Equilibrium Theory for Pressure-Swing Adsorption. 4. Optimizations for

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Ind. Eng. Chem. Res. 1998, 37, 2516-2528

Equilibrium Theory for Pressure-Swing Adsorption. 4. Optimizations for Trace Separation and Purification in Two-Component Adsorption Giuseppe Pigorini Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442

M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Box 1604, Station B, Nashville, Tennessee 37235

This paper applies local equilibrium theory to analyze pressure-swing adsorption with short dimensionless contact times for systems with two adsorbable components present in trace amounts in an inert carrier gas. We are interested in finding optimal operating conditions for both separation and purification problems with nonlinear isotherms. We consider Langmuirian systems only; however, several concepts illustrated in this paper apply irrespective of the type of isotherm. By minimizing appropriate objective functions, at the periodic state and for the purification process, the purge step is found to be of short duration, corresponding to low volumetric purge-to-feed ratios and a high degree of incomplete regeneration. In the separation process, higher or lower degrees of regeneration may be appropriate depending on whether the goal is to enrich the purge in the impurities or to recover the light component as product. Introduction Pressure-swing adsorption (PSA) is an increasingly popular method for separating or purifying gas mixtures. It is commonly used in light gas separation and in the drying of air. Newer applications include solvent recovery and ultrapurification. In recent years, it has been recognized that pressure-swing adsorption can be extended broadly to adsorbates of moderate and low volatility, including mixed adsorbates. For purifications, recovery of mixed solvents used in spraying, coating, and drying operations can be accomplished by pressure-swing adsorption. For separations of components in carrier gas, pressure-swing adsorption can be employed as shown in our previous paper (Pigorini and LeVan, 1997). Economically viable systems are expected to be those in which the components are originally present in the vapor phase or those for which distillation is difficult because of low relative volatilities (e.g., mixed xylenes) or true azeotropic mixtures. For these, pressure-swing adsorption can provide a competing technology for the temperature-swing adsorption and simulated moving-bed liquid-phase adsorption processes currently used. Pressure-swing adsorption relies on the physical phenomenon that at a certain pressure different gas species are attracted and retained on the surface of an adsorbent material with differing concentrations in such a way that separation of a mixture can be attained between the gas and adsorbed phases (Ruthven, 1984; Yang, 1987). During the high-pressure step of a PSA cycle for separation or purification, the gas mixture is fed to the adsorption bed and some degree of saturation of the adsorbent material with the adsorbates occurs. During * Author to whom correspondence should be addressed. Telephone (615) 322-2441. Fax: (615) 343-7951. E-mail: [email protected].

the subsequent regeneration step conducted at a lower pressure, the bed is purged with light product in inert gas or pure inert gas and the adsorbates are partially removed from the bed. However, since the partial pressures of the adsorbates are related to the adsorbedphase concentrations, the partial pressure of the inert gas is much lower during the purge step. Thus, the gasphase mole fractions of the adsorbed species increase with respect to those in the feed mixture, and the gas phase is enriched in those components (Ruthven et al., 1994). This series of papers applies local equilibrium theory to PSA cycles. The principal assumption in equilibrium theory is that at each cross section in the bed equilibrium exists between the gas and adsorbed phases, so that gas-phase and adsorbed-phase concentrations are related through adsorption isotherms. This assumption implies that mass transfer between phases is infinitely fast, i.e., that there are no resistances to mass transfer. Although this assumption is never entirely true, it is, nevertheless, reasonable in many instances of practical interest. Also, while making the mathematical analysis of the problem easier, it provides an asymptotic result that is useful in understanding the dynamics of the cycle. Throughout this series we have applied the method of characteristics to solve the material balances for the adsorbed species. This has allowed us to gain better insight into how the concentration profiles in the bed evolve in time and space and to derive analytical contributions to the solution of the systems we have studied. In part 3 of this series (Pigorini and LeVan, 1997) we analyzed separation and purification problems in pressure-swing adsorption for a system of two adsorbable components present in trace amounts in an inert carrier gas. For the separation problem the light component breaks through and is present in the product during

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Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2517

much of the feed step. For the purification problem both components are retained in the bed with no elution of either one; i.e., the product is pure inert carrier gas. For both problems the goal in part 3 was to find a priori the periodic configuration of the system. In part 3, for the separation problem the bed was uniformly loaded in the two adsorbates at the end of the feed step. During the purge with pure inert gas, two zones of varying concentration separated by a plateau of both components formed. Each of these zones corresponded to a simple wave of different type. For the purpose of part 3, the bed was purged until one of the two waves completely exited the bed and the concentration profile consisted of a plateau adjacent to a region of nonuniform concentration. The bed was subsequently fed countercurrently, and two shocks eventually formed, with light component rolled up between them and retrieved as the product. In the purification problems considered in part 3 the feed was discontinued when the fast-moving shock had exited the bed, leaving a blocklike concentration profile in the bed. Purge with pure inert gas was carried out until the interaction between the two simple waves was resolved and only one simple wave was left in the bed. Depending on the set of parameters, we showed that the component that was left in the bed at the end of the purge step once the cyclic periodic state was attained could be either the light or the heavy component. This paper deals with separation and purification problems where we do not impose any limitation on the duration of the purge step. In other words, we do not require that one of the simple waves that sweeps through the bed during the purge step exits the bed before we start the feed step. Rather, we carry out the purge step for as little a time as is considered suitable. The fact that we are not constrained by the duration of the purge step means that our analysis can be made more general because this additional flexibility allows us to explore different purge-to-feed ratios and to search, among all possible cycle configurations, for the cycle that is economically optimum. We will show that in both problems it is possible to choose the purge time in such a way that the economic return is maximized. Given that the feed step is carried out in the separation and purification problems until respectively the slow or the fast shock just breaks through, what really makes the difference is the purge step. In this paper, we allow for the purge to be discontinued when both of the simple waves are still partially in the bed. In the purification problem, the interaction between simple waves may not be resolved, in which case, when the bed is fed countercurrently with the feed mixture, it is necessary to treat the interaction between a stream of constant composition and a nonuniform initial concentration profile created by this interaction of simple waves. As of now no papers have been published that address this type of interaction. In part 3 of this series, for instance, the method of characteristics was used to analyze the interaction of a feed of constant composition and a nonuniform concentration profile, which was due to one simple wave only, however. That analysis owed much to the analysis of Rhee et al. (1989). Mathematical Problem Material Balances. The feed consists of two adsorbable components in inert gas. We restrict our

analysis to adsorbates that are present in trace amounts and adsorb at least moderately strongly, meaning partition ratios are large compared to unity. As in the previous parts, we consider an isothermal bed, as is appropriate for separation and purification processes involving trace concentrations. This allows us to simplify the material balances since the gas velocity in the bed can be assumed constant and no accumulation term in the fluid phase is necessary. We assume that, compared to feed and purge steps, the pressurization and blowdown steps are of comparatively much shorter duration and of less impact on the bed profile, based on the fact that we deal with systems characterized by large partition ratios (typically greater than 102); thus, the volume of gas involved in the pressurization and blowdown of the bed is very small compared to that used during the feed and purge steps (LeVan, 1995). We nondimensionalize the material balances following what was done in part 3. We obtain

∂q* ∂c* 1 1 ( )0 ∂τ ∂ζ ∂c* 2 2 1 ∂q* ( )0 η ∂τ ∂ζ

(1)

ζ is a dimensionless axial coordinate ranging from 0 at the bed inlet for feed to 1 at the bed outlet, and τ is a dimensionless time analogous to a throughput parameter. The ( sign accounts for the fact that we reverse the direction of flow in the bed every half cycle, with velocity changing sign from positive for feed to negative for purge. Variables are defined in the notation. Equilibrium between phases is described using the binary Langmuir isotherm which has the dimensionless form (Pigorini and LeVan, 1997)

q*1 )

c* 1 R1 + (1 - R1)c* 1 + (R1/R2)(1 - R2)c* 2

(2)

where the constant separation factor Ri is given by

Ri )

1 1 + Kici,ref

(3)

We limit the analysis to isotherms that are concave downward, i.e., with Ri < 1. Our system will be entirely characterized once three parameters are provided: the two constant separation factors Ri and the ratio of partition ratios η ≡ Λ1/Λ2. The component with the lower value of the ratio Ri/Λi will be referred to as component 1 or the heavy component. The selectivity between species 1 and 2 is constant and is given by

R1,2 )

q1 c2 Q1K1 Λ1 R2 ηR2 ) ) ) >1 c1 q2 Q2K2 R1 Λ2 R1

(4)

Because of the way τ is defined, if we were to feed heavy component in inert gas into a clean bed, breakthrough would occur at τ ) 1. Numerical Solution. We solve eq 1 by the method of characteristics. This involves mapping the solution between a hodograph plane (c* 1 vs c* 2) and a physical plane (τ vs ζ). This mapping is instrumental in showing how the concentration profiles in the bed at different times form as a consequence of wave interactions as we feed or purge. The background for this approach is

2518 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998

given in the works by Courant and Friedrichs (1948) and Rhee et al. (1989), and some details can be found in part 3 of this series. We introduce the dependent variable vector

()

c* 1 Y h ) c* 2

(5)

in order to be able to put eq 1 in the array form

C

∂Y h ∂Y h + )0 ∂τ ∂ζ

where C ) B-1A

(6)

with arrays A and B defined in part 3. We determine the eigenvalues σ+ and σ- of array C, with the convention that |σ+| > |σ-|, and the left eigenvectors, which satisfy the condition hl(C ) σ(hl(. The two sets of characteristics in the physical plane are referred to as C+ and C-, with their slopes in that plane given by

dτ/dζ ) σ(

along C(

(7)

In the hodograph plane a Γ( characteristic is the image of a C- characteristic from the physical plane. The slope of a Γ( characteristic is, in general, given by

n( dc*1 )dc* m 2 (

(8)

where hl( ) (m(, n(). Equations 7 and 8 allow us to trace characteristics in the two planes and in doing so to generate the concentration profiles in the bed at different times. Because of the Langmuirian nature of our system, the Γ( are straight lines. When simple waves cannot exist physically, shocks form. Their slopes in the physical plane are determined by performing a material balance around the region of discontinuity in concentration and are given by 1 2 1 ∆q* dτ ∆q* ) ) dζ ∆c* η ∆c* 1 2

(9)

Due to the Langmuirian nature of the system, the image in the hodograph plane of a C( characteristic and of a S( shock coincide exactly and we refer to both as the Γ- characteristic. We determine the periodic position of the shock by performing an analysis suggested by Davis et al. (1988). We introduce an “effluent ratio” E defined by

E)

mol of 2/mol of 1 removed during purge c* 2/c* 1 in feed

(10)

This monitors the ratio of the amount of each adsorbate removed during the purge step at different purge times, for a given initial concentration profile in the bed, relative to the composition of the feed stream. The idea is that, at the periodic state, in order for the overall cyclic material balances on the two species to be satisfied, the amounts of the two components removed during the purge step must be in the same ratio as the concentrations in the feed, or, in other words, E must equal unity. In part 3 we illustrated the types of wave interactions that take place in the problems studied there. In this paper, in the purification problem, an additional, more complex type of interaction occurs and is the topic of the next section.

Theory for Interaction between a Constant State and a Hybrid Wave. The main mathematical contribution of this section is the treatment of how a shock path is affected by a region of interaction of simple waves. We are unaware of any prior treatment of such a problem. This discussion is necessary in order to address the purification problem, which in terms of mathematical complexity is more difficult to solve than the separation problem. When simple waves interact in the physical plane, the characteristics become distorted and do not have a straight path. In part 3 it was straightforward to locate in the hodograph plane the point representative of the feed and the Γ wave representative of the concentration profile in the bed at the end of the purge step. Once that was done, it was possible to determine the speed of the shocks that formed during the feed step by successively considering the concentrations the feed was interacting with and that lie along the initial concentration wave in the hodograph plane. However, if the interaction between the two simple waves is still occurring and we stop purging and start feeding countercurrently, waves are not as clear in the hodograph plane. For this case it is not possible to determine a priori, based only on the concentration profile in the bed at the end of the purge step, the concentrations that the feed is going to interact with before actually finding the shock path. The explanation for this is that the shocks that form when we feed proceed across the region of wave interaction and progressively sweep through the bed, but meanwhile the wave interaction itself evolves in such a way that, as we keep feeding, the concentrations that are found in the bed and that the shocks interact with are not the same as those at the beginning of the feed step. In the purification problem that we illustrated in part 3, we purged until only one simple wave was left in the bed. Each characteristic for this type of wave is a straight line in the physical plane and carries a constant concentration until the shock catches up with it. As we feed the bed, each characteristic of the wave travels in the physical plane and at each time the spatial location of the concentration that it is carrying changes. However, the set of concentrations that are carried by the entire wave does not change; only the positions of the concentrations in the set change. Since no interaction is initially taking place, there is no reason for concentrations different from the ones carried by the wave to appear in the bed. The interaction between simple waves is reversible as we start feeding and exists for as long as the faster shock does not capture the waves. Therefore, the portion of the interaction region that is not yet captured by the faster shock evolves as we feed and produces concentrations different from the ones that were in the bed at the beginning of the feed step. Because of that, we do not know in advance, before the feed step starts, what concentrations the feed stream will be interacting with, and we are unable to locate the interacting waves in the hodograph plane a priori. It is therefore necessary to interactively use the hodograph and physical planes at each step of the calculation in order to generate the shock paths in the physical plane and the Γ wave in the hodograph plane. In part 3 this did not occur because the interaction had been resolved. A general treatment of this interaction follows. Suppose that the feed is represented by point F in the


a

b

Figure 1. Hodograph plane illustrating the two possible ways of interaction of a hybrid wave with the characteristic net in the purification problem: (a) hybrid wave intersecting a Γ-; (b) hybrid wave intersecting a Γ+.

hodograph planes shown in Figure 1. Also, suppose that the concentration profile in the bed at the end of the purge step is the result of a region of interaction between two simple waves of different type, i.e., C+ and C-. This region of interaction is a characteristic net, and the points where characteristics of different types meet are nodes of the net. As we discussed in part 3, the image of a C+ (C-) wave in the physical plane is a Γ- (Γ+) wave in the hodograph plane. Let point H in Figure 1 represent the composition of the effluent stream at the end of the purge step, i.e., the first composition that the feed interacts with. Since the region in the physical plane close to ζ ) 0 is where the interaction of waves is taking place, H does not really belong to a pure simple wave. According to the procedure outlined in part 3, as F interacts with H we expect two shocks to form, with the composition G between the two shocks being given by the intersection in the hodograph plane of the Γ+ through H and the Γthrough F. The first portion of the S+ shock will connect compositions F and G, whereas that of the S- shock will connect compositions G and H. The shock speeds that are determined by applying the shock condition (eq 9), allow us to draw a small portion of the path of each

shock in the physical plane. As we start feeding, the region of interaction of the two simple waves in the physical plane is altered in that, for times greater than the time corresponding to the end of the purge step and until it is captured by the shocks, that region is the mirror image in the physical plane of the interaction during the purge step. This is similar to what was shown in part 3; each of the slopes of the characteristics not yet captured by the shock has the same absolute value that it had during the purge, with only the sign changing because of the change in the direction of the flow. We consider the intersection of the small segment of the fast shock that we have drawn with the first characteristic in the net that it encounters. The approach to the solution consists of considering the two interacting C- and C+ waves as being decomposed into a collection of segments of C- and C+ characteristics between the nodes of the net and of treating each of these fragments according to its type as if it were part of a pure wave of that type. In other words, as we generate the shock path in the physical plane, we successively consider the interaction of the shock with the appropriate portions of C+ and C- characteristics that it intersects with as it proceeds. Two possibilities exist for an intersection. If the segment intersected is part of a C+ (Γ-) characteristic, as represented by segment AB of the hodograph plane in Figure 1a, and we call the intersection P1, then that characteristic is transmitted by the S- shock and the slopes of both shocks are affected. In fact, if we draw the Γ+ characteristic through P1 and the Γ- characteristic through F and call their intersection G1, in general G1 will be different from G (in addition to the fact that P1 is different from H). If instead the segment intersected is part of a C- (Γ+) characteristic, as represented by segment AB of the hodograph plane in Figure 1b, and we call the intersection P2, then that characteristic is canceled by the S- shock, but still the slope of both shocks is affected because P2 lies on a different Γ+ than H. From the theory outlined in part 3, only a transmitted characteristic C+ can affect the slope of the S+ shock. However, in this problem, when we move from H to either P1 or P2, we are not moving along a pure Γ+ or a pure Γ- wave but rather along a hybrid Γ wave which is a blend of the two and which could therefore be decomposed into a stepwise path along a succession of portions of Γ+ and Γ- characteristics. This is the reason the slope of both shocks is actually affected by the Ccharacteristic. This complex hybrid Γ wave has the character and properties of both a Γ+ and a Γ- since it is the interaction of two waves, one from each of the two families of waves. It is a curved line, unlike pure Γ+ and Γ- simple waves, which, for Langmuirian systems like the one we are studying, are straight lines. The procedure that we have followed in dealing with the interaction between the feed and the hybrid Γ wave is, therefore, as follows. When the S- shock crosses a C+, the slopes of both shocks are affected. The local slopes of the two shocks can be evaluated directly once the intermediate concentration G1 is found. When the S- shock crosses a C-, the slope of the S- shock changes and can be determined once the intermediate concentration G2 is found. The slope of the S+ shock changes too, however, according to the discussion above; since the C- that is responsible for that change in slope is


canceled by the S- shock, we take as the slope of the S+ shock the average of the slope at this C- and the slope corresponding to the concentration intercepted on the next C+ that it cancels. The average is extended to more than one slope if more than one C- is intercepted before the shock intersects a C+. Once the shocks have swept through the region of wave interaction, they still interact with a portion of the simple wave before the blocklike initial concentration profile in the bed is restored. That portion of simple wave only carries one component and has its image in the hodograph plane lying along the axis corresponding to that component. At this point, the analysis simplifies to the one outlined in part 3.

(11)

where c*2F ) 1 because of the way we have nondimensionalized concentrations and c*2G is given by eq 36 of part 3. c*2G is a function of R1, R2, and η, so that once those parameters are assigned, its value is fixed and can be determined a priori. In order to derive the expression for the enrichment of the heavy and light components, we perform a material balance on each of them. We can say that all the heavy component that is removed from the bed during the purge step has to be supplied during the feed step, or

jpurge τ1 cfeed 1 (τ3 - τ1) ) c 1

Examples In this section we illustrate a cycle for separation and a cycle for purification and show how the dimensionless duration of the cycle and other operative parameters affect the three objective functions commonly used in the literature and in industrial practice to gauge the efficiency of an adsorption process, notably recovery, enrichment, and productivity. Recovery of a species is defined as the ratio of the net quantity of pure species produced to the total quantity of species fed to the process during a cycle (Knaebel and Hill, 1985). Enrichment of a species is defined as the ratio of the average mole fraction of the species in the purge effluent to the mole fraction of the species in the feed (Knaebel and Hill, 1985). Adsorbent productivity is defined as the flow rate of product per unit mass of adsorbent (Kayser and Knaebel, 1989). Separation Problem. We are interested in finding the optimal operating conditions for a separation problem. We know that it is not necessary to completely remove one of the waves from the bed in order for the cycle to separate some of the light component from the feed. Reducing the purge time and therefore the volumetric purge-to-feed ratio γ often moves the system toward an economic optimum because less pure inert carrier gas needs to be used. When measured in terms of τ, the purge time must be equal to or greater than the feed time. In the example illustrated in part 3 the purge time was much longer than the feed time because it was difficult to completely remove the fast simple wave from the bed due to isotherm nonlinearity. In this example we have chosen the duration of the purge step based on economic considerations as will appear from the following discussion. We consider only regeneration with inert. Regeneration with light component product would give a single transition system. We can consider three dimensionless times and use these as a basis for our consideration of cycle optimization. We let τ1 be the duration of the purge step, τ2 be the time during the cycle when the light component first breaks through as we feed the bed, and τ3 be the time corresponding to the end of the feed step and of the cycle. The amount of light product that is recovered from the bed every cycle is directly proportional to the time between the breakthrough of the two shocks that form during the feed step as we showed in part 3, i.e., to τ3 - τ2. It is fed as a mixture to the bed between τ1 and τ3. It is therefore straightforward to express recovery in terms of quantities that we can evaluate from the analysis that follows. It turns out that

τ3 - τ2 c* 2G τ3 - τ1 c* 2F

R)

(12)

Therefore, expressing concentrations in terms of partial pressures, we can say that

E1 )

yjpurge 1 yfeed 1

)

Π γ

(13)

where γ ) τ1/(τ3 - τ1) is the purge-to-feed ratio for the separation problem and Π ) Ph/Pl is the ratio of the high pressure in the feed step to the low pressure in the purge step. As for the light component, the feed must compensate for what is removed in the purge effluent and for what is produced during the feed. The material balance is therefore feed (τ3 - τ1) ) cprod jpurge τ1 c2,in 2,out(τ3 - τ2) + c 2

(14)

Now, we know that cfeed 2,in ) c2,ref, as defined in part 3, whereas

cprod 2,out )

ξ0-

k c2,ref ) ψc2,ref +h

(15)

as can be easily inferred from eq 36 of part 3 and from the way concentrations have been nondimensionalized. ψ is a function of R1, R2, and η, so that once that set of parameters is chosen, cprod 2,out can be determined. Equation 15 allows us to put eq 14 in the form feed c2,in [ψτ2 - τ1 - (ψ - 1)τ3] ) cjpurge τ1 2

(16)

This means that the enrichment of the light component is given by

E2 )

yjpurge 2 yfeed 2

)

[ψτ2 - τ1 - (ψ - 1)τ3] Π τ1

(17)

Concerning productivity, a comment is due on how we have nondimensionalized times. Our definition of τ incorporates the interstitial velocity of the gas in the bed, v. This velocity is related to the gas flow rates once the operating pressures are specified. Therefore, in general, v in the feed step differs from v in the purge step. In a two-bed system, part of the product recovered during the feed step in one bed is used to purge the other bed, so that there is a relationship between the two velocities as dictated by a material balance on flow rates and by operative considerations. Our analysis so far has been focused on single-bed systems, though. There-


fore, there is no real constraint on the ratio of the two velocities, and this ratio can be assigned a value. We define

χ≡

| |

vpurge >1 vfeed

(18)

as the ratio of the two velocities. This ratio is useful in the subsequent analysis. Based on the definition given above, it is straightforward to say that productivity is proportional to (t3 - t2)/t3, with t3 - t2 being proportional to the amount of product and t3 being the duration of the cycle (with its inverse being the number of cycles per unit time). When expressing the duration of the cycle in nondimensional times, however, we have to take into account that the purge and feed steps are characterized by different values of v and that therefore the duration of the purge step is actually proportional to τ1 + χ(τ3 - τ1). This means that, in general, productivity in the separation process will be

P∝

τ3 - τ 2 τ1 + χ(τ3 - τ1)

(19)

We now illustrate an example of separation and show how to use the objective functions introduced above to select near-optimal operating conditions. We consider a system characterized by R1 ) 0.2, R2 ) 0.1, and η ) 5. For the purpose of this example we have chosen a purge time of τ1 ) 0.2. The physical plane in Figure 2b illustrates that when we stop purging, the C- simple wave is still mostly in the bed and the C+ wave is entirely in the bed. A plateau region exists between the two simple waves. The composition of that plateau is given by point P in the hodograph plane in Figure 2a (c* 1 ) 0.054, c* 2 ) 0.0). We discuss in part 3 the details on how to draw this plane. In the hodograph plane the C- wave is represented by the straight segment PH and carries both the light and heavy components, whereas the C+ wave is represented by the segment PB and carries the heavy component only. Of course, segment PH lies on a Γ+ and PB on a Γ-, since they are images of part of the C- and of the whole C+ wave, respectively. It often happens that as a compromise between different economic goals the optimal purge time lies between the time when the C- wave first reaches ζ ) 0 during the purge step and the time when it is completely out of the bed; i.e., by referring to the notation in part j 2, 3, Fb0a02 < τ < Fb0a12. Once we choose τ ) τ1 ) Fb0a j < a1 since F and b0 are known we can evaluate a0 < a (see eqs 24 and 26 of part 3, where b0 ) b(ξ0+) and ξ0+ is given by eq 32). Having evaluated a j , we can determine ξ- from eq 25 of part 3. Using eqs 18 and 20 of part 3, with ξ+ ) ξ0+, we can find the concentrations that are pushed back to ζ ) 0 at τ1. The concentration profile at the beginning of the purge step is illustrated by curve a of Figure 3. We purge from right to left with pure inert gas and stop purging at time τ ) τ1 ) 0.2, i.e., after the fast simple wave has reached ζ ) 0 but before it is entirely out of the bed. The concentration that reaches ζ ) 0 at the end of the purge step is identified by point H (c* 1 ) 0.222, c* 2 ) 0.178). As we start feeding, the feed F interacts with what is left of the C- simple wave, and a S- shock forms. This shock cancels the C- wave, because it is a wave of the same type, and it moves in the bed at a nonuniform speed because it encounters different concentrations

Figure 2. Separation problem with R1 ) 0.2, R2 ) 0.1, and η ) 5: (a) hodograph plane; (b) physical plane for τ1 ) 0.2.

carried by that wave. Therefore, in the physical plane (Figure 2b), its path is curved and becomes straight only when it hits the plateau region. When the shock catches up with the C+ wave, it splits into a S- shock that transmits the wave and a S+ shock that cancels the transmitted wave and restores the initial concentration profile in the bed. As soon as the slow-moving shock S+ reaches the bed outlet at ζ ) 1, the feed is discontinued. Light component in inert gas is collected between the time that the fast and slow shocks break through, at τ2 and τ3, respectively. Figures 3 and 4 show concentration profiles in the bed at different times as we purge and feed. In Figure 3 it is clear that we progressively remove light and heavy


Differentiating eq 21 with respect to a, we obtain

dτ dζ ) -2Fab0(1 - ζ) + Fa2b0 da da

(22)

We can rearrange eq 20 and put it in the form

dζ dτ ) Fb0a0a da da

(23)

By subtracting eqs 22 and 23, we get a first-order differential equation where the unknown function is ζ ) ζ(a). This equation can be solved with the condition that, for ζ ) 0, a ) a j , and this yields

ζ)1Figure 3. Concentration profiles during purge in the separation problem for τ1 ) 0.2: (a) τ ) 0.0, (b) τ ) 0.005, (c) τ ) 0.01, (d) τ ) 0.02, (e) τ ) 0.05, (f) τ ) 0.2. Solid lines represent component 1 (heavy) and dashed lines component 2 (light).

[

components from the bed and that at the end of the purge step, at time τ1 ) 0.2, profile f, the bed still contains some light and heavy components. This is different from what we encountered in part 3, where for the same system the bed only contained some heavy component at the end of the purge step. In Figure 4 the concentration profile labeled a is the same as that labeled f in Figure 3, because at time τ ) 0.2 we stop purging and start feeding the bed from left to right. The shock front progressively advances in the bed and cancels the wave, as is apparent in profiles b-e. We know that the portion of the C- wave that is left in the j bed at τ1 is characterized by the Riemann invariants a e a e a1 and b ) b0. Its characteristics have equation τ ) σ-(a)(ζ - 1), where σ-(a)) -Fa2b0. At τ ) τ1, each of these characteristics is at a position in the bed given by ζh(a) ) 1 + τ1/σ-(a). We know that the S- shock has slope

dτ/dζ ) Fb0a0a

(20)

The generic equation of the C- characteristics, for τ > τ1 and until they are captured by the shock, is given by τ - τ1 ) -σ-(a)(ζ - ζh(a)), which can also be expressed as

τ ) 2τ1 - Fa2b0(1 - ζ)

(21)

a0 - a j a0 - a

2

(24)

along the shock path between ζ ) 0 and ζ ) ζH. Substituting this expression for ζ ) ζ(a) into eq 23 and integrating the resulting first-order differential equation in τ with the condition that, for a ) a j , τ ) τ1, we obtain τ as a function of a along the S- shock between τ ) τ1 and τ ) τH in the form

j )2 τ ) τ1 + Fa0b0(a0 - a

Figure 4. Concentration profiles during feed in the separation problem for τ1 ) 0.2: (a) τ ) 0.2, (b) τ ) 0.21, (c) τ ) 0.22, (d) τ ) 0.24, (e) τ ) 0.26, (f) τ ) 0.276, (g) τ ) 0.282. Solid lines represent component 1 (heavy) and dashed lines component 2 (light).

( )

a0 - 2a 2

(a0 - a)

-

a0 - 2a j

]

(a0 - a j )2

(25)

The coordinates of point H in the physical plane (Figure 2b) are determined by setting a ) a1 in the expressions for ζ and τ along the shock. For our system, ζH ) 0.918 and τH ) 0.265. The coordinates of point M can be determined by considering the interaction of the straight shock between H and M, given by τ - τH ) Fb0a0a1(ζ ζH), and the first characteristic of the C+ wave that it encounters. This characteristic is given by τ ) 2τ1 Fa1b02(1 - ζ). For our system, ζM ) 0.961 and τM ) 0.269. As is illustrated in Figure 2b, at M the shock front catches up with the C+ wave and it splits into two shocks. Once we know the coordinates of point M, by performing the same analysis as in part 3, we can find τ2 as

τ2 ) τM + Fa0a1b1(1 - ζM)

(26)

In our system, the faster shock S- transmits the wave and reaches the bed outlet at τ2 ) 0.276. At τ ) τ2 ) 0.276 the interaction between the transmitted wave and the slower shock S+ is not resolved yet, as is shown by profile f of Figure 4. From that profile it appears that when the faster shock exits the bed, the slower shock has not canceled the transmitted wave and that the concentration profile is given by the S+ shock, followed by a region of varying concentration, which is due to the C+ wave, and then by a region of constant composition that stretches to ζ ) 1. The latter region contains the light component that rolled up between the two shocks and that is retrieved as product between τ2 and τ3. The composition of the plateau region is given by point G in Figure 2a (c*1 ) 0.0, c*2 ) 1.168). In particular, in profile f the shock is located at ζ ) 0.982 and across it c*1 varies from 1 to 0.184, whereas c*2 varies from 1 to 1.137. By the time the shock reaches ζ ) 0.988, the transmitted simple wave has been completely canceled, and for 0.988 e ζ e 1, we have the plateau concentrations. As for the time τ3 at which the slower shock hits ζ ) 1, we can determine it by following the procedure that we suggested in part 3, which is based


a

Figure 5. Recovery of light component in the separation problem as a function of purge time.

b on a material balance. In particular, starting with a uniformly saturated bed, we numerically integrate eq 1 for 0 e ζ e 1 and 0 e τ e τ1 to obtain the concentration profile in the bed at the end of the purge step and therefore evaluate the amount of each component left in the bed. We then note that to completely saturate the bed and bring it back to the configuration it was in at the beginning of the feed step, we need to feed the bed with the feed mixture for as long as is required to supply the amount of material that was removed during the purge step. If we perform the material balance on the heavy component, by integrating the first of eq 1 (with + sign) over the bed length and between τ1 and τ3 we get

(27)

Figure 6. Enrichment of heavy (a) and light (b) components in the separation problem as a function of purge time and pressure ratio Π.

which is the same as eq 37 of part 3, with q*1,feed-binary given by eq 38 of part 3. In our system the slower shock exits the bed at τ3 ) 0.282. For the separation problem it is not necessary to use the effluent ratio E that we mentioned in the treatment of the numerical solution to determine the periodic behavior of the bed. Figure 5 shows recovery for our system as a function of the duration of the purge step. For our system, c*2G is 1.168. It is apparent that recovery increases monotonically with τ1. This is in agreement with what can be found in the literature (Ivo and Pavel, 1993). It is worthwhile to comment on the fact that recovery is zero at τ1 ) 0.0124, when γ ) 1 or, in other words, when the faster simple wave has just hit ζ ) 0. For such a cycle no light component is recovered between τ2 and τ3 because these two times coincide. As a matter of fact, as we start feeding, at ζ ) 0 the feed stream meets a concentration front which has its same composition and therefore it moves with it. Therefore, at τ ) τ2 ) 2τ1, when the mirror image of the characteristic carrying the feed composition reaches ζ ) 1, the initial concentration profile in the bed is completely restored, with the shock forming only at ζ ) 1. This implies that the splitting of the shock does not take place and hence no light component rolls up. For τ1 > 5, which corresponds to complete regeneration, recovery reaches a constant value. This can be explained by noting that no matter how long we purge, the cost of the purge stream does not appear in the objective function and since the bed is ultimately completely cleaned out, when we start

feeding it has the same initial condition, regardless of the duration of the purge step. Parts a and b of Figure 6 show the enrichment of the heavy and light components, given by eqs 13 and 17, respectively. The latter is evaluated with ψ ) 1.168 which is determined from eq 15 for our system. Enrichment is higher in shorter cycles and for higher pressure ratios. Moreover, at any duration of the purge step, enrichment of the heavy component is higher than that of the light component. Figure 7 shows that productivity goes through a maximum at a time corresponding to incomplete purge of the bed. When the duration of the purge time is chosen, productivity is lower at higher values of χ. Based on these results, we find that equilibrium theory gives us different results for the optimal operating conditions in a separation process. If we are interested in recovering the light component, we are better off with cycles of longer duration. If, however, our goal is to enrich the feed stream in the components present in trace amounts, then we want the cycle to be short. In our illustration we have considered a system with the volumetric purge-to-feed ratio γ ) 2.44, which is much smaller than that of the separation problem in part 3, where we had γ ) 7.28. Purification Problem. We start with a bed that is loaded nonuniformly, with a blocklike concentration profile consisting of two plateau regions of different concentrations adjacent to each other. The plateau closest to the feed end of the bed contains both the light

q* 1,feed-binary -

∫0 q*1 dζ ) τ3 - τ1 1


Figure 7. Adsorbent productivity in the separation problem as a function of purge time and velocity ratio χ.

and heavy components, whereas the rest of the bed contains pure light component. Our goal is to find the optimal operating conditions for the cycle. In the purification process the product is pure inert carrier gas. For objective functions, we can identify two cases. First, we consider that part of the pure inert carrier gas that is recovered during the feed step between τ1 and τ2 is used to regenerate the bed during the purge step. For this case recovery can be defined as

R)

τ2 - τ1 - (Pl/Ph)τ1 γ )1τ2 - τ1 Π

(28)

which is developed based on our assumption of a trace concentration of impurities and no accumulation in the gas phase. The purge-to-feed ratio for the purification problem is γ ) τ1/(τ2 - τ1). Equation 28 shows clearly that the pressure ratio Π plays an important role in the recovery of inert, whereas it has no role in the recovery of the light component in the separation process. Unlike in the separation problem, in the purification process enrichment is defined for both components as

E1,2 )

yjpurge 1,2 yfeed 1,2

)

(

)

τ2 - τ1 Ph Π ) τ1 Pl γ

(29)

because only the inert gas is recovered during the feed step. In a fashion similar to that of the separation process and considering that in this case part of the inert produced during the feed step is used to purge the bed, adsorbent productivity is given by

P∝

τ2 - τ1 - τ1/Π τ1 + χ(τ2 - τ1)

(30)

Second, we can consider the case in which the bed is purged with an inert gas that is different from the product and which has no cost. For this case recovery is unity, and enrichment does not change with respect to the previous case and is still given by eq 29. However, productivity is now given by

P∝

τ2 - τ1 τ1 + χ(τ2 - τ1)

(31)

Figure 8. Purification problem with R1 ) 0.1, R2 ) 0.2, and η ) 5, for τ1 ) 0.2. Plot of the effluent ratio E as a function of the duration of the purge step for various initial positions of the shock in the bed.

We choose a system characterized by the parameters R1 ) 0.1, R2 ) 0.2, and η ) 5, which is the same system studied in part 3. We purge the bed with pure inert carrier gas from ζ ) 1. Two simple waves form and they intersect each other, thereby interacting and becoming distorted. In part 3 the purge step was discontinued when one of the waves had completely exited the bed and only one simple wave, carrying the heavy or the light component, was left in the bed. In this paper, rather than having an initial condition that is carried by a simple wave, we have an initial condition that is partly the result of the interaction between the two simple waves. Our goal is to select an appropriate purge time which is presumably lower than the time in part 3. Once the time is chosen, we study the cyclic periodic behavior of the bed. Here again we are trying to find the periodic position of the shock and therefore the depth of penetration of the slow shock into the bed at the end of the feed step. We want to show that the blocklike concentration profile at the end of the feed step is restored. In Figure 8 we show plots of the ratio E as a function of the purge time for different initial positions of the shock at the beginning of the purge step. This allows us, once the purge time is fixed, to determine the periodic position of the shock at the end of the feed step by considering the curve that crosses E ) 1 at the desired purge time. We consider a purge time τ1 ) 0.2, which is close to an economic optimum, as will be shown later. From Figure 7, we find that for τ1 ) 0.2 the curve that yields E ) 1 corresponds to ζ1 ) 0.225, which is therefore the periodic position of the slow shock at the end of the feed step for this example. Figure 9a is the hodograph plane for this system. P represents the composition of the purge stream, F the composition of the feed stream, H (c* 1 ) 0.204, c* 2 ) 0.269) the concentration of the stream that is pushed back to ζ ) 0 at the end of the purge step, and K (c*1 ) 0.0, c* 2 ) 1.189) the concentration of the plateau closest to ζ ) 1 at the end of the feed step, when the blocklike concentration profile is restored. Figure 9a can be used to illustrate that the Γ wave passing through H is actually a hybrid wave, a point we developed earlier. Note that the Γ- simple wave through H is a straight line that would intersect the c*2 axis at M′ (c*1 ) 0.0, c*2 ) 0.293). The Γ wave through H does not coincide with that wave. In particular, it intersects the c* 2 axis at M (c* 1 ) 0.0, c* 2 ) 0.269), and all


a

Figure 10. Concentration profiles during purge in the purification problem for τ1 ) 0.2: (a) τ ) 0.0, (b) τ ) 0.005, (c) τ ) 0.02, (d) τ ) 0.1, (e) τ ) 0.2. Solid lines represent component 1 (heavy) and dashed lines component 2 (light).

b

the time τ2 and therefore the duration of the feed step by performing a material balance on either of the adsorbates. Specifically, once the purge step has been chosen and the periodic position of the shock at the end of the feed step has been evaluated by monitoring the effluent ratio E and finding the value of ζ1 for which E ) 1, we know what concentration profile to expect in the bed at the end of the feed step. By numerically integrating the set of equations (1), we can determine the concentration profile at the end of the purge step, and we can therefore evaluate the amounts of the light and heavy components that are still in the bed. If we decide to perform a material balance on the light component, by integrating the second of eq 1 (with + sign) for 0 e ζ e 1 and 0 e τ e τ1, then we obtain

1 [q* ζ + (1 - ζ1)q j 2*] ) τ2 - τ1 η 2,feed-binary 1

(32)

where

q* 2,feed-binary )

Figure 9. Purification problem with R1 ) 0.1, R2 ) 0.2, and η ) 5: (a) hodograph plane; (b) physical plane for τ1 ) 0.2.

concentrations along the wave, other than those at H, differ from those of the pure Γ- wave. The rest of the concentration profile that the feed interacts with is given by the portion of the true Γ+ between M and P. Thus, when the region of interaction in the bed is resolved, the shocks interact with what is left in the bed, which in this example is a portion of the C- (Γ+) wave. This is apparent in the physical plane (Figure 9b). As we start feeding at τ ) τ1 ) 0.2, the Sshock transmits the C+ characteristics of the net and cancels the C- characteristics. Once it moves past that region of interaction, it cancels the residual portion of the C- wave and exits the bed at τ2. Just like in the separation problem and by analogy to what was illustrated in part 3, it is straightforward to determine

R1 R1 + R2(1 - R1)

(33)

The only unknown in eq 32 is τ2; for our system we find τ2 ) 0.3058. The slower shock cancels all of the transmitted C+ characteristics before τ2. The last transmitted characteristic is intercepted at R (ζ ) 0.21, τ ) 0.296). Figures 10 and 11 show concentration profiles in the bed at different times as we purge and feed. From profile e in Figure 10 it is apparent that both components are left in the bed at the end of the purge step. Profile a in Figure 11 coincides with profile e in Figure 10. As we feed, the gradual waves are captured and the blocklike profile is restored. Since the feed step ends at τ2 ) 0.3058, for this system we obtain a purge-tofeed ratio of γ ) 1.89, which is much smaller than the value of γ ) 5.79 that we found for the same system in part 3. Figure 12 shows that, for the case where we purge with the inert gas that we produce during the feed step, recovery is higher when we choose short cycles and high pressure ratios. At each pressure ratio the purge time has an upper bound above which recovery becomes negative; i.e., we are using more inert to purge the bed than we are producing during the feed step.


Figure 11. Concentration profiles during feed in the purification problem for τ1 ) 0.2: (a) τ ) 0.2, (b) τ ) 0.22, (c) τ ) 0.24, (d) τ ) 0.27, (e) τ ) 0.3058. Solid lines represent component 1 (heavy) and dashed lines component 2 (light).

Figure 14. Adsorbent productivity as a function of purge time and pressure ratio Π with χ ) 1 for the purification problem with purging using product (first case).

Figure 12. Recovery of inert in the first case of the purification problem as a function of purge time and pressure ratio Π.

Figure 15. Adsorbent productivity as a function of purge time and velocity ratio χ for the purification problem with purging using an auxiliary inert gas (second case).

Figure 13. Ratio of enrichment of the light and heavy components to the pressure ratio and periodic position of the shock as a function of the duration of the purge step in the purification problem.

Figure 13 shows the ratio of enrichment to pressure ratio plotted versus the duration of the purge step τ1. It is apparent that short cycles yield higher enrichments. In general, different purge times correspond to different periodic positions of the shock in the bed at the end of the feed step, and this is also shown in Figure 11. For purge times lower than τ1 ) 0.03, we have E1,2/Π ) 1. This is because for such purge times the

interacting waves do not reach ζ ) 0, so that during the feed step no shocks form and for all τ1 e τ e τ2 the physical plane is the mirror image with respect to τ ) τ1 of the region in the plane characterized by 0 e τ e τ1. This means that τ2 - τ1 ) τ1 or γ ) 1. It is interesting to note that as the purge time increases the periodic position of the shock ζ1 increases also for τ1 < 0.61. At τ ) 0.61, ζ1 is maximum and it has the value ζ1 ) 0.2625. For τ1 > 0.61, as the purge time increases the periodic position of the shock shifts toward lower values of ζ, until at τ1 ) 1.737 it is at ζ1 ) 0.2432. This corresponds to a bed which is completely regenerated during the purge step. For 0 e τ1 e 0.03, ζ1 is indeterminate, meaning that, regardless of the position of the shock at the beginning of the purge step, the blocklike concentration profile is restored. We have dashed both curves for τ1 > 1.737 because there is no point in purging the bed longer than is required for it to be completely regenerated. Figure 14 illustrates productivity for the first case with χ ) 1. It is apparent that at higher pressure ratios productivity is higher and that once we fix Π the purge time has an upper bound, above which we consume more inert gas than we produce. Figure 15 shows productivity in the second case. It is higher for faster cycles and at lower velocity ratios. Unlike in the first case, there is no restriction on the duration of the purge step.


Discussion Our goal in this paper has been to optimize cycles for separation and purification problems. We also wanted to develop a better understanding of cycle behavior with short cycle times. It should be clear that when we refer to short cycles and short contact times, we are actually referring to the value of the nondimensional time τ. Short values of τ do not necessarily correspond to a short dimensional time t of the cycle. Rather, if the cycle lasted for a very short t, then some of the assumptions made in this paper, notably the negligibility of the pressurization and blowdown steps and the existence of equilibrium between the adsorbed and gas phase at every cross section in the bed would not be valid. It is interesting to note how profitability behaves in the asymptotic limit of very short cycles, i.e., cycle time or purge time approaching zero. In part 3 we showed that when we purged until the fast-moving simple wave was completely out of the bed, the volumetric purge-tofeed ratio was very high, well above what would be within normal industrial practice, but light component was removed from the bed for a long enough time. As the duration of the purge step is reduced, the volumetric purge-to-feed ratio decreases, which is what we want to achieve, but the amount of light component that we retrieve from the bed also decreases. In the limit, if we discontinue purging when the fast-moving simple wave has just reached ζ ) 0, we get a volumetric purge-tofeed ratio of unity, but a shock only forms at ζ ) 1 and no splitting occurs. Therefore, no light component is recovered. This implies that, in order to maximize product recovery, regeneration of the bed should be complete. If, however, we are interested in high enrichment, then fast cycles are the best option. Finally, incomplete regeneration yields maximum adsorbent productivity, as should be expected. In the purification problem we have found that the highest economic return occurs when γ ) 1. In this case no shocks form, and we are just moving waves back and forth in the bed but at different pressures and therefore at different mole fractions of inert gas. Thus, if what we are interested in is recovering a solvent, then we can say that a system operating at γ ) 1 would be optimal. In fact, if enriched purge is our product, we want to make sure that we use little inert gas to purge the bed, so that the degree of enrichment, which is related to the pressure ratio between the feed and purge steps, is maximized. We have shown how to find the duration of the feed step corresponding to the selected duration of the purge step in both problems. We have also illustrated the procedure that can be used to locate the periodic position of the shock in the bed in the purification problem. By considering different purge times and the significant times corresponding to the feed step, it is possible to perform on any system, in a straightforward way, an optimization analysis like the one we illustrated, once the three parameters R1, R2, and η that characterize the system are provided. In this paper we have analyzed optimal operating conditions for separation and purification processes of Langmuirian systems. Many of the concepts in our analysis can be generalized to other systems. In particular, at the periodic state for a purification process with no breakthrough, the effluent ratio must equal 1, irrespective of the type of isotherm (favorable or unfavorable). For favorable isotherms, the periodic concen-

tration profiles at the end of the feed step should be qualitatively the same as those shown in this paper (i.e., uniform for separation and blocklike for purification), and for purification processes the position of the intermediate shock can be evaluated using the approach based on the effluent ratio. By performance of a material balance, it is possible to determine the duration of the feed step that restores the periodic concentration profile once the duration of the purge step is chosen. Hence, it is possible to vary the duration of the purge step and to search for optimal operating conditions, as was done in this paper for Langmuirian systems. It is important to point out that, as the overall duration of the cycle decreases, the importance of the pressurization and blowdown steps becomes more significant, so that an accurate analysis should evaluate the impact of those steps on the cycle. Cycle optimizations could be affected by the inclusion of such contributions. With pressure changes, concentration shifts are possible with corresponding changes in the characteristic plane. For this case, it would no longer be valid to assume that during the feed step the characteristics that are not intercepted by a shock are the mirror image of those in the purge step. Conclusions We have used the method of characteristics as a guide to solve the problems of separation and purification in pressure-swing adsorption with short dimensionless contact times. Our goal has been to optimize the cycles from an economic standpoint. We have defined objective functions for both problems and have selected optimal operating conditions in order to maximize the economic return. We have shown that in the purification problem short cycles are preferred economically over longer cycles, whereas in the separation problem the optimal duration of the cycle depends on our goal. Specifically, short cycles are best if we want a purge enriched relative to the feed and long cycles are preferrable if we want to recover the light component. In part 3 of this series the purge step ended when one of the waves had completely exited the bed. In this paper, since we are looking for an economic optimum, we have relaxed this constraint, so that, in general, at the end of the purge step part or all of both waves are in the bed. This means that at the end of the regeneration step neither of the components has been completely removed. This makes our analysis of the problems of separation and purification for two-component adsorption more general. We have contributed a mathematical explanation of how to deal with the interaction between a shock and a region of interaction between two simple waves. This was a necessary step in the solution of the purification problem, where such an interaction occurred. By adopting the methods outlined in this paper, or similar methods, it should be possible to obtain quick estimates for near-optimal operating conditions for pressure-swing adsorption systems used for separation and purification. Our analysis is restricted to systems for which adsorbates are present in trace amounts in an inert carrier gas, so that velocity variations are negligible. As a consequence, it cannot be extended as such to bulk separations. Many of the concepts illustrated in this paper apply to non-Langmuirian systems also and are


useful in predicting the periodic state concentration profiles and in exploring optimal operating conditions. Acknowledgment The authors are grateful to the U.S. Army ERDEC for financial support. Notation c ) fluid-phase concentration, mol/m3 C( ) characteristic lines for simple waves in the physical plane E ) effluent ratio, eq 10 Ei ) enrichment of component i k ) ratio of unit cost of purge and feed in the separation problem Ki ) Langmuir isotherm parameter for component i, m3/ mol L ) bed length, m P ) adsorbent productivity Pl,h ) low and high pressure in the purge and feed step, respectively q ) adsorbed-phase concentration, mol/kg Qi ) Langmuir monolayer capacity for component i, mol/ kg R ) product recovery Ri ) separation factor for the isotherm for component i, eq 3 S( ) characteristic lines for shocks in the physical plane t ) time, s v ) interstitial velocity, m/s z ) axial coordinate, m Greek Letters R1,2 ) separation factor γ ) purge-to-feed ratio Γ( ) characteristic lines in the hodograph plane ) void fraction of packing ζ ) z/L, dimensionless axial coordinate ζ1 ) periodic position of the shock in the purification problem η ) ratio of partition ratios for heavy and light components Λi ) partition ratio for component i

Fb ) bulk density of packing, kg/m3 τ ) |v|t/Λ1L, dimensionless time τ1 ) dimensionless duration of the purge step τ2 ) dimensionless duration of the feed step in the purification problem; dimensionless time for breakthrough of the light component in the separation problem τ3 ) dimensionless duration of the feed step in the separation problem χ ) ratio of interstitial velocities in the purge and feed steps

Literature Cited Courant, R.; Friedrichs, K. O. Supersonic Flow and Shock Waves; Interscience: New York, 1948. Davis, M. M.; McAvoy, R. L.; LeVan, M. D. Periodic States for Thermal Swing Adsorption of Gas Mixtures. Ind. Eng. Chem. Res. 1988, 27, 1229-235. Ivo, R.; Pavel, D. Pressure Swing Adsorption: Analytical Solution for Optimum Purge. Chem. Eng. Sci. 1993, 48, 723-734. Kayser, J. C.; Knaebel, K. S. Pressure Swing Adsorption: Development of an Equilibrium Theory for Binary Gas Mixtures with Nonlinear Isotherms. Chem. Eng. Sci. 1989, 44, 1-8. Knaebel, K. S.; Hill, F. B. Pressure Swing Adsorption: Development of an Equilibrium Theory for Gas Separations. Chem. Eng. Sci. 1985, 40, 2351-2360. LeVan, M. D. Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655-2660. Pigorini, G.; LeVan, M. D. Equilibrium Theory for Pressure Swing Adsorption. 3. Separation and Purification in Two-Component Adsorption. Ind. Eng. Chem. Res. 1997, 36, 2306-2319. Rhee, H.-K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations: Volume II. Theory and Application of Hyperbolic Systems of Quasilinear Equations; Prentice-Hall: Englewood Cliffs, NJ, 1989. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley: New York, 1984. Ruthven, D. M.; Farroq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH Publishers: New York, 1994. Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Stoneham, MA, 1987.

Received for review October 20, 1997 Revised manuscript received February 26, 1998 Accepted March 4, 1998 IE970731G

Equilibrium Theory for Pressure-Swing Adsorption. 4. Optimizations for

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