Simple Criterion for Adsorbent Selection for Gas Purification by

concentration just leaving the adsorber at the end of the purge step, is employed for selecting the best adsorbent and the optimal bed layering. This ...
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Ind. Eng. Chem. Res. 2005, 44, 1914-1921

Simple Criterion for Adsorbent Selection for Gas Purification by Pressure Swing Adsorption Processes Jong-Ho Park† and Ralph T. Yang*,‡ Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, Korea Institute of Energy Research, 71-2 JangDong Yuseong-gu Deajeon, 305-343, Korea

A simple criterion is derived for the selection of adsorbent as well as optimal bed layering for layered beds for the removal of a single trace impurity by pressure swing adsorption (PSA). The theory is based on equilibrium theory under the assumptions of isothermality and frozen solid approximation during pressure changing steps. A working capacity for the adsorbent, defined as the difference between the amount adsorbed under the feed condition and that at the impurity concentration just leaving the adsorber at the end of the purge step, is employed for selecting the best adsorbent and the optimal bed layering. This working capacity depends on the adsorption capacity, isotherm nonlinearity, and operating conditions such as the purge-to-feed ratio. Thus, in selecting the best adsorbent and determining the optimal bed layering, all these factors should be accounted for. The sorbent selection criterion is also applicable for more general bulk purification processes such as H2 PSA. However, care must be taken when weak sorbents are involved because the frozen solid approximation is no longer valid. Introduction The pressure swing adsorption (PSA) process has emerged as a major gas separation and purification process because it is useful for a wide variety of applications. Trace impurity removal by PSA is one of the applications for PSA processes.1,2 Separation of gas mixtures is achieved by either preferential adsorption for one of the components (equilibrium separation) or due to a difference in uptake rates (kinetic separation). In the case of equilibrium separation, the performance of the PSA process is largely dependent on the adsorption characteristics of the adsorbates on a specific adsorbent. Various kinds of adsorbents such as zeolite, activated carbon, and silica gel are available commercially. None of the adsorbents mentioned above has the same adsorption affinity for a given component. Therefore, selecting a proper adsorbent for a specific application is most important for the development of PSA processes. It has long been thought that a good adsorbent should have a high adsorption capacity and good regenerability for the adsorbate of interest.3 The regenerability is a term defining how well the adsorbent is regenerated by purging with a inert gas or the light component of the mixture. When an isotherm is of the favorable type, it is unfavorable for desorption. The more it is favorable for adsorption, the more it becomes unfavorable for desorption. Often the adsorbent that holds the higher adsorption capacity also yields a more favorable isotherm. So the adsorbent showing a high adsorption capacity suffers from poor regenerability. Because of these two conflicting tendencies, it is not straightforward to determine which adsorbent is more suitable for a specific PSA process. Despite the importance of the selecting or screening of the adsorbents, few theoretical studies have been * To whom correspond should be addressed. E-mail: [email protected]. † Korea Institute of Energy Research. ‡ University of Michigan.

done because of the complexity of the PSA process. Up to date, adsorbent selection is carried out by rigorous process simulation or by using a heuristic rule. Kumar4 performed a theoretical analysis on the H2 PSA process with a mathematical model and investigated the effects of the isotherm shapes on the performance. Using the simulation, he could determine which isotherm is best for the H2 PSA. Chue et al.5 compared the two adsorbents, activated carbon and zeolite, for the recovery of carbon dioxide from flue gas. Their results also showed that the performance of the process was dependent on the adsorbent. Despite the completeness of the mathematical model, it does not give any insight on the selection of the best adsorbent. Rege and Yang6 have proposed a simple and semiempirical parameter for sorbent selection based on empirical results on PSA processes. They showed the usefulness of the simple parameter for air separation. Jain et al.7 also proposed a heuristic parameter for the sorbent selection. The parameter was not the same as that proposed by Rege and Yang.6 They also showed the usefulness of their parameter for air separation. Though the heuristic rules are valid in some cases, the validity of the rules is not guaranteed for other systems because they lack a rigorous theoretical basis. The equilibrium theory of PSA, although based on the unrealistic assumption of instant equilibrium between the gas and the solid phases, has provided much insight into the PSA processes. The equilibrium theory is mostly applied for the system where the adsorption isotherms are linear. An analytical solution for the general PSA cycles, which include all the basic cycle steps such as the co-current and counter-current depressurization steps, is available for the binary system where the uncoupled linear isotherms are valid.8 The disadvantage of the above theory is that it does not take into account of the nonlinearity of the isotherms that most commercial adsorbents show. Analytical solutions for nonlinear isotherms are available for single and binary systems9-14 under the assumption of the frozen solid

10.1021/ie049105r CCC: $30.25 © 2005 American Chemical Society Published on Web 02/11/2005

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1915

approximation. One of the interesting results from equilibrium theory for nonlinear systems has been given by Pigorini and LeVan.9 They have derived an analytical solution of a PSA process using layered beds for removal of trace impurities. They arrived at the conclusion that even in the process for the removal of a single component a layered bed could be beneficial depending on the adsorption affinities and capacities of the two adsorbents. To find the optimum bed layering, they used a graphical method. No analytical expression for the optimum layering was given. We found that, by extending their analysis, a useful and simple analytical expression can be obtained which can be used for the optimum bed layering and also the selection of the adsorbent. Thus, we extend the analysis of Pigorini and LeVan9 to give a simple criterion for sorbent selection and optimal bed layering for the removal of a single trace impurity. The applicability of the criterion will be demonstrated with the examples treated by Pigorini and LeVan. Moreover, its applicability to a more general and complex case such as H2 purification will be assessed.

Figure 1. Schematic diagram of (a) the characteristic lines and (b) the solid-phase loading at the end of the purge step (hatched area corresponds to the amount desorbed during the purge step).

The dimensionless form of the above equation is

Theory A simple criterion for the selection of adsorbent is derived on the basis of equilibrium theory. We start by analyzing the performance of the layered beds of different adsorption affinities as Pigorini and LeVan have done. Here, their results are briefly reviewed in the following section and a simple criterion for the selection of the adsorbent suitable for a given purification application will be derived. The equilibrium theory relies on the assumption that spontaneous equilibrium between the gas and the solid phases is achieved. In addition, one more assumption that is not always true in the general PSA processes is introduced for simplification of the analysis: the frozen solid approximation. Here one assumes that the solidphase concentration of the impurity does not change during the pressure changing steps. When the impurity is adsorbed strongly and is present in a trace amount and the pressure change is accomplished in a much shorter time than the diffusion time constant, the frozen solid approximation becomes a reasonable assumption. The frozen solid approximation allows us to exclude the pressure changing steps from the analysis so that it greatly simplifies the analysis. Several other assumptions were introduced that are generally true in the case of purification of trace impurities. These include the constant gas velocity inside the bed, isothermal operation, and negligible pressure drop in the adsorber. With the above assumptions, a material balance on the adsorbate is written for each layer:

∂qi ∂c + V ) 0, i ) 1, 2 Fbi ∂t ∂z

(1)

where qi represents the solid-phase concentration in equilibrium with the gas-phase concentration ci and Fbi is the bulk density of layer i. The gas-phase accumulation of the impurity is neglected in eq 1 because the solid-phase accumulation is much greater in the general PSA processes.

∂q*i ∂τ

(

∂c* )0 ∂ζ

(2)

where

c* ) c/cf

(3)

q*i ) Fbiqi/(Fb1q1f)

(4)

Λ1 ) Fb1q1f/cf

(5)

ζ ) z/L1

(6)

τ ) |V|t/(Λ1L1)

(7)

The ( sign in eq 2 is included to account for the reversed flow directions occurring in every half cycle. Here, “+” is applied for the adsorption half-cycle and “-” is applied for the desorption half-cycle. The characteristic length, L1, is the length of the first layer. The characteristic concentration is the concentration of the impurity in the feed. It should be noted that the amount adsorbed is normalized by the adsorption capacity of the first layer at the feed condition. Therefore, the dimensionless amount adsorbed of the first layer, q*1, ranges from 0 to 1, but that of the second layer, q*2, is not 1 at the feed condition because it is assumed that the two adsorbents have different adsorption affinities. Because the characteristic length of the system is set as the bed length of the first layer, the dimensionless length, ζ, at the interface of the two adsorbents is 1. The cycle analysis begins with the bed where a concentration shockwave is fully developed at the end of the adsorption step. It is known that a dispersive wave is developed for isotherms of the favorable type during the purge step. The characteristic velocity of a certain concentration is given by Figure 1a depicts the

∂q*i ∂τ )∂ζ ∂c*

(8)

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characteristic lines that evolve during the purge step. Because the adsorption affinities of the impurity for the two adsorbents are different, the characteristic lines associated with specific concentrations are refracted at the interface of the two adsorbents. The time for a specific concentration wave to arrive at the boundary of the two layers in the purge step is given by

dq*2

τ* ) (ζa - 1)

(9)

dc*

The distance that a specific concentration wave penetrates in the first layer during the rest of the purge step is

∂q*1 γτa - τ* )ζ-1 ∂c*

(10)

When eqs 9 and 10 are combined, the position of a specific concentration wave at the end of the purge step is

ζ)1-

γτa - (ζa - 1) dq*2/dc* dq*1/dc*

(11)

where γ is the volumetric purge-to-feed ratio (at their respective pressures) and τa is the dimensionless adsorption step time. If we denote the concentration that arrives at the feed end as c*0, one can obtain the following relationship from eq 11 by letting ζ ) 0:

(

ζa ) 1 + γτa -

| )( | ) dq*2

dq*1 dc*

c0

*

dc*

-1

c0

*

(12)

With the above equation one can predict the final position of the shockwave of the feed step at the cyclic steady state. Another condition that should be satisfied at the cyclic steady state comes from the overall material balance. That is, at the cyclic steady state the amount of impurity desorbed during the purge step must equal that fed during the adsorption step. The amount desorbed during the purge step is given by

∆)

∫01(1 - q*1,d) dζ + ∫1ζ (η - q*2,d) dζ a

(13)

where η is the adsorption capacity ratio of the two adsorbents at the feed condition, that is, Λ2/Λ1. The hatched area in Figure 1b represents the amount desorbed during the purge step. The above integration can be performed with respect to the solid-phase loading in each layer.

∆)

∫01(1 - ζ) dq*1 + ∫0η(ζa - ζ) dq*2

(14)

Substituting eqs 9 and 10 into eq 14 as intergands and using the fact that the amount of the impurity fed during adsorption is τa, one gets the following equation which should be satisfied at the cyclic steady state:

τa ) 1 + (ζa - 1)(η - q*2,0) - q*1,0 + γτac*0

(15)

Here q*1,0 and q*2,0 are the amounts adsorbed in the first and second layers which are in equilibrium with c*0, the concentration of the impurity that arrives at

the feed end at the end of the adsorption step. Equations 12 and 15 are the governing equations that determine the process performance of the layered bed. If any two of the four unknowns among ζa, c*0, τa, and γ are specified, then the other two unknowns are determined by eqs 12 and 15. Optimization The recovery and throughput are the important performance criteria for the PSA processes. The throughput parameter is defined as follows.

φ)

τa |v|ta L1 ) ζa Λ1L1 L

(16)

Therefore, throughput equals the number of bed volumes (bed on the total bed length) of gas fed to the bed during the feed step. We are interested in the optimal bed layering which maximizes the recovery at a given throughput or throughput at a given recovery. The recovery of the inert component is proportional to the enrichment of the impurity obtained in the purge step. The highest enrichment of the impurity for a single adsorbent bed is obtained at a minimal purge-to-feed ratio if all the assumptions introduced here are valid.11 The same is true for the layered bed. Therefore, finding the maximum recovery at constant throughput is the same as finding minimum purge-to-feed ratio at constant throughput. The optimum bed layering that gives the maximum recovery at constant throughput should satisfy following condition.

∂γ ∂ζa

|

φ

)0

(17)

Another objective function of concern is to find a bed layering which maximizes the throughput at a fixed purge-to-feed ratio. The condition for this maximization can be represented as follows.

|

∂φ )0 ∂ζa γ

(18)

Equations 17 and 18 are not independent of each other. If all the assumptions introduced here are valid, the impurity is desorbed only during the purge step. Therefore, the amount of the impurity that can be treated during the adsorption step should be equal to that desorbed during the purge step. At a given purgeto-feed ratio, the amount desorbed is proportional to the enrichment. Therefore, similarly to the recovery, the maximum throughput at a constant purge-to-feed ratio is obtained when the enrichment of impurity is the highest. Therefore, the two optimization problems are the same in the sense that they are seeking a bed layering which gives the highest enrichment of impurity. Detailed theoretical justification for the statement that the two optimization problems are exactly the same is given elsewhere.8 Because the two optimization problems above are exactly the same, one would reach the same conclusion whatever equation is used. We will use eq 18 for convenience. To find the maximum throughput at a fixed purge-to-feed ratio, Pigorini and LeVan9 used a graphical method. That is, they first fixed two of the

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four variables, for example, τa and ζa, and then determined the two remaining variables, c*0 and φ, by solving eqs 12 and 15 simultaneously. If an explicit expression for the minimum is available, it will greatly simplify the selection of adsorbent or bed layering. To solve the optimization problem explicitly, eqs 12 and 15 are reformulated by substituting eq 16 into eqs 12 and 15. Rearranging the resulting equations, one gets the following equations.

|

dq*2,0

φ)

φ)

(ζa - 1)

c0*

dc* γζa

+

|

dq*1,0 dc*

c0*

1 + (ζa - 1)(η - q*2,0) - q*1,0 ζa(1 - γc*0)

(19)

(20)

Differentiating eq 20 with respect to ζa at a constant γ and rearranging the equation, one gets the following equation.

|

-(1 - γc*0)[1 - q*1,0 - (η - q*2,0)] ∂φ (21) γ ) ∂ζa [ζa(1 - γc*0)]2 Above the critical purge-to-purge ratio, that is, when γ > 1, the value of γc*0 is always less than 1. The proof of the above statement will be given in a qualitative manner. The enrichment of the impurity from a single adsorbent is given by the following.11

E ) Rc*avg ) R/γ

(22)

where R is the ratio of the adsorption pressure to desorption pressure, PH/PL. Though the above equation is derived for a single adsorbent bed, it is equally applicable for layered beds because in cyclic steady state the amount of the impurity fed during the adsorption step should equal that desorbed in the purge step. Therefore, the product of the average concentration of the impurity and the purge-to-feed ratio, γc*avg, is always 1. The concentration of the impurity during the purge step decreases monotonically, so that c*0, which arrives at the feed end at the end of the purge step, is always less than the average concentration of the impurity. Therefore,

γc*0 < γc*avg ) 1

(23)

Consequently, for the throughput to reach maximum, the following equation should be satisfied when γ is above the critical purge-to-feed ratio.

1 - q*1,0 ) η - q*2,0

(24)

The left- and right-hand sides of eq 24 can be regarded as a working capacity of each layer. However, the meaning of the working capacity is different from the conventional one where the working capacity of the adsorbent is defined as the difference of the adsorption amount between the adsorption and desorption pressures.5 In eq 24, the working capacity of an adsorbent is defined as the difference of the amounts adsorbed at the feed concentration and the concentration arriving at the feed end at the end of the purge step. If there is a c*0 which satisfies eq 24, then there can be an optimum bed layering which maximizes the throughput.

As shown in eq 24, the final expression that determines the optimum bed layering is surprisingly simple. The above criterion can be used with any forms of the isotherms as long as the isotherm is favorable. The method with which to apply eq 24 in determining the optimal layering and in selecting the best adsorbent will be demonstrated in the following section, using the example of Pigorini and LeVan.9 Before going to applications, it is worth mentioning why the optimal layer exists even for the removal of a single component. The adsorbate that exhibits favorable isotherms forms dispersive concentration waves during the purge step. Thus, at the end of the desorption step a broad concentration front of the impurity develops in the adsorber. In the subsequent adsorption step, the major role of the product end is to retain the impurity of the lower concentration. For the removal of impurity of low concentration, a stronger adsorbent with a higher adsorption capacity is needed. This leads to the layered beds in which a strong adsorbent is packed behind the weak adsorbent. Applications Optimal Bed Layering. We first demonstrate how to apply eq 24 for determining the optimal bed layering using the example of Pigorini and LeVan.9 Even though the theory derived above could be used for any type of favorable isotherms, the Langmuir isotherm will be used for illustration. The Langmuir isotherm for layer i is

qi )

QiKici 1 + Kici

(25)

The equilibrium factor (R) for the Langmuir isotherm is given by

Ri )

1 1 + Kicf

(26)

The dimensionless amount adsorbed in each layer is

q*1 )

c* R1 + (1 - R1)c*

(27)

q*2 )

ηc* R2 + (1 - R2)c*

(28)

First, assume that the equilibrium factors of the two adsorbents, R1 and R2, and the adsorption capacity ratio, η, are 1, 0.1, and 2, respectively. For a dilute impurity and weak sorbate-sorbent interactions as in layer 1, the value of R approaches 1 and the isotherm is usually linear. To find optimum bed layering, we need to solve eq 24. The quadratic equation can be solved readily, but, here, to see the behavior of the isotherms, we plot the working capacities of the two adsorbents, 1 - q*1,0 and η - q*2,0, against c*0 in Figure 2. Because R1 for the first layer is 1, the working capacity decreases linearly with c*0. As shown in Figure 2, the curves cross each other at c*0,max, which satisfies eq 24. Below this value, the working capacity of the second layer, η - q*2,0, is larger than that of the first layer, 1 - q*1,0. In the region where eq 19 is positive, the throughput increases with ζa as long as the operating conditions allow the value of c*0 to be lower than c*0,max. On the contrary,

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Figure 2. Working capacity, η - q*i, with the concentration of impurity which reaches the feed end at the end of the purge step. Figure 4. Throughput as a function of 1/ζa () L1/L) at different purge-to-feed ratios (R1 ) 1, R2 ) 0.1, η ) 2). The optimum bed layerings from eqs 30 and 31 are shown together.

Figure 3. Relation between c*0 and ζa at a specific purge-to-feed ratio. (R1 ) 1, R2 ) 0.1, η ) 2). The feasible range of c*0 is where ζa is larger than 1.

the throughput would decrease with increasing ζa if the operating conditions are such that c*0 > c*0,max. For this example, c*0,max ) 1/9. Our goal is to find the optimum bed layering, ζa, at a fixed purge-to-feed ratio, γ. Let us select several values of γ, that is, 1.3, 1.4, 1.5, and 1.6. In a real process, not all values of c*0, from 0 to 1, are allowed for a given value of γ. This can be understood with eq 23. According to eq 23, the value of c*0 must be less than 1/γ. The exact range of c*0 allowed for a given γ can be obtained from eqs 19 and 20. By equating these two equations and rearranging the resulting equation, the value of ζa is given by

ζa ) 1 -

(1 - γc*0) dq*1/dc - γ(1 - q*1) (1 - γc*0) dq*2/dc - γ(1 - q*2)

(29)

By definition of ζa, it should be larger than 1. The values of ζa as a functions of c*0 for the four values of the purge-to-feed ratio are shown in Figure 3. There are two regions. One is where ζa is larger than 1 and the other is where it is less than 1. The values of c*0 where ζa is less than 1 are not allowed under these operating conditions. The range of c*0 allowed with the four values of the purge-to-feed ratios involves c*0,max, 1/9. Therefore, for this example there exist optimum bed layerings. According to Figure 3, c*0 decreases with decreasing length of the second layer. As ζa approaches 1, c*0 approaches 0. This means that as the length of the second layer decreases, complete regeneration of the whole bed is obtained at all purge-to-feed ratios. This is obvious because the first layer has a linear isotherm. For the linear isotherm, complete regeneration is achieved with the purge-to-feed ratio of 1. In this example, because the purge-to-feed ratio is higher than

1, complete regeneration is accomplished even for a strong adsorbent. The behavior of throughput with the length of the second layer can be seen in Figures 2 and 3. When ζa is close to 1, that is, when the length of the second layer is minimal, c*0 becomes close to 0. If this is the case, the working capacity of the second layer is higher than that of the first layer so that the throughput increases with increasing length of the second layer, according to eq 21. Equation 21 also stipulates that if the length of the second layer exceeds a certain value when c*0 ) c*0,max, then the throughput decreases with increasing length of the second layer. So, ζa at which c*0 ) c*0,max is the optimum length for the second layer. The optimum ζa and the corresponding throughput can be obtained by substituting the value of c*0,max and eq 24 into eqs 19 and 20. For this example, the throughput and the optimum bed length are given by

φ)

8γ 9-γ

ζa(φ - 5) ) -4

(30) (31)

Optimal Bed Penetration and Throughput. The optimum ζa and the corresponding φ as functions of the purge-to-feed ratio are shown in Figure 4 together with the results obtained from the graphical methods. It is clearly seen that the optimum ζa and φ are wellpredicted by the equation. Next, let us consider a system where the throughput increases or deceases monotonically with the addition of a strong adsorbent. With this example, we will demonstrate how to select the best adsorbent. We will again use one of the examples treated by Pigorini and LeVan.9 Suppose that the equilibrium factors of the weak and strong adsorbents, R1 and R2, are 0.1 and 0.01, respectively. And suppose that we have three strong adsorbents for which the adsorption capacities under the feed conditions are 2×, 5×, and 10× higher than that of the weak adsorbent. That is, η of the strong adsorbents are 2, 5, and 10, respectively. We then examine how the throughput of the layered beds varies at a fixed purge-to-feed ratio of 1.2. The variations of the working capacity with c*0 are calculated from the Langmuir isotherm using the parameters mentioned previously, and the results are shown in Figure 5. As the adsorption capacity of the

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1919 Table 1. Comparison of the Performance for H2 Purification Using Adsorbents of Different Langmuir Constantsa,b b0,CH4, atm-1

QPc

RH2d (%)

ypue (%)

9.38 × 10-4 2.81 × 10-4 0.94 × 10-4 0.31 × 10-4

0.49 0.82 1 0.80

72 82 85 80

41.9 50.2 52.1 45.6

a The comparison is done at the optimum performance. b Isotherms for CH4: q ) qsBp/(1 + Bp) where B ) b0 exp(q/RT); qs ) 1.51 mmol/g, and q ) 5347 cal/mol for all adsorbents. c QP: relative net product produced. d RH2: hydrogen recovery. e ypu: concentration of methane at the end of the purge step.

Figure 5. Working capacities of adsorbents. Inset is the magnification of the curves between 0.5 and 0.6.

Figure 6. Variation of c*0 as a function of ζa at the purge-to-feed ratio of 1.2 (R1 ) 0.1, R2 ) 0.01).

strong adsorbent increases, the value of c*0,max increases and the region where the working capacity of the strong adsorbent is higher than that of the weak adsorbent is widened accordingly. This means that as the adsorption capacity of the stronger adsorbent increases, the possibility that the stronger adsorbent is a better adsorbent increases despite its poor regenerability. Comparing the working capacities of the strong adsorbents, one can find that the adsorbent which has the highest adsorption capacity always gives higher working capacities. Therefore, the adsorbent with higher adsorption capacity is always the better adsorbent among the adsorbents which have the same equilibrium factor. Throughput. Let’s examine how the throughput changes by layering the bed with the weakest adsorbent and one of the stronger adsorbents. As in the first example, not all the values of c*0 shown in Figure 5 are allowed when the purge-to-feed ratio is 1.2. To determine which adsorbent is better or how the throughput changes with ζa, we have to find the range of c*0 allowed with the purge-to-feed ratio of 1.2. It is again obtained from eq 29, and the results are shown in Figure 6. The range of c*0 allowed is much narrower than that in the previous example. Regardless of the value of ζa, the minimum value of c*0 allowed is 0.57. This corresponds to the value at which the numerator is 0 and the numerator depends only on the isotherm parameters of the first layer. Therefore, the lower limits of c*0 for three cases are the same. The upper boundary of c*0 is the value where the denominator becomes 0. So, the upper boundaries for the three cases are all different because the denominator depends on the isotherm parameters of the strong adsorbent. The range of c*0

allowed under the specific operating conditions for the three adsorbents is between 0.57 and 0.6. The working capacities of each adsorbent within this range are shown in the inset of Figure 5. The two adsorbents whose adsorption capacities are 2 and 5 times higher than the weaker adsorbent have smaller working capacities, while the adsorbent with the highest adsorption capacity also has the greatest working capacity. These results imply that, while adding the adsorbent with adsorption capacity ratios of 2 and 5 to the weak adsorbent always reduces the process throughput, adding the adsorbent having an adsorption capacity ratio of 10 always enhances the throughput. These trends are also observed by Pigorini and LeVan.9 Thus, one may conclude that the adsorbent with the adsorption capacity ratios, η, of 2 and 5, is worse than the weak adsorbent, but the adsorbent with an adsorption capacity of 10 is better than the weak adsorbent. Application to the H2 Purification Process It is interesting to test the applicability of the selection criterion given above to more general PSA processes such as bulk purification processes because they are more frequently encountered in industry. In those processes, the isothermal and frozen solid approximation which are the basic assumptions employed in this work are no longer valid. Hydrogen purification is a typical example for bulk purification. Kumar4 has compared several adsorbents for H2 purification from a 75% H2/25% CH4 mixture with a general mathematical model which accounts for the non-isothermal and nonequilibrium effects. He performed a rigorous simulation by fixing other equilibrium parameters except the Langmuir constant of CH4 to investigate the role of the isotherm shape on process performance. Kumar tested four different Langmuir constants. The parameters are listed in Table 1 together with the corresponding process performance. The process configuration studied by Kumar is a typical H2 PSA which consists of four beds and eight PSA steps.4 As shown in Table 1, the adsorbent with the Langmuir constant of 0.94 × 10-4 shows the best performance. We will now show how closely the theory can be used to predict the best adsorbent for hydrogen purification. Hydrogen is actually not inert. However, because the adsorption of hydrogen is negligible, it is assumed to be inert. The first step is to plot the working capacity with reduced concentration as in the above examples. The plot is shown in Figure 7. The adsorption capacities of the adsorbents are normalized with respect to the weakest adsorbent for which the Langmuir constant is 0.31 × 10-4. It should be noted that the ordinate of

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Figure 7. Dependence of the working capacity of various adsorbents on the final effluent concentration of impurity.

Figure 7 is the reduced pressure, not the reduced concentration. If we replace the reduced concentration in the equations developed above with reduced pressure, all the equations can be applied to this system. Because the adsorption pressure and the concentration of methane for this system are 21 and 25%, respectively, the reference pressure, pref, is 5.25 atm. The exact value of the purge-to-feed ratio is not specified in Kumar’s paper. Instead, the concentrations of methane that just arrive at the feed end at the end of the purge step are known, which is between 42 and 52% as shown in Table 1. Converting this to the reduced pressure, it ranges from 0.08 to 0.1 if the desorption pressure is assumed to be 1 atm. The working capacities of the adsorbents within this range are known. The order of the working capacity in this range is

b0(0.94 × 10-4) > b0(0.31 × 10-4) > b0(2.81 × 10-4) > b0(9.38 × 10-4) The adsorbent with the highest working capacity shows the best performance. The order of the performance obtained from simulation agrees well with the order given above if we rule out the adsorbent for which b0 is 0.31 × 10-4. The reason that the criterion based on a trace impurity purification can closely predict the performance will be explained in a qualitative manner. The criterion developed here is based on the frozen solid approximation, which is not valid for bulk purification especially when the partial pressure of the impurity is higher than the desorption pressure. If this is the case, some of the impurity must be desorbed during the blowdown step. However, the frozen solid approximation can still be valid during the co-current depressurization step if the final pressure of the co-current depressurization step is higher than the partial pressure of the impurity in the feed and the impurity is strongly adsorbed. According to Park et al.,8 if the selectivity is high enough, the throughput was not affected so much by co-current depressurization to a certain pressure, which is roughly the same as the partial pressure of the strong adsorbate in the feed. The invariant of the throughput indicates that the concentration wave movement during the co-current depressurization step is not large. So, the situation is close to the frozen solid approximation. Next, during the blow-down step, some of the impurity will be desorbed but the steplike concentration wave front is not disturbed. That is, the whole bed would have the same total concentration after the blow-down step. In the subsequent purge step, a specific concentration of the impurity starts to move toward the feed end and

Figure 8. Schematic diagram of the solid-phase concentration profile at the end of (a) blow-down and (b) purge steps for the frozen solid limit (FS) and for bulk purification (B) cases. For the bulk purification case, the frozen solid approximation is applied for the co-current depressurization step (the reason is explained in text).

the velocity is given by eq 2 if the variation of the interstitial velocity of gas is negligible. Therefore, the final solid-phase concentration profile for bulk separation would be the same as that obtained with the frozen solid approximation. This situation is depicted schematically in Figure 8. Consequently, the amounts desorbed in the frozen solid limit and in bulk separation would be almost the same if the purge amount is the same for the two cases. If the frozen solid approximation can be applied for the co-current depressurization step for bulk purification, then the throughput of bulk purification would be close to that obtained under the frozen solid approximation for all steps. This is why the criterion developed here predicts well the order of the adsorbent selection for bulk purification. But the criterion does not predict well the performance order for the adsorbent with b0 ) 0.31 × 10-4 which is worse than that of the adsorbent with b0 ) 2.81 × 10-4. This may be because the adsorbent with b0 ) 0.31 × 10-4 is so weak that the frozen solid approximation for the cocurrent depressurization step simply does not apply. According to Park et al.,8 the throughput of the process is greatly affected by the co-current depressurization if the selectivity is low. But if the selectivity is high enough, the throughput would not be affected much by co-current depressurization as long as the final pressure of the co-current depressurization step is above a certain pressure, which is roughly the same as the partial pressure of the strong adsorbate in the feed. The selectivity in the system treated by Park et al. 8 does play the same role as the equilibrium factor here in the sense that the two are the origin which makes the concentration profile during the adsorption self-sharpening. Thus, if the equilibrium factor is low, then the change of the throughput with depressurization may be significant and the frozen solid approximation for the co-current depressurization step becomes invalid. If the movement of the concentration wave front during the co-current depressurization step is significant, then we need a more realistic model which accounts for the movement. If we rule out the weakest adsorbent, the order of the ranking of adsorbents is well predicted by the theory despite the assumptions made. For the case of a weak adsorbent, care must be taken because the frozen solid approximation becomes invalid. In that

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1921

case, a more general theory which accounts for variation of the solid loading during the pressure changing steps is required.

t ) time, s V ) interstitial velocity, m/s z ) axial coordinate/distance, m Greek Symbols

Conclusions A simple criterion for the selection of adsorbent and optimal bed layering for the removal of single trace impurity by the PSA process is derived from the equilibrium theory in conjunction with the isothermal, frozen solid approximation for the pressure changing steps. The simple criterion is based on the working capacity of the adsorbent. But the exact meaning of the working capacity is different from the conventional one in that the working capacity is defined here as the difference between the amount adsorbed under the feed condition and that at the concentration which arrives at the feed end at the end of the purge step. If two adsorbents show the same working capacity at some concentration of impurity and that concentration is with the range of the PSA operating conditions, a layered bed of two adsorbents is better than a single adsorbent bed. If the working capacity of one of the adsorbents is always higher than the other under the operating conditions, then a single adsorbent bed with the higher working capacity is better than layered beds of two adsorbents. Even though the criterion is derived under a number of simplifying assumptions, it can also be applied to the more general nonisothermal, nonequilibrium PSA cycle but care should be taken when applying to a weak adsorbent for which the frozen solid approximation becomes invalid. Acknowledgment J.H.P. is grateful to Korea Science and Engineering Foundation (KOSEF) for partial funding of his work at University of Michigan. Notation c ) gas-phase concentration, mol/m3 cavg ) dimensionless average concentration of impurity in the purge effluent c* ) dimensionless gas-phase concentration c*0 ) dimensionless gas-phase concentration which arrives at the feed end at the end of the purge step c*0,max ) dimensionless gas-phase concentration where the working capacity of the two adsorbents becomes equal cf ) gas-phase concentration in feed, mol/m3 E ) enrichment ratio of the impurity Ki ) Langmuir isotherm parameter for layer i, m3/mol L1 ) length of the first layer, m L ) total length of the bed, m qi ) amount adsorbed in layer i, mol/kg q1,f ) amount adsorbed of impurity in the first layer at the feed condition, mol/kg q*10) dimensionless amount adsorbed of impurity in the first layer at the concentration of c*0 q*20 ) dimensionless amount adsorbed of impurity in the second layer at the concentration of c*0 Qi )Langmuir monolayer capacity for layer i, mol/kg Ri ) equilibrium factor for isotherm for layer i, defined by eq 26

R ) ratio of adsorption pressure to purge pressure ∆ ) dimensionless amount of adsorbate removed by purge  ) void fraction of the column φ ) throughput γ ) volumetric purge-to-feed ratio at their respective pressures η ) ratio of adsorption amount in the first layer to the second layer Λi ) partition ratio in layer i Fbi ) bulk density of packing in layer i, kg/m3 τ ) dimensionless time τa ) dimensionless adsorption time τ* ) dimensionless time when a specific concentration reaches the layer interface τp ) dimensionless purge time ζ ) dimensionless axial coordinate ζa ) depth of penetration at the end of the adsorption step

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Received for review September 14, 2004 Revised manuscript received November 13, 2004 Accepted November 15, 2004 IE049105R