Equilibrium Theory Analysis of a Pressure-Swing Adsorption Cycle

Ind. Eng. Chem. Res. 2003, 42, 3381-3390

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SEPARATIONS Equilibrium Theory Analysis of a Pressure-Swing Adsorption Cycle Utilizing an Unfavorable Langmuir Isotherm. 2. Approach to Periodic Behavior Karen D. Daniel and James A. Ritter* Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, South Carolina 29208

An isothermal equilibrium theory analysis of a simple two-step pressure-swing adsorption (PSA) process utilizing an adsorbate-adsorbent system that exhibits an unfavorable Langmuir isotherm was carried out. Analytical expressions that directly describe the approach to periodicity as a function of the cycle number were obtained and used to derive expressions for all of the important process performance indicators that change with the cycle number for the case with breakthrough of the heavy component into the light product (i.e., γ < 1). This PSA process goes through four unique stages on the approach to the periodic behavior that are marked by different shock and simple wave interactions. The most unusual wave interaction that was found occurs between two simple waves and a shock wave that manifests and disappears before the periodic state is reached. A case study with a H2-metal hydride system that exhibits an unfavorable Langmuir isotherm was used to reveal this unique behavior. This system was also used to show which process performance indicators are a function of the cycle number and how they change as periodicity is approached. Introduction series,1

a novel pressure-swing In part 1 of this adsorption (PSA) process utilizing an adsorbateadsorbent system that exhibits an unfavorable Langmuir isotherm was shown to produce a very high purity light product and a heavy product enriched to the pressure ratio. This latter result was surprising because the enrichment of the heavy component for a favorable Langmuir isotherm is typically far below the pressure ratio. The dilution effect of the purge gas that is used to regenerate the column and the fact that a favorable Langmuir isotherm gives rise to a simple spreading wave during desorption both contribute to this low enrichment. However, because an unfavorable Langmuir isotherm gives rise to a simple spreading wave during the feed step and a shock wave during the purge step,2 heavy-component enrichments equal to the pressure ratio can be achieved with a PSA process utilizing this kind of adsorbate-adsorbent isotherm.1 Isothermal equilibrium theory was used in part 1 of this series1 to establish the periodic state directly and analytically for this novel PSA process, and simple expressions were obtained that describe all of the important process performance indicators at the periodic state, with and without breakthrough of the heavy component into the light product. These expressions should be very useful for carrying out feasibility studies because they represent the upper thermodynamic limit for the performance of this kind of PSA process. They * To whom correspondence should be addressed. Tel.: (803) 777-3590. Fax: (803) 777-8265. E-mail: [email protected].

should also be useful for educational and training purposes because much insight can be gained about PSA in general from such simple analytical relationships. In this regard, it would be very insightful to develop analytical expressions that describe the wave dynamics and process performance on the approach to the periodic state. Very few studies have considered the nonperiodic behavior of a PSA process, especially from the point of view of obtaining analytical expressions. A notable exception is the work by LeVan,2 in which the nonperiodic behavior was analyzed in terms of equilibrium theory to show the buildup of the heavy-component heel on the approach to periodicity for a favorable Langmuir isotherm; however, the unfavorable Langmuir isotherm case, as well as the process performance indicators, was not considered.2 Therefore, the objective of this work is to give an interesting overview of the equilibrium theory analysis that describes the approach to periodicity for a PSA process utilizing an unfavorable Langmuir isotherm. The analysis done in part 1 of this series1 is extended here to establish simple analytical expressions that describe the wave dynamics and the process performance indicators as a function of the cycle number during the approach to periodicity. The isothermal equilibrium theory developed for evaluation of periodic states of solvent vapor recovery systems by PSA, first by LeVan2 and later extended by Ritter and coworkers3-7 and then most recently by Daniel and Ritter,1 is used to analyze this nonperiodic behavior. Only the case with breakthrough of the heavy component into the light product is considered because without

10.1021/ie020958y CCC: $25.00 © 2003 American Chemical Society Published on Web 06/07/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003

Table 1. Definition of Variables Used in the Analysis3 description

equation

dimensionless adsorbed-phase concentration of the adsorbate dimensionless fluid-phase concentration of the adsorbate dimensionless time dimensionless axial coordinate stoichiometric depth of penetration for the first feed step (m) Langmuir isotherm characteristic invariant

breakthrough the periodic behavior corresponds to the first cycle.1,2 A brief case study is carried out with the same H2-metal hydride unfavorable Langmuir isotherm used in part 1 of this series.1 This system is used to reveal some unique wave dynamic features associated with this kind of PSA process-isotherm combination that manifest and then disappear as the periodic state is approached. This system is also used to show which process performance indicators are a function of the cycle number and how they change as periodicity is approached. Theory Equilibrium theory accounts only for mass conservation and hence ignores mass-and heat-transfer resistances and other nonideal phenomena.8 The assumptions associated with the equilibrium theory developed here include instantaneous local equilibrium between the fluid and adsorbed phases, isothermal operation, no axial dispersion, and no axial pressure drop. Because the feed contains a very low concentration of adsorbate in an inert carrier gas, the fluid-phase velocity is assumed to be constant throughout the column. Fluidphase accumulation of the adsorbate is also negligible because of the large partition ratio between the adsorbed and fluid phases. Another important assumption concerns freezing of the adsorbed and fluid phases during pressurization and blowdown. This assumption simplifies the PSA cycle to a two-step process consisting of only feed and purge steps.9 The frozen solid-phase approximation also requires the feed mole fraction to be less than the inverse of the pressure ratio.3 Finally, this analysis is limited to purging with only pure carrier gas, as was done elsewhere.1-3 Taking these assumptions into account, eqs 1-12 and 15-21 of Subramanian and Ritter3 apply. Equations 20 and 213 are two first-order ordinary differential equations that describe the inverse concentration velocity as a function of the fluid-phase concentration for simple and shock waves, respectively, as

dτ/dσ ) (Ra2

(1)

dτ/dσ ) -RaLaR

(2)

The parameters in these equations and other pertinent variable definitions are given in Table 1. This analysis is limited to an adsorbate-adsorbent system that exhibits an unfavorable Langmuir isotherm (R > 1). This analysis is also limited to a process with a volumetric purge-to-feed ratio (γ) of less than 1. With γ < 1, breakthrough of the adsorbate into the light product will occur, but the heavy product will not be unnecessarily diluted with the purge gas.1 As explained in part 1 of this series,1 a unique feature of this system is that once breakthrough of the adsorbate into the light product occurs during the feed step, a discontinuity

q/a ) qa/qref c/a ) ca/cref τ ) t/tf

σ ) z/L L ) vtfcref/Fbqref q/a ) c/a/[R + (1 - R)c/a] a ) 1/(c/aR - R - c/a)

(shock) will develop immediately at the beginning of the purge step. Expressions that describe the bed profiles during the feed and purge steps on the approach to periodicity are now derived. During the approach to the periodic state, the feed and purge steps will go through four different stages, exhibiting different end-of-step bed profiles in each stage. In the first stage, breakthrough of hydrogen into the light product has not occurred yet, so both the end-of-step feed and purge profiles contain only a simple wave. This stage will occur only if R < σb (i.e., no breakthrough on the first feed step). Before breakthrough occurs, each cycle will consist of only a simple wave moving back and forth through the bed. Once breakthrough of adsorbate into the light product occurs, the second stage has been reached. At this point, the feed step will consist of two simple waves, but one of these waves will be completely pushed out of the bed by the end of the feed step, so the end-of-step feed profile will only show one simple wave. The purge profile in this stage will contain a simple wave and a shock wave. After several more cycles, the second simple wave will not be completely pushed out of the bed, so the profile at the end of the feed step will consist of two simple waves, which is the third stage. During the purge step in this stage, the shock wave will completely consume one of the simple waves and part of the second simple wave, leaving a simple wave and a shock wave in the bed at the end of the purge step. In each purge step, the shock wave will continue to grow until a shock of the fullest strength (i.e., a shock between the feed concentration and the purge concentration) develops. This marks the beginning of the fourth stage. Subsequent feed steps will always contain only one simple wave, and a shock of the fullest strength will develop by the end of all subsequent purge steps. Eventually the breakthrough concentration and the location of the shock wave at the end of the purge step will no longer change appreciably, meaning that the periodic state has been reached. The bed profiles will also remain the same during this stage once the periodic state has been reached. Stage One. Each stage in the approach to the periodic state must be analyzed separately to develop equations that describe the feed and purge profiles. The first stage includes all cycles prior to the first cycle with breakthrough. End-of-step feed and purge profiles in this stage are shown by cycle 1 in Figure 1. If R > σb, breakthrough will occur on the very first feed step and the bed will skip the first stage. Otherwise, starting with an initially clean bed, the profile during the first feed step is obtained by integrating eq 1 with the initial conditions σ ) 0 and τ ) 0. Performing this integration and simplifying give

σf,1(a) ) τ/Ra2

(3)

where τ changes from 0 to 1 during the feed step. The

Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3383

concentration during each feed step in stage one, which is given by

σf,i(a) )

1 [τ + (i - 1)(1 - γ)] Ra2

(7)

where i is the cycle number. Equation 7 is easily modified to represent end-of-step feed profiles by substituting τ ) 1. Likewise, a comparison of eqs 4 and 6 leads to a general expression for the position of each concentration during each purge step in stage one, as shown by

σp,i(a) )

1 [i - (i - 1)γ - τ] Ra2

(8)

Substituting τ ) γ in eq 8 gives the position of each concentration at the end of each purge step in stage one. Equations 7 and 8 are used to map the concentration bed profiles during the feed and purge steps in the first stage. Recall that the end of the first stage is marked by breakthrough of the adsorbate into the light product. The cycle when this first occurs is found by substituting a ) -1/R, σ ) σb, and τ ) 1 into eq 7 and solving for i, which yields

ibt,1 )

Figure 1. End-of-step bed profiles during the approach to periodicity for the H2-MmNi4.2Al0.8 system, where i is the cycle number: (a) end of the feed step; (b) end of the purge step.

first end-of-step feed profile, which is the initial condition for the subsequent purge step, is found by substituting τ ) 1 into eq 3. Integrating eq 1 with σ ) 1/Ra2 and τ ) 0 as the initial conditions gives the position of each concentration during the first purge step as

σp,1(a) )

1 (1 - τ) Ra2

(4)

where τ changes from 0 to γ during the purge step. The first end-of-step purge profile is found by substituting τ ) γ into eq 4. Integrating eq 1 again with σ ) (1 γ)/Ra2 and τ ) 0 as the initial conditions gives the position of each concentration during the second feed step as

σf,2(a) )

1 (τ + 1 - γ) Ra2

(5)

Again, the end-of-step feed profile is found by substituting τ ) 1 into eq 5. Using σ ) (2 - γ)/Ra2 and τ ) 0 as the initial conditions and integrating eq 1 give

1 (2 - γ - τ) σp,2(a) ) Ra2

(6)

Now, the end-of-step purge profile is obtained by simply substituting τ ) γ into eq 6. A comparison of eqs 3 and 5 leads to a general expression for the position of each

Rγ - σb

(9)

R(γ - 1)

where the subscript bt,1 refers to the first cycle with breakthrough. Stage Two. The result from eq 9, rounded up to the nearest whole number, is the cycle when the second stage begins. End-of-step feed and purge profiles in this stage are shown by cycle 4 in Figure 1. During this stage, the breakthrough concentration is higher than the concentration that was just consumed by the shock wave at the end of the purge step. In terms of the characteristic invariant, this is expressed as a/i < ap,i-1, where ap,i-1 is the characteristic invariant that was just consumed by the shock wave at the end of the purge step of the previous cycle (because a is always negative, a higher fluid-phase concentration corresponds to a more negative characteristic invariant). In other words, a/i was part of the simple wave in the previous purge step, not part of the shock wave. This is the defining characteristic of the second stage. Starting with the first cycle with breakthrough, the purge step will contain both a simple and a shock wave. For the feed step, eq 7 still applies, but only for the first part of the feed step. Once the simple wave reaches the end of the bed, two expressions are now needed to map the concentration profiles. The dimensionless time when the simple wave reaches the end of the bed, τb0, is found by substituting σ ) σb and a ) -1/R into eq 7 and solving for τ. Once the simple wave reaches the end of the bed, the complete expression for the profile during the feed step is

{

1 [τ + (i - 1)(1 - γ)] σf,i(a) ) Ra2 σb

-1 e a e a* a* < a e -1/R (10)

where a* represents the concentration that has just reached the end of the bed and changes from -1/R (at τ ) τb0) to a/i (at τ ) 1) where τb0 is the dimensionless

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time when c* ) 0 reaches the bed exit. Substituting a ) a* and σ ) σb into eq 10 and rearranging give

x

a* ) -

τ + (i - 1)(1 - γ) Rσb

{

-1 e a e am am < a e -1/R (12)

The first expression in eq 12 describes the simple wave portion of the profile during the purge step and is the same as eq 8. The constraint am in eq 12 is the characteristic invariant that joins the simple and shock waves, and it changes from a/i at the beginning of the purge step to ap,i at the end of the purge step. To obtain expressions for ap,i and σs,i(τ) in eq 12, eq 2 is rewritten as

dσ dτ ) -RaaR da da

(13)

which describes the shock wave portion of the profile during the purge step. The derivative of the right-hand side of the first expression in eq 12 (with τ ) γ) with respect to a is substituted into eq 13, yielding an expression with only two variables, τ and a, which is then integrated. By using the simple wave equation to obtain an expression for dσ/da and then substituting this expression into the equation that describes the shock wave, the resulting expression, τs,i(a), will give the dimensionless time when the shock wave overtakes each concentration in the simple wave. Taking the derivative of eq 12 and substituting it into eq 13 yield

dτ 2[τs,i(a) - i + γi - γ] ) da a(Ra + 1)

(14)

Integrating eq 14 with the initial conditions τ ) 0 and a ) a/i gives

τs,i(a) )

(i - γi + γ)(a/i - a)(a/i + 2Ra/i a + a) 2 a/2 i (Ra + 1)

σs,i(a) )

(11)

The characteristic invariant that just reaches the end of the bed at the end of the feed step, a/i , is found by substituting τ ) 1 into eq 11. A shock wave will form immediately between aR and a/i at the beginning of the purge step. The shock wave is traveling faster than the simple wave, so the shock wave will gradually overtake the simple wave while becoming decelerated. The position of each concentration during the purge step is found by solving for the locations of the simple and shock waves simultaneously over the purge step. First, τ ) 1 is substituted into the first expression in eq 10. Then eq 1 is integrated over the purge step, with this modified eq 10 and τ ) 0 as the initial conditions. This integration yields

1 [i - (i - 1)γ - τ] σp,i(a) ) Ra2 σs,i(τ)

expression in eq 12 and simplified to give

(15)

The concentration that is just consumed by the shock wave at the end of the purge step, ap,i, is determined by substituting a ) ap,i and τs,i(a) ) γ in eq 15 and solving for ap,i. Equation 15 is substituted into the first

(i - γi + γ)(Ra/i + 1)2 2 Ra/2 i (Ra + 1)

(16)

Equations 15 and 16 are parametric equations that describe the shock path as it consumes the simple wave during the second stage in the approach to the periodic state. The location of the shock wave at the end of the purge step is found by substituting a ) ap,i into eq 16. Equations 10 and 12 are used to generate the concentration bed profiles during the feed and purge steps in stage two. The end-of-step feed and purge concentration profiles can also be plotted using eqs 10 and 12 (with τ ) 1 and a* ) a/i in eq 10 and τp(a) ) γ and am ) ap,i in eq 12). Stage Three. During each purge step, the shock wave continues to grow, thereby consuming a higher concentration than that in the previous purge step. The third stage begins when the shock wave has grown large enough so that the breakthrough concentration is lower than the concentration that was just consumed at the end of the previous purge step (a/i > ap,i-1). This means that there are now two simple waves present in the bed at the end of the feed step, as shown by cycles 20 and 50 in Figure 1. One simple wave extends from the feed concentration to ap,i-1, while the other simple wave extends from ap,i-1 to a/i . Adding the second simple wave to eq 10 gives

{

1 [τ + (i - 1)(1 - γ)] Ra2 σf,i(a) ) τ + σ s,i-1 Ra2 σb

-1 ea < ap,i-1 ap,i-1 e a e a* a* < a e -1/R (17)

which is the profile during the feed step in stage 3. Again, a* ranges from -1/R (at τ ) τb0) to a/i (at τ ) γ). Similar to stage two, τb0 is calculated by substituting a ) -1/R and σ ) σb into the second expression in eq 17 and solving for τ. Only the first two expressions in eq 17 are needed during the first portion of the feed step. Once the leading simple wave reaches the end of the bed (at τb0), all three expressions in eq 17 are used to create bed profiles. An expression for a* is obtained by substituting a ) a* and σf,i(a) ) σb into eq 17, which leads to

x

a* ) -

τ R(σb - σs,i-1)

(18)

which is an expression for the breakthrough concentration during the third stage. Substituting τ ) 1 into eqs 17 and 18 gives the end-of-step feed profile and the maximum breakthrough concentration, respectively. Again, a shock wave will form immediately between aR and a/i at the beginning of the purge step, and it will overtake the simple waves while becoming decelerated. The parametric equations for the shock path during the purge step are derived in the same manner as outlined above. First, eq 1 is integrated over the purge step, with eq 17 (end-of-step version) and τ ) 0 as the initial


{

conditions. This integration yields

1 [i - (i - 1)γ - τ] Ra2 2 σp,i(a) ) -τ + 1 + σs,i-1Ra Ra2

-1 e a < ap,i-1 ap,i-1 e a e am am < a e -1/R (19)

σs,i(τ)

which is the profile during the purge step. Again, the constraint am changes from a/i to ap,i during the purge step. Once am is equal to or greater than ap,i-1, the middle expression in eq 19 is no longer used. The profile at the end of the purge step is given by the first and last expressions in eq 19, with τ ) γ. Taking the derivative of the simple wave expressions in eq 19 and substituting them into eq 13 yield

{

2(τ - i + γi - γ) a(Ra + 1) dτ ) 2(τ - 1) da a(Ra + 1)

-1 e a < ap,i-1 (20) ap,i-1 e a e

a/i

The first expression in eq 20 is integrated with the initial conditions a ) ap,i-1 and τ ) τap, where τap is the time it takes the shock wave to reach the concentration ap,i-1. The second expression is integrated with the initial conditions a ) a/i and τ ) 0. Performing the integration yields

{

Figure 2. Evolution of the bed profiles during cycle 20 for the H2-MmNi4.2Al0.8 system: (a) during the feed step; (b) during the purge step.

τs,i(a) ) 2

2

[(i - γi + γ)(ap,i-1 (1 + 2Ra) - a (1 + 2Rap,i-1)) + a2τap(Rap,i-1 + 1)2]/[ap,i-12(Ra + 1)2] a/2 i (1

2

+ 2Ra) - a (1 +

2Ra/i )

2 a/2 i (Ra + 1)

-1 e a < ap,i-1 ap,i-1 e a e a/i

(21)

Now a ) ap,i-1 is simply substituted into the second expression in eq 21 to find a value for τap. Next, τ ) γ and a ) ap,i is substituted into the first expression in eq 21 and solved for ap,i. Then eq 21 is substituted into eq 19 and simplified to give

{

σs,i(a) ) (i - γi + γ - τap,i)(Rap,i-1 + 1)2 Rap,i-12(Ra + 1)2 2 (Ra/i + 1)2 + Ra/2 i σs,i-1(Ra + 1) 2 Ra/2 i (Ra + 1)

-1 e a < ap,i-1 ap,i-1 e a e a/i (22)

Equations 21 and 22 are parametric equations that describe the shock path as it consumes the simple wave during the third stage in the approach to the periodic state. The location of the shock wave at the end of the purge step is given by the first expression in eq 22, with a ) ap,i. Figure 2a shows the progression of bed profiles, obtained by using eq 17, during the feed step of cycle 20. These profiles clearly show the presence of two simple waves at the end of the feed step. To calculate the bed profiles during the feed step for a given cycle, first calculate a/i and ap,i-1 using eqs 18 and 21, respectively. Next pick a value of τ between 0 and 1 and

use eq 17 to calculate σ for a range of a values from -1 to -1/R. Figure 2a shows the results of this type of calculation for τ values of 0, 0.25, 0.5, 0.75, and 1. Figure 2b shows the progression of bed profiles, obtained with eq 19, during the purge step in cycle 20. These profiles show that the shock wave continues to grow until it has completely overtaken one of the simple waves and part of the second simple wave. To calculate the bed profiles during the purge step for a given cycle, first calculate τap using the second expression in eq 21 (with a ) ap,i-1). Next, pick a value of τ between 0 and γ. If τ < τap, use the second expression in eq 21 to calculate the characteristic invariant that links the simple and the shock waves, am, at this dimensionless time. If τ > τap, use the first expression in eq 21 to calculate am. Next use this am to calculate σs,i at this specific τ (using eq 22). Once am and σs,i are known, use the first two expressions in eq 19 (with a range of a values from -1 to am) to map the simple wave at this τ. End-of-step feed and purge profiles are obtained in a similar manner, except τ ) 1 and a* ) a/i are substituted into the feed expressions and τ ) γ and am ) ap,i are substituted into the purge expressions. Stage Four. The shock wave will continue to grow with each purge step, until eventually a shock of the fullest strength develops, which is the beginning of the fourth and final stage. Cycle 130 in Figure 1 shows the presence of a shock of the fullest strength at the end of the purge step. During the fourth stage, a shock of the fullest strength connects the purge concentration (a ) -1/R) and the feed concentration (a ) -1). In this stage, every concentration in the feed profile was a part of the

3386


shock wave in the previous purge step. Eliminating the first expression in eq 17 and adjusting the characteristic invariant limits give

{

τ + σs,i-1 σf,i ) Ra2 σb

-1 e a e a*

(23)

a* < a e -1/R

which is the profile during the feed step in the fourth stage. Again, the second expression is only used once the simple wave has reached the end of the bed, and a* changes from -1/R (at τ ) τbt) to a/i (at τ ) 1). Equations 21 and 22 are easily modified for cycles in which a shock of the fullest strength develops by eliminating the first expression. This yields

τs,i(a) )

σs,i(a) )

2 / a/2 i (1 + 2Ra) - a (1 + 2Rai ) 2 a/2 i (Ra + 1)

2 (Ra/i + 1)2 + Ra/2 i σs,i-1(Ra + 1) 2 Ra/2 i (Ra + 1)

(24)

(25)

Equations 24 and 25 apply to all concentrations between a/i and -1. The dimensionless time and position of the shock wave of the fullest strength (τsfs and σsfs) during the first purge step in stage four are found by substituting a ) -1 into the first expressions in eqs 21 and 22. For all subsequent purge steps, a ) -1 is substituted into eqs 24 and 25 to find τsfs and σsfs. Once the shock of the fullest strength develops, the shock will travel with a constant velocity of -1. Substituting aL ) -1 and aR ) -1/R into eq 2 and then integrating with the initial conditions τ ) τsfs and σ ) σsfs yield

σs,i(τ) ) σsfs,i + τsfs,i - τ

(26)

Eliminating the first expression in eq 19 and adjusting the characteristic invariant limits give

σp,i(a) )

{

-τ + 1 + σs,i-1Ra2 Ra2 σs,i(τ)

periodic state is reached. Therefore, the purity and enrichment of the heavy product will always be equal to ya,fR and R, respectively. The recovery of the heavy product also remains constant (and equal to γ). This is a very interesting result that suggests that a highly enriched heavy product can be produced essentially at the onset of cycling of this kind of PSA process, even when starting up the process with both of the beds filled with only a pure light component, as an example. During stage one, the light product will, of course, be 100% pure. However, once stage two begins, the purity of the light product will decrease, until it reaches the periodic state purity. During stage one, before breakthrough of the adsorbate into the light product, all of the adsorbate that is not recovered in the heavy product remains in the bed at the end of the purge step, forming a heel that grows with each cycle. The growth of this heel can be seen in Figure 1b. Once stage two is reached, some of the unrecovered adsorbate is lost in the light product during the feed step while the rest is added to the heel in the bed at the end of the purge step. At the beginning of stage two, only a small portion of the unrecovered adsorbate leaves the bed with the light product. As the cycle number increases, however, more of the unrecovered adsorbate is lost in the light product. This causes the purity of the light product to decrease as the cycle number increases. Eventually the amount of unrecovered adsorbate added to the heel in the bed is so insignificant that the profile at the end of each purge step is essentially the same, and the periodic state has been reached. As explained by Daniel and Ritter,1 the light-product impurity is found by determining the average adsorbate concentration in the light product. Equations 34 and 35 of Daniel and Ritter1 give the dimensionless concentration exiting the bed during the feed step as a function of time and the time at which breakthrough first occurs at the periodic state, respectively. These equations are easily converted to apply to each cycle in the approach to the periodic state by changing σa to σp,i-1, which gives

-1 e a e ap,i-1 ap,i-1 < a e -1/R (27)

which is the profile during the purge step in the fourth stage, where σs,i(τ) is given by eq 26. Once a shock of the fullest strength has developed, only the second expression in eq 27 is used to calculate bed profiles during the purge step. Substituting τ ) γ in eq 26 gives the dimensionless position of the shock wave at the end of the purge step. Equations 24-27 apply for all subsequent cycles, even once the periodic state has been reached. Process Performance Indicators Expressions for the important process performance indicators, such as the heavy-product enrichment, recovery, and purity and the light-product recovery and purity, at the periodic state for this adsorbateadsorbent system are given in part 1 of this series.1 If γ < 1, the heavy-product performance indicators do not change during the approach to periodicity because the concentration of the adsorbate in the heavy product is always equal to the feed concentration, even before the

/ ) ca,bt

tR - xtRtf(σb - σp,i-1) t(R - 1)

(28)

and

ti,b0 )

tf(σb - σs,i-1) R

(29)

Equations 28 and 29 apply to all cycles in which breakthrough occurs, but σp,i-1 will be different for each stage. In eq 29, σs,i-1 is given by eq 8 (with τ ) γ) for the first cycle with breakthrough, by eq 16 (with a ) ap,i) for all other cycles in stage two, by the first expression in eq 22 (with a ) ap,i) during stage three, and by eq 25 (with a ) ap,i) during stage four. In eq 28, during stage two there will be two expressions for σp,i-1 because some concentrations that break through were part of a simple wave in the previous purge step and some concentrations were part of the shock wave in the previous purge step. In this stage, σp,i-1 is given by eq 12 (with τ ) γ, am ) ap, and i ) i - 1). The second expression in eq 12 will be used from ti,b0 until the time (ti,ap) when the concentration ap,i-1 reaches the bed exit. During this time, all of the concentrations that exit the bed were part of the shock wave in the previous purge


step. Making the appropriate substitutions into eq 28 gives / ca,bt,sh

)

tR - xtRtf(σb - σs,i-1)

(30)

t(R - 1)

/ where σs,i-1 is given by eq 16 with a ) ap,i-1, a/i ) ai-1 , and i ) i - 1. From ti,ap to tf, the concentrations exiting the bed were part of the simple wave in the previous purge step, so the first expression in eq 12 is used to find σp,i-1. Substituting the first expression in eq 12 into eq 28 gives

tR / ) ca,bt,sw

x (

tRtf σb -

)

(i - 1)(1 - γ) Rai,b2

t(R - 1)

(31)

Substituting the definition of ai,b (from Table 1 with c/a / / ) ca,bt ) into eq 31 and solving for ca,bt give / ) ca,bt,sw

R[tf(i - 1)(γ - 1) - t] + xtfσbR[t + tf(i - 1)(1 - γ)] (1 - R)[t + tf(i - 1)(1 - γ)]

(32) which is used during stage two from t ) ti,ap to tf. The average adsorbate concentration in the light product can then be found using

∫0

ti,b0

/ ca,LP,avg

)

0 dt +

∫t

ti,ap

i,b0

/ ca,bt,sh dt +

∫t

tf

i,ap

simplifying give / ca,LP,avg )

R + σb - σs,i-1 - 2xR(σb - σs,i-1) (36) R-1

which applies to stages three and four. Equations 3436 can be converted to mole fraction units by multiplying by yf. It is easy to show that the recovery of the light product, as defined by eq 31 of Daniel and Ritter,1 is not a function of the cycle number because it includes the adsorbate that breaks through during the feed step as part of the light product. However, because the PSA cycle considered here (and also in part 1 of this series1) utilizes only a pure inert purge, once stage two begins, only the fraction of the light product that is produced during the feed step as pure inert is actually suitable for use as the purge gas because breakthrough of the adsorbate will contaminate the rest of the light product. This means that only the pure inert carrier gas coming out of the bed between time 0 and ti,b0 can be used as purge in the purge step. Therefore, it is useful to define a second recovery, Recpure inert, which is the ratio of the difference between the pure inert that leaves the bed during the feed cycle and the pure inert that enters the bed during the purge cycle to the inert that enters the bed during the feed cycle. This definition leads to

vfAPH vfγAPL RcT RcT vfAPH (1 - yf) RcT

τi,b0(1 - yf) Recpure inert )

/ ca,bt,sw dt

∫0t 1 dt f

(37)

which can be simplified to

(33) Recpure inert )

Performing the integration and simplifying give / ca,LP,avg ) {2xR[x(σb - σs,i-1)tf(xti,ap - xti,b0) +

tfxσb(i - iγ + γ) - xσbtf[ti,ap + tf(1 - γ)(i - 1)]] + R(ti,b0 - tf)}/[tf(1 - R)] (34)

which applies to stage two. For the first cycle in stage two, eq 34 is modified by setting ti,ap ) ti,b0 because all concentrations that exit during the first cycle in stage two were part of a simple wave in the previous purge step. This simplifies eq 34 to

σ a e σb -

xσbtf[ti,b0 + tf(1 - γ)(i - 1)]] + R(ti,b0 - tf)}/

During stages three and four, all of the concentrations that exit during the feed step were part of the shock wave in the previous purge step. Equation 30 still applies to these stages, but σs,i-1 will be given by the first expression in eq 22 during stage three and by eq 26 (with τ ) γ) in stage four. Equation 33 is modified for stages three and four by substituting ti,ap ) tf, which eliminates the last integral in the numerator, because all concentrations that break through during these stages were a part of the shock wave in the previous purge step. Performing the integration of this modified eq 33, substituting in the expression for ti,b0, and

(38)

where ti,b0 is given by eq 29. Clearly, this recovery is a function of the cycle number. Equation 38 is used to calculate the recovery of the pure inert during the approach to the periodic state. Equation 38 shows that it is possible to have a recovery of pure inert that is less than zero, meaning that there is not enough pure inert produced during the feed step for use in the purge step. Therefore, for this PSA process to be feasible, the righthand side of eq 38 must be greater than (or equal to) zero. Substituting eq 29 into eq 38 and rearranging give

/ ) {2xR[tfxσb(i - iγ + γ) ca,LP,avg

[tf(1 - R)] (35)

ti,b0 γ tf (1 - yf)R

γR (1 - yf)R

(39)

as the condition that must be met for sufficient pure inert to be produced during the feed step for use in the purge step. Case Study A brief case study is carried out to show the unique wave interactions that exist during the approach to the periodic state (as discussed above) and to reveal how the performance indicators change as a function of the cycle number, including the effect of the feed time (i.e., cycle time), on the approach to the periodic state. The specific system of interest is the same H2-MmNi4.2Al0.8 metal hydride system studied in part 1 of this series;1

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Figure 3. H2-MmNi4.2Al0.8 adsorption isotherm at 308 K. Points: experimental data. Solid line: CSF correlation. Table 2. Operating Conditions and Parameters for the Hydrogen-Metal Hydride Design and Case Studies1 yf Vf Fb Lb db γ PH

m3 STP/min kg/m3 m m atm

0.05 44.5 2000 6.00 0.60 0.80 29.6

PL T tf R qs b

atm K s mol of H2/kg atm-1

Figure 4. Average mole fraction of H2 in the light product during the approach to periodicity for the H2-MmNi4.2Al0.8 system.

1.56 308 550 5.45 -0.372 -0.552

the typical s-shaped adsorption isotherm (symbols) at 308 K10 for this system is depicted in Figure 3, along with the Langmuir isotherm (solid line) fitted to only the unfavorable portion of the data (because significant changes in loading can be achieved simply by cycling pressure in this unfavorable region). The Langmuir isotherm parameters are given in Table 2. In the design study in part 1 of this series, a hydrogen mole fraction in the feed (yf) of 0.05 was imposed, which limited the pressure ratio to be less than 20, and to stay on the concave portion of the adsorption isotherm, the high (feed) pressure was limited to 30.0 atm. Because a pure heavy product was desired, the pressure ratio was set at 19, which resulted in a low (purge) pressure of 1.56 atm. Other chosen process conditions and selected parameters are specified in Table 2. The resulting periodic state performance indicators for this hydrogenmetal hydride PSA process were a heavy product containing 95 mol % H2, which corresponded to a heavycomponent enrichment of 19.0 and a recovery of 80.0%. The light product contained 1 mol % H2 (which was specified), at a recovery of 95.8%.1 With these conditions set, this periodic state design study is now extended to include the approach to the periodic state. The equations developed above were used to produce the feed and purge profiles seen in Figure 1. For this process, breakthrough of H2 into the light product occurs during the second cycle. During the fifth cycle, the double-wave profile at the end of the feed step is seen for the first time because the breakthrough concentration during this cycle was part of the shock wave in the fourth purge step. Two simple waves are present at the end of each feed step from cycle 5 to cycle 116 (stage three). Figure 2 shows the bed profiles during cycle 20 for both the feed and purge steps. At the beginning of the feed step in cycle 20, there is just a simple wave and a shock wave in the bed. During the feed step, however, the shock wave spreads out into a simple wave and only a portion of this simple wave exits the bed, leaving two simple waves at the end of the feed step. The bed profiles during the purge step in cycle 20 show

Figure 5. Recovery of pure inert carrier gas as a function of the cycle number for the H2-MmNi4.2Al0.8 system.

that the shock wave develops immediately at the bed exit and continues to grow during the purge step until it has consumed one of the simple waves. By the end of the purge step, there is again just one simple wave and a shock wave in the bed. A shock of the fullest strength develops during the purge step of cycle 116 (and all subsequent purge steps). From this cycle on, the feed step once again contains only one simple wave. Although the approach to the periodic state is asymptotic, by cycle 130 the position of the shock wave at the end of the purge step (σs) changes by less than 0.003%. The process has essentially reached the periodic state at this point. Figures 4 and 5 show the purity of the light product and the recovery of the pure inert during the approach to periodicity, respectively. These are the only process performance indicators that change with the cycle number; recall that the heavy-product purity, enrichment, and recovery are not functions of the cycle number. In contrast, the light-product purity and the recovery of the pure inert both change significantly with the cycle number during stages one and two and the first part of stage three. Around cycle 60 (halfway through stage three), however, the change in the process performance indicators with subsequent cycles is less significant and they start to approach the periodic state values. Figure 5 shows that the recovery of the pure inert decreases with the cycle number until it reaches 31%, the periodic value. Although this is a low recovery, the heavy product is the stream of interest in this case,


Figure 6. Effect of cycle time on the approach to the periodicity for the H2-MmNi4.2Al0.8 system.

so the PSA process is feasible as long as the recovery of pure inert is not less than zero. Also, note that if the specified purity level of the light product at 1 mol % H2 is not acceptable, it is a simple matter to carry out a different PSA process design with a lower value specified for the maximum impurity; refer to part 1 of this series.1 It was shown in part 1 of this series that changing the cycle time does not change the process performance indicators, as long as the maximum allowed value of γ is greater than 0.8.1 However, changing the cycle time does effect how quickly the process approaches the periodic state, as shown by Figure 6. Increasing the feed time from 550 to 1000 s does not change the process performance indicators mentioned above, but the process will essentially be at the periodic state after only 58 cycles, compared to the 130 cycles needed to reach the periodic state at the feed time of 550 s. This interesting result is understood by considering the definition of the periodic state. The periodic state is reached when the net amount of heavy component added to the bed during the feed step (i.e., the amount of heavy component fed into the bed minus the amount of heavy component that leaves the bed with the light product during the feed step) equals the amount of heavy component removed during the purge step. Of these three quantities, only the amount of heavy product that leaves the bed with the light product is a function of the cycle number, as explained earlier. This amount is initially zero and gradually increases, with each subsequent cycle, to the periodic state value once breakthrough occurs. Increasing the feed time allows the concentration profile in each feed step to penetrate further into the bed, causing breakthrough of the heavy component into the light product to occur at an earlier cycle. Therefore, the amount of heavy component lost in the light product will reach the periodic state value after fewer cycles. Conclusions In part 1 of this series, the periodic behavior of a simple two-step PSA process utilizing an unfavorable Langmuir isotherm was analyzed in detail using isothermal equilibrium theory. A complete analytical description of the process was obtained, including expressions for all of the important process performance indicators. This initial work was extended here (i.e., in

part 2 of this series) by analyzing the approach to the periodic state. Again, a complete analytical description of the process was obtained as a function of the cycle number, including expressions for all of the important process performance indicators that change with the cycle number. The equilibrium theory analysis revealed a unique interaction between two simple waves and a shock wave during the approach to periodicity. This interaction was not seen when the periodic state was analyzed directly because it was shown here that the feed and purge profiles go through four unique stages during the approach to periodicity. The first stage includes all cycles before breakthrough of the heavy component into the light product occurs. In this stage, there is only one simple wave during both the feed and purge steps. The second stage begins when breakthrough of the heavy product occurs. There are two simple waves present during the feed step in this stage, but one of the simple waves is completely pushed out of the bed during the feed step. The purge step contains one simple wave and one shock wave. The third stage begins when the second simple wave is not pushed out of the bed during the feed step. Therefore, the profile at the end of the feed step in this stage consists of two simple waves. During the purge step, the shock wave consumes all of one simple wave and part of the second simple wave, leaving just one simple wave and a shock wave at the end of the purge step. The fourth stage begins once a shock of the fullest strength has developed during the purge step. In this stage, there is again just one simple wave in the end-of-step feed profile. During the purge step in this stage the shock wave completely consumes the simple wave, leaving a shock of the fullest strength at the end of the purge step. The bed remains in this stage for all remaining cycles. Eventually, the breakthrough concentration and the location of the shock wave at the end of the purge step no longer change appreciably, marking the periodic state for the process. Expressions for the purity of the light product and recovery of pure inert were developed as a function of the cycle number. It was shown that these performance indicators change significantly during stages one and two and the first part of stage three and then slowly approach the periodic state values. Interestingly, the heavy-product purity, enrichment, and recovery are not functions of the cycle number. In part 1 of this series, it was shown that the cycle time does not effect the heavy-product enrichment or the recovery or purity of the heavy and light products; however, this work found that increasing the cycle time causes the process to reach the periodic state faster. Overall, the expressions developed in parts 1 and 2 of this series should be very useful for carrying out preliminary designs and feasibility studies because the analysis represents the best possible separation that can be achieved with this kind of PSA process. They should also be very insightful for training and educational purposes because the expressions presented in this series provide a complete description of the PSA process not only at the periodic state but also during the approach to periodicity. In fact, a unique PSA process simulator, based on the very interactive LabVIEW platform from National Instruments, is being developed with the expressions developed in parts 1 and 2 of this series.

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Acknowledgment Funding provided by the MeadWestvaco Charleston Research Center, the Separations Research Program at the University of Texas at Austin, the NSF GK-12 Program through award DGE-008642, and an NSF Graduate Research Fellowship to K.D.D. is greatly appreciated. Nomenclature a ) characteristic invariant a* ) characteristic invariant that reaches the end of the bed during the feed step a/i ) characteristic invariant that reaches the end of the bed at the end of the feed step in cycle i aL ) characteristic invariant on the left side of the shock am ) characteristic invariant that represents the concentration where the simple and shock waves meet; changes from a/i to ap during the purge step ap,i ) characteristic invariant that is just consumed by the shock wave at the end of the purge step in cycle i aR ) characteristic invariant on the right side of the shock A ) cross-sectional area of bed (m2) b ) Langmuir isotherm parameter (atm-1) ca ) fluid-phase concentration of the adsorbate (mol/m3) c/a ) dimensionless fluid-phase concentration of the adsorbate c/a,R ) dimensionless fluid-phase concentration on the right side of the shock cref ) fluid-phase concentration of the adsorbate in the feed (mol/m3) i ) cycle number ibt,1 ) first cycle with breakthrough of the heavy component into the light product L ) stoichiometric depth of penetration for the first feed step (m) Lb ) bed length (m) PH ) feed pressure (atm) PL ) purge pressure (atm) qa ) adsorbed-phase concentration of the adsorbate (mol/ kg) q/a ) dimensionless adsorbed-phase concentration of the adsorbate qref ) adsorbed-phase concentration of the adsorbate in equilibrium with cref (mol/kg) qs ) Langmuir isotherm parameter (mol/kg) R ) isotherm separation factor RC ) universal gas constant (m3 atm/mol K) ti,b0 ) time at which breakthrough first occurs at the periodic state (s) t ) time (s) tf ) feed duration (s) T ) temperature (K) Vf ) volumetric feed flow rate (m3 STP/min) yf ) mole fraction of adsorbate in the feed yLP ) mole fraction of adsorbate in the light product z ) axial coordinate (m)

Greek Letters ) void fraction of the adsorbent γ ) volumetric purge-to-feed ratio γmax ) maximum allowed volumetric purge-to-feed ratio v ) superficial velocity (m/s) τ ) dimensionless time τap ) dimensionless time it takes the shock wave to reach the concentration ap,i-1 Fb ) bulk density of the adsorbent (kg/m3) σ ) dimensionless axial coordinate σb ) dimensionless bed length Subscripts a ) adsorbate f ) feed step i ) cycle number p ) purge step sh ) shock wave avg ) average bt ) breakthrough LP ) light product sh ) shock wave sfs ) shock of the fullest strength

Literature Cited (1) Daniel, K. D.; Ritter, J. A. Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle Utilizing and Unfavorable Langmuir Isotherm. 1. Periodic Behavior. Ind. Eng. Chem. Res. 2002, 41, 3676. (2) LeVan, M. D. Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655. (3) Subramanian, D.; Ritter, J. A. Equilibrium Theory for Solvent Vapor Recovery by Pressure Swing Adsorption: Analytical Solution for Process Performance. Chem. Eng. Sci. 1997, 52, 3147. (4) Subramanian, D.; Ritter, J. A. Equilibrium Theory for Binary Solvent Vapor Recovery by Pressure Swing Adsorption: Conceptual Process Design for Separation of the Lighter Component. Chem. Eng. Sci. 1998, 53, 1295. (5) Liu, Y.; Subramanian, D.; Ritter, J. A. Theory and Application of Pressure Swing Adsorption for the Environment. Stud. Surf. Sci. Catal. 1998, 120, 213. (6) Ritter, J. A.; Liu, Y.; Subramanian, D. New Vacuum Swing Adsorption Cycles for Air Purification with the Feasibility of Complete Clean-Up. Ind. Eng. Chem. Res. 1998, 37, 1970. (7) Subramanian, D.; Ritter, J. A.; Liu, Y. Equilibrium Theory for Solvent Vapor Recovery by Pressure Swing Adsorption: Analytic Solution with Velocity Variation and Gas-Phase Capacity. Chem. Eng. Sci. 1999, 54, 481. (8) Ruthven, D. M.; Farooq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH Publishers: New York, 1994. (9) 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. (10) Mongole, M. N.; Balasubramaniam, R. Effect of Hydrogen Cycling on the Hydrogen Storage Properties of MmNi4.2Al0.8. Int. J. Hydrogen Energy 2000, 25, 55.

Received for review December 4, 2002 Revised manuscript received April 29, 2003 Accepted April 29, 2003 IE020958Y

Equilibrium Theory Analysis of a Pressure-Swing Adsorption Cycle

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