Ind. Eng. Chem. Res. 1995,34, 2655-2660
2655
Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment M. Douglas LeVan Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442
Local equilibrium theory allows us to determine many interesting features involving the application of pressure swing adsorption for purification, enrichment, and solvent recovery. I n this paper, a simple local equilibrium model is constructed and solved analytically for the periodic state and its development. A single component is adsorbed and the adsorption isotherm is nonlinear. We consider the effect of isotherm nonlinearity, the minimum bed depths required for operation a t the critical volumetric purge-to-feed ratio and at higher purge-to-feed ratios, the direct determination of periodic states, the development of periodic states, and the extent of accumulation of adsorbate in beds for favorable isotherms.
Introduction Solvent recovery by pressure swing adsorption (PSA) is a fairly new technology. It is applied t o raise the gasphase mole fraction of the solvent by eliminating much of a diluting, inert carrier gas. The process cycle involves the purification of a large-volume gas stream followed by enrichment of the solvent in a small-volume purge stream. The solvent can then be recovered economically by some other means, such as condensation. Purification processes are similar; they involve enrichment but lack the final recovery step. Local equilibrium theory, developed in the 1940s and enhanced considerably by Helfferich and Klein (1970),can be used t o shed considerable light on the performance of these processes, as is shown in this paper. These processes differ considerably from typical PSA bulk gas separation processes. For purification and enrichment, followed possibly by recovery, the feed is generally dilute in the adsorbable component, and cycle times are typically longer than those encountered in bulk gas separation. Moreover, in purification and enrichment applications, adsorption occurs to high loadings with rapid rates, and effects of pressurization and depressurization are diminished because of the small fraction of the cycle time over which they occur. Thus, with reasonable accuracy, the cycle can be treated as two constant-pressure steps: a high-pressure feed step and a low-pressure purge step. Relatively few papers have been published on properties of complete adsorption cycles, in comparison to the large number published on behavior of individual cycle steps. For the former, regeneration is incomplete; at the end of regeneration, a heel of adsorbate remains in the bed. Some of the early research on properties of complete cycles contrasted efficiencies of cocurrent and countercurrent regeneration in purge sweep cycles for linear isotherms (Tan and Spinner, 1971, 1972) and nonlinear isotherms (Bunke and Gelbin, 1975, 1978; Gelbin and Bunke, 1979; Ortlieb et al., 1981; Gelbin et al., 1983). With increased interest in PSA, numerous studies have been published on global properties of cycles, evaluating product purity, product recovery, exhaust enrichment, and adsorbent productivity. Matz and Knaebel(1988) applied local equilibrium theory to a PSA cycle to examine effects of incomplete purge for separation of two components with adsorption equilibrium described by linear isotherms. Kayser and Knaebel (1989) considered a cycle with uncoupled, nonlinear isotherms for the case of complete purge, in which
composition fronts pass completely through beds during cycle steps. The PSA study most closely related to this paper is that of Ritter and Yang (19911, who adsorbed dimethyl methylphosphonate on activated carbon in a PSA cycle for air purification and vapor recovery. We will adopt many of their assumptions in this paper. For some time we have been interested in the evaluation of periodic states (cyclic steady states) of adsorption cycles. Some of these efforts have been largely analytical (Davis and LeVan, 1987), as is the present paper, and some have been numerical (Davis et al., 1988; LeVan, 1990; Croft and LeVan, 1994ab). This paper applies local equilibrium theory to a simple PSA cycle for purification and enrichment to determine properties of the periodic state. The theory allows us to determine the periodic state directly, with no iteration, and provides a simple, powerful method to understand the effects of changes in process parameters. We examine the effect of isotherm nonlinearity, minimum bed depths required for operation at various volumetric purge-to-feed ratios, and the development of periodic states.
Underlying Theory We consider a PSA cycle with adsorption of one component present in the high-pressure feed at low mole fraction. The low mole fraction allows us to assume that the cycle operates isothermally with velocities that are independent of composition. Purge is countercurrent with pure carrier gas. For adsorption of a solvent-type species, the partition ratio is large and fluid-phase accumulation of the adsorbing species is negligible. Furthermore, because feed and purge steps can be much longer than pressurization and blowdown steps for this cycle, we assume that concentration profiles do not change during the pressurization and blowdown steps. We describe adsorption equilibrium using the constant separation factor isotherm. The problem just described mimics that considered by Ritter and Yang (1991);they treated adsorption rates using a linear driving force approximation and solved their mathematical model numerically. In contrast, we assume that local equilibrium between adsorbed phase and fluid phase exists at each cross section in the bed. Furthermore, there is no dispersion and axial pressure gradients are negligible. The material balance on the adsorbable component is
0888-5885/95/2634-2655$09.0Qf0 0 1995 American Chemical Society
2656 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
We let the interstitial velocity u be positive for the feed step and negative for the countercurrent purge step. Appropriate dimensionless variables are
c*
= Ci/Cref
I; ZIL z
€lUlt/A.L
(3)
(5)
(6)
Here, Cref is the concentration of the adsorbable component in the high-pressure feed, qref is the corresponding adsorbed-phase concentration, A is the partition ratio, and z is a dimensionless time. We let L be the stoichiometric depth of penetration of the adsorption front into a clean bed for the first feed step. Because of this definition, for a feed step z will run from 0 to 1.(And by setting z = 1in eq 6, L can be calculated.) Similarly, for the purge step, z will run from 0 to y , the volumetric purge-to-feed ratio. Thus, beginning with a clean bed, the adsorption front penetrates to 5 = 1on the first feed step if the isotherm is linear or favorable (concave downward) and deeper if the isotherm is unfavorable (concave upward). For any other feed step for a favorable isotherm, the depth of penetration relative t o that for the first feed step will be equal to the value of 5 reached by the front. Introducing these variables into the material balance, and assuming that the partition ratio A is much greater than unity and, consequently, that the fluid-phase accumulation term can be neglected, gives
(7) where, because of the reversal in flow direction, the plus sign corresponds to the feed step and the minus sign to the purge step. Two types of solutions exist to this equation. Application of the method of characteristics gives the solution for a simple wave (gradual wave)
again with the plus sign corresponding to the feed step. It should be noted that the inverse of dddc is a dimensionless concentration velocity. If this solution is not physically plausible, then the solution is a shock (abrupt transition) described by (9)
where the differences are taken between endpoints of the shock. A combined wave can exist with both gradual and abrupt parts, and a shock can capture a gradual wave, as shown below. The constant separation factor isotherm is given by (Vermeulen et al., 1984)
'*
=R
+
-* L" (1- R)c*
(10)
0 0
0.2
0.4
0.6
0.8
1.0
C'
Figure 1. Constant separation factor isotherms (eq 10). Isotherm classifications are favorable ( R l),linear ( R = 11, and unfavorable ( R > 1). The dashed line is the slope dq*/dc*l+1 shown for R = 0.1. The inverse of this slope gives the depth of penetration of adsorbate into the bed at the end of the feed step for the periodic state at the critical volumetric purge-to-feed ratio (eq 14).
Several isotherms, corresponding t o different values of the separation parameter R , are shown in Figure 1.For R < 1 the isotherm is favorable, for R = 1 it is linear, and for R > 1 it is unfavorable. Solely on the basis of eq 6 above, we can assess the effect of changes in the capacity of the adsorbent on periodic behavior. Capacity is reflected only in the partition ratio A which is contained in the dimensionless time t. On the basis of the definition oft, we reach the obvious conclusion that if the partition ratio were increased, say doubled, then either the cycle step times could be doubled or the bed depth could be halved.
Direct Determination of Periodic States To simplify the analysis, we consider the bed t o be long enough to recover all of the solvent. The feed enters the bed at 5 = 0; the purge enters at the other end of the bed, which will be at a large value of 5. Furthermore, we will assume now, and later show, that for the case of the favorable isotherm the bed contains only a shock at the end of the feed step and only a simple wave at the end of the purge step. Construction for Purge Step for Favorable Isotherms. To solve for the periodic state directly, we begin by assuming that the cycle is operating at the periodic state. Referring to Figure 2, the feed passes from left to right, and at the end of the feed step, the bed contains the shock located at 5a. Within the region 0 < I; < the bed is loaded uniformly at q* = 1. The clean purge enters from the right. Equation 8 is solved to give the characteristics shown in the bottom of Figure 2. Along each of these, concentrations q* and c* are constant. Stopping the characteristics at t = y gives the concentration profile labeled q d in the top of the figure. The shaded region labeled "Area = A" indicates the amount of adsorbate removed from the bed during purge. This area is given by
ca,
A = L ' 5( l - q $ ) d
