Ind. Eng. Chem. Res. 1993,32, 1686-1691
1686
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 (PSAJprocesses. Suzuki (1985) has provided a simple technique to calculate the cyclic steady state for two-step PSA where the steps of pressurization and blowdown are assumed to have negligible effect on concentration profiles and the system is isothermal. 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 ozonedepleting compounds such as chlorofluorocarbons, adsorption 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.
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-phasefilm coefficient, k,, is written as k, = 15D/r2 (4) 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.
Adsorption Equilibrium
Development of the CCF Model
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:
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
(5) ( b + P)l/t where m is the saturation loading in mol/kg, P is the adsorbate partial pressure in Torr, and 0 I t I 1. Figure 1shows isotherm plots for different adsorbates displaying
0888-588519312632-1686$04.00/0
4=
mP
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 eq 5, T = 298 K b, Torr' 1.8091 0.7794 0.2698 0.0401 0.0077 842.4 4031.5 735.13 20.68 470 1986 1.7875 1.9836 0.2242 0.0401 0.0401 0.0401
LDF coeff eq 4 k,, 8-1 kN,8-1
mol wt, g/mol t m,mol/kg ref 18.1 a 86 0.2632 R31&BPL 200 0.275 5.97 0.012 b R113-BPL 187 0.2237 7.51 0.009 b HEX-BPL 102 0.4467 5.19 0.0024 b DMMP-BPL 124 1.0 3.95 0.005 C C&-PIT 16 0.773 6.03 d C&-BPL 16 1.0 5.93 e C&-BPL 16 0.9107 2.22 2x104 d,f C2&-PIT 28 0.503 7.25 d CzHd-PIT 28 1.0 4.31 g 8c C2&-PIT 28 1.0 6.08 8 9a C2&-CMS 28 0.288 6.04 5x106 d,f 9b C2&-5AZ 28 1.0 2.33 0.19 h 10 HEX-SIL 102 0.4467 10.0 11 HEX-SIL 102 0.4467 1.0 12a HEX-BPL 102 1.0 5.19 12b HEX-BPL 102 0.22 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, trichlorc1,2-trifluoroethane;DMMP, dimethyl methylphosphonate; HEX, hexanol; BPL, PIT, activated carbons; 5AZ, 5A zeolite; SIL, silicalite; CMS, carbon molecular sieve (Takeda). no. 1 2 3 4 5 6 7a 7b 8a Bb
I
I-+=[: !
't gt
system' R 22-BPL
'
I
1 1 _ _ _ - _ _ _ _
6000
(7)
.8
I
4000
CH(Z=L) pressure ratio
We treat the dilute, single-component system in the 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 1and 2, the rate terms need to be specified. To do this, we first invoke the assumption of zero net accumulation in the solid phase at the periodic state. Therefore,
3 -
2000
C,(Z=L) =
that the total pressure _ _ _ _ _ _ _ . _ _ _ _ _ . . - inert - . . -carrier - - - - -such -
4-
0
for the low-pressure step:
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 11show 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 = 1assumes 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.
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
Boundary Conditions Appropriate boundary conditions for PSA have been specified by Ruthven (1984). for the high-pressure step:
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 approach 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
eq 24 can be rewritten as
Combining eqs 12 and 13 reveals
Assumingwe 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
-
TI =
(16)
Using the known boundary condition eq 7 and the purification condition, eq 15, gives C1 0 and CL = aCH
where
(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
T, =
($ + l)?
(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 singlesolute 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
