Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle

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Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle Utilizing a Favorable Langmuir Isotherm: Approach to Periodic Behavior D. Jason Owens, Armin D. Ebner, and James A. Ritter* Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, South Carolina 29208, United States ABSTRACT: An isothermal equilibrium theory analysis of a simple two-step pressure-swing adsorption (PSA) process utilizing an adsorbate−adsorbent system that exhibits a favorable Langmuir isotherm was carried out. Analytic expressions, either simple or recursive, were obtained that describe process operation and process performance during the approach to periodicity. These expressions are a function of cycle number and various process parameters. A recursive relationship for the dimensionless penetration depth for each cycle was determined and, although no closed form is readily available (and likely does not exist), the recursive relationship is easily applicable in any spreadsheet program. All other expressions were derived as functions of the penetration depth, thus lending a full analysis to the capabilities of a spreadsheet program. The analysis is primarily focused on the case of no breakthrough, because of the fact that breakthrough forces the system prematurely to periodicity and is therefore trivial for the approach analysis. The resulting expressions were used to examine process performance upon the approach to periodicity for several hypothetical systems, and the effects of various parameters on the number of cycles required to reach a periodic, or virtually periodic, state were examined. From an understanding or educational point of view, this analysis clearly shows how the so-called “heel in the bed”, i.e., the adsorbate loading remaining in the bed after the end-of-purge step, forms on the very first cycle and continues to increase cycle after cycle until periodicity is attained. This buildup of the heel in the bed is characteristic of all PSA processes, with a slower buildup resulting from a more nonlinear isotherm or a smaller purge-to-feed ratio.

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INTRODUCTION In part I of this work,1 Subramanian and Ritter carried out an isothermal equilibrium theory analysis of a simple two-step pressure swing adsorption (PSA) system exhibiting a fully convex, or favorable Type I (Langmuir) isotherm. Based on the work of LeVan,2 they developed analytic expressions for the direct prediction of the performance of a PSA process at the periodic state with and without breakthrough of the heavy component into the light product. The expressions derived by Subramanian and Ritter1 predicted not only the dimensionless penetration depth at the end of the feed step and the bed profile at the end of the purge step, as in the analysis of LeVan,2 but they also predicted the periodic process performance in terms of the heavy component enrichment and recovery, and light product impurity. An analysis similar to that of Subramanian and Ritter1 was carried out by Daniel and Ritter,3 but for a fully concave, or unfavorable Type III isotherm. They derived analytic expressions for the direct prediction of the performance of a PSA process at the periodic state, but only for the case when breakthrough of the heavy component into the light product occurs. When breakthrough of the heavy component into the light product does not occur, the periodic state corresponds to the first cycle, as mentioned by LeVan.2 In subsequent work, Daniel and Ritter4 derived analytic expressions that directly describe the approach to periodicity as a function of cycle number for this unfavorable case, of course, with breakthrough of the heavy component into the light product. The resulting analytic expressions described all the important process © 2012 American Chemical Society

performance indicators for any cycle number during the approach to periodicity. LeVan2 also described how a PSA process approaches periodicity for the favorable isotherm case using a simple recursive analysis based on physical arguments when starting from a clean bed. He showed how the penetration depth changes on the approach to periodicity for different purge-tofeed ratios and favorable isotherm concavities. However, he did not develop any analytic expressions that describe how the process performance changes during the approach to periodicity. Therefore, the objective of this work is to extend the analyses of LeVan2 and Subramanian and Ritter1 by deriving analytic expressions, simple and recursive, that describe the wave dynamics and the process performance as a function of the cycle number during the approach to periodicity, with and without breakthrough.

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THEORY This analysis considers the purification of a nonadsorbing carrier gas containing a small concentration of an absorbing impurity by a PSA process. The usual equilibrium theory assumptions apply, including isothermal operation, negligible axial dispersion and pressure gradients, and instantaneous local equilibrium. Velocity changes in the column are also neglected Received: Revised: Accepted: Published: 13454

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The results of this analysis are all in terms of the characteristic invariant, a, which is defined in terms of the dimensionless fluid-phase concentration of the adsorbate, ca*, as shown in Table 1. The values of ca* are bounded between 0 and 1. At c*a = 0, a = −1/R and at c*a = 1, a = −1. Thus, the values of the characteristic invariant are bounded between −1/R and −1, respectively. While the characteristic invariant and its bounds are useful during the development of the analysis, their actual interpretation becomes somewhat abstract. Thus, to more easily interpret the results visually (i.e., graphically), it is often useful to rewrite the equations in terms of the dimensionless adsorbed-phase concentration of the adsorbate, q*a . Solving the Langmuir isotherm for c*a and substituting the result into the definition of the characteristic invariant yields, after simplification:

due to the low concentration of the adsorbing impurity. Because of high partition ratios between the gas and adsorbed phases, the rate of accumulation of the adsorbate in the gas phase is neglected. Also, in this simple analysis, the pressure changing steps are not considered, implying that the concentration profiles of the gas and adsorbed phases remain unchanged or frozen during these steps. This is known as the frozen solid-phase approximation.5 These assumptions reduce the PSA cycle to a simple two-step process consisting only of the feed and purge steps, as analyzed by LeVan,2 Subramanian and Ritter,1 and Daniel and Ritter.3,4 The frozen solid-phase approximation also restricts the feed mole fraction to a value less than the inverse of the pressure ratio.1 Finally, this analysis is valid only for systems with constant separation factors with values less than one, i.e., for a fully convex, favorable Langmuir isotherm. Taking the aforementioned assumptions into account, eqs 1−12 and eqs 15−21 of Subramanian and Ritter1 apply. Equations 20 and 21 of Subramanian and Ritter1 are two firstorder ordinary differential equations (ODEs) that describe the inverse concentration velocity as a function of the fluid-phase concentration for simple and shock waves. They are given, respectively, as dτ = ±Ra 2 dζ

(1)

dτ = RaLaR dζ

(2)

a=

Table 1. Definitions of Variables Used in the Analysis

τ=

t tf

dimensionless axial coordinate

ζ=

z L νt f c ref

L=

qref ca c ref

ca* =

qa* =

∫ζ

ζ(a)

−Ra 2 dζ

a,i

(4)

Substituting eq 3 into eq 5 and solving for qa* yields 1−

ca* R + (1 − R )ca*

qa*(ζ ) =

1 R= 1 + bcref a=

dτ =

ρb qref

where

characteristic invariant

γ

Solving this integral for ζ(a) results in an equation for the penetration depth in the bed, as a function of the characteristic invariant, a: γ ζ(a) = ζa , i − (5) Ra 2

qa

qa* =

stoichiometric depth of penetration for the first feed step (m) Langmuir isotherm

∫0

equation

dimensionless adsorbed-phase concentration of the adsorbate dimensionless fluid-phase concentration of the adsorbate dimensionless time

(3)

R

Approach to Periodicity−Purge Steps. The feed step generally consists of a wave interaction between a simple expansive wave and a shock wave. However, the shock wave overtakes the simple wave in a determinable amount of time, resulting in a fully developed shock profile by the end of the feed step. The purge step consists only of a simple expansive wave and is a function of the penetration depth reached by the feed step, ζa. During the approach to the periodic state, the value of ζa increases with each iteration (as shown in the next section). The penetration depth reached by the shock wave at the end of the ith feed step is denoted by ζa,i. The profile at the end of the ith purge step is determined from eq 1, by taking the negative sign for the purge step and integrating across the purge step:

where R is the Langmuir isotherm constant separation factor, a measure of isotherm nonlinearity through the Langmuir affinity parameter, as defined elsewhere1,2 and shown in Table 1. The

description

(1 − R )qa* − 1

Rγ ζa , i − ζ

1−R

(6)

The two end-points of the simple wave remaining in the bed at the end of the purge step, which are useful for interpretation and further developments, can be easily calculated from eqs 5 and 6. First, inspection of eq 6 shows that the function becomes asymptotic as ζ approaches ζa,i and is, in fact, negative. Thus, the physical interpretation of eq 6 is only valid between ζ = 0 and the root of the equation (i.e., the value of ζ at which q*0,i = 0). While this root can be calculated from eq 6 fairly easily, it can also be calculated by evaluating eq 5 at a = −1/R (which corresponds to qa* = 0). This value, denoted as ζ0,i, is given by

1 ca*R − R − ca*

parameters in these equations and other pertinent variable definitions are also provided in Table 1. In eq 1, the positive and negative signs represent the feed and purge step expansive waves, respectively, and the implicit positive sign in eq 2 implies the possibility of a shock wave in only the feed step. This behavior is a result of the fully favorable isotherm (R < 1) considered in this analysis. Since eqs 1 and 2 are the governing equations, they can be used to directly describe periodicity, as done by Subramanian and Ritter,1 or the approach to periodicity, as shown below. Similarities in the ensuing analysis with those of Subramanian and Ritter1 are noted throughout.

ζ0, i = ζa , i − Rγ

(7)

This result is important for several reasons. First, this result stipulates that all adsorbate is removed from the portion of the bed described by ζ0,i ≤ ζ ≤ ζa,i, or that the bed is fully cleansed for these values of ζ. Furthermore, this result stipulates that, 13455

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necessarily, ζ0,i < ζa,i. This suggests that the end-of-purge profile is independent of breakthrough. The other useful quantity obtainable from eqs 5 and 6 is the concentration of the adsorbate present at the front of the bed at the end of the purge step (i.e., the amount of adsorbate at ζ = 0). These values are obtained by substituting ζ = 0 into eq 5 and solving for a (resulting in ai*) and into eq 6 (resulting in q0,i * ): ai* = −

γ ζa , iR

1− q0,*i =

ζa = 1 + ζsfs − τsfs

Determining the position and time at which the shock of fullest strength first develops proves to be more difficult. For the first feed step, the feed flows into an entirely clean bed. Thus, instead of wave interaction, there exists only a shock wave of the fullest strength, resulting in initial conditions of τsfs = ζsfs = 0. These initial conditions stipulate that ζa,1 = 1, which is independent of the values of R and γ. This shock wave moves a distance z = L, as defined in Table 1. For each subsequent feed step, the initial condition of the feed is the heel remaining at the end of the previous purge step, which is described in the previous section. Thus, the feed profile initially involves an interaction between a simple expansive wave and a shock wave. However, the shock wave proceeds down the bed at a faster rate than the simple wave and overtakes the simple wave in a determinable amount of time. During the part of the feed step with wave interaction, eqs 1 and 2 must be satisfied simultaneously. The solution to these equations is derived as follows. First, eq 1 is integrated with initial conditions τ = 0 and eq 5 (the heel remaining at the end of the purge step) and final conditions τ(a) and ζ(a), as follows:

(8)

Rγ ζa , i

1−R

(11)

(9)

Notice that eq 9 is identical (aside from the index indicator) to eq 20 in the work by LeVan,2 which he obtained by using physical arguments. Thus, the heel remaining in the bed at the end of the ith purge step is fully described by the parameters of the system and ζa,i, the penetration depth of the shock wave from the ith feed step. Furthermore, this heel serves as the basis for the subsequent, (i+1)th, feed step. Figure 1 shows representative end-of-feed and end-of-purge profiles for the ith feed and purge steps, respectively.

∫0

τ(a)

dτ =

∫ζ

ζ(a)

a , i − 1− γ / Ra

2

Ra 2 dζ

(12)

The solution to this integral is τ(a) = Ra 2ζ(a) − Ra 2ζa , i − 1 + γ

(13)

This equation is then differentiated with respect to the characteristic invariant, a, to obtain an expression for dτ/da as a function of dζ/da, as dτ dζ = 2Ra(ζ(a) − ζa , i − 1) + Ra 2 da da

(14)

Thus, eq 14 represents a solution for eq 1 that can be used as a constraint in the solution to eq 2. First, the chain rule is applied to eq 2, resulting in derivatives with respect to a. In addition, aR is replaced with the variable a since the heel from the previous purge step exists as the right-hand initial conditions for the shock wave. These modifications result in Figure 1. (a) End-of-feed profile for the ith feed step and (b) end-ofpurge profile for the ith purge step.

dτ dζ = RaLa da da

It should be noted that aL remains a constant. Equations 14 and 15 now both contain expressions for dτ/da as a function of dζ/ da. Substituting eq 14 into eq 15 results in a simple, separable differential equation that is given by

Approach to Periodicity−Feed Steps. With the ith purge step completely defined as a function of the penetration depth from the ith feed step, the difficulty in analyzing the approach lies in determining how far the fully developed shock wave penetrates into the bed during each feed step. With a fully developed shock wave, aL = −1, which corresponds to the gasphase concentration c*a = 1 on the left side of the shock wave. Similarly, aR = −1/R, which corresponds to the gas-phase concentration ca* = 0 on the right side of the shock wave. Substituting these values into eq 2 results in a linear relationship between τ and ζ, described by

dτ = dζ

(15)

2 1 da = dζ aL − a ζ − ζa , i − 1

(16)

Integrating eq 16 with the initial condition ζ(ai−1 * ) = 0 results in one-half of the parametric description of the wave down the bed. After rearrangement and simplification, this is given by ζi(a) =

(10)

The penetration depth is found by integrating eq 10 with initial conditions of τsfs and ζsfs, indicating the time and position where the shock of fullest strength first develops, and final conditions of τ = 1 and ζ = ζa, resulting in an equation for ζa, as

ζa , i − 1(a − ai*− 1)(a + ai*− 1 − 2aL) (a − aL)2

(17)

The other half of the parametric description of the wave is found by substituting eq 17 into eq 13. After substitution and simplification, this yields 13456

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Article

Rζa , i − 1aL(ai*− 1 − a)(aLa + aLai*− 1 − 2aai*− 1) (a − aL)2

ζa,periodic

(18)

Equations 17 and 18 are identical (aside from the index indicators) to eqs 25 and 26 from Subramanian and Ritter.1 However, instead of using these equations to determine the conditions for periodicity, these equations are used to describe the feed step in terms of parameters from the previous cycle. From eqs 17 and 18, ζsfs,i and τsfs,i, the respective time and position where the shock of fullest strength first forms, are determined by making the following substitutions: aL = −1, a = −1/R, and a*i−1 as defined in eq 8. Making these substitutions, substituting ζsfs,i and τsfs,i into eq 11, and making the appropriate simplifications results in a recursive expression for ζa,i that is given by ζa , i =

which is exactly eq 28 from Subramanian and Ritter,1 except with restrictions. Aside from the cases of Rγ ≥ 1, periodicity is approached slowly and is, in fact, never actually reached, as is expected from an equilibrium-type system, as noted by LeVan.2 Thus, to analyze and compare systems, some criterion must be adopted to quantify when a system has essentially reached the periodic state, even if the truly periodic conditions are not fully satisfied. Since the periodic state is defined based on the differences, or lack thereof, between two consecutive cycles, it makes sense to set a criterion based on the change from one cycle to another. For this study, the system is assumed to be virtually periodic when the percent difference between two consecutive penetration depths is equal to 0.001%, i.e., when

2 Rγ R(γ − 1) + 1 1 ζa , i − 1 − ζa , i − 1 + 1−R 1−R 1−R (19)

⎛ ζa , i − ζa , i − 1 ⎞ ⎜⎜ ⎟⎟ = 0.00001 ζa , i ⎝ ⎠

Ideally, a closed formula describing the penetration depth as a function of process parameters and cycle number is desirable. Unfortunately, it does not appear that a closed form for this recursion exists. However, a spreadsheet program, such as MS Excel, can be used to easily calculate the penetration depth for each cycle using eq 19 and the knowledge that ζa,1 = 1. The purge step following each feed step can then be modeled using eqs 5−9, since each of these is a function of ζa,i. Before proceeding with further analysis, it is useful to evaluate eq 19 in detail. First, it is necessary to ensure that ζa,i ≥ ζa,i−1, since the shock wave for the ith feed step should penetrate at least as far as the previous shock wave and, for the most part, farther. This is accomplished by applying the first derivative test to eq 19 to determine the intervals on which eq 19 is increasing, decreasing, or neither. Performing this calculation shows that eq 19 is increasing when Rγ < 1, which is usually the case. However, when this condition fails, further investigation is required. For these cases, it is sufficient to examine only the first feed and purge step, for which ζa,1 = 1. For the case of Rγ = 1, inspection of eqs 7 and 9 show that ζ0,1 = q0,1 * = 0, meaning that all the adsorbed material is purged out of the bed, i.e., the bed is restored to a clean (adsorbate-free) state. Similarly, for Rγ > 1, ζ0,1 < 0 and q*0,1 < 0, also representing an entirely clean bed. For these cases, the subsequent feed step is identical to the initial feed step, meaning that ζa,i = 1 for all values of i. Thus, eq 19 can be modified to account for all cases as follows: ⎧⎛ ⎛ 2 Rγ ⎞ ⎪⎜ 1 ⎞⎟ζa , i − 1 − ⎜ ⎟ ζa , i − 1 Rγ < 1 ⎝ ⎠ ⎪ ⎝1 − R ⎠ ⎪ 1−R ζa , i = ⎨ R(γ − 1) + 1 ⎪ + 1−R ⎪ ⎪1 Rγ ≥ 1 ⎩

⎧ (1 − R )(γ − 1) + γ − 2 (1 − R )(γ − 1)γ ⎪ Rγ < 1 ⎪ R =⎨ ⎪ ⎪1 Rγ ≥ 1 ⎩ (21)

(22)

Although this criterion is somewhat arbitrary, it allows for a quantitative comparison of various systems and is easy to evaluate when using a spreadsheet to calculate the penetration depths. In addition, it is possible to calculate a virtual periodic penetration depth by combining eqs 19 and 22 and solving for the penetration depth. However, the result is quite complicated and is more easily determined by inspection from a spreadsheet.

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PROCESS PERFORMANCE INDICATORS The common PSA process performance indicators include purity or enrichment, recovery, feed throughput or productivity, and power. In this analysis, as done elsewhere,1 only the heavy component enrichment and recovery were considered. For the analysis with no breakthrough, which is analyzed first below, only the enrichment of the heavy component was considered because the recovery is always 100%. For the analysis with breakthrough treated in the next section, both enrichment and recovery were considered. The following analysis parallels the derivations of Subramanian and Ritter,1 but accounts for the cycle number upon approach to periodicity. First, the differential equation governing the propagation of the expansive wave is obtained as a function of the dimensionless concentration parameters and the volumetric purge-to-feed ratio as (modified from eq 30 of Subramanian and Ritter1) γ dζ =− ⎛ dqa* ⎞ dt t f ⎜ dc * ⎟ ⎝ a⎠

(20)

(23)

Integrating this equation from the beginning of the purge step (t = 0) to a representative time t yields

In addition, eq 20 can be used to determine the penetration depth required for periodic restoration. Obviously, if Rγ ≥ 1, the periodic penetration depth is ζa,periodic = 1. However, for Rγ < 1, the periodic penetration depth can be found by setting ζa,i = ζa,i−1 = ζa,periodic in eq 20 and solving for ζa,periodic. This results in the following equation:

ζ = ζa , i −

13457

γt ⎛ dq * ⎞ t f ⎜ dca* ⎟ ⎝ a⎠

(24)

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penetrated depth for a given feed step is greater than the bed length. Intuitively, this suggests that the case of breakthrough has relatively little bearing on the approach to periodicity. A quick examination of the periodic dimensionless penetration depth (from eq 21) is enough to determine if breakthrough eventually occurs in the system. If ζa,periodic > ζb, where ζb is the dimensionless bed length, breakthrough eventually occurs. However, the approach to breakthrough is identical to the approach to periodicity, and all the theory and equations previously discussed apply. The cycle number when breakthrough first occurs, i.e., when ζa,i > ζb, is denoted ibt. Thus, for the case of breakthrough (Rγ > 1 is inherent), eq 20 can be redefined as

An equation relating the dimensionless bulk phase concentration that just exits the bed and time is found by setting ζ = 0 and substituting the derivative of the Langmuir isotherm: ca*(1 − R ) γt 1 − = ζa , it f R + (1 − R )ca* [R + (1 − R )ca*]2

(25)

The exiting concentration, as a function of time, is found by solving eq 25 for ca*, yielding Rγζa , it f t − Rγt

ca* =

γt(1 − R )

(26)

At the beginning of the purge step, the shock profile that exists at the end of the previous feed step is pushed out of the bed as a simple expansive wave. The maximum velocity for the simple wave occurs for the highest concentration existing as part of the shock wave, which is known to be ca* = 1. Thus, the purge effluent has a dimensionless bulk-phase concentration of c*a = 1 until the time at which that concentration crosses the penetration depth reached by the preceding feed step, ζa,i. This time is found from eq 24, as t0, i

ζa , it f ⎡ dqa* ⎤ ⎢ ⎥ = γ ⎢⎣ dca* ⎥⎦

= ca*= 1.0

⎧⎛ ⎛ 2 Rγ ⎞ ⎪ ⎜ 1 ⎟⎞ζa , i − 1 − ⎜ ⎟ ζa , i − 1 i < ibt ⎝ ⎠ ⎪ ⎝1 − R ⎠ ⎪ 1−R ζa , i = ⎨ R(γ − 1) + 1 ⎪ + 1−R ⎪ ⎪ζ i ≥ ibt ⎩ b

On the approach to breakthrough, the system can be fully modeled as prescribed above. Once breakthrough occurs, the system is forced prematurely into a periodic state, since breakthrough occurs on every subsequent step. Furthermore, the end-of-purge profile is identical for each step (since ζb provides the initial conditions for the purge step), meaning that the same amount of gas “breaks through” on each subsequent feed step. Thus, the system has reached periodicity and eqs 40− 59 from Subramanian and Ritter1 apply; no further analysis is required.

Rζa , it f γ

(27)

Thus, the dimensionless bulk phase concentration exiting the bed is a piecewise-defined function given by ⎧1.0 0 ≤ t ≤ t0, i ⎪ ⎪ ca*(t ) = ⎨ Rγζa , it f t − Rγt ⎪ t0, i < t ≤ t f ⎪ γt(1 − R ) ⎩

■

CASE STUDY With a recursive relationship for the penetration depth (eq 20) and equations for the performance indicators based on the penetration depth (eqs 29 and 30), a given system can be modeled quickly in a spreadsheet. Table 2 shows an example of the calculations, generated by Excel, for the first 20 cycles of a system with parameters R = 0.1, γ = 2.0, and α = 15.0. The parameters and the indices are input manually, in addition to the penetration depth for the first feed cycle (ζa,1 = 1.0 for all values of R and γ). The remaining values in column B are generated from eq 20. The values in column C represent the percent difference between the penetration depths for consecutive feed steps. This column is used to determine when virtual periodicity is reached based on eq 22. It should be noted that, while this example only presents the first 20 feed steps, virtual periodicity is not yet reached for this system. Columns D and E are generated using eqs 29 and 30, respectively. A similar spreadsheet can be used to investigate how different values of R and γ affect the approach to periodicity. Figures 2 and 3 show the aforementioned process performance indicators as a function of cycle number for representative parameters. Values of 0.1 and 0.01 are shown for the Langmuir isotherm constant separation factor, R, while purge-to-feed volumetric ratios (γ) of 1 and 2 are considered. A pressure ratio of α = 15 was chosen arbitrarily and is used only in calculation of the average enrichment. These representative values allow for a good understanding of how each affects the approach to periodicity, including how many cycles it takes to reach a virtual periodic state. The dotted line on each graph represents the true periodic value. The periodic value for the dimensionless

(28)

The average dimensionless bulk phase concentration exiting the bed during the ith purge step is then found by integration, as t

* i= cavg,

∫0 f ca*(t ) dt tf

∫0 dt

=

2 Rγζa , i − R(ζa , i + γ ) γ(1 − R )

(29)

The average enrichment during the ith purge step (Ei) is defined as the ratio of the mole fraction of the adsorbate in the heavy product averaged over the ith purge step to the mole fraction of the adsorbate in the feed, which is given by Ei =

* i PHcavg, PL

(31)

⎡ 2 Rγζ − R(ζ + γ ) ⎤ a,i a,i ⎥ * i = α⎢ = αcavg, ⎥⎦ ⎢⎣ γ(1 − R ) (30)

Unfortunately, this result does not simplify as nicely on the approach to periodicity as it does at periodicity (see eq 39 from Subramanian and Ritter1). However, with a spreadsheet set up to calculate the depth of penetration for each feed step, the average dimensionless bulk phase concentration exiting the bed and the average enrichment during the ith purge stage can be easily calculated. Analysis with Breakthrough. The extension of the above theory to allow for breakthrough of the adsorbate into the light product is restricted to the case of a pure carrier gas purge, as done elsewhere.1 Under the simplifications of the above theory, breakthrough occurs when the bed is not long enough to contain the shock wave that forms and propagates during the feed step. Mathematically, this occurs when the projected 13458

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Table 2. Example MS Excel Spreadsheet with Calculations of Process Performance Variables 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

A

B

R= γ= α=

0.1 2.0 15.0

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ζa,i 1.000 1.340 1.560 1.715 1.826 1.908 1.970 2.016 2.051 2.078 2.098 2.114 2.126 2.136 2.143 2.148 2.153 2.156 2.158 2.160

C

D

E

100 × (ζa,i − ζa,i−1)/ζa,i

c*avg,i 0.330 0.390 0.423 0.444 0.459 0.469 0.477 0.482 0.487 0.490 0.492 0.494 0.495 0.496 0.497 0.498 0.498 0.499 0.499 0.499

Ei 4.954 5.844 6.344 6.664 6.884 7.039 7.153 7.236 7.299 7.346 7.382 7.409 7.430 7.446 7.458 7.468 7.475 7.481 7.485 7.489

0.2535 0.1415 0.0899 0.0610 0.0431 0.0312 0.0230 0.0171 0.0129 0.0098 0.0074 0.0057 0.0044 0.0034 0.0026 0.0020 0.0015 0.0012 0.0009

penetration depth is calculated using eq 21, and the periodic values for the average bulk-phase concentration and the average enrichment are calculated using eqs 29 and 30, respectively. Now, with the values for the dimensionless depth of penetration calculated for each cycle number, the end-of-feed and end-of-purge curves can be generated for each cycle. Figure 4 shows the bed profiles at the end-of-feed and end-of-purge steps for the 1st, 2nd, 3rd, 20th, and 45th cycles of a system with a Langmuir isotherm constant separation factor of R = 0.1 and volumetric purge-to-feed ratio of γ = 2. The figure shows that the dimensionless penetration depth increases the most during the initial cycles. While the profiles for the 1st, 2nd, and 3rd cycles are easily distinguishable, the profiles for the 20th and 45th cycles virtually overlap, suggesting very little change over this vast number of cycles. This figure also illustrates the concept of virtual periodicity: based on eq 22, this system reaches its virtual periodic state at i = 45, although very little changes in the previous 20+ cycles. Even after a large number of virtually periodic cycles, the dimensionless penetration depth remains slightly less than the true periodic value (although the difference, for this system, is very small). Thus, Figure 6 (shown later in this work) not only provides a visual for the approach to periodicity, it also validates eq 22 as a sufficient measure of the cycle at which a system reaches its virtual periodic state. The results in Figure 4 also show the slow buildup of the socalled “heel in the bed” on the approach to periodicity. Figures 2 and 3 show the concomitant increases in the dimensionless penetration depth, average bulk-phase concentration, and average enrichment. This heel is the dimensionless loading remaining in the bed after the end-of-the purge step. For the case with no breakthrough into the light product, the physical

Figure 2. Process performance indicators as a function of cycle number for γ = 1.

meaning of periodicity is when the amount of adsorbate entering the bed during the feed step of each cycle equals the amount of adsorbate exiting the bed during the purge step of each cycle. This is when the heel stops growing and remains the same cycle after cycle. It is important to emphasize from an understanding or educational point of view that this slow buildup of the heel on the approach to periodicity is characteristic of all PSA processes. Upon further inspection of Figures 2 and 3, it is easy to see that the approach to periodicity is much slower for the smaller purge-to-feed ratio (i.e., the number of cycles to periodicity for γ = 1 is an order of magnitude greater than for γ = 2). It is also 13459

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Figure 4. End-of-feed (shock wave) and end-of-purge (simple wave) profiles for R = 0.1, γ = 2, and cycles 1, 2, 3, 20, and 45.

R and γ, the periodic performance observed in actuality would be quite different from that modeled by the equations presented here. Because of the great effect that the purge-to-feed ratio has on the approach to periodicity, it is useful to further investigate the effects. Using the same representative Langmuir isotherm constant separation factors, R = 0.1 and R = 0.01, the volumetric purge-to-feed ratio was varied from the critical value, γ = 1.0, to a value of γ = 2.5 in increments of 0.05. The number of cycles to periodicity was then recorded by inspection from the spreadsheet using eq 22 as a guide; the results are shown in Figure 5. For both cases, there is a significant drop in the number of cycles to periodicity at values of γ greater than the critical value (i.e., γ > 1.0). This phenomenon is a result of the definition of the critical volumetric purge-to-feed ratio, which is the value needed to result in q0* = 1.0 at the end of the purge step at the periodic state. As the system approaches periodicity with these parameters, the bed approaches full saturation just at the entrance of the bed (exit of the purge step), which results in a decreased equilibrium driving force. Thus, although the system remains relatively far from its true periodic operation, the change between each cycle becomes minimal, reaching the virtual periodic state previously mentioned. However, even the slightest increase from the critical value results in significantly less cycles to periodicity, as shown in Figure 5. The function relating the volumetric purge-to-feed ratio to the cycles to periodicity is a monotonically decreasing function, suggesting that the higher the ratio, the quicker periodicity is reached. However, after the initial dropoff from the critical value, the rate of change is relatively small, suggesting that increasing the value of γ to high values does not offer a significant advantage in terms of the approach to periodicity (note that it may offer an advantage in terms of other quantities that are functions of γ). Figure 6 shows a three-dimensional representation of the effects of R and γ on the number of cycles to virtual periodicity. This figure clearly shows the trends of decreased values of R and γ, leading to more cycles to virtual periodicity with concomitant slower buildup of the heel. It is worth pointing out that other PSA process parameters and other cycle steps not considered in this work might also affect the buildup of the heel. In some cases, a large heel might be present in the bed at the end of the last regeneration step and, in other cases, only a small heel might remain in the bed. A very rare case would be one with no heel remaining in the bed at the end of regeneration. In this rare case, the bed would be

Figure 3. Process performance indicators as a function of cycle number for γ = 2.

evident that the approach to periodicity is faster for greater values of R, i.e., more linear isotherms, although the difference is not as extreme as for the purge-to-feed ratio. These same trends are also true for the growth or buildup of the heel in the bed. Another feature of Figure 2 is that for the critical purge-tofeed ratio of γ = 1 and the constant R = 0.01, the system does not reach the periodic state even after 10 000 cycles. However, the percent difference between the 9999th cycle and the 10000th cycle is only 0.00034%, meaning that, based on eq 22, the system is well into virtually periodic operation (in fact, the percent difference reaches 0.001% after the 4484th cycle). Thus, it is reasonable to assume that, for relatively low values of 13460

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Figure 5. Effect of volumetric purge-to-feed ratio (γ) on number of cycles to periodicity.

Based on these observations, expressions were derived for all relevant quantities on a given cycle as a function of the dimensionless penetration depth achieved during the feed step of that cycle. The feed step of the PSA system is characterized by an interaction between a simple expansive wave and a shock wave. For each feed step, the shock wave overtakes the simple wave in a determinable amount of time, and determination of that time becomes the challenge in modeling the approach to periodicity. For the very first feed step, it was determined that the dimensionless penetration depth always has a value of one, regardless of the parameters. This results because, with an entirely clean bed, only the shock wave enters and propagates down the column quite predictably. It was shown that, for certain systems (i.e., for systems where the product between the Langmuir isotherm constant separation factor and the volumetric purge-to-feed ratio is less than or equal to one), the purge step fully cleans the bed, resulting in a trivial periodic restoration system (i.e., just one cycle). However, for most relevant systems, the wave interaction is a function of the characteristic heel, i.e., the adsorbate loading, remaining in the bed at the end of the previous purge step. The propagation of the wave interaction down the column was modeled parametrically by simultaneously solving the governing differential equations for both the simple expansive wave and the shock wave. Using these parametric equations, expressions for the dimensionless time and position at which the shock wave overtakes the simple wave (i.e., when the shock of fullest strength is formed) were obtained. An expression for the dimensionless penetration depth achieved by the end of the feed step was then determined by integrating the differential equation governing the shock wave over the remainder of the feed step. Ideally, the expression for the penetration depth for a given feed step would be obtained as a function of cycle number. However, the expression derived is a recursive relationship in quadratic form; the penetration depth for a given cycle is a somewhat complicated function of the penetration depth achieved during the previous cycle. While a concerted effort was made to derive a simple closed formula for the penetration depth as a function of cycle number, no formula was found. It is hypothesized that no simple closed formula exists, although one

Figure 6. Effects of R and γ on the number of cycles to virtual periodicity.

returned to its initial state, containing only the light component after the completion of just one cycle and thus achieving the periodic state after just one cycle. This complete cleanup case has been treated by Ritter et al.,6 using a similar analysis.

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CONCLUSIONS In previous work, the periodic behavior of a simple two-step PSA process utilizing a favorable Langmuir isotherm using isothermal equilibrium theory was analyzed. In that study, a complete analytical description of the process was obtained, including expressions for all of the important process performance indicators. That work was extended here by analyzing the system on the approach to periodicity. Based on the behavior of the system, it is a given that the end of each feed step contains a shock of the fullest strength propagating down the column. The extent to which the shock proceeds down the column, known as the penetration depth, is a function of the end-of-purge profile from the previous purge step. However, the end-of-purge profile for a given purge step is determined by the penetration depth of the previous feed step. 13461

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L = depth of penetration of the square profile in the first feed step (m) Lb = length of the adsorbent bed (m) PH = high pressure of the feed step (kPa) PL = low pressure of the purge step (kPa) qa = adsorbed-phase concentration of the species a (mol/kg) qa* = dimensionless adsorbed-phase concentration q0* = dimensionless adsorbed-phase concentration at the head (at ζ = 0) of the heel qref = adsorbate concentration in equilibrium with the feed concentration cref (mol/kg) R = Langmuir isotherm constant separation factor t = time (s) tf = feed duration (s) z = axial coordinate (m) za = depth of penetration of the shock at the end of the feed step (m)

may very well be derived at a later time. If a closed formula exists, it is likely extremely complicated. As such, it was determined that the best method for modeling the approach to periodicity is using a spreadsheet to calculate the penetration depth for each cycle, and an example of such a spreadsheet is given. Because expressions for all the other modeling parameters and process performance indicators were derived as functions of the penetration depth, the spreadsheet can be easily used to model the entire approach to periodicity. Furthermore, the case of breakthrough was investigated, although this case is somewhat trivial with regard to the approach to periodicity: before breakthrough, the approach is identical to any other system, and once breakthrough occurs, the system is forced automatically into a periodic state. Overall, the expressions derived in this analysis should be useful for carrying out preliminary design and feasibility studies as it represents the best possible separation that can be achieved with this type of simplified PSA process. They should also be useful for training and educational purposes as the expressions provide a complete description of the PSA process not only at the periodic state but also upon the approach to periodicity. For example, from an understanding or educational point of view, this analysis clearly shows how the so-called “heel in the bed” forms on the very first cycle and continues to increase cycle after cycle until periodicity is attained. This buildup of the heel in the bed is characteristic of all PSA processes, with a slower buildup resulting from a more nonlinear isotherm or a smaller purge-to-feed ratio.

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Greek Letters

AUTHOR INFORMATION

Corresponding Author

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*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

α = pressure ratio γ = volumetric purge-to-feed ratio ζ = dimensionless axial coordinate ζ0 = dimensionless axial coordinate reached by ca* = 0 at the end of the purge step ζa = dimensionless depth of penetration of the shock at the end of the feed step ζsfs = dimensionless coordinate for formation of the shock of fullest strength ν = superficial velocity (m/s) τ = dimensionless time τsfs = dimensionless time for formation of the shock of fullest strength ρb = bulk density of the absorbent (kg/m3)

REFERENCES

(1) 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. (2) LeVan, M. D. Pressure swing adsorption: equilibrium theory for purification and enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655. (3) Daniel, K. D.; Ritter, J. A. Equilibrium Theory Analysis of a Pressure-Swing Adsorption Cycle Utilizing an Unfavorable Langmuir Isotherm. 1. Periodic Behavior. Ind. Eng. Chem. Res. 2002, 41, 3676. (4) Daniel, K. D.; Ritter, J. A. Equilibrium Theory Analysis of a Pressure-Swing Adsorption Cycle Utilizing an Unfavorable Langmuir Isotherm. 2. Approach to Periodic Behavior. Ind. Eng. Chem. Res. 2003, 42, 3381. (5) 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. (6) Ritter, J. A.; Liu, Y.; Subramanian, D. New Vacuum Swing Adsorption Cycles for Air Purification with the Feasibility of Complete Cleanup. Ind. Eng. Chem. Res. 1998, 37, 1970.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support provided by ExxonMobil through the Process Science and Technology Center, an industrial sponsored center with participants from the University of Texas at Austin, the University of South Carolina, and the Texas A&M University.

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NOMENCLATURE a = characteristic invariant aL = characteristic invariant corresponding to the left-hand state of the shock aR = characteristic invariant corresponding to the right-hand state of the shock a* = characteristic invariant at the head (at ζ = 0) of the heel A = cross-sectional area of the bed (m2) b = Langmuir isotherm affinity parameter (kPa−1) ca = fluid-phase concentration of species a (mol/m3) cavg * = time-averaged dimensionless concentration of the purge effluent c*a = dimensionless fluid-phase concentration cref = fluid-phase feed concentration (mol/m3) E = time-average enrichment of the solvent vapor in the purge effluent i = cycle number ibt = first cycle with breakthrough of the heavy component into the light product 13462

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