electric charges, dipoles, a n d magnetism constitute different “phases” in addition to the gas phase. literature Cited
(1) Ergun, D., Chem. E q . Progr. 48, 89 (1952). (2) Ferron, J. R., Watson, C. C., Ibid., Symp. Ser. 58, 79 (1962). (3) Fuchs, N. A., “The Mechanics of Aerosols,” pp. 159, 181, Macmillan? New Yorlr, 1964. (4) Goldstein, H., “Claasical Mechanics,” p. 58, Addison-Wesley, Reading, Mass., 1950. (5) Herne: H., Intern. J . Air Pollution 3, 26 (1960). (6) Kennard, E. H., “Kinetic Theory of Gases,” p. 280, McGrawHill, New York, 1938 (7) Kraemer, H. J., Johnstone, H . F., Znd. E n g . Chem. 47, 2426 (1955). (8) Lamb, H., “Hydrodynamics,” Dover Publications, New York, 1932. (9) Langmuir, I., Blodgsett, K. B., General Electric Rept. RL-225 (1945); J . Meteor. 5, 175 (1948). (10) Lin, C. C., Quart. J . A@. M a t h . 1, 43 (1943). (11) Peskin, R . L.. “Diffusivity of Small Suspended Particles in Turbulent Fluids,” Heat Transfer and Fluid Mechanics Institute. 1960. (12) Peskin, R. L., “Some Effects of Particle-Particle and ParticleFluid Interaction in Two-Phase Flow Systems,” Ph. D. thesis, Princeton University, Princeton, N. J., 1959. (13) Ranz. IV., IL’ang, J., Znd. Eng. Chem. 44, 1371 (1952). (14) Richardson, E. G., “‘Ultrasonic Physics,” Elsevier, New York, 1952. (151 Schlichting, H., “Eloundary Layer Theory,” McGraw-Hill, h e w York, 1960. (16) Sherwood, T. K., .Pigford, R. L., “Absorption and Extraction.” p. 237, McGraw-Hill, New York, 1952. (17) Soo, S. L., A.I.Ch.LI. J. 7 , 384 (1961).
(18) Soo, S. L., “Boundary Layer Motion of a Gas-Solid Suspension,” Proc. Symposium on Interaction between Fluids and Particles, p. 50, Inst. of Chem. Engrs., London, 1962. (19) Soo, S. L., “Gas-Solid Flow,” Proc. Symposium on Singleand Multi-Component Flow Processes, Rutgers Engineering Centennial, Rutgers University, New Brunswick, N. J., May 1, 1964 (in press). (20) Soo, S. L., IND.ENG.CHEM.FUNDAMENTALS 1, 33 (1962). (21) Ibtd., 3, 75 (1964). (22) Soo, S. L., Phys. Fluids 6 , 145 (1963). (23) Soo, S. L., Dimick, R. C., “Experimental Study of Thermal Electrification of a Gas-Solid System,” Multi-Phase Flow Symposium, p. 43, ASME, 1963. (24) Soo, S. L., Peskin, R. L., “Statistical Distribution of Solid Phase in Two-Phase Turbulent Motion,” Proiect SQUID Tech. Rept. PR-80-R (ONR) (1958). (25) Soo. S. L.. Trezek. G. J.. Dimick. R. C.. Hohnstreiter, G. F.. ‘ I k . ENG.CHEM.FUNDAMENTALS 3, 98 (1964). (26) Stuetzer, 0.M., Phys. Fluids 5, 534 (1962). (27) Tangren, R. F.. Dodge, C. H., Seifert, H. S., J . Appl. Phys. 20, 637 (1949). (28) Tchen. C. M . , “Mean Value and Correlation Problems ‘ Connected with the Motion of Small Particles in a Turbulent Fluid,” dissertation, Delft, Martinus Nijhoff, The Hague, 1947. (29) Timoshenko, S., “Theory of Elasticity,” p. 339, McGrawHill, New York, 1934. (30) White, J. A., “Industrial Electrostatic Precipitation,” Addison-\Yesky, Reading, Mass., 1963. RECEIVED for review September 16, 1964 ACCEPTED April 19, 1965 Work sponsored by Project SQUID, supported by the Office of Naval Research, Department of the Navy, under contract Nonr 3623 (S-6), NR-098-038. Reproduction in full or in part is permitted for any use of the United States Government.
CON CENT R A T ION POLAR E A T ION EFFECTS
IN A REVERSE OSMOSIS SYSTEM W I L L I A M
N . G I L L
Clarkson College of Technology, Potsdam, N. Y . C H I T l E N AND
DALE W .
ZEH
Department of Chemical Engineering and Metallurgy, Syracuse University, Syracuse, A’.
Y.
An analysis was made to determine the polarization of salt concentration a t the membrane boundary in a continuous reverse osmosis system. The results indicated that earlier studies based on constant transverse water velocity along the membrane surface tend to overestimate this phenomenon and consequently give a conservative evaluation of the economic feasibility of reverse osmosis for desalination purposes. When variable wall velocity is taken into consideration, the boundary condition at the membrane surface for the diffusion equation is nonlinear. A perturbation method is used to solve this nonlinear problem such that results can b e related directly to parameters defined in terms of operating variables.
c
effort has been directed toward the study of desalination of sea water in recent years. Reverse osmosis has been considered as a possible method primarily because of its simplicity. T h e essence of a reverse osmosis ONSIDERABLE
system involves the use of selective membranes which permit the passage of \\ater but not salt through them. In practical design, the process probably will utilize a stack of flat membranes srparated by narrow passages through which brine and water nil1 flow in alternate sections as shown in F i”m r e 1. O n e of the most seriow problems encountered in a continuous reverse osmosis system i:, the build-up of salt concentration a t the \Val1 of the membrane along the axial direction as a result
water
(low pressure)
tttfttttttfttltt brine
-*
(high pressure)
‘ : r?/.!..’/ ?, ,?/ !A : !/+/ /+/M/+/ 4
t + + + + + + & & I
-,,,,.,,,,,,,, )/
t
wsotet
+
+ - f
hdw+p!e/strL)+
, + /
brine
+
,? ! 5 ? /
’
‘
+
(high pressure)
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,~ t I t O t t t t t t t t t + t + k i t t t ( t 4
+ +
woter ~i~~~~ 1.
(low pressure)
Schematic arrangement
of
osmosis
system for desalination VOL. 4
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NOVEMBER 1 9 6 5
433
of the transverse flow of water through the membrane. Since the osmotic pressure is approximately directly proportional to the concentration, this results in a reduction of the effective driving force and hence a reduction in the rate of separation into pure water and salt. Where this phenomenon is very pronounced, the use of reverse osmosis for desalination could become economically impractical. Quantitative studies of the salt build-up phenomenon have been made recently by Sherwood, Brian, and Fisher (3) and Dresner (2). T o carry out their studies, a number of assumptions were made, one being that the transverse velocity across the membrane is constant along the longitudinal direction. One purpose of this note is to demonstrate that this assumption tends to exaggerate the effect of salt build-up and therefore underestimate the productive capacity of the system. I n certain cases this estimation of salt concentration might lead to a n erroneous conclusion regarding the economic feasibility of the system, unless results are interpreted carefully. The primary purpose of the present study is to develop a method for solving the nonlinear problem involved when one studies continuous constant pressure reverse osmosis systems, so that performance can be predicted explicitly in terms of operating variables. I t will be shown that concentration polarization is substantially greater for constant water flux than in constant pressure systems. This is a pleasant result, since in practice continuous reverse osmosis processes operate a t essentially constant pressure so far as diffusional processes are concerned. Analysis
The dimensionless diffusion equation for the case of laminar flow between two parallel membranes can be written as:
where
(xf, 0)
=
0
(4)
bY+
);(
X+
V,C+(x+, 1)
- (x+, 1) =
by+
where the last boundary condition is obtained on the basis that the salt flux a t the wall is zero since the membrane is impermeable to salt. V , is the transverse velocity of water a t the membrane boundary and is given by the following expression
v, = A [ a - Po,]
(6)
where PG,is the effective osmotic pressure corresponding to the wall concentration C(x, a ) . A linear relationship between concentration and osmotic pressure can be assumed and this gives
(7) If Equations 6, 7, and 5a are combined, the result is
and
7r
Equation 1 together with Equations 3, 4, and 5b gives a complete description of the problem considered here and its solution yields expressions for the concentration distribution. However, in a rigorous sense, Equation 1 cannot be solved independently and its solution has to be determined simultaneously with the appropriate equation of motion. This would naturally introduce enormous complexity. O n the other hand, if the flow is assumed to be fully developed and if the transverse velocity is small in comparison with the average velocity in the main flow, the velocity distribution of Berman (7) can be used as an approximation. This is given by u+ = 3 [l
- Sxf][I - y+21
2
x+
x
2
a
y+ =
(3
- Y+*)
(94
(9b)
Berman’s expressions were obtained on the basis of constant transverse velocity a t the wall and in Equations 9a and 9b this is assumed to be V,(O). T h e transverse velocity actually varies along the axial direction and is always equal to or less than Vm(0). Positive radial convection in Equation 1 tends to depress C+(n+, 1). Thus, assuming V, = V,(O) in Equation 9b reduces predicted polarization, but this effect should be small. O n the other hand, the correct choice of V, in the boundary condition (Equation 5a) is important since it markedly affects the results obtained. Assume a series solution of Equation 1 in the following form
r a
a = half distance between membranes
2 m
C+(x+, y + ) =
V,(O), the transverse velocity a t
x =
various parameters given above. This quantity can be expressed unambiguously and explicitly in terms of operating variables as Ar[1 - B2] and therefore is useful for defining system parameters. T h e boundary conditions are given as C+(O,y+) = 1 434
l&EC FUNDAMENTALS
(3)
Ck(X+,
y + ) Bzk
k=O
0, is used to define the
From Equations 10 and 1, one has
a ax + (u+C,)
+ s-8+ ” ( V+C, -
T h e boundary conditions are
CO(O,Y+) = 1
ff
”)
hY+ = 0
(10 )
For the inlet region, the asymptotic solution to the same problem was found by Dresner (2) to be
C,(x, 1) = 1
+ [: B l x + ] +
{
5 1 and
fk ( x + )
- exp
[- (3
B ? x + ) ~ / ~ ] } (21a)
is given as and for small values of SBiZx+a n alternate expression was also given :
k = 1, 2,
.,.
(15)
or k-1
5(.+)=
C,(X’, 1)
C:k-t-l(x+,
1)
+fk-l(X+)
- Ck-l(X+,
1)
j=o
T o obtain a solution for C k , first assumefk = $ = constant, and then it can easily be shown that Ck =
#[l
C,(X+,1) = 1
+
1.536
[SBi2~+]1/3
---r
(21b)
Y3
Equations 21a and 21b agree reasonably well for values of C,+(l, x + ) less than 1.5, and since Equation 21b is considerably simpler than Equation 21a, it will be used to obtain closed form expressions for the Ck’s which are accurate for SBl2x+/3 less than about 0.02. Combining Equations 15, 16, and 21b, after straightforward
- C,(x+,y+)]
However, in the present problem $ is not a constant but depends on the longitudinal distance, x + , Thus, by using Duhamel’s principle, the correct expression for Ck is found to be
Consequently, the concentration distribution is given as
C+(x+,y+) = C,(x+,y+)
-
and the wall concentration becomes C+(x+, 1) =
T h e explicit expression for C + ( x + , 1) or C+(x+, y + ) , therefore, depends on the zeroth-order solution, Co(x+, y+) of Equation 10. Clearly, Co(x+,y + ) is the solution of Equation 1 if the wall velocity, V,,, in Equation 5a is assumed constant. T h e case of constant V, was precisely the problem considered by Dresner (2) and Sherwood, Brian, and Fisher (3). According to Sherwood, Brian, and Fisher, C, is given as
n=l
and
n=l
Numerical values of D,,p,, and Y,u p to n = 8 are given by Sherwood et al. ( 3 ) . ‘Theoretically, this expression can be substituted into Equation 16 for the evaluation of higher order solutions. However, expressions of the type of Equation 20 are generally unsuitable for numerical calculation for small values of Sx+, and to test this the numerical value of C, a t x + = 0 was computed by using the eight available eigenvalues and eigenfunctions and was found to be 1.38084 rather than the correct value of unity. Since the value of Sx+ will be small in practical reverse osmosis systems, it was decided that Equation 20 is not suitable for the evaluation of the Ck’s. VOL. 4
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435
? .4
!.O
1.6
I .2
).8
).4
3
Figure 2.
Values of expansion coefficients,
P3 =
(Kz
Ct, vs. z
+ 2K1 + L P+ 2L1 + Mz + 2M1 f N z f 2N1) B ( 2 ,
Ps =
A3 (2M6 + 2KzL4+ AT7+ L32f
2N6
:)
+ 2K1M, f
(28c)
i2.0
1.8
4
8, = 0.25
-
c
*;- 1.6 3;
1.4 0,=0.50 . L
2
0
0.I
0.2
0.3
Figure 3.
0.4
il
0.5
0.6
I
Surface concentration vs. z
0.8
(
9
(2%)
B ( m , n), the beta function, is defined as
where
r(m)
is the gamma function.
The results given above are useful only for small values of z, say z 5 0.4 or 0.5, but the most interesting and perhaps the most important region is that a t larger distances downstream where polarization effects are more pronounced. I n order to obtain data valid for larger values of z , solutions for the Ck's were continued numerically by using Equations 15, 16, and 21a. Numerical values of C o to Cj for various values of z are presented in Figure 2. For small z the C,, k 2 1, determined here are all negative, but for z > 0.6 the series is alternating. Clearly, the magnitude of the coefficients increases rapidly beyond z equal to about 0 75. For practical purposes. this limits the use of the perturbation solution for C+(x+, 1) to values of z 5 1.0unless very small values of Bz are of interest. Discussion
It became obvious th,at the surface concentration C+(xT, 1) will always be less tha1-i CObecause the algebraic sum of the m
C,B,' is always negative.
series
This shows that the analysis
i=1
carried out previously, which assumed constant V,, overestimated the concentration polarization and, therefore, gives a more pessimistic picture regarding the possible use of reverse osmosis for desalination purposes.
Figure 4. Volumetric production rate as function of 62 and system length, I
VOL.
4 NO. 4 N O V E M B E R 1 9 6 5 437
T o demonstrate this point, some particular examples are considered here. For sea water with a salt concentration of 3.5%, P,,(O) = 380 p s i . , so that the total operating pressures corresponding to BZ = 0.15, 0.25, and 0.5 are approximately 2500, 1500, and 750 p.s.i. For these conditions, the wall concentration was computed from Equation 10 together with the numerical values of the C k ' s and is shown on Figure 3. If the constant wall velocity in Dresner's analysis is assumed to be V,(O), the present results for B Z = 0 correspond identically with his solution and one can compare constant water flux and constant pressure systems by using the data on Figure 3. For example, with B Z = 0.5, a t z = 0.65, the actual increase in wall concentration turns out to be approximately 50% of that for the constant flux case. The slopes of the various curves on Figure 3 indicate that differences between concentration polarization results, based on analyses of constant flux and constant pressure systems, will increase markedly for larger values of z . This trend suggests the need for solutions valid for larger values of z than are the solutions developed here. When viewing the results on Figure 3, it should be remembered that the maximum possible value of C+(x+, 1) is 1/Bz, because then the water flux will be identically zero. When constant flux and pressure results are compared, there is some ambiguity regarding the proper constant to choose for V,. The choice of V,(O) is reasonable because this analysis applies mainly for relatively small z. However, from a practical viewpoint, the constant flux analysis would apply to real systems better if some average value were used to describe the constant V,. Since the water velocity decreases along the wall, the predicted wall concentration, based on the work of Dresner and Sherwood et al., would be somewhat lower if an average value for V, is used. However, since the problem is nonlinear, a proper average involving the constant flux results cannot be chosen a priori. The objective of any reverse osmosis system is to produce fresh water. Therefore, the most important quantity to predict is the rate of water production for a given lengrh of membrane, This is a simple matter, because the volume of
Table I.
SBI'X+ 3
k = O
0.00005 0.0005
0.000062 0.000659 0.00135 0,00347 0.00722 0.0111 0.0151 0.0233 0,0317 0,0494 0.0680 0.0876 0.108 0.129 0.151 0.173 0.196 0,244 0.294 0.346 0.40 0.456 0.602
0,001
0.0025 0.0050 0.0075 0,010 0.015 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.10 0.12 0.14 0.16 0.18
0.20 0.25
438
I&EC FUNDAMENTALS
k = l
-0.000005 -0.000017 - 0.000081 - 0.000268 - 0.000540 -0.000889 -0.001 80 -0.00297 -0.0061 -0.0105 -0.0164 -0.0237 -0.0326 -0.0431 -0,0553 -0.0693 -0.103 -0.144 -0.194 -0.252 -0.319 -0.528
water produced per unit time per unit width of a single membrane surface is given as
Q
=
sx
V, dx
0
Combining Equation 30 with Equations 6 and 7 the resulting expression in dimensionless form is given as
where
4
SBl2
= 1.536 L
J
J
Numerical values of Q + for several values of B Z are shown on Figure 4. If Equation 10 is substituted in Equation 31 the result is
J
k=O
J O
Since the Ck's, based on either Equation 21a or 21 b, depend on a only, the integrals in Equation 32 can be calculated once and for all. The first six of these integrals are tabulated in Table I, and these results, together with Equation 32, enable one to calculate Q+ for any desired operating conditions. Conclusions
This work shows that the performance of a constant pressure reverse osmosis system can be predicted explicitly in terms of operating variables and concentration polarization is somewhat less pronounced than if a constant water flux boundary condition is assumed. Clearly, further analyses of more general geometries and flow conditions are needed. Additional solutions valid for larger values of salt concentration a t the wall should be very useful for future design purposes when higher capacity membranes are available. Acknowledgment
The authors have profited from the useful suggestions of L. Dresner of Oak Ridge National Laboratory, who read the manuscript carefully.
Values of Integrals
k = 2
-0.000005
-0.000015 -0.000066 - 0.000201 - 0.000378 -0.000587 -0.00107 -0.00159 -0.00260 -0.00352 -0.00405 -0.00383 -0.00252 0.000286 0.005 0.0120 0.035 0.0729 0.130 0.210 0.318 0.743
k = 3
- 0.000004 -0.00001 3 - 0.000053 -0.000152 - 0.000271
-0.00040 -0.000671
- 0.00094 - 0.00148 - 0.0020
-0.00256 -0.00336 -0.00476 - 0.00728 -0.0116 -0,0186 - 0.0452 -0.098 -0.191 -0.344 -0.576 -1.69
k = 5
k = 4
- 0.000004 -0.000010 - 0.000044 - 0.000117 - 0,000197 - 0.000281 - 0.000445 - 0.000601 - 0.00088 -0.00113 -0.00133 - 0.00138 -0.00106 0,000125 0.00296 0.00869 0.0365 0.106 0.255 0.537 1.03 3.96
-
-
0.000003 0.000010 0.000036 0.000090 0.000146 0.000203 0.000309 0.00041 0.000595 0.000759 0.00091 0.00109 0.00148 0.0025 0.00509 0.011 0,0455 0.152 0.42 1.01 2.18 10.8
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
U
c
= functions defined by Equation 15
fh
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
Li,L z , La, Ld
26
ATl,-Y*, Sa,
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 A\74,
Sj,
Yn01) Y
Y+ 2 CY
= = = =
direction eigenfunction in Equation 19 transverse distance measured from center of channel y / u , dimensionless transverse distance defined by Equation 23
D
= defined as 2 a V w( 0 ) = eigenvalue in Equation = gamma function
19
= 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, M a s s .
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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