350
Ind. fng. Chem. Process Des. Dev. 1985, 2 4 , 350-358
Analytical Design Equations for Multicomponent Reverse Osmosis Processes by Spiral- Wound Modules Ravl Prasad and Kamalesh K. Slrkar" Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
Analytical expressions have been developed to predict the membrane channel length of a spiral-wound module for achieving a given fractional solvent recovery if the dilute feed has two highly rejected solutes. Extension to a large number of noninteracting solutes is straightforward. For a single-solute system, the species flux expressions used reduce to a two-membrane-permeation parameter model compared to the more exact model requiring three-membrane parameters. Design equations are valid for turbulent feed flow and a range of chopped laminar flow conditions. Analytical expressions to predict permeate solute concentrations have also been developed for turbulent flow. For laminar flow conditions, a simple Integral has to be numerically integrated to predict permeate concentration of any solute. Accuracies of these analytical design equation predictions have been compared with those from a numerical solution of the exact set of equations. Analytically predicted membrane channel lengths are usually within 10% of those obtained numerically. The predicted permeate concentrations are always higher than the correct values, and the predictive accuracies are somewhat lower than in length calculations.
Introduction Reverse osmosis (RO) is being increasingly considered as an economical replacement for existing separation processes for water purification, water reuse, and solute concentration (Sapakie et al., 1982; Weeks et al., 1981; and Whalen et al., 1981). Such exercises require computer simulations of a large number of interconnected processes and units. Simple and accurate analytical design equations for each unit are a virtual necessity for such preliminary process design studies on computers. Numerical design procedures and results for single-solute RO with spiral wound modules are available in the work of Ohya and Sourirajan (1971) and Tweddle et al. (1980). A particular example is its application by Ohya and Taniguchi (1975). Sirkar et al. (1982) have developed simple analytical design equations to predict membrane length and permeate solute concentration in a spiral-wound module for an RO process with a dilute solution of a single highly rejected solute species. These were carried out for turbulent flows and a range of chopped-laminar flow conditions. Most feed solutions in practical RO have more than one solute. Analfiical design procedures for RO processes with a multisolute feed are therefore expected to be very useful. Recently, Prasad and Sirkar (1984) have developed analytical design equations for a multisolute feed fed to a tubular RO module. Such modules are of interest in handling fouling feed streams. Spiral-wound and hollow fiber RO modules are, however, overwhelmingly used when clean, nonfouling process streams are available. In this paper, we develop simple analytical design equations and procedures for multisolute feeds to spiral-wound RO modules. This analysis is restricted to dilute feeds with solutes that are noninteracting and highly rejected by the membrane. Conditions of flow include turbulent as well as a range of chopped-laminar flow conditions. The technique used here for the development of the design equations for turbulent flow is similar to that used by Prasad and Sirkar (1984) for tubular systems in turbulent flow. But there are some basic differences. In tubular systems, mass transfer coefficient continuously varies along membrane length and therefore for moderate to high water recoveries, split module calculation procedures are necessary. A spiral-wound module will rarely
have a very high water recovery, the latter being usually less than 0.20 to 0.30. Thus split-module calculations are not necessary. In addition, mass transfer coefficients can be assumed to be constant. For chopped laminar flow regime (Solan et al., 1971; Sirkar et al., 1982), the technique used here has no analogue in tubular systems which are always operated in turbulent flow. Existence of spacer screens in spirallywound membrane channels facilitate the development of simple design equations exactly as in Sirkar et al. (1982). Flux Expressions and Polarization Relations for Multisolute RO For the reverse osmosis transport of a solute and solvent through a membrane, Soltanieh and Gill (1981) have shown that the preferential sorption and capillary transport model is mathematically equivalent to the solution diffusion model. Malalyandi et al. (1982) have extended the preferential sorption capillary transport model to a system of multiple solutes, and water being transported through the membrane and experimentally verified it. Prasad and Sirkar (1984) have also used this model developed by Malalyandi et al. (1982) in their analysis of reverse osmosis separation of two or more noninteracting solutes from water in a tubular module. It is to be noted, however, that for a single-solute system, these flux expressions have only two membrane-permeation parameters in them. This is to be contrasted with the more exact irreversible thermodynamic formulations which require three membranepermeation parameters for a single-solute system. For two noninteracting solute species in a feed solution, the flux expressions for the solvent species 1 and solute species 2 and 3 under conditions of concentration polarization can be written (following Malalyandi et al., 1982) as follows (1) N1r = A [ w - "wall + spermeatel
We now assume that (1)the total molar density C of the solution remains constant, which yields
0196-4305/85/1124-0350$01.50/00 1985 American Chemical Society
cij= CX,
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
351
[+
(10)
(2)
x 2 2 =x21
and (2) osmotic pressures of dilute solutions can be expressed by the linear relations H w a ~=
b C w d = b[C22 + C.321;
rpermeate
=
bcpermeate
1
+
N1rP1 kl2
...I
--
b[C23r + C33rl (2a) Use now Brian's (1966) film theory expression separately for concentration polarization for each solute species. Cross-diffusion terms are neglected following Srinivasan and Tien (1970) although the solvent flux couples the wall concentration for each solute X21
r2
exp [NlrVdk121 + [1 - r2I ex~CNlrVl;/kd
(3)
_-
ex~[NlrVdkd r3 + [1 - f31 exP[Ni~Vi/kld
(34
x 2 2 _ -
x32
x3i
under conditions such that
These conditions are always ensured by currently available RO membranes being used under typical RO process conditions with turbulent feed flow regime. The solvent flux expression 8 can therefore be simplified using expressions 10 and 10a to the following N1. =
Here klz and k13 are the mass transfer coefficients for solute species 2 and 3 and r2 and r3 are the membrane solute rejections defined as xi3r
( i = 2, 3, ...)
ri=l--;
Xi2
(4)
If we assume pure cross flow in the spiral module for relating local permeate concentrations to the fluxes (Ohya and Sourirajan, 1971; Ohya and Taniguchi, 1975; Sirkar et al., 1982), we have
If, on the other hand, the flow conditions on the feed side is laminar or chopped laminar, Sirkar et al. (1982) have shown that the concentration polarization expressions 9 and 9a can be approximated by x 2 2
[
(N.7)' + ...I
=x211 + N1rV1 -+ y2 kl2
-
(12)
under the limitations For specified operating conditions, eq 1-5a can be solved numerically to yield the exact local values of N,,, N2,, N3r, c221 c32, C23r, C33r, r29 and r3* Approximate analytical design equations can be obtained only on the basis of some assumptions and approximations. We assume now that the membranes are highly rejecting. Therefore, for a two-solute system r2 N 1 and r3 N 1
(6)
,,.?, Nlr;>I,,
X +1z l
2klzX+21
(31)
where This may be simplified to
[ (; &) & 91"' +
+
2(
-
(31a)
the latter being merely a nondimensional form of expression 18 for NlrVl. Similarly, for solute species 3, the averaged concentration in the permeate is given by
where
-->
-(; [ (; &) &
523 =
+
k13
+
1 SX+3 1
+
+
2(
- ;)]1'2
(32a)
and
If the range of validity of the above equations is to be increased to Nl,Pl/k12 I1.5 and NlrVl/k13I 1.5, the functions t l , t2, and t3 would have to be substituted respectively by tl', t2', and t3' defined by eq 18b and 18d. The integrals 31 and 32 have not been solved analytically because of their complexity. They can be easily evaluated numerically with minimum time and effort. Furthermore, this procedure can be extended without difficulty to more than two solutes. For a system of two solutes (i = 2 and 3) and solvent (i = l ) , it is not necessary to evaluate both (C+23r)avand Evaluation of any one by integrating the relevant
where L+ to be used corresponds to eq 24 since we are dealing with chopped laminar flow. However, eq 34 is valid in general and could be used for turbulent flow as well. Therefore, evaluation of either (C+23r)av or (C+33r)aV is enough since, L+ and A being known, the other can be evaluated from eq 34. Exact Numerical Design Procedure The utility of the above simple design equations can be evaluated by comparing their predictions with those obtained from a numerical solution of the exact set of flux equations and conservation equations. This is done as follows. (a) The governing material balance eq 15 for solvent and two equations for i = 2 and 3 from eq 15a for solute i are expressed in finite difference form using the forward difference approximation. (b) The values of the local fluxes of the solvent and solute species (Nl,,N%,N3J as well as X22, x32, X23r, x3311 r2,r3 are found by iteratively solving the eq 1, la, lb, 2, 2a, 3,3a, 4,5, and 5a by a Newton-Raphson scheme. Inputing the above data in the finite difference form to eq 15 for the solvent and to eq 15a for two solutes i = 2 and 3 will allow us to calculate the values of Dx, XZ1,and X31 at a location dl from entrance. In this manner, one can march forward by first determining local values of Nl,, Nir, X i 2 ,Xiartand ri at location 1 for i = 2 and 3 and then Dx, X 2 1 ,X31 at location 1 + dl. The procedure is continued till the total volumetric rate of production of permeate reaches the value specified by the required A. For the detailed equations, see a thesis by Prasad (1983). Although the feed conditions such as AP,feed concentrations, and membrane properties are not needed for the dimensionless analytical results derived in the previous sections, these had to be suitably assumed for the exact numerical solution. The final value of the length of membrane channel and permeate concentrations obtained from the numerical schemes were made dimensionless using the feed conditions assumed for the calculations.
356
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
Table 1. Comparison of Multisolute Analytical Predictions with Exact Numberical Solution for a Spiral-Wound Module with Turbulent Feed Flow length, L+ feed 3000 ppm NaC1; 2000 ppm MgS04; O2 (NaC1) = 0.01; 83 (MgS04) = 0.0007; v = 5.0
3000 ppm NaC1; 2000 ppm MgC1,; O2 (NaC1) = 0.01; 03 (MgCl2) = 0.0056; 17 = 5.0
25000 ppm NaC1; 10000 ppm MgS04;B2 (NaC1) = 0.01; 8 3 (MgS04) = 0.007; 7 = 5.0
a
Using eq 22.
permeate comDositions exact numerical analyt.b
Y A exact 0.1 0.2 0.269
analyt" 0.2421
(C+Br)av (C+33r)av (c+23r)av (C+33r)sv 0.127 X lo-' 0.901 X 0.157 X lo-' 0.115 X lo-*
0.1 0.1 0.2 0.3
0.368 0.638 0.418 (0.35) 0.632
0.137 X lo-' 0.975 X 0.165 X lo-' 0.117 X 0.152 X lo-' 0.107 X 0.218 X lo-' 0.16 X
0.17 X lo-' 0.206 X lo-' 0.19 X lo-' 0.276 X lo-'
0.1 0.2 0.269
0.251
0.130
lo-' 0.736 X
0.160 X lo-' 0.96
0.1 0.1 0.2 0.3
0.38 0.65 0.520 0.782 (0.32)
0.139 X 0.167 X 0.174 X 0.258 X
lo-' lo-' lo-' lo-'
0.173 X lo-' 0.211 X lo-' 0.219 X lo-' 0.338 x lo-'
0.3 0.5 0.3 0.3
0.3 0.5 0.3 0.3
0.4041 0.6905 0.4394 (2.53) 0.639
0.4082 0.691 0.538 0.74 (3.5)
0.1 0.3 0.403 (2.5)
X
0.783 X 0.943 X 0.980 X 0.14 x lo-'
0.360 (0.34) 0.135 X lo-' 0.95 X
0.2 0.2 0.3139 (2.55) 0.288 (0.36) 0.148 X lo-' 0.105 X 0.2 0.3 0.4708 (2.98) 0.444 (0.38) 0.161 X lo-' 0.114 X 0.3 0.3 0.572 (2.67) 0.573 (0.31) 0.198 X lo-' 0.142 X
0.167
X
0.128 X 0.152 X 0.17 X 0.2 X
loF2
X
0.102 X 10-1 0.124 X lo-' 0.130 X lo-' 0.2 x 10-1
lo-' 0.123 X low2
0.18 X lo-' 0.135 X 0.2 X 10-1 0.148 X 0.263 X lo-' 0.188 X loT2
Using eq 28 and 29; ( ) indicates CPU time in seconds.
Further details and computer programs for the above computations are available in Prasad (1983).
Predictive Accuracy of the Analytical Design Equations The predictive accuracies of the analytical design equations developed here are checked by comparing the L+ and permeate compositions obtained therefrom against those obtained by an exact numerical solution procedure outlined in the previous section. In the case of turbulent flow, eq 22,28, and 29 have been used to predict L+,(Par), and (C+,), respectively. This has been done for three dilute feeds with the following compositions: (1)3000 ppm NaCl and 2000 ppm MgSO,; (2) 3000 ppm NaCl and 2000 ppm MgCI2; (3) 25000 ppm NaCl and 10000 ppm MgS04. These solutes and composition levels were selected because a considerable amount of membrane and solute data exists for practical membranes in such a system ranging from seawater to brackish water. While the NaC1-MgS04-H20 system did not have any common ions for the two salts, the NaCl-MgCI2-H20 does have a common ion. I t is known that mixed electrolytes form an interacting system. However, Eliash and Bennion (1976) have shown that the ion-ion interaction increases the rejection of, say, MgC12 in the presence of NaCl by only about 1.5% for not too concentrated a solution. One can also infer from the data in Sourirajan (1970) that the interaction effects are minimal for high rejection membranes and dilute systems. Thus while the brackish systems are truly dilute, questions can be raised about the applicability of the equations for a seawater system. In the cases considered here, the maximum value of A used for seawater is 0.3. Commercial applications of RO rarely exceed this A value for a given spiral-wound module. Thus the solution stays dilute. Hence the interaction effects are minimized and noninteracting solute assumption may be used. Further, our objective is to demonstrate the quality of predictions of design equations. The above selection of solute and feed concentrations is relevant only for the computation of the dimensionless multisolute parameters tl, t2,t3,t4,t6,and t6. The calculations have been done for suitable values of y and A and the results are shown in Table I. It can be seen that as in the case of a single-solute system studied by Sirkar et al. (1982), the analytical eq 22 usually underpredicts the
value of L+ obtained numerically. The maximum error in L+ is 9.7% for a value of y = 0.1 and A = 0.2. Note that the errors with systems of higher y are considerablylower. Such systems are encountered quite frequently. We have included a few illustrative estimates of CPU times to indicate that on an average 7-9 times less time is required for these analytical L+ calculations (with a minor loss of accuracy). Analytical expressions for the permeate concentrations given by eq 28 and 29 always overpredict those obtained by numerical solutions. The extent of overprediction varies between 20% to 30% and is somewhat on the high side. But since RO modules are designed to satisfy a certain maximum level of concentration of solutes in the permeate, the error, in the prediction of permeate compositions, will tend to make safe designs. In the case of chopped-laminar flow in a spiral-wound module eq 24,31, and 32 were used to predict L+,(C+23r)av and (C+33r)av,respectively. The limits of applicability of these equations are indicated by assumptions 12b due to approximations 12 and 12a. Equations 31 and 32 were integrated by using Simpson's rule. The calculations have been carried out for the same set of feed compositions used in the turbulent flow computations. The computed results are shown in Table 11. This table exhibits the trend previously observed in Sirkar et al. (1982) for a single-solute system. The values of L+ predicted analytically are generally higher than those obtained by an exact numerical solution. The error in L+ prediction is usually around 10% except for the extreme case of y = 0.2 and A = 0.3 for a seawater type of system where the error is 22%. The permeate compositions predicted are again higher than those obtained by the exact numerical procedure, but again, this error would tend to make safer designs. The CPU time required for the exact numerical solutions were significantly higher than those required for the analytical solutions. For example, for y = 0.2 and A = 0.3 the exact numerical procedure consumed 1.78 s compared to 0.57 s for the analytical solutions. The difference in the two times is somewhat less than in the turbulent flow case due to the necessity of numerically integrating both the integrals for the permeate concentrations. For chopped laminar flow, where the approximations 13 and 13b have been used, calculations for L+ have also been done by changing tl, t2,and t3to t;, t i , and t i . The results are also shown in Table 11. It can be seen that the values
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 357
Table 11. Comparison of Multisolute Analytical Predictions with Exact Numerical Solution for a Spiral-Wound Module in Chopped Laminar Flow permeate compn, (C+23r)avand length, L+ feed 3000 ppm NaC1; 2000 ppm MgSOa; O2 (NaCl) = 0.01; (MgS04)=-0.000?; 7 = 1 0
3000 ppm NaCl; 2000 ppm MgCl,; 82 (NaCl) = 0.01; 63 (MgC12) = 0.0056; 7 = 1.0
25000 ppm NaC1; 10000 ppm MgS04;O2 (NaC1) = 0.01; 63 (MgSOJ = 0.0007; 7 = 1.0
"Using eq 24 with approximations 12 and 12a. eq 31, 32 with approximations 12 and 12a.
0.1
A 0.3
exact numer. 0.4708
0.1
0.5
0.2
analyt"
(C+33J."
0.4353
analytb 0.477
0.7847
0.77
0.848
0.3
0.47
0.531
0.55
0.3
0.3
0.678
0.701
0.72
0.1
0.3
0.4709
0.453
0.4687
0.1
0.5
0.785
0.809
0.834
0.2
0.3
0.6278
0.669
0.6829
0.1
0.3
0.4708
0.436
0.447
0.2
0.2
0.3139
0.37
0.396
0.2
0.3
0.4709
0.578
0.618
0.3
0.2
0.4708
0.533
0.534
y ~
exact numer. 0.1315 X lo-'
analytc 0.166 X lo-'
0.932 X 0.158 X lo-' 0.1123 X lo-' 0.14 X lo-' 0.101 x 10-2 0.177 X lo-' 0.126 X
0.139 X 0.199 x 10-1 0.168 X 0.18 x 10-1 0.13 X 0.212 x 10-1 0.143 X
0.133
0.169 X lo-'
X
lo-'
0.750 X lo-' 0.16 X lo-' 0.901 x 10-2 0.172 X lo-' 0.972 X
0.168 X 0.204 X 0.202 x 0.206 X 0.199 x
lo-' lo-' 10-1 lo-' 10-1
0.129 X lo-'
0.173 X lo-'
0.917 X 0.138 X lo-' 0.9808 X low3 0.149 X lo-' 0.105 X 0.17 X lo-' 0.124 X
0.13 X 0.173 X 10-1 0.138 X 0.189 X lo-' 0.15 X 0.21 x 10-1 0.17 X
eq 24 with approximations 13a, 13b, and t i , t i , t i instead of t,, t2,and t,. cUsing
of L+ thus obtained do not differ significantly from those obtained by using approximations 12 and 12a. This behavior can be explained as follows. The level of concentration polarization indicated by the value of q = 1.0 used in this table is such that approximations 12 and 12a are highly accurate. Improvements in the flux approximations resulting from eq 13 and 13b are not sufficiently greater. Further, other approximations are necessary before the design equations are developed. Consequently L+ values obtained by two different approximations are quite close. This wab also observed for a single solute case (Sirkar et al., 1983). We are not reporting calculations carried out at smaller values of 7 (e.g., 9 = 0.75) reflecting higher levels of concentration polarization in the chopped laminar regime. Generally the analytical results overpredict the length by 12-20%. Conclusions For a dilute feed with two noninteracting, highly rejected solutes, simple analytical design equations have been developed to predict the membrane channel length of a spiral-wound module required to achieve a given water recovery for turbulent as well as a range of chopped laminar feed flow conditions. Extensions of these results to three or more highly rejected solutes are straightforward. The analytical expressions predict membrane channel length within 10% of the values obtained from exact numerical solutions of the governing equations for almost all the design examples investigated. Simple analytical design equations have also been developed to predict permeate solute concentrations with turbulent feed flow. For a range of chopped laminar feed flow conditions, permeate solute concentrations appear as integrals to be evaluated numerically. The predicted permeate concentrations are always higher than the exact values and the predictive accuracies are somewhat lower than in length calculations.
Nomenclature A = pure water permeability constant, g-mol/cm2-s-atm b = constant defined by osmotic pressure relation 2a C = total molar density of solution, g-mol/cm3 Cilf = molar concentration of species i in feed, g-mol/cm3 CilL = molar concentration of species i in high-pressure solution leaving the module of length L , g-mol/cm3 Cij = molar concentration of species i at location j , i = 1 for solvent, i = 2, 3, 4 for solute species; j = 1 feed stream, j = 2 high-pressure membrane water interface, j = 3 permeate stream, g-mol/cm3 Ci, = permeate molar concentrationof species i at any location with concentration polarization, g-mol/cm3 DiM/KiS = solute transport parameter of species, i, cm/s F,, F2 = defined by eq 17a and 17b l / h = total membrane surface area per unit volume of the feed channel facing active membranes, cm-' kli = mass transfer coefficient of species i in the high-pressure solution, cm/s Ki = ratio of solute i concentration in brine over solute i concentration in membrane with which the brine is in contact 1 = axial distance along the membrane from inlet of feed brine channel, cm L+ = dimensionless length of membrane module defined by 21 h N1, = solvent flux at any location 1 with concentration polarization, g-mol/cm2-s Ni, = flux of solute species i through the membrane with concentration polarization, g-mol/cm2-s AP = applied pressure difference across membrane at any location, 1, atm aPf = value of hp at entrance of module, atm ri = solute rejection of species i defined by (4) t l , t 2 ,t3, t l , tat t6 = defined by eq 17c, 17d, 18a, 29e, 29f, 32b, respectively tl', t i , t i = defined by 18b, 18c, and 18d, respectively uxf, D ~ fZL , = bulk velocity on the high-pressure side at the entrance of module. at anv location I along the module, as it leaves the module, cmis
Ind. Eng. Chem. Process Des. Dev. 1985, 24,358-364
358
Vl = partial molar volume of solvent cm3/g-mol V, = total volumetric permeate production rate from module of length, L , cm3/s W = width of membrane perpendicular to the mean flow direction, cm Xij.= mole fraction of species i at location j : j = 1feed stream, J = 2 high pressure membrane solution interface, j = 3 permeate stream X, = mole fraction of species i in the permeate at any location 1 with concentration polarization
Ellash, B. E.; Bennion, D. N. Chem. Eng. Prog. Symp. Ser. 1978, 73(166),
166. Jeffers, J. D.; Kranc, S.C.; Carnahan, R. P. Ind. Eng. Chem. Process Des. D ~ V .i m , 22, 687. Malalyandi. P.; Matsuura, T.; Sourirajan, S. Ind. Eng. Chem. Process Des. D e v . 1982, 27, 277. Ohya, H.; Taniguchi, Y. Desalination 1975, 16, 359. Ohya, H.; Sourlrajan, S. "Reverse Osmosis System Specification and Performance Data for Water Treatment"; Thayer School of Englneerlng. Dartmouth College, Hanover, NH, 1971. Prasad, R.; Sirkar, K. K. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 320. Prasad, R. M. Eng. (Chemical) Thesis, Stevens Institute of Technology, Hoboken, NJ, 1983. Sapakie, S. F.; Hanson, M. C.; Renshaw, T. A. Chem. Eng. Prog. 1982, 78(5), 33. Sirkar. K. K.; Dang, P. T.; Rao, G. H. Ind. Ens. Chem. Process Des. Dev. 1982. 27, 5171 Sirkar, K. K.; Dang, P. T.; Rao, G. H. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 687.Sirkar, K. K.; Rao, G. H. Desallnation 1978, 27, 99. Solan, A.; Wingorad, Y.; Katz. U. Desallnatlon 1971, 9 , 89. Sobnleh, M.; 0111, W. N. Chem. Eng. Commun. 1981, 72, 279. Sourirajan, S. "Reverse Osmosis"; Logos Press: London, 1970. Srinivasan, S.; Tien, C. Desalination 1970, 7, 133. Tweddle, T. A.; Thayer, W. L.; Matsuura, T.; Hsleh, F. U.; Sourirajan, S.Desallnation 1980, 32, 181. Weeks, M. C.; Rogers, A. N.; May, S. C.; Houle, E. H. Chem. Eng. Prog. 1981, 77(7), 31. Whalen, R. C.; Fosberg, T. M.; Mukhopadhyay, D.; O'Neail, T. M. Chem. Eng. Prog. 1981, 77(10), 63.
Greek Letters = osmotic pressure of solution, eq 2a, atm y, y1 = dimensioniess feed osmotic pressures of solute species 2 and 3 defined by eq 21e and 29b, respectively A = fractional water recovery defined by eq 21f 02, O3 = dimensionless permeability of solutes 2 and 3 defined T
by eq 28a and 29g, respectively dimensionless mass transfer coefficients defined by eq 21c and 29c, respectively 6 = effective membrane thickness 7, q1 =
Subscripts av = averaged quantity from module
f = refers to high pressure feed at module inlet
i = species i L = refers to conditions at end of module r = real feed channel mixing conditions indicating existence
Received for review February 8, 1984 Accepted June 11, 1984
of concentration polarization L i t e r a t u r e Cited
R.P. wishes to acknowledge the summer support provided by the Filtration Society, New York Chapter. This work was presented in part at the AIChE Diamond Jubilee Meeting, Washington,DC,
Bennlon, D. N.; Pintauro, P. N. "Membrane Transport Fundamentals", Paper 4d presented at 89th Natlonal AIChE Meeting, Portland, OR. Aug 1980. Brlan, P. L. T. "Desalination by Reverse Osmosis", Merten U.. Ed.; MIT Press: Cambrklge, MA, 1966; p 161.
Nov 4, 1983.
Equilibrium Theory for Adiabatic Desorption of Bulk Binary Gas Mixtures by Purge Shlvajl Slrcar and Ravl Kumar' Air Products and Chemicals, Inc., Allentown, Pennsylvania
78 705
A local equilibrium theory for adiebatic desorption of bulk binary gas mixtures by isobaric purge is developed. The model is used for evaluating the effect of adsorption selectivity, the strength of sorption, and the purge gas composition on the desorption process. Examples of the desorption of CO, from mixtures with CH, N,, and H, are considered. Analytical equations to describe isothermal desorption of binary Langmuir adsorbates are derived. Desorption profiles obtained from isothermal and adiabatic models are compared. I t is demonstrated that (a) desorption of a more strongly adsorbed species (component 1) by purging with a less strongly species (component 2) is more efficient when component 1 is less selectively adsorbed; (b) an adsorbent which exhibits higher adsorption capacity and selectivity for component 1 also requires more purge gas to clean the column and thus the adsorbent may not be preferred for separation by PSA process: (c) composition of the purge gas is important only to determine the level of residual component 1 at the end of the purge step: and (d) assumption of column isothermalky during the purge process can severely underestimate the quantity of the purge gas requirement of an adiabatic column.
Pressure swing adsorption processes for separation of gas mixtures usually consist of sequential cyclic steps of adsorption a t super ambient pressure and desorption by (a) pressure reduction to ambient level followed by (b) isobaric purging at near-atmospheric pressure (Smith, 1983). The purge step is carried out with a gas mixture which is rich in the less strongly adsorbed species of the feed gas mixture. The purge gas is typically produced as the column effluent during the adsorption step, a portion 0196-4305/85/1124-0358$01.50/0
of which is used for purging, and the remainder is withdrawn as the primary product gas. Thus the quantity of purge gas required is a critical variable for determining the overall efficiency of the separation process. Theoretical studies of desorption by purging are few. An excellent summary of the subject is given by Basmadjian et al. (1975a).. The general problem is to simultaneously solve the mass and heat balance (partial differential) equations during the purging step using appropriate 0 1985 American Chemical Society
