origin on the inner surface of the membrane. See Fig. 7. Flux J3w of dissolved molecules consists as before of two parts: convective transport coupled with water flow and diffusive flux
percmeate
"trr4p3o dp(y) 8/zL dy
(101)
and the dependencep(y) is calculated according to Eq. (93); the dependence ofJ3w(X ) is given by Eq. (100). The concentration in the permeate Cp is now determined in terms of fluxes of water and dissolved molecules. According to [ 13] this concentration is equal to the ratio Cp= {total amount of dissolved molecules permeated through the hollow fiber walls}/{total amount of water permeated through the hollow fiber walls } or L
f J3w(x)dx
~ / / / / / / / / / / / / / / / / / / / A
°1
ca(x)
s
"~
feed
CpX
Fig. 7. Local coordinate with origin on the inner surface of the membrane wall.
o L
f V3w(x)dx 0
(102)
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
269
The integrals in this equation can be calculated using Eqs. (9) and (54)
with a boundary condition in the same dimensionless units
L
c(O) = 1
L
fV3w(x)dx=
1 (dQ3dx 2~-Tri,J "-~ O
0
1 --
L
-[Q3(L) 27rr i
- Q3(0)
]
(103)
]
(104)
L
f J3w(x)dx = - --
2~'r~J dx O
o
(109)
In the case under consideration, Eq. (108) does not have a singular point at y = 0 and can be easily solved by the Runge-Kutta method using iterations. Concentration in the permeate Cp is expressed using concentration c( 1 ) at the hollow fiber end, which can be found in the process of iteration. Results of calculations according to Eqs. (108) to (109) is shown in Fig. 4.
1 -- - - - [ C 3 ( L ) Q 3 ( L
27rri
) - C3(0)Q3(0)
5. Conclusions
Substituting Eqs. (103) and (104) into definition (102) the following expression for concentration in the permeate is obtained
C3(L)Q3(L) -
COO3(0) Q3(L) - Q3(0)
Cp =
(105)
Introducing dimensionless concentrations c---C3/Co and % = Cp/Co and using Eq. (101), Eq. (105) gives
dp
C(1)~yyly=
1--~yly=O
co= ~y,>,=,-~y,,,=o
(lO6)
Dependence o f p ( y ) is given by Eq. (93). After some rearrangement of Eq. (106) Cp= ( F + 1) - c( 1)F
(107)
where F - ( 1 - ~cosh o-) / [ ( ~c+ 1 ) (cosh o" - 1) ]. It is important to notice that concentration in the permeate is determined by an unknown concentration c(1) at the end of the hollow fiber, which can be calculated in the process of solution only. Applying Eq. (54) to the case under consideration, let us introduce the following dimensionless groups M= KP3o(H- h)/O b and w = ashDb/(H- h)Ds. Using these notations, Eq. (54) can be recast as d -~y[C(y)p' (y) ] =
An analytical model that provides a method to optimize performance of hollow fiber membranes is developed from fundamental principles. Fiber performance is defined in terms of a balance between fiber productivity and fiber selectivity. Fig. 1 illustrates the dynamic interplay of these coupled phenomena. A dimensionless length parameter is designated as the independent variable to gage how fiber performance changes as a function of fiber length. Other independent variables such as operating pressure, fiber packing density, and fiber diameter are studied in Part II of this series of papers. The sensitivity of fiber performance to these variables will be illustrated with appropriate plots.
6. List of symbols
b c C D F
c(y) - ceexp ( - p ( y ) M ( w + 1) ) ~l~[ _ e x p ( - p ( y ) w M ) ] + e x p ( - p ( y ) M w ) [ 1 - e x p ( - p ( y ) M ) ] as
M see definition after Eq. (107) ~ dimensionless number p dimensionless pressure diffusion coefficient P pressure ( 1 - ~ c o s h ~r)/ q dimensionless flow [ (~+ 1) (cosh rate
,r-l)]
(108)
fl o'2p(y)
see definition Eq. (69) dimensionless concentration concentration
f(y)
see definition Eq. (25) see definition after Eq. (67)
Q flow rate r radial coordinate
V.M. Starov et al. / Journal of Membrane Science 103 (1995) 257-270
270
g ( y ) see definition after h H J K
Eq. (86) skin layer thickness thickness of hollow fiber wall dissolved molecule flux permeability
L
fiber length
y Z
x/L r--ri
R rejection coefficient or selectivity u axial coordinate U potential of interaction v radial velocity w see definition after Eq. ( 1 0 7 ) x axial coordinate
Greek symbols a
departure o f convective velocity of dissolved molecules from convective velocity o f water. see definition Eq. ( 4 1 )
7 8(ro/rl)'f~(X) A re~to> 1 ratio of imposed pressure tr dimensionless length parameter; see definition after Eq. ( 5 0 ) /~ viscosity productivity
X { [ ( T + l ) / T ] / ( r o / r i ) } °'s distribution coefficient
Subscripts 1 2 3 i o ext w s b 0 f p Re Pec
exterior region (outside the fiber) in the m e m b r a n e (fiber wall) interior ( l u m e n ) region inner outer outer b o u n d a r y (free surface) at the outside fiber wall skin layer support layer at the origin at the axial coordinate L in the permeate Reynolds number Peclet n u m b e r
Superscript * integration constant
Acknowledgements The authors are grateful to the State of Texas A d v a n c e d Technology Program and The University of Texas at Austin Separations Research Program for the financial support of this project.
References [ 1] G. Belfort, Membrane modules: Comparison of different configurations using fluid mechanics, J. Membrane Sci., 35 (1988) 245-270. [21 M. Soltanieh and W.N. Gill, A note on the effect of fiber length on the productivity of hollow fiber modules, Chem. Eng. Commun., 22 (1983) 109-113. [3 ] J.J. Hermans, Hydrodynamics of hollow fiber reverse osmosis modules, Membrane Dig., 1(3) (1972) 45. [4] J.K. Lawson, J.D. Bashaw, and T.A. Orofino. Hollow fiber technology for advanced waste treatment, ASME J., April (1973) 1-13. [5] B. Bansal, A Theoretical and Experimental Study of Hollow Fiber Reverse Analysis, Clarkson College of Technology, Ph.D. Dissertation, 1973. [6] W.N. Gill and B. Bansal, Hollow fiber reverse osmosis systems analysisand design. AIChE J., 19 (1973) 823-831. [7] B. Bansal and W.N. Gill, Theoretical experimental study of radial flow hollow fiber reverse osmosis, AIChE Symp. Ser., 70 (1974) 136-149. [8] M.S. Dandavati, M.R. Doshi, and W.N. Gill, Hollow fiber reverse osmosis: experiments and analysis of radial flow systems, Chem. Eng. Sci., 30 (1975) 877-886. [9] M. Soltanieh, Experimental Study of Hollow Fiber Modules in Desalination Processes, State University of New York at Buffalo, Ph.D. Dissertation, 1979. [ 10] M.R. Doshi, W.N. Gill, and V.N. Kabadi, Optimal design of hollow fiber modules, AIChE L, 23 (1977) 765-768. [ I 1] PerraasepEngineering Manual, E.1.DuPont de Nemours, 1982. [ 12] N.V. Churaev and V.M. Starov, The theory of reverse osmosis separation of solutions on hollow fiber membranes, L Colloid Interface Sci., 89 (1982) 77-85. [ 13] VM. Starov and A.M. Torkunov, Theory of reverse osmosis separations of solutions of hollow fibers in a continuous flow, Khimiya i Tekhnologiya Vody, 13 (1990) 195-201. [ 14] J. Happel, Viscous flow relative to array of cylinders, AIChE J., 5 (1959) 175. [15] E.M. Sparrow and A. Locffler, Longitudinal laminar flow between cylinders arranged in regular array, AIChE J., 5 (1959) 325. [ 16] E.M. Sparrow, A.L. Loeffler, and H. Hubbard, Heat transfer to longitudinal laminar flow between cylinders, ASME J. Heat Trans., 415 (1961).