Equilibrium Theory for Pressure Swing Adsorption. 3. Separation and

2306

Ind. Eng. Chem. Res. 1997, 36, 2306-2319

Equilibrium Theory for Pressure Swing Adsorption. 3. Separation and Purification in Two-Component Adsorption Giuseppe Pigorini and M. Douglas LeVan* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442

The periodic behavior of an adsorption bed that is alternatively fed with a binary mixture in inert gas and purged countercurrently, but incompletely, with pure inert gas is analyzed using local equilibrium theory. The bed is isothermal with the two adsorbable components present in trace amounts and adsorption equilibrium described by a binary Langmuir isotherm. The purge step ends as soon as one of the components has been completely removed from the bed. For this nonlinear system, the periodic state is found analytically, without iteration, and feed and purge times are obtained directly. Both separation and purification problems are analyzed. A system is completely defined by three parameters: the constant separation factor for each component and the ratio of partition ratios of the components as pure feeds. Different combinations of these parameters yield wave interactions among simple waves and shocks, which are resolved numerically, and corresponding bed profiles that are quite dissimilar. Introduction Pressure swing adsorption (PSA) is used in many industrial applications ranging from the separation of a gaseous mixture into its components to the purification of an inert carrier gas by the elimination of impurities, possibly followed by recovery of these impurities if they have some commercial value (Ruthven, 1984). It is based on the physical phenomenon that the amount of a species that is adsorbed by an adsorbent increases as its partial pressure is raised. Since different species are adsorbed in different proportions, this is a valuable method for obtaining a certain degree of separation of a feed mixture. In a PSA cycle for separation or purification, the adsorption bed is initially fed at high pressure with the feed mixture of adsorbable components in inert gas and then purged with the light product or, for purification, possibly pure inert gas at low pressure. This means that the partial pressure of the inert carrier gas, which makes up the bulk of the gas phase, drops on depressurization, whereas the partial pressures of the adsorbable components remain roughly the same since they are related to adsorbed-phase concentrations through adsorption isotherms. This paper is concerned with the a priori prediction of the periodic state (cyclic steady state) of an adsorption cycle. This is in contrast to the use of many mathematical models, which are cycled until a periodic state is approached closely. Considerable research has been done recently on cycles and their periodic behavior. Knaebel and Hill (1985) have studied complete PSA cycles with linear isotherms, applying the theory of characteristics to analyze all four steps of the basic cycle. Matz and Knaebel (1988) have applied local equilibrium theory to a PSA cycle with incomplete regeneration for the separation of a binary mixture with adsorption equilibrium described by linear isotherms. Kayser and Knaebel (1989) have studied a cycle with complete regeneration for a system described by uncoupled, * To whom correspondence should be addressed. Present address: Department of Chemical Engineering, Vanderbilt University, Box 1604, Station B, Nashville, TN 37235. Tel: (615) 322-2441. Fax: (615) 343-7951. E-mail: mdl@ vuse.vanderbilt.edu. S0888-5885(96)00716-6 CCC: $14.00

nonlinear isotherms. Davis and LeVan (1987) have used equilibrium theory to study complete adiabatic temperature swing adsorption cycles. Croft and LeVan (1994) have presented a numerical method for the direct determination and stability analysis of periodic states of adsorption cycles. LeVan (1995) has solved analytically for the periodic state of a PSA cycle where a single component is adsorbed and the adsorption isotherm is nonlinear. In many cases the assumption is made that the adsorbed phase is “frozen” during pressurization and blowdown, in the sense that the rapid change in total pressure does not affect adsorbed-phase concentrations. For systems which have large partition ratios, such as those considered here, the pressurization and blowdown steps use considerably less gas than the feed and purge steps and therefore this assumption is reasonable. See remarks concerning this point in part 2. It is understood that this assumption may be suspect when analyzing cycles for separation of light gases, as first shown by Flores Fernandez and Kenney (1983). Beginning with Shendalman and Mitchell (1972), equilibrium theory has been a common method for the analysis of PSA cycles. Equilibrium theory leads to a rigorous mathematical treatment which gives insight and generally provides the solution for best possible performance of a process (Yang, 1987; Ruthven et al., 1994). In part 1 (LeVan, 1995), equilibrium theory was applied to a system with one adsorbable component to solve the material balance on that component. The focus was on the determination of the periodic behavior of the adsorption bed in a purification and enrichment process. This paper is the extension of part 1 to a system with two adsorbable components. As in part 1, we consider only the feed and purge steps of a PSA cycle and make the assumption that the other two steps, pressurization and blowdown, 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, in the range 100-10 000, so that the volume of gas involved in the pressurization and blowdown of the bed is negligible compared to that required during the feed and purge steps. Examples consider a separation process and two purification processes. For the separation process, the bed is fed © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2307

until both adsorbable components have broken through. Prior to breakthrough of the second component, only the first component is present in the effluent. For the purification processes, the bed is fed until the more volatile of the two components breaks through. The unique aspect of this paper is that we explore analytically the behavior of a PSA cycle described by coupled nonlinear isotherms and involving incomplete regeneration of the bed. We also perform a numerical solution to generate the concentration profiles in the bed at any specified time during the cycle. These profiles could be obtained by analytical means, but not in a straightforward way. Equation Set Material Balances. We consider isothermal, fixedbed adsorption of two components present at low mole fractions in a nonadsorbing carrier gas. Adsorption isotherms are nonlinear, specifically concave downward, as in literature examples cited in part 2. Partition ratios are large, allowing the neglect of fluid-phase accumulation terms in material balances. Following the classical local equilibrium theory, mass transfer rates are infinitely fast and dispersion is nonexistent. Local equilibrium exists between the adsorbed and fluid phases at each cross section in the bed. Material balances are

∂qi ∂ci + v ) 0 Fb ∂t ∂z

i ) 1, 2

∂c* ∂q* 1 1 ( )0 ∂τ ∂ζ

(1)

(2)

∂c* 1 ∂q* 2 2 ( )0 η ∂τ ∂ζ where q* i ) qi/qi,ref, c* i ) ci/ci,ref, τ ) |v|t/(Λ1L), ζ ) z/L, Λi ) Fbqi,ref/ci,ref, and η ) Λ1/Λ2. The ( sign and the absolute value of velocity account for the fact that the flow direction in the bed is reversed every half cycle step. We take v to be positive during the feed step and negative during the purge step. For the reference value ci,ref, we use the concentration of component i in the feed to the bed for the feed step. The reference value qi,ref is the adsorbed-phase concentration in equilibrium with ci,ref only (i.e., as a pure component). Isotherms. We describe adsorption equilibrium using the binary Langmuir isotherm

i ) 1, 2

(3)

(5)

where the constant separation factor Ri is given by

Ri )

1 1 + Kici,ref

(6)

The subscripts in eq 5 are interchanged for component 2. Without loss of generality, the component with the lower value of the ratio Ri/Λi is component 1, which we will also refer to as the heavy component. This means that R1 < ηR2. Considering that the separation factor, or selectivity, between species 1 and 2 is defined as

R1,2 )

q1 c2 c1 q2

(7)

from eqs 3, 4, and 6 it is straightforward to show that

R1,2 )

Q1K1 Λ1 R2 ηR2 ) ) Q2K2 R1 Λ2 R1

(8)

Therefore, the condition R1 < ηR2 is equivalent to R1,2 > 1. In Langmuirian systems R1,2 is constant; hence no selectivity reversal can take place.

QiKici,ref 1 + Kici,ref

Manipulation of eq 3 gives the dimensionless form

Simple Waves. We solve eqs 2 by the method of characteristics. This classical method is particularly effective for solving a system of equations of our type (a pair of first-order, quasi-linear, hyperbolic partial differential equations), and the graphical representation of the solution in both the hodograph plane (c* 1 vs c* 2) and physical plane (τ vs ζ) helps understand how the concentration profiles form as a consequence of wave interactions. In terms of the dependent variable vector

()

c* 1 Y h ) c* 2

(4)

(9)

eqs 2 can be written

∂Y h ∂Y h +B )0 ∂τ ∂ζ

A

(10)

where

A)

(

∂q*1/∂c* ∂q*1/∂c* 1 2 (1/η)∂q* /∂c* (1/η)∂q* 2 1 2/∂c* 2

)

B)

(

(1 0

)

0 (1 (11)

Premultiplying matrix A by the inverse of matrix B (which is B itself) gives

∂Y h ∂Y h + )0 ∂τ ∂ζ

C

where Qi is the monolayer capacity of component i. We recognize that this equation satisfies the Gibbs adsorption isotherm only for the case Q1 ) Q2 (e.g., LeVan and Vermeulen, 1981). Equation 3 gives for qi,ref

qi,ref )

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

Theory for Numerical Solution

with variables defined in the notation section. Nondimensionalization gives

QiKici qi ) 1 + K1c1 + K2c2

q* 1 )

where

C ) B-1A

(12)

The eigenvalues σ+ and σ- of matrix C can be determined, where |σ+| > |σ-| by convention, as well as the left eigenvectors, which satisfy hl(C ) σ(hl(. The two sets of characteristics in the physical plane are designated C+ and C-. From the theory, the slopes of the characteristics in the physical plane are

dτ ) σ( dζ

along

C(

(13)

2308 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

We know that a C+ is the image of a Γ- and a C- is the image of a Γ+, Γ+ and Γ- being the symbols assigned to the characteristics in the hodograph plane. In the hodograph plane, the characteristics have the local slopes

dc*1 n( )dc*2 m(

along

Γ(

(14)

where hl( ) (m(, n(). Equation 14 allows a general construction of the family of solutions in the hodograph plane without reference to the physical plane. Equation 13 then can be used to map a particular solution into the physical plane. For our problem, owing to the particular behavior of Langmuirian systems, the Γ( characteristics are linear (Rhee et al., 1989). Therefore, it is relatively straightforward to map Γ( from the hodograph plane to C- in the physical plane. Shocks. Shocks form when simple waves cannot physically exist. Because the system considered is described by favorable (i.e., concave downward) isotherms, we expect simple waves to be generated only during the purge step and we expect shocks to be formed only during the feed step. The shock speed is determined by a material balance around the shock, and it is a function of the gas and adsorbed-phase concentrations before and after the shock. The expression for the reciprocal of the shock speed for our system, written in nondimensional form, is

1 ∆q* 1 2 dτ ∆q* ) ) dζ ∆c*1 η ∆c* 2

(15)

In the hodograph plane, again owing to the particular nature of Langmuirian systems, a shock will be a straight line, and this line will overlay the line for a simple wave (Ruthven, 1984). Usually, in the literature, shocks are designated ∑( in the hodograph plane and S- in the physical plane. However, because shocks and simple waves are indistinguishable in the hodograph plane, we will denote them both by Γ(. Since waves generally appear in pairs connected at a plateau region, whenever such a plateau region occurs between two constant states (e.g., a constant feed condition and a constant initial condition), its composition is the same regardless of whether it is connected to these constant states through shocks or simple waves. For the physical plane, if a shock travels between two constant state regions (including a plateau), then its slope is constant, and its image will be a straight line. On the other hand, if the concentrations behind or ahead of the shock change in time and space, the velocity of the shock will change and so will its slope in the physical plane, so that its image there is no longer a straight line. Wave Interactions. When a cycle step begins with the bed loaded nonuniformly, a wave interaction is likely to occur. Different kinds of interactions can take place when two waves (shocks or simple waves) meet. In dealing with the numerical analysis of wave interactions, the hodograph plane is essential since in that plane the slopes of the characteristic lines do not change when they meet, unlike in the physical plane. In general, we may have S+ and S- shocks and fans of C+ and C- characteristics, and the pattern of interaction depends on whether we have a shock-shock, simple

Figure 1. Hodograph plane (first column), physical plane (second column), and range of Riemann invariants (third column): (a) transmission of two simple waves; (b) cancelation of a wave by a shock of the same kind; (c) transmission of a wave by a shock of different kind.

wave-shock, or simple wave-simple wave interaction as well as on whether the waves are of the + or - type. In the present work, we encounter interactions between C+ and C- simple waves, between a C+ simple wave and an S+ shock, and between a C+ simple wave and an Sshock. For the interaction between two simple waves of different kinds (C+ and C-), transmission occurs with the two waves passing through each other, as shown in Figure 1a. In the physical plane each characteristic gets distorted and becomes straight again only when the interaction is complete. The interaction speeds up the C- wave and slows down the C+ wave. For the interaction between a shock and a simple wave of the same kind (C+ and S+), as shown in Figure 1b, cancelation occurs since the simple wave does not get transmitted through the shock. The shock on the other hand changes speed upon interacting with the wave and so in the physical plane its path is curved. Only when the interaction is complete does it resume a constant speed. Finally, the interaction between a shock and a simple wave of different kinds (C+ and S-) is characterized by the transmission of the simple wave through the shock, as shown in Figure 1c. The characteristics change slope abruptly upon crossing the shock, whereas the shock gradually changes slope as it meets with characteristics that carry different values of concentration. Theory for Analytical Contribution Langmuirian systems lend themselves to a relatively straightforward mathematical analysis owing to the fact that the characteristics are straight lines in the hodograph plane. This allows us to deal with wave interactions relatively easily and in doing so to make analytical contributions to certain parts of the full solution, including the a priori prediction of the periodic state. The following analysis relies on the mathematical development of Rhee et al. (1989), which can be consulted for details. We provide only a sketch of the solution procedure and the major results. Simple Waves. During the purge step, when v < 0, eqs 2 can be written


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

(16)

∂c* 1 ∂q* 1 ∂q* 2 2 ∂c* 1 2 ∂c* 2 )0 ∂ζ η ∂c*1 ∂τ η ∂c* ∂τ 2 Each of these equations can be transformed to give dτ/ dζ, the reciprocal of the concentration velocity, and then those equations can be set equal to one another to eliminate this reciprocal quantity. After substitution of the Langmuir isotherm and rearrangement, we obtain

( )

c* 2

( )

dc*1 2 dc*1 - (k + c* - hc* 1 - hc* 2) 1 ) 0 dc* dc*2 2

(17)

where

h )

η(1 - R2) 1 - R1

and

k)

R1 - ηR2 1 - R1

(18)

Equation 17 is a Clairaut equation, which has as its solution a family of straight lines Γ( with parabolic envelope. The straight lines are given by

c*1 ) ξ(c*2 -

kξ( ξ( + h

where

ξ( )

( )

dc* 1 dc*2 (

(19)

Note that ξ is the slope of the characteristics in the hodograph plane. Then, from the definitions of h and k in eq 18 and the restriction to favorable isotherms (Ri < 1), we have that (i) if R1 < ηR2, then k < 0, h > 0 giving the envelope shown in Figure 2, and (ii) if R1 ) ηR2, then k ) 0 and characteristics Γ( have equation c*1 ) ξc* 2. The equality in (ii) corresponds to the condition R1/ Λ1 ) R2/Λ2, or equivalently R1,2 ) 1. If this condition is verified, then no separation can be achieved between the two adsorbable components, since the system is essentially azeotropic at all compositions. In the hodograph plane, the coordinates of the intersection point of generic Γ+ and Γ- characteristics are

c* 1 )

-kξ-ξ+

c* 2 )

(ξ- + h)(ξ+ + h)

hk (20) (ξ- + h)(ξ+ + h)

In the (ζ, τ) plane, the slopes of the characteristics are given by

(

σ( ) (

)

∂q*1 1 ∂q* 2 - ξ( ∂c* η ∂c* 1 1

(21)

Substituting the expressions for the isotherms, eq 5, into eq 21 and simplifying give for the slopes of the characteristics in the physical plane

C+: σ+ ) (Fab2

(22)

C-: σ- ) (Fa2b

(23)

F ) ηR1R2

(24)

where

a ) a(ξ-) )

ξ- + h R1h + ηR2ξ-

(25)

Figure 2. Envelope of the characteristics in the system where R1 < ηR2. Figure corresponds to examples 1 and 3: R1 ) 0.2, R2 ) 0.1, and η ) 5. Negative concentrations are shown but, of course, are not feasible.

b ) b(ξ+) )

ξ+ + h R1h + ηR2ξ+

(26)

The ( on the right hand side of eqs 22 and 23 accounts for flow in the positive or negative direction in the bed. Shocks. We know that S+ is the image in the physical plane of a Γ- shock and S- that of a Γ+ shock. As a consequence, along S+, a will be constant, and along S-, b will be constant. The slope of the shock is a function of the value of the appropriate parameter (a or b) across it, i.e., of the states to the left and to the right of the shock. Thus we have

S+: σj +(bl,br;a) ) (Fablbr

(27)

S-: σj -(al,ar;b) ) (Fbalar

(28)

Just as for simple waves, the ( accounts for flow in the positive or negative direction in the adsorption bed. Wave Interactions. It is possible to deal analytically with the different kinds of wave interactions that are encountered in this problem, because a and b are Riemann invariants (Courant and Friedrichs, 1948), meaning that they have certain properties of constancy in regions of interaction. The types of wave interactions that occur in this paper were shown in Figure 1. Referring again to that figure with emphasis now on the third column, for the transmission of simple waves, shown in Figure 1a, we know that a C+ is the image of a Γ- and a C- is the image of a Γ+. Therefore, a is constant along a C+ and b is constant along a C-. Along each C+ characteristic in the interaction region though, b keeps its value and a changes. Analogously, each Ccharacteristic carries into the interaction region a specific value of a and allows b to change. Therefore, in the interaction region b is constant along a C+ characteristic and a along a C-. During the interaction


between the two simple waves the parameter a increases along each C+, whereas b decreases along each C-, so that both sets of characteristics change slope in the physical plane. The equations that describe how the characteristic lines change slope upon meeting are, from eq 13,

∂τ ∂ζ ) σ+(a,b) ∂a ∂a

C+:

(29)

∂τ ∂ζ ) σ-(a,b) C : ∂b ∂b -

By cross differentiation and combination, this system can be reduced to a second-order partial differential equation that can be solved for τ and ζ on the characteristic net for given boundary conditions (values of τ and ζ) along two of the sides of the net. Boundary conditions are obtained by solving the first-order linear ordinary differential equation that occurs when considering the system between the differential equation for the border C+ (C-) characteristic and the C- (C+) simple wave (described by the parameter a (b)) that is entering the interaction region. For the cancelation of a C+ simple wave by an S+ shock, shown in Figure 1b, the C+ carries a constant value of b and a changes along it. The value of b changes from characteristic to characteristic, so that the slope of the S+ shock that interacts with the simple wave (the slope being a function of the values of b on both sides of the shock) changes gradually and is a function of the local value of b carried by the intersecting characteristic. Therefore, the system to solve, from eqs 13 and 14, is

C+: τ - θ(b) ) σ+(b)(ζ - ψ(b)) dτ ) σj +(b) S : dζ

(30)

+

where θ(b) and ψ(b) are the given values of τ and ζ for the various characteristics along a given line. Differentiating the first equation with respect to b and combining it with the second one yield a first-order linear ordinary differential equation that can be solved for the shock path S+. Finally, for the transmission of a C+ simple wave through an S- shock, shown in Figure 1c, the parameter a remains constant on either side of the shock, whereas the parameter b, invariant across the shock, varies along the shock path, which therefore has a oneparameter representation in terms of b (just as in the previous case). What changes here with respect to the previous case is the expression for the slope of the shock, since it is of a different type. The system of equations is

C+: τ - θ(b) ) σ+(b)(ζ - ζ(b)) S-:

dτ ) σj -(b) dζ

(31)

Just as before, by differentiation and rearrangement the system becomes a single ordinary differential equation that can be solved for the shock path S-. Problem Definition and Method of Solution The purification and separation problems are defined below, and some results are described qualitatively. In

both problems, three parameters need to be provided in order to specify the system. These are R1, R2, and η. In the separation problem, we start with a bed saturated with both components at the feed concentrations and purge it countercurrently with pure inert gas. We stop purging when we have removed all of the light component and we are left with some pure heavy component in the bed. At this point, we reverse the flow direction and start feeding the feed mixture. At the bed outlet we get first pure inert gas as a product and then light component in the inert gas (the product), and then the bed saturates at uniform loadings with the feed mixture again, restoring the initial condition. In the purification problem, the initial concentration profile in the bed prior to purging consists of a region uniformly loaded with light component close to ζ ) 1 and a region in equilibrium with feed along the rest of the bed. We purge with pure inert gas until one of the two components is completely removed from the bed. As we will show, this can be the heavy or the light component, depending on the set of parameters. In a manner similar to that for the separation problem, we then feed the feed mixture countercurrently. Here, too, the initial effluent will be pure inert gas. However, in this case we are interested in retrieving only pure inert gas, so we discontinue feeding at the time that we would start getting the light component as product. This will leave the bed with a concentration profile identical to the one that we started with, provided that we are at the periodic state, which has the asymptotic depth of penetration of the heavy component into the bed (i.e., for the infinite cycle). As was mentioned in the previous section, the approach to the numerical solution of the system of partial differential equations through the method of characteristics relies on the fact that in the hodograph plane characteristics stay straight even when they cross. This allows us to determine easily the coordinates of the nodes of the net that is generated in the hodograph plane by intersecting C+ and C- characteristics. Once the nodes are identified, it is necessary to determine the eigenvalues of matrix C at each node because this allows us to draw the nonstraight characteristics in the physical plane emanating from that node. In the interaction region, we let the slopes of the characteristics in the physical plane (which vary as they meet with different characteristics of the other type) remain constant in the interval between two nodes, and these are taken to be the averages of the slopes evaluated at both ends of the intervals. This allows us to generate the characteristics of varying slope. A similar approach is followed when dealing with the interaction between a shock and a simple wave of different types. The speed of the shock is a function of the states between which it moves; hence when it crosses one characteristic its slope changes and stays constant until it crosses the next discretized characteristic of the wave. In the interval between the intersections with two characteristics, the slope is the average of the slopes that correspond to the jump in the concentrations carried by the two characteristics. When the interaction between a shock and a simple wave of the same type is considered, the change in slope of the shock is accounted for by following the same procedure as in the previous case with the only difference in the physical plane being that the characteristics do not get transmitted. Adding more nodes to regions of intersection in the hodograph plane, thus refining the extent of the discretization, and mapping


to the physical plane assure convergence of the solution in the physical plane. Our approach to the solution of the problem is to predict the periodic profile in the bed, then to run one purge step and one feed step in order to restore the initial condition. In the purification process, the bed is not uniformly loaded initially. A shock is present and a region with only light component is assumed close to ζ ) 1. We are interested in determining the cyclic profile in the limit as the number of cycles goes to infinity. Rather than actually solve for several adsorption-desorption cycles, we predict the final position of the shock analytically. At the periodic state, the proportion in which the two components are removed from the bed during the purge step will be the same as that in which they are fed to the column during the feed step. By imposing this condition, we are able to predict analytically the periodic position of the shock by just considering the purge step, specifically the initial condition and the final condition, when one component has just been completely removed. Examples We consider one example for separation and two for purification. The purification examples differ by whether, at the periodic state, some of the heavy component or some of the light component is left in the bed at the end of the purge step. Example 1. Separation. This first example deals with a system defined by R1 ) 0.2, R2 ) 0.1, and η ) 5. Parts a and b of Figure 3 show respectively the hodograph plane and the physical plane for this case. In what follows, we will consider the purge step first and determine at what time the simple wave that carries both components completely exits the bed. This is the time at which we start feeding. As we feed, an S- shock develops and moves down the bed. When this shock interacts with the simple wave that is left in the bed by the purge, it transmits it and, in doing so, its speed is changed. However, when the interaction is over, it becomes straight again, so that it is straightforward to determine at what time it reaches the bed outlet. The concentrations and velocities of the transmitted wave are such that when it interacts with the feed a new shock forms, of the S+ type, which cancels the wave and restores the initial concentration profile. A simple material balance on the heavy component will be performed to determine the time at which this slower shock exits the bed. Purge Step. We start purging a uniformly loaded bed with pure inert gas. At the end of the purge step, all of the light component is removed and the bed still contains a heel of heavy component. As soon as the purge with pure inert gas begins, two simple waves separated by a plateau develop at ζ ) 1, as shown in Figure 3b. The faster moving wave carries both the light and the heavy component, whereas the slower wave carries only the heavy component. In the hodograph plane, Figure 3a, the feed stream is identified by point B (c*1 ) c* 2 ) 1) and the purge stream by point F (c* 1 ) c* 2 ) 0). F is characterized by ξ- ) -∞ and ξ+ ) 0. For B, the values of ξ+ and ξ- are roots of eq 17, which can be rearranged to

()

1 - η ( x(1 - η)2 + 4η(1 - R1)(1 - R2) ξ0+ (32) ) 2(1 - R1) ξ0-

Figure 3. Example 1. Separation problem with R1 ) 0.2, R2 ) 0.1, and η ) 5: (a) hodograph plane; (b) physical plane.

P is the intersection point (plateau) between the Γwave through F and the Γ+ wave through B. It lies on the c* 1-axis and is characterized by ξ- ) -∞ and ξ+ ) ξ0+. Figure 2 shows clearly that c* 1P < -k, and therefore that -k is an absolute upper bound on the plateau concentration of the heavy component during the purge step, the point characterized by c* 1 ) -k and c* 2 ) 0 being the watershed point for this system. From eq 20, we have

c* 1P )

-kξ0+ ξ0+ + h

c* 2P ) 0

(33)


Figure 4. Concentration profiles during purge in example 1: (a) τ ) 0; (b) τ ) 0.01; (c) τ ) 0.02; (d) τ ) 0.05; (e) τ ) 0.6; (f) τ ) 1.645. Solid lines represent component 1 (heavy), and dashed lines, component 2 (light).

which gives c* 1P ) 0.054 and c* 2P ) 0. In what follows, we will let a0 ) a(ξ0-), b0 ) b(ξ0+), a1 ) a(-∞), and b1 ) b(0). Figure 4 shows the concentration profile in the bed at different times during the purge step. Flow is from right to left. The formation of the plateau becomes apparent in profiles d through f. We stop the purge as soon as the faster wave has completely exited the bed. At this point the bed still contains the plateau and gradual wave of pure heavy component, as shown by profile f. The time τ1 at which the first simple wave exits the bed during the purge is determined from eqs 13 and 23 to be

τ1 ) (σ-)P(ζ - 1)|ζ)0 ) Fa2b|P ) Fb0a12

(34)

For our system, we have τ1 ) 1.645 for the end of the purge step. The C+ characteristics have equation τ ) σ+(b)(ζ - 1) from eq 13. At τ ) τ1, ζ*(b) ) 1 + τ1/σ+(b) gives the location in the bed of the composition carried by the C+ characteristic defined by parameter b. This is used to determine where the slower wave is when we start feeding again. Feed Step. We start feeding, and the feed captures the heavy component that is left in the bed after the purge. The light component breaks through first, and then the bed becomes uniformly loaded, in equilibrium with the feed, back to the state it was in before purge. The bed is fed in the opposite direction to the purge with the feed mixture. Figure 5 shows the concentration profile at different times as we feed. Profile a coincides with profile f in Figure 4. Owing to the presence of a plateau extending to ζ ) 0 and to the favorable isotherm, one shock develops initially. This is clear in profiles b and c, where the shock is followed by what is left of the plateau P and by a portion of the slower wave. This shock moves in the positive direction and transmits the simple wave that is left in the bed. The concentrations and velocities of the transmitted wave are such that when it interacts with the feed, another shock forms, starting at point H where the Sshock first interacts with the C+ wave. This phenomenon of formation of a second shock can be explained easily with the support of the hodograph plane, Figure 3a. As the feed (point B) starts interacting

with the plateau region (point P), only one shock forms since the Γ+ characteristic through P goes through B also, and therefore there is no need to move along the Γ- through B before that Γ+ is met. Practically, this means that no S+ forms because the composition of the plateau that would lie between the S- and the S+ shocks is that corresponding to the feed B itself. However, suppose that rather than interacting with a plateau of composition P, the feed interacted with a uniform region of composition P′ (or F, for that matter), as shown in Figure 3a. In that case, two shocks would form, an S+ shock between B and B′ (or G) and an S- shock between B′ (or G) and P′ (or F), B′ (or G) being the composition of the plateau between the two shocks. Since feed B actually interacts with plateau P first and then with all concentrations between P and F, it follows that as long as it meets with the plateau concentration, only one shock will form and as soon as the concentrations that it interacts with move away from P and toward F, two shocks will be required to connect the feed concentration with the concentrations carried by the C+ simple wave in Figure 3b. As is shown in Figure 3b, of these two shocks, the faster one transmits the gradual wave (the two waves being of different type), whereas the slower one cancels the transmitted wave (the two waves being of the same type). When the faster shock reaches ζ ) 1, the interaction between the transmitted wave and the slower shock is not yet over. The coordinates of H in the physical plane are determined by finding the point of intersection of the S- shock through (0, τ1) and the first C+ characteristic, which has the slope Fa1b02. Then, taking straight lines through (ζ*(b), τ1) of slope -σ+(b), and considering the equation for the S- shock through (ζH, τH), we get a system of equations analogous to eqs 31, characterized by parameter b. Differentiating both equations with respect to b, subtracting one from the other, and solving the resulting equation, it is possible to get an expression for ζ ) ζ(b) along HM. Substituting this back into one of the two equations we can obtain an expression for τ ) τ(b). Point M is characterized by setting b ) b1. Between M and (1, τ2), considering eq 13, written for a shock, and eq 28, we have τ - τM ) Fa0a1b1(ζ - ζM), so that we get τ2 ) τM + Fa0a1b1(1 - ζM). Simplifying, we obtain

τ2 )

Fa1[a0b1(b0 - a0) + a1b1(b0 + a0) - 2a0a1b0] (b1 - a0) (35)

which gives τ2 ) 1.799 for breakthrough of the light component. The plateau closest to ζ ) 1 contains pure light component. From eq 20, the composition of plateau G is given by

c* 1G ) 0

c* 2G )

k ξ0- + h

(36)

or c* 1G ) 0 and c* 2G ) 1.168. Note the roll-up of the light component in Figure 5 (and in many subsequent figures). Feeding is carried out until the slower shock reaches ζ ) 1 at τ ) τ3 and reestablishes the initial bed profile. This allows us to retrieve the light component that is “trapped” between the two shocks at τ2 and τ3. To obtain τ3, a material balance is performed on the heavy component. We integrate the first of eqs 22 (with + sign) over the bed length and between τ1 and τ3. This


Figure 5. Concentration profiles during feed in example 1: (a) τ ) 1.645; (b) τ ) 1.68; (c) τ ) 1.71; (d) τ ) 1.795; (e) τ ) 1.83; (f) τ ) 1.871. Solid lines represent component 1 (heavy), and dashed lines, component 2 (light).

gives us

q*1,feed-binary -

∫01q*1 dζ ) τ3 - τ1

(37)

which states that between τ1 and τ3 enough heavy component must be fed into the bed in addition to what is still adsorbed at the end of the purge step in order for the bed to be fully loaded again. From eq 5, considering the feed composition, we obtain

q*1,feed-binary )

R2 R1 + R2(1 - R1)

(38) –

Once we know the amount of adsorbate left in the bed at the end of the purge step (or equivalently the amount removed during the purge step), eq 37 can be solved to yield τ3 ) 1.871 as the time required for breakthrough of the light component and restoration of the initial condition. Example 2. Purification with Total Removal of Light Component during Purge. The solution to this problem is developed in a general fashion. Important times and distances are determined at the end when calculations for the closed cycle can be performed. Unlike in example 1, in this example and the next one, the initial condition of the bed at the start of the purge step is not obvious. Specifically, we must determine the depth of penetration of component 1, the heavy component, into the bed at the end of the feed step. As relations are derived for these calculations, correct corresponding figures will be shown and some numbers will be used to locate regions on figures. It should be understood that these correct figures and numbers are available only after the cycle calculations are complete. In this example, we consider a system defined by R1 ) 0.1, R2 ) 0.2, and η ) 5. Parts a and b of Figure 6 show respectively the hodograph and physical planes. As we will show analytically, the periodic position of the shock at the end of the feed step is ζ1 ) 0.2545. In what follows, we will first determine the time at which the wave that carries both components exits the bed during the purge step, leaving only one component in the bed. In order to do that, we need to solve for the interaction region that develops when the two simple waves that form at the points of discontinuity along the

Figure 6. Example 2. Purification problem with R1 ) 0.1, R2 ) 0.2, and η ) 5: (a) hodograph plane; (b) physical plane.

bed meet. This means that we need to derive the coordinates of the nodes in the net in order to determine at what time and for what value of the appropriate Riemann invariant the wave that carries both components is out of the bed. Once that time is known, we evaluate the position in the bed of the only remaining wave and analyze its interaction with the shocks that form when we start feeding. This part of the analysis is in many respects analogous to what has been done in example 1. Purge Step. In this purification problem, the initial concentration profile, shown in Figure 7, consists of a region of pure light component for large ζ and a region in equilibrium with the feed all along the rest of the bed. We purge until the light component is completely


(b1 - a0) ζY ) 1 - b1(1 - ζ1) (b1 - a1)2

(40)

(b1 - a0) τY ) Fa12b12(1 - ζ1) (b1 - a1)2


removed from the bed and only some of the heavy component is still adsorbed. As purge begins, a fast centered wave emerges at ζ ) 1 and a slow one at the point of discontinuity along the bed (ζ ) ζ1). Point G in the hodograph plane is obtained as the intersection of a Γ- wave from B (the feed) and a Γ+ wave from F (the purge). G is on the + c* 2-axis, since Γ through F is the c* 2-axis. We know that FG is always a Γ+ (described by a, with constant b) with image C-, and GB is always a Γ- (described by b, with constant a) with image C+. Equation 36 gives the composition of G (c* 1 ) 0, c* 2 ) 1.189). In the physical plane, before interacting and getting transmitted through each other, the characteristics have equations

τ ) σ- ) -Fa2b1 C-: ζ-1

a0 e a e a1

τ ) σ+ ) -Fa0b2 ζ - ζ1

b0 e b e b1

In much the same way, along JW, all C+ characteristics that originate from point (ζ1,0) intersect the Ccharacteristic having a ) a0. It is thus possible to set up a system of equations between the generic equation for C+, with parameter b, and the equation that expresses the change in slope of C-. Differentiating both equations with respect to b, subtracting one from the other, and rearranging, we can solve the resulting ordinary differential equation to give ζ ) ζ(b). This expression can be used to get τ ) τ(b). Inside the interaction region, where the two simple waves go through each other, we know that b is constant along C+ and a along C-. Therefore, rearranging eqs 29, we obtain

C+:

1 1 ∂τ ∂ζ ∂τ ) )∂a σ+(a,b) ∂a Fab2 ∂a

C :

1 1 ∂τ ∂ζ ∂τ ) )- 2 ∂b σ-(a,b) ∂b Fa b ∂b

-

(41)

Differentiating the first equation with respect to b and the second one with respect to a and subtracting, we get a second-order partial differential equation in τ that can be treated in a way similar to that by Rhee et al. (1989). Using as boundary conditions the values of τ ) τ(a) along JY and τ ) τ(b) along JW, we obtain

τnet(a,b) )

[

Fa2b2(b1 - a0)(1 - ζ1) (a - b)2

1-

]

(39)

(a0 - a)(b1 - b) 2 (42) (a - b)(a0 - b1)

As outlined previously, we need to solve for the interaction net that forms when the two simple waves meet, in order to be able to evaluate the Riemann invariant and the relevant time when the wave that carries both components exits the bed. In order to do that, we have to determine the spatio-temporal paths of two of the sides of the interaction net, so that we have the boundary conditions that are required to solve the second-order partial differential equation that describes the net. In Figure 6b, the coordinates of J, the first intersection point, are obtained by solving the system of eqs 39 with a ) a0 and b ) b1. Along JY, all C- characteristics intersect one C+, so that a system of equations is set up between the generic equation for C-, with parameter a, and the equation that expresses the change in slope of C+. Differentiating both equations with respect to a, subtracting one equation from the other, and rearranging yield an ordinary differential equation that can be solved for ζ ) ζ(a). This expression can be used to get τ ) τ(a). Point Y, where the component closest to ζ ) 1 changes from light to heavy, is obtained by setting a ) a1, giving

Along a generic C- in the net, a is constant. So substituting τ ) τnet from this equation into the second of eqs 41, setting a ) a1 ) constant (which applies along YK), and integrating between Y and a generic point on the C- through Y (between Y and K), we obtain

C+:

ζ - ζY ) (1 - ζ1)

[

2D + N(3b - a1) 3(a1 - b)3

+

]

2a1b1 - a0b1 - a12 (a1 - b1)2

D ) a0a12 + a12b1 - 2a0a1b1

(43)

N ) 2a0a1 + 2a1b1 - a0b1 - 3a12 We solve eq 43 with ζ ) 0 for bK ) b* to get the value of the Riemann invariant b that corresponds to point K. Once b* is determined, it is substituted into τnet(a1,b) to obtain

τ1 )

Fa12(b*)2(b1 - a0)(1 - ζ1) (a1 - b*)2


[

1-

]

(a0 - a1)(b1 - b*) (44) 2 (a1 - b*)(a0 - b1) In reality, with all the sets of parameters that we have considered, it happens that b0 < b* < b1; i.e., the net is never entirely in the bed at the periodic state. For our system, we find τ1 ) 0.991 for removal of the light component (see Figure 6b). Figure 7 shows the concentration profile at various times as we purge. In profile b there is still a portion of plateau G in the bed, which disappears shortly thereafter. As both waves are gradually pushed toward ζ ) 0, eventually only the heavy component is left in the bed, as is apparent from profile f. The C+ characteristics that exit the net satisfy

τ - τnet(a1,b) ) -Fa1b2(ζ - ζYK(b))

(45)

which is obtained by combining eqs 13 and 22. Setting τ ) τ1 and ζ ) ζ*(b), we obtain net

ζ*(b) ) ζYK(b) +

τ

(a1,b) - τ1 Fa1b2

(46)

which represents the position in the bed at time τ1 of the concentration carried by the C+ defined by parameter b, as it has already gone through the interaction region. Once the faster gradual wave, which carries both the light and the heavy component, completely exits the bed, the purge is discontinued. Feed Step. The bed is now fed from ζ ) 0 with the feed mixture. The feed interacts with the heavy component left in the bed in such a way that the light component rolls up and by the time it breaks through the end of the bed the initial stepwise concentration profile has been restored. The pattern of interaction between the feed and what is left in the bed at the end of the purge step resembles that for example 1. One noteworthy difference is that there is not a plateau in the bed once the faster wave is gone, which means that two shocks, rather than one, emerge from ζ ) 0, τ ) τ1. Once again, the faster shock transmits the only residual gradual wave in the bed, whereas the slower shock cancels it. In this problem the feed step is interrupted as soon as the faster shock exits the bed, so that both the light and the heavy component are trapped in the bed and pure inert gas is eluted. In the purification process this gives a carrier purified of all adsorbable components and, for a recovery process, a purge product enriched in the adsorbable components. Figure 8 shows the concentration profile at different times as we feed. In profile b, next to the slower shock, the bed still contains a portion of the wave that has been transmitted through the faster shock. This wave is followed by plateau G and by the faster shock. Writing the equation of the straight lines through (ζ*(b), τ1) of slope Fa1b2 and the equation of the S- shock, differentiating the equation for the fan of C+ exiting the net with respect to b, and rearranging, we get a linear first-order ordinary differential equation that is solved with the condition that for ζ ) 0, we have b ) b*. It is thus possible to obtain ζ along KM and, by setting b ) b1, ζM, which is

Figure 8. Concentration profiles during feed in example 2: (a) τ ) 0.9907; (b) τ ) 1.05; (c) τ ) 1.1; (d) τ ) 1.1624. Solid lines represent component 1 (heavy), and dashed lines, component 2 (light).

ζM )

∫b*b

1 (b1 - a0)2

1

[

2ζ*(b)(b - a0) + b(b - a0)

]

dζ*(b) db db (47)

Once this is done, we know that, from eqs 13 and 28,

dτKM dζKM ) Fa0a1b db db

(48)

Therefore, we can evaluate τKM and, by setting b ) b1, τM, which is

τM ) τ1 + Fa0a1

∫b*b

1

[

]

2ζ*(b) + b(dζ*/db) - 2ζKM(b) (b - a0)

b db (49)

At this point, now that we have point M, we have, again from eqs 13 and 28,

τ2 ) τM + Fa0a1b1(1 - ζM)

(50)

For our system, this gives τ2 ) 1.162 for breakthrough of the light component and restoration of the initial condition. In much the same way, considering the equation of the S+ shock and the fan of C+ characteristics that gets transmitted through the S- shock, differentiating the equation for the fan with respect to b, and rearranging, we get a linear ordinary differential equation to be solved with the condition that b ) b* for ζ ) 0. This allows us to get ζ along KS (see Figure 6b) and, by setting b ) b1, ζS. By following the same procedure as for the S- shock, it is possible to get τ along KS and τS. Beyond point S, combining eqs 13 and 27 yields the equation of the S+ shock as

τ ) τS + Fa0b0b1(ζ - ζS)

(51)

Profiles c and d show how the shocks move to the right as time advances and how at τ2 the initial concentration profile is restored. Determination of the Periodic Position of the Shock. In order to determine a priori the periodic


position of the shock, it is necessary to perform a material balance on the light component, i.e., the component that is completely removed from the bed during the purge step. We know that between τ1 and τ2, enough of the light component must have been fed into the bed in order to reestablish the initial concentration profile. Therefore, integrating the second of eqs 2 (with + sign) over the bed length and time, we find

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

(52)

where qj 2* is the adsorbed-phase concentration corresponding to point G, with composition given by eq 36 and q*2,feed-binary given by eq 38. Moreover, combining eqs 49 and 50, we have

τ2 - τ1 )

∫b*b

Fa0a1

1

[

]

2ζ*(b) + b(dζ*/db) - 2ζKM(b) (b - a0)

b db +

Fa0a1b1(1 - ζM) (53)

Equation 52 has two unknowns, b* and ζ1. When coupled with eq 43 (with ζ ) 0) it yields a system of two equations in the two unknowns. In order to be meaningful, the solution to this system must be such that 0 < ζ1 < 1 and b0 < b* < b1. If this is not the case, then example 3 accounts for what is actually occurring in the system. By this process we obtained ζ1 ) 0.2545, which has been used throughout the example. The material balance given in eq 52 is based on the assumption that the interaction between the simple wave and the slower shock S+ is complete by the time the faster shock breaks through at ζ ) 1. Many different sets of parameters have been considered when numerically generating the characteristics for this problem. In all cases this assumption has been correct. Example 3. Purification with Total Removal of Heavy Component during Purge. This example differs from the previous example in that here the component that is removed completely from the bed during the purge is the heavy component. This is because the first wave to exit the bed is C+. All of the analysis that gave us the behavior of the interaction between C+ and C- simple waves is still valid here. We have chosen a system characterized by R1 ) 0.2, R2 ) 0.1, and η ) 5. For this system, we will find ζ1 ) 0.401. Parts a and b of Figure 9 show respectively the hodograph plane and the physical plane at the periodic state. Here again, we need to solve for the interaction region first in order to determine the value of the appropriate Riemann invariant and the relevant time when the wave that carries both components exits the bed. Then, we start feeding and analyze the interaction of the feed mixture with what is left in the bed at the end of the purge step. Since the faster shock cancels the simple wave, the slower shock will have a straight path. Purge Step. We purge the bed until we completely remove the heavy component and a heel of light component is still adsorbed. As we stop purging when the first wave exits the bed (in this case the C+ wave), a portion of the C- wave is still left in the bed. In order to find τ1, using a procedure similar to that outlined in example 2, we obtain, for ζ ) ζ(a) along JK, the equation

Figure 9. Example 3. Purification problem with R1 ) 0.2, R2 ) 0.1, and η ) 5: (a) hodograph plane; (b) physical plane.

(b1 - a0) 1 - ζ ) b1(1 - ζ1) (b1 - a)2

(54)

It is necessary to solve this equation for a* when ζ ) 0. This gives

a* ) b1 - xb1(b1 - a0)(1 - ζ1)

(55)

Then, since along any C- of the GF simple wave we have τ ) Fa2b1(1 - ζ), on the basis of eqs 13 and 23, we find that (when ζ ) 0)

τ1 ) Fb1[b1 - xb1(b1 - a0)(1 - ζ1)]2

(56)


carried by the C- defined by parameter a. At this point, we take the equation for straight lines through (ζ*(a), τ1) of slope Fa2b1 and the equation of the S- shock, write both in terms of a, and combine them to get a linear ordinary differential equation that is solved with the condition that ζ ) 0 when a ) a*. This yields

ζKM ) 1 -

(

)

a* - a0 a - a0

2

As for τ, along KM we have

τKM ) 2τ1 - Fa2b1


a* e a e a1

(58)

)

(59)

(

a* - a0 a - a0

2

Point M is characterized by a ) a1. Between M and (1, τ2), combining eqs 13 and 28, we find τ - τM ) Fa0a1b1(ζ - ζM), which gives

τ2 ) 2τ1 - Fa1b1(a1 - a0)

(

)

a* - a0 a1 - a0

2

(60)

For our system, we find τ2 ) 0.861 for breakthrough of the light component and restoration of the initial condition. The S+ shock simply connects the plateau composition to the feed composition. Its equation is

τ ) Fb1[a0b0ζ + (b1 - xb1(b1 - a0)(1 - ζ1))2]

Figure 11. Concentration profiles during feed in example 3: (a) τ ) 0.7177; (b) τ ) 0.73; (c) τ ) 0.77; (d) τ ) 0.82; (e) τ ) 0.8608. Solid lines represent component 1 (heavy), and dashed lines, component 2 (light).

For our system τ1 ) 0.718 for removal of the heavy component. Figure 10 shows concentration profiles at different times as we purge. The heavy component has been completely removed from the bed in profile f. Feed Step. This time the feed interacts with the light component left in the bed. This component rolls up and by the time it breaks through the end of the bed the initial concentration profile is restored. When we countercurrently feed B back into the bed, the C- wave gets intercepted and canceled by the Sshock of the same kind. Figure 11 shows bed profiles during the feed step. In profile b, the slower shock is followed by plateau G and then by the faster shock. This is followed by a portion of the wave that has not yet been canceled by the shock. The C- simple wave that is left in the bed is characterized by b ) b1 and a* e a e a1. At τ ) τ1, from τ ) Fa2b1(1 - ζ), which is a combination of eqs 13 and 23, we have

ζ*(a) ) 1 -

τ1 Fa2b1

(57)

which gives the location in the bed of the composition

(61)

Profiles c through e in Figure 11 illustrate how the shocks move to the right and in so doing how they cancel the residual wave and restore the initial concentration profile. Determination of the Periodic Position of the Shock. In order to determine a priori the periodic position of the shock, it is necessary to perform a material balance on the heavy component, i.e., the component that is completely removed from the bed during the purge step. Between τ1 and τ2, enough of the heavy component must have been fed into the bed to reestablish the initial concentration profile. Therefore, considering the first of eqs 2 (with + sign) and integrating it over the bed length and time, we obtain

(62)

q* 1,feed-binary ζ1 ) τ2 - τ1 Now, from eqs 5 and 6a, we have

q* 1,feed-binary )

R2

(63)

R1 + R2(1 - R1)

τ2 - τ1 ) τ1 - Fa1b1(a1 - a0)

(

)

a* - a0 a1 - a0

2

(64)

where τ1 is given by eq 56 and a* by eq 55. Thus, eq 62 is one equation in one unknown, ζ1. This equation is second order and a solution is meaningful only if 0 < ζ1 < 1. If, given the set of parameters for the system, a value in that range is not found, then the behavior of the bed is as outlined in example 2, where the light component is the component that is totally removed during the purge step. For the example, we found ζ1 ) 0.401. Discussion The first paper of this series analyzed the periodic behavior of a bed that is fed with only one adsorbable


component and inert gas and purged with pure inert gas. This paper extends the analysis to a binary mixture and studies the two problemssseparation and purification. The three parameters R1, R2, and η are required to specify the system. It has been recognized that it is not R, nor Λ, that by itself specifies the behavior of one component relative to the other, but rather the ratio of the two. In this context, mixtures characterized by components with the same value of the constant separation factor R and by different partition ratios Λ or by the same partition ratio and different constant separation factors can be analyzed by considering the respective values of R/Λ. We have referred to the component with the lower value of R/Λ as the heavy component. The ratio of these ratios has been shown to represent the selectivity between the two components. In the separation process, the choice of the set of parameters only affects the behavior of the bed in a qualitative way, since during the purge step the light component is the one that is removed completely, no matter what combination of parameters is picked. The combination will only affect the time for the purge and the feed step, both as absolute values and as values relative to one another. On the other hand, the purification process is different in that the component that is totally removed during the purge step can be either the light or the heavy component, depending on the pattern of wave interaction and ultimately on the choice of the set of parameters. However, at the end of the feed step the component that breaks through first is always the light component. If the two components are characterized by the same value of R/Λ, then there is no way to distinguish between them because the system is effectively azeotropic and in Langmuirian systems R1,2 is constant (i.e., unity at all concentrations). In the separation process, the plateau between the two centered waves disappears as does one of the waves. We only have a C- centered wave that connects the regenerated region to the region that is still saturated. This wave carries both components in equal proportions. Once it exits the bed, the bed is completely regenerated, so that the feed will introduce only one shock into the bed. It is recognized that in all three examples the times chosen for ending the purge steps (i.e., when all of one component has just been removed from the bed) are arbitrary and different termination criteria could have been used, possibly based on economic considerations. In particular, a lower ratio of purge time to feed time would improve performance. However, the analytical approach to the determination of the periodic state that we used should provide useful guidelines on solving a similar problem where the criterion for stopping the purge step is different. Simple expressions for the time required to completely remove the region in the bed that is saturated can be easily determined from the solution provided for all three cases. A way of evaluating the periodic position of the shock in the two purification cases has been provided. Once a set of parameters is assigned, it is necessary to determine which of the two cases will occur in practice, i.e., whether the light or the heavy component will be completely removed during the purge step. In order to make the determination, the value of ζ1 for the two cases needs to be calculated and the case that yields a value 0 < ζ1 < 1 is the solution. It is suggested that the value of ζ1 in example 3, which is easier to determine, be

evaluated first in order to determine quickly and in a straightforward manner which of the two cases will occur. The system has been analyzed in order to check if multiple periodic states could exist, given the nonlinearities present. However, no multiple periodic states have been found for the examples shown or for many sets of parameters that have been considered. Conclusions The periodic behavior of an adsorption bed with two components adsorbed from inert gas has been studied for a separation process and two purification processes. We have been able to solve for the periodic concentration profiles in the bed and to locate a priori the periodic position of the shock at the end of the feed step for the purification problem. We have derived analytical expressions for the times necessary to carry out the regeneration and feed steps in the processes. We have shown that each component is best characterized by a value of R/Λ and that for the heavy component this ratio is lowest. In the separation process, the bed was alternatively fed until saturated and then purged for as long as is required to completely remove the light component from the bed. No separation can be achieved if the ratio R/Λ is the same for the two components since the system has a selectivity of unity. In the purification processes, the bed is fed until the pure light component begins to break through at the bed outlet. At that point, the bed is purged until it contains only one component. This can be either the heavy or the light component, depending on the set of parameters that describes the system. Acknowledgment The authors are grateful to the U.S. Army ERDEC for financial support. Nomenclature a ) negative Riemann invariant, eq 25 b ) positive Riemann invariant, eq 26 c ) fluid-phase concentration, mol/m3 C( ) characteristic lines for simple waves in the physical plane Ki ) Langmuir isotherm parameter for component i, m3/ mol L ) bed length, m q ) adsorbed-phase concentration, mol/kg Qi ) Langmuir monolayer capacity for component i, mol/ kg Ri ) separation factor for the isotherm for component i, eq 6 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( ) characteristic lines in the hodograph plane ) void fraction of packing ζ ) dimensionless axial coordinate ζ1 ) periodic position of the shock in the purification problems η ) ratio of partition ratios for heavy and light components Λi ) partition ratio for component i Fb ) bulk density of packing, kg/m3

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2319 τ ) dimensionless time τ1 ) dimensionless duration of the purge step τ2 ) dimensionless duration of the feed step in purification problems τ3 ) dimensionless duration of the feed step in separation problems

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Received for review November 11, 1996 Revised manuscript received March 11, 1997 Accepted March 17, 1997X IE960716W

X Abstract published in Advance ACS Abstracts, May 1, 1997.

Equilibrium Theory for Pressure Swing Adsorption. 3. Separation and

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