Equilibrium Theory for Pressure Swing Adsorption. 5. Separation and

Ind. Eng. Chem. Res. 1999, 38, 2439-2449

2439

Equilibrium Theory for Pressure Swing Adsorption. 5. Separation and Purification in Multicomponent Adsorption Giuseppe Pigorini Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442

M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235

We analyze the periodic behavior of an adsorption bed that is alternatively fed with a dilute ternary mixture of adsorbable components in an inert carrier gas and purged countercurrently and incompletely with pure inert gas. We rely on local equilibrium theory to determine the asymptotic wave character of the process. We consider one purification and two separation problems and illustrate (i) how the duration of the feed step can be determined on the basis of the duration of the purge step and (ii) where intermediate shocks are located at the end of the feed step for the periodic state. We show how the approach used for a three-component system can be generalized to a system with n adsorbable components. Partial analytical contributions to the solution of one of the separation problems are provided to illustrate how the method of characteristics can, even for such complex systems, provide insight into the types of wave interactions taking place in the bed. Introduction We have been interested for some time in predicting the periodic behavior of pressure swing adsorption (PSA) cycles for systems where gas species present in trace amounts in an inert carrier gas are adsorbed in layered or unlayered beds. For systems of two adsorbable components we have considered both separation and purification processes.1 For all of the systems that we have studied, we have performed optimization analyses which allow us to choose the best cycle configuration.2 This paper concludes our line of this investigation. In it we want to extend the study of separation and purification processes to multicomponent systems; that is, we want to extend the methods illustrated in our previous papers to systems where more than two trace components are being adsorbed. We predict the concentration profile at the end of the feed step and perform a complete cycle, which in our simplified model consists of a purge and feed step only, to show that the predicted periodic state concentration profile is restored and is therefore indeed the correct one.3 The assumption that the adsorbates are present in truly trace amounts allows us to neglect any change in gas velocity in the bed and to neglect the energy balance, because for such systems the rate at which energy is released upon adsorption is not significant. Furthermore, pressurization and blowdown steps are treated using frozen adsorbed-phase profiles, as in our previous papers. We consider cycles in which none of the adsorbates are completely removed during the purge step. In the past, researchers have considered the asymptotic behavior of PSA cycles where adsorbates were completely removed during the purge step. This made the analysis * Author to whom correspondence should be addressed. Vanderbilt University, Box 1604, Station B, Nashville, TN 37235. Telephone: (615) 322-2441. Fax: (615) 343-7951. E-mail: [email protected].

simpler because the feed mixture was then fed to a clean bed, and a straightforward Riemann problem was solved either numerically or in some cases with partial analytical contributions.4 In this series, we have illustrated how to determine the periodic state without any restriction on the duration of the cycle. This has allowed us to vary the duration and to look for economically preferred cycles. In all systems we are able to predict the periodic concentration profile in the bed at the end of the feed step and the duration of the feed step once we fix the duration of the purge step. In our previous papers, we have relied on the theory of characteristics to analyze the cycle and to obtain some analytical contributions to the solution. It is possible to derive important analytical contributions to the solution of a three-component system also, and in theory, this can be extended to systems with n adsorbable components. In order to focus attention on the approach followed to determine a priori the periodic behavior of the cycle, we have limited the analytical contribution to portions of one of the examples, which is the most straightforward. In the other examples we solve the equations that govern our system numerically and show how our predictions of the periodic state based on equilibrium theory are confirmed by the numerical results. Our approach to the determination of the periodic state of PSA cycles is isotherm-independent because it only involves closure of the material balance for each adsorbate. The general dynamics of the feed and purge steps are also qualitatively the same for all systems in which adsorption is described by favorable isotherms.2 In this paper we restrict our analysis to Langmuirian systems. Equation Set Material Balances. We consider a system with n adsorbable components present in trace amounts in an

10.1021/ie9806949 CCC: $18.00 © 1999 American Chemical Society Published on Web 05/13/1999

2440 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

inert carrier gas. We assume that velocity is constant, since adsorption of trace gases is considered. We neglect gas-phase accumulation on the basis of the assumption that the partition ratio Λ is much greater than unity for all adsorbable components. The nondimensionalized material balance equations have the form3

∂q/i

∂c/i

1 ( )0 ηi ∂τ ∂ζ

QiKici (i ) 1, 2, ..., n) (2) 1 + K1c1 + K2c2 + ... + Kncn

where Qi is the monolayer capacity of component i. Reference concentrations are related by

qi,ref )

QiKici,ref 1 + Kici,ref

(3)

with ci,ref taken to be the feed concentration of component i. Manipulation of eqs 1 and 2 gives in dimensionless form

q/i

c/i /Ri

) n

1+

(4)

(1 - Rj)c/j /Rj ∑ j)1

where the constant separation factor Ri is given by

Ri )

1 1 + Kici,ref

(5)

Components are ordered in such a way that 1 is the heavy component and n is the light component. This also means that1

R1 R2 Rn < < ... < Λ1 Λ2 Λn

(6)

or, for a three component system like the one we study in the example section, we have R1 < η2R2 and R2 < (η3/η2)R3. Numerical Solution We are interested in both purifications and separations. In the purification problem we want to remove from an inert carrier gas the adsorbable components present in trace amounts in the feed mixture, so that the product during the feed step is the pure inert gas. We want to show for the periodic state of a purification process that at the end of the feed step the bed is characterized by a blocklike concentration profile, similar to that for a system of two adsorbable components in Parts 31 and 4.2 For a system of n adsorbable components in inert gas we want to be able to predict for the feed step the positions of n - 1 shocks at the

/

/

/

1

2

n

dτ ∆q1 1 ∆q2 1 ∆qn ) /) ) ... ) dζ ∆c η2 ∆c/ ηn ∆c/

(1)

where q/i ) qi/qi,ref, c/i ) ci/ci,ref, τ ) |v|t/(Λ1L), ζ ) z/L, Λi ) Fbqi,ref/ci,ref, and ηi ) Λ1/Λi, for i ) 1, 2, ..., n. The ( sign and the absolute value of velocity account for the fact that the flow direction is reversed every half cycle step. Isotherms. We describe adsorption equilibrium using the multicomponent Langmuir isotherm

qi )

time that the first shock breaks through the bed outlet. The composition of the concentration plateaus between these shocks can be determined by imposing the shock condition. The reciprocal of the shock speed written in nondimensional form is

(7)

The gas-phase and adsorbed-phase concentrations are related through the Langmuir isotherm, eq 4. Thus, the problem of determining the compositions of the plateaus between shocks amounts to solving a system of nonlinear algebraic equations, eqs 4 and 7. To position the shocks in the bed at the end of the feed step, we use the concept of the effluent ratio.2,5 Generalizing to a system with n adsorbable components gives n - 1 effluent ratios Ej defined as

Ej )

mol of j/mol of 1 removed during purge (8) cj/c1 in feed

where j ) 2, ..., n. We assign initial tentative values ζk to the positions of the shocks, k ) 1, ..., n - 1, and we know ζn ) 1 for the shock breaking through. Then, we purge countercurrently to the feed flow direction with pure inert gas for a chosen duration. We determine the amount of each adsorbable component that is removed from the bed and compute the n - 1 effluent ratios. The tentative positions of the shocks are the correct ones if at the end of the purge step the n - 1 effluent ratios are all unity. If this check is not satisfied, then the positions of the shocks are changed and the purge step is re-run. We change the positions of the shocks by applying Newton’s method. This requires that we determine the effect on all effluent ratios of changing the (r) location of each shock separately. If we call e(r) j ) Ej 1 the deviation from unity of effluent ratio j, then from Newton’s method for iteration r + 1 we have

) e(r) e(r+1) j j +

|

n-1

∂e(r) j

k)1

∂ζk

∑

ζ*ζk

dζk

(9)

Considering that we can evaluate the vector e(r) and the matrix ∂e(r)/∂ζ, with e(r+1) ) 0, we can determine the vector dζ and then the new values of ζk for iteration r + 1. In the separation problems, we want to separate one or more lighter components in inert carrier gas from the rest of the feed mixture. When dealing with separation of a gas mixture of n components, in general it might be desirable to retain components 1, ..., k in the bed while producing a product stream or product streams containing as much as possible of components k + 1, ..., n. In this case, if the k components are left completely in the bed at the end of the feed step, along with certain amounts of the lighter components k + 1 to n as determined by adsorption equilibrium and equilibrium theory dynamics, then k - 1 shocks must be located by defining Ej, j ) 1, ..., k - 1, and by imposing the condition that at the periodic state all of these effluent ratios must be unity. Therefore, the approach that we follow to solve a separation problem is the same as that for the purification problem, only the number of shocks to be located is smaller.

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2441

For the numerical contributions, we solve eq 1 by discretizing it with backward differences and by solving the resulting system of ordinary differential equations using ODEPACK,6 a Gear’s method solver. Problems Considered We consider two separation problems and one purification problem. For a gas mixture with traces of three adsorbable components, we illustrate separation of the light components from the heaviest component in the first example and separation of the lightest component from the heavy components in the second example. The third example addresses the purification of the inert carrier gas by retention of all of the adsorbable components in the bed during the feed step. Thus, the examples consider in order the breakthrough of two, one, and then no components. We explain the mathematical background that is required to derive analytical contributions to the solutions in the first separation problem, that is, separation of the light components from the heaviest component. This is the easiest example of the three to solve analytically. The other two problems could also be partially solved analytically by the same method with a more involved analysis. For those problems, we have chosen to focus on the numerical method used to predict the periodic state, which can be applied to general isotherm relations. Example 1: Separation of Light Components from the Heaviest Component. In this example, we analyze a PSA cycle in which two light adsorbable components of a three-component mixture in inert gas are separated from the heaviest component. During the feed step, the light components in inert gas are retrieved at one end of the bed, and during the subsequent purge step, the purge stream is enriched in the heaviest component. We provide the mathematical background that is necessary to give partial analytical contributions to the solution of the problem. The complete theoretical background can be found in refs 7 and 8. Three-component and multicomponent adsorption systems can be treated from a mathematical standpoint in a manner similar to that for the development in Parts 31 and 42 for a system of two adsorbable components. In general, once the parameters that characterize the system are provided, it is necessary to determine the values of ωk, the Riemann invariants for our system of n adsorbable components; these are equivalent to a and b for the two-component system of Part 3. In particular, we need to solve the equation n

Akq/k

∑γ

k)1

k

-ω

)1

(10)

for ω, where γk ) l/(ηkRk) and Ak ) (1 - Rk)/(Rkηk), in order to determine the value of these invariants for each region of constant concentration. Each region is characterized by a set of n values of ω. Equation 10 expresses the coherence condition for a system of n adsorbable components.9 The Riemann invariants are used to determine the slopes of the characteristics belonging to the various simple waves, the concentrations carried by simple waves, and the speeds of shocks. We have considered a system in all examples characterized by R1 ) 0.1, R2 ) 0.5, R3 ) 0.8, η2 ) 0.8, and η3 ) 1.5. We assume, and later show, that at the end of

Figure 1. (a, top) Physical plane for example 1 with heels of all components left in the bed at the end of the purge step. (b, bottom) Magnification of the portion of part a at high values of τ and ζ.

the feed step the bed is uniformly loaded in the three components. As we start purging the bed with pure inert gas, countercurrent to the feed, three simple waves originate from ζ ) 1, separated by plateaus, and these waves connect the purge concentration to the feed concentration. Along each wave only one of the Riemann invariants ω varies; the others stay constant. The invariant that changes is the one corresponding to the component that disappears across that wave, which also gives the name to the relevant wave; that is, the Cj wave is the simple wave on one side of which component j disappears. Figure 1a shows the physical plane for the separation problem. We have chosen a duration of the purge step of time τ1 ) 0.1. Solving eq 10 given the purge concen-


tration (state R, c/1R ) c/2R ) c/3R ) 0), we obtain ωp1 ) γ1 ) 10, ωp2 ) γ2 ) 2.5, and ωp3 ) γ3 ) 0.833. Similarly, for the feed concentration (state B, c/1B ) c/2B ) c/3B ) 1), we obtain ωf1 ) 3.496, ωf2 ) 1.126, and ωf3 ) 0.470. Each plateau region is characterized by the relevant set of ω parameters. Across a Sk shock or a Ck wave, all ω’s stay constant except for ωk. Thus, simple wave C1 is described by parameter ω1; this parameter varies from the value ωp1 to ωf1 while connecting the purge concentration to the region of uniform concentration F between this wave and the next one, C2. Plateau F is therefore characterized by F(ωf1, ωp2, ωp3). Along wave C2, we have ω1 ) ωf1, whereas ω2 changes from ωp2 to ωf2. This means that plateau P is characterized by P(ωf1, ωf2, ωp3). Finally, along wave C3 both ω1 and ω2 stay constant, whereas ω3 changes from ωp3 to ωf3. We stop purging the bed before any of the simple waves have been completely removed from the bed. We have shown previously that low purge-to-feed ratios often give an economic optimum.2 Following the purge step, we start feeding the bed from left to right; a S3 shock forms upon interaction of the feed mixture with the C3 wave, as shown in Figure 1a. This shock has a curved path until it reaches the plateau between the C3 and C2 waves, where its path becomes straight. As soon as the shock meets the C2 wave, it splits into a S3 shock and a S2 shock. The S3 shock transmits the C2 wave, whereas the S2 shock cancels it. Both shocks have a curved path. Their paths become straight when they interact with constant states. In particular, S3 interacts with plateau F between waves C2 and C1, whereas S2 interacts with the region of constant concentration G that rolls up between the C2 and C1 waves that are transmitted by shock S3. This is shown in Figure 1b, which is a magnification of the portion of Figure 1a at high values of τ and ζ. Plateau G contains only the light and heavy components. This behavior is interesting, since the component characterized by the intermediate value of the R/Λ ratio is not present in this region of uniform concentration, whereas the other two are. Considering that we have shown that adsorbates can be ordered with respect to each other on the basis of the value of that ratio, the most strongly adsorbed component being the one with the lowest such ratio, one would expect that components that are next to each other in this classification would have a tendency to stay together, which is not what happens here. Since G is separated from state B by shock S2, it is given by G(ωf1, ωp2, ωf3); that is, only ω2 changes with respect to B. Wave C1 is transmitted by shock S3, whereas shock S2 splits upon meeting with C1, as is shown in Figure 1b. Two shocks form, a S2 shock which transmits the wave and a S1 shock that cancels it and that ultimately restores the initial uniform concentration profile in the bed. Plateau M is given by M(ωp1, ωf2, ωf3), since it is separated from B by shock S1, and plateau H is defined by H(ωp1, ωp2, ωf3) because it is across S2 from M. The slope of the S3 shock is given by5

(dζdτ)

S3

)

ω1ω2 l r ωω γ1γ2γ3 3 3

(11)

with the varying ω being the one that characterizes the simple wave that the shock meets and the other ω’s taking values that change according to which portion

of the shock path we consider. Specifically, for the S3 shock we have ωl3 ) ωf3 because all regions to the left of S3 have the same value of ω3 as the feed. Between points A and C, we have ω1 ) ωf1, ω2 ) ωf2, and ωr3 varies according to C3. Between C and E, since the shock is across shock S2 from state B, ω2 changes with C2 and ω1 ) ωf1 and ωr3 ) ωp3. Finally, between E and ζ ) 1, ω1 changes with C1 and ωr3 ) ωp3 and ω2 ) ωp2. Similar considerations apply to shock S2, which has a slope given by

(dζdτ)

)

S2


(12)

Between C and J, ωr2 varies according to the transmitted C2, whereas, between J and W, we have ωr2 ) ωp2 and ω1 varies as it interacts with C1. Moreover, ω3 ) ωf3 and ωl2 ) ωf2. As for shock S1, its slope is given by

(dζdτ)

)

S1


(13)

with ωr1 varying with C1 and ωl1, ω2, and ω3 taking on values corresponding to those for feed B. Simple waves behave in a similar fashion as they interact with shocks. Wave C3, which will be canceled by shock S3, has slope

(dζdτ)

)(

C3

ω1ω2 2 ω γ1γ2γ3 3

(14)

and is described by parameter ω3. The + sign corresponds to the feed step, and the - sign, to the purge step. For this wave, we have ω1 ) ωf1 and ω2 ) ωf2. The slope of wave C2 is given by

(dζdτ)

ω1ω3 2 )( ω C2 γ1γ2γ3 2

(15)

with appropriate values for ω1 and ω3 depending on the region that the wave is in. Specifically, before meeting up with shock S1, we have ω1 ) ωf1 and ω3 ) ωp3, whereas in between shocks, after crossing shock S3, ω1 remains constant and ω3 ) ωf3. Wave C1 interacts with all three shocks. Its slope is given by

(dζdτ)

ω2ω3 2 )( ω C1 γ1γ2γ3 1

(16)

with the values of ω2 and ω3 depending on the region the wave is in. In particular, before crossing shock S3, we have ω2 ) ωp2 and ω3 ) ωp3. Upon crossing S3, we have ω3 ) ωf3 and, as the wave crosses S2, ω2 ) ωf2. It is worth noting that, with respect to the separation problems illustrated in Parts 31 and 4,2 the fact that we are dealing with a three-component system means that during the feed step two shocks split, rather than only one, and one plateau (G) forms and later disappears, so that the constant concentration plateau region does not reach the bed outlet. Concentrations of the plateau regions are determined as follows. As a general rule, upon crossing shock Sk from the region of feed concentration B, component k disappears and the other two components have concentrations related to the feed concentration by the expres-


sion8

and

c/i

)

1 - γi/ωpk

c/,f i

1 - γi/ωfk

(17)

with i * k. In particular, plateau P(c/1 ) 0.5425, c/2 ) 0.4631, c/3 ) 0) contains components 1 and 2, plateau G(c/1 ) 0.3807, c/2 ) 0, c/3 ) 2.5647) contains components 1 and 3, and plateau M(c/1 ) 0, c/2 ) 2.6325, c/3 ) 1.2024) contains components 2 and 3. Similarly, upon crossing another shock Sj, component j disappears and the concentration of the remaining component is given by

c/i ) c/,f i

1 - γi/ωpk 1 - γi/ωpj 1 - γi/ωfk 1 - γi/ωfj

(18)

Thus, plateau F(c/1 ) 0.2065, c/2 ) 0, c/3 ) 0) contains only component 1, whereas plateau H(c/1 ) 0, c/2 ) 0, c/3 ) 3.0840) contains only component 3. Between plateaus separated by a simple wave Cm, component m appears/ disappears. Across that wave, the concentrations of components will vary on the basis of the multiplying factor

1 - γi/ωm

(19)

γi/ωfm

1-

For instance, we have shown that, between shocks S2 and S3, wave C1 separates plateau G, which contains components 1 and 3, from plateau H, which contains only component 3. Therefore, component 1 will disappear along wave C1 as we go from G to H. Moreover, that portion of C1 lies across shock S2 first and then across shocks S2 and S3 from state B. Therefore, the concentrations of components 1 and 3 on that portion of C1 will be a function of ω1 and will be given by

c/i ) c/,f i

1 - γi/ωp2 1 - γi/ω1 1 - γi/ωf2 1 - γi/ωf1

(20)

with i ) 1, 3. To illustrate the development of an analytical expression for τ2, the breakthrough time for the first shock (see Figure 1a), given the purge time τ1, we rely on knowledge of simple waves and shocks. In our example, we have a purge time of τ1 ) 0.1. Simple wave C3 is described by eq 14. Using that equation, this purge time j 3 ) 0.7275 e ωp3 when corresponds to the value ωf3 e ω we set ζ ) 0. At τ ) τ1, the generic characteristic belonging to this wave has reached position ζ′ ) 1 j 3 e ω3 e ωp3 and a ) ωf1 τ1/(aω23) in the bed, with ω f ω2/γ1γ2γ3. As we start feeding the bed countercurrently with the feed mixture, the slopes of the characteristic lines change sign. These characteristics are described by τ ) 2τ1 - aω23(1 - ζ). Differentiating this equation and eq 11 with respect to ω3, subtracting one from the other, and integrating the result between ζ ) 0 and ζ, we find along portion AB of shock S3

ζ)1-

(

ωf3 - ω j3

ωf3

)

- ω3

2

ω j 3 e ω3 e ωp3

(21)

τ ) τ1 +

2b(ωf3

-ω j 3)

[

ωf3 - 2ω3

2

2(ω3 - ωf3)2

-

ωf3 - 2ω j3

]

2(ω j 3 - ωf3)2 (22)

with b ) aωf3. The coordinates of point B can be obtained by setting ω3 ) ωp3 in eqs 21 and 22. For our system, we find ζB ) 0.4968 and τB ) 0.1340. As for point C, it is the intersection of the first C2 characteristic (corresponding to ω2 ) ωf2), for which the slope has changed sign at τ ) τ1, and the straight shock S3 through B. Substitutions give ζC ) (τB - 2τ1 + a˜ b˜ ζB)/(a˜ - b˜ ) with a˜ ) ωf1(ωf2)2/γ1γ2 and b˜ ) ωf1ωf2ωf3/γ1γ2. Furthermore, it follows that τC ) 2τ1 - a˜ (1 - ζC). For our system, we obtain ζC ) 0.7220 and τC ) 0.1507. The portion of the S3 shock between C and D is evaluated by following the same procedure, that is, by differentiating the equation of the shock and that of the deflected C2 wave with respect to ω2, subtracting one equation from the other, and integrating between ωf2 (point C) and the generic ω2. We obtain

ζ ) 1 - (1 - ζC)

(

)

ωf3 - ωf2

2

ωf2 e ω2 e ωp2

ωf3 - ω2

and

τ ) τC +

2b h (ωf3

-

ωf2)2(1

[

- ζC)

ωf3 - 2ω2

2(ω2 - ωf3)2

(23)

-

ωf3 - 2ωf2

]

2(ωf2 - ωf3)2

(24)

where b h ) ωf1ωf3/γ1γ2. Point D is characterized by ω2 ) p ω2. For our system, we find ζD ) 0.9710 and τD ) 0.1746. Point E is the intersection between shock S3 and the first C1 characteristic. Its coordinates are given by ζE ) (τD - 2τ1 + a′ - b′ζD)/(a′ - b′) and τE ) 2τ1 - a′(1 - ζE), with a′ ) (ωf1)2/γ1 and b′ ) ωf1ωf3/γ1. For our system, we have ζE ) 0.9805 and τE ) 0.1762. Finally, the portion between E and F of shock S3 is given by

ζ ) 1 - (1 - ζE)

(

τ ) τE +

-

2

ωf1 e ω e ωp1

ωf3 - ω1

and

2b*(ωf3

)

ωf3 - ωf1

ωf1)2(1

[

- ζE)

ωf3 - 2ω1

2(ω1 - ωf3)2

(25)

-

ωf3 - 2ωf1

]

2(ωf1 - ωf3)2

(26)

where b* ) ωf3/γ1. Point F is characterized by ω1 ) ωp1. For our system, we obtain ζF ) 0.9980 and τF ) 0.1803. The time τ2 when shock S3 reaches ζ ) 1 is given by τ2 ) τF + ωf3(1 - ζF) ) 0.1813. In order to determine the time τ4 at which shock S1 reaches ζ ) 1, it is sufficient to perform a material balance on the heavy component. Specifically, since no heavy component is retrieved from the bed outlet


Figure 2. Concentration profiles during purge for example 1 at τ1 ) 0.1: (a) τ ) 0.0; (b) τ ) 0.02; (c) τ ) 0.04; (d) τ ) 0.06; (e) τ ) 0.1. The subscripts 1-3 indicate the various components, 1 being the heavy and 3 the light component.

Figure 3. Concentration profiles during feed for example 1 at τ1 ) 0.1: (a) τ ) 0.1; (b) τ ) 0.13; (c) τ ) 0.165; (d) τ ) 0.178. The subscripts 1-3 indicate the various components, 1 being the heavy and 3 the light component.

between τ1 and τ4, the bed must be fed long enough to restore the initial uniform concentration profile, thereby replenishing the heavy component that was removed during the purge step. Integrating eq 1 (with + sign) for i ) 1 over the bed length and between times τ1 and τ4 gives

a function of time during the purge step and check to determine if at the end of the purge step E2 is equal to unity within a certain tolerance. If it is not, we repeat the analysis by adjusting the position of the shock using Newton’s method until the aforementioned condition is met. For this system we have chosen a purge time of τ1 ) 0.2. For this, the periodic position of the shock is ζ1 ) 0.527. Figure 4 shows the periodic concentration profile at the end of the feed step or at the beginning of the purge step. In Figure 5 we plot numerically generated concentration profiles in the bed during the purge step. As we start purging the bed from right to left with inert gas, the carrier interacts with a two-component region, specifically with the region to the right of the shock, which contains components 2 and 3 (see Figure 5b and c). Since the purge stream, which contains no adsorbates, interacts with a portion of the bed with two adsorbates, two simple waves form at the point of discontinuity at ζ ) 1, and they connect the purge stream concentration to the plateau concentration to the right of the shock. The two simple waves are separated by a plateau of component 2, which can be seen on the right side of Figure 5b. Another simple wave forms at the point of discontinuity corresponding to the position of the intermediate shock. In general, we expect complex interactions to take place between the three simple waves that form as we start purging. As the figure shows, at the end of the purge step, heels of all three adsorbates are still left in the bed. In Figure 6 we plot concentration profiles in the bed during the feed step in order to show that at the end of the feed step the initial concentration profile is restored with the shock positioned at ζ1 ) 0.527 at the periodic state. As we feed the bed countercurrently to the purge with feed mixture, we expect three shocks to form as

q/1,feed -

∫01q/1 dζ ) τ4 - τ1

(27)

/ where q1,feed is obtained from eq 4 with c/i ) ci,feed ) 1. For our system we obtain τ4 ) 0.1868. Figure 2 shows concentration profiles in the bed at different times as we purge. It is apparent that three simple waves separated by two plateaus form as we start feeding pure inert carrier gas from ζ ) 1. At the end of the purge step, heels of all three components are left in the bed. Figure 3 illustrates concentration profiles at different times during the feed step. Shocks form and gradually capture the simple waves that are left in the bed at the end of the purge step until eventually, at τ4 ) 0.1868, the initial concentration profile has been restored. Example 2: Separation of the Lightest Component from Heavy Components. In this example, the lightest component is separated from the two heavier components. The system that we consider is the same as that in example 1. Since only component 3 is retrieved from the mixture during the feed step, we again expect the periodic concentration profile in the bed at the end of the feed step to be blocklike, with one shock at some intermediate position which is to be determined. In order to locate the position, we fix the duration of the purge step in our cycle and the periodic position of the shock in the bed at the end of the feed step. We then generate a plot of the effluent ratio E2 as


Figure 4. Periodic concentration profile at the end of the feed step for example 2.

the feed interacts with a different set of concentrations at ζ ) 0, as shown in Figures 6. The interactions between shocks and simple waves, with the related transmissions and cancellations, make the shock paths curved. The light component in inert gas is recovered as product between the first and second shocks. As soon as the second shock reaches the bed outlet, we discontinue feeding. In order to determine the time at which the second shock reaches the bed outlet, given the duration of the purge step, we perform a simple material balance. For this, we need to know the concentration of the plateaus that form as we feed. For our system, to the left of the shock we have all three components, with c/1 ) c/2 ) c/3 ) 1, whereas to the right of the shock the concentrations are c/1 ) 0, c/2 ) 2.6325, and c/3 ) 1.2024, as for plateau M in example 1. Thus, to determine when the initial concentration profile is restored, we perform a material balance on the heaviest component, which yields

q/1,feedζ1 -

∫01q/1 dζ ) τ3 - τ1

(28)

For our system we find τ3 ) 0.3736. Example 3: Purification Problem. In this example, we remove three components present in trace amounts in a carrier. We consider the same system as we have used for our separation examples. In this problem, we have to locate two shocks in the bed at the end of the feed step. We predict that the profile in the bed is blocklike, with the composition of the plateaus being determined by applying the shock condition, eq 7. The plateau to the left has c/1 ) c/2 ) c/3 ) 1, the one in between the two shocks has c/2 ) 2.6325 and c/3 ) 1.2024 (the same as for plateau M in example 1), and the one to the right has c/3 ) 3.084 (the same as for plateau H in example 1). We pick the duration of the

Figure 5. Concentration profiles during purge for example 2 and for (a, top) component 1, (b, center) component 2, and (c, bottom) component 3, at τ1 ) 0.2. Times are (1) τ ) 0.0, (2) τ ) 0.04, (3) τ ) 0.08, (4) τ ) 0.12, and (5) τ ) 0.2. 1 is the heavy component and 3 the light component. ∆ζ ) 5 × 10-4.


Figure 7. Periodic concentration profile at the end of the feed step for example 3.

purge step and the position of the two shocks and then plot the two effluent ratios, E2 and E3, to determine if they both equal unity at the end of the purge step. For τ1 ) 0.3 this condition is satisfied by ζ1 ) 0.6233 and ζ2 ) 0.8611. Figure 7 shows the periodic concentration profile at the end of the feed step or at the beginning of the purge step. In Figure 8 we plot concentration profiles in the bed at different times during the purge step. At the beginning of the purge step we expect one simple wave to form at each point of concentration discontinuity in the bed, including at ζ ) 1. These waves will in general interact, and the interactions will be complex. Figure 9 shows concentration profiles in the bed at different times during the feed step. As we feed countercurrently, shocks form and gradually capture the simple waves and restore the blocklike concentration profile. In order to determine a priori the duration of the feed step, given the duration of the purge step, a material balance can be performed on any component. If we choose the heaviest component, then we have

q/1,feedζ1 -

∫01q/1 dζ ) τ2 - τ1

(29)

For our system, we obtain τ2 ) 0.544. Discussion

Figure 6. Concentration profiles during feed for example 2 and for (a, top) component 1, (b, center) component 2, and (c, bottom) component 3, at τ1 ) 0.2. Times are (1) τ ) 0.2, (2) τ ) 0.2087, (3) τ ) 0.2174, (4) τ ) 0.2347, and (5) τ ) 0.3736. 1 is the heavy component and 3 the light component. ∆ζ ) 5 × 10-4.

General Behavior. Each of the three examples has treated one purge time only. The general behavior for the full range of purge times can be considered. Figure 10 shows the duration of the feed step as a function of the duration of the purge step for the three examples that we have illustrated. Curve A corresponds to example 1, in which two components break through and all shocks reach the bed outlet. It shows that the feed step time progressively increases as we purge the bed


Figure 8. Concentration profiles during purge for example 3 and for (a, top) component 1, (b, center) component 2, and (c, bottom) component 3, at τ1 ) 0.3. Times are (1) τ ) 0.0, (2) τ ) 0.06, (3) τ ) 0.12, (4) τ ) 0.18, and (5) τ ) 0.3. 1 is the heavy component and 3 the light component. ∆ζ ) 5 × 10-4.

Figure 9. Concentration profiles during feed for example 3 and for (a, top) component 1, (b, center) component 2, and (c, bottom) component 3, at τ1 ) 0.3. Times are (1) τ ) 0.3, (2) τ ) 0.3122, (3) τ ) 0.3244, (4) τ ) 0.3488, and (5) τ ) 0.544. 1 is the heavy component and 3 the light component. ∆ζ ) 5 × 10-4.


Figure 10. Plot of the duration of the feed step as a function of the duration of the purge step for examples 1 (curve A), 2 (curve B), and 3 (curve C). Numbers indicate number of components breaking through in that region. ∆ζ ) 1 × 10-2.

more, until it stays constant at τfeed ) τ4 ) 0.889 for τpurge ) τ1 > 1/R1 ) 10, which corresponds to complete regeneration of the bed. Curve B corresponds to example 2, with one component breaking through. Feeding times are shorter than those in example 1, and the bed is cleaned out earlier. The feed time for complete purge is τfeed ) τ3 ) 0.6356. Finally, curve C corresponds to example 3, with no breakthrough, and shows that feed time for complete purge is τfeed ) τ2 ) 0.4705. From Figure 10 we find the following. If the duration of the feed step is longer than τfeed ) 0.6356, then two or three components break through at the periodic state depending on whether the duration of the purge step is to the right or to the left of curve A. If the duration of the feed step is shorter than τfeed ) 0.6356 and longer than τfeed ) 0.4705, then three components break through at the periodic state if the duration of the purge step is to the left of curve B, two break through if it is between curve B and curve A, and one breaks through if it is to the right of curve A (long purge time). Finally, if the duration of the feed step is shorter than τfeed ) 0.4705, then three components break through if the duration of the purge step is to the left of curve C, two break through if it is between curve C and curve B, one breaks through if it is between curve B and curve A, and none break through for purge times to the right of curve A. Blocklike Profile. In Parts 3 and 4 of this series we showed several examples in which the blocklike concentration profile was restored at the end of the feed step, but we did not provide general reasoning to show that this will occur irrespective of the set of parameters that describe the system. Here we want to provide such reasoning, albeit not an analytical proof but deductive in nature, which is valid for systems with any number of trace adsorbates. Basically, we want to illustrate why in the purification problem, at the periodic state and at the end of the feed step, the shocks that form and capture the waves that are in the bed at the end of the purge step do so in a complete way, so that when the faster shock hits the bed outlet the other shocks have

already canceled all waves and have restored a blocklike concentration profile. We begin by noting that all cycles of practical interest have a duration of the purge step intermediate between two values that correspond to limiting cases. These are dimensionless purge time τpurge, equal to dimensionless feed time τfeed, and long purge time, corresponding to complete regeneration of the bed. These are discussed separately below. First, as we start purging a bed that has reached its periodic state, simple waves form at each point of concentration discontinuity, in a manner similar to what we showed for the binary system in Parts 3 and 4. These waves in general interact as they meet each other while moving in the bed, and in so doing they get distorted. For systems described by simple favorable isotherms, no shocks form during the purge step. If we stop purging as soon as the faster simple wave or an interaction of waves reaches the bed inlet, and we start feeding the bed countercurrently with the feed mixture, we know that the characteristics in the physical plane will be the mirror images of the ones that formed during the purge step. Moreover, since the feed mixture is fed to a bed that at ζ ) 0 is in equilibrium with it, no shock will form that captures these waves but rather the waves will gradually get sharper, until the characteristics will eventually converge onto the points of discontinuity from where the waves originated at the beginning of the purge step. In other words, for this limiting case the shocks will form and they will form at the end of the feed step only, with the feed step having a duration equal to that of the purge step (or, in other words, with a volumetric purge-to-feed ratio γ of unity). The other limiting case of interest is the one where the bed is entirely regenerated during the purge step; that is, it is clean at the end of the purge step. In this case, as we start feeding countercurrently with the feed mixture, a number of shocks equal to the number of adsorbable components will form and they will attain maximum strength from the beginning of the feed step. Thus, if we consider a duration of the purge step such that we do not completely clean the bed out but at the same time we consider a volumetric purge-to-feed ratio γ greater than unity, the shocks will form from the beginning of the feed step, unlike in the limiting case characterized by γ ) 1. However, these shocks will not be as strong as they would be if the bed had been completely regenerated during the purge step, because they are interacting with waves that are still in the bed. Thus, if the blocklike concentration profile is restored in the weakest case (γ ) 1), this will also happen a fortiori when shocks form from the beginning of the feed step. Actually, because the shocks are stronger and sweep through the bed faster, the duration of the feed step is shorter than that of the purge step (which is why γ is greater than unity). This argument carries over to the other limiting case, with complete regeneration of the bed, in which the shocks are strongest from the beginning of the feed step. Conclusions We have illustrated how to determine the periodic configuration of PSA cycles for separation and purification of a mixture of three adsorbable components in inert gas. The approach is general and can be applied to systems of n adsorbable components. We have as-


sumed that the components are present in trace amounts and that heat effects upon adsorption are negligible. Partial analytical contributions to one of the examples have been given to illustrate how the theory of characteristics, which we have extensively relied upon in previous papers as a guide to the study of several systems, can be useful for more complex systems also. All that is required to analyze a particular system is the set of parameters by which it is described and the duration of the purge step. Our analysis allows us to predict a priori the concentration profile in the bed at the end of the feed step and thus the location of the shocks that form as we start feeding and the duration of the feed step. This in turn makes it possible to perform optimization analyses in a manner similar to what we illustrated in two of our previous papers,1,10 because it is possible to determine how the volumetric purge-to-feed ratio γ changes as we change the duration of the purge step. We can therefore choose the duration of the purge step in such a way that certain optimization functions (such as those mentioned in Part 4 of this series) are maximized. In the separation problem, which we have illustrated in more detail, we have shown that, for a threecomponent system with heels of all components left in the bed at the end of the purge step, two shock splittings take place and one plateau region forms that does not reach the bed outlet and that contains the light and the heavy components only. In an n-component system, with heels of all components left in the bed at the end of the purge step, n - 1 shock splittings take place and n - 1 shocks reach the bed outlet before the nth shock restores the initial uniform concentration profile. More than one plateau region with the same peculiarity observed for a three-component system, that is, containing components with nonsubsequent R/Λ values, should form and not reach the bed outlet. The approach to the determination of the periodic state is not restricted to Langmuirian systems. Although significant analytical contributions to the prediction of cycles for systems governed by a generic favorable isotherm may not be possible, the effluent ratio concept extends naturally to all systems, regardless of the isotherm by which they are described. Also, for systems described by favorable isotherms, concentration profiles during the purge and feed steps should be qualitatively similar to the ones illustrated in this paper. Acknowledgment The authors are grateful to the U.S. Army ERDEC for financial support. Notation c ) fluid-phase concentration, mol/m3 C ) simple wave in the physical plane E ) effluent ratio, eq 8 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 isotherm for component i, eq 5 S ) shock in the physical plane t ) time, s v ) interstitial velocity, m/s z ) axial coordinate, m Greek Letters ) void fraction of packing ζ ) z/L, dimensionless axial coordinate ζi ) periodic position of shock i ηi ) ratio of partition ratios for heavy (1) and light (i) components Λi ) partition ratio for component i Fb ) bulk density of packing, kg/m3 τ ) |v|t/(Λ1L), dimensionless time τ1 ) dimensionless duration of purge step τ2 ) dimensionless duration of cycle in purification problem; dimensionless time for breakthrough of component 3 in examples 1 and 2 τ3 ) dimensionless duration of cycle in example 2 τ4 ) dimensionless duration of cycle in example 1 ω ) generalized Riemann invariant

Literature Cited (1) Pigorini, G.; LeVan, M. D. Equilibrium Theory for Pressure Swing Adsorption. 3. Separation and Purification in TwoComponent Adsorption. Ind. Eng. Chem. Res. 1997, 36, 23062319. (2) Pigorini, G.; LeVan, M. D. Equilibrium Theory for Pressure Swing Adsorption. 4. Optimizations at Short Cycle Times for Trace Separation and Purification in Two-Component Adsorption. Ind. Eng. Chem. Res. 1998, 37, 2516-2528. (3) LeVan, M. D. Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655-2660. (4) 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. (5) Davis, M. M.; McAvoy, R. L.; LeVan, M. D. Periodic States for Thermal Swing Adsorption of Gas Mixtures. Ind. Eng. Chem. Res. 1988, 27, 1229-1235. (6) Hindmarsh, A. C. Odepack, A Systematized Collection Of Ode Solvers. In Scientific Computing; Stepleman, R. S., et al., Ed.; North-Holland: Amsterdam, 1983; pp 55-64. (7) 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. (8) Courant, R.; Friedrichs, K. O. Supersonic Flow and Shock Waves; Interscience: New York, 1948. (9) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley: New York, 1984. (10) Pigorini, G.; LeVan, M. D. Equilibrium Theory for Pressure Swing Adsorption. 2. Purification and Enrichment in Layered Beds. Ind. Eng. Chem. Res. 1997, 36, 2296-2305.

Received for review November 4, 1998 Revised manuscript received March 25, 1999 Accepted April 7, 1999 IE9806949

Equilibrium Theory for Pressure Swing Adsorption. 5. Separation and

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