Simulation of rapid pressure swing adsorption and reaction processes

Chemical Engineering Science, Printed in Great Britain.

Vol.48,

No. 18, pp. 317353186, 1993.

SIMULATION

0009-2509/93 $6.00 + 0.M) C] 1993 l%rgamon Press Ltd

OF RAPID PRESSURE SWTNG ADSORPTION AND REACTION PROCESSES

E. ALPAY,+ C. N. KENNEY and D. M. SCOTT of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K.

Department

(First received 28 October 1992; accepted in revised form 4 March

1993)

Abstract ~-Ageneral model for non-isothermal adsorption and reaction in a rapid pressure swing process is described. Several numerical discretisation methods for the solution of the model are compared. These include the methods of orthogonal collocation, orthogonal collocation on finite elements, double orthogonal collocation on finite elements, and cells-in-series. Computationally, orthogonal collocation on finite elements is found to be the most efficient of these. The model is applied to air separation for oxygen production. Calculations confirm the formation of a concentration shock when an adsorbent bed is pressuriscd with air. The form and propagation of the shock over short times is found to be in excellent agreement with the exact similarity transformation solutions derived for an infinitely long bed. For air separation, novel experimental measurements, showing an optimum particle size for maximum product oxygen purity, are accurately described by the model. Calculations indicate that a poor separation results from ineffective pressure swing for beds containing very small particles, and from intcaparticle diffusional limitations for beds containing very large particles. For adsorption coupled with reaction, finite rate and reversible reactions are considered. These include both competitive and non-competitive reaction schemes. For the test case of a dilute reaction A+ B + 3C, with B the only adsorbing species, bed pressurisation calculations are found to be in excellent agreement with the solutions obtained by the method of characteristics.

1.

INTRODUCTION

Pressure swing adsorption (PSA) constitutes an important process for gas separation; examples include the commercially established multibed processes, the single-bed rapid-cycle (RPSA) processes, and pressure parametric pumping [see Ruthven (1984) and Yang (1987)]. Of these, the RPSA processes are the simplest in configuration, and yet offer relatively high adsorbent productivities for some small-scale, low purity separations (Jones and Keller, 1981; Pritchard and Simpson, 1986; Hart and Thomas, 1991). As illustrated in Fig. 1, the basic RPSA cycle consists of two steps: feed gas pressurisation and countercurrent depressurisation. A single bed of small particles, typically 200-700 pm in diameter, is employed, and rapid pressurisation and depressurisation steps are applied, typically 0.1-0.5 Hz. The combination of a rapid cycle with small particles leads to steep and cyclically varying pressure gradients within the bed. The pressure dynamics within the bed is such that the product end pressure remains approximately constant, thus allowing continuous product release. At the other end of the bed, gas is alternately fed and released, eliminating the need for any external purge gas supply as is often used in multibed PSA processes. A modified form of the RPSA process has been proposed by Jones and Keller (1981). In this process, relatively short pressurisation times ( < 1 .O s) and long

+Present address: Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, U.K. 3173

depressurisation times (5-20 s) are used. In addition, a third step, referred to as the delay step, is incorporated into the cycle prior to the onset of depressurisation. In the delay step, the feed gas supply is interrupted for a short period of time (typically 0.5-3.0 s), allowing the gas in the bed to depressurise into the product delivery line. When applied to air separation, the modified RPSA process was shown by Jones and Keller to give higher oxygen yields than the basic RPSA process; this, however, was at the expense of higher feed gas pressures. More interestingly, for the production of 90% (v/v) oxygen, with 40% of the feed oxygen recovered as product, the adsorbent productivity of the modified RPSA process was measured to be about 5 times that of an equivalent two-bed PSA process. Adsorptive reactors couple the operations of adsorption and chemical reaction into a single process. These have advantages over conventional catalytic reactors in that improved reactor performance may be attained, such as higher conversions (with no restrictions on the equilibrium conversion), and improved selectivities. For example, for reversible reactions with + C), separation of the multiple products (e.g. A =B product components may limit the extent of the backward reaction, improving the overall reactant conversion. In turn, improved conversions permit a lower reactor operating temperature, thus giving energy savings, and reducing the degree of catalyst deactivation due to, for example, coke deposition. Finally, the possible combination of reaction with product separation offers process intensification. One of the first examples of an adsorptive reactor is the chromatographic reactor (Roginskii et al., 1961;

E. ALPAY

3174 pmduct

pressurise withfeed ga.s

product

dspressurisc to exhaust

Fig. 1. A schematic diagram of the basic RPSA process.

Magee, 1963). Later examples include real and simulated moving bed processes, in which the countercurrent contact of the bed with the reaction and product gases maximises separation and suband reactor performance (Takeuchi sequently Uraguchi, 1977; Petroulas et al., 1985; Fish and Carr, 1989). Neither the chromatographic nor the moving bed processes are, however, suitable for large-scale or continuous operation. Until recently, cyclic chemical operations have been restricted to either separation or reaction. Examples of the latter include studies in which the concentration of feed reactant to a catalytic reactor is periodically varied to give improved conversions or selectivities, see Kevrekidis (1986) for a general review. The first published account of a cyclic process with adsorption and reaction was given by Vaporciyan and Kadlec (1987); a theoretical study of an RPSA-based reactor was carried out. This was later tested experimentally by Vaporciyan and Kadlec (1989) for the oxidation of CO over a mixture of platinum-supported catalyst and zeolite 5A adsorbent. For such a reaction, it was found that CO2 production could be increased by up to 2 times the plug flow reactor production. More recently, novel simulation studies of a two-bed PSAbased reactor have been carried out by Kirkby and Morgan (1991), for which separation enhancement in the presence of reaction was indicated. In a similar study by Chatsiriwech et al. (1992), enhanced reactor performance in the presence of PSA-based separation was reported. Unlike the chromatographic and moving bed processes, PSA- and RPSA-based reactors provide a continuous product stream, and are suitable for relatively large-scale operation. The existence of many design parameters associated with pressure-swing-based adsorbers or reactors precludes any intuitive process design. Computer models of such processes arc therefore necessary for design and optimisation purposes. Whilst several RPSA models have appeared in the literature (e.g. Turnock and Kadlec, 1971; Kowler and Kadlec, 1972; Verelst and Baron, 1986; Vaporciyan and Kadlec, 1987, 1989; Doong and Yang, 1988; Jianyu

rt al.

and Zhenhua, 1990), there is no comprehensive model which describes RPSA coupled with reaction. Furthermore, almost all RPSA or RPSA with reaction (RPSAR) models to date employ one or more of the usual simplifying assumptions of negligible mass transfer resistance, simple gas flow dynamics, or isothermal operation. Indeed, some of the computational schemes employed in the solution of these models, e.g. the method of characteristics, necessitates such assumptions. Other computational schemes, such as the cells-in-series solutions, are often computationally demanding [see, for example, Kowler and Kadlec (1972) and Doong and Yang (1988)]. The difficulties in the simulation of such cyclic adsorption processes arise from the non-linear nature of the system, and the inherent solution discontinuities or steep gradients. A further complexity arises from the step changes in gas pressure at the feed end of the bed, causing periodic reversals in the gas flow direction within the bed. High-order approximations or fine grid discretisations are usually required in the numerical solution of such problems. These methods, however, often result in a large system of stiff equations, with subsequent problems of numerical instability and long computational time. The method of characteristics provides one means of accommodating solution discontinuities; characteristic solutions of simple models are often viewed as asymptotes for models of higher complexity. Other numerical methods, some of which are to be described below, can accommodate steep gradients but not necessarily discontinuities. For adsorption in a packed

bed, the inclusion of axial dispersion, or backmixing, is a convenient way of eliminating any discontinuities in concentration, and hence making such numerical methods viable. In reality, of course, such axial dispersion will arise from local Auctuations in flow velocities, and molecular and turbulent (eddy) mixing. In this paper, a general model for the simulation of non-isothermal adsorption and reaction in a rapid pressure swing process is reported. The model allows the simulation of the basic two-step process, and the modified three-step process of Jones and Keller (1981). As a comparative study, various discretisation methods are employed in the numerical solution of the model. These include spatial discrctisation using polynomial approximations, low-order polynomial approximations on a finite element grid and cellsin-series. In addition, a global method, based on loworder polynomial approximations for derivatives in both space and time, is employed. As a test case, the simulation program is applied to air separation over zeolite .5A adsorbent. Calculations are compared with the exact bed pressurisation solutions of Scott (1991), and with experimental measurements on a small-scale RPSA unit (Alpay et al., 1992). In addition, as a second test case, adsorption coupled with the finite rate and reversible reaction AeB + 3C is considered. Such a reaction scheme depicts the industrially important dehydrogenation reactions, examples of which include ethane dehydrogenation to ethylene, and

Simulation of rapid pressure swing adsorption

methylcyclohexane dehydrogenation to toluene. In order to avoid equilibrium limitations, these reactions are currently carried out at high temperatures, albeit at the expense of catalyst deactivation due to coke deposition. Cyclic adsorptive reactors, however, may allow lower temperatures of operation, whilst maintaining high product yields. In this work, calculations for the above-mentioned reaction scheme are compared with the results of Chatsiriwech et al. (1992), which were obtained using the method of characteristics.

and reaction processes

where the convective derivative is DT 3T -_=-+u-_. Dt dt

2. THE

RPSAR

MODEL

5T

(5)

dx

For an ideal gas pg = P/RT, T[dP/dT],z = P and u = nRT/AP. The thermal conduction flux, J,, is given by Fourier’s law J,=

The principal assumptions made in the development of the model are now listed:

3175

- kgg.

Neglecting the viscous dissipation of energy, and applying eqs (4H6) to a packed bed with adsorption and reaction (Sircar et al., 1983; Rota and Wankat, 1990), gives

(1) Ideal gas law obeyed.

(2) Ideal solution behaviour in the gas phase.

(3) Gas thermophysical properties independent of temperature. (4) Axially dispersed plug flow. (5) Gas flow described by (a) Darcy’s law or (b) the Ergun equation. (6) Adsorption isotherms described as (a) linear and independent or (b) non-linear and multicomponent. (7) Adsorption rate described as (a) instantaneous or (b) mass transfer limited. (8) Bed of uniform voidage and particle size. reactions of finite rate. (9) Pseudo-homogeneous (10) Simple power law kinetics for the reaction rate. thermal equilibrium between (11) Instantaneous the gas and solid phases. (12) Wall heat transfer described by an overall heat transfer coefficient. (13) No radial pressure, temperature or concentration gradients. Based upon these assumptions, component and total material balances for the packed-bed adsorber and reactor yield

a

Er-

at

1 a (CUi)

=

-

-

-

A hi

+

Dx

-

ax2

(CYi)

-p+ri

(1)

and

respectively. The total bed void fraction, st, includes the intraparticle and interparticle voids, and is defined by E, = Eb + E&.(1- Eb)

(3)

where E* and .ap are the bed (interparticle) and intraparticle void fractions, respectively. The one-dimensional energy equation for non-isothermal and compressible flow is (Bird et al., 1960)

c

i=1

H,,Q-$%-

c

T.).

(7)

The overall heat transfer coefficient, he. accounts for heat transfer from the bed to the wall, through the wall, and to the atmosphere. Gas flow through a packed bed can be described by the Ergun correlation (Ergun, 1952) dP dx=

- J,u - Jku2

where J, and Jk are parameters corresponding to the viscous and kinetic pressure loss terms, respectively. For low particle Reynolds numbers (i.e. < 5), the kinetic contribution to pressure loss becomes negligible, and eq. (8) reduces to Darcy’s law. Semi-empirical relationships for J, and Jk have been derived by Ergun (1952) as

(9) J

=

k

a2 (nYi)

aqi NC

NC

- Pb 1 H,,xi= 1

‘%(l - &b)i% 13

.

The constants CL and /3 were determined experimentally to be about 150 and 1.75, respectively. Macdonald et al. (1979) report that for a wide range of unconsolidated media, a = 180 gives improved predictions of the pressure and flow relationship. This value has been used in the present work. Isotherm data for a given adsorbent are usually available for pure gas adsorbates, and in some cases for gas mixtures. Coadsorption effects may be of importance for the latter case. Several models have been proposed for the prediction of pure and multicomponent isotherms, reviews of which are given by Breck (1974), Ruthven (1984) and Yang (1987). Of these, the Langmuir-type models are among the simplest in form, and yet are quite adequate in predicting both physical and chemical adsorption. Furthermore, in contrast to other adsorption equilibrium models, the

E. ALPAY

3176

Langmuir-type models have both theoretical and empirical bases. The basic Langmuir equation for a pure gas adsorbate is 1 _=

4.. liPi

_1

( >‘+’

Or 47

qi* =lf

qs,4 Pi

Yr;

&exp[

-2(=)]

4.7,li

1+

Pi

z

(13)

ljpj

At low adsorbate concentrations (or partial pressures), the Langmuir and extended Langmuir equations simplify to the linear form 4: = q9,1iPi = mini

(14)

where mi is usually referred to as the adsorption isotherm gradient. The rate of adsorption of component i is given by dqJdr. When mass transfer (or diffusional) limitations to adsorption are negligible, the gas and adsorbed phase concentrations are in instantaneous local equilibrium (ILE), and the rate of adsorption is then simply

w

a4i

at

Equation (20) may be applicable to ultrarapid-cycle processes, or processes in which intraparticle diffusional limitations are very large. Two general reaction schemes are considered in this work. The first is a non-competitive reaction of the form SACA + sacs=

k,l

sccc krb

f saca

kJ,

kr/ = SjJcB, krb

&CA

ti I&

s&C.

When mass transfer limitations are important, the linear driving force approximation of Glueckauf and Coates (1947) is often employed [see, for example, Verelst and Baron (1986) and Liow and Kenney (1990)]:

8% - = ki(qr - qi) at

(22)

The rate constants k, or k:, and krb or kib refer to the forward (left to right) and backward (right to left) reactions, respectively; s represents a stoichiometric coefficient. The effect of temperature on the reaction rate constants is described by Arrhenius’ law. For the non-competitive reaction scheme, it is convenient to relate the rate of reaction of each component to the reaction advancement, X, of a base component b, where Xh = rhlvh and v is a stoichiometric

(23)

coefficient defined as

vi = - si (for i a reactant) vi = + si (for i a product).

at

(21)

and the second is a competitive reaction of the form SACA

j=l

-x-.

where t: is the cycle half-time for the cyclic adsorption and desorption process, for equal adsorption and desorption times (Glueckauf, 1955). Under normal RPSA operation, 8,. > 0.1; for 0, < 0.1, Alpay and Scott (1992) have shown that

(12)

where the subscript b refers to a known base value. In eq. (12), the H,, are assumed to be independent of temperature and the adsorbed phase concentration. For small temperature fluctuations, the square-root dependence is often neglected (Chihara and Suzuki, 1983; Sircar et (II., 1983), as has been done in this work. For a gas mixture of N, components, eq. (11) can be extended to

qT =

Equation (16) is a time-independent expression for the adsorption rate, which acts as a computationally convenient approximation to the diffusion equation within a spherical particle. The validity of eq. (17) is however limited to

(11)

where qsi is the saturation value of qf, and Ii is the reciprocal of pi when half the adsorbent surface is occupied by i and the rest of the surface is vacant. The value of qsr depends on the surface area occupied by one molecular layer of i, and is therefore independent of temperature. The temperature dependence of li is given as [see, for example, Rhee et al. (1970, 1972)] ::;‘:,=

et al.

The rate of reaction then given by

of a particular

component

r; = viXb

(24) i is (25)

and the net change in moles due to reaction, Av, by Av = z

vi.

i=f

(26)

For simple power law kinetics, r,, can be expressed as (17) fi

=,

=

%(l - &b)& pbmiRT

*

(18)

rb = ff$(k.,C”;Cz

- krbCzCz).

(27)

For the competitive reaction scheme with simple power law kinetics, the rate of reaction of A, B and

Simulation of rapid pressure swing adsorption and reaction processes C can be written as

(la) rA = rh + r;7

(28)

Incompressible

3177

gas flow through a valve

n = C”&ZZ

(36)

where c, is the valve sizing coefficient, and P, and Pd are the pressures directly upstream and downstream to the valve, respectively; or (lb) gas flow at a constant delivery rate, n,,, n = n,+

where rb = k,b(cY,p

- k,,(C_v,P

(31)

(2) The product gas concentration

(37) of component

i

and r& = k;,(Cyc)“’ - k;,(CyA)S:.

(32)

Very limited studies for the measurement of axial dispersion in packed-bed adsorbers or reactors have been carried out. A rapid adsorption or reaction rate is generally expected to lead to an additional contribution to axial dispersion due to transport through the solid. For a wide range of particle sizes, however, an approximate value of the axial dispersion coefficient can be obtained from the correlation of Langer et al. (1978). For the adsorption rate described by ILE theory, 2 + N, initial conditions must be given: the initial bed pressure and temperature, and the gas phase mole fractions of N, components. For mass transfer limited adsorption, as described by the LDF approximation, initialisation of the adsorbed phase concentrations yields an additional N, conditions. In this work, the initial bed pressure and temperature are set as the exhaust pressure, P,, and feed gas temperature, T,, respectively. Gas phase mole fractions of the N, components can be arbitrarily chosen; typically, these are set as the feed gas composition values. The N, adsorbed phase concentrations are initialised according to the describing adsorption isotherm, such that equilibrium between the gas and adsorbed phases is set. Boundary conditions are now presented for each step of the RPSAR process; the Danckwerts (1953) boundary conditions are applied where appropriate. The boundary conditions at the onset of pressurisation for the feed end of the bed (x = 0) are: (1) A step change in the feed gas pressure P = P,. (2) The inlet concentration

Dx

(33) Zofcomponent

i

$ (CYi) =t (Vi - Yif)

where yi, is the feed gas mole fraction of component (3) The inlet gas temperature kxg=%(T-

Tf)

where T, is the feed gas temperature

(K).

The boundary conditions at the onset of pressurisation for the product end of the bed (x = L) are:

(3) The product gas temperature dT

-

dx

= 0.

(39)

The boundary condition at the onset of depressurisation for the feed end of the bed (x = 0) is given as P = P,.

(40)

The product end conditions (x = L) are as for those of the feed gas pressurisation step listed above. The boundary conditions for the delay step for the feed end of the bed (x = 0) are given as dT dP -~--_--_0, dx dx

dy, dx

(41)

The product end conditions (x = 1) are as for those of the feed gas pressurisation and countercurrent depressurisation steps listed above. Finally, it is important to note that in some studies of RPSA, a product delivery vessel (or reflux volume) is incorporated into the process design [see, for example, Pritchard and Simpson (1986) and Jianyu and Zhenhau (1990)]. Such a vessel acts as a pressure buffer during the short feed and long waste cycle, thus providing a constant delivery rate of product. One consequence of using a product delivery vessel is the possible back flow of product gas into the bed. Such back flow may, of course, be desirable in that it enhances adsorbent regeneration during the depressurisation step of the process (cf. countercurrent purge in multibed PSA processes). As mentioned earlier, for a basic (two-step) RPSA process, the product end pressure of the bed remains approximately constant, thus eliminating the need for any pressure buffer. Mathematical details for the inclusion of a product delivery vessel into the RPSAR model are given by Alpay (1992). 3. NUMERICAL

METHOD

3.1. Overview The above model requires the simultaneous solution of the total material balance, eq. (2), N, - 1 component material balances, eq. (l), the energy balance, eq. (7), and, when accounting for intraparticle diffusional limitations, the LDF approximation, eq. (16). This yields a total of NR = N, + 2 simultaneous rate equations to be solved. For various simplified

E. ALPAY et al.

3178

versions of the model, such as isothermal operation or ILE, the number of equations can, of course, be reduced. Nevertheless, no analytical solutions exist, and so a numerical solution is required. The numerical solutions of the RPSAR model arc based upon the method of lines, and a global method. The former involve the spatial discretisation of the governing partial differential equations (PDEs) to a system of ordinary differential equations (ODES), which are subsequently solved using a standard library integration algorithm. In this work, the spatial discretisation methods of orthogonal collocation orthogonal collocation on tinite elements (OC),

(OCFE) and cells-in-series (CTS) are employed. In the global method, discretisation of both space and time derivatives is carried out, based on OCFE discretisation in the space domain and OC discretisation in the time domain. Brief descriptions of these methods, with emphasis on their application to the RPSAR model, are now given; further mathematical details are given by Alpay 3.2.

(1992).

Discretisation

methods

(OC). In the OC method, the entire space domain is divided into a number of grid points, the collocation points, and an interpolation polynomial for each dependent variable chosen at thcsc. The collocation points are not equidistant, but are chosen to be the roots of an orthogonal polynomial, such as the Jacobi polynomials. Having set up the polynomial approximations for each dependent variable, the spatial derivatives of these cun be readily calculated. These polynomial approximations of the spatial derivatives reduce the governing PDEs to a system of ODES to be integrated in time. The choice of collocation points as the roots of an orthogonal polynomial leads to solutions which approach the accuracy of the well-known Galerkin method, see Villadsen and Michelsen (1978). The application of OC to non-linear problem is, however, much easier than the Galerkin method, in that no numerical evaluation of intermediate integrals is required. In this work, Lagrange interpolation polynomials were employed, and the collocation points chosen as the roots of the Legendre polynomials. Algorithms for the calculation of the roots of these polynomials, and for the first and second derivatives of the Lagrange interpolation polynomials, are readily available (Villadsen and Michelsen, 1978). Having specified the number of collocation points, J, the resulting system of simultaneous ODES consisted of J x NR equations. These equations were solved using the NAG FORTRAN integration algorithm DOZEBF. D02EBF is based on the variable order, variable steplength Gear’s method, and is suitable for a stiff system of non-linear ODES. Where steep profiles are present in the solution, as is inherent to the RPSA and RPSAR processes, a very large number of collocation points will be required to maintain the integration accuracy. There are, howOrthogonal

collocation

ever, two main disadvantages of collocation points:

in using a large number

(i) the system of discretised equations will be large, thus requiring a large amount of computing time in their integration, and (ii) the associated high-order imations for the spatial gradients ical instabilities

[see, for example,

polynomial approxmay lead to numerSegall

rf al. (1984)].

In order to avoid these problems, Carey and Finlayson (1975) suggest extending the OC method to finite elements (i.e. OCFE), a brief description of which is now given.

on finite elements edlocation Orthogonal (OCFE). In the OCFE method, the problem domain is divided into a number of elements (or zones), and orthogonal collocation applied to each of these. Loworder polynomial approximations are employedwithin each element, whilst a large number of elements maintains the overall accuracy of the solution. Since the first derivatives of the Lagrange interpolation formulae are not continuous, continuity of the solution and its first derivative must be imposed at the element boundaries. A detailed procedure for such continuity calculations is given by Finlayson (1980). In this work, having specified the number of collocation points per element, J’, and the total number of elements, K, the resulting system of simultaneous ODES consisted of (J’ x NR) x K equations. These, as for the OC method, were solved using the NAG FORTRAN integration algorithm DO2EBF. In addition, equal-sized elements were employed, with no element size optimisation attempted; possible optimisation methods for the element sizes are however described by Carey and Finlayson (1975).

Double (DOCFE).

orthogonal

collocation

on jinite

elements

The double orthogonal collocation (DOC) method [see Villadsen and Sorensen (1969)] is similar to OC, except that discretisation is applied to both the space and time derivatives. The method thus reduces a PDE into a system of simultaneous algebraic equations (AEs), which can then be solved by a standard library algorithm at every time step (say At). In doing so, the necessity of solving sets of mixed algebraic and differential equations is avoided, the solution of which may be computationally demanding due to stiffness and convergence problems [see, for example, Hassan et al. (1987)]. It is interesting to note that provided At is small enough, the accuracies of the DOC and OC methods are identical, since the accuracy is governed by the number of spatial discretisation points, J. In this work, DOC was extended to finite elements (referred to as DOCFE), such that OCFE discretisation was applied to the space domain, and OC discretisation to the time domain. Having specified J’ and K, and the number of collocation points in the time domain, I’, the resulting system of AEs then consisted of [.I’ x (I’ + 1) + 61 x Na x K equations. At

Simulation

of rapid pressure swing adsorption

every time increment, these were solved using the NAG FORTRAN Library routine C05NBF, which uses a Powell Hybrid method of solution. For given J’ and K, and provided that At is small enough, or I’ large enough, the accuracy of the DOCFE and OCFE methods are expected to be identical.

Cells-in-series (CZS). In the CIS method, the domain is divided into a series of well-mixed cells. The well-mixed assumption eliminates any spatial gradients within a cell, but a spatial gradient is maintained from one cell to the next. Thus, the governing PDEs of the model can be discretised for each cell, and the resulting simultaneous system of ODES for all cells solved by a standard library integration algorithm. Solution accuracy increases with the number of cells, as does the plug flow condition for gas flow in a packed bed. Usually, axial dispersion within the bed is described by the choice of a suitable number of cells [see Levenspiel (1972)], eliminating the need for the axial dispersion terms appearing in the governing equations. In this work, having specified Ncen cells, the resulting system of simultaneous ODES Ncell x N, equations. These were again the NAG FORTRAN integration D02EBF. 4.

4.1.

RESUI,TS

AND

consisted of solved using algorithm

and reaction processes

3179

a similar solution for both these models is to be expected. The different numericai methods cussed.

adsorption will

now

OCFE: For the OCFE method, convergence and CPU time measurements are shown in Fig. 3(a) and (b), respectively. These are given as a function of the number of elements, K, and collocation points per element, J’. As for the OC method, measurements were carried out for both the TLE and LDF models. General trends in Fig. 3 are as for the OC method. For a given convergence error, however, the CPU time requirements of the OCFE method are significantly less than that for the OC method. For example,

(a>

and CZS

Computing efficiency is assessed in terms of the solution convergence and the requirements of computational time. In this work, a converged solution is defined as that which is consistent to two decimal places in the cyclic equilibrium product gas composition (v/v%), irrespective, for example, of any further increases in the number of collocation points or elements. Specifically, the product gas composition at the end of the pressurisation step is used. For any calculation, the degree of solution convergence is then measured as the absolute error between the cyclic equilibrium gas composition at the end of the pressurisation step, and that of the converged solution. The associated computer processing time or central processing unit (CPU) time, is measured in terms of the time per cycle at the cyclic equilibrium. The computer efficiency analysis is considered for a two-step isothermal RPSA air separation process (see Section 4.2 below). Linear and independent adsorption isotherms were chosen, but calculations for both ILE and LDF models carried out. The base case process parameters shown in Table l(a) were selected, these being typical of a small-scale air separation unit. For both ILE and LDF models, converged solutions are attained for calculations based on the OCFE method with a discretisation grid of J’ = 5 and K = 10, i.e. 68.83 and 68.80% product oxygen for the JLE and LDF models, respectively. Due to the small particle diameter chosen for this analysis, and thus the small time constants for the diffusional processes,

be dis-

OC: For the OC method, convergence error and CPU time measurements are shown in Fig. 2(a) and (b), respectively. These are given for a range of collocation points, J, and for both ILE and LDF models. As expected, an increase in J leads to an improvement in convergence for both models, this at the expense of CPU time. For the LDF model, however, a relatively slow convergence is observed; this is attributed to the additional stiffness imposed into the system of equations when employing a rate equation for adsorption.

DISCUSSION

Comparison of the OC, OGFE, DUCFE discretisation methods

rate

5

10

15

20

25

number of collocation

points, J

of collocation

points. J

number

Fig. 2. Convergence and CPU time analyses for the OC discretisation method: (a) absolute convergence error; (h) CPU time requirements. Calculations are shown for the ILE (0) and LDF (m) models, and for the design conditions listed in Table l(a).

E.

3180 1”

ALPAY

et d. 100.0

16)

80.0

Q Q 2 c 5

60.0

a 8 % 3

40.0

4 number of elements, K

20.0

l/(number of cells)

Fig. 4. Cyclic equilibrium 0s purity (0 0) and CPU time calculations (m-0) for the CIS discretisation method. Calculations are shown for the ILE model, and for the design conditions listed in Table l(a).

be solved, and the small and fixed At employed (cf. the variable step length feature of D02EBF).

5

Fig. 3. Convergence and CPU time analyses for the OCFE discretisation method: (a) absolute convergence error;(b) CPU time requirements. Calculations are shown for the ILE ( 0 ) and LDF (0) models, and for the design conditions listed in Table l(a). In each set of results, the three boxes refer to J = 3, 4 and 5, from left to right, as indicated.

for a convergence error of approximately 0.1, and for the ILE model, the OC method (with J = 20) requires about 25 s cycle- 1 CPU time, whereas the OCFE method (with J’ = K = 4) about 8 s cycle-‘. Furthermore, for the LDF model, and for the range of investigation of J, the OC method does not achieve a convergence error less 0.1, whereas such an error can be readily achieved with the OCFE method. DOCFE: As a test case, calculations using the DOCFE method were carried out for the ILE model, and for J’ = I’ = K = 4. The time increment, At was then set as 1 x lo- ’ s, smaller values of At had a negligible effect on the calculated solutions. At the cyclic equilibrium, and at the end of the pressurisation step, an oxygen purity of 68.97% was calculated (cf. 68.94% for OCFE), indicating computational consistency between the DOCFE and OCFE methods. The DOCFE method, however, required a CPU time of 37.2 s cycle- I, i.e. about 5 times that required by OCFE. For I’ = 2 and I’ = 3, even smaller At’s were required, resulting in only small improvements in the CPU time requirements. Such large CPU time requirements can be attributed to the intrinsic inefficiency of the algorithm COSNBF when compared to D02EBF, the relatively large number of equations to

CZS: Calculations using the CIS method were carried out for the TLE model. In Fig. 4, the cyclic equilibrium oxygen purity, at the end of the pressurisation step, and CPU time requirements (s cycle- ‘). are shown as a function of the number of cells. Solution convergence for this method is relatively slow, and subsequently demanding on CPU time. As mentioned above, similar observations were made by Kowler and Kadlec (1972) and Doong and Yang (1988). To save on computational time, however, the latter workers extrapolated results to an infinite number of cells; any such extrapolation method is not obvious from the results shown in Fig. 4. In terms of computing efficiency, the above calculations illustrate the superiority of the OCFE method over that of the OC method. The DOCFE method was found to be computationally slow when compared to either single collocation method, but all collocation methods were superior to CTS discretisation. In all subsequent RPSA or RPSAR calculations, therefore, the OCFE method was employed. A discretisation grid of J’ = 4 and K = 5 was chosen, this, even for non-isothermal or reacting systems, yielding convergence errors less than 0.2. Such errors were considered compatible with those due to the experimental measurement of gas composition. In addition to convergence analysis, component material balance checks were carried out for each process step at the cyclic equilibrium. Such material balances accounted for the net adsorbate input or output to the process, and the net gas and solid phase accumulation. For the OCFE calculations with J’ = 4 and K = 5, component balances were met to within a 10% error for all calculations, and typically to within 2% error for the air separation calculations described above.

Simulation of rapid pressure swing adsorption and reaction Processes

4.2. Air separation studies: pressurisation of a semiinjinite bed Scott (1991) used a similarity transformation to solve the equations governing the pressurisation of a semi-infinite bed of adsorbent with a binary gas mixture. This was possible by introducing the assumptions of ILE for adsorption, an isothermal pro-

cess, linear and independent adsorption isotherms and the plug flow of gases with no axial dispersion. Pressure distribution within the bed was however accounted for using Darcy’s law. For such a process, gas compositions within the bed were shown to be functions of the transformation variable

where < and 0 are dimensionless ables, respectively, i.e.

space and time vari-

e=E

(43)

e=+!$ ”

(44)

0

and where the subscript 0 represents an arbitrary reference parameter. The method was specifically applied to air separation over zeolite 5A adsorbent, for which the formation of a concentration shock was indicated. In the present work, the RPSAR model was applied to the bed pressurisation process, and gas phase composition profiles compared with those calculated by Scott (1991). In order to do this, the design parameters and initial conditions listed in Table l(b) were chosen.

3181

A bed length of 4 m was found adequately to approximate a semi-infinite bed. The axial dispersion coefficient was defined either by the correlation of Langer et al. (1978) (i.e. D, - 1 x lo-’ m2 s-l), or as a fixed and small value of 1 x lo- 5 mz s-l. The latter led to gas flows which approximated plug flow. The bed was initially set at atmospheric pressure and equilibrated with air. The reference parameters P, and x0 were set as 1.0 bar and 4.0 m, respectively. Oxygen profiles, as a function of 0, are shown in Fig. 5(a) for D, - 1 x lo-’ m’s_’ and in Fig. 5(b) for D, = 1 x 10~5mzs~1, respectively. These are given for times 2.5,5.0 and 7.5 s after the onset of pressurisation. Also shown is the exact shock position calculated by Scott (1991). The following observations are noted: (i) the overlap of the calculated profiles at times 2.5, 5.0 and 7.5 s after the onset of pressurisation, indicating the correct propagation of the shock, (ii) the accurate prediction of the shock position when compared to the exact solution, (iii) improved agreement with the exact solution as axial dispersion within the bed is reduced, albeit at the expense of some numerical instability. In addition to the shock position, the shock profile was also found to be in excellent agreement with that calculated by Scott (1991) [not shown in Fig. 5(a) and

(WI. 4.3. Air separation studies: RPSA As for the bed pressurisation studies described above, RPSA air separation over a bed of zeolite 5A

Table 1. Summary of the design conditions

4 b-4 4 (rm)

J, (Nsme4) L (m) mix 106(mol mo, mN, mA mu mc

m2N-‘kg-‘):

PJ (bar) n,x 104(mols-1) Process step times (s) Pressurisation Depressurisation Delay T(K) Voidages: Cb EP Pb (kg m-? TP

0.05 200 5x 105 1.0

0.05 n/a 2 X 105 4.0

0.05 100 to 1000 ea. (91

1.43 3.08

1.72 5.16

1.43 3.08

n/a u/a 0.0 0.98 0.0

1.84 4.15

2.00 n/a

2.12 4.15

10.00 n/a

1.5 1.5 0.0 290.0 0.35 0.55 800.0 3.0

10.0 0.0 0.0 300.0 0.4 0.55 700.0 n/a

i.0‘

1.5 1.5 0.0 290.0 0.34 0.55 800.0 3.0

1 X 105 1.0

10.0 0.0 0.0 473.0 0.75 0.00 750.0 n/a

E. ALPAY et al.

3182

0.0

0.5

I

.o

1.5

2.0

2.5

3.0

8

ively [see, for example, Bird et al. (1960)]. At ambient conditions, these are very similar in magnitude (i.e. ‘), suggesting that both mechan- lx10~5m2s isms are important in controlling the intraparticle diffusion. The effective intraparticle diffusion coeffiby a resistorscient, D,, can then be approximated in-parallel analogy, such that

0.45

0.0

Transport mechanisms within the macropores constitute molecular and Knudsen diffusion, surface diffusion [see Doong and Yang (1987) and Kapoor and Yang (1991)], and hydrodynamic or convective, transport. Due to the negligible amount of adsorption onto the macroporous matrix of zeolites, surface diffusion can be neglected. Furthermore, pressure gradients across an individual particle are small, and so is the convective transport within the macropores. For molecular and Knudsen diffusion, diffusion coefficients can be determined using Chapman-Enskog kinetic theory and Knudsen’s diffusion law, respect-

0.5

1.0

1.5

2.0

2.5

3.0

0

Fig. 5. A comparison of OCFE and similarity transformation (Scott, 1991) solutions for the pressurisation of a bed of zeolite5A adsorbent with air. Calculations for the former are shown for the cases in which axial dispersion is described by (a) the Langer er al. (1978) correlation and (b) a fixed dispersion coefficient of 1 x 10e5 m’s_‘. Other design conditions are listedin Table 1 (b); the time after the onset of pressurisation is displayed.

adsorbent was studied. Such an adsorbent preferentially retains nitrogen, concentrating oxygen in the gas phase, and subsequently in the product stream. At low feed gas pressures (< 3 bar), the process is to a good approximation isothermal, and described by linear and independent adsorption isotherms. Diffusional limitations for adsorption may however be important. Due to the bidisperse structure of the zeolite pellets, i.e. zeolite crystals bound together in a macroporous matrix, it is necessary to consider both macropore and intracrystalline diffusion mechanisms for these, some discussion of which is now given. Wicke-Kallenbach measurements using zeolite 5A crystals (Ruthven, 1990) indicate that the intracrystalline diffusion coefficients for oxygen and nitrogen are of the order of 1 x IO- ’ ” m2 s- ’ at ambient temperatures. Assuming a crystal radius of 1 pm, which is typical for commercial zeolite 5A pellets, a characteristic time constant for intracrystalline diffusion is given as approximately 0.01 s, suggesting that this diffusion is fast enough to be neglected as a transport resistance. Such a conclusion is supported by Haq and Ruthven (19X6), whose study of adsorption and diffusion of oxygen and nitrogen in zeolite SA adsorbent indicated the dispersion of the chromatographic response to be controlled mainly by axial dispersion, with some contribution from macropore diffusion.

where gp is the intraparticle void fraction, zp is the particle tortuosity factor and D, and Dk are diffusion coefficients for molecular and Knudsen diffusion, respectively [see, for example, Yang (1987)]. For zeolite SA particles, cp - 0.55 and 2 < zp < 5 (say ~~ - 3). In this work, RPSA air separation calculations were compared to measurements carried out on a small-scale air separation unit, details of which will be given by Alpay et al. (1992b). In particular, the experiments concentrated on the effect of particle size on the separation performance of the unit. The separation performance was assessed in terms of the cyclic equilibrium product oxygen purity, the fractional recovery of feed oxygen as product, and the adsorbent productivity (i.e. molts of oxygen produced per unit mass of adsorbent per unit time). Process parameters are summarised in Table l(c). The model had no free parameters. In particular, (i) the viscous

pressure

loss term, I,,

was defmed by

eq. (9), (ii) adsorption isotherm gradients were derived from pure gas isotherm data supplied by the adsorbent manufacturer (Bayer GMBH), (iii) the LDF mass transfer coefficient was calculated from eqs (17), (18) and (4S), (iv) the axial dispersion coefficient was calculated from the correlation of Langer et nl. (1978). An example of measured and calculated values of the cyclic equilibrium product oxygen purity, as a function of particle size, is shown in Fig. 6. The following observations are noted: (i) an optimum particle size for maximum product oxygen purity was measured, this being accurately predicted by the LDF model, but not the ILE model, (ii) the expected agreement of the LDF and ILE calculations for beds containing very small particles,

Simulation

of rapid pressure

swing adsorption

and reaction

processes

assumed to contain pure inert carrier design parameters are summarised in tional features necessary in repeating of Chatsiriwech et al. (1992) include

3183 gas only. Other Table l(d); addithe calculations the following:

(i) a modification of the boundary condition given by eq. (33), so as to give a linear ramp in the feed gas pressure during the pressurisation step, i.e. a linear ramp from P, to P, in the time tf. (ii) the use of a small viscous dissipation term, J,, so as to approximate the assumption of no spatial

L”.”

0

200

600 800 400 pziKle diameter(nlicrons)

IrnM

Fig. 6. Measured and calculated values of the cyclic equilibrium aroduct 0, twritv for the RPSA air sewration nrocess. Calc&tions are Ghowk for both the ILE aAd LDF models, and for D, = 1 x lo-’ mz s-’ (- - -) and D, described by the Lange; et nl. correlation (A). Other design conditions are listed in Table I(c).

since diffusional limitations for small particles are negligible, (iii) the significant effect of axial dispersion on the cyclic equilibrium solutions. The results indicate that the separation capabilities of the RPSA air separation process for beds containing very large particles are constrained by intrapartitle diffusional limitations. For zeolite 5A adsorbent, a combination of molecular and Knudsen diffusion mechanisms adequately describes the rate controlling intraparticle diffusion. For beds containing very small particles, the gas flow resistance of the bed is high, resulting in ineffective pressure swing within the bed, and subsequently a loss in the separation performance. The effects of pressure drop and intrapartitle diffusional limitations thus result in an optimum particle size for maximum product oxygen purity.

pressure gradients within the bed, (iii) the use of a small and fixed axial dispersion coefficient of 1 x 10m5 mz s-l, so as to approach the assumption of plug flow. In Fig. 7(aj(c), calculated gas phase composition profiles are shown for components A, B and C correspondingly, these at 3 and 9 s after the onset of the first pressurisation step of the process. Excellent agreement with the solutions of Chatsiriwech et al. (1992) are indicated, any discontinuities in the composition profiles are however eliminated by the effects of axial dispersion. 5. CONCLUSlONS

AND DISCUSSION

4.4. Adsorption coupled with reaction Simulation studies have been carried out by Chatsiriwech et al. (1992) to evaluate the performance of a two-bed PSA-based reactor. The model assumptions of these workers include ILE for adsorption, the plug flow of gases with no axial dispersion, no spatial pressure gradients within the bed, dilute reactants and simple power law kinetics for the reaction rate. In addition, linear pressurisation or depressurisation rates during the pressure changing steps of the process were assumed. Such a process is described by a system of hyperbolic PDEs, suitable for solution by the method of characteristics.

A general model for the simulation of non-isothermal adsorption and reaction in a rapid pressure swing adsorption process has been developed. The method of lines, in whch spatial gradients were discretised using the method of orthogonal collocation on finite elements, was found adequately to accommodate the steep pressure and concentration gradients inherent to RPSA. In terms of computing efficiency, this discretisation method was found to be superior to the methods of non-finite element orthogonal collocation and cells-in-series, and the global method of double orthogonal collocation on finite elements. In tests of the model, bed pressurisation calculations were found to be in excellent agreement with the exact and asymptotic solutions of Scott (1991) and Chatsiriwcch et al. (1992), respectively. For RPSA air separation, novel experimental measurements, showing an optimum particle size for maximum product purity, were accurately described, with no free parameters. The calculations illustrate the importance of mass transfer limitations. Details of the experiments, and further theoretical analysis, will be given by Alpay et al. (1992). The potential advantage of coupling adsorption and reaction processes has been mentioned above. The results presented in Fig. 7(aHc) illustrate the

In this work, the solutions of Chatsiriwech et al. (1992) were compared to the general RPSAR model, when reduced to the simple case of a moving wave front. The reaction A= B + 3C was chosen as a test case, with k,, = 0.1444 s- ’ and krb = 0.0120 (m3 mol-‘)3 s. The feed gas to the bed was considered as a dilute mixture of A (i.e. 5% v/v) in an inert carrier; component B was set as the only adsorbing species. Initially, the bed was set at atmospheric pressure, and

computational accuracy of the RPSAR simulation program. For a two-bed PSA-based reactor, and for certain reaction schemes, subsequent cyclic steadystate calculations have indicated possible improvements in equilibrium conversions, details of which are to be published at a later date. In future work, a comparison of PSA- and RPSA-based reactors will be carried out. This will concentrate on the dehydrogem&ion-type reactions mentioned earlier.

E.

3184

ALPAY

0.06 C (a)

J

et al.

adsorption and reaction processes, D. Chatsiriwech for providing the data mentioned above. and the SERC for financial support. NOTATION

A CO C Cm C”,

0.01

CS

0.00 U.0

0.2

0.1

0.6

0.8

Ic-’

1.0

x

(b) -

0.2

0.0

0.4

0.6

0.8

1.0

x

4 4 D, 0,

H, ff, I’ J

0.10 . 0.08

9.0

Cc)

9

’

J’

-

0.06 -

JC Jk

0.04 -

J, k

0.0

0.2

0.4

0.6

0.8

1.0

x

of OCFE (symbols) and characteristic [lines; Chatsiriwech et al. (1992)] solutions for the pressurisation of a bed with a dilute reactant A. These are shown for Fig. 7. A comparison

the dilute reaction A-.- B + 3C, the design conditions listed in Table 1(d), and for (a) component A, (b) component b and

(c) component C; the time after the onset of pressurisation is displayed.

k, k,,, b kx K 1 L m n ad

Computational savings on non-isothermal calculations were obtained by imposing the assumption of instantaneous thermal equilibrium between the gas and solid phases. Preliminary calculations and measurements by Alpay (1992) suggest that such an assumption is only applicable for large cycle times, i.e. %=0.5 s. Future experimental and theoretical studies of non-isothermal RPSA air separation (e.g. wide-bed studies), will ascertain the conditions for which the thermal equilibrium assumption is applicable. Acknowledgments-E. A. would like to thank N. F. Kirkby and L. S. Kershenbaum

for useful discussions on cyclic

bed cross-sectional area, mz valve sizing coefficient gas concentration, mol m- 3 gas heat capacity at constant pressure or volume, J mol ’ K 1 adsorbent (solid) heat capacity, J kg- ’

NC

N SC,, NR P 4 4*

column internal diameter, m particle diameter, m effective gas diffusion coefficient, mz sd- ’ modified effective gas diffusion coefficient [eq. (18)], m2 s-l Knudsen diffusion coefficient, mz s ’ molecular diffusion coefficient, rn’ s - ’ axial dispersion coefficient based on the area available for Bow, m2 s-l overall heat transfer coefficient, W m2 K-1 heat of adsorption, J mol- I heat of reaction, J mol- ’ number of collocation points (time domain) number of collocation points (space domain) number of collocation points (space domain) per element thermal conduction flux, W me2 kinetic pressure loss term for gas flow in a packed bed, N sz m 5 viscous pressure loss term for gas flow in a packed bed, N s m e-4 LDF mass transfer coefficient, s-l gas thermal conductivity, W m- ’ K-’ reaction rate constants, see main text axial bed thermal conductivity, W m- 1 K-’ number of finite elements Langmuir constant, N- ’ m* bed length, m adsorption isotherm gradient, m2 mol N-‘kg-’ gas flow rate, mol s-l product delivery rate, mol s - ’ number of components number of cells number of rate equations total pressure, N m- ’ adsorbed phase concentration, mol kg- i equilibrium adsorbed phase concentration, mol kg-’ saturation value of q* reaction

rate, mol m - ; S‘Y

particle radius, m gas constant, J mol-’ K-’ stoichiometric coefficient time, s

kg- ’

Simulation of rapid pressure swing adsorption and reaction processes

tS

cycle half-time,

T u

temperature, K superficial gas velocity, m s-l space coordinate, m reaction advancement term, mol m gas phase mole fraction

; Y

Greek

s

3s _ 1

letters constants interparticle (or bed) void fraction intraparticle void fraction total void fraction dimensionless time variable [eq. (44)] dimensionless cycle half-time [eq. (IY)] transformation variable for space and time [eq. (42)] particle shape factor gas viscosity, N s mm2 normalised stoichiometric coefficient dimensionless space coordinate [eq. (4311 bed bulk density, kg m- 3 gas density (mole basis), molmm3 gas density (mass basis), kg m -’ particle tortuosity factor shear stress for the viscous dissipation of energy, J m - 3

reference parameter ambient conditions component identifiers base value component i oxygen and nitrogen product upstream and downstream Abbreviations AE CIS CPU DOCFE ILE LDF oc OCFE ODE PDE PSA RPSA RPSAR

algebraic equations cells-in-series computer processing

unit

double orthogonal collocation on finite elements instantaneous local equilibrium linear driving force orthogonal collocation orthogonal collocation on finite elements ordinary differential equation partial differential equation pressure swing adsorption rapid pressure swing adsorption rapid pressure swing adsorption and reaction

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3185

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