Equilibrium Theory for Pressure Swing Adsorption ... - ACS Publications

2296

Ind. Eng. Chem. Res. 1997, 36, 2296-2305

SEPARATIONS Equilibrium Theory for Pressure Swing Adsorption. 2. Purification and Enrichment in Layered Beds Giuseppe Pigorini and M. Douglas LeVan* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442

Equilibrium theory is applied to the analysis of the periodic performance of an isothermal pressure swing adsorber in which the bed is composed of two layers of different adsorbents. The feed consists of a single adsorbable component in inert gas, and the purge is pure inert gas. The adsorbents differ in capacities for the adsorbable species and in the shapes of isotherms, which can be linear or concave downward. We solve for the periodic concentration profile and for the depth of penetration of the shock into the second layer during the feed step. The solution is analytical and relies on the theory of characteristics. The periodic state is determined directly, without iteration. Examples consider the effects of isotherm shapes and capacities, the direct determination of the periodic states, and the optimum layering of beds. Introduction Pressure swing adsorption (PSA) is utilized to separate or purify a gas stream by feeding it at high pressure to an adsorption bed and subsequently purging the bed at low pressure with product or inert gas. Comprehensive analyses and reviews of PSA are readily available (Ruthven, 1984; Yang, 1987; Suzuki, 1990; Ruthven et al., 1994); they deal with all aspects of this process, show how the cycle can be modeled, and illustrate the major industrial applications. Tondeur and Wankat (1985) have reviewed the possible configurations of a PSA cycle with emphasis on positive and negative aspects. The various configurations are all modifications of a basic cycle that consists of four steps. The adsorbent is chosen on the basis of the feed composition as well as on economic considerations. The concept of using more than one adsorbent in a PSA bed, beyond a need to remove heavy impurities in a feed (e.g., water), is relatively new. This paper is concerned with the analysis of the periodic behavior of a PSA bed composed of two contiguous layers of different adsorbents and used to adsorb isothermally a single component present in trace amounts in an inert carrier gas. The use of beds in series is virtually equivalent to the layering of one bed, but it adds some volume of gas in between beds. The selection of the two adsorbents is an attempt to exploit the best features of each with regard to capacity for the adsorbing component and nonlinearity of the isotherm. For example, an adsorbent with a strongly concave-downward (favorable) isotherm, such as a zeolite or activated carbon in many applications, is not easily regenerated, giving it a small working capacity. Similarly, an adsorbent with a more linear isotherm and potentially greater working capacity, such as alumina or silica gel, often cannot give purification to desired levels. Thus, * 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: [email protected]. S0888-5885(96)00715-4 CCC: $14.00

it may be possible to choose an adsorbent with a more linear isotherm and larger working capacity for the feed end of a bed to remove the bulk of the impurity and to choose an adsorbent with a strongly nonlinear isotherm for the downstream section to almost completely eliminate the impurity from the gas stream. This may allow a more economically efficient cycle than is possible with a single adsorbent. Few papers have been published on layered beds in adsorption processes. Klein and Vermeulen (1975) have analyzed the cyclic performance of layered beds for binary ion exchange. They considered a layer of resin with a favorable isotherm and a layer with an unfavorable isotherm. They were concerned with determining the best ratio of thicknesses of the two layers for optimal regenerant efficiency and resin utilization. Recently, Chlendi and Tondeur (1996) have considered the adsorption of a three-component mixture carried by an inert gas in a layered PSA bed consisting of activated carbon followed by zeolite. They showed that when crossing the boundary between the two layers, each characteristic or shock gets “refracted”, since its slope changes owing to the different adsorptive capacities of the layers. Moreover, for multicomponent adsorption, the presence of two layers of adsorbent complicates the analysis of the system in that, during the feed step, a shock on crossing the boundary between the two layers may or may not split into two or more distinct shocks. Other complex behavior can be observed when waves generated during the purge step interact with what is left in the bed at the end of the feed step. Ultimately, the justification for these behaviors should be sought in the need for the coherence condition to be satisfied everywhere in the bed, including at the boundary between layers of different adsorbent. The work described in this paper relies on equilibrium theory. This is a powerful tool that allows us to solve many problems related to PSA processes, ranging from separation to purification operations. It is based on the assumption that there is no resistance to mass transfer between the fluid phase and the adsorbed phase, so that © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2297

the degree of separation of each component between the two phases is based entirely upon equilibrium considerations. This paper builds on part 1 of this series (LeVan, 1995). There, equilibrium theory was applied to the purification and separation of a mixture with one adsorbable component fed to a bed containing one adsorbent. The main objective of this paper is to use equilibrium theory and the theory of characteristics to develop some useful relations that give insight into the periodic behavior of layered beds in adsorption operations. Most of the sections in the present work parallel analogous sections of part 1, which should be consulted for reference. Underlying Theory We consider a PSA cycle with isothermal adsorption of an impurity present in trace amounts in the highpressure feed. Both Liu and Ritter (1996) and Mahle et al. (1996) have recently considered heat effects in adsorption processes of a kind similar to the ones described in this paper and have shown that the assumption of isothermal operation should not be taken lightly and that temperature variations can affect in a significant way the concentration profiles and the process dynamics. Also, Chihara and Suzuki (1983) have shown that a small temperature swing is often present in the PSA cycle and that its effect on the extent of separation is not negligible. However, it is true that in the absence of adsorption of carrier gas and of thermal effects associated with compression and decompression of the carrier gas, the process approaches isothermal operation as the mole fraction of the adsorbable component in the feed approaches zero; under these circumstances, energy given off by adsorption is transferred to the relatively large masses of solid adsorbent and carrier gas and has an insignificant effect on raising temperature. It is this trace system in which we are interested. As literature examples, the temperature rise during an adiabatic feed step for the system of Ritter and Yang (1991) in which DMMP is adsorbed from air on activated carbon can be estimated to be less than 0.5 K. Similarly, the same temperature rise for the system of Davis and LeVan (1987) in which benzene at 10% of its saturation concentration is adsorbed from nitrogen on activated carbon is less than 2 K. Of course, small beds often operate closer to isothermally than adiabatically (because of the mass of the column wall and energy losses), and a bed can be thermostated to achieve nearly isothermal behavior. We make several common assumptions. We take the velocity to be independent of the amount adsorbed, since the adsorbable component is present in the feed at a low mole fraction. We ignore any pressure drop through the bed. For adsorption of a solvent-type species the partition ratio is large, so that the fluid-phase accumulation term in the material balance can be neglected. As in part 1, we consider only a high-pressure feed step and a countercurrent low-pressure purge step with pure inert gas. The pressurization and blowdown steps are not accounted for because we analyze systems characterized by high partition ratios (in the range 10010 000), such that the volume of gas required to pressurize or blowdown the bed from one pressure to the other is negligible compared to the volume of gas used during the feed and purge steps. For example, it takes less than one bed volume of inert gas measured at the

high pressure to pressurize the bed (because of the skeletal matrix of the adsorbent and a non-zero initial pressure), whereas the number of bed volumes passed into the bed during the feed step can be on the order of the partition ratio. Thus, in this kind of process the pressure-changing steps do not affect the concentration profiles in the bed on a major scale (Yang, 1987), and we assume that during pressurization and blowdown the adsorbed phase is “frozen” in place and that its distribution in the bed is not altered in a significant way. We apply equilibrium theory, assuming that there is equilibrium between the fluid phase and adsorbed phase at each cross section in the bed and ignoring axial dispersion. Therefore, mass transfer between the two phases is considered to take place infinitely fast, without any resistance. We allow both general adsorption equilibrium and, in examples, describe equilibrium using the constant separation factor isotherm (see LeVan, 1995); each layer will be characterized by its own constant separation factor and a ratio of reference capacities will appear as a parameter. A material balance on the adsorbable component is written for each of the layers. At the interface between the layers, there is a discontinuity in the adsorbed-phase concentration profile because, owing to the abrupt change from one isotherm to the other, two different adsorbed phase concentrations are in equilibrium with the same gas-phase concentration. The material balance for layer i is

Fbi

∂qi ∂c + v ) 0 ∂t ∂z

i)1, 2

(1)

where qi represents the adsorbed-phase concentration in equilibrium with gas-phase concentration c and Fbi is the bulk density of the adsorbent in layer i. Nondimensionalization gives

∂q* ∂c* i ( )0 ∂τ ∂ζ

(2)

c* ) c/cref

(3)

q* i ) Fbiqi/(Fb1q1,ref)

(4)

Λ1 ) Fb1q1,ref/cref

(5)

ζ ) z/L1

(6)

τ ) |v|t(Λ1L1)

(7)

where

The ( sign in eq 2 and the absolute value of velocity in eq 7 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. cref is the gas-phase concentration of the adsorbable component in the high-pressure feed, and q1,ref represents the adsorbed-phase concentration in equilibrium with cref in the first layer of the bed. Λ1 is the partition ratio for the first layer. Unlike what was done in part 1, where the system had no characteristic length, we now nondimensionalize the axial coordinate with respect to L1, the length of the first layer. As a result, for the purge step, τ will vary between 0 and τp, the dimensionless purge time, and for the feed step, between τp and τp + τa, where τa is the dimensionless adsorption time.

2298 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

A few additional nondimensional variables pertaining to the cycle are used below. We introduce a volumetric purge-to-feed ratio γ, which is related to the cycle step times by γ ) τp/τa. The depth of penetration of the adsorption front after any cycle or at the periodic state will be denoted ζa, as in part 1; for a process begun with a clean bed, ζa will generally increase from cycle to cycle since we are considering incomplete regeneration. For the time being, we assume that the second layer can extend to infinity. The determination of the periodic depth of penetration will allow us to find a value for L1/L as a function of the volumetric purge-to-feed ratio γ, L being the total length of bed required to contain the wave. Since we are studying layered beds, we assume now that ζa > 1, i.e., that the depth of penetration of the adsorption front lies somewhere in the second layer of adsorbent. We will derive the conditions for this to happen below. If it is not the case that ζa > 1, then the analysis in part 1 for an unlayered bed applies. In what follows we will develop relations that pertain to general favorable isotherms and linear isotherms. However, to demonstrate solutions, we will apply the more general relations to the particular case of the Langmuir or constant separation factor isotherm, with each layer having its own value of the constant separation factor. As indicated in eq 4, we have chosen to nondimensionalize the adsorbed-phase concentrations with respect to the reference adsorbed-phase concentration for the first layer, q1,ref. Thus, if the system is Langmuirian, the isotherm for layer i is

qi )

QiKic 1 + Kic

(8)

QiKicref 1 + Kicref

(9)

Then, letting

qi,ref ) and

1 1 + Kicref

(10)

q* 1 )

c* R1 + (1 - R1)c*

(11)

q*2 )

ηc* R2 + (1 - R2)c*

(12)

Ri ) we obtain

and

Fb2q2,ref Λ2 ) Fb1q1,ref Λ1

dτ ∆q* i ) dζ ∆c*

(14)

where the differences are taken between the two states that are connected by the shock or contact discontinuity. The shock or contact discontinuity is refracted (Chlendi and Tondeur, 1996) as it crosses the interface between the two layers. As we start purging with pure inert gas, with flow from right to left, a simple wave forms or a contact discontinuity exists. The slopes of the characteristic lines in the (ζ, τ) plane are given by

dq*i dτ )( dζ dc*

(15)

where the minus sign corresponds to the purge step and the plus sign applies to the characteristics during the feed step before they are intercepted by the shock. Because we have different isotherms in the different layers, even though a line carrying a particular gasphase concentration across the interface between the two layers will be continuous, the slope of the line will be discontinuous. It will be more difficult to regenerate a layer having a high partition ratio or a more strongly concave-downward isotherm, and we consequently expect steeper characteristics for one of the layers. In going through the interface between the two layers of adsorbent, each characteristic is refracted. For the special case of the Langmuir isotherm, in the first layer, we have from eqs 11 and 15

R1 dτ )( dζ [R1 + (1 - R1)c*]2

(16)

whereas in the second layer, using eq 12, we obtain

where we have introduced the parameter

η)

the bed during the feed step, capturing any remnants of the simple wave left in the bed at the end of the purge step. For the special case of the linear isotherm, these waves are contact discontinuities, although the equations for simple waves and shocks still apply. Since we are concerned with only one adsorbable component carried by an inert gas, within the method of characteristics we have no hodograph plane and the physical plane is used to follow the paths in time and space of different concentrations in the bed (Rhee et al., 1989). We feed the bed from left to right in figures which follow. During the feed step, the solution is a shock or contact discontinuity for which the reciprocal speed is given by

ηR2 dτ )( dζ [R2 + (1 - R2)c*]2

(17)

Direct Determination of Periodic States

(13)

which is a measure of the adsorptive capacity of the second layer relative to that of the first. The shape of the isotherm for different values of R is illustrated in Figure 1 of part 1. We limit our analysis to those systems characterized by R e 1, i.e., linear and concavedownward (favorable) isotherms. For a general concave-downward isotherm, it can easily be shown (see below) that a simple wave will form during the purge step and a shock will sweep through

We make the assumption that the bed is long enough to recover all of the adsorbable component. The feed enters the bed at ζ ) 0, and the purge enters at a large value of ζ. Construction for Purge Step. (a) General Considerations. We assume that the cycle is operating at its periodic state. Thus, the amount of the adsorbable component that is removed from the bed during the purge step is equal to the amount that is fed to the bed during the feed step. As mentioned earlier, at the end of the feed step we expect the shock to be somewhere


we can follow the progression of any concentration c* crossing the interface between layers 2 and 1 and retained in the bed in two steps, as shown in Figure 2. We let τj be the value of τ when the concentration of interest is at ζ ) 1, the interface between the layers. Thus, for 0 < τ < τj, the specified concentration will be in layer 2 and for τj < τ < τp it will be in layer 1. Of course, each concentration c* will have its own value of τj. Thus, given the concentration c* at ζ ) 1 at τ ) τj, the slope of the characteristic for this concentration in the second layer, given by eq 15, is

dq* τj 2 )1 - ζa dc* Figure 1. Concentration profiles at the end of the feed step (dimensional and nondimensional). Flow for the feed step is from left to right.

(19)

Similarly, in the first layer, the same gas-phase concentration, retained in the bed at time τp ) γτa, is carried along a characteristic of slope

dq* γτa - τj 1 )ζ-1 dc*

(20)

If we eliminate τj between eqs 19 and 20, we get

ζ)1-

γτa - (ζa - 1) dq* 2/dc* dq*1/dc*

(21)

which gives the position at time τp ) γτa of concentration c* left in layer 1, regardless of whether or not the process is at the periodic state, when we stop feeding with a shock at ζa. It corresponds to eq 12 of part 1, which is simpler because it accounts for only one layer of adsorbent. Considering the bed inlet, by setting ζ ) 0 in eq 21, we obtain Figure 2. Construction in the physical plane for the purge step. Flow is from right to left. A generic concentration c* crosses the interface between layers at τ ) τj. The concentration reaching ζ ) 0 at τ ) τp ) γτa is denoted c*0.

in the second layer, so that at the periodic state the bed is uniformly loaded in the first layer with q* 1 ) 1 and in the second layer for 1 < ζ < ζa with q*2 ) η, as shown in Figure 1. The conditions for this to be the case are developed below for the feed step. As we start purging, a centered wave appears at ζ ) ζa and moves to the left, as shown in Figure 2. Characteristics in this problem will always be straight lines. Each characteristic carries a given concentration, denoted generally by c*, and has a slope given by eq 17, which changes to that given by eq 16 if it crosses the interface between the two layers of adsorbent. We stop the characteristics at τp ) γτa, the end of the purge step. We denote the concentration reaching the bed inlet, ζ ) 0, at this time by c* 0, as shown in Figure 2. The amount of adsorbate removed is given by

∆)

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

(18)

where q* 1,d and q* 2,d represent adsorbed-phase concentrations in the first and second layers at the end of the purge step. At the periodic state we have ∆ ) τa, meaning that we remove as much adsorbate during the purge step as we fed during the feed step. Equation 18 parallels eq 11 of part 1, in which ∆ had a value of unity owing to a different nondimensionalization of lengths and times. To evaluate the integrals in eq 18, concentration profiles at τp must be determined. For the first integral,

[

ζa ) 1 + γτa -

| ]( | )

-1 dq* dq* 1 2 dc* c*0 dc* c*0

(22)

At the periodic state, this expression tells us that the depth of penetration of the shock can be obtained from the dimensionless purge time and from the slopes of the two isotherms at c* 0, the concentration that just reaches ζ ) 0 at the end of the purge step. Away from the periodic state, it gives the concentration that is pushed back to ζ ) 0 during the purge step when the shock has penetrated to ζ ) ζa. Equation 22 is a generalization of eq 13 of part 1. (b) Condition for Blocklike Profile at End of Feed Step. We want to determine the general conditions under which the blocklike profile shown in Figure 1 will form at the end of the feed step for the periodic state. We know that during the feed step, until they are captured by the shock, the characteristics will be the mirror images of those for the purge step and eq 15 will apply. For Langmuirian systems, from eqs 12 and 19, the characteristic that carries c* 0 ) 0 during the feed step reaches the interface between the two layers at τ* ) η(ζa - 1)/R2 and lies within the first layer for an additional time τ˜ ) τp - τ*. During the feed step, if it is not intercepted by the shock, then that characteristic lies within the first layer for 0 < τ < τ˜ . At τ ) τa, the end of the feed step, it reaches ζ* ) ζa + R2(τa - τp)/η. In order to ensure that this characteristic is captured by the shock before the end of the feed step (and therefore that the blocklike structure is re-established), we need to impose the condition ζ* e ζa. This requires τa e τp, i.e., γ g 1. This means that no matter what


values we choose for R1, R2, and η, the blocklike structure always forms at the end of the feed step for the periodic state if γ g 1, and no other profile can exist. Based on the same considerations, this result can be generalized to all systems described by any combination of linear and favorable isotherms. Critical Volumetric Purge-to-Feed Ratio. The critical volumetric purge-to-feed ratio is the minimum value for the volumetric purge-to-feed ratio that is required to ensure that the adsorbable component does not escape from the downstream end of a bed during the feed step, or in other words that the distance that the adsorption front can travel during any feed step after repeated cycles is finite and lies within the bed. Our critical purge-to-feed ratio differs from that of Chan et al. (1981), who defined it as the ratio of volume of purge to volume of feed necessary to completely purge the bed. A consequence of our definition is that the purge is as rich in the adsorbable component as it can be, and purging is discontinued when the concentration of the effluent is on the brink of decreasing, or when the simple wave that forms when we start purging reaches ζ ) 0. In particular, the concentration in the effluent during the entire purge step will be equal to the concentration fed into the bed during the feed step. Thus, the critical volumetric purge-to-feed ratio for this isothermal system is γ ) 1 (i.e., τp ) τa). In this case, the concentration in the purge stream is c* ) 1 all of the time. Therefore, from eq 22, we obtain for the minimum ζa

[

ζa ) 1 + τa -

| ]( | )

dq* dq* 1 2 dc* c*)1 dc* c*)1

1 (τ - R1) ηR2 a

(23)

(24)

for the minimum ζa, which is analogous to eq 15 of part 1. Our analysis pertains only to ζa > 1. Imposing this condition on eq 24 gives the requirement

τa > R1

(25)

in order for the adsorbate to penetrate into the second layer during the feed step at the critical volumetric purge-to-feed ratio. This is an important result, since it gives us a criterion for determining the lower boundary for the adsorption time during the cycle. If the adsorption time is chosen in such a way that this condition is satisfied, then the periodic position of the shock lies within the second layer, or, in other words, both layers are being used and the layered bed is performing its function. Greater Volumetric Purge-to-Feed Ratios. (a) General Considerations. When we increase the volumetric purge-to-feed ratio above unity, the periodic position of the shock inevitably shifts toward the feed end of the bed. The characteristics that stay within the second layer during the purge step satisfy

( )

ζ ) ζa - γτa

-1 dq* 2 dc*

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

(26)

which is obtained by integrating eq 15, written for the

(27)

where qi,0* is the adsorbed-phase concentration in equilibrium with c* 0 in layer i. At the periodic state, we have ∆ ) τa and γ > 1. From eq 27, it is apparent that when γ ) 1 (and, consequently, c* 0 ) 1), then ζa is finite, whereas if γ < 1, then ζa ) +∞ and the adsorption front penetrates indefinitely into the bed. For γ > 1, the periodic position of the shock ζa is given by eq 22. (b) Conditions for Validity. At the end of the feed step, a shock will be in the second layer (ζa > 1). From eq 22, this requires

γ>

|

1 dq*1 τa dc* c*0

(28)

at the periodic state. This condition is more general than the one given by eq 25 because it includes the former condition in a special case. Of course, c* 0 is itself a function of γ as well as of τa, so that eq 28 is not as simple a condition on γ as it may at first appear. In order to reduce eq 28 to a relationship between γ (or τp) and τa, we combine eq 27, written with ∆ ) τa, and eq 22. Solving eq 27 for ζa yields

-1

which corresponds to eq 14 of part 1. For the Langmuir isotherm, substituting the values for the derivatives obtained from eqs 11 and 12 into eq 23, we obtain

ζa ) 1 +

second layer, between the limits τ ) 0 and τp ) γτa. For the characteristics that cross the interface, the expression for ∆ given by eq 18 is still valid, and eq 21 still holds for the first layer. Therefore, integrating eq 18 by parts and using eqs 21 and 26 gives

ζa )

+ q* - γτac* τa - 1 + η - q* 2|c* 1|c* 0 0 0 η - q* 2|c* 0

(29)

Setting eq 29 equal to the right hand side of eq 22, we get a relationship between c*0 and γ once τa is assigned (together with isotherm parameters). If we combine this equation in c*0 and γ with eq 28 written with an equal sign, we get the value of cj*0 ) cj0*(τa) that is pushed back to ζ ) 0 in the limiting case where the shock penetrates only to ζ ) 1 during the feed step. The governing equation is

j0* τa - 1 + q* 1|cj0* - c

|

dq* 1 )0 dc* cj0*

(30)

Once an adsorption time is chosen, this equation can be solved for cj0* and the required γ j is determined from eq 28, written with an equal sign. At this point, with τa being fixed, we have for γ j > γ > 1 that the depth of penetration at the end of the feed step is ζa > 1, whereas for γ > γ j > 1 it is ζa < 1. (c) Langmuir Isotherm. When we substitute the relevant expressions derived from the Langmuir isotherm into eq 30, we obtain

(1 - τa) + x(1 - τa)/(1 - R1) cj0* ) R1 τa + R1 (1 - τa)

(31)

We are interested in a value 0 e cj0* < 1. By imposing this condition on eq 31, we find that τa must be within the range

R1 < τa e 1

(32)

The upper boundary on τa comes naturally from the analysis of the domain of existence of cj0* in eq 31, but it


can also be explained easily on physical grounds. If τa > 1, the adsorption front will penetrate to ζa > 1 on any feed step, including the first feed step for an initially clean bed. Once the value of cj*0 is determined, this can be substituted into eq 28, written with an equal sign, giving

γ j)

R1 1 τa [R + (1 - R ) cj*]2 1 1 0

(33)

Then, at the periodic state, considering that 0 e cj*0 < 1 and, from eq 32, R1 < τa e 1, we have

1 < γj e

1 R1

(34)

In comparing results for the case of γ ) 1 with those for γ > 1, we find that in the former we only have one value of τa (and hence of τp) such that ζa ) 1 at the periodic state (i.e., τa ) R1), whereas in the latter an infinite number of combinations of τa and τp are possible that yield the same periodic configuration at the end of the feed step. The significance of the previous analysis is as follows. When dealing with the case γ ) 1, we were led to the conclusion that if we want the periodic position of the shock at the end of the feed step to fall in the more strongly adsorbing layer, then we must consider values of τa such that τa > R1. In particular, a shock at ζa > 1 would require an adsorption time

and

ζa ) 1 +

Examples We consider three types of examples: (i) role of isotherm shapes and capacities, (ii) determination of periodic states, and (iii) optimizations. Reasons for layering beds suggest that we place some emphasis on a more strongly concave-downward isotherm for the second layer (R2 < R1). Role of Isotherm Shapes and Capacities. (a) Example 1: R1 ) R2 ) 1. We begin with a simple system characterized by two linear isotherms but different capacities for the layers (η * 1). The slopes of the characteristics during the purge step, obtained from eqs 16 and 17, are single-valued in each layer. This means that during the purge step, no simple waves will form, but rather contact discontinuities will exist. From eq 22, the periodic position of the shock is given by

1 ζa ) 1 + (γτa - 1) η

R1 < τa e [1 + η (ζa - 1)]

(36)

where the upper boundary is derived from eq 27 written for c*0 ) 0. As for the corresponding value of γ at the periodic state, this can be determined by picking a value of τa in the above interval and solving eqs 27 and 22 simultaneously for 0 e c* 0 < 1 and the corresponding γ. The admissible lower value of τa depends on the value of c*0. For instance, if we consider c* 0 ) 1 (which implies γ ) 1), we find that ζa is given by eq 24, and if we want ζa > 1, then we must consider τa > R1. However, if we pick c* 0 ) 0 and solve eqs 27 and 22, then at the periodic state we obtain

(

)

τa - 1 1 1 + γ) τa R1 R2

(37)

(39)

Thus, in order for the adsorption front to penetrate into the second layer of adsorbent at the periodic state we need

(35)

Equation 35 is derived directly from eq 24. If the feed step lasts for the time τa given in eq 35, then the given value ζa corresponds to the minimum periodic depth of penetration of the shock in the second layer. However, the bed can be at a periodic state with the same τa and a higher ζa, because the amount purged would still equal the amount fed to the bed. Similarly, for γ > 1, the lower boundary on τa given by eq 32 applies, whereas the upper boundary is just related to the periodic position of the shock. In general, if we want the shock at the periodic state to be at ζa > 1, then we have

(38)

so that for ζa > 1, we need τa > 1.

γ> τa ) R1 + ηR2 (ζa - 1) > R1

τa - 1 η

1 τa

(40)

Then, if we consider eq 27, we find that for this system we have τa ) γτa, or γ ) 1; i.e., only a critical volumetric purge-to-feed ratio can yield a periodic configuration (unless the bed is completely regenerated and clean purge eluted, which would occur for this case with γ > 1). Moreover, we have c* 0 ) 1. This result is not surprising since there are no simple waves but only contact discontinuities that move in the system in a blocklike fashion during both feed and purge steps. Equation 40 coupled with the fact that γ ) 1 implies that since we want to use both layers of adsorbent, we need to consider τa > 1, which is in agreement with eq 25. (b) Example 2: R1 ) R2. From eq 22, written using the Langmuir isotherms given by eqs 11 and 12 with R1 ) R2 ) R but allowing for different partition ratios (η * 1), we obtain 2

1 γτa[R + (1 - R) c* 0] + η ηR

ζa ) 1 -

(41)

At the periodic state, the length of the second layer is given by

ζa - 1 )

(

)

2 γτa[R + (1 - R)c* 0] 1 -1 R η

(42)

and is therefore inversely proportional to η. We know that during the purge step the slope of the characteristic that carries the generic concentration c* in the second layer is given by eq 17, taken with the minus sign. Hence, the intercept of such a characteristic with ζ ) 1


Figure 3. Region of operability for example 3 (R1 ) 1, R2 < 1). The shaded region corresponds to a cycle that uses both layers of adsorbent and for which the bed is not cleaned completely during the purge step. Above this a region for complete removal of the adsorbate by the purge (c*0 ) 0) is shown.

is given by

τj )

(

)

2 γτa[R + (1 - R)c* 0] R -1 R [R + (1 - R)c*]2

(43)

Since τj does not depend on η, it can be inferred that if R1 ) R2, then the first layer just acts as if it were deeper, regardless of η (i.e., the concentration that bleeds back from the second layer into the first layer during the purge step is not a function of the ratio of the capacities of the two adsorbing materials). Periodic States. (a) Example 3: R1 ) 1, R2 < 1. We consider a first layer with a linear isotherm and a second layer with a concave-downward isotherm. By combining eqs 22 and 27, we obtain

c* 0 ) 1 - xτa(γ - 1)/[(1 - R2)(γτa - 1)]

(44)

It is interesting to note that this expression does not contain the relative capacity parameter η. Thus, once the adsorption time τa and the volumetric purge-to-feed ratio γ are fixed, the concentration that at the periodic state is pushed back to ζ ) 0 is determined independently of η. We are interested in periodic states that involve both layers of adsorbent. From eq 44, we get the condition γ g 1/τa (together with γ g 1). Imposing c* 0 g 0, we find γ e (τa - 1 + R2)/τaR2. If we want the periodic depth of penetration of the shock to be at a certain value of ζa, eq 36 gives us an upper bound for τa. Using these results, we can map out the region of operability in the (τa, γ) plane as shown in Figure 3. Hence, beginning with a desired depth of penetration ζa and an adsorption time τa in the admissible area of Figure 3, eq 44 can be used to determine c* 0 ) c* 0(γ), which is substituted into eq 22, which for a Langmuirian system has the form

ζa ) 1 + (γτa - 1)

2 [R2 + (1 - R2)c* 0] ηR2

(45)

to get a value for γ and thus c* 0. (b) Example 4: R1 ) 0.1, R2 ) 0.01, η ) 2, τa ) 0.25, γ ) 1.2. In this example, we illustrate the main features of a cycle that has reached its periodic state. The value of τa would saturate only one-fourth of the first layer in a clean bed.

Figure 4. Physical plane for example 4 (R1 ) 0.1, R2 ) 0.01, η ) 2, τa ) 0.25, γ ) 1.2). The purge step is in the lower part of the figure with flow from right to left, and the feed step in the upper part with flow from left to right. The lower region with uniform c* ) 1 is spanned in both layers by characteristics (not shown) parallel to the lower one shown. The region with c* ) 1 above τ ) 0.3 is spanned by characteristics that have a slope of the same absolute value as those in the lower constant-concentration region, but of opposite sign.

The periodic state is found directly. By simultaneously solving eqs 22 and 27, we obtain c* 0 ) 0.572 and ζa ) 1.5884. A check of eq 28 shows that it is satisfied, indicating that the shock will be in the second layer at the end of the feed step. To show the dynamics of the cycle, we begin at the end of the feed step, with a blocklike concentration profile. Equations 21 and 26 are used to get the concentration profile during the purge step. The construction in the physical plane is shown in the lower part of Figure 4; concentrations are constant along the straight lines in each of the two layers, and the adsorbed-phase concentration is discontinuous at ζ ) 1. For this cycle step, as we start purging from right to left with pure inert gas, a simple wave develops at τ ) 0 and ζ ) ζa and expands to the left. As it crosses the interface between the two layers, each characteristic, corresponding to a certain value of fluid-phase concentration, is refracted. The purge step ends at τp ) γτa ) 0.3, when the predetermined concentration, c*0 ) 0.572, reaches ζ ) 0. Figure 5 shows bed profiles of adsorbedphase concentration during the purge step. The feed step requires the combined use of eq 14 and eqs 21 and 26, which are modified to account for the change in flow direction from the purge step; the characteristic lines in the physical plane during the feed step have positive slope. As shown in the upper part of Figure 4, the simple wave is intercepted and canceled by the shock which forms at the feed end of the bed at τ ) 0.3 (the start of the step). The simple wave is fully captured at τ ) 0.5491 and ζ ) 1.5881, just before the end of the feed step, which occurs at τ ) 0.55. Bed


to operate the cycle. In terms of our notation, we fix φ and vary L1/L (or ζa) to minimize γ. The condition for this minimization is therefore

|

∂γ ∂ζa

)0

(47)

φ

A second objective function is maximization of the throughput parameter with the volumetric purge-to-feed ratio fixed. This maximizes the adsorption time for a bed of fixed length or, equivalently, minimizes total bed length at fixed adsorption time. Mathematically, this is described by

|

∂φ )0 ∂ζa γ Figure 5. Bed profiles for the purge step in example 4: (a) start of the purge step; (b) simple wave reaches the interface between layers; (c) simple wave starts exiting the bed; (d) simple wave partially removed; (e) end of the purge step (τp ) 0.3).

(48)

These two optimization problems are not independent of each other. Rather, they are really the same one. This is clear from the identity

|

| |

∂φ ∂γ ∂φ )∂ζa γ ∂ζa φ∂γ

(49) ζa

So, the same bed structure will satisfy both eqs 47 and 48. To find the optimal value of ζa and to generate figures below, we adopt the following procedure. Our periodic states are uniquely defined once we assign adsorption equilibrium (R1, R2, and η) and the fixed-bed cycle parameters (φ, γ, and ζa). In eqs 22 and 27, we make the substitution τa ) φζa, which gives

(

ζa ) 1 + γφζa -

| )( | )

-1 dq* dq* 1 2 dc* c*0 dc* c*0

(50)

φζa ) 1 + (ζa - 1)(η - q* 2,0) - q* 1,0 + γφζac* 0 (51) Figure 6. Bed profiles for the feed step in example 4: (a) start of feed step; (b) shock in the first layer; (c) shock at the interface between layers; (d) shock in the second layer; (e) end of the feed step (τa ) 0.25, τ ) 0.55).

profiles for the feed step are shown in Figure 6. At the end of the feed step, the blocklike structure existing at the beginning of the purge step has been fully reestablished. Optimizations. Example 5. We consider here two cases of a single example on the optimal layering of adsorption beds based on equilibrium theory. As a preliminary notion, we know that for a periodic state at particular values of ζa and γ, cycle step times and depths of layers will all be directly proportional to one another (i.e., if we double the adsorption and purge times, then the length of each layer will double too, and the ratio L1/L remains constant). We introduce the nondimensional group

φ≡

τa |v|ta L1 ) ζa Λ1L1 L

(46)

which is a throughput parameter, equal to the number of empty bed volumes (based on total bed length) of gas fed to the bed during the feed step. Different objective functions are of interest. One such function corresponds to the minimization of the volumetric purge-to-feed ratio for a cycle with a fixed throughput parameter (or fixed adsorption time and bed length). With this we seek to layer the bed in such a way as to minimize the amount of purge gas necessary

Thus, with adsorption equilibrium specified, we have two equations in four unknowns (φ, γ, ζa, and c* 0). It is possible, in theory, to solve the optimization problems analytically, simply imposing the condition that γ be minimum or φ (or L) be maximum. This constraint increases the number of equations by one, so that a system of three algebraic equations needs be solved. However, we have resorted to the simpler procedure of fixing either φ or γ, then picking values of ζa (in 0 e ζa-1 e 1) and solving eqs 50 and 51 for c* 0 (in 0 e c* 0 e 1) and the remaining γ or φ. This gives a clear picture of how the system behaves and helps discriminate between cases where extrema are well-defined and cases where they are not. Case A: R1 ) 1, R2 ) 0.1. Figure 7 shows the profile of γ as a function of ζa-1 with parameter φ for a system with η ) 2. The curves have minima, giving the optimal combinations of layer thicknesses to minimize the volumetric purge-to-feed ratio (or purge time) for systems with fixed adsorption times and bed lengths. As a general trend, we note that an increase in φ causes an increase in the minimum γ. Also, as φ is increased, the minimum γ shifts toward lower values of ζa-1. For much higher values of φ, the lowest value of γ is obtained by considering a bed made up of the adsorbent with the more favorable isotherm only. However, this value of γ may be too large to be of practical interest. On the other hand, as φ is decreased, the minimum gradually disappears on the right of the plot and γ monotonically decreases as ζa-1 increases, indicating that for those values of φ, a bed made up of the


Figure 7. Behavior of the purge-to-feed ratio γ as a function of bed composition for case A of example 5 (R1 ) 1, R2 ) 0.1, η ) 2). Different values of the throughput parameter φ are considered.

Figure 9. Behavior of the throughput parameter φ as a function of bed composition for case A of example 5 (R1 ) 1, R2 ) 0.1, η ) 2). Different values of γ are considered.

Figure 8. Behavior of the purge-to-feed ratio γ as a function of bed composition for case A of example 5 (R1 ) 1, R2 ) 0.1, φ ) 0.2). Different values of η are considered.

Figure 10. Behavior of the throughput parameter φ as a function of bed composition for case B of example 5 (R1 ) 0.1, R2 ) 0.01, γ ) 1.2). Different values of η are considered.

adsorbent characterized by the less favorable isotherm requires the lowest purge-to-feed ratio and is therefore preferable. Figure 8 shows γ as a function of ζa-1 with parameter η for a system characterized by φ ) 0.2. The most noticeable feature of this figure is the monotonic variation of γ with respect to ζa-1. From the figure it can be inferred, as a general trend, that an increase in η, for a system with a first layer characterized by an isotherm with a higher constant separation factor, causes the volumetric purge-to-feed ratio to decrease. The analysis of systems characterized by higher values of φ has shown that as φ is increased the curves tend to shift toward higher values of γ. Also, γ increases faster for the bed richer in the component with the less favorable isotherm than for the bed richer in the component with the more favorable isotherm. Therefore, for relatively low values of η, the curves can easily pass through minima, whereas for higher values of η, they show that γ can increase as ζa-1 increases. As was mentioned previously, though, what makes the presence of minima of γ at higher values of φ unappealing is the actual numerical value of γ, which is much greater than values normally found in industrial applications. Figure 9 shows curves of φ as a function of ζa-1 for different values of γ for the system characterized by R1 ) 1, R2 ) 0.1, and η ) 2. The figure illustrates that when the purge-to-feed ratio is increased, the throughput parameter φ increases

and that for all values of γ considered, a maximum exists for a bed made up of an appropriate combination of the two adsorbents. However, this will not always be the case. As general trends in comparing curves, as we increase γ, curves shift upward, and as we increase η, it turns out that for this set of parameters a bed made up of only the adsorbent with the more favorable isotherm will be the one with the higher throughput. On the other hand, as we decrease η, the adsorbent with the less favorable isotherm tends to yield the higher throughput. Case B: R1 ) 0.1, R2 ) 0.01. Figure 10 shows φ as a function of ζa-1 for γ ) 1.2. It is apparent that for this combination of parameters the adsorbent with the more favorable isotherm is to be preferred when η ) 10, whereas when η ) 2 or η ) 5 a bed containing more of the adsorbent with the less favorable isotherm yields a higher throughput. Thus, for large η, the higher capacity of the second adsorbent more than offsets its more favorable isotherm. For lower values of η the reverse is truesthe adsorbent with the less favorable isotherm is preferred. Conclusions Adsorption beds may be layered because of equilibrium factors or mass transfer concerns. We have considered only equilibrium factors. Efforts to promote favorable mass transfer characteristics and reduce


pressure drop may suggest that a bed be layered, for example, with large particles near the feed inlet and small particles of the same adsorbent near the purge inlet. Equilibrium theory gives insight on the effect of layering adsorbents of different types. We have considered several systems and analyzed effects of parameters that characterize them to select desirable combinations of adsorbents and best operating conditions. Restrictions on adsorption time and volumetric purgeto-feed ratio have been identified. We have focused on practical problems such as the minimization of the purge-to-feed ratio, which is related to product recovery, and the minimization of bed length or maximization of throughput, which determines adsorbent productivity. Equilibrium theory for PSA does show that in some cases a combination of two adsorbents is to be preferred over the use of one adsorbent alone. This means that the usefulness of layering adsorbents can be inferred even from a simple equilibrium model that assumes no resistance to mass transfer. In restricting ourselves to realistic values of parameters, in some cases our analysis has shown that one adsorbent is better than a combination, and that there is no advantage to layering. The approach developed here can easily be applied to systems with more than two layers. Acknowledgment The authors are grateful to the U.S. Army ERDEC for financial support. We thank John J. Mahle for suggesting this problem to us. Notation c ) fluid-phase concentration, mol/m3 Ki ) Langmuir isotherm parameter for layer i, m3/mol L1 ) length of the first layer, m L ) total length of the bed, m qi ) adsorbed-phase concentration in layer i, mol/kg Qi ) Langmuir monolayer capacity for layer i, mol/kg Ri ) separation factor for isotherm for layer i t ) time, s v ) interstitial velocity, m/s z ) axial coordinate, m Greek Letters γ ) volumetric purge-to-feed ratio ∆ ) adsorbate removed by purge ) void fraction of packing ζ ) dimensionless axial coordinate ζa ) depth of penetration at end of feed step

η ) relative capacity parameter, eq 13 Λi ) partition ratio in layer i Fbi ) bulk density of packing in layer i, kg/m3 τ ) dimensionless time τa ) dimensionless adsorption time τp ) dimensionless purge time φ ) throughput parameter

Literature Cited Chan, Y. N. I.; Hill, F. B.; Wong, Y. W. Equilibrium Theory of a Pressure Swing Adsorption Process. Chem. Eng. Sci. 1981, 36, 243-251. Chihara, K.; Suzuki, M. Simulation of Nonisothermal Pressure Swing Adsorption. J. Chem. Eng. Jpn. 1983, 16, 53-61. Chlendi, M.; Tondeur, D. Dynamics of Two-Adsorbent Beds with Flow-Reversal for Gas Separation. Fundamentals of Adsorption; LeVan, M. D. Ed.; Kluwer Academic Publishers: Boston, MA, 1996; pp 187-194. Davis, M. M.; LeVan, M. D. Equilibrium Theory for Complete Adiabatic Adsorption Cycles. AIChE J. 1987, 33, 470-479. Klein, G.; Vermeulen, T. Cyclic Performance of Layered Beds for Binary Ion Exchange. AIChE Symp. Ser. 1975, 71 (152), 6976. LeVan, M. D. Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655-2660. Liu, Y.; Ritter, J. A. Pressure Swing Adsorption-Solvent Vapor Recovery: Process Dynamics and Parametric Study. Ind. Eng. Chem. Res. 1996, 35, 2299-2312. Mahle, J. J.; Friday, D. K.; LeVan, M. D. Pressure Swing Adsorption for Air Purification. 1. Temperature Cycling and Role of Weakly Adsorbed Carrier Gas. Ind. Eng. Chem. Res. 1996, 35, 2342-2354. Rhee, H.-K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations: Volume I. Theory and Application of Single Equations; Prentice-Hall: Englewood Cliffs, NJ, 1989. Ritter, J. A.; Yang, R. T. Pressure Swing Adsorption: Experimental and Theoretical Study on Air Purification and Vapor Recovery. Ind. Eng. Chem. Res. 1991, 30, 1023-1032. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley: New York, 1984. Ruthven, D. M.; Farroq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH Publishers: New York, 1994. Suzuki, M. Adsorption Engineering; Elsevier: Amsterdam, 1990. Tondeur, D.; Wankat, P. C. Gas Purification by Pressure Swing Adsorption,. Sep. Purif. Met. 1985, 14 (2), 157-212. Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Stoneham, MA, 1987.

Received for review November 11, 1996 Revised manuscript received March 11, 1997 Accepted March 17, 1997X IE9607154

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

Equilibrium Theory for Pressure Swing Adsorption ... - ACS Publications

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