Ind. Eng. Chem. Process Des. Dev. 1982, 2 1 , 517-527
517
Approximate Design Equations for Reverse Osmosis Desalination by Spiral-Wound Modules Kamalerh K. Slrkar’ and Phuong T. Dang Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
Goruganthu H. kao Indrkm Organic Chemicals limited, Khopoli, Maharastra 4 10202, Indk
For a given fractional water recovery in RO desallnation, simple analytical design equatlons have been developed to predict the channel length of a spirabwound membrane modub for turbulent as well as laminar flow in the brine channels with spacers. A simple arralytlcal expression to estknate the averaged permeate solute concentration has also been developed. The predicted dimensionless channel length and the averaged permeate solute concentration are qbiie close to the numerical results available in the tables of Ohya and Sourirajan except when membrane solute rejection Is b w and tl?e fractional water recovery is high. These debign equations predict quite well the fractlond water recovery and permeate solute cobcentratlons measured by Ohya and Taniguchi with a ROGA-4006 spiral-wound module. ThEi design equations for turbulent mixing conditians are valid for (NirV , / k , ) I0.2, while those for chopped laminar flow concentration polaflzatiqn conditions are valid for ( N , , V , I k , ) 5 1.0.
Introduction The rapid growth of reverse osmosis (RO) as a commercially attractive microsolute separation process suggests renewed investigations of the current design methods for developing simpler d e s i b procedures. Among the three membrane configurations used commercially in RO processes, spiral-wound, hollow fiber, and tubular, simple analytical desigh equations ahd simplified numerical design procedures b v e already beeh developed by S i r h and b o (1981) for straight-through tubular modules. The reverse osmosis process industry is, however, increasingly relying on spiral-wohd and hollow fiber modules if a clean nonfouling feed ia available. We have, therefore, directed our attention in this paper to developing approximate analytical design equations for reverse osmosis separation by spiral-wound modules. The primary difficulty in simplifying existing design procedures for RO processes is caused by the presence of concentration polarization. Under such conditions, the solvent flux expression is a fuhction of the wall solute concentration which, in turn, is a function of the solvent flux throdgh a complicated transcendental relation. Consequently, considerable iteration is needed to obtain the solvent flux from a given membrane tube in tubular RO processes (McCutchan and Goel, 1974; Harris et al., 1976). Rao and Sirkar (1978) have, however, shown that an explicit solvent flux expression can be developed for a membrane tube in RO since turbulent flow conditions usually exist and the brine side solute mass transfer coefficient is large. In fact, such an explicit flux expression and other suitable assumptions enabled Sirkar and Rao (1981) to develop approximate analytical design equations and sirppler numerical schemes for designing straightthrough tubular modules for RO processes with high rejection membranes. Development of analytical design equations is also directly affected by the nature of the mass transfer coefficient for solute transport in brine channels. The solute transport coefficient is controlled by the flow conditions that exist in the brine side channels of a spiral-wound module. The flow conditions in turn are significantly influenced by the presence of Vkxar screens or similar 0196-4305/a2/1 i21-0517$01.25/0
“tubulence-promoting” (or “mixing-promoting”) devices. Depending on the feed velocity, one can have turbulent or “chopped” laminar (Spiegler, 1981) flow in such an arrangement with corresponding mass transfer coefficients. The “chopped” laminar flow concept is exemplified by the studies of Solan et al. (1971) of mass transfer in a spacer net in an electrodialysis cell. They proposed that the concentration boundary layer starts developing at the beginning of each mesh step of length equal to the screen mesh size. After traversihg the length equal to the screen mesh size, the concentration profile undergoes a sudden partial mixing at the location of the next screen wire beyond which the concentration profile starts developing again to repeat the procese. Solan et al. (1971) have further pointed out that one could also represent this saw-tooth development of laminar concentration boundary layer by a constant boundary layer thickness model simply by intergrating it over the screen mesh size. This is tantamount to assuming a constant nondeveloping laminar mass transfer coefficient along the flow channel. Particularly relevant in this connection is the investigation by Ohya and Taniguchi (1975) of desalination by a ROGA-4000 spiral-wound module. They have indicated that the performance of such a module containing spacer screens could be adequately modeled by a nondeveloping constant laminar flow mass transfer coefficient. Such an assumption is singularly useful in developing analytical design equations. If, due to a d u c h higher feed velocity and Vexar screens, turbulent flow mass transfer conditions exist in the brine feed channels, one can similarly assume a constant hass transfer coefficient with a significantly reduced concentration polarization (Kremen, 1977). The level of concentration polarization in reverse osmosis desalination depends primarily on the solvent flux level vis-a-vis the brine side salt mass transfer coefficient. It will be briefly shown here that one can develop explicit flux expressions even €or certain ranges of laminar flow in feed brine channels with mixing elements. Under such conditions the ratio of the solvent flux level to the salt mass transfer coefficient is considerably higher than that under turbulent flow conditions. With the explicit flux expressions of Rao and Sirkar (1978) being valid for turbulent 0 1982 American Chemical Society
518
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
flow conditions, we have now a range of flow conditions for which explicit flux expressions are available. In this paper, we have utilized these flux expressions for developing simple analytical design equations for reverse osmosis desalination in a spiral-wound module of the ROGA-4000 type for turbulent as well as certain range of laminar flow conditions. The predictive accuracy of these design equations has been checked with the results obtained by Ohya and Sourirajan (1971) from the simultaneous numerical solution of a coupled set of one algebraic and two ordinary differential equations. The predictive capacity of the proposed design equations of this work has also been checked with the data available in Ohya and Taniguchi (1915) for laminar flow conditions in a ROGA4000 spiral-wound module. The simple analytical design equations of this paper are to be contrasted with the existing procedures. If the values for the parameters of a given problem were fortuitously exactly equal to those used for the development of a given table in Ohya and Sourirajan (1971), then no particular advantage is available with the present set of design equatipns. However, more commonly, the parameters will have values different from those given in the tables of Ohya and Sourirajan (1971). Considerable interpolation would then be necessary; witness the involved calculations in Ohya and Taniguchi (1975) to get predictions for their expimental system from Ohya and Sourirajan (1971). No such complications are present when the proposed analytical design equations are used in the range of parameter values where it is accurate. A number of plant design or plant optimization studies for desalination plants have been conducted (Singh and Cabibo, 1980, Cabibo et al., 1979). These plants use a large number of processing w i t s beside spiral-wound modules. Computer studies of such plants for many preliminary process design and optimization purposes would be considerably simplified if analytical design equations with reasonable accuracy are available. The utility of the present work is to be viewed from such a context. The proposed relations are useful in addition for developing equations for an ideal reverse osmosis stage needed for efficiency-based design procedures (Sirkar and Rao, 1979). Simplified Flux Expressions and Polarization Relations for High Rejection RO Membranes. An exact irreversible thermodynamics based analysis of solvent and solute flux expressions in RO desalination requires three membrane parameters, L,, 0, and o,for the given solvent-solute-membrane combination (Spiegler and Kedem, 1966). On the other hand, a large amount of practical information on membrane performances is available currently in terms of two-parameter solution-diffusion models or in terms of two-parameter models treatable as solution-diffusion models. It has been shown (Podall, 1975) that for high rejection reverse osmosis membranes, twoparameter solution-diffusion models adequately represent the membrane behavior. In fact, Sirkar and Rao (1981) have also used a two-parameter solution-diffusion model for solvent and solute flux expressions in a tubular membrane. This is especially true for cellulose acetate membranes. We adopt the two-parameter Kinlura-Sourirajan (1967) formalism for the solvent and solute flux expressions. Although it is based on the preferential-sorption-capillary flow mechanism of reverse osmosis, mathematically its flux expressions are essentially identical with those of a twoparameter solution-diffusion model (see Soltanieh (1980) for a comparative analysis)
(2) = (&M/K6) [CXZ - CXz3,l We assume that the total molar density of the solution, C, is constant throughout the module. Further, the osmotic pressures are expressed by the linear relations (3) ~ ( C 2 2 )= bC,, = bCX,,; ~(C23,)= bC23, = bCX23, These are considered quite reasonable for dilute feed solutions encountered in reverse osmosis desalination etc. For the concentration polarization relationship, we adopt the film model proposed in Kimura-Sourirajan (1967) (quite similar to an earlier one proposed by Brian (1966)) Nlrv1 = ki(1 - X23r) In [(X22 - xz3J/(Xzi - XzdI (4) N2r
In order to simplify the above relations, we first assume that the membranes under consideration have high salt rejection so that X,, 0.2. If ( N l r ~ l / k=J 1, then the following truncated form of exp(N1,Vl/kJ in eq 8
will merely cause an error of less than 2% in X Z 2for a-given
Xzl. A still further truncated expansion of exp(N1,V1/kl) in eq 8 leading to
x 2 2
=
x21[1 + NlrV1 +
(?)’$I
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 519
(11)
will introduce about 9% error in X22 if (Nl,Vl/-kJ = 1. Thus for reverse osmosis conditions where (NlrVl/kJ I 1, one could use either eq 10 or eq 11for the concentration polarization relation. The use of eq 11 will apparently entail somewhat larger error than the use of eq 10 for (Nl,Vl/kl) I 1; however subsequent expressions using eq 11 will be considerably simpler than those using eq 10. More details will be presented in Sirkar et al. (1982) to evaluate the accuracy of the quadratic and cubic approximations against the exact results. We now substitute these simplified concentration polarization relations in the solvent flux eq 5. First eq 9 is substituted and on rearrangement, we get A(AP - bCX21) (turbulent flow) (12) N1r = (1 bAX2,/kl)
+
where we have assumed CVl r 1. This expression for solvent flux will be used here for spiral-wound systems under turbulent flow conditions in the feed channel. Similarly, substitution of eq 12 and eq 9 in solute flux eq 6 leads to L
which is an explicit flux expression for solute to be used under turbulent flow conditions in spiral-wound membrane channels. We next consider the explicit flux expressions for solvent and solute under laminar flow conditions. It will not be possible to consider the whole range of laminar mass transfer conditions as pointed out before. We consider here conditions under which (NlrVl/kl) I 1 for which we substitute the quadratic approximation eq 11 of eq 8 in eq 5 and obtain the following quadratic in (NlrVl)
10 in eq 5 and get the following cubic equation in N1,Q1 for (NlrVl/kl)I 1
’
(1 - &)6kf’
= 0 (17)
An expression for N2, could also be developed using the solution for Nlr from eq 17. Details will be presented in Sirkar et al. (1982). Simple Design Equations for the Length of Spiral-Wound Membrane Channel. A spiral-wound membrane module of the ROGA-4000 type may be simply conceived of as two parallel thin channels of feed brine with feed brine spacers in between two active membrane surfaces in a given channel for a total of four membranes (Ohya and Taniguchi, 1975). Figure 1indicates very briefly the relevant details of such a module. Define (l/h) as the total active membrane surface area divided by the total volume of the feed channel space facing active membrane. The quantity h in this case is actually half of any brine channel height. Following Ohya and Taniguchi (1975), we may write the following tws mass balances for the solvent and the solute, respectively, for a feed channel control volume of differential length dx in the direction of mean flow d(o,Cli)/dx = -Nl,/h (18)
In Ohya and Taniguchi (1975), Cll is assumed equal to C, which is quite realistic for dilute solutions dealt with in RO desalination. Therefore with C being assumed constant along the module (see eq 3), eq 18 is simplified to
Consistent with our assumption (a) based on X23, + = -
&) (. t) [ +
+
(1 - 7)
1
X+ZIL(1 - YX+21L) (34) For definition 31, although we use 77 in this paper, the symbol A0 is also mentioned here due to its extensive use in the tables of Ohya and Sourirajan (1971). It is important to note here that relation (22) suggests (1-
ln
We should note in passing that eq 34 is restricted to the condition that yX+2u < 1; otherwise the osmotic pressure will equal the driving pressure at a location of the module where the concentration process has created the condition of yx+21L = 1. We must recognize here that turbulent flow or laminar-turbulent transition is likely to be achieved in a spiral-wound module with, say Vexar screens at an empty channel Reynolds number lower than usual. For example, Bray (1966) reports that brine side pressure drop starts deviating from laminar type of dependence at apparent Reynolds numbers less than 100. However, the pressure drop along the flow path is not likely to be large except for very low pressure operations. To illustrate, Ohya and Taniguchi (1975) report a maximum pressure drop of 30 psi for feed pressures around 500 psig in a ROGA-4000 spiral-wound module with water recoveries between 0.05 to 0.44. Thus the assumption of AP A€', used for integrating the integral in eq 27 is not likely to introduce large errors. The numerical solution of Ohya and Taniguchi (1975) also used this assumption. It follows therefore that the variation of A from Af along the module will be very small. Thus both A and AP may be assumed constant along the module at the values Af and APp Due to substantial permeation along the module, bulk velocity decreases with x and so will kl,the salt mass transfer coefficient. Although Ohya and Taniguchi (1975) have inferred the existence of a certain type of correlation between kl and ud etc. in the laminar region, none is known for the turbulent as well as the laminar region in the spiral-wound module with, say, Vexar screens. We therefore assume a constant value of k1 for the whole module-a procedure also followed by Ohya and Taniguchi (1975). Next we substitute expression 15 for N1, into eq 26 for obtaining estimates of L+ under laminar flow conditions with (Nlrvl/kl)5 1. The resulting relation is
=
AbX21 Further the fractional water removal from the module, A, is defined by
I
+
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 521
To integrate, we m u m e as before that the values of A , AP, and kl remain constant along the module length. Specifically A = Af, AP = AP, and the nondimensional quantities 77 and defined respectively by eq 31 and 32 become constant parameters for the problem. These assumptions me identical with those of Ohya and Taniguchi (1975). We therefore nondimensionalize eq 35 and obtain for nondimensional membrane channel length
MEMBRANE
FEED B R I N E CHRHNEL GAP
, BACKING
MATERIAL
M E S H SPACER
\--. GLUE
YX+21) +
[
( 1 + i ) + 2 ( L - 1 YX+21 ) ] 1 / 2 / YX+21
PRODUCT WATER*,
LINE
@ ,
(36) The analytical expression for the integral on the right-hand side of eq 36 is as shown in eq 37, where Al = (7 + 1)'
A2 =
(7
+ Bql + q12
+ 1)' + Bqz + 42'
(374 (37b)
B = 2(1 + 17 + (1/77))
(37c)
B1 = 1 + (77/Y)
(374
B2 = 1 + (O/Y)/X+ZlL
(374
B1 - (1 + d = Bz - (1 + 77)
-
MEMBRANES; iI T
PRODUCT WATER S I D BACKING MATERIAL
E
7
Figure 1. Spiral-wound module (ROGA-4000 type).
quite general. If the. total volumetric permeate flow rate from the module is Vp and the averaged permeate solute concentration is (C23r)av,then
q1 =
(37f)
JLN2r4W dx = Vp(C23r)av
q2
(37g)
Now substitute for Czz in eq 6, the expression obtainable from eq 5 and obtain
+ 1)2+ Bql + q12 A2 = (77 + 1)' + Bqz + qZ2
Al = (17
(37b) (37d)
=1+
(374
(77/Y)/X+2IL
B1 - (1 + 77) q 2 = B2 - (1 + 77) q1 =
(38)
(37d
+ t + (1/77)) B1= 1 + h / Y )
B=20
B2
4
(37c)
Use this expression for N2,and an expression for dx from eq 23 in the integral of eq 38 to get
(37f)
(37d Note that eq 34a and 34b are to be used along with these equations if a relation between X+zlLor L+ and A is needed. We are not using here the solution of the cubic eq 17 for Nlrto obtain another estimate of L+ for NlrVl/kl I 1. Neither for the sake of brevity do we propose here to extend the validity of L+ expression to conditions where (NlrV1/kl) I 1.5. A Simple Design Equation for Estimating Average Solute Concentration in the Permeate. For a given membrane in a particular spiral-wound module subject to a set of operating conditions, the solute concentration in the permeate will be estimated now. To this end, we consider, as before, a spiral-wound module of the ROGA4OOO type with two parallel thin feed brine channels having a total of four membranes each of width W. The nondimensional result to be derived ultimately is independent of the number of parallel channels in the module and is
Assume, as before, AP,A , and klto be constants. Integrate and obtain the following expression for (C23,),
In terms of nondimensional quantities, A and L+ defined earlier, eq 41 may be rearranged if we recognize that A = Vp/(Uxf)2(W2h) (42) for the given spiral-wound module. Thus (43)
522
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
Table I. Comparison of Predictions from Design Equationsa,b with Those of Ohya and Sourirajan (1971) for Turbulent Conditions in a SDird-Wound Module and A = 0.2, 0.3 % error % error in A Y e L+a L+C in L' (C23r)*avb (C23r)+avc (C23r)'av 0.2 q=5
0.1
0.0 0.006 0.01 0.02 0.03
0.2302
0.2306 0.2303 0.2301 0.2296 0.2291
-0.19 -0.06 + 0.03 + 0.24 + 0.46
0 0.009 0.0151 0.0301 0.045
0 0.0091 0.01508 0.0296 0.0437
0 -1.1 +0.01 + 1.68 -t 3.33
0.2
0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03
0.2690
0.2699 0.2690 0.2684 0.2670 0.2656
-0.32 -0.00 + 0.24 +0.76 + 1.28
0 0.01036 0.01726 0.0345 0.05178
0 -0.32 +0.82 +2.90 + 4.93
0.3211
0.3224 0.3202 0.3188 0.3155 0.3124
-0.40
0 0.01039 0.01712 0.0335 0.0492 0 0.0120 0.0198 0.0385 0.0562
0.0 0.006 0.01 0.02 0.03
0.5065
0.3
0.5
0.3 q=5
0.5095 0.4961 0.4887 0.4723 0.4584
+ 2.05 + 3.51 + 6.75 + 9.50
0.3492 0.3489 0.3485 0.3476 0.3468
-0.16 -0.07 + 0.04 + 0.30 + 0.53
0 0.00973 0.01622 0.03244 0.04866
0 0.0098 0.0162 0.0318 0.0468
+0.18
+ 0.07 + 0.32 + 0.94
-0.22
0 0.01128 0.0188 0.0376 0.0564
0 0.0113 0.0186 0.0363 0.0532
0 -0.09 + 1.06 + 3.40 + 5.60
0 0.01343 0.02238 0.04471 0.06715
0 0.01332 0.02188 0.04234 0.0615 5
0 0.02199 0.03665 0.0733 0.11
0 0.02096 0.03373 0.06257 0.0880
0.0 0.006 0.01 0.02 0.03
0.3487
0.2
0.0 0.006 0.01 0.02 0.03
0.4128
0.4137 0.4125 0.4115 0.4089 0.4064
0.3
0.0 0.006 0.01 0.02 0.03
0.5015
0.5021 0.4993 0.4966 0.4904 0.4846
0.0 0.006 0.01 0.02 0.03
0.8498
0.1
0.5
0.8422 0.8225 0.8052 0.7689 0.7397
-t
1.75
+ 2.72 -0.40
+ 1.55 -0.13
+ 0.43 + 0.98 + 2.21 + 3.36 + 0.89 + 3.21
+ 5.25
+9.52 +12.95
a L+ is obtained from eq 34 with X + 2 1 L obtained from A by eq 34b. From Ohya and Sourirajan (1971). obtained from eq 34.
where L+ is defined by eq 26 and is independent of the regime under consideration. Define 6 (D~M/K~)/U,* (44) and obtain the following nondimensional form of eq 43 (45)
where (C23r)+ev
0
0 0.0121 0.02019 0.04038 0.06057 0 0.0184 0.0365 0.0613 0.0919
+ 0.29 + 0.73
(C23r)av/cZlf E ( ( x 2 3 r ) a ~ / ~ 2 l f )= (X23r)+ev
(46)
Since the derivation of eq 45 is based on eq 5 , 6 , 2 3 , and 38, it is valid for values of NlrVlobtained from either eq 12 or eq 15 or eq 17. Consequently, L+ in eq 45 is not tied to a specific approximation in the estimate of NlrVl.For turbulent flow conditions with (NlrVl/kJ5 0.2, eq 34 for L+ should be used for given X+zlLor A. For other flow conditions such that (NlrVl/kl) I1.0, eq 36 or 37 for L+ should be used for given X+zuor A. The calculation method for (C23r)av adopted here is independent of the flow direction of the permeate vis-a-vis the feed since N,, depends only on Cm and not on C23r (by
(C23r)+av is
0 0.0178 0.0289 0.0545 0.0775
+ 0.80 + 1.69 t4.52 t7.17 0
+ 3.1 +5.64
+ 11.06
+ 15.64 0 -0.60
+ 2.00
+3.75
0
+ 0.82 + 2.25 + 5.43
+8.34 0 +4.69 + 7.97 +14.64 +19.98
obtained from eq 45 with L'
eq 6). Similarly, N1,depends only on CZzand not on C23r (by eq 5). In reality, the solute concentration of the permeate at any given location in the spiral will be determined by the integrated permeate quality from the outermost spiral to the location under consideration. However, for high solute rejection membranes, consequent errors due to our adopted calculation procedure are expected to be small. An exact analysis is currently under our consideration. Predictive Accuracy of the Proposed Design Equations. The usefulness of the proposed design equations can be determined in two ways. Ohya and Sourirajan (1971) have tabulated the results of their extensive computer calculations for a series of values for the parameters y, 6, A6 (=q) at various levels of fractional water recovery (i.e., A). One can calculate by using either eq 34 and 45 or eq 37 and 45 the predicted values of L+ and (C23r)+avand compare these with the computed values available in the tables of Ohya and Sourirajan (1971). One could, in addition, compare the data in Ohya and Taniguchi (1975) with the predictions from eq 37 and 45 for laminar flow systems. Since L+ is known for their experimental arrangement, eq 37 may be used to determine X+zlLand the corresponding A and compared with the
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 523 Table II. Comparison of Predictions from Design Equationsapb with Those of Ohya and Sourirajan (1971)for Turbulent Conditions in a Spiral-Wound Module and A = 0.4, 0.5 % error % error in A Y e L+a L' in L ' (C23r)+avb (C23r)'avc (c23r)'av 0 0.4713 -0.16 0.0 0.0 0.0 0.4705 0.4 0.1 0.01058 0.01063 -0.47 0.4707 -0.04 0.006 q=5 + 0.42 0.01763 0.01756 0.4702 +0.07 0.01 + 2.28 0.0352 0.03443 0.4688 + 0.37 0.02 + 4.30 0.0529 0.05066 0.4675 0.03 + 0.64 0.0 0 0.5674 -0.18 0.0 0.2 0.0 0.5664 +0.21 0.01248 + 0.13 0.01245 0.5656 0.006 +1.46 0.5638 0.02079 + 0.45 0.02049 0.01 0.04159 +4.08 0.5595 + 1.21 0.03989 0.02 +6.51 0.5555 0.06238 0.05832 + 1.92 0.03 0 0.7053 -0.04 0.0 0.0 0.3 0.0 0.7050 + 1.32 0.7003 + 0.67 0.01525 0.01505 0.006 + 3.03 0.6955 0.0254 0.02465 0.01 + 1.35 + 6.72 0.6843 + 2.94 0.05084 0.04742 0.02 + 10.05 0.6743 0.07626 0.06859 0.03 +4.36 1.3350 0 0.0 1.3657 + 2.25 0.0 0.0 0.5 + 6.49 0.02897 + 8.94 1.277 0.006 0.02638 0.04166 0.01 + 9.79 + 13.72 0.04828 1.232 0.07484 0.09657 1.148 + 22.50 0.02 + 15.94 + 28.82 0.1031 0.03 0.1448 1.087 + 20.40 -0.17 0.0 -0.01 0.01168 0.01946 + 0.12 0.03892 + 0.47 0.05839 + 0.81 -0.16 0.0 0.2 0.7354 +0.26 0.01412 0.02354 + 0.67 0.04708 + 1.62 0.07062 + 2.52 0.9510 t0.13 0.3 0.0 + 1.16 0.01804 0.0300 + 2.12 0.0601 + 4.28 0.09020 + 6.15 a L+ is obtained from eq 34 with Xf21L obtained from A by eq 34b. (C23r)+av is obtained obtained from eq 34. From Ohya and Sourirajan (1971). 0.5 n = 5
0.1
0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03
0.5973
0.5983 0.5974 0.5966 0.5945 0.5925 0.7366 0.7335 0.7305 0.7235 0.7169 0.9498 0.9400 0.9308 0.9103 0.8925
measured A in the experiments of Ohya and Taniguchi (1975). Equation 45 is to be used next to predict (C23r)av using known values of L+, 8 (estimated by Ohya and Taniguchi, 1975), and predicted value of A (from eq 37). The prediction may be compared with the values measured by Ohya and Taniguchi. Consider Tables I and I1 containing results calculated from eq 34 and 45. Since eq 34 is valid for (NlrVl/kl) 5 0.2 and the only values in Ohya and Sourirajan's (1971) extensive tables relevant for this condition are q = AB = 5 and m, calculations in Tables I and I1 are based on q = A8 = 5. No calculations are being presented for q = A0 = corresponding to no concentration polarization since eq 34 of this work reduces to eq 15 of Ohya and Taniguchi (1975) valid for Ohya and Sourirajan's (1971) set of equations for q = a. The results have been presented for values of A = 0.2, 0.3 in Table I and for A = 0.4 and 0.5 in Table 11. For each A, five values of y = 0.10,0.20,0.30, 0.40, and 0.5 were used to calculate L+. For each y value, calculations have been carried out to determine (C23r)+av for values of 8 = 0.00,0.006,0.01,0.02, and 0.03. The value of 8 = 0.03 corresponds to a membrane which is not that tight and has a high value of (D2,/K6) and therefore low salt rejection. Against the calculated value of L+for a given A, y, and q = 5, the values of L+ available in the tables of Ohya and Sourirajan (1971) are given. Note that different values of 8 were used in Ohya and Sourirajan (1971) for L+ calculation so that for one L+ value calculated by eq 34, there are five L+ values from these authors' tables
0.0 0.01170 0.01933 0.03782 0.05554 0.0 0.01403 0.02306 0.0447 0.06511 0.0 0.01763 0.02875 0.0547 0.07852
0 -0.19 + 0.68 + 2.84 +4.88 0
+ 0.67 + 2.04
+ 5.03 +7.80 0
+ 2.27 +4.38 + 8.97 + 12.95
from eq 45 with L'
corresponding.to particular values of 8. Further, no calculations are being presented for the conditions A = 0.5, y = 0.5 when osmotic pressure of feed becomes equal to the driving pressure. It is clear from the two columns in Tables I and I1 on the percent error that the predictions of the simplified design eq 34 and 45 are quite accurate over a wide range of values y, A, and 8. It is also obvious that these errors increase in general as the values of A, y, and 8 increase. For L+,the errors are less than 10% unless we are dealing with a value of y = 0.5, 8 I 0.02, and A I 0.4. For low water recoveries, regardless of the level of y and 8,the L+ errors are very low. The magnitudes of errors are somewhat larger for (C23r)+av.The errors q e less than 10% except when y = 0.5,O I 0.02 for any of the A used in the calculation. However, the design predictions are safe since a higher permeate concentration is obtained from eq 45. The same safety feature exists in L+prediction when the errors are large. For concentration polarization more severe than (Nl,Pl/kl) I0.2, calculations of L+ and (C23r)+av (E (Xar)+av) have been carried out with eq 37 and 45. These are valid for (NlrVl/kJ I1.0. Specifically, in Tables I11 and IV, we report the results of calculations based on 9 = A8 = 1 since these can be compared with the corresponding results available in Ohya and Sourirajan (1971). As before, the results have been presented for values of y = 0.1, 0.2, 0.3, 0.4,and 0.5 and for A values of 0.2, 0.3, 0.4, and 0.5. For each combination of y, A, and q, five
524
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
Table 111. Comparison of Predictions from Design Equationsa,bwith Those of Ohya and Sourirajan (1971) for Laminar Flow Conditions in a Spiral-Wound Module and A = 0.2, 0.3 % error % error in is Y 0 L' a L' in L' (C23r ) + av ( C23r) + av (C23r I + av I_______
0.2
0.1
0.1 0.006 0.01 0.02 0.03
0.2584
0.2628 0.2616 0.2608 0.2598 0.2571
-1.68 -1.23 -0.92 -0.54 0.50
0 0.01752 0.0292 0.0584 0.0876
0 0.0185 0.0304 0.0589 0.0857
0 -5.36 -4.02 -0.90 2.18
0.2
0.0 0.006 0.01 0.02 0.03
0.3270
0.3361 0.3329 0.3308 0.3260 0.3216
-2.70 -1.77 -1.14 0.31 1.69
0 0.01905 0.0317 0.0635 0.0952
0 0.0199 0.0327 0.0630 0.0912
0.00 -4.59 -3.04 0.71 4.41
0.3
0.0 0.006 0.01 0.02 0.03
0.4147
0.4294 0.4224 0.4179 0.4078 0.3989
-3.43 -1.83 -0.77 1.69 3.96
0 0.0214 0.0357 0.0715 0.1073
0 0.0222 0.0363 0.0693 0.0994
0 -3.64 -1.65 3.22 7.90
0.5
0.0 0.006 0.01 0.02 0.03
0.7185
0.7481 0.7123 0.6922 0.6508 0.6182
-3.95 0.88 3.81 10.41 16.23
0 0.0311 0.0518 0.1037 0.1555
0 0.0308 0.0493 0.0902 0.1255
0.0 0.92 5.21 14.93 23.96
0.1
0.0 0.006 0.01 0.02 0.03
0.3938
0.4008 0.3987 0.3974 0.3941 0.3911
-1.74 -1.23 -0.90 -0.07 0.69
0 0.01876 0.03127 0.06254 0.0938
0 0.01978 0.0324 0.0627 0.09117
0 -5.15 -3.73 -0.40 2.89
0.2
0.0 0.006 0.01 0.02 0.03
0.5062
0.5210 0.5152 0.5116 0.5031 0.4953
-2.83 -1.74 -1.04 0.63 2.21
0 0.02063 0.03438 0.06875 0.10313
0 0.02155 0.0353 0.06773 0.09771
0 -4.29 -2.62 1.51 5.54
0.3
0.0 0.006 0.01 0.02 0.03
0.6546
0.6787 0.6654 0.6572 0.6386 0.6224
-3.54 -1.61 --0.38 2.52 5.19
0 0.0236 0.0394 0.0788 0.1182
0 0.0244 0.0397 0.0753 0.1075
0 -3.09 -0.81 4.68 9.98
0.5
0.0 0.006 0.01 0.02 0.03
1.2252
1.2740 1.189 1.145 1.059 0.9947
-3.83 3.05 7.01 15.7 23.18
0 0.0371 0.0617 0.1234 0.1850
0 0.0356 0.0564 0.1012 0.139
0 3.81 9.37 21.90 33.13
q = l
0.3 r,=l
a L' is obtained from eq 37 with X + 2 1 L obtained from 4 by eq 34b. obtained from eq 37. From Ohya and Sourirajan (1971).
values of 6, 0.0, 0.006, 0.01, 0.02, and 0.03 were used. The columns for percent error in L+ as well as (C23r)+av indicate a trend observed earlier with Tables I and 11. The predictions of the design eq 37 and 45 are quite accurate over a large range of values of y, A, and 8. The error in L+ prediction is in general less than that for (C23r)+av. Further for a given A, the errors increase to significant levels when y increases to 0.5 and 6 is in the range of 0.02 to 0.03 corresponding to low rejection membranes. Obviously for a high A, high 6, and high y reverse osmosis, deviation from predictions will be large since the assumptions made in the analysis will introduce significant errors. However, as long as 6 is low corresponding to high rejection membranes, the errors are quite low. These levels of error are certainly adequate for preliminary design purposes. This is especially true when it is realized that A values in practice for a given spiral-wound module rarely exceed 0.20, and a number of modules are used in series to achieve high water recoveries. Recognizing further that the proposed design equations for L+ and (C23r)+avare simple andytical expressions, advantages are obvious both in terms of lack of complexity as well as enormous savings of computer time when compared with the numerical scheme proposed in Ohya and Sourirajan (1971) for solving
(C23r)+av is
obtained from eq 45 with L'
the governing differential and algebraic equations. Calculations carried out with 7 = A6 = 2 (but not presented here) indicate similar behavior between the predictions and those available in Ohya and Sourirajan (1971). In fact, the errors are smaller since the level of concentration polarization is lower than q = A6 = 1. The previous comparisons were based on the design mode; given the feed and operating conditions how do the predicted values of L+ and (C23r)+av from the proposed design equations compare with those predicted by the more exact numerical results of Ohya and Sourirajan (1971). We adopt a different basis of comparison for laminar flow condition data of Ohya and Taniguchi (1975). For a given feed and operating conditions and given L+ for the given spiral-wound module, what values of A and (C23r)av are predicted by eq 37 and 45, respectively, and how they compare with the experimental data assuming values of uw*, (D2M/KS),and kl are known from Ohya and Taniguchi's analysis. The experimental conditions and the values of various parameters for the ten runs of Ohya and Taniguchi (1975) are summarized in Table V. We use the analytical solution eq 37 and a NewtonRaphson technique to determine the value of X+211,which makes the value of the integrated expression on the
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
525
Table IV. Comparison of Predictions from Design Equationsa*bwith Those of Ohya and Sourirajan (1971) for Laminar Flow Conditions in a Spiral-WoundModule and A = 0.4, 0.5 % error in
% error A
7
0
0.4 q = 1
0.1
0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03
0.2
0.3
0.5
0.5
0.1
q = l
0.2
0.3
0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03 0.0 0.006 0.01 0.02 0.03
L' 0.5352
0.7019
0.9328
2.0162
0.6854
0.9233
1.280
in L' -1.84 -1.24 -0.86 0.05 0.94
L I C
0.5453 0.5420 0.5399 0.5350 0.5303 0.7233 0.7138 0.7079 0.6942 0.6819 0.9681 0.9443 0.9300 0.8981 0.8710 2.098 1.85 1.746 1.569 1.448 0.6991 0.6941 0.6910 0.6836 0.6767 0.9529 0.9375 0.9280 0.9063 0.8872 1.33 1.285 1.259 1.204 1.159
(C23r)+avb
(C23r)+avc
(C~rI+av
-2.95 -1.66 -0.84 1.11 2.94 -3.65 -1.22 0.31 3.86 7.09 -3.90 8.99 15.48 28.5 39.24
0 0.0203 0.0338 0.0676 0.1014 0 0.0226 0.0377 0.0754 0.1132 0 0.0266 0.0444 0.0888 0.1332 0 0.0485 0.0808 0.1616 0.2424
0 0.0213 0.035 0.0675 0.0978 0 0.0235 0.0385 0.0736 0.1057 0 0.0272 0.0442 0.083 0.1178 0 0.0435 0.06739 0.1169 0.1573
0 -4.87 -3.40 0.19 3.76 0 -3.88 -2.02 2.59 7.12 0 -2.25 0.44 6.9 13.06 0 11.10 19.92 38.26 54.12
-1.96 -1.26 -0.81 0.26 1.28 -3.10 -1.51 -0.50 1.88 4.07 -3.75 -0.38 1.68 6.32 10.45
0 0.0222 0.03707 0.07415 0.1112 0 0.0254 0.04233 0.0846 0.127 0 0.0312 0.052 0.1040 0.1560
0 0.02332 0.03822 0.07344 0.1061 0 0.02627 0.0428 0.0813 0.1162 0 0.0314 0.0506 0.09386 0.1317
0 -4.61 -3.0 0.97 4.83 0 -3.31 -1.16 4.15 9.29 0 -0.71 2.72 10.82 18.47
a L+ is obtained from eq 37 with X + 2 1 L obtained from A by eq 34b. obtained from eq 37. F'rom Ohya and Sourirajan (1971).
(C23r)+avis
obtained from eq 4 5 with L'
Table V . Membrane Properties and Feed Conditions of Ohya and Taniguchi (1975) exptl run no.
A P ~atm ,
KA-1-3 KA-1-4 KA-1-5 KA-1-6 KA-2-5 KA-2-6 KA-2-7 KA-2-8 KA-2-9 KA-2-10
34.0 28.2 34.0 28.2 34.0 34.0 34.0 28.2 28.2 28.2
u W * x 104, cm/s 6.77 5.51 7.05 5.93 3.90 4.08 4.37 3.17 3.91 4.10
Y
0.061 0.0735 0.0495 0.0595 0.068 0.068 0.068 0.081 0.081 0.081
right-hand side equal to L+. This value of X+21Land the corresponding value of A (from eq 34b) is reported in Table VI. Further, (C23r)avis then calculated from eq 45 for the predicted value of A and reported in this table in ppm units. This table also contains the measured values of A and (Car), (in ppm units). One notices that the maximum error in predicting A is less than about 2.5%. Since A values range between 0.05 to 0.44, such a predictive accuracy is quite gratifying. The low values of y for the experiments contributed significantly to this low error level. Although the levels of error in (CZ3Javprediction are higher, the errors are generally less than 10%. Further, the predicted (Car)avvalue is higher than the experimental value suggesting that the design errs toward safety since reverse osmosis separation plant designs specify a maximum level of solute in the permeate. Finally, we must
(D2M/KS x 105 2.4 2.3 2.3 2.1 2.2 2.6 3.1 2.5 3.1 3.8
I) =
he
0.7533 0.9970 0.8652 0.9949 1.4872 1.1029 0.9611 0.9098 1.4066 2.1951
PPm 2600 2600 2100 21 00 2850 2850 2850 2850 2850 2850
CZlfl
mention that the design equations are quite effective since 9i values in these experiments go to very high levels, as much as 0.09 (run KA-2-10),indicating quite open membranes (for which these design equations were not developed).
Discussions It is possible to develop design equations with improved predictive accuracy by relaxing some of the assumptions. The resulting equations will most certainly require a numerical effort since analytical expressions are unlikely. In this connection we should point out that relaxing the assumptions of a constant aP = AP,is most difficult as has been pointed out for tubular modules by Sirkar and Rao (1981a). Further, although the axial variation of k,can be handled through the bulk velocity variation (see tubular
526
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
Table VI. Comparison of Predicted Fractional Water Recoverya and Permeate Concentration with Experimental Data of Ohya and Taniguchi (1975)
KA-1-3 KA-1-4 KA-1-5 KA-1-6 KA-2-5 KA-2-6 KA-2-7 KA-2-8 KA-2-9 KA-2-10
0.1615 0.0875 0.1725 0.0785 0.0812 0.209 0.425 0.051 0.229 0.440
1.1539 1.0789 1.1760 1.0852 1.0900 1.2670 1.7631 1.0545 1.3004
1.8046
0.1334 0.0731 0.1496 0.0785 0.0826 0.2107 0.4328 0.0517 0.2310 0.4459
0.130 0.073 0.1465 0.0785 0.0812 0.209 0.425 0.051 0.229 0.440
+ 2.54 +0.137 t 2.07 + 0.00 + 1.695 tO.808 + 1.800 + 1.351 f 0.870 t 0.425
319 280.2 211.9 198.55 331.3 488.6 707.1 339.2 531.1 809.7
exptl
% error in
260.0 260.0 198.0 198.0 315.0 440.0 620.0 310.0 470.0 660.0
+ 18.45 t7.2
+ 6.56
+ 0.278 +4.92 t 9.97 + 12.3 + 8.63 t 11.50 + 18.5
A has been determined from eq 34b with X+21Ldetermined from eq 37 for given L'. (C23r)+avdetermined from eq 45, L' and eq 34b and converted to (CZ~,),, in ppmby the following formula for NaCl and H,O system: (C23r)av (ppm) = [(58.5/18)106 x (C23r)+av x ( C 2 1 f ) l / [ C - (C23r)+av(C21f)t (58.5/18) (C23r)tav(C21f)l. ' From Ohya and Taniguchi (1975).
module analysis by Sirkar and Rao, 1981a), there exists no correlation for kl in spiral wound modules with spacers. The recent studies of ultrafiltration in a spiral wound module by Light and D a n (1981) did not develop any mass transfer correlation. The suggested dependence of kl on Uzf by Ohya and Taniguchi (1975) uses on overall value of kl based on the performance of the whole module and, although usable in our analysis, is inappropriate. On the other hand, it would certainly be worthwhile to investigate the possibility of using 3-parameter models or models of the solution-diffusion-imperfection type to improve the design equation prediction of the salt concentration in the permeate. The recent study by Soltanieh (1980) in a thesis on modeling hollow fiber systems for desalination would certainly point in that direction. An additional approach that can significantly improve the predictive accuracy of the design equations is to incorporate suitably the effect of salt leakage through membranes by modifying eq 21 and 22. This is being explored elsewhere (Sirkar and Dang, 1982). Recommended Design Procedures Turbulent Flow: For Membranes with High Solute Rejection. Given: 4,b, Czif, (C23r)avi ( D ~ M / K h, ~ )kl, , Uxf, W , APf, A. Determine N1, with Af, CZlf,kl, and APf from eq 12. Check (NlrVl/kl) 50.2. Determine L+ from eq 34 with X+21Ldefined by eq 34 b, y defined by eq 32, and q defined by eq 31. Obtain the value of L from definitions 25 and 27. Using the L+ determined above, obtain (C23,), from eq 45 and definitions 44,46, and (D2,/K6). Compare it with given (C23r)av. Design is reasonable if calculated (C23r)av is greater. Maximum overdesign of L by 10% for y I0.4,8 5 0.02, A 5 0.4. Maximum overprediction of (C23r)avby 15% for y I0.3, 8 I0.02, A I0.5. For lower values of y, similar accuracy in L and (C23r),v can be achieved with higher values of 8. For higher values of y, similar accuracy can be achieved with lower values of 0. As A decreases, accuracy increases. Laminar Flow: For Membranes with High Solute Rejection. Given: An b, C21f, (C23r)ev, ( D ~ M / K h, ~ )ki, , Oxf, @,, A.
w,
Determine N,, with Af, CZlf,kl, and AP,from eq 15. Check (NlrVl/k,) 5 1.0. Calculate L+ from eq 37 with X+zlLdefined by eq 34b, y defined by eq 32, and q defined by eq 31.
Using definitions 25 and 27, obtain L from estimate of L+. Use estimate of L+ in eq 45 to obtain (C23,), from definitions 44,46, and (Dm/KS). Compare it with given (C&Iav Design is reasonable if calculated (Car)avis greater. Maximum overdesign of L by about 10% for y 5 0.4, 8 5 0.02, A 5 0.4. Maximum overprediction of (C23,), by 15% for y 50.3, 8 I 0.02, A I0.5. For lower values of y, similar accuracy in L and (C23,), can be achieved with higher values of 8. For higher values of y, similar accuracy can be achieved with lower values of 8. As A decreases, accuracy increases. Frequently, membrane performance is reported in terms of solute rejection and A , the pure water permeability constant. A simple way to estimate (D2,/K6) in this case for high rejection membranes is as follows 1 - (XZ3,/Xz2) solute rejection = R = 1 - (C23,/C,,) (47) Using eq 6 and the crossflow assumption, namely (Rao and Sirkar, 1978) N2r/Nlr = x23r/(1 - X23r) E X23r (48) obtain NlrX23r = ( D Z M / K ~ ) C ( X Z ~ ) (49) Therefore (50) 22
Thus N1, being known from eq 12 or 15, (D,M/K6) may be determined from known values of R. Should one decide to check the regime with (uw*/kl) instead of (NlrVl/kI),the same limits will provide higher accuracy. Calculate (uw*/kl) from Af, hpf, VI, and kl by using eq 25. If (uw*/kl) I 0.2, follow the procedure for turbulent flow. If 0.2 I (vw*/kJ Il.0, follow the laminar flow design procedure. Since (N,,V,) will always be less than uw* (except for a pure solvent feed) checking with (uw*/kl) will always lead to a somewhat higher accuracy. Under normal operating conditions, the A values for a single spiral-wound module rarely exceeds 0.2 so that the design accuracy of the above procedures is high. For A > 0.2 with a few modules in series, stepwise design will ensure high accuracy. Availability of kl correlation for channels with mixing screen is, however, necessary regardless of the design procedure. Conclusions Simple analytical expressions for the channel length of a spiral-wound membrane module and the average con-
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 527
centration of solute in the permeate thereof have been obtained for reverse osmosis separation of microsolutes from dilute solutions with high rejection membranes for a required fractional water recovery. Separate expressions developed for both quantities for either turbulent or laminar mas~ transfer conditions have been found to be quite accurate when compared with the results of the exact numerical scheme of Ohya and Sourirajan (1971). Deviations increase as the membrane becomes more open and the fractional water recovery increases to half and beyond. The expressions for laminar flow conditions are restricted to (N,,Pl/k,) I1.0 which provides a broad range useful for practical reverse osmosis. Reasonable accuracy in predicting the observed fractional water recovery and permeate solute concentration in the laminar flow spiral-wound reverse osmosis experiments by Ohya and Taniguchi (1975) suggests considerable confidence in the equations developed. Nomenclature A = pure water permeability constant, g-mol/cm2 s atm b = constant defined by osmotic pressure relation 3 C = total molar density of brine solution, g-mol/cm3 CZlf= salt concentration of brine fed to spiral-wound module, g-mol/cm3 CZlL= salt concentration of concentrated brine leaving spiral-wound module of length L , g-mol/cm3 C23, = permeate salt concentration at any location of module under conditions of concentration polarization, g-mol/cm3 (CDJav = averaged permeate salt concentration from module of length L for a fractional water recovery of A, g-mol/cm3 (C23r)+av= defined by (C23r)av/C21f C , = concentration of species s at location j for any x ; s = 1 for water; s = 2 for salt; j = 1 feed stream; j = 2, high pressure side membrane-water interface; j = 3, permeate stream, g-mol/cm3 D2M = diffusivity of salt in membrane phase, cm2/s (D2M/K6) = salt transport parameter, cm/s (l/h) = total active membrane surface area per unit volume of the feed channel facing active membranes, cm-' kl = salt mass transfer coefficient in feed brine, cm/s K = ratio of salt concentration in brine over salt concentration in membrane with which the brine is in contact L = total length of a membrane from channel entrance, cm L+ = dimensionless length of a membrane in spiral-wound module, defined by eq 24 Lp = hydraulic permeability of RO membrane, cm/s atm N1, = solvent flux at any location x with concentration polarization, g-mol of B20/cm2s N2, = solute flux throtlgh membrane at any location x with concentration polarization, g-mol of solute/cm2 s AP = applied pressure difference across membrane at any x , atm APf = value of AP at the entrance of the module, atm R = solute rejection, defined by eq 47 D,f, ox, oxL = bulk velocity of brine in feed channel of module at the entrance to the module, at any location x along the module and as it leaves the module of length L , respectively, cm/s u,* = pure water permeation rate through membrane at feed location values of Af and APf, defined by eq 25, cm/s TI = partial molar volume of solvent, cm3/g-mol Vp = total volumetric desalinizedwater production rate from the spiral-wound module, cm3/s W = the width of membrane in module perpendicular to the mean flow direction, cm x = axial distance along the length of the membrane from inlet of feed brine channel, cm
X , = mole fraction of species s at location j for any x ; s = 1 for water; s = 2 for salt; j = 1, feed stream; j = 2, high pressure membrane solution interface; j = 3, permeated stream Xzlf,XzlL = solute mole fractions corresponding to salt concentrations CZlf,CZlL,respectively, with Csj = CX, X+zl = normalized bulk solute mole fraction of feed at any x , defined by eq 29 X+zu,= normalized bulk solute mole fraction of concentrated brine at module outlet defined by eq 30 X23r = solute mole fraction in permeate at any location with concentration polarization (XmJaV = solute mole.fraction in total permeate from module corresponding to V p (or A) (X23r)+av = defined by [ ( X Z ~ ~ ) ~ ~ / X Z I ~ I Greek Letters r ( C , ) = osmotic pressure of salt solution of molar salt concentration C,, eq 3, atm
dimensionless feed osmotic pressure, eq 32 A = fractional volumetric water recovery, eq 34b and eq 42 S = effective membrane thickness in Kimura-Sourirajan (1967) analysis u = reflection coefficient 0 = dimensionless solute permeability, eq 44 A0 = dimensionless mass transfer coefficient, eq 31 7 = dimensionless mass transfer coefficient, eq 31 y =
Subscripts f = refers to high pressure feed at module inlet j = 1, feed stream; j = 2, high pressure side wall; j = 3,
permeate stream
L = refers to conditions at the end of the module r = real feed channel mixing conditions indicating existence
of concentration polarization
s = s = 1 for water; s = 2 for solute or salt
Literature Cited Bray, D. T. "Desalination by Reverse Osmosis"; Merten, U., Ed.; MIT Press: Cambridge, MA, 1968; p 203. Brian, P. L. T. "Desalination by Reverse Osmosis"; Merten, U., Ed.; MIT Press: Cambridge, MA, 1966; p 161. Cabibo, S. V.; Guy, D. B.; Ammeriaan, A. C.; KO, A,; Singh, R. NWSIA J . 1979, 6(2), 17. Harris, F. L.; Humphreys, 0. B.; Spiegler, K. S. "Membrane Separation Processes"; Meares, P., Ed.; Elsevier: Amsterdam, 1978; p 122. Kimura, S.; Sourirajan, S. AIChE J . 1967, 13, 479. Kremen, S. S. "Reverse Osmosis and Synthetic Membranes"; Sourirajan. S., Ed.; National Research Council of Canada: Ottawa, 1977; p 371. Light, W. G.; Tran, T. V. I d . En$. Chem. Process Des. Dev. 1981, 20, 33. McCutchan, J. W.; Goel, V. Desalination 1974, 1 4 , 57. Ohya, H.; Sourirajan, S. "Reverse Osmosis System Specification and Performance Data for Water Treatment"; Thayer School of Engineering, Darmouth College, NH, 1971. Ohya, H.; Taniguchi, Y. Desalhaffon 1975, 16, 359. Podall, H. E. "Recent Developments in Separation Science"; Li, N. N., Ed.; CRC Press: Cleveland, OH, 1975; Vol. 11, p 174. Rao, G. H.; Sirkar, K. K. Desallnetion 1978, 27, 99. Singh, R.; Cabibo, S. V. Desalination 1960, 32, 261. Sirkar, K. K.; Dang. P. T.; Rao, G. H. "Explicit Flux Expressions in Reverse Osmosis Desalination", manuscript under preparation, 1982. Sirkar, K. K.; Dang, P. T. "Improved Design Equations for Reverse Osmosis Desalination wlth Spiral Wound Modules", to be presented at the World Filtration Congress 111, Sept 13-17, 1982, Downingtown, PA. Sirkar, K. K.; Rao, G. H. Ind. Eng. Chem. Process Des. D e v . 1#61. 20, 116. Sirkar, K. K.; Rao. G. H. "Progress in Separation and Filtration", Wakeman, R. J., Ed.; Elsevier: Amsterdam, 1979; p 293. Sirkar, K. K.; Rao, G. H. Desalination, submitted for publication, 1981a. Solan, A.; Winograd, Y.; Katz, U., Desalination 1071, 9 , 89 (brought to our attention by Professor K. S. Spiegier). Soltanieh, M. Ph.D. Thesis, Chemical Engineering Department, State University of New York at Buffalo, Buffalo, NY, 1980. Spiegier, K. S.; Kedem, 0. Desalination 1966, 1 , 31 1. Spiegler, K S., Department of Mechanical Engineering, University of California, Berkeley, CA 94720, personal communication, 1981.
Received for review March 26, 1981 Accepted March 8, 1982
