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Ind. Eng. Chem. Res. 1998, 37, 154-158
On the Pseudomultiplicity of Pressure Swing Adsorption Periodic States Narasimhan Sundaram and Ralph T. Yang* Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109
Pseudomultiplicity of periodic states in pressure swing adsorption (PSA) is illustrated with a simple cycle. Pseudomultiplicity in PSA is defined by the existence of at least two different periodic states for a single PSA operating specification, such as fixed sorbent productivity, while product purity or product recovery may vary. Non-isothermal behavior, number of components, isotherm nonlinearities, and cycle complexity are considered requirements for conventional multiplicity in PSA, defined by different periodic states depending only on the initial condition of the beds, for the same input variables. The examples presented here, obtained with a simple model, show that pseudomultiplicity in PSA is neither a consequence of system nonlinearities described above nor the result of wave interactions due to cycle step complexity. Introduction The steady state of cyclic pressure swing adsorption (PSA) processes is the state used to design and study their operation (Yang, 1997; Ruthven et al., 1994). This state may be computed in several ways, which may or may not involve transients (see, for example, LeVan and Croft, 1991; Smith and Westerberg, 1992). Efficient methods for direct determination of periodic steady states as well as mapping regions with different bifurcation diagrams are available for reactor theory (Khinast and Luss, 1997) and should be applicable to PSA studies. It is possible in some cases that the PSA steady state may be reached only asymptotically. It is also possible that PSA multiple steady states may exist for a fixed set of operating conditions, over a particular range of one or more of these conditions. This is the conventional definition of multiplicity, prevalent in reactor engineering as described by Gavalas (1966), Luss and Amundson (1967), Aris (1969), Iooss and Joseph (1980), and Jensen and Ray (1982). Such an example of multiplicity in PSA has been presented by Kikkinides et al. (1995). Except for this work, generally, unique PSA steady states were expected and found by performing simulations of the process after reaching a cyclic steady state. An alternate definition of multiplicity in PSA given by Suh and Wankat (1989) is based on specifying a single operational characteristic. PSA systems attaining distinct periodic states while meeting this constraint were considered as examples of multiplicity. An elegant and detailed stability analysis of the two-bed PSA process proposed by Suh and Wankat (1989) for separating a two-component mixture was performed by Croft and LeVan (1994b). In this process, each half-cycle consists of four steps. As continuous feed to bed 1 occurs, bed 2 simultaneously undergoes the four sequential steps of cocurrent blowdown, countercurrent blowdown, countercurrent purge, and countercurrent pressurization with the product. The complete cycle thus consists of eight steps (beds 1 and 2 interchange for the next half-cycle) and uses three pressures (high, intermediate, and low). For fixed sorbent productivity, * Author to whom correspondence should be addressed.
two different periodic states, distinguished by different bed profiles, were found such that each displayed the same purity. Product recovery and system conditions such as feed velocity and purge/feed ratio were different for each state. We consider this a case of pseudomultiplicity as opposed to the conventional multiplicity in PSA found by Kikkinides et al. (1995). In what follows, it is shown that such examples of pseudomultiplicity occur even when the cycle is not complicated. In contrast to the eight-step process a simple two-step process will be used, consisting only of a feed step and a purge step in two beds (Figure 1) with two operating pressures (high and low), approximated by analogy to the countercurrent contactor as given by Suzuki (1985). This permits direct convergence on the steady state, and the governing equations may be manipulated easily. This model, while having extreme assumptions such as isothermal operation and small throughput, has previously been shown to capture all the qualitative aspects of more rigorous transient computations, even for bulk gas separations (Farooq and Ruthven, 1990). The separation is discussed in the context of air separation for nitrogen production using molecular sieve carbon (MSC). This process is widely used in industry and is based on differing diffusivities of O2 and N2 in MSC (Yang, 1997). Model Equations Following the development of Suzuki (1985), consider a binary isothermal system of components A and B. The mass balance of component A at the steady state for the high-pressure and low-pressure steps is given by
CH d(uHYAH) CL d(uLYAL) kA ) ) - (QAH - QAL) (1) dz dz 4 The overall balance for both the high pressure and low pressure steps is
kA kB CH d(uH) CL d(uL) ) ) - (QAH - QAL) - (QBH dz dz 4 4 QBL) (2)
S0888-5885(97)00522-8 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/05/1998
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Figure 1. Schematic of a two-step PSA cycle with relevant stream conditions marked.
In these balances the assumption of zero net accumulation in the solid phase is used to rewrite the linear driving force (LDF) rate (Farooq and Ruthven, 1990). This permits the compact form of eqs 1 and 2. Here kA and kB are the LDF rate constants obtained from diffusivities using (1 - )ΩDi/(R2) and Ω is the correction factor for the LDF constant for rapid cycles (Kapoor and Yang, 1989; Yang, 1997). Di are assumed to be constant and independent of concentration and pressure so that ki are the same for high- and low-pressure steps. In eqs 1 and 2, subscript H refers to high pressure (feed) and L to low pressure (purge). The binary Langmuir isotherms are used to describe the mixed gas adsorption, expressed for species A during the feed or high-pressure step as
QAH KACHYAH ) QAM 1 + KACHYAH + KBCHYBH
(3)
KA and KB are Langmuir equilibrium constants and QAM and QBM are saturation capacities. The Langmuir model requires these capacities be equal for thermodynamic consistency. The continuity condition stipulates that CH and CL are constant during the two steps. In all subsequent analyses CL is based on a desorption or purge pressure of 1 atm. The principal boundary condition is to satisfy YAH ) YAL at the product end, since purge is with the product. The second boundary condition is for the velocity at the inlet of the purge bed which is at z ) L. Refer to Figure 1. This is given by the product of G, the purge to feed velocity ratio and the feed inlet velocity (uOH or V). uOH ) V/2 where u refers to the countercurrent analogy, V is actual feed interstitial velocity, and the factor 2 is needed for mass-balance consistency. Binary Concentration Dependent Diffusivities We can also use the LDF model with coupling terms. For this purpose we rewrite the linear driving force model including the off-diagonal or cross terms, to represent the concentration dependent diffusivities in binary diffusion as
dqA ) kA(qA* - qA) + kAB(qB* - qB) dt dqB ) kBA(qA* - qA) + kB(qB* - qB) dt
(4)
Such an extended LDF model has previously been used
for PSA by Sircar (1991). One way to obtain the rate constants in eq 4 is to fit binary uptake data. Here we use concentration dependent diffusivities which are found from the multicomponent isotherm using irreversible thermodynamics (Yang et al., 1991). In this way the simple LDF model is extended to include the effects of binary diffusion and the effect of changing total pressure. For a binary Langmuir isotherm Di are given by
QM - qB* DA ) DA0 , QM - qA* - qB* q A* DAB ) DA0 QM - qA* - qB* qB* DBA ) DB0 , QM - qA* - qB* QM - qA* DB ) DB0 (5) QM - qA* - qB* The LDF constants ki of eq 4 are obtained from these diffusivities using k ) ΩD/R2. For rapid cycling especially, correlations exist to enable this relation to be used properly (Nakao and Suzuki, 1983). We will use Ω ) 15 in subsequent results. In eq 5 the intrinsic diffusivities Di0 are the limiting, pure component values. The structure of eqs 4 permits these equations to describe overshoot in concentration uptake curves at a constant pressure and temperature such as that shown by Dominguez et al. (1988). Illustrating the Periodic States The recovery-purity diagram for a fixed productivity and either fixed P or L/V can be constructed using an iterative method to satisfy the boundary conditions in Figure 1. The fixed productivity constraint holds over a range of operating conditions. The system is isothermal air separation on MSC, with a diffusivity ratio around 46. Figure 2 shows that for fixed productivity and pressure ratio P, both [L/V,G] have to vary to generate the recovery-purity diagram (Rota and Wankat, 1990). In Figure 2, we have used both a concentration independent diffusivity (crosses) as well as the coupled mass transfer model (circles). It is clear that in both cases the recovery-purity diagram shows two branches and the existence of two different states possessing the same productivity, pressure ratio, and purity. These states
156 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
Figure 2. Nitrogen recovery-purity diagram for the system air (21% O2-79% N2) on molecular sieve carbon. O2 ) A, N2 ) B. Langmuir QAM ) QBM ) 0.002 64 mol/cm3, KA ) KB ) 3500 cm3/ mol. LDF rate kA ) 0.0405 1/s, kB ) 0.000 885 1/s, PL ) 1 atm, L ) 1 m. Fixed P ) 2, productivity ) 0.0123 s-1, L/V increases with recovery.
Figure 3. Nitrogen recovery-purity diagram for the system air (21% O2-79% N2) on molecular sieve carbon. Same conditions as Figure 2 but kB ) 0.008 85 1/s and productivity ) 0.0177 s-1.
are, however, characterized by the purge-feed ratio G, space time L/V, and recovery. This is similar to that reported for eight-step isothermal PSA by Suh and Wankat (1989) and later confirmed by Croft and LeVan (1994a,b). Figure 3 shows the same system as Figure 2, but with a lower diffusivity ratio ) 4.6. Once again for fixed productivity, the pseudomultiplicity is discernible in both kinetic models. The coupled mass-transfer model shows higher purities in both Figures 2 and 3, which may be evidence of codiffusion as defined by Yang et al. (1991). Figure 4 shows the profiles at the end of the feed step for two states of Figure 1, with coupled mass transfer, corresponding to L/V ) 40 and 60 s. It is clear from Figure 4 that the two periodic states representing the single fixed operating specification of productivity display distinctly different profiles and recoveries for the same pressure ratio and purity. Uniqueness of these periodic states constitutes an example of pseudomultiplicity, as defined by Croft and LeVan (1994a,b). Figure 5 shows that the two-step process with a fixed
Figure 4. Concentration profiles at end of feed step for system of Figure 2 with concentration dependent diffusivities.
Figure 5. Nitrogen recovery-purity diagram for the system air (21% O2-79% N2) on molecular sieve carbon. O2 ) A, N2 ) B. Langmuir QAM ) QBM ) 0.002 64 mol/cm3, KA ) KB ) 3500 cm3/ mol. LDF rate kA ) 0.0405 1/s, kB ) 0.000 885 1/s, PL ) 1 atm, L ) 1 m. Fixed L/V ) 25 s, productivity ) 0.024 98 s-1, P decreases with recovery.
L/V ) 25 s possesses pseudomultiplicity when [P,G] are varied for a fixed productivity. Diffusivity was independent of concentration in this example. Figure 6 shows that for this system the resulting profiles at the end of the feed step are distinct and unique. Discussion A simple solution to the two-step PSA periodic states with velocity variation is used to illustrate pseudomultiplicity while meeting a fixed productivity constraint. Since several operating parameters vary between the states, these artificial multiplicities appear to be only functions of possible solutions that exist for the constrained mass balances. In conventional treatments of multiplicity in the steady states of chemical reactors, Aris (1979) discusses the utility of the simplest model that is sophisticated enough to serve the purpose. Such a model would then provide an understanding of the minimum conditions under which multiplicity would be manifested. In cyclic adsorption processes, generally, our understanding is through numerical simulations, whether via transient
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 157
Figure 6. Concentration profiles at end of feed step for system of Figure 5 with no concentration dependence of diffusivity.
simulations converging on a steady state or through direct determination, accelerated in a variety of ways. In either case, the task of finding examples of multiplicity is somewhat hampered by sensitivity to initial conditions and model assumptions and by numerical considerations. In addition, there is a large choice in the operating variables and conditions for such cyclic processes. Over certain ranges of these variables, it is possible that multiple periodic states may not appear. Indeed, in the example described by Kikkinides et al. (1995), multiplicity was exhibited only over a narrow range of feed velocities. These multiple states were dependent on the initial bed conditions. Further, nonisothermal behavior, three components with at least one component being adsorbed very strongly with a nonlinear isotherm, and an eight-step process were required to define the multiple-state region. These conventional multiplicities were also found in the odd numbers expected from reactor theory as shown by Gavalas (1966). In the present work, which deals with pseudomultiplicities in cyclic PSA processes, only two stable states were detected. We have attempted to delineate the simplest system for which this kind of multiplicity occurs in keeping with the approach of Aris (1979). The scheme of Suh and Wankat (1989) was an eight-step PSA process and the pseudomultiplicity was detected using transient numerical simulations. Subsequently, Croft and LeVan (1994b) verified these pseudomultiplicities for the eight-step system using an accelerated convergence method to find the periodic state and also performed a detailed stability analysis of the pseudomultiple periodic states. Each step in a PSA cycle creates a different initial bed condition for the subsequent step, which then operates under different flow and pressure conditions. With an eight-step process, in contrast to a two-step process, there are additional initial conditions set up within a complete cycle, increasing the possibility for wave interactions as the cycle progresses. This could be one reason for conventional multiplicity in PSA, where the periodic state is known to be sensitive to the initial condition (Yang, 1997; Ruthven et al., 1994; Kikkinides et al., 1995). In addition, numerical methods, although necessary to study the eight-step process or other complicated PSA cycles, can sometimes introduce uncertainty and additional complexity to the
steady-state analysis. For example, as Suh and Wankat (1989) discovered, several trial runs were needed to approach the periodic state using transient simulations, while holding sorbent productivity constant. Similarly, for a PSA purification process with a strongly nonlinear adsorption isotherm, Croft and LeVan (1994a) found an extremely slow approach to the periodic state. Other model assumptions such as axial dispersion (Sundaram, 1995) can also introduce complexity both to the intrinsic wave interaction and the numerical schemes. With respect, however, to pseudomultiplicity in PSA as defined here, the simple model used shows that even a two-step process can display this particular kind of multiplicity. The method does not rely on any complicated numerical techniques. In addition, the algebra for equal Langmuir constants is reducible to that for linear isotherms. An isothermal system with only two operating pressures or alternately with only two operating space times, was used to generate these states. Therefore, cycle complexity, non-isothermal behavior, and nonlinearity or coupling of isotherms do not appear to be required to generate these pseudomultiplicities. Acknowledgment This research was supported by the NSF Grant CTS9520328. Notation B: Henry constant, cm3/mol CH: total concentration in high-pressure steps, mol/cm3 CL: total concentration in low-pressure steps, mol/cm3 D: diffusivity ratio, DA0/DB0 Di: diffusivity, m2/s G: purge-feed ratio ki: linear driving force rate constant of component i, 1/s Ki: Langmuir constant of component i, cm3/mol, eq 3 L: length of bed, m L/V: space time, s P: pressure ratio, P ) PH/PL PH: high pressure, atm PL: desorption pressure, 1 atm QiM: Langmuir isotherm saturation capacity, mol/cm3 u: velocity, m/s U: normalized velocity, u/L, 1/s V: interstitial velocity of actual PSA cycle, m/s Yi: gas phase mole fraction of component i z: distance along bed, m Greek Symbols : bed porosity Ω: correction factor for diffusivity in LDF model
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Sircar, S. Pressure Swing Adsorption: Research Needs by Industry. In Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds.; Engineering Foundation: New York, 1991; pp 815843. Suh, S. S.; Wankat, P. C. Combined Cocurrent-Countercurrent Blowdown Cycle in Pressure Swing Adsorption. AIChE J. 1989, 35, 523. Sundaram, N. Axial Dispersion Effects on Gas Purification Pressure Swing Adsorption Periodic Steady State. Chem. Eng. Commun. 1995, 133, 207. Suzuki, M. Continuous Counter-Current Flow Approximation for Dynamic Steady State Profiles of Pressure Swing Adsorption. AIChE Symp. Ser. 1985, 81 (242), 67. Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Boston, 1987; reprinted by Imperial College Press: London, 1997. Yang, R. T.; Chen, Y. D.; Yeh, Y. T. Prediction of Cross-Term Coefficients in Binary Diffusion: Diffusion in Zeolite. Chem. Eng. Sci. 1991, 46, 3089-3099.
Received for review July 15, 1997 Revised manuscript received September 30, 1997 Accepted October 3, 1997X IE970522W
X Abstract published in Advance ACS Abstracts, December 15, 1997.
