Nomenclature
V+
x
= membrane constant defined by Equation 6
A
= half distance between membranes B I , B2 = constant defined by Equations 2g and 8 B(m, n) = beta function = concentration of salt C,(O.y+) = concentration of salt a t inlet = kth-order solution of perturbation solution DS = diffusion coefficient of salt in brine D?l = coefficients of Equation 20
X+
= v/ Vu(0), dimensionless transyerse velocity = distance along longitudinal direction = x / u , dimensionless distance along longitudinal
direction
U
c
fh
= functions defined by Equation 15
K I , K2
=
coefficients defined by Equation 24 = coefficients defined by Equation 25 M I , M z , Ma, Mb, M E ,M 6 = coefficients defined by Equation 26 ATl,-Y*, S a , A\74, S j , A T a , AV7,.V8 = coefficients defined by Equation 27 P I , Pz,Pa, P1, Pj, Pg, P7, Ps, Pg,Plo = coefficients defined by Equation 28 Po, = osmotic p r m u r e (P,J0 = osmotic pressure at concentration C, = volumetric rate of production defined by Equation Q 30 = parameter defined by Equation 2f S U = velocity along longitudinal direction = buik velocity a t x = 0 0 U+ = u / U , dimmsionless velocity along longitudinal direction = velocity along transverse direction U = transverse water velocity across membrane VU Vu(0) = transverse velocity a t x = 0
Li,L z , La, Ld
Yn01) Y 2
= eigenfunction in Equation 19 = transverse distance measured from center = y / u , dimensionless transverse distance = defined by Equation 23
CY
=
Y+
= =
= = =
of channel
D
defined as 2 a V w( 0 ) eigenvalue in Equation 19 gamma function total pressure drop across membrane constant defined in Equation 31
literature Cited (1) Berman, A. C., J . Appl. Phys. 24, 1232 (1953). (2) Dresner, Lawrence, “Boundary Layer Build-up in the De-
mineralization of Salt Water by Reverse Osmosis,” Oak Ridge National Laboratory, Rept. 3621 (May 1964). (3) Sherwood, T. K., Brian, P. L. T., Fisher, R. E., “Salt Concentration at Phase Boundary in Desalination Processes,” Desalination Research Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Rept. 295-1 (1963). RECEIVED for review October 17, 1964 ACCEPTED April 22, 1965 IYork performed with the financial support of the Office of Saline Water under contract No. 14-01-0001-401. Numerical calculations supported in part by National Science Foundation Grant GP1137.
CON CENT RATIO N PO LA RIZATI 0 N I N REVERSE OSMOSIS DESALINATION WITH VARIABLE FLUX AND INCOMPLETE SALT REJ ECT IO N IP , L . T , B R I A N ,
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
A finite-difference solution is obtained for salt concentration polarization in reverse osmosis with laminar flow of brine between parallel flat membranes. The analysis accounts for the permeation flux falling off as the salt concentration at membrane surface builds up, and the effect of incomplete salt rejection is also included. The results are compared with the constant-flux solution, and it is found that the average polarization over the length of the membrane is very nearly the same’for the two cases, if the average permeation fluxes are equal. This formed the basis of a simplified design procedure, based upon the constant-flux solution, which accurately predicts polarization effects upon pressure drop requirements and product water salinity. N WATER
desalinatiori by reverse osmosis, potable water is
I removed from a saline solution by permeation through a
semipermeable membrane which rejects the dissolved salts more or less completely. The convective flow of the solution carries salt u p to the membrane surface, and since the salt is rejected by the membrme it must diffuse back into the bulk saline solution. Thus the salt concentration a t the membrane surface builds up to a value exceeding the bulk salt concentration until the back diffuijion of salt produced by this concentration gradient just counterbalances the convection of salt to the membrane surface by iihe water flowing through the membrane.
This salt concentration polarization has several effects which are detrimental to the desalination process. First of all, the osmotic pressure which must be overcome is that corresponding to the salt concentration a t the membrane surface, and concentration polarization results in this effective osmotic pressure exceeding the osmotic pressure of the bulk saline solution. I n addition, the concentration polarization has a detrimental effect upon the salinity of the product water, because this salinity will generally increase when the salt concentration a t the membrane surface in the saline solution increases. Likewise, the useful life of the osmotic membrane is often shortened by increased salinity of the saline water, and conVOL. 4
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centration polarization will aggravate this effect. Finally, most saline waters contain a t least one salt a t a concentration level which is within a factor of 2 of its solubility limit. Thus, excessive concentration polarization will often cause some salt to concentrate a t the membrane surface to its solubility limit, causing precipitation of this salt a t the membrane surface. The problem of concentration polarization at the surface of a reverse osmosis membrane has been discussed by Merten (3), Merten, Lonsdale, and Riley ( 4 ) , and Sherwood. Brian, Fisher, and Dresner ( 5 ) . Theoretical solutions are presented (5)for concentration polarization effects for both turbulent and laminar flo\\ in round tubes and for laminar flon in a t\vodimensional channel. These results show that concentration polarization effects are not insignificant when the water flux rate is approximately 10 gallons per day per sq. foot, and become increasingly important a t higher water flux rates. Thus, as membranes are developed which permit operation a t higher \\ater flux rates, concentration polarization Mill become increasingly important and will be a major consideration in the design and optimization of reverse osmosis systems. The theoretical analyses for laminar flow (5) were based upon the assumptions that the water flux rate through the membrane N as uniform, independent of longitudinal position, and that the salt rejection by the membrane was complete. These idealizations are approached in desalination of brackish waters of low salinity. but in such cases the effects of concentration polarization are much less detrimental than in desalination of brines of high salinity. In desalination of sea water, for example, the idealizations of uniform water flux and complete salt rejection are not realized. Thus, this work extends the theoretical analysis ( 5 ) for laminar flow between flat osmotic membranes to assess the effects of water flux variation and of incomplete salt rejection upon concentration polarization at the membrane surface. Theoretical Model
The problem analyzed is concentration polarization in a saline solution flou ing in a two-dimensional channel between flat parallel osmotic membranes. The continuity equation for salt conservation can be written in dimensionless form as
[
a ( CC) + 4 vc ax aY
__
=
0
This equation considers convection in the longitudinal direction and diffusion and convection in the transverse direction, but neglects longitudinal diffusion. The normalized salt concentration, C, is the local salt concentration divided by the salt concentration in the brine feed a t the channel inlet, and U and V represent the normalized velocity components in the longitudinal and transverse directions, respectively. T h e dimensionless coordinates, X and Y , represent dimensionless distances in the longitudinal and transverse directions, respectively; X would be numerically equal to the fractional water removal a t a given longitudinal position if the water flux through the membrane were to remain constant a t its value a t the channel inlet. The normalized diffusion coefficient, a,, is defined as the diffusion coefficient divided by the channel half width and the flux through the membrane a t the channel inlet, The boundary conditions to be imposed upon the solution of Equation 1 are At
X
=
0, any Y : C = 1
At Y = 0, any:
440
l&EC FUNDAMENTALS
ac
X-
dY
=
0
(2)
(3)
At Y = 1, a n y X :
cyo
(E)
=
RVC
(4)
Equation 2 represents the assumption that the salt concentration is uniform a t the channel inlet, and Equation 3 implies symmetry with respect to the midplane ; this latter condition is based upon the assumption that the t\vo osmotic membranes forming the channel walls are identical in properties and are therefore permeating a t equal rates. Equation 4 relates salt diffusion and convection at the membrane surface to the salt rejection, R, defined as 1 minus the ratio of the salt flux through the membrane to the product of the permeation velocity a t the membrane surface and the salt concentration a t the membrane surface. T o integrate this differential equation, the velocity field must first be specified. Berman (7) has obtained a solution for the velocity field for the case in which the water flux through the membrane is uniform.
The quantity .\'I is a permeation Reynolds number, based upon the half width of the channel and the permeation velocity, and \+ill generally be so small in reverse osmosis desalination applications that the terms involving it can be neglected. With this simplification. Equation 5 reveals that the longitudinal velocity varies in a parabolic fashion with transverse position just as it does Lvhen there is no flux through the wall, although in the present case the average velocity does vary with longitudinal position as water is withdrawn through the membrane. Similarly, the transverse velocity is simply equal to the transverse velocity at the channel wall multiplied by a cubic function of the transverse position. A solution for the velocity field has not been obtained for the general case in which the flux through the membrane varies with longitudinal position. But when the permeation Reynolds number, )TI, is small, the effect of a constant permeation flux is simply to change the average longitudinal velocity with longitudinal position; the parabolic profile is not distorted. Thus, it seems reasonable to assume that the longitudinal velocity profile can be represented by 3 2
u = -73 (1
- Y')
(7)
even when the permeation flux varies with longitudinal position, provided that the permeation Reynolds number is suitably small. Similarly, it seems reasonable to assume that the transverse velocity can be approximated by u = uw
();
(3
-
Y2)
where uw is the permeation velocity a t the local longitudinal position. Thus, the following analysis is based upon the assumption that the velocity field can be approximated by Equations 7 and 8. This implies the additional assumption that the parabolic velocity profile is already developed a t x = 0. Written in dimensionless form, these equations become
L7 =
(4)
v = v,
(1
- A)(l
(3 -
(3
- Y')
- Y*)
The quantity V, is the local value of the permeation velocity divided by the permeation velocity a t the channel inlet. A is the fractional water removal, obtained by integrating the permeation velocity with respect to longitudinal position.
x VwdX’
A =
The permeation flux must now be related to the salt concentration a t the membrane surface. Merten (3) reports that the permeation velocity can be expressed by U,
=
K(AP
- AT)
(12)
where AP is the pressure drop across the membrane, AT is the difference between the osmotic pressures of the saline solution a t the membrane surface and the product water, and K is the membrane permeability constant. Assuming that the osmotic pressure is directly proportional to the salt concentration and that AP is essentially constant, Equation 12 may be written as
vu = 1 - P-f
(13)
T h e constant 8, defined as
is a measure of the fraction of the pressure driving force which is represented by the osmotic pressure of the feed solution. T h e concentration polarization, y, is defined as y
c, - 1
(1 5)
and thus in Equation 13 represents the decrease in the permeation rate due to the concentration polarization a t the membrane surface. The analysis that follows is for a brine containing a single salt with a constant diffusion coefficient, and the volume change upon mixing solutions of different salt concentrations is assumed to be negligible. In considering membranes with incomplete salt rejection, it has been assumed that the salt rejection, R, is constant. This is a simplification; most reverse osmosis membranes probably show a salt rejection which decreases as the salt concentration a t the membrane surface increases. Severtheless, the manner in which salt rejection varies with salt concentration has not been studied in detail nor well characterized, and thus the assumption of a constant salt rejection seems to be most appropriate a t this time. It should be adequate in order to indicate the effect of incomplete salt rejection in a general way. Solution of Differential Equation. T h e system of equations represented by Equations 1 through 4, 9 through 11, and 13 was solved by a finite-difference method on a digital computer. A description of the finite-difference method, a copy of the Fortran program, and instructions for using the program are presented by Brian (2, Appendix), The degree of convergence of the finite-difference solution could be judged by making standard convergence tests, in which the increments in the longitudinal and transverse distances were decreased, and also by comparison with the results of Sherwood, Brian, Fisher, and Dresner ( 5 ) for the case in which the permeation. flux was constant and the salt rejection was unity. Based upon these considerations, the finite-difference solutions are believed to be convergent to within less than 0.5%.
tion flux does not fall off as the concentration polarization builds up, and thus this is the case of a uniform water flux. The finite-difference solutions obtained with P = 0 and R = 1 are presented in Figure 1, plotted in the same manner as Figure 3 of ( 5 ) . T h e results for values of CY = 0.0677, 0.27, and 0.50 are in excellent agreement with the infinite series solution presented in (5), and indeed the generation of the present finite-difference results required considerably less computer time than did the generation of the infinite series solutions. Thus, this problem represents a n example in which a finitedifference solution of the partial differential equation is more readily obtained than is the infinite series solution. Shown for comparison, as the dashed curve. is the approximate solution for the entrance region given by Equations 23 and 25 in (5). The curve for each value of CY peels a\\ay from the entrance region solution as it approaches the limiting ’.far-downstream” polarization corresponding to that value of CY. Figure 1 shows that the approximate solution ne11 represents the entrance region solution, although the approximate solution is somewhat low a t low values of the abscissa in Figure 1. T h e excellent agreement betxveen the finite-difference solution obtained in the present work and the infinite series and approximate solutions reported previously lends considerable confidence regarding the accuracy of these solutions. Since Figure 1 contains the solution for a greater number of values of CY than had been reported previously, it should be very useful for quickly obtaining the concentration polarization of a reverse osmosis membrane with a uniform permeation flux and complete salt rejection. Effect of Variable Flux. T h e effect of a variable permeation flux is illustrated by the results of calculations performed for CY, = 0.27, = 1, R = 1. T h e value of a, = 0.27 corresponds to a permeation flux a t the inlet end of the membrane equal to 10 gal./(day)(sq. ft.) if the membrane spacing is 0.1 inch-that is, if h = 0.127 cm.-or it corresponds to a permeation flux of 100 gal./(day)(sq. ft.) if the membrane spacing is only 0.01 inch. The value of p equal to unity corresponds to a n applied pressure drop across the membrane equal to twice the osmotic pressure of the feed brine; thus, a t the inlet end of the channel, half of the pressure driving force is expended in overcoming the osmotic pressure of the feed solution and half is expended in overcoming the membrane permeation resistance. These computations are for a membrane whichc ompletely rejects the salt. Figures 2, 3, and 4 present the results of the calculation for this case. The solid curve in Figure 2 shows the manner in which the salt concentration a t the membrane surface increases with
450.0677
0.27
k I
Results and Discussion
A number of computer runs were made with a value of = 0. For this case, according to Equation 13, the permea-
Figure 1 . rejection
Solution for constant flux and complete salt
VOL. 4
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441
L L A - u J
I .o .'0
0.1
0.2
0.4
0.3
0.5
0.6
0.7
X
Figure 2. Concentration a t membrane surface vs. longitudinal position
-
= 0.27,6 = 1, R = 1
_...._.__ ~~=0.933,P=O,R=1
longitudinal position. The salt concentration increases rapidly a t first because of the high permeation flux near the channel inlet. Then, as the concentration a t the membrane surface approaches twice the feed concentration, the permeation flux falls off drastically and the concentration curve flattens out. At a value of X = 0.656, the salt concentration at the membrane surface has reached a value of 1.797 times the feed concentration. The solid curve in Figure 3 displays the manner in which the permeation flux falls off with increasing longitudinal position. The values of V , are readily obtained from the values of C, by use of Equation 13. At a value of X = 0.656, the permeation flux has fallen to approximately 20% of its value a t the channel inlet. The solid curve in Figure 4 presents the fractional water removal, A, as a function of longitudinal position. X is numerically equal to the fraction of the water that would have been removed if the permeation flux had remained constant a t the inlet value. Figure 4 shows that A = 0.19 when X = 0.656, and thus only 19% of the water was removed in this membrane instead of the 6670 which would have been removed if the concentration polarization had not caused a reduction in the permeation rate. The average value of the normalized permeation rate up to X = 0.656 is given by
yw
0.19 -_ _ = 0.2895 0.656
-
and thus the average value of a is 0.27 0.2895
s=--
-
0.933
Comparison with Constant Flux Solution. Since the solution for a constant permeation flux and complete salt rejection is well generalized according to Figure 1, it should be useful to compare the results of the variable flux solution with those for a constant flux and to inquire into the utility of the constant 442
I&EC FUNDAMENTALS
0 0
0.2
0.1
Figure 3. position ~
0.3 X
0.4
0.5
0.6
0.7
Permeation flux vs. longitudinal
-- - - - - -- -
cyo CY
= 0.27, 6 = 1, R = 1 = 0.933, 6 = 0, R = 1
flux solution for designing a variable flux system. The most logical way to compare the two solutions is a t a common value of the average permeation flux over the entire membrane. Thus, the constant flux solution was generated for a = 0.933, /3 = 0, R = 1. This corresponds to a case in which the permeation flux is constant a t a value equal to 29y0 of the permeation flux a t the channel inlet for the variable flux example considered previously. The dashed line in Figure 3 represents the permeation flux for this case. The dashed curve in Figure 2 presents the salt concentration a t the membrane surface as a function of longitudinal position (normalized with the value of uwo for the variable flux problem). Near the channel inlet the concentration polarization for the constant flux case is lower than that for the corresponding variable flux case because the permeation flux is lower in the constant flux case. At larger values of X approaching 0.656, the reverse is true. The dashed line in Figure 4 presents the fractional water removal as a function of longitudinal position for the constant flux example. At a value of X = 0.656, 19% of the water has been removed in both example problems. Figure 2 reveals that the concentration polarization behavior in the constant flux example is different from that in the variable flux example, as would be expected. Indeed, a t X = 0.656, C, has slightly exceeded 2 in the constant flux example, and this could never happen in the variable flux example because this would correspond to a negative value of the permeation flux, according to Equation 13. Nevertheless, it is logical to inquire into how the average polarization in the constant flux example compares with that in the variable flux example and how useful might the constant flux solution be for designing a membrane system with a variable flux. Thus, for the constant flux example shown in Figures 2, 3, and 4, the average value of the concentration polarization up to X = 0.656 was computed according to
0 5
0 0
0.1
Figure 4. position
0.2
0.3
X
0.4
0.5
0.6
0.7
Water removal vs. longitudinal
- #za= 0.27,9, = 1 , R = 1 I-..-..--cy = 0,933, p = 0,R =
1
and the resulting value was found to be 0.6725. This value can be used with Equati.on 13, modified to the average form
in order to form the basis of a design procedure based upon the constant flux solution. Inserting values of 0.2895 and 0.6725 for p, and 7,r'espectively, in Equation 19, the calculated value of is 1.057, which is 6% greater than the actual value. This corresponds to a required pressure drop equal to 1.95 times the osmotic pressure of the feed solution, instead of the actual value of exactly twice the osmotic pressure of the feed solution. This example sugger,ts that the constant flux solution is useful for preliminary design calculations of the effect of concentration polarization when the permeation flux is in fact variable according to E,quation 13. A constant flux solution for a given average permeation flux can be used to obtain the average polarization according to Equation 18. Equation 19 can then be used to calculate the pressure drop required to give that permeation flux (on the average) in the presence of the concentration polarization. For the example presented above, this approximate design calculation predicts a required pressure drop only 2.5% lower than the actual pressure drop required. Furthermore, the conditions in the example problem presented a rather severe test of this approximate design procedure, corresponding to a situation in which the permeation flux fell off by a factor of approximately 5 owing to the concentration polarization. At lower values of A in the example case cited, the permeation flux has not fallen off as much, and the approximate design procedure is even more accurate. This approximate design procedure has been tested for several other conditions. One such condition is cyo = 0.5, /3 = 1, R = 1, and A = 0.274. At this water removal, the permeation flux had fallen to of its value a t the channel
1.0 0
1
l 0.1
1
l 0.2
l
l
l
0.3 X
l 1 l I l I 0.4 0.5 0.6 0.7
Figure 5. Concentration a t membrane surface vs. longitudinal position
-cyo = 0.27,p = 1, R = 0.9 ._ ....-..cy = 0.838, = 0,R = 0.9
inlet. Another condition tested is cio = 0.27, p = 0.5, R = 1, and A = 0.359. At this point, V , had fallen to 0.26. For both of these conditions, the approximate design procedure was found to be as accurate as for the example cited previously, the calculated value of p being within 6% of the actual value. Thus, it is concluded that this approximate design procedure, based upon the constant flux solution, should be useful in predicting the excess pressure drop requirements due to concentration polarization a t the reverse osmosis membrane. T h e integration required to obtain the average value of y according to Equation 18 can conveniently be avoided by simply approximating the average value of y for the constant flux calculation by the value of y a t the midpoint of the channel. For example, in the case shown in Figures 2 through 4, a n approximation to the average value of y can be obtained by reading the dashed curve in Figure 2 a t a value of X = 0.328. This yields a value of 0.702, and inserting this into Equation 19 with 8, = 0.2895 yields a calculated value of 13 equal to 1.01, a n error of only 1yo as compared to 670 when the integral average value of y was computed according to Equation 18. I n all examples which were examined, the use of the concentration polarization a t the midpoint of the channel in place of the integral average value resulted in a more accurate calculation of according to Equation 19; since it is also much simpler, it is the preferred method of calculation. Effect of Salt Rejection. T h e solid curve in Figure 5 depicts the salt concentration buildup as a function of longitudinal position for a case in which cyo = 0.27, p = 1, and R = 0.9. These conditions are similar to those for the example shown in Figure 2, but in the present case the membrane has a 90% salt rejection. Furthermore, while p is equal to unity for both cases, in the present case this corresponds to a n over-all pressure drop equal to 1.8 times the osmotic pressure of the feed solution instead of twice this osmotic pressure. Figure 5 shows that, a t X = 0.656, the salt concentration a t the membrane surface has built up to a value equal to 1.768 times the feed concentration, and therefore the permeation VOL. 4
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443
10,
-4
i
~2~0.0677
I
t
1
A
0.1 0.001
0.0I
A/3a2
Figure 6.
Solution
for constant flux, R = 0.9
Figure 7.
vm
and, a t X = 0.656, (c,/c,) is found to be 0.164. The dashed curve in Figure 5 shows the result of a computation for cy = 0.838, p = 0, and R = 0.9. This represents the case of a constant flux membrane operating at the same average flux as that for the variable flux case shown as the solid curve in Figure 5. As in the variable flux case, the constant flux case is for a membrane with a 90% salt rejection. The comparison between the constant flux results and the variable flux results is seen to be similar to that shown in Figure 2. Using Equation 18, the value of a t X = 0.656 for the constant flux example was found to be 0.651. Using this value together with = 0.322 in Equation 19 yields a calculated value of p equal to 1.04, only 4% greater than the actual value. The value of y a t the midpoint of the channel, X = 0.328, is read from the dashed curve in Figure 5 as 0.682. Using this value instead of 4 in Equation 19 yields a calculated value of p = 0.994, only 0.6y0 lower than the actual value. For this example calculation the use of the constant flux solution to calculate the required pressure drop for the membrane is just as accurate when the salt rejection is 90% as in the preceding example with complete salt rejection. The constant flux solution can also be used to predict the salinity of the product water. For the special case in which the permeation flux is constant, Equation 20 simplifies to - =
(1 - R)(Y
+ 1)
(21)
60
Using 4 = 0.651 in Equation 21 results in a value of 0.165 for (cJc,), which is within O.6Y0 of the actual value. If, instead of using the integral average value of y, the value of y a t the channel midpoint is used, the resulting value of ( c B / c 0 ) is 0.168, which is 2.5% greater than the actual value. Utility of Approximate Design Method. The preceding examples show that the proposed approximate design method, based upon the constant flux solution, yields accurate predictions of the effects of concentration polarization on the required pressure drop and on the salinity of the product water. The method is simple, and a digital computer is not required. Therefore, the method should be very useful, a t least for making preliminary design and optimization calculations. 444
I&EC FUNDAMENTALS
I
10
IO0
A/3a2
flux has fallen to 2370 of its value a t the channel inlet. At this point A = 0.211, and thus = 0.322 and E = 0.838. The salinity of the product water can be computed by the formula
GP
0.I
Solution for constant flux, R = 0.8
Furthermore, the general conclusion that the average polarization is relatively insensitive to the manner in which the permeation flux varies would be expected to apply to cases not treated in this study-for example, the tubular membrane analyzed in ( 5 ) . Consider also a brine containing a mixture of salts with different rejection values; the computer program developed in this study is not applicable to this case without considerable modification. But the approximate procedure can be applied to each salt separately to determine its average polarization, and the extra pressure drop requirement and the product water salinity can be computed from the average polarization of each salt. The problem of a salt rejection varying with concentration or the problem of osmotic pressure not being exactly proportional to concentration would be handled in a similar manner. T o furnish all requirements for the approximate design method, it would be useful to provide a generalized description of the constant flux solution for incomplete salt rejection similar to that given in Figure 1 for the case of complete salt rejection, Figures 6 and 7 present the calculated results for a number of different values of cy and for salt rejection values of 90 and 80%, respectively. The variables plotted in Figures 6 and 7 are the same as those plotted in Figure 1, but the ordinate, defined as (1 f y ) ( l
-
A)
-1
(22)
does not have the same physical significance when the salt rejection is incomplete as it does for complete salt rejection. When salt rejection is complete, r represents the concentration polarization relative to the mixing-cup average salt concentration, as described in (5). For incomplete salt rejection r as defined by Equation 22 does not have this interpretation because of the salt which leaked through the membrane. Nevertheless, P as defined by Equation 22 is a more convenient variable to work with for the case of incomplete salt rejection because it is simply related to y . Furthermore, the use of this variable results in bringing together the curves for different values of cy into a common curve for the entrance region, facilitating interpolation with respect to cy. Thus, the method of plotting shown in Figures 6 and 7 represents a convenient way of expressing the numerical solutions obtained for concentration polarization of a constant flux membrane with incomplete salt rejection. Conclusions
Finite-difference solutions have been presented for the concentration polarization of a reverse osmosis desalination
membrane with a variable permeation flux and incomplete salt rejection. When these solutions are compared with the solutions for a constant flux membrane operating a t the same average permeation flux, it is found that the concentration polarization is greater for the variable flux case near the channel entrance and lower near the channel exit. The average polarization for the constant flux case is very nearly equal to that for the variable flux case, even under the relatively severe conditions examined in which the polarization caused the permeation flux to fall off by a factor of 5 from the channel entrance to the channel exit. This formed the basis of a simple design procrdure in which the constant flux solution can be used to predict i he effects of concentration polarization upon the required pressure drop and upon the product water salinity for a variable flux membrane. This simple procedure, together with generaliied graphs of the constant flux solution \vhich are presented, should be useful in the design and optimization of reverse osmosis processes. More accurate calculations of the concentration polarization can be performed by using the finite-difference method developed in this study, described in the appendix to (2). Acknowledgment
The machine computations were performed a t the Massachusetts Institute of Technology Computation Center, and the author is grateful for the use of the center’s facilities. This study is part of a research program on desalination supported by the Office of Saline LVater, U. S. Department of the Interior. Nomenclature c
=
salt concentration, g./cc.
c = c c:,
D = molecular diffusion coefficient of salt, (sq. cm.)/sec. h
= half xvidth of channel, cm.
K = membrane permeability constant, (cm./sec.)/p.s.i.
v = hu,./u
-.r
Ts = salt flux through membrane, g./’(sq. cm.) (sec.) R = fractional salt rejection = 1 - iVs/ceu, u
.ii
=
average value of u over channel a t a given value of x , cm./sec.
u
= u/a,
u
=
5,
y
= average value of urn over channel length, cm./sec. = u/uwo
x
=
Ve
X y Y
velocity component in y-direction, cm./sec.
average value of Vu over channel length longitudinal distance from channel inlet, cm. = (u~,/U,) (x/h) = transverse distance from channel midplane, cm. = yjh =
GREEK D,‘u,h for constant permeation flux D, Gw,h s = D,%,h P = R x o / ( 4 P - RT0) r = defined by Equation 22 Y = c, - 1 4 = defined by Equation 18 A = fractional water removal at a given longitudinal position AP = pressure drop across membrane, p.s.i. Air = difference in osmotic pressure across membrane, p.s.i. v = kinematic viscosity of solution, (sq. cm.)/sec. x = osmotic pressure, p.s.i. f f = ff, =
SUBSCRIPTS o = channel inlet-Le., x = 0 p = product water, mixed average over length of membrane w = channel wall-Le., membrane surface SUPERSCRIPT ’ = dummy variable in definite integral literature Cited (1) Berman, A. S., J . Appl. Phys. 24, 1232 (1953). (2) Brian, P. L. T., M.I.T. Desalination Research Laboratory, Rept. 295-7 (May 1965). (3) Merten, Ulrich, IND. END. CHEM. FUNDAMENTALS 2, 229 (1 963). (4) Merten, Ulrich, Lonsdale, H. K., Riley, R. L., Ibid., 3, 210 (1964). (5) Sherwood, T. K., Brian, P. L. T., Fisher, R. E., Dresner, L., Ibid.,4, 113 (1965).
RECEIVED for review May 7, 1965 ACCEPTED September 3, 1965
= velocity component in x-direction, cm./sec.
LIQUID P E R M E A T I O N T H R O U G H PLASTIC F I L M S R
. B.
L 0 N G , Central Basic Research Laboratory, Esso Research and Engineering Co., Linden, N . J.
Liquid permeaition i s apparently a special case of ordinary diffusion and can be explained by a classical diffusion model. The exponential concentration dependence of diffusivity leads to equations which are very sensitive to the concentration of liquid in the upstream side of the film. Furthermore, the liquid concentration gradient through the film shows that essentially all the resistance to diffusion i s at the downstream edge of the film. The imodel has been experimentally tested for permeation of hydrocarbons through polypropylene film and predicts the observed effect of downstream pressure as well as solvent absorption rate for bulk plastic. HE selective permeation of gases through plastic films has Tbeen \vel1 documented in the literature and the possibility of using this phenomenon to carry out practical separations More rehas frequently been suggested (2, 4, 77-73). cently Binning et al. ( 7 ) reported that permeating mixtures from the liquid phase on one side of the film to the vapor phase
on the other side has good commercial separation potential. He called this process liquid permeation. However, this process is not yet understood. Binning has suggested a liquid permeation model based on the existence of two zones in the polymer film, where the upstream zone exists as a highly swollen liquid solution and occupies the major portion of the VOL. 4
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