1686 Ind. Eng. Chem. Res. 1993,32, 1686-1691

A Noniterative Solution for Periodic Steady States in Gas Purification Pressure Swing Adsorption

Narasimhan Sundaram

Geocenters Znc., 10903 Indian Head Highway, Fort Washington, Maryland 20744

An analytic solution for periodic steady state concentration profiles in a pressure swing adsorption purification process was obtained from the countercurrent flow analogy. The system is a dilute, isothermal, single component in inert carrier with a general nonlinear adsorption equilibrium isotherm. Diffusion in the solid phase is rate controlling and axial dispersion is neglected. The solution permits the calculation of minimum bed requirements for complete purification without iteration. The effect of isotherm shape, mass transfer, pressure ratio, and purge-feed ratio in the purification of air contaminated with common chlorofluorocarbons is studied theoretically.

The cyclic steady state is a useful way to characterize pressure swing adsorption (PSAJ processes. Suzuki (1985) has provided a simple technique to calculate the cyclic steady state for two-step PSA where the steps of pres- surization and blowdown are assumed to have negligible effect on concentration profiles and the system is iso- thermal. For negligible pressure gradients and axial dispersion, an analogy to the countercurrent flow (CCF) contactor allows the governing equations to be solved simultaneously for the feed (high pressure) step and the purge (low pressure) step. For air purification of trace contaminants these assumptions are reasonable. When cycle time is reduced, the CCF model is adequate for initial PSA design (Hirose and Minoda, 1986), although for small particles pressure gradients may not be negligible (Wankat, 1986).

Farooq and Ruthven (1990) have extended the CCF model to the case of bulk, binary gas mixtures where velocity is assumed to vary with adsorption. Jacobian analysis was used to aid convergence of the numerical integration. Levan and Croft (1991) have presented a general numerical scheme using direct determination to converge on the periodic steady state. A similar method has been suggested by Smith and Westerberg (1992). Numerical schemes will continue to be the method of choice in transient PSA evaluations as well as bulk, adiabatic separations and have also been used to obtain information on the multiplicity of steady states in PSA (Farooq et al., (1988) and thermal swing adsorption (TSA) (Levan, 1990).

With increased interest in the abatement of ozone- depleting compounds such as chlorofluorocarbons, ad- sorption based technologies such as PSA may offer energy efficient alternatives for these and other purification processes. This article presents an analysis of the periodic steady state based on the CCF model for representative challenges of such contaminants. Isotherm nonlinearity for heterogeneous adsorbents is accounted for. Pressure ratio, purge-feed ratio and mass transfer are investigated.

Development of the CCF Model

For a single adsorbing component assuming negligible dispersion and pressure gradients, isothermal operation, and dilute concentrations so that the system velocity is constant, we can write the material balances at the periodic state for high- and low-pressure steps as

In eqs 1 and 2, UH and UL are interstitial velocities and the factor j3 on the rate term accounts for the equivalence of the CCF model with the actual PSA cycle. 13 is given by FE/(FE + PU). When feed and purge steps are of equal duration, j3 = (1 - j3) = 112 (Suzuki, 1985; Yang, 1987). Following Farooq and Ruthven (1990), the mass-transfer rate can be written using the linear driving force (LDF) model as

Ski = k,[q* - q] dt (3)

Generally, solid diffusion is believed to be controlling in most PSA applications (Yang, 1987). The solid-phase film coefficient, k,, is written as

In Table I, k,values are reported for several systems. Some of these systems, such as system 9b (Hassan et al., 19851, may possess large contributions from axial dispersion. In this case, the LDF coefficient will be reduced to k* This technique has been used by Chihara et al. (1978). This "lumping" of effects has also been used by Ritter and Yang (1991b) for system 5.

k, = 15D/r2 (4)

Adsorption Equilibrium

Myers (1989) presents a number of useful isotherm expressions for microporous adsorbents, among which is the Toth expression. The Toth expression can be written as an implicit function in P or q but does not approach the limit of saturation vapor pressure at maximum loading. It possesses singularities in higher derivatives, and the parameters are temperature dependent. Nevertheless it is a useful expression for heterogeneous, microporous adsorbents.

Toth isotherm:

(5)

where m is the saturation loading in mol/kg, P is the adsorbate partial pressure in Torr, and 0 I t I 1. Figure 1 shows isotherm plots for different adsorbates displaying

m P ( b + P)l/t 4 =

0888-588519312632-1686$04.00/0 0 1993 American Chemical Society

Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1687

Table I. Toth Isotherm and Mass-Transfer Parameters Toth equilib LDF coeff

eq 5, T = 298 K eq 4 no. system' mol wt, g/mol t b, Torr' m, mol/kg k,, 8-1 kN, 8-1 ref 1 R 2 2 - B P L 86 0.2632 1.8091 18.1 a 2 R31&BPL 200 0.275 0.7794 5.97 0.012 b 3 R113-BPL 187 0.2237 0.2698 7.51 0.009 b 4 HEX-BPL 102 0.4467 0.0401 5.19 0.0024 b 5 DMMP-BPL 124 1.0 0.0077 3.95 0.005 C 6 C&-PIT 16 0.773 842.4 6.03 d 7a C&-BPL 16 1.0 4031.5 5.93 e 7b C&-BPL 16 0.9107 735.13 2.22 2x104 d, f 8a C2&-PIT 28 0.503 20.68 7.25 d Bb CzHd-PIT 28 1.0 470 4.31 g 8c C2&-PIT 28 1.0 1986 6.08 8 9a C2&-CMS 28 0.288 1.7875 6.04 5 x 1 0 6 d, f 9b C2&-5AZ 28 1.0 1.9836 2.33 0.19 h 10 HEX-SIL 102 0.4467 0.2242 10.0 11 HEX-SIL 102 0.4467 0.0401 1.0 12a HEX-BPL 102 1.0 0.0401 5.19 12b HEX-BPL 102 0.22 0.0401 5.19 a Sundaram et al. (1993). * Mahle and Friday (1991). Ritter and Yang (1991b). dvalenzuela and Myers (1989). e Cheng and Hill (1985).

f Chihara et al. (1978). 8 Sircar (1984). h Haesan et al. (1985). i R22, dichlorodifluoromethane; R318, perfluorocyclobutane; R113, trichlorc- 1,2-trifluoroethane; DMMP, dimethyl methylphosphonate; HEX, hexanol; BPL, PIT, activated carbons; 5AZ, 5A zeolite; SIL, silicalite; CMS, carbon molecular sieve (Takeda).

4 -

3 -

I ' I

!::+[+=+-I _ _ _ - _ _ _ _ _______._____. . - - . . - - - - - - -

' t 1 g t 1

.8 I

0 2000 4000 6000 8000 40000 12000 14000 16000 18000 20000 c - 4/.'

Figure 1. Isotherms for adsorbate-adsorbent systems 2-5 and 8-11 of Table I.

character from linear to very favorable. Table I lists the Toth equilibrium parameters and LDF rate coefficients along with the data sources. For example, systems 4,10, and 11 show how adsorbent character is changed, reflected by a change in maximum capacity and also by the different slopes. This could occur, for example, to activated carbon in the presence of water. Ritter and Yang (1991a) use isotherms similar to systems 4 and 11, to describe silicalite and activated carbon with a Langmuir form. System 8 shows the same data set (C2HrPIT) correlated using the Toth isotherm (8a) and the Langmuir plots for low- and high-pressure regions (8b,c) obtained by Sircar (1984). t = 1 assumes the adsorbent isnot heterogeneous, and Sircar (1984) has shown this may not be true for BPL carbon. System 12 shows parameters when t varies, keeping b and m equal to the values for system 4.

Boundary Conditions

specified by Ruthven (1984). Appropriate boundary conditions for PSA have been

for the high-pressure step:

for the low-pressure step:

CH(Z=L) C,(Z=L) = pressure ratio (7)

We treat the dilute, single-component system in the inert carrier such that the total pressure is constant at the high or low value during the feed or purge step, respectively. We therefore use concentrations in place of mole fractions. In eqs 1 and 2, the rate terms need to be specified. To do this, we first invoke the assumption of zero net accumu- lation in the solid phase at the periodic state. Therefore,

Combining eq 8 with eq 3, assuming that solid diffusion controls and using the frozen solid-phase approximation, we can write

With these rate terms and when feed and purge steps are of equal duration, eqs 1 and 2 become

Using equilibrium expressions such as eq 5, these coupled equations may be solved using a Runge-Kutta technique with or without Jacobian analysis. This numerical ap- proach has been used by Suzuki (1985) and Farooq and Ruthven (1990). Here the problem is treated differently.

1688 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993

Calculating the Periodic State Profiles Combining eqs 12 and 13 reveals

Assuming we are interested only in cases where the product concentration is zero, Le., purification of contaminated gas streams, then

CH(Z=L) = cL(z=L) = 0 (15) Integrating eq 14 gives

CL = 'YCH + C1 (16) Using the known boundary condition eq 7 and the purification condition, eq 15, gives C1 - 0 and

CL = aCH (17) where

Equations 12 and 13 can be decoupled using eq 17, and the following analysis will show how this can be integrated for the Toth isotherm, eq 5. Assuming the LDF coefficient does not depend on pressure, we can write eq 12 as

Using eq 5 and the ideal gas law with eq 19 gives

or

where

Therefore deu

Using eq 17 gives dC,

where K = a-t. Assuming short cycle time and low throughput PSA,

for an isothermal, trace, single-component system, it is possible to expand the power term in the square brackets of the denominator of eq 24.

We use only three terms in the expansion, and these are sufficient to describe the entire polynomial accurately for all the combinations of parameters we are studying. Thus

eq 24 can be rewritten as

where

TI = ($ + l)?

T, = (i + 1 ) ( i + 2)- 6

After some algebra, eq 25 can be integrated to give a relation between CH and Z that satisfies eqs 6 and 15.

Valenzuela and Myers (1989) have compiled single- solute adsorption equilibria data for several adsorbent/ adsorbate pairs and also present correlations using the Toth and Unilan isotherms. For the majority of systems, t is <1, a characteristic of heterogeneous adsorbents. In fact in subsequent multicomponent equilibria calculations, this is a required convergence condition. Also, l / t is generally not an integer. We conduct the following analysis for general t I 1. Equation 25 can be shown to consist of combinations of the integral

Since l i t 2 1, we can successively reduce the power in the integrand by removing powers of 4 + 1 from the numerator of eq 26. This is carried out until the remainder integral has a power such that 0 I -(l /t - n ) I 1. Therefore,

(27)

For example if t = 0.2237, l i t = 4.47, and n = 5. The remainder integral can be written as a hypergeo-

metric series (Gradshteyn and Ryzhik, 1980). It is easier to bound the integrand in eq 27 with the term l/($. The singularity at the upper limit of pure product is logarithmic in nature. Especially for strongly adsorbed and low vapor pressure (low b values in eq 5) compounds, the effect of this term on the bed length required for approach to pure product is minimized. Consider the special case, t = 1, or the Langmuir isotherm. The exact solution is

For example, selecting system 5 from Table I, with PR = 4, PF = 0.5, flow rate = 150 standard L/min (SLPM), and UH = 200 cmis, we can calculate the coefficient of Z on the right-hand side of eq 28 to be 77. Therefore the product concentration can be reduced by several orders of mag- nitude with no exceedingly large increase in the value of 2. Conversely, for systems 6 ,7 , or 8, with the same flow rate, we can expect the reduction of the product concen- tration to require large changes in 2, as we approach complete clean-up. Clearly, the value of the LDF coef- ficient directly affects Z in eq 25 or eq 28 (Yang, 1987).

Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1689 I I 4000 L

3600 - 3200 - 2800 - -. 2400

I 2000 k

j2O0 800 I 400 1

o c t l I

0 7 14 25 28 35 42 49 54 63 70 2 - C.

Figure 2. Periodic steady s t a b profiiea at end of feed step for systems 2,3,4, 12a, and 12b of Table I. PR and PF vary for system 4.

For increased mass transfer, the coefficient wi l l be larger and the sensitivity to the singularity further reduced.

The integration of eq 25 for the PSA feed step profile can now be accomplished,. given the basic form eq 27 and replacing the integrand mth the term l/#. To do this we can reduce eq 25 to various combinations of terms involving eq 27. This manipulation is simple, using partial fractions. For t = 0.503, l / t = 1.98, n = 2, the solution for CH with the three-term expansion we chose in eqs 24 and 25 is given in eq 29 where the first three terms on the left-hand side are also evaluated from #FEED to 0. There is no

a--1 t

iteration required, and eq 29 and similar expressions for other values of t will be used to study the effect of the important operating variables in the short cycle PSA process, namely, pressure ratio, purge-feed ratio, feed concentration, k,, and the shape of the isotherm expressed by t, b, and m.

In the following figures unless specified otherwise, the bed inner diameter = 5.3 cm, feed flow rate = 150 slpm, p~ = 520 kg/m3, t = 0.4, and T 298 K.

Results and Discussion In Figure 2, the periodic steady state profiies for systems

2,3,4, and 12a,b are shown for PR = 4, PF = 0.5, and a feed concentration at atmospheric pressure of 4000 mg/ m3. We use the values as reported in Table I. Figure 2 suggesta, for instance, that a 35-cm bed length of carbon could produce uncontaminated product for PR = 8, but would require an additional 25 cm at PR = 4 for system 4 at a PF = 0.5. Figure 3 shows the increased bed lengths required for different PR and PF values, when feed

1 - FLOW RATE . 150 SLPM P a . 4

- 9 - P F . 0 5

8 -

7 -

squares - numerical S O I U ~ I O ~ -

- .- r 3

1

2001 1 O ,10000 ,

25 50 75 100 125 150 175 200 225 250 1 - 0

Figure 3. Effect of feed concentrations on periodic steady state profiies for systems 3 and 4. Concentrations in mg/m* at 1 atm.

concentrations are increased. In Figures 2 and 3, we use the values of k , = k N , neglecting axial dispersion contri- butions. Following Chihara et al. (1978) the effect of axial dispersion may be studied by changing k , to k . In these calculations, therefore, the bed lengths should $e regarded as minimum requirementa for the PSA conditions chosen.

This is a theoretical study. However, a comparison to experimenta may be made with the work of Ritter and Yang (1991a,b). Some of their experiments were run with purge-feed fractions such that the contaminant DMMP appeared in the product. For these profiles eq 29 may not be applicable. If we are interested in complete clean-up and in using shorter cycles, then eq 29 may be applied. Ritter and Yang (1991a) have also compared silicalite and activated carbon as PSA adsorbenta to clean air contam- inated by DMMP. The model chosen was isothermal and with no axial dispersion. The LDF coefficient was used as a parameter. However, the Langmuir isotherm chosen did not match well with the measured isotherm data. Further, no maas-transfer information is available for DMMP on silicalite. In Table I, systems 4, 10, and 11 exhibit rectangular shape and differing capacities.

Equation 29 can be used to calculate L m , the bed length required to reduce the product concentration to an arbitrary degree. As discussed earlier, for low vapor pressure compounds, the bed length does not increase significantly for an order of magnitude reduction in concentration. We choose the ratio of the contaminant concentration in product to feed, ~ ( Z = L . , , ) / ~ F E E D , as 1VZ2 and calculate L m for this value. Figure 4 shows the variation of L, for systems 4,10, and 11 and represents the principal efficacy of eq 29 over the Runge-Kutta method of iterative solution. This is due to the analytic solution which permita a study of the effect of isotherm shape and operating parameters. For comparison, Figure 3 shows two numerical solutions obtained from the Runge- Kutta solution, for system 4. The iterative method of solution requires a guess of CL(Z=O). The integration of eqs 12 and 13 is then performed, and CL(Z=L) is checked to see if it satisfies eq 7. To generate the solutions presented in Figure 3, we used eq 17 to provide the initial guess, although the numerical method would require several trials before it arrived at this value. Any other value as the initial guess fails to satisfy eq 7. Equation 17, of course, is the starting point for the analytic development. As is seen in Figure 3, the numerical and analytic solutions are indistinguishable. To resolve the profiles at lower concentrations, an extremely small step size was required in the numerical solution. The analytic solution matches the numerical solutions even for these concentrations.

1690 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993

200 1 180 I-

140

120 1 100 1

FEED, FLOW

I 4000 rnplrn' ' RATE. IS0 SLPM 1

-1

4

- . 4 j + $0 --- 11

....

A

i -1

i

i -. 0 3 6 9 12 IS 18 21 24 27 30

PI...."I. ratio - PR Figure 4. Effect of pressure ratio and purge-feed ratio on L, for systems 4, 10, and 11.

i b4,.S FEED . 2'18 pDm o t 3 atm FLOW RATE. 0 7 4 SLPM LDF ccefl . 0 0 0 5 V s e c PR . 3

I 7 -

6 -

5 &

P -

3 -

:F I 1

0 6 4 128 192 2 5 6 32 3 8 4 4 4 8 512 576 6 4 z - n

Figure 5. Periodic steady state profiles for systems 4,5,8a, and Bb. PSA conditions for systems 4 and 5 from Ritter and Yang (1991b).

Figure 5 compares periodic state profiles for systems 4 and 5 at two different purge-feed ratios with the same pressure ratio and overall LDF coefficient. System 5 uses the Langmuir parameters from Ritter and Yang (1991b) for DMMP on BPL. For this system Figure 5 shows the CCF solution requires a length of about 4.2 cm, while Ritter and Yang (1991b, Figure 6) showed a steady state profile penetrated about 6.2 cm for PF = 0.41. They used a finite difference numerical scheme with the frozen solid-phase approximation. The difference in the profiles can be ascribed to the longer cycle times for which the numerical simulation was run. From Figure 5 and system 5, we also see that, at the lower PF value of 0.34, a 52-cm bed is required for complete clean-up, assuming the same k,g, value. In Figure 5, we also show the profiles for systems 8a and 8b, which are Toth and Langmuir forms for the same operating conditions. The difference in these profiles is due to the poorer Langmuir fit, which, according to Sircar (19841, severely underestimated the data. Since the calculations are sensitive to the value of k,g, , Figure 6 was calculated to show the effect on L, for systems 3-5. The variation of k, is over the same range used by Ritter and Yang (1991b).

Conclusions The complete clean-up condition has value in the

purification of contaminated gas streams. Under this condition, the PSA steady state is a useful design tool. For isothermal operation with short cycle times and dilute, low vapor pressure contaminants, this PSA steady state

A r . 3.4 FEED. 4000 mo/m' O t 1 otm FLOW RATE. 1.50 SLPM PR . 4 PF - 0 5

4 0 501 30 20 I '0 t n L

I r 5 FEED. 218 ppm FLOW RATE m 074 SLPM PR 3 P F . 0414

I 1 1 0 001 002 003 004 005 006 007 008 009 01

*. - 1/..s Figure 6. Effect of LDF ratecoefficient on L, for systems 3-5. PSA conditions for system 5 from Ritter and Yang (1991b).

can be rapidly estimated. Minimum bed requirements at different pressure ratios and purge-feed ratios are cal- culated. The method uses no iteration and may prove useful for evaluating existing PSA designs and the effect of changes in operating conditions on these designs. This analysis is also relevant when simply adding adsorbent to contain the wave may not be possible for PSA applications where bedsize restrictions arestringent, e.g., militaryfilters for warfare agents.

The Tothisotherm, ageneral form of nonlinear isotherm that is suitable for heterogeneous, microporous adsorbenta Myers (1989), is used. The Langmuir isotherm is a special case of this form.

The technique can also be extended to two components, where both adsorbing species compete, yet are dilute with respect to the inert carrier, The CCF model used shows that, with an initially clean bed, there is only one steady state for the two-step, isothermal PSA process, despite the nonlinear isotherm.

Nomenclature b = Toth isotherm constant, torrt C = gas-phase concentration, mg/m3

C1 = integration constant, eq 16 D = micropore diffusivity, cm2/s FE = feed step duration, s i = integer counter, eq 27 I = integral, defined in eq 26 k, = linear driving force (LDF) rate coefficient, s-l K = defined in eq 24 L = bed length, cm L, = bed length for approach to complete clean-up, cm m = Toth isotherm saturation loading, mol/kg MW = molecular weight of adsorbate, g/mol n = integer used in eq 27 P = partial pressure of adsorbate, Torr PF = purge-feed fraction, eq 18 PR = pressure ratio PU = purge step duration, s q = adsorbed-phase concentration, mol/kg q* = adsorbed-phase concentration in equilibrium with

r = particle radius, cm R = gas constant = 0.06236 Torr ma/(rnol/K) t = Toth isotherm constant, 0 I t I 1 T = temperature, K TI, T2 = defined in eq 25 U = interstitial velocity in an actual PSA cycle, cm/s

= dimensionless gas-phase concentration, eq 22

C, mol/kg

Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1691

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Mahle, J. J.;Friday,D.K. Axiddupmioneffectaon the breakthrough behavior of favorably adsorbed vapors. In Adsorption Processes for Gas Separation, Meunier, F., Levan, M. D., Eds.; CNRS-NSF Gif-Yvette, 1991; Vol. 5, No. 17, p 157.

Myers, A. L. Theories of adsorption in micropores. In Adsorption: Science and Technology, Rodrigues, A. E.,Levan, M. D., Tondeur,

D., EMS.; Kluwer: Dordrecht, 1989; pp 15-17. Ritter, J. A.; Yang, R. T. Air purification and vapor recovery by

pressure swing adsorption: A comparison of silicalite and activated carbon. Chem. Eng. Commun. 1991a, 108,289.

Ritter, J. A.; Yang, R. T. Pressure swing adsorption: Experimental and theoretical study on air putiftcation and vapor recovery. Znd. Eng. Chem. Res. 1991b, 30,1023.

Ruthven, D. M. Principles of adsorption and adsorption processes; Wiley: New York, 1984, p 365.

Sircar, S. Adsorption of gases on heterogeneous adsorbents. J. Chem. SOC., Faraday Trans. 1 1984,80, 1101.

Smith, 0. J. nT; Westerberg, A. Acceleration of cyclic steady state converaenceof meesureswinx adsomtionmcdels. Znd . Em. Chem.

Z = axial distance, cm Greek Letters a = feed to purge velocity ratio f l = FE/(FE + PU), ratio of feed, purge step durations e = bed voidage p = bed density, kg/m3

= defined in eq 25

Subscripts H = high pressure, feed step L = low pressure, purge step 0 = overall value FEED = feed value B = bed value PRODUCT = product value

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eauilibria of chloro-fluorocarbon refrigerants on activated carbon”;

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Technical report in preparation, 1993;ERDEC, Aberdeen Proving Ground, MD.

Suzuki, M. Continuous counter-current flow approximation for dynamic steady state profile of pressure swing adsorption. AZChE Symp. Ser. 1985,81 (242), 67.

Valenzuela, D. P.; Myers, A. L. Adsorption equilibrium data handbook; Prentice-Hall: Englewood Cliffs, NJ, 1989.

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Yang, R. T. Gas separation by adsorption processes; Butterworths:

Received for review January 22, 1993 Revised manuscript received April 23, 1993

Accepted May 5,1993

Stoneham, MA, 1987.