Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle

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Ind. Eng. Chem. Res. 2002, 41, 3676-3687

Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle Utilizing an Unfavorable Langmuir Isotherm. 1. 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 PSA process utilizing an adsorbate-adsorbent system that exhibits an unfavorable Langmuir isotherm has been carried out. Analytic expressions that directly describe the periodic state have been obtained and used to derive expressions for all of the important process performance indicators, with and without breakthrough of the heavy component into the light product. A design study with a H2-metal hydride system revealed that the enrichment of this kind of PSA system ideally is always equal to the pressure ratio, regardless of whether breakthrough occurs. The breakthrough case compared to the no-breakthrough case caused the recovery of the light product to increase, but at the expense of the heavy-product recovery and light-product purity both decreasing below 100%. A parametric study revealed the effects of the most important process parameters on the process performance indicators and some subtle features that appear to be unique to this PSA process-isotherm combination. The design expressions developed here should be very useful for rapid feasibility studies, as the results represent the best possible separation that can be achieved with this kind of PSA process; they should also be insightful for training and educational purposes. Introduction Conventional PSA cycles,1 which are designed for the most part to produce a very pure light product via stripping reflux,2-4 are limited by the high-to-low pressure ratio when it comes to enriching the heavy component.5 In reality, the enrichment of the heavy component is far below the pressure ratio as a result of two factors: the dilution effect of the purge gas that is used to regenerate the column for the ensuing cycle and the fact that the adsorption isotherm of the heavy component is typically a favorable Langmuir isotherm, which gives rise to a simple spreading wave during desorption. Clearly, it would be desirable to produce not only a very high-purity light product, but also a heavy product enriched at least to the pressure ratio. A novel PSA cycle that combines a conventional PSA cycle with an adsorbate-adsorbent system that exhibits an unfavorable Langmuir isotherm is introduced here for that purpose. The isothermal equilibrium theory first developed by LeVan6 and later extended by Subramanian and Ritter7 is used to analyze this PSA cycle. This unique analysis is based on a simple two-step feed and purge process because the frozen solid-phase assumption is invoked during the pressurization and blowdown steps.8 This analysis also gives rise to analytic relationships that directly describe the periodic-state wave dynamics for both the favorable and unfavorable forms of the Langmuir isotherm.6 However, only the favorable form of the Langmuir isotherm has been analyzed in terms of establishing expressions for all of the PSA process performance indicators with and without breakthrough of the heavy component into the light product.5,7 * Corresponding author. Phone: (803) 777-3590. Fax: (803) 777-8265. E-mail: [email protected].

Therefore, the objective of this work is to carry out an analysis similar to that done by LeVan6 and Subramanian and Ritter,7 to establish expressions for the periodic-state wave dynamics and all of the PSA process performance indicators for a single adsorbable component that exhibits an unfavorable Langmuir isotherm in an inert carrier gas. Two cases are considered: the simpler case is when there is no breakthrough of the heavy component into the light product, and the complicated case is when there is breakthrough. A brief design study is carried out with a H2-metal hydride system that exhibits an unfavorable Langmuir isotherm. This system is then used to perform a parametric study. These studies reveal some unique features of this kind of PSA process-isotherm combination and the feasibility of developing an actual PSA process that can concentrate a dilute feed stream, with very high purities and recoveries of both the light and heavy components. Theory Equilibrium theory accounts only for mass conservation and ignores transport phenomena such as masstransfer resistance; consequently, in some cases, it results in analytical solutions to the governing material balance equations.5-7 The assumptions associated with equilibrium theory 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. Fluid-phase accumulation of the adsorbate is also negligible, because of the large partition ratio between the solid phase and the bulk phase. Another important assumption concerns freezing of the adsorbed

10.1021/ie0107035 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/28/2002

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3677 Table 1. Definition of Variables Used in the Analysis7 description

only eq 1 needs to be solved to describe the concentration profiles within the column. For the feed step, eq 1 can be integrated to obtain a characteristic trajectory for each concentration. Performing this integration, subject to the initial conditions τ(a) ) 0 and σ(a) ) σp, where σp is the location of each concentration at the end of the previous purge step, yields

equation

dimensionless adsorbed-phase concentration of the adsorbate

q/i )

qi qref

dimensionless fluid-phase concentration of the adsorbate

c/i )

ci cref

dimensionless time

τ)

t tf

dimensionless axial coordinate

σ)

z L

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

L)

vtf cref Fbqref

Langmuir (constant separation factor, CSF) isotherm

q/i )

characteristic invariant

a)

σf(a) )

c/i R + (1 - R)c/i

1 (c/i R - R - c/i )

and fluid phases during pressurization and blowdown. This assumption simplifies the PSA cycle into a twostep process consisting of only feed and purge steps.5,6,8 The frozen solid-phase approximation also requires the feed mole fraction to be less than the inverse of the pressure ratio of the cycle.7 Also, this analysis is limited to purging with only pure carrier gas, as done elsewhere.7 Taking these assumptions into account, eqs 1-12 and 15-21 of Subramanian and Ritter7 apply. Equations 20 and 217 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 ) (Ra2 dσ

(1)

dτ ) -RaLaR dσ

(2)

The parameters in these equations and other pertinent variables are defined in Table 1. The major difference in this paper is that an adsorbate-adsorbent system with an unfavorable Langmuir isotherm (R > 1) is considered. Accordingly, eqs 1 and 2 show that, as the absolute value of the characteristic invariant a decreases (i.e., as the fluid-phase concentration decreases), the concentration velocity increases. This means that the highest fluid-phase concentration will travel at the slowest speed. Therefore, for an unfavorable isotherm, the feed step will always consist of a simple wave, but the purge step will consist of a simple wave only if there is no breakthrough of the adsorbate during the highpressure feed step. If breakthrough of the adsorbate occurs, a discontinuity (shock) will develop immediately at the beginning of the purge step. Mathematical Model without Breakthrough When there is no breakthrough of the adsorbate during the high-pressure feed step, both the feed and the purge steps consist of only simple waves. Therefore,

τ + σp Ra2

(3)

During the feed step, τ will vary from 0 to 1, and during the purge step, it will vary from 0 to γ (the volumetric purge-to-feed ratio).6 Equation 1 can also be integrated with the initial conditions τ(a) ) 0 and σ(a) ) σf, where σf is the location of each concentration at the end of the previous feed step. This integration, using the negative sign to signify that the resulting characteristic trajectories are for the purge step, yields

σp(a) ) σf -

τ Ra2

(4)

With unfavorable isotherms, lower concentrations travel faster and will therefore penetrate farther down the column during the feed step. If the column is clean at the beginning of the feed step, the subsequent purge step will consist of a self-sharpening simple wave, meaning that the slope of the concentration profile will gradually become steeper as it travels back toward the feed end of the column. At the end of the purge step (τ ) γ), a shock wave of the fullest strength (vertical concentration profile) develops exactly at the bed inlet when γ ) 1. To determine the conditions necessary for periodic restoration, γ and σp must be specified. A qualitative analysis of γ leads to the conclusion that, if γ < 1, each successive feed step will result in the concentration profile shifting further down the column, and eventually breakthrough will occur. If γ ) 1, the purge step will stop just as a shock wave of the fullest strength is developed. If γ > 1, the purge step will continue past the point where the shock wave of the fullest strength develops, and the heavy product will be contaminated with pure carrier gas. These conclusions can be shown mathematically by considering the very first feed step. Substituting τ ) 1 and σp ) 0 into eq 3 gives

σf(a) )

1 Ra2

(5)

which is the dimensionless position of each concentration at the end of the first feed step. For the subsequent purge step, eq 5 is used in conjunction with eq 4, with τ ) γ, to give

σp(a) )

1 γ 2 Ra Ra2

(6)

Equation 6 clearly shows that, if γ < 1, then σp(a) > 0, and therefore, in subsequent feed steps, each concentration will penetrate farther down the column, until breakthrough occurs. For the case when γ > 1, eq 6 shows that σp(a) < 0, meaning that each concentration is pushed out of the column, resulting in contamination of the heavy product. However, when γ ) 1, then σp(a) ) 0, and each purge step will result in a completely clean bed, without any unnecessary contamination of

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Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002

the heavy product. Requiring γ ) 1 allows eq 3 to be simplified to

σf(a) )

τ Ra2

(7)

The concentration that will travel farthest down the column is a ) -1/R, and to avoid breakthrough, the column must be long enough to contain this concentration. Setting τ ) 1, a ) -1/R, and σf(a) ) σb in eq 7 shows that, when γ ) 1, σb > R will avoid breakthrough. Process parameters such as recovery, purity, and enrichment can now be calculated for this process. The recovery of the heavy product is defined as the ratio of the change in the amount of adsorbate in the solid phase over the purge step to the amount of adsorbate in the solid phase after the first feed step.7 Obviously, because each purge step results in complete cleanup, the recovery of the heavy product is 100%. The recovery of the light product is defined as the ratio of the difference between the number of moles of carrier gas used during the feed and purge steps to the number of moles of carrier gas used in the feed step. Mathematically, this is represented by

RecLP )

nLP,f - nLP,p PH - PL 1 ) )1nLP,f PH R

(8)

where R is defined as the pressure ratio (PH/PL). The light-product impurity is defined as the ratio of the number of moles of adsorbate to the total number of moles in the light product. Obviously, when there is no breakthrough of adsorbate, the light product will be 100% pure. An analogous definition for the purity of the heavy product results in

yi,HP )

ni,HP yi,f PH ) ) yi,fR nHP PL

(9)

Enrichment of the heavy product is defined as the ratio of the mole fraction of the adsorbate in the heavy product during the purge step to the mole fraction of adsorbate in the feed.7 Because the dimensionless concentration of the adsorbate in the heavy product is always c/i ) 1, the heavy-product concentration is always equal to the feed concentration. Using this fact, the enrichment of the heavy product simplifies to

E)

PH )R PL

(10)

The final process performance parameter considered here is the bed capacity factor (BCF), which is defined as the fraction of the total bed capacity utilized at the periodic state.7 Without breakthrough, each purge step results in complete cleanup, so the periodic-state feed step is identical to the first feed step. Therefore, when there is no breakthrough, the BCF is calculated from

BCF )

L 1 ) Lb σb

(11)

Mathematical Model with Breakthrough When breakthrough of the adsorbate occurs in the light product, the purge step will involve both a selfsharpening simple wave and a shock wave. As discussed

above, breakthrough will always occur if γ < 1. If γ < 1 and σb < R, then breakthrough will occur on the very first feed step. In this case, it is convenient to define a* as the characteristic invariant (or concentration) that just reaches the end of the bed during the feed step at the periodic state. At the beginning of the purge step, a shock wave will form instantaneously. On the left side of the shock wave, the fluid-phase concentration will correspond to a*, and on the right side of the shock wave, the fluid-phase concentration will be 0 (aR ) -1/ R). Because a* and aR are now connected as part of a shock wave, they will travel with a common velocity. If the concentrations on the left and right sides of the shock are known, the shock-wave velocity can be evaluated using eq 2. The result of these two concentrations being connected by a shock wave is that aR will travel more slowly and aL will travel more quickly than if they were part of a self-sharpening simple wave.9-11 Because the shock wave travels faster than the simple wave on the left, it will gradually consume the simple wave while being decelerated. Once the shock wave has completely consumed the simple wave, a shock wave of the fullest strength will exist, connecting states aL ) -1 (c/i ) 1) and aR ) -1/R (c/i ) 0). From this point on, the shock wave will propagate toward the feed end of the column with a constant velocity. If the purge step ends before the shock wave reaches the end of the column, a square heel will be left in the column as an initial condition for the subsequent feed step. The dimensionless position of this heel is defined as σa. The characteristic trajectories during the feed step are found using eq 3, which becomes

σf(a) )

τ + σa Ra2

(12)

in the context of σa. Setting τ ) 1, a ) a*, and σf(a) ) σb in eq 8 and solving for a* yields

a* ) -

x

1 R(σb - σa)

(13)

During the purge step, where the shock and simple waves meet, eqs 1 and 2 must both be satisfied. An equation for the simple wave that exists at the beginning of the purge step is found by setting τ ) 1 in eq 12. This yields

σf(a) )

1 + σa Ra2

(14)

where a varies from a* to -1. Using τ ) 0 and eq 14 as initial conditions, eq 1 is integrated to obtain

σ(a) ) -

τ(a) 2

Ra

+

1 + σa Ra2

(15)

which is an equation that relates the dimensionless position and time for each concentration in the simple wave. Equation 15 is used to obtain an expression for the derivative of σ with respect to a, so that eq 2 can be rewritten as a function of only τ and a and then integrated to obtain τ(a) for the shock wave. First, eq 2 is rewritten as

dσ dτ ) -RaaR da da

(16)

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3679

where a varies from a* to -1 and aR ) -1/R. Taking the derivative of eq 15 with respect to a and substituting it into eq 16 gives

[

1 dτ dτ 2 (τ - 1) ) - RaaR da Ra3 Ra2 da

]

(17)

Solving for dτ/da yields

2aR(τ - 1) dτ )da a(a - aR)

Alternatively, eq 21 can be rearranged to give the required γ for periodic restoration as a function of σa, σb, and R

(18)

γ)

τ(a) )

a*2(a - aR)2

(19)

Equation 19 was derived in a manner similar to eq 26 of Subramanian and Ritter,7 but it yields a different form because of the type of isotherm used in this analysis. The parametric equation for the position of the shock wave is found by substituting eq 19 into eq 15 to obtain

σ(a) )

(aR - a*)2 + σaRa*2(a - aR)2 Ra*2(a - aR)2

(20)

Equations 19 and 20 are used to map the shock path through the column. To determine the value of σa required for periodic restoration, eq 2 is evaluated from the point where the shock wave of the fullest strength develops to the end of the purge step. Once the shock wave of the fullest strength develops, the right-hand side of eq 2 will be constant, allowing it to be integrated directly. The initial conditions for the integration are the time and position at which the shock of the fullest strength first develops (τsfs and σsfs). These values are found by setting a ) -1 and aR ) -1/R and substituting eq 13 into eqs 19 and 20. The final conditions for the integration are τ ) γ and σ ) σa. Performing the integration and solving for σa yields

γmax )

1 + σb - 2xRσb 1-R

(24)

If values greater than γmax are used, the shock wave is propagated out of the column, needlessly diluting the heavy product with pure carrier gas. The process performance parameters are derived in a similar manner as when there is no breakthrough. All of the same definitions apply, but in this case, the recovery of the heavy product will not be 100%, because some adsorbate is lost in the light product during the feed step. At the end of a feed step, the column will be saturated at q/i ) 1 up to a position σ1. After σ1, the amount of solid-phase adsorbate will decrease until it reaches the value corresponding to a* at the bed exit. An expression for q/i can be obtained as a function of position after the feed step by integrating eq 10 with the initial conditions τ ) 0 and σ ) σa and final condition τ ) 1. Integrating and solving for q/i yields

q/i,f )

σ - σa - xR(σ - σa) (σ - σa)(1 - R)

(25)

The amount of solid-phase adsorbate at the end of the feed step can then be found using

∫σσ q/i,f dσ

Qf ) qref AFbLσ1 + qref AFbL

b

1

(26)

where σ1 is found from eq 12 with τ ) 1 and a ) -1. Because the heel at the end of the purge step has a square profile, the amount of solid-phase adsorbate at the end of the purge step is

σa )

Qp ) qref AFbLσa

[γ(R - 1) + 1] + σb[γ(R - 1) + 1 - 2R] - 2σbxR(R - 1)(1 - γ)

(23)

Equation 23 further restricts the range of reasonable γ values for a given process. For fixed R and σb, the maximum value of γ allowed is calculated by setting σa equal to 0 in eq 23 to give

which can be integrated subject to the initial conditions τ ) 0 and a ) a* to obtain

a*2(a - aR)2 - a2(a* - aR)2

1 + σb - σa - 2xR(σb - σa) 1-R

(27)

2

γ(R - 1) + 1 - 2R - 2xR(R - 1)(1 - γ)

The amount of solid-phase adsorbate at the end of the very first feed step is given by1

(21) Ritter.7

which is similar to eq 28 of Subramanian and From eq 21, it is evident that, if γ > 1, the equation for σa will not be physically realistic. When γ ) 1, eq 21 simplifies to

σa ) σb - R

(22)

However, it was shown earlier that, when γ ) 1, breakthrough only occurs if R > σb, which would result in a negative value for σa according to eq 22. This means that, if γ ) 1, the shock wave is propagated out of the column, needlessly diluting the heavy product with pure carrier gas. Therefore, only γ values of less than 1 are reasonable when there is breakthrough.

Q1 ) qref AFbL

(28)

Using eqs 26-28, the recovery of the heavy product is given by

RecHP )

Qf - Qp 1 + σb - σa - 2xR(σb - σa) ) Q1 1-R

(29)

Comparing eq 29 with eq 23, it is clear that the recovery of the heavy product simplifies to

RecHP ) γ

(30)

Obviously, when breakthrough occurs, the maximum γ allowed for a given process also determines the maxi-

3680


mum heavy-product recovery for that process. The expression for recovery of the light product is derived in the same manner as the case without breakthrough. The final expression is

RecLP )

PH - γPL γ )1PH R

(31)

Comparing all of the recovery equations for the two cases shows that the recovery of the light product will increase when there is breakthrough, and this recovery increases with decreasing γ, whereas the heavy-product recovery decreases with decreasing γ. The light-product impurity is found by determining the average adsorbate concentration in the light product. During the feed step at the periodic state, an equation for the dimensionless concentration exiting the bed as a function of time is found using eq 17 of Subramanian and Ritter.7 Integrating this equation with the initial conditions σ ) σa and τ ) 0 and final conditions σ ) σb and τ ) τ(t) gives

τ(t) )

(c/ibR - R - c/ib)2

tf R(σb - σa) (c/ibR - R - c/ib)2

tR - xtRtf(σb - σa) t(R - 1)

tf(σb - σa) R

)

∫0t

0 dt +

b0

(34)

(35)

∫tt c/ib dt f

b0

∫0t 1 dt f

(36)

Performing the integration, simplifying, and converting to units of mole fraction gives

yi,LP ) yf

(R + σb - σa - 2xR(σb - σa)) R-1

(37)

Comparing eqs 23 and 37, it is clear that the lightproduct impurity can be simplified to

yi,LP ) yf(1 - γ)

b

1

σb

(39)

BCF )

σa + γ σb

(40)

The expressions for the purity and enrichment of the heavy product are the same as for the case with no breakthrough and are given by eqs 9 and 10.

The average adsorbate concentration in the light product (the light-product impurity) can then be found using / ci,LP,avg

∫σσ q/i,f dσ

where σ1 is the location of c/i ) 1 at the end of the feed step and σb is the dimensionless bed length. σ1 is found using eq 12, and q/i,f is given by eq 25. Performing the integration in eq 39 and simplifying yields

(33)

which is an equation describing the dimensionless concentration exiting the bed during the feed step as a function of time, once breakthrough occurs. The time (tb0) required for breakthrough to occur is found by substituting c/ib ) 0 into eq 33, which gives

tb0 )

σ1 +

(32)

Rearranging eq 33 to solve for c/ib(t) gives

c/ib(t) )

With breakthrough, the expression for the BCF becomes

BCF )

R(σb - σa)

where c/ib is the concentration exiting the bed at each time t. Equation 32 is converted into real time using the definitions of τ and L, which yields

t)

Figure 1. H2-MmNi4.2Al0.8 adsorption isotherm at 308 K. Points: experimental data. Solid line: Langmuir (CSF) correlation.

(38)

Design Study A brief design study is carried out here to show the feasibility of a stripping reflux PSA process for concentrating dilute streams for a system that exhibits an unfavorable Langmuir isotherm. The process performance is analyzed for both the no-breakthrough and breakthrough (BT) cases. The specific system being considered is the H2-MmNi4.2Al0.8 metal hydride system at 308 K;12 the typical s-shaped adsorption isotherm (symbols) for this system is depicted in Figure 1, along with the Langmuir (CSF) isotherm (solid line) fitted to only the unfavorable portion of the data. It must be emphasized that, with respect to a PSA process, an unfavorable Langmuir isotherm represents this sshaped isotherm system well, because significant changes in loading can be achieved simply by cycling pressure in the unfavorable region. Cycling over higher pressures past the inflection point and into the favorable region of the isotherm would not contribute much to the loading change and complicate the analysis considerably. This system is also unique because only H2 adsorbs on the metal hydride, which makes all other components such as CO2, CH4, etc., the light-inert components. In this study, a H2 mole fraction in the feed (yf) of 0.05 and a H2 maximum light-product impurity (yLP) of 0.01 are imposed. As mentioned above, yf ) 0.05 limits the pressure ratio to less than 20, and to stay on the concave portion of the adsorption isotherm, the high (feed) pressure is limited to 29.6 atm. The pressure ratio is thus set at 19, which results in a low (purge) pressure

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3681 Table 2. Operating Conditions and Parameters for the Design Study

Table 4. Parameter Ranges for the Parametric Case Study

parameter

units

value

parameter

range studied

range for γmax > 0.8

yf vf Vf Fb Lb db PH PL T tf yLP qs b

m/s m3 STP/min kg/m3 m m atm atm K s mol of H2/kg atm-1

0.05 0.10 44.5 2000 6.00 0.60 29.6 1.56 308 550 0.01 -0.372 -0.552

γ R yf PH (atm) Vf (m3 STP/min) Lb (m) tf (s)

0.18-1.0 9.18-19.0 0.013-0.050 0.730-29.6 20-150 1.8-10.5 300-2000

>0.00124 >0.730 25.6-142 1.88-10.31 316-1750

Table 3. Calculated Process Parameters and Performance Indicators

γ R σa L Lb/db E RecHP RecLP yH2,LP yH2,HP

units

w/o BT

w/ BT

m % % -

1.00 5.45 0.00 0.97 10.0 19.0 100 94.7 0.00 0.95

0.80 5.45 4.23 0.97 10.0 19.0 80.0 95.8 0.01 0.95

Parametric Study

of 1.56 atm. Other chosen process conditions are specified in Table 2. With these conditions set, the remaining process conditions and performance indicators can be calculated. The CSF isotherm parameter, R, is calculated from b using the one-to-one correlation between the Langmuir and the CSF isotherm, i.e.

R)

1 1 + byf PH

analysis, such as heat and mass transfer and blowdown effects. Nevertheless, these simple expressions can be used to evaluate the feasibility of a stripping reflux PSA process rapidly for a system that exhibits an unfavorable Langmuir isotherm, such as many H2-metal hydride systems. If this analysis shows that a proposed PSA process is economically challenged, there would be no need to pursue a more rigorous analysis, as the performance from such an analysis would necessarily be worse.

(41)

L is then calculated from the definitions of τ and L, with τ ) 1 and t ) tf. Once R and L are known, a bed length (Lb) is specified that results in σb > R, so that there is no breakthrough when γ ) 1. For the case with breakthrough, eq 24 is used to determine γmax ) 0.995. The only restriction on γ is that it is less than γmax, so γ ) 0.8 was chosen for the case with breakthrough. Now, σa can be calculated using eq 21, and the recovery, purity, and enrichment can be calculated using the appropriate equations given above. The results of these calculations are summarized in Table 3. The results of the no-breakthrough design study show that a 5 vol % H2 stream can be enriched 19 times with a recovery of 100%. This can be achieved while producing a 100% pure light component with a recovery of 94.7%. The columns are also reasonably sized. For the breakthrough case with 1 vol % H2 allowed in the light product, although the light-component recovery increases, the H2 recovery decreases significantly to 80%, which outweighs the beneficial effects associated with breakthrough. At this point, it is worth mentioning that this remarkably good separation and enrichment of the heavy component (in this case, H2) represents an upper thermodynamic limit on the performance that can be achieved with this system. In reality, the separation and enrichment would not be nearly as good, because of nonideal effects that are not taken into account in this

A parametric study is carried out to determine the effect of the main process parameters on the heavyproduct enrichment and the recovery and purity of the heavy and light products when there is breakthrough. The same H2-metal hydride system shown in Figure 1 is used here, also. From eqs 9, 10, 30, 31, and 38, it is easy to see that only γ, R, and yf will have a direct effect on the recovery, enrichment, and purity performance indicators. Equation 40 shows that these parameters will also change the BCF. When not being varied, the values of these parameters are set at γ ) 0.8, R ) 19.0, and yf ) 0.05. The ranges of these parameters are given in Table 4, and the values of the other process parameters are given in Table 2. Figure 2 displays the effect of γ on the PSA process performance indicators for the case with breakthrough. The purity of the light product (LP), recovery of the heavy and light products, and BCF all change with γ. The light-product purity and the heavy-product (HP) recovery decrease significantly as γ decreases, whereas the BCF increases and the light-product recovery increases only slightly as γ decreases. In general, when the bed size has been specified, a smaller BCF is preferred as long as the feed throughput remains the same, because this gives more room for deviations from the predicted periodic state. The feed throughput, θf, is defined as the volume of feed processed per unit mass of adsorbent per unit time during the adsorption step,5 i.e.

θf )

vf FbLb

(42)

Because the feed throughput will not change with γ, a smaller BCF (corresponding to a larger value of γ) is preferred. Therefore, smaller values of γ should only be chosen for a process in which the recovery of the light product is the most important performance indicator. Clearly, for the specific system considered here, with the goal being H2 separation and enrichment, a value of γ as close to unity as possible should be chosen. To change the pressure ratio (R), PL is held constant while PH is varied. To remain on the unfavorable portion of the isotherm, PH is also kept below 29.6 atm. Figure 3 shows the effect of R on the process performance with

3682


Figure 2. Effect of the volumetric purge-to-feed ratio (γ) on the PSA process performance indicators for the case with breakthrough.

Figure 3. Effect of the pressure ratio (R) on the PSA process performance indicators for the case with breakthrough. R was varied by changing PH while keeping PL constant.

breakthrough. The enrichment, heavy-product purity, and light-product recovery all increase as R increases, whereas the heavy-product recovery and light-product

purity are independent of the pressure ratio, and the BCF decreases as R increases. Equation 42 shows that the feed throughput is independent of the pressure ratio,


so smaller BCF values are preferred. However, as discussed earlier, for the frozen solid-phase approximation to be valid, R cannot be greater than the inverse of the feed mole fraction (a mathematical, not physical, constraint). Therefore, the value of R should be as high as possible, but within the constraints of the system and also of the economics of the compression and/or evacuation costs. Figure 4 reveals the effect of yf on the process performance with breakthrough. Only the purities of the heavy and light products and the BCF change with yf. The purity of the light product and the BCF increase as yf decreases, whereas the purity of the heavy product decreases as yf decreases. Again, there is no change in the feed throughput with yf. Therefore, if a maximum allowable light-product impurity is set for a given process, this determines the maximum allowable feed mole fraction, which, in turn, determines the maximum possible purity of the heavy product and also the maximum allowable pressure ratio that can be used in this analysis. From eq 38 and the definitions of τ and σb, it is clear that Lb, tf, Vf, and PH will all have indirect effects on the enrichment, recovery, and purity performance indicators because they change the value of γmax (eq 24). These parameters do effect the BCF, but as long as they are varied in a way that γmax > 0.8, no effect is observed on the other performance indicators. However, if values of these parameters are chosen such that γmax < 0.8, then a lower value of γ will need to be chosen, and the other performance indicators will change accordingly. Therefore, a brief study is done to show how γmax changes with these parameters. The ranges of the parameters studied are given in Table 4, along with the acceptable ranges for a process with γ ) 0.8. Figure 5A-C shows that Vf, Lb, and tf have both upper and lower limits of acceptable ranges. For example, Figure 5A shows that the maximum allowed γ will only be greater than 0.8 (the value chosen for this parametric study) if Lb is between 1.88 and 10.31 m. If the chosen bed length is outside this range, the maximum acceptable γ will be less than 0.8. Thus, to avoid needlessly diluting the heavy product with purge gas, a smaller γ value will have to be used for the process. As discussed earlier, decreasing γ significantly decreases the heavyproduct recovery and the light-product purity while only slightly increasing the light-product recovery. Therefore, it is best to select a bed size that results in γmax > 0.8. Figure 5D displays the effect of PH on γmax. Unlike the parameters in Figure 5A-C, PH has only a lower limit. As long as PH is kept above 0.73 atm, the γmax will be above the γ chosen for the process. The BCF is the only performance indicator that is directly affected by the process parameters PH, Vf, Lb, and tf;; hence, it is used here to clarify further the optimum values of these parameters. Figure 6 shows the effect of these parameters on the BCF; for example, the BCF decreases as tf, Vf, or PH increases, and the BCF increases as Lb increases. The BCF was defined earlier as the fraction of the total bed volume that is utilized at the periodic state. When the bed length changes (Figure 6A), both the contaminated bed volume and the total bed volume change, but the uncontaminated bed volume remains the same. This result is given in Figure 7, which shows the end-of-feed and end-ofpurge concentration profiles for two different bed lengths. Therefore, an increase in Lb will result in a larger

Figure 4. Effect of the feed mole fraction (yf) on the PSA process performance indicators for the case with breakthrough.

fraction of the bed being utilized (higher BCF), but this is of no benefit to the process, because the volume of the bed that is uncontaminated does not change. Moreover, because the bed length does not change the

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Figure 5. Effect of the (A) bed length (Lb), (B) feed time (tf), (C) volumetric feed flow rate (Vf), and (D) feed pressure (PH) on the maximum γ allowed for the process.

other performance indicators (within the range studied here), a smaller bed length is preferred for economic reasons. Smaller bed lengths are also supported by eq 42, because decreasing the bed length increases the feed throughput. Thus, although Figure 5A shows that any bed length between 1.88 and 10.31 m is acceptable for this process, it is better to choose bed lengths closer to 1.88 m.

Figure 6. Effect of the (A) bed length (Lb), (B) feed time (tf), (C) volumetric feed flow rate (Vf), and (D) feed pressure (PH) on the bed capacity factor (BCF) for the process.

The interesting and counterintuitive result that the BCF decreases with increasing tf or Vf (Figure 6B and C) is understood with the results presented in Figure 8, which shows the end-of-feed and end-of-purge concentration profiles for two different volumetric feed flow rates. When the volumetric feed flow rate is doubled, the distance that the shock wave travels during the


Figure 9. General effluent concentration breakthrough curve during the feed step.

Figure 7. Effect of the bed length (Lb) on the end-of-feed and end-of-purge concentration profiles.

study. However, as mentioned, the upper limit for the feed pressure is set by the adsorbate isotherm. The feed pressure must be low enough to keep the process on the unfavorable portion of the isotherm. The change in BCF shown in Figure 6A-D indicates that it is possible for a parameter to change the BCF (which means that the periodic-state values such as σa and a* have changed) without changing the other performance indicators. For example, changing the feed time results in a different maximum breakthrough concentration but the same average light-product purity. This result can be explained by considering the definition of a periodic state. A PSA process has reached a periodic state when the change in the amount of adsorbate in the bed over the feed step equals the change in the amount of adsorbate in the bed over the purge step. This is represented mathematically by

ni,in,f - ni,out,f ) ni,out,p - ni,in,p

Figure 8. Effect of the volumetric feed flow rate (Vf) on the endof-feed and end-of-purge concentration profiles.

purge step also doubles (because the volumetric purge flow rate also increases). Therefore, at higher volumetric feed rates, the concentration front is closer to the feed end of the bed at the beginning of the feed step. During the feed step, the concentrations will be moving faster, but there is more room for the shock wave to spread out into a simple wave. Figure 8 shows that, when the shock wave has more room to spread out into the simple wave, the concentration profile at the end of the feed step is not as steep, which results in a larger uncontaminated bed volume at the end of the feed step. As mentioned above, a larger uncontaminated bed volume (with the same total bed volume) causes a decrease in the BCF and leaves more room for deviations from the predicted periodic state. The feed throughput will also increase as Vf increases, further supporting the use of a higher volumetric feed flow rate. Note that increasing tf at constant Vf is similar to increasing Vf at constant tf, so the conclusions are the same for the effect of tf on the BCF. Combining these data with the results from the γmax parametric study suggests that a feed time close to (but not longer than) 1750 s and a volumetric feed flow rate close to (but not higher than) 142 m3 STP/ min should be chosen for this process. Figure 6D shows that decreasing PH (along with PL, to keep R constant) causes the BCF to increase. Thus, higher values of PH are preferred, and there is no upper limit for this parameter according to the γmax parametric

(43)

The purge is done with pure inert, so there is no adsorbate coming into the bed during the purge step (ni,in,p ) 0). For the specific process considered here, the dimensionless fluid-phase concentration of the purge is always equal to 1, because the shock wave is never pushed out of the bed. This means that the purge concentration always equals the feed concentration. This equality, expressed in terms of moles instead of fluidphase concentration, is

ni,out,p ni,in,f ) Vp Vf

(44)

Using the definition of γ, eq 44 is rewritten as

ni,out,p ) ni,in,fγ

(45)

Because γ is specified as always being less than 1 when there is breakthrough, the amount of adsorbate leaving during the purge step will always be less than the amount of adsorbate entering during the feed step. Therefore, for a periodic state to be reached, this difference between the feed and purge steps must equal the amount of adsorbate that leaves as breakthrough during the feed step. Substituting eq 45 into eq 43 and simplifying yields

ni,out,f ) ni,in,f(1 - γ)

(46)

The amount of adsorbate that leaves during the feed step is calculated from the breakthrough curve. Figure 9 shows a general breakthrough curve during the feed step, where t0 is the time when breakthrough first occurs and tf is the feed time. The average dimensionless

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Figure 10. Effluent concentration breakthrough curves for different pressure levels (same light-product purity during the feed step).

concentration of the breakthrough gas (i.e., the area under the curve divided by the total feed time) multiplied by the feed concentration and the volumetric feed flow rate gives the total amount of adsorbate that leaves during the feed step. This is represented mathematically by / ni,out,f ) cfVfci,avg,LP ) ni,in,f(1 - γ)

(47)

and can be simplified to / )1-γ ci,avg,LP

(48)

Because c/i ) yi/yf, eq 48 is equivalent to the definition of the light-product purity (eq 38). Therefore, eq 38 is not simply an equation used to calculate a performance indicator, it is an expression that must be satisfied for the process to be at a periodic state. Changing the feed time, pressure level, or volumetric feed flow rate will change the shape of the breakthrough curve, but because the process is at a periodic state, the lightproduct purity will not change with these parameters. Figure 10 shows breakthrough curves for two different pressure levels. One process has a high pressure of 29.6 atm, and the other process has a high pressure of 9.87 atm (both have R ) 19, with other parameters given in Table 2). Although the breakthrough curves clearly have different shapes, the processes are at a periodic state so the light-product purity is the same in both cases (i.e., the areas under the two curves are equal). Figure 10 clearly shows that a process can have different periodicstate values for a* and σa (and therefore different BCF values) and still have the same light-product impurity. Conclusions The equilibrium theory analysis carried out here showed that it is possible to achieve high enrichments and recoveries of the heavy product with a simple twostep PSA process using an adsorbate-adsorbent that exhibits an unfavorable Langmuir isotherm. The resulting analytical expressions can be used for feasibility studies and the design and development of heavycomponent upgrade systems as shown here with a brief design study using the H2-MmNi4.2Al0.8 metal hydride system that included the effects of H2 breakthrough into the light product. The equilibrium-theory-based expressions also represent the upper thermodynamic limit on the performance that can be achieved with this kind of PSA process; in reality, the performance will always be

worse. Hence, a feasibility study can be used to judge rapidly whether a PSA process of the kind studied here is economically challenged. A parametric study was also performed to determine how changing γ, R, yf, Lb, tf, Vf, and PH affects the performance indicators. The enrichment, recovery, and purity were only affected by γ, R, and yf. To optimize the performance of the PSA system, γ should be as close to unity as possible, and R should be as close to the inverse of the feed mole fraction as is economically feasible. The feed mole fraction will determine the lightproduct impurity (with higher feed mole fractions corresponding to higher light-product impurities). In general, smaller bed lengths are preferred, with the minimum possible bed length being determined by γmax for the system (if a bed length results in γmax < γ, a larger bed length will need to be used). Vf and tf should be as high as possible, and the upper limits of the acceptable range of these parameters are determined by the γmax for the system. PH should also be as high as possible, but in this case, the upper limit is determined by the adsorption isotherm and economic considerations. Future work will include the development of an Excelbased PSA calculator based on the simple analytical expressions developed in this work. This calculator will be very useful for carrying out preliminary design and feasibility studies. It will also be useful for training and educating students and practitioners. Acknowledgment Funding provided by the Westvaco Charleston Research Center, the Separations Research Program at the University of Texas at Austin, and the NSF GK-12 Program through Award DGE-008642 is greatly appreciated. Nomenclature a ) characteristic invariant a* ) periodic characteristic invariant that reaches the end of the bed at the end of the feed step aL ) characteristic invariant on the left side of the shock aR ) characteristic invariant on the right side of the shock A ) cross-sectional area of the bed (m2) b ) Langmuir isotherm parameter (atm-1) ci ) fluid-phase concentration of the adsorbate (mol/m3) c/i ) dimensionless fluid-phase concentration of the adsorbate c/ib ) dimensionless fluid-phase concentration of the adsorbate exiting the bed during the feed step c/i,L ) dimensionless fluid-phase concentration of the adsorbate on the left side of the shock c/i,LP,avg ) average dimensionless fluid-phase concentration of the adsorbate in the light product c/i,R ) dimensionless fluid-phase concentration of the adsorbate on the right side of the shock cref ) fluid-phase concentration of the adsorbate in the feed (mol/m3) db ) bed diameter (m) E ) enrichment of the heavy product L ) stoichiometric depth of penetration for the first feed step (m) Lb ) bed length (m) nLP,f ) amount of carrier gas used during the feed (mol) nLP,p ) amount of carrier gas used during the purge (mol) PH ) feed pressure (atm) PL ) purge pressure (atm)

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3687 qi ) adsorbed-phase concentration of the adsorbate (mol/ kg) q/i ) dimensionless adsorbed-phase concentration of the adsorbate q/i,f ) dimensionless adsorbed-phase concentration of the adsorbate at the end of the feed step qref ) adsorbed-phase concentration of the adsorbate in equilibrium with cref (mol/kg) qs ) Langmuir isotherm parameter (mol/kg) Q1 ) amount of adsorbate in the solid phase at the end of the first feed step (mol) Qf ) amount of adsorbate in the solid phase at the end of the feed step (mol) Qp ) amount of adsorbate in the solid phase at the end of the purge step (mol) R ) isotherm separation factor RecHP ) recovery of the heavy product RecLP ) recovery of the light product t ) time (s) tb0 ) time at which breakthrough begins during the feed step (s) tf ) feed duration (s) T ) temperature (K) Vf ) volumetric feed flow rate (m3 STP/min) yi,f ) mole fraction of adsorbate in the feed yi,HP ) mole fraction of adsorbate in the heavy product yi,LP ) mole fraction of adsorbate in the light product z ) axial coordinate (m) Greek Letters R ) high-to-low pressure ratio ) 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 Fb ) bulk density of the adsorbent (kg/m3) σ ) dimensionless axial coordinate σ1 ) periodic-state dimensionless axial coordinate reached by c/i ) 1 at the end of the feed step

σa ) periodic-state dimensionless axial position of the shock wave at the end of the purge step σb ) dimensionless bed length σf ) dimensionless axial coordinate during the feed step σp ) dimensionless axial coordinate during the purge step

Literature Cited (1) Skarstrom, C. W. Use of Adsorption Phenomena in Automatic Plant-Type Gas Analyzers. Ann. N. Y. Acad. Sci. 1959, 72, 751. (2) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley & Sons: New York, 1984. (3) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: London, 1997. (4) Ruthven, D. M.; Farooq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH Publishers: New York, 1994. (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) LeVan, M. D. Pressure Swing Adsorption: Equilibrium Theory for Purification and Enrichment. Ind. Eng. Chem. Res. 1995, 34, 2655. (7) 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. (8) 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. (9) Rhee, H. K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations; Prentice Hall: Englewood Cliffs, NJ, 1986; Vol. I. (10) DeVault, D. The Theory of Chromatography. J. Am. Chem. Soc. 1943, 65, 532. (11) Ritter, J. A.; Yang, R. T. Equilibrium Theory for HysteresisDependent Fixed-Bed Desorption. Chem. Eng. Sci. 1991, 46, 563. (12) 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 August 24, 2001 Revised manuscript received April 24, 2002 Accepted May 1, 2002 IE0107035

Equilibrium Theory Analysis of a Pressure Swing Adsorption Cycle

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