On the formulation of linear driving force approximations for adsorption and desorption of multicomponent gaseous mixtures in sorbent particles

Separation and Purification Technology 24 (2001) 343– 367 www.elsevier.com/locate/seppur

On the formulation of linear driving force approximations for adsorption and desorption of multicomponent gaseous mixtures in sorbent particles Atanas Serbezov a,1, Stratis V. Sotirchos a,b,* b

a Department of Chemical Engineering, Uni6ersity of Rochester, Rochester, NY 14627, USA Institute of Chemical Engineering and High Temperature Processes, Foundation of Research and Technology Hellas, P.O. Box 1414, 26500 Patras, Greece

Received 24 January 2001; received in revised form 28 March 2001; accepted 29 March 2001

Abstract A generalized linear driving force (LDF) approximation for multicomponent adsorption-based separations is formulated in this study. It is based on the three-parameter dusty-gas model and accounts for bulk diffusion, Knudsen diffusion, Knudsen flow and viscous flow in the sorbent particle. No restrictions are imposed on the type of adsorption isotherm. The generalized LDF approximation equations are derived using global or piecewise continuous polynomial approximations for the intraparticle partial pressure profiles and applying the subdomain method of weighted residuals to the mass balance equations. Similar expressions are derived using the Fickian diffusion model and the Fickian diffusion/convection model. The validity of the various approximations is examined by comparing their predictions to the exact solution of the corresponding intraparticle models. Since the generalized LDF approximation permits the inclusion of convective coupling among the various species, it can satisfactorily predict the overshoot in the uptake of the less adsorbable/faster moving species in a multicomponent mixture. This phenomenon cannot be described by existing LDF approximations, since their derivation is based on transport and adsorption models that do not account for the coupling of the fluxes of the components of the multicomponent mixture. For short cycle times, the accuracy of the approximation increases with the degree of the polynomial. However, a generalized LDF approach, based on a fourth degree global polynomial approximation, may suffice in most applications. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Linear driving force approximation; Pressure swing adsorption; Dusty-gas model; Multicomponent mass transport; Method of weighted residuals

* Corresponding author. E-mail address: [email protected] (S.V. Sotirchos). 1 Present address: Department of Chemical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN 47083, USA. 1383-5866/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S1383-5866(01)00146-0

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Nomenclature Be B Cij DK DS Di, j Hi J K Lij Ni PT pi p qi r rp R1 Rg Rp t t1/2 T yi

effective permeability of the adsorbent particle (m2) matrix defined in Eq. (19) and Eq. (20) coefficients defined in Eq. (24) Knudsen transport coefficient (m2/s) solid diffusion coefficient defined in Eq. (39) (m2/s) binary diffusion coefficient for gases i and j (m2/s) adsorption equilibrium constant in Eq. (9) Jacobian matrix of the adsorption isotherm LDF matrix defined in Table 2 LDF coefficients defined in Eq. (22) molar flux of species i relative to stationary coordinates (kmol/m2·s) total pressure (Pa) partial pressure of species i (Pa) vector of the partial pressures (Pa) equilibrium solid phase concentration of species i (kmol/m3) radial coordinate in the particle (m) pore radius (m) radius of inner subdomain ideal gas law constant (J/mol·K) radius of the adsorbent particles (m) time variable (s) half cycle time (s) temperature (K) mole fraction of species i

Greek letters ratio R1/Rp h k parameter used in Eq. (36) (s − 1) mp porosity of the adsorbent particles p tortuosity factor q dimensionless time u parameters used in Eq. (41) and Eq. (42) v viscosity of the gaseous mixture (kg/m·s) Superscripts e ‚ – *

effective quantities approximated quantities averaged quantity reference quantity in Table 1

A. Serbezo6, S.V. Sotirchos / Separation/Purification Technology 24 (2001) 343–367

1. Introduction The mathematical modeling of many adsorption-based processes involving porous sorbents usually leads to a complex system of partial differential equations with two sets of spatial coordinates: one space variable (or more than one for multidimensional models) for the adsorbent bed and another for the adsorbent particle. The mathematical model can be significantly simplified if the particle uptake rate (which is obtained by a differential mass balance in the particle) can be approximated and replaced by an expression that does not involve the spatial coordinates in the particle. The most widely used uptake rate approximation in the literature is the linear driving force (LDF) approximation, which assumes that the uptake rate of a species in the particle or pellet of the porous sorbent is proportional to the difference between the concentration (gas phase or solid phase) of that species at the outer surface of the particle and its average concentration in the interior of the particle or pellet. The LDF approximation was first proposed by Glueckauf and Coats [1]. Based on a comparison with experimental data, the dimensionless value of the LDF proportionality coefficient was set equal to 14 for pellets of spherical geometry, but on the basis of the asymptotic expansion of the general solution of the diffusion equation, Glueckauf [2] showed that a value of 15 should be applied for dimensionless times \0.1. The solution of the diffusion equation was used by various investigators to examine the adequacy of the LDF approximation for pellets exposed to environments characterized by cyclically changing conditions. For rapid cyclic adsorption or desorption with dimensionless cycle times B 0.1, Nakao and Suzuki [3] compared the LDF approximation with the numerical solution of the diffusion equation and proposed a graphical correlation from which the LDF coefficient can be determined as a function of the dimensionless cycle time. Analytical expressions for this dependence were later derived by Alpay and Scott [4] using Fourier series and by Carta [5] using Laplace transforms. In most studies dealing with the adequacy of the LDF approximation for cyclic process, square

345

wave concentration perturbations at the particle surface have been employed, but Kim [6] and Sheng and Costa [7] dealt with the development of LDF approximations for perturbations of different form (e.g. sinusoidal). An alternative way to derive LDF expressions is to assume a certain form of the intraparticle concentration profile and then solve numerically the diffusion equation to obtain a relation between the uptake rate and the concentration difference (driving force). Methods of weighted residuals [8], such as collocation, the subdomain method and the Galerkin method, are usually used to discretize the diffusion equation in the spatial coordinate using the assumed concentration profile. Applying such an approach, Liaw et al. [9] showed that the value of 15 for the LDF coefficient corresponds to the assumption of a parabolic profile. Do and Rice [10] found that the parabolic profile assumption gave large errors for short cycle times and proposed a fourth-degree approximation. Since such an approximation involved two parameters, an extra variable (the concentration at the center of the particle) was introduced and as a result, the LDF approximation involved the simultaneous solution of two first-order ordinary differential equations. Yao and Tien [11–13] and Kikkinides and Yang [14] also showed that by using higher order polynomials in the profile approximation, more accurate uptake rate expressions might be obtained. Because of the great simplification that the use of the LDF approximation introduces into the combined bed-pellet model of an adsorptionbased separation process, there is still a lot of activity in the literature on the development of LDF approximations using the two approaches outlined above and the investigation of the performance of these approximations in different situations of transient adsorption and desorption. Recent studies have addressed, among other subjects, the effects of the type of the approximation for the intraparticle concentration profile, the effects of surface diffusion, the effects of external mass transport resistance and the effects of the mode of concentration perturbation [13,15–19]. However, in all of these studies and those mentioned above, LDF approximations are derived

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for adsorption of a single component in particle or pellet of a porous sorbent. The derivation is based on the use of the Fick’s first law of diffusion for a single species as the flux model for mass transport in the interior of the porous medium, and thus, when applied to multicomponent systems, they cannot account for multicomponent transport effects. Using the dusty-gas model to describe the coupling and the interactions of the fluxes of different species during their transport in the porous particles, we have shown in past studies [20,21] that even in binary systems these interactions lead to pronounced maxima (during adsorption) and minima (during desorption) in the uptake versus time curve of the species with the smaller effective transport resistance (fast moving or weakly adsorbing species). These results were in agreement with experimental observations reported in the literature [22,23]. To describe the multicomponent interactions, Marutovski and Bu¨ low [22] proposed to formulate an LDF approximation based on the generalized Fick’s law. This approximation involved a matrix of LDF coefficients, with the off-diagonal terms reflecting the mutual influence of the mixture components on their simultaneous intraparticle mass transfer. It correctly predicted the maxima and minima in the uptakes, but it required experimental data in order to determine the LDF coefficients. A semi-empirical LDF approximation based on the dusty-gas model was formulated by Mendes et al. [24], but it was applied to binary mixtures with linear or Langmuir adsorption isotherms without addressing the issue of the multicomponent effects. A class of generalized LDF approximations applicable to multicomponent systems is formulated in this study. Our approach is based on the subdomain method of weighted residuals. No restrictions are imposed on the number of the components, the mass transport model, the adsorption isotherm, the type of perturbation or the degree of the polynomial approximation. Specific LDF expressions are derived for arbitrary multicomponent mixtures for cases in which the intraparticle mass transport is described by the Fickian diffusion model, the

Fickian diffusion/convection model or the dustygas model, the adsorption isotherm is linear or nonlinear and the intraparticle partial pressure profiles are represented by parabolic or quartic (global or piecewise continuous) polynomials. The numerical method we employ can be used with any piecewise continuous polynomial representation of the intraparticle partial pressure profile and therefore, it can be used to obtain the exact numerical solution of the intraparticle multicomponent adsorption/desorption problem. This method has been employed successfully by Serbezov and Sotirchos [25] to discretize the pellet model in a rigorous particle-bed model for multicomponent adsorption/desorption in packed beds under cyclic conditions, similar to those encountered in PSA applications. In general, the proposed generalized LDF approximations have the form of a system of ordinary differential equations for the vector of the average concentrations (loadings) of the species in the pellet and the vectors of some other average concentrations (auxiliary variables). The right-hand side of the system of differential equations has, in vector form, the compact form of the product of an LDF matrix and a vector of driving forces (differences) involving the average concentrations in the pellet (overall and auxiliary) and the concentrations at the external surface. In its simplest form (i.e. for parabolic concentration profiles), the generalized LDF approximation involves only the vector of the average concentrations in the pellet and has a form similar to that of the approximation proposed by Marutovski and Bu¨ low [22]. Results are presented for total pressure (at the external surface) change cycles of a broad range of periods and the effects of the mass transport model and the degree of the polynomial approximation on the generalized LDF approximation are investigated by comparing the exact solution of the intraparticle mass balance partial differential equations [20] with the solution of the generalized LDF approximation. The results show that the generalized LDF approximation can reproduce satisfactorily, both qualitatively and quantitatively, multicomponent transport effects in the porous sorbent.

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2. Theory

2.2. Generalized LDF approximation

2.1. Particle model

We apply the method of weighted residuals to the mass balance equation in the particle. We use the symbol ‚ to denote the approximated quantities and form the residual

Consider a spherical particle of an isotropic porous adsorbent immersed in a multicomponent gaseous mixture of constant composition subjected to a cyclic change of the total pressure. The process is isothermal and the operating conditions ensure the validity of the ideal-gas law. The adsorbed and fluid species within the pores are assumed to be locally in equilibrium with each other at all times. The mass balance equation for each species in the multicomponent mixture is mp (pi 1 ( 2 (qi + (r Ni ) + (1 −mp) =0 RgT (t r 2 (r (t

(1)

The boundary conditions are: At r =0 At r =Rp

(Ni )r = 0 =0

(2)

(pi )r = R p =(yi )r = R pPT

(yi )r = R p = const

Xi (r)=

mp is the porosity of the adsorbent particle, pi is the partial pressure of species i, T is the temperature, Rg is the ideal gas law constant, Ni is the molar flux of species i in radial direction relative to stationary coordinates, qi is the equilibrium solid phase concentration of species i, r is the radial coordinate in the particle, Rp is the radius of the particle, yi is the mole fraction of species i and PT is the total pressure. The first boundary condition (Eq. (2)) stems from considerations of symmetry. The second boundary condition (Eq. (3)) arises from the fact that the particle, being much smaller than its enclosing environment, cannot alter the composition of the latter. Thus, at the outer surface of the adsorbent particle, the mole fractions of the components remain constant while their partial pressures change in accordance with the total pressure. The initial condition for each half-cycle is the final condition of the previous one. At the onset of the process, the bulk phase inside the particle has uniform composition, the same as the surrounding mixture.

(4)

We choose to work with the subdomain method of weighted residuals which requires that the average value of the residual over a number of spherical regions in the particle be zero. The specific number of these spherical regions is equal to the number of the coefficients encountered in the approximation of each pi minus the number of the boundary conditions on each variable. For an arbitrary region we have

&

R

r 2Xi (r) dr

0

= (3)

mp (pˆi 1 ( 2 (qˆi + 2 (r N. i )+ (1− mp) RgT (t r (r (t

mp R gT

&

R

r2

0

+ (1− mp)

(pˆi dr+ (t

&

R

0

&

R

0

( 2 (r N. i ) dr (r

(qˆ r 2 i dr =0 (t

(5)

R is an arbitrary position in radial direction in the range Rp ] R\0. Eq. (5) can be rewritten in terms of quantities averaged over a sphere with radius R as mp ((p¯i )R 3 + 3 RgT (t R

&

R

0

( 2 ((q¯i )R (r N. i ) dr + (1−mp) (r (t

=0

(6)

The averaged quantities are denoted using the overbar symbol – and are calculated from the expression ’R =

1 VR

&

V

’ dV =

3 R3

&

R

’r 2 dr

(7)

0

VR is the volume of the sphere with radius R. Applying the boundary conditions to the second term in Eq. (6), we obtain the following result: mp d(p¯i )R 3 d(q¯i )R + (N. i )R + (1− mp) =0 RgT dt R dt

(8)

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348

(N. i )R denotes the molar flux of species i at r = R. In order to solve Eq. (8) and obtain the averaged partial pressures of the individual components, models for the intraparticle mass transport and the adsorption isotherms are needed. Their purpose is to replace the terms (N. i )R and (d(q¯ i )R )/dt by more specific mathematical expressions that will enable their evaluation under a specific set of variables. For a linear adsorption isotherm, i.e. q¯ i =

Hi p¯ i RgT

(9)

Eq. (8) simplifies to d(p¯ i )R 3 RgT =− (N. i )R dt R [mp +(1 −mp)Hi ]

(10)

For a nonlinear the adsorption isotherm, we can use a numerical quadrature formula to obtain an approximation for (d(q¯ i )R )/dt. An easier, but less accurate approach is to assume that the average fluid phase concentration and the average solid phase concentration obey the same functional relationship as the local quantities, the form of which is known from the adsorption isotherm. Using the chain rule of differentiation, Eq. (8) can be written as n mp d(p¯ i )R 3 ((q¯ i )R d(p¯ j )R + (N. i )R +(1 −mp) % RgT dt R ¯ j )R dt j = 1 ((p

=0

D eF,i (pi RgT (r

n

mp d(p¯ )R 3 I +(1− mp)J + (N. )R =0 R gT dt R



n

d(p¯ )R 3 mp =− I +(1 − mp)J dt R RgT

(12)

−1

(N. )R

(13)

J is the Jacobian matrix of the adsorption isotherm. There are several models for the representation of the intraparticle mass transport that are commonly used in literature. A comprehensive discussion of these models is provided by Serbezov and Sotirchos [20,21]. The simplest mass transport model is the well-known Fickian diffusion model, given by

(14)

D eF,i is the effective Fickian diffusion coefficient. When the intraparticle viscous flow is important, the Fickian model can be complemented by D’Arcy’s law. The resulting model is known as the Fickian diffusion/convection model: Ni = −



(pi B epi (PT 1 D eF,i + R gT (r v (r



(15)

B e is the effective permeability of the porous medium and v is the viscosity of the mixture. In compact vector–matrix form, the Fickian diffusion/convection model can be rewritten as N= −



n

1 Be (p diag(D eF,i)+ pe T RgT v (r

(16)

where diag(D eF,i)is a diagonal matrix with elements equal to D eF,i, p is the vector of partial pressures and e T is the transpose of an n-dimensional vector that has all elements equal to unity. A mass transport model that is recognized to be more rigorous than those mentioned above is the dusty-gas model, [26–28]:



B epi (PT (pi + (r vD eK,i (r







yj Ni − yi Nj N + ei e Di, j D K,i j"i

= − RgT %

(11)

or in compact vector–matrix form



Ni = −

(17)

D eK,i is the effective Knudsen diffusion coefficient of component i and Dei, j is the effective binary diffusion coefficient of component i in component j. The formulae for the calculation of the parameters in the dusty-gas model are listed in Table 1. The dusty-gas model is implicit with respect to the fluxes when used in the form given by Eq. (17). However, it can be cast in a vector–matrix form that is explicit with respect to the fluxes as N= −



n

1 Be (p B − 1 + pe T RgT v (r

(18)

where B is an [n× n] matrix whose elements are: yj 1 + e e D K, j j " i Di, j

Bi,i = %

(19)

A. Serbezo6, S.V. Sotirchos / Separation/Purification Technology 24 (2001) 343–367

Bi, j = − j"i

yj Dei, j

(20)

Using Eq. (14), Eq. (16) or Eq. (18) for N in Eq. (13) gives the following general equation for the variation of the average partial pressures over the [0, R] domain:

 

d(p¯ )R (p =(K)R dt (r

(21)

R

The forms of matrix K for the various models that we have developed above are given in Table 2. To develop an LDF approximation starting from Eq. (21), we replace the driving force for transport into the [0, R] domain, ((p/(r)R, with a linear combination of the differences of the partial pressures at the external surface from the average partial pressures in the subdomains employed in the construction of the approximation. Specifically, for subdomain [0, Ri ] we set

  (p (r

=

Ri

1 % L [(p)R p −(p¯ )Rj ] Rp j ij

(22)

349

For N subdomains, [0, Ri ], i= 1,…, N, the LDF approximation that is thus constructed consists of N sets of ordinary differential equations of the general form d(p¯ )Ri 1 N % Lij [(p)R p − (p¯ )Rj ] = (K)Ri dt Rp j = 1

(23)

with RN = Rp. Since the evaluation of the quantities in the matrices (K)Ri that result from the transport flux model is done using the partial pressures at Ri, Eq. (23) are complemented by N− 1 sets of equations giving the concentrations at Ri, i= 1,…, N− 1, as functions of (p)R p and (p¯ )Rj. These equations have the form N

(p)Ri = (p)R + % Cij [(p¯ )Rj − (p)R p] p

(24)

j=1

Therefore, an LDF approximation based on N subdomains is completely described by Eq. (23) and Eq. (24) given the values of two set of dimensionless coefficients (N 2, Lij and N(N− 1) Cij ).

Table 1 Formulae used in the computation of the effective transport coefficients Effective transport coefficients Coefficient Dei, j

D eK,i

B ep v

Equation Dei, j =D* i, j

   P*T PT

T T*

Reference

3/2

S1

T *3(Mi+Mj )/(2Mi Mj ) D*i, j = 2.628×10−3 2 P*| T i, jdDi, j mp S1 = pB T 1/2 e D K,i = D*K,i S2 T*

   

T* DK,i * =97r*p M i m p rp S2 = pK r*p v =ni= 1 1

1/2

Present (Ref. [35])

yivi nj= 1yjbi, j

Semiempirical formula of Wilke (Bird et al., Ref. [34])

    n

Mi 1+ Mj

8 2 mpr p Be= pV 8 bi, j =

Chapman-Enskog equation (Bird et al., Ref. [34])

−1/2

1+

vi vj

1/2

Mj Mi

1/4 2

350

Mass transport model

Adsorption isotherm Linear adsorption isotherm

Fickian diffusion model 1 (p N= − diag(D eF,i) RgT (r



n

Fickian diffusion/con6ection model 1 (p Be diag(D eF,i)+ pe T R gT v (r

N=−



n

Dusty-gas model (p 1 Be N= − B−1+ pe T (r RgT v



Nonlinear adsorption isotherm

n

K=

1 3 diag diag(D eF,i) mp+(1−mp)Hi Rp

K=

3 1 diag R mp+(1−mp)Hi

n

K=

1 3 diag mp+(1−mp)Hi R

n

 

3 K = [mpI+RgT(1−mp)J]−1diag(D eF,i) R Be T pe v

diag(D eF,i)+ Be

B−1+

v

pe T

n

n

K=



3 Be [mpI+RgT(1−mp)J]−1 diag(D eF,i)+ pe T R v



n

3 Be K = [mpI+RgT(1−mp)J]−1 B-1+ pe T R v

n

A. Serbezo6, S.V. Sotirchos / Separation/Purification Technology 24 (2001) 343–367

Table 2 Generalized LDF matrices for different mass transport models and adsorption isotherms

Global quartic approximation (Eq. (31)) h= R1/Rp l21 l22 l11 l12 c11 c12

0.3 −3.22 12.616 2.111 −3.492 0.868 0.222

0.6 −6.51 17.76 4.844 −6.494 0.25 1.11

Piecewise continuous quadratic-quartic approximation (Eq. (32)) 0.8 −20.576 36.687 4.28 −2.591 −1.37 3.01

0.3 −3.235 12.652 7.071 −11.724 0.873 0.211

0.6 −7.01 18.74 9.145 −12.924 0.342 0.931

0.8 −25.271 43.917 15.02 −18.131 −0.923 2.321

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Table 3 Values of Lij and Cij in Eq. (29) and Eq. (30) for global quartic and piecewise continuous quadratic-quartic approximations of the partial pressure profiles

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352

The values of these coefficients depend on the approximation used for the partial pressure profiles and the boundaries of the spherical subdomains, i.e. the Ri /Rp. The approximation of each profile should involve N parameters (one for each subdomain) in addition to two parameters that are fixed by the boundary condition at the surface (that is, the value of the partial pressure at the external surface) and the boundary condition at the center. In the following sections, we will develop approximation for global parabolic, global quartic and piecewise continuous quadratic (parabolic)-quartic approximations of the partial pressure profiles. Fig. 1. Domains in the adsorbent particle.

2.3. LDF equations for global parabolic approximation For a parabolic profile we have pi = a0,i + a2,i r 2

(25)

This form of partial pressure profile satisfies the boundary condition at the center of the particle — since it is an even function or r. If we use (pi )R p and (p¯ i )R p as parameters in the approximation for pi instead of a0,i and a2,i, it can be shown that pi is expressed as





 

5 3 5 r2 pi = (pi )R p − + +(p¯ i )R p 2 2 R 2p 2

1−

r2 R 2p



(26)

 

Then, (pi (r

Rp

=

5 [(p ) − (p¯ i )R p] Rp i R p

(27)

and introducing this expression in Eq. (23), we find that the LDF approximation for parabolic partial pressure profiles has the form d(p¯ )R p 5 [(p)R p −(p¯ )R p] = (K)R p Rp dt

(28)

(K)R p is the LDF matrix evaluated at the outer surface of the particle (see Table 2). From Table 2 it can be seen that if we use the simple Fickian diffusion model with linear adsorption isotherm, the LDF matrix becomes diagonal and the Glueckauf LDF approximation is obtained. The LDF equations for the different species are independent of each other and can be solved

analytically. For all other cases, the LDF equations have to be solved numerically as a system of ordinary differential equations. However, this should not be perceived as a drawback of the generalized linear driving force approach because the LDF equations are always used as a part of a complex mathematical model (e.g. in conjunction with the partial differential equations for the adsorbent bed), the solution of which is obtained numerically. This means that, in practice, the LDF equations are always solved numerically regardless of whether an analytical solution is available or not. Therefore, by using rigorous mass transport models one can improve dramatically the performance of the linear driving force approximation without making it computationally more involved than the simple Glueckauf LDF approximation.

2.4. Higher order LDF approximations The LDF approximation that was developed in the preceding section was based on the simplest possible representation of the partial pressure profiles in the interior of the particles (pellets). Approximations with much better accuracy can be constructed by using higher order approximations of the partial pressure profiles. The next higher order family of LDF approximations from that based on parabolic partial pressure profiles would involve three parameters and therefore, two subdomains should be employed. We take the subdomains to be spheres with radii Rp and R1, as shown

A. Serbezo6, S.V. Sotirchos / Separation/Purification Technology 24 (2001) 343–367

in Fig. 1. The LDF approximation is given by Eq. (23) and Eq. (24) for N =2. Specifically, we have: Æd(p¯ )R p Ç Ã dt à à à d(p¯ )R 1 Ã Ã È É dt 1 (K)R p 0 = Rp 0 (K)R 1





n

L11((p)R p −(p¯ )R 1) + L12((p)R p −(p¯ )R p) L21((p)R p −(p¯ )R 1) + L22((p)R p −(p¯ )R p)

n

(29)

(pi )R 1 =(pi )R p +C11((pi )R p −(p¯ i )R 1) +C12((pi )R p −(p¯ i )R p)

(31)

and (ii) another based on a piecewise continuous approximation of the partial pressure profiles, with a quadratic piece in the [0, R1] region and a quartic piece in the [R1, Rp] region, viz. pi =

!

a4,i +a6,i r 2 +a8,i r 4 R1 5r 5 Rp a0,i +a2,i r 2 0 5r 5 R1

equations to express aj,i as functions of (p¯ i )R p, (p¯ i )R 1 and (pi )R p and then casting the equations for ((p/(r)R 1, ((p/(r)R p and (p¯ )R 1 in the form of Eq. (22) and Eq. (24). The two-piece, piecewise continuous polynomial approximation involves five parameters, but the requirements of continuity of the partial pressures and their derivatives at the r=R1 breakpoint yields two equations that are used to eliminate two of these parameters. Appendix A describes the procedure used to determine the elements of matrices L and C for a global polynomial approximation of degree N in r 2.

(30)

LDF approximations for two three-parameter approximations of the partial pressure profiles were developed in this study: (i) one based on a global quartic approximation of the partial pressure profiles, viz. pi = a0,i + a2,i r 2 +a4,i r 4

353

(32)

The values of Lij and Cij for selected values of R1/Rp (h) for the LDF approximations that are based on the above representations are given in Table 3. The data of Table 3 show that the values of the LDF coefficients depend rather strongly, not only on the form of the polynomial approximation of the partial pressure profiles, but also on the value of R1/Rp. Serbezov [29] has derived expressions giving a set of coefficients that are equivalent Lij and Cij as functions of R1/Rp for the global quartic and piecewise continuous quadratic-quartic approximations of Eq. (29) and Eq. (30). However, the coefficients of Table 3 can be computed in a straightforward manner by using Eq. (29) or Eq. (30) to derive equations giving (p¯ i )R p, (p¯ i )R 1 and (pi )R p in terms of aj,i (starting from the definitions of these quantities), inverting this set of linear

3. Results and discussion In order to investigate the applicability of the proposed generalized LDF approximation to multicomponent systems, we apply the generalized LDF equations to obtain the dynamic response of a single sorbent particle immersed in a binary mixture of oxygen and nitrogen subjected to cyclic pressurization and depressurization and compare the results with the solution of a rigorous intraparticle distributed-parameter model. The sorbent particle is assumed to be spherical with radius Rp = 10 − 3 m, uniform pore size rp = 10 − 7 m and porosity mp = 0.6. The composition of the surrounding mixture stays constant all the time and the partial pressures of the components at the outer surface of the particle follow the changes of the total pressure. The molar ratio between oxygen and nitrogen in the mixture is taken to be 21:79, and the operation is assumed to occur at m300 K. We have used three different patterns of total pressure cycling, in all of which we have alternated between pressurization and depressurization with equal times. 1. Stepwise pressurization and stepwise depressurization. Pressurization: PT = PT,high,

05 t5 t1/2

(33)

Depressurization: PT = PT,low,

05 t5t1/2

(34)

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354

where t1/2 denotes the half-cycle time. 2. Linear pressurization and exponential depressurization. Pressurization: PT =PT,low +

PT,high −PT,low t, t1/2

0 5 t 5t1/2

(35)

Depressurization: PT =PT,low +(PT,high −PT,low)e − kt,

0 5t 5 t1/2 (36)

3. Sinusoidal total pressure change. PT =PT,low + +

PT,high −PT,low 2

 

n

PT,high − PT,low 2y t sin t − cycle 2 tcycle 4

(37)

The stepwise pressurization and depressurization (Eq. (33) and Eq. (34)) are often used to test and validate LDF approximations, but in real PSA processes the adsorbent particles are seldom exposed to such conditions. The stepwise pressurization and depressurization are the harshest boundary conditions possible and the argument is that if the LDF approximation is valid under these conditions, it will be applicable to all other cases. However, if the LDF approximation performs poorly during stepwise pressurization and depressurization, there is not much to be said about the performance of this approximation under typical PSA conditions. Typically in PSA operations, the pressurization is carried out by compressors at constant molar flow rate. It can be shown that if the mass transport limitations are negligible, the total pressure in the adsorbent bed increases linearly with time. For depressurization, experimental results show that the total pressure in the column decreases exponentially [30]. Therefore, the cycle with linear pressurization and exponential depressurization (Eq. (35) and Eq. (36)) is a more realistic test case than stepwise pressurization and stepwise depressurization.

In some PSA processes, such as rapid PSA, the pressurization and the depressurization are carried out by connecting the adsorbent bed to surge tanks kept at constant pressure. However, the step changes in the total pressure at the ends of the column are smeared out because of the mass transport limitations in the packed bed. Therefore, a sinusoidal change in the total pressure (Eq. (37)) at the outer surface of the adsorbent particles is more representative of the conditions inside the adsorbent bed during rapid PSA. We assume that the adsorption process takes place at the surface of the pore walls and that the adsorption isotherms are linear (unless specified otherwise) with dimensionless adsorption coefficients of 15 for nitrogen and five for oxygen. By choosing linear adsorption isotherms in our discussion, we eliminate the coupling due to competitive adsorption. In this way, we can study the multicomponent transport effects by themselves because the model equations are coupled only through the multicomponent mass transport models. However, the generalized LDF approximation is not limited to linear adsorption isotherms, and in Table 2 we have formulated the appropriate equations for the nonlinear case. A brief discussion of the application of the method to nonlinear isotherms is given at the end of this section. The formulae used for the calculation of the transport coefficients are summarized in Table 1. In order to determine the structural parameters of the porous medium, the latter is approximated by an array of long cylindrical capillaries randomly oriented in the three-dimensional space. The tortuosity factors are obtained as orientationally averaged quantities with the assumption that the mass transport in the overlap regions among different capillaries affects negligibly the overall mass transport. Tomadakis and Sotirchos [31] showed that this assumption cannot be justified at intermediate and high porosities. The tortuosity factors which they report for porosity mp = 0.6 are slightly smaller than the value of 3 adopted here, but this has an insignificant effect on the results of the present study. A comprehensive discussion of the intraparticle distributed parameter models and their numerical solution can be found elsewhere [20,21].

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3.1. Characteristic time constant in multicomponent systems When the mass balance in the adsorbent particle is transformed in dimensionless form, the ratio (D e/R 2p), which has units of s − 1, multiplies the real time variable to make it dimensionless De q= 2 t (38) Rp In this equation, D e is an effective mass transport coefficient. If there is only one component (single equation), there is no ambiguity in choosing D e. In multicomponent systems, however, the different species have different mass transport coefficients and the definition of the dimensionless time must be given explicitly. For the Glueckauf LDF approximation, which is not valid for dimensionless times B0.1, it is the smallest transport coefficient that has to be used in the definition of the dimensionless time because if the applicability criterion is satisfied for this species, it will automatically be satisfied for the rest. Since the Glueckauf LDF approximation has been derived in terms of the solid phase concentration, the transport coefficient which is used in the applicability criterion is the so called solid diffusion coefficient which for linear adsorption isotherms is related to the fluid phase diffusion coefficient by the expression D es,i =

D eF,i [mp + (1− mp)Hi ]

which, however, is strictly valid only for equimolar counterdiffusion of traces of species i diffusing in species j. The values of the transport coefficients for nitrogen in a binary mixture of oxygen and nitrogen are shown in Fig. 2 as functions of the pore radius at constant total pressure and temperature. The values for oxygen are similar since the molecular weights and hence the Knudsen transport coefficients of oxygen and nitrogen are very close. Fig. 3 shows the values of the solid diffusion coefficient of nitrogen as function of the pore radius at constant total pressure and temperature. Since the effective Knudsen diffusivity is independent of pressure (see Table 1), the difference in the values of the solid diffusion coefficient of nitrogen at the high and low limits of the operating pressure range decreases with decreasing pore size. For large pore sizes the Fickian transport coefficient approaches asymptotically the value of the bulk diffusion coefficient and thus the ratio of the solid diffusion coefficient of nitrogen at low pressure to that at high pressure approaches the high to low pressure ratio.

(39)

Eq. (39) suggests that the solid diffusion coefficient and, therefore, the dimensionless time based upon this coefficient account not only for the transport rate but also for adsorption rate differences. A composition independent value is commonly employed for D es,i, but D eF,i is usually a strong function of the composition of the mixture and of the fluxes themselves. For binary mixtures, an approximation to the Fickian transport coefficient D eF,i can be determined from the Bosanquet formula [32] 1 1 1 = e + e e D F,i Di, j D K,i

355

(40)

Fig. 2. Values of the transport coefficients of N2 in a binary mixture of O2 and N2 as functions of the pore radius rp at constant total pressure PT =0.35 MPa and constant temperature T= 300 K.

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3.2. Effects of the mass transport model

Fig. 3. Values of the solid diffusion coefficient of N2 in a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) as functions of the pore radius rp at constant total pressure (PT = 0.1 MPa and PT = 0.35 MPa) and constant temperature T= 300 K.

As explained earlier, if the Glueckauf LDF approximation is applied to multicomponent systems, the definition of the dimensionless time must be based on the smallest solid diffusion coefficient in the mixture. For the binary mixture of oxygen and nitrogen the values of the Fickian transport coefficients are close, and since the adsorption coefficient of nitrogen is three times bigger than the adsorption coefficient of oxygen, the solid diffusion coefficient for nitrogen is smaller than the solid diffusion coefficient for oxygen. Furthermore, the solid diffusion coefficient of nitrogen at high pressure is smaller than the solid diffusion coefficient of nitrogen at low pressure. Consequently, the solid diffusion coefficient of nitrogen at high pressure is used to define of the dimensionless time in the applicability criterion for the Glueckauf LDF approximation. For 0.35 MPa, 300 K, and the other parameters used in our calculations, q = 0.152t with the time t in s.

As pointed out earlier, almost all of the existing LDF approximations are based on the Fickian diffusion model which is the simplest of all mass transport models. In this case, the LDF matrix is diagonal, that is, there is no coupling among the LDF equations. In the presence of intraparticle total pressure gradients, the Fickian diffusion model produces inaccurate results because Knudsen flow is underestimated and viscous flow is not taken into consideration [20]. Part of this deficiency is corrected in the Fickian diffusion/convection model, which uses D’Arcy’s law to account for the viscous fluxes of the components. The generalized LDF equations based on the Fickian diffusion/convection model are coupled through the viscous fluxes of the individual components since these fluxes are proportional to the total pressure gradient (Eq. (15)). When the dusty-gas model is employed, the fluxes of the various species and hence the LDF equations, are coupled, not only through the viscous flow process, but also through the bulk diffusion and Knudsen flow processes as well. Figs. 4–6 show a comparison between the predictions of the different generalized LDF models with parabolic approximation for the intraparticle partial pressure profiles and the exact solution of the corresponding distributed parameter models from which they result. Shown on these figures is the dimensionless amount of oxygen adsorbed from a binary mixture of oxygen and nitrogen for the first two cycles. The reference quantity used to make the plots dimensionless is the equilibrium amount of oxygen adsorbed in the particle when the particle is equilibrated with the surrounding environment at high operating pressure. The duration of the cycle is 3.29 s which has a dimensionless value of 0.5. Fig. 4 is for stepwise pressurization and stepwise depressurization, Fig. 5 is for linear pressurization and exponential depressurization with k=3 s − 1 and Fig. 6 is for sinusoidal total pressure change. The low limit of the operating pressure range is 0.1 MPa and the high limit is 0.35 MPa. The results shown in Figs. 4–6 are typical for systems of slow moving/strongly adsorbable

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species and fast moving/weakly adsorbable species in which there is an overshoot in the solid phase concentrations of the fast moving/weakly adsorbable species. Specifically, for a certain period during the pressurization cycle the amount of species adsorbed exceeds the maximum amount that can be adsorbed at steady state, that is, the amount which is adsorbed when the particle is equilibrated with the surrounding environment at the high operating pressure. The main cause for this is the coupling of the mass transport processes of the various species through flow driven by total pressure gradients. Similar results have been observed experimentally by Hu et al. [23] and Asaeda et al. [33].

Fig. 5. Dimensionless amount of O2 adsorbed in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 =15 and HO2 =5) as function of dimensionless time for the first two cycles of a long cycle time process with linear pressurization and exponential depressurization (k =3 s − 1) at constant temperature T= 300 K.

Fig. 4. Dimensionless amount of O2 adsorbed in a particle (Rp = 10 − 3 m, rp =10 − 7 m, mp = 0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) as function of dimensionless time for the first two cycles of a long cycle time process with stepwise pressurization and stepwise depressurization at constant temperature T= 300 K.

For the binary system of oxygen and nitrogen, the mass transport coefficients of the two species have similar values. Thus, the overshoot in the solid phase concentration of oxygen is caused by its smaller adsorption constant which makes its intraparticle mass transport to be effectively faster than that of nitrogen. The faster transport of oxygen or, equivalently, the higher rate of retention of nitrogen on the gas–solid surface can lead to a situation where the partial pressure of oxygen in some region of the interior of the particle is higher than the partial pressure of oxygen at the external surface of the particle. Since local equilibrium is assumed to occur at all points in the intraparticle space, this can make the average uptake of oxygen larger than the equilibrium

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value at the surface conditions. This effect is present in all the examples, but it is most clearly seen in Fig. 4 in the case for stepwise pressurization. Because of its faster transport, oxygen is the main contributor to the total pressure equilibration, but once this happens, diffusion becomes the primary mode of transport with oxygen diffusing outward and nitrogen inward. Thus, the average uptake of oxygen goes through a maximum and after this, decreases to its equilibrium value. In Figs. 5 and 6 the total pressure changes continuously during the cycle, and as a result, the average uptake of oxygen is at all times larger than the equilibrium value at the external surface. An interesting observation is that the average amount of oxygen varies with time in almost the same

fashion as the pressure at the external surface, but with larger relative slope in the linear case and larger relative amplitude in sinusoidal mode. We see in Fig. 4 that two of the generalized LDF approximations (those based on the dustygas model and on the Fickian diffusion/convection model) predict dynamic responses that are qualitatively similar to that given by the exact solution of the multicomponent transport and adsorption model. Since we have linear adsorption isotherms, there is no competitive adsorption and, in order to explain the differences between the different models and the resulting LDF approximations, we have to consider only the transport effects. The reason for the failure of the Fickian diffusion model and the Glueckauf LDF approximation to describe the overshoot in the average uptake of oxygen is the inability of the Fickian diffusion model to account for the coupling between the mass transport fluxes of the species in a multicomponent system. For a stepwise pressurization or depressurization the Glueckauf LDF approximation predicts that the average solid phase concentrations increase or decrease monotonically to their equilibrium values according to the expression (q¯ i )R p = (qi )R p  Ci exp(− ui t)

(41)

Ci and ui are positive numbers, the negative sign applies during pressurization and the positive sign applies during depressurization. Since the Glueckauf LDF matrix is diagonal, each ui is proportional to the corresponding diagonal element. On the other hand, for the generalized LDF approximation based on the Fickian diffusion/convection model, the LDF matrix is a full matrix and the solution for the average solid phase concentrations of the two species has the form (q¯ i )R p = (qi )R p + Ci,1 exp(− u1t)+ Ci,2 exp(− u2t) (42)

Fig. 6. Dimensionless amount of O2 adsorbed in a particle (Rp = 10 − 3 m, rp =10 − 7 m, mp = 0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) as function of dimensionless time for the first two cycles of a long cycle time process with sinusoidal variation of the total pressure at constant temperature T= 300 K.

u1 and u2 are proportional to the eigenvalues of the LDF matrix. For nitrogen, which is the more adsorbable species in the system, Ci,1 and Ci,2 have one and the same sign and the averaged solid phase concentration of nitrogen increases or decreases monotonically. For oxygen, Ci,1 and Ci,2 have different signs and as a result, the averaged

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solid phase concentration of oxygen predicted by the approximation exhibits a maximum in agreement with the real behavior of the system (Fig. 4). We see in Fig. 2 that at the conditions of Figs. 4– 6 (rp =10 − 7 m), the viscous transport coefficient (B ePT/v) is about one order of magnitude larger than the Fickian diffusion coefficient D eF,i. This is why the inclusion of viscous flow in the Fickian diffusion model has a very strong effect in the predicted response. For a smooth change in the pressure of the surrounding gas phase, the generalized LDF approximations based on the Fickian diffusion/convection model and the dusty-gas model produce almost identical results (Figs. 5 and 6), which practically coincide with the exact solutions. This result indicates that it is the off-diagonal terms in the LDF matrix that result from the viscous flow that are mainly responsible for the differences between the Glueckauf LDF approximation and the LDF approximation based on the dusty-gas model. As the pore size decreases, the viscous transport coefficient decreases faster than the Knudsen transport coefficient (Fig. 2) and this decreases the importance of the off-diagonal terms. Thus, for pore sizes smaller than :10 − 8 m, the difference between the Fickian diffusion model and the other models are very small, even for instantaneous (stepwise) pressurization or depressurization. For stepwise pressurization (Fig. 4), the peak in the solid phase concentration of oxygen that is predicted by the dusty-gas model is higher than that shown by the Fickian diffusion/convection model. This difference results from the representation of the Knudsen flow. Knudsen flow flux is equal to ( −yi D eK,i(9PT/RgT)) and is correctly described by the dusty-gas model, but in the Fickian diffusion/ convection model it is represented as (− yi D eF,i(9PT/RgT)) (see Ref. [20]). It is seen in Fig. 2 that for pores 10 − 7 m in radius the Fickian diffusion coefficient is much smaller than the Knudsen diffusion coefficient and therefore, the Fickian diffusion/convection model underestimates the flow caused by the total pressure gradient. For the case of smooth variation in the surface conditions (Figs. 5 and 6), the total pressure gradients are small and as a result the effects of Knudsen flow are not as important as for stepwise pressurization or depressurization.

359

Fig. 7. Dimensionless amount of O2 adsorbed in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 =15 and HO2 =5) as function of dimensionless time for the first five cycles of a short cycle time process with stepwise pressurization and stepwise depressurization at constant temperature T =300 K.

3.3. Effects of cycle time Figs. 4– 6 present results for long cycle times, that is, times that are long enough to reach equilibrium at the end of each half cycle. Figs. 7–9 compare the predictions of the generalized LDF approximations based on the dusty-gas model with parabolic and quartic (R1 = 0.8 Rp) intraparticle partial pressure profiles with the exact solution for short cycle times for the first five cycles. The duration of each cycle is 0.13 s which in dimensionless time is 0.02, about an order of magnitude below the limit of applicability of the Glueckauf LDF approximation. Fig. 7 is for stepwise pressurization and stepwise depressurization, Fig. 8 is for linear pressurization and

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exponential depressurization with k = 75 s − 1 and Fig. 9 is for sinusoidal total pressure change. The operating conditions are the same as in Figs. 4– 6. The results of all three figures indicate that a generalized LDF approximation based on the quartic partial pressure profile yields responses that are in excellent agreement with the exact solution of the multicomponent model. However, the accurate representation of the averaged quantities does not necessarily imply an accurate representation of the partial pressure profiles. Figs. 10 and 11 compare the partial pressure profiles of oxygen that are extracted from the parabolic and the quartic LDF approximations with these found from the exact solution at the middle and at the end of the second cycle for the pulsing schemes of

Fig. 9. Dimensionless amount of O2 adsorbed in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 =15 and HO2 =5) as function of dimensionless time for the first five cycles of a short cycle time process with sinusoidal variation of the total pressure at constant temperature T =300 K.

Fig. 8. Dimensionless amount of O2 adsorbed in a particle (Rp = 10 − 3 m, rp =10 − 7 m, mp = 0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) as function of dimensionless time for the first five cycles of a short cycle time process with linear pressurization and exponential depressurization (k= 75 s − 1) at constant temperature T =300 K.

Figs. 7 and 9. The value of the lower limit of the operating pressure (0.1 MPa) is used to make the partial pressures dimensionless. In all cases, the partial pressure profiles based on the quartic approximation are much closer to the exact solution than the parabolic partial pressure profiles. Significant differences between the quartic partial pressure profiles and the exact solution are encountered mainly close to the center of the particle. However, in a fast cycle the central region of the particle contributes insignificantly to the average loading and this is reflected in the very good agreement that exists between the oxygen loading of the quartic LDF approximation and the exact result in Figs. 7 and 9. We have used only the dusty-gas model for intraparticle mass transport to obtain results for

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short cycles because as Figs. 10 and 11 show the profiles of the various species do not reach equilibrium at the end of each half cycle. As we saw in Figs. 4– 6, there are small differences between the predictions of the rigorous multicomponent model (based on the dusty-gas model) and the Fickian diffusion/convection model only for large pressurization times (for stepwise pressurization) or low rates of pressure change. These conditions are far from being satisfied for small cycle times.

3.4. Effects of the form of polynomial representation It was noted earlier that for an even polynomial approximation of the partial pressure profiles of degree 2N we need N domains. The simplest case

Fig. 11. Partial pressure profile of O2 in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) in a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 =15 and HO2 =5) at the middle (q =0.3) and at the end (q =0.4) of the second cycle of a short cycle time process with sinusoidal variation of the total pressure (Fig. 9) at constant temperature T= 300 K.

Fig. 10. Partial pressure profile of O2 in a particle (Rp = 10 − 3 m, rp = 10 − 7 m, mp = 0.6) in a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) at the middle (q = 0.3) and at the end (q = 0.4) of the second cycle of a short cycle time process with stepwise pressurization and stepwise depressurization (Fig. 7) at constant temperature T = 300 K.

is N= 1 which corresponds to a parabolic profile. For this case, the calculations are based on a single LDF matrix evaluated at the outer surface of the particle. This means that the dynamics of the averaged partial pressures and of the averaged solid phase concentrations are determined solely by the dynamics of the partial pressures at the outer surface of the particle. At the outer surface of the particle, the response of the partial pressures is faster than at the interior points because of the intraparticle mass transport limitations. Therefore, the generalized LDF approximations with parabolic intraparticle partial pressure profiles predict faster dynamic responses. This can be clearly seen in the depressurization branches in Figs. 7–9. For a generalized LDF approximation based on a quartic profile, we use two domains and therefore, the dynamics of the averaged solid phase concentrations are influenced by two LDF

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matrices: the first one evaluated at the outer surface of the particle and the other at some interior location. Because we use information both from the outer surface and from the interior of the particle, the predicted response is more accurate. This is particularly important for processes with short cycle times, such as those shown in Figs. 7 – 9, in which the quartic intraparticle partial pressure approximation clearly performs much better than the parabolic partial pressure approximation. For processes with long cycle times, there is no need to use the LDF equations based on the quartic profile approximation. As seen in Figs. 4 – 6, the generalized LDF approximation based on the parabolic profile produces results almost identical to those obtained by the exact solutions of the corresponding distributed parameter models. For global approximations that require the use of more than one subdomain, that is, for those involving quartic or higher polynomials, the performance of the resulting LDF approximation depends strongly on the points used to define the subdomains. This is also the case for the piecewise continuous polynomial approximations. Of course, in the last case, the results are also influenced by the breakpoint sequence that defines the subintervals over which the various polynomial pieces are defined. Fig. 12 presents results on the dependence of the response of the global quartic and the piecewise quartic/parabolic LDF approximations on the location of the breakpoint for the two subdomains and the two subintervals. The operating conditions are the same as in Fig. 7. In Fig. 13, the partial pressure profiles of oxygen extracted from the corresponding LDF approximations in Fig. 12 are compared to the exact profile at the middle of the fifth cycle (q = 0.9). It is seen from these two figures that for the presented approximations, better results are obtained when R1 is closer to the outer surface of the particle than to its center. For the global quartic approximation, the best location appears to be in the vicinity of h= R1/Rp =0.8 and this is why this value has been employed in the majority of the results presented in this study. (See Table 3 for the values of Cij and Lij for the values of h used to obtain the results of Fig. 13.)

3.5. Quaternary system with nonlinear adsorption isotherm Fig. 14 presents results from the application of the generalized LDF approximation to a multicomponent (quaternary) system with nonlinear adsorption isotherms. Presented in this figure are the dimensionless amounts of the various components of the mixture (CO2, H2, CH4 and N2) adsorbed in the particle during the first two cycles of stepwise pressurization and stepwise depressurization. The composition of the mixture is maintained constant at a ratio of 0.4:0.2:0.15:0.25 for

Fig. 12. Dimensionless amount of O2 adsorbed in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) from a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 =15 and HO2 =5) as function of dimensionless time for the first five cycles of a short cycle time process with stepwise pressurization and stepwise depressurization at constant temperature T =300 K.

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the literature. The reference value used to make the adsorbed quantity for each species dimensionless is the amount of that species that is adsorbed when the particle is equilibrated with the surrounding environment at the high operating pressure. The lower operating pressure is 0.1 MPa and the higher operating pressure 0.4 MPa. Hydrogen is not only the least adsorbable species — see the parameters of the Langmuir adsorption isotherm that are given above — but also the fastest moving species; its Knudsen diffusion coefficient is approximately five times larger than the Knudsen diffusion coefficient of carbon dioxide ( MCO2/MH2). As a result, a pronounced overshoot and a pronounced undershoot are present in the variation of the average uptake of hydrogen with time during pressurization and depressurization, respectively, in Fig. 14. The species with the next largest effective rate of transport in the particle is N2 and as the results of Fig. 14 show, this species also exhibits overshoot in the variation of its uptake by the particle during

Fig. 13. Partial pressure profile of O2 in a particle (Rp = 10 − 3 m, rp = 10 − 7 m, mp = 0.6) in a binary mixture of O2 and N2 with linear adsorption isotherms (HN2 = 15 and HO2 = 5) at the middle of the fifth cycle (q=0.9) of a short cycle time process with stepwise pressurization and stepwise depressurization (Fig. 12) at constant temperature T=300 K.

CO2:H2:CH4:N2. The duration of the cycle is 1 s and the adsorption equilibrium is taken to be the extended Langmuir isotherm qi = qm,i

bi pi

(43)

n

1+ % bj pj j=1

with qm,CO2 =qm,H2 = qm,CH4 =qm,N2 =1.78 kmol/m3 bCO2 =12.5 MPa − 1; bCH4 =47.5 MPa

−1

;

bH2 =0.14 MPa − 1 bN2 =0.98 MPa

−1

The values for the parameters of the isotherm were chosen on the basis of data compiled from

Fig. 14. Dimensionless amounts of H2, CO2, N2 and CH4 versus dimensionless time for adsorption/desorption in a particle (Rp =10 − 3 m, rp =10 − 7 m, mp =0.6) from a quaternary mixture with Langmuir adsorption isotherms for the first two cycles of a process with stepwise pressurization and stepwise depressurization at constant temperature T =300 K.

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pressurization and undershoot during depressurization. Both the overshoot and the undershoot in the variation of the adsorbed amounts of H2 and N2 are satisfactorily described by the LDF approximation for global quartic approximation of the profiles with h =0.8. The LDF model also performs well for the profiles of CO2 and CH4. Of these two species, the amount of CO2 varies monotonically with time both during pressurization and depressurization and this is most provably a consequence of the fact that CO2 is the species that combines the least diffusion coefficient and the largest adsorption constant. The CH4 amount adsorbed in the pellet presents a weak local minimum and a weak local maximum at about the time where the hydrogen profile attains its maximum during pressurization. It eventually reaches values that are above (during pressurization) or below (during pressurization) the equilibrium value at the ambient pressure and composition and it relaxes slowly towards the latter values. The assumption that we have made in deriving the generalized LDF approximation in the case of nonlinear adsorption isotherms is that the average quantities, partial pressures and solid phase concentrations, follow the same functional relationship as the local quantities. In the context of a numerical quadrature, this approach is equivalent to using a single point quadrature formula and assuming that all partial pressures equal their corresponding average values at that point. Because of the different forms of the partial pressure profiles of the two species, this is definitely not the case, but the results in Fig. 14 show a very good agreement between the generalized LDF approximation and the exact solution. However, much better results can be obtained by using a quadrature formula to compute (d(q¯ i )R /dt).

4. Summary and conclusions A generalized approach was presented for developing LDF approximations in multicomponent systems, and it was applied to develop specific LDF expressions. Three different mass transport models have been employed: the Fickian diffusion

model, the Fickian diffusion/convection model and the dusty-gas model. Both linear and nonlinear adsorption isotherms were considered and global parabolic, global quartic, and piecewise continuous polynomial approximations were employed to represent the intraparticle partial pressure profiles. The obtained generalized LDF approximations consist of coupled systems of ordinary differential equations, which in general have to be solved numerically. The predictions of the LDF approximations we developed were compared with the results given by the exact solution of the corresponding models. It was found that for low rate of pressure change or large pressurization or depressurization times for stepwise pressure change a satisfactory agreement between the exact solution and the LDF approximation can be obtained for parabolic partial pressure profiles. However, for small cycle times, higher order or piecewise continuous polynomial approximations, such as global quartic, have to be employed. One of the most interesting features of the developed approximations is that they allow for the inclusion of convective effects and are capable of describing the dynamic overshoot that is observed experimentally and predicted theoretically by diffusion/ convection models of the fast moving/weakly adsorbable species in a multicomponent mixture during pressure pulsing. This behavior cannot be reproduced by the conventional Glueckauf LDF formula that treats the transport of each species independently. The approach that we formulated for the construction of generalized LDF approximations is not limited only to the mass transport mechanisms investigated in the present study, that is, bulk diffusion, Knudsen diffusion, Knudsen flow and viscous flow. If, for example, surface diffusion is an important mass transport mechanism, it can be added to the mass transport model and a new generalized LDF approximation can be derived. Furthermore, if the sorbent particle cannot be treated as isothermal, this generalized approach can be used to construct nonisothermal LDF approximations. The incorporation of the particle (or pellet) model that results after the application of the LDF approximation into the

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overall particle-bed model is straightforward since, as it was pointed above, the LDF approximation reduces the particle model to a set of ordinary differential equations. The subdomain method of weighted residuals that was used as basis for the development of the multicomponent LDF approximations can be employed with any global or piecewise continuous approximations of the partial pressure profiles in the pellets. Serbezov and Sotirchos [25] used this method to discretize the partial pressure profiles in the interior of sorbent pellets in a numerical study of the behavior of a rigorous multicomponent, nonisothermal particle-bed model they formulated for adsorption/desorption processes. For piecewise continuous, quadratic (second degree) approximations of the partial pressure profiles, they found that results that were obtained from the overall particle-bed model did not change significantly when the partial pressure profiles were represented with more than four quadratic pieces. For a small number of polynomial pieces (one or two), the computed results could differ significantly from those found from the exact numerical solution. For example, in the application of the particle-bed model to air separation, it was found that the representation of the partial pressure profiles with one quadratic piece (parabolic approximation) predicted oxygen mole fractions in the product that were by : 25% lower than those predicted by the exact numerical solution. In view of these results, it is recommended that the use of the multicomponent LDF approximations that are based on more involved representations of the partial pressure profiles (such as the quartic and parabolic-quartic for the LDF approximations developed in this study) be preferred over that of the approximation based on the parabolic partial pressure profiles.

Appendix A. Elements of the matrices L and C for a global polynomial approximation For a polynomial approximation of pk pk = a0,k +a2,kr 2 + a4,kr 4 +···+ a2N,kr 2N

(A1)

365

we have

(pk )R p = a0,k + a2,kR 2p + a4,kR 4p + ···+a2N,kR 2N p (A2)

(pk)Ri = a0,k + a2,kR 2i + a4,kR 4i + ···+a2N,kR 2N i

(A3)

(p¯ k)Ri 3 3 = a0,k + a2,kR 2i + a4,kR 4i + ··· 5 7 +

3 a R 2N 2N+ 3 2N,k i

(A4)

Subtracting Eq. (A4) from Eq. (A2) and Eq. (A2) from Eq. (A3) and casting the resulting equations for i= 1,…, N and i= 1,…, N− 1, respectively, in matrix form we get Æ (pk)R p − (p¯ k)R 1 Ç Ã (p ) − (p¯ ) à k Rp k R2 à Ã= — à à È(pk)R − (p¯ k)R É p

Æ ÃR2 −3 R2 à p 5 1 à à R 2p − 3 R 22 5 à à — à à 2 3 2 ÃR p − 5 R N È Æ a2,k Ç Ãa à 4,k à à à — à Èa2N,k É

N

3 R 4p − R 41 7

···

3 R 4p − R 42 7

···

—

··

3 R 4p − R 4N 7

···

·

Ç 3 à R 2N 1 à 2N+ 3 à 3 2N à R 2N − R p 2 2N+ 3 à à — à 3 2N 2N à Rp − RN à 2N+ 3 É R 2N p −

(A5)

A. Serbezo6, S.V. Sotirchos / Separation/Purification Technology 24 (2001) 343–367

366

Æ (pk)R 1 −(pk)R p Ç Ã (p ) −(p ) à k R2 k Rp à Ã= — à à È(pk)R −(pk)R p É N−1 Æ R 21 −R 2p à R 2 −R 2 2 p à — à ÈR 2N − 1 −R 2p

Æ a2,k Ç Ãa à 4,k à = Mà à — à Èa2N,k É

R 41 −R 4p

···

R 42 −R 4p

··· ·· · ···

— R 4N − 1 −R 4p

2N R 2N Ç 1 −R p 2N à R 2N −R 2 p à — à 2N É R 2N N − 1 −R p

Æ a2,k Ç Ãa à 4,k à à à — à Èa2N,k É

(A6)

or

(A7)

Æ (pk)R 1 − (pk)R p Ç Æ a2,k Ç Ã (p ) − (p ) à Ãa à k Rp à k R2 à = H à 4,k à — à à à — à È(pk)R É Èa2N,k É −(p ) k Rp N−1

(A8)

with RN =Rp. Differentiating Eq. (A1), we get in matrix form for the driving forces ((pk/(r)Ri

    (pk (r (pk (r —

Ç Ã Ã Ri à à Ri à à à à à RN É

  (pk (r

Æ 2R1 4R 31 ··· à 2R 4R 32 ··· =à 2 ·· — à — · È2R 2N 4R 3N ···

−1 2NR 2N Ç Æ a2,k Ç 1 2N − 1 à à 2NR 2 a à à à 4,k à — Ãà — à 2N − 1 É È 2NR N a2N,k É

Eq. (A9) and Eq. (A7) give Æ Ã Ã Ã Ã Ã Ã Ã Ã Ã È

   

Ç Ã Ã Ri Ã Æ (pk)R p − (p¯ k)R 1 Ç Ã Ã (p ) − (p¯ ) à k R2 −1 à k Rp à Ri à =MF — à à à à È(pk)R − (p¯ k)R É p N (pk à à (r RN É (pk (r (pk (r —

 

(A10)

Æ (pk)R p − (p¯ k)R 1 Ç Æ a2,k Ç Ã (p ) −(p¯ ) à Ãa à k R2 à k Rp à = F à 4,k à — à à à — à È(pk)R −(p¯ k)R É Èa2N,k É p N

Æ Ã Ã Ã Ã Ã Ã Ã Ã Ã È

(A9)

Similarly, from Eq. (A8) and Eq. (A7) we get Æ (pk)R 1 − (pk)R p Ç Æ (pk)R p − (p¯ k)R 1 Ç Ã (p ) − (p ) à à (p ) − (p¯ k)R à k Rp 2 à k R2 à =HF − 1 à k R p à — — à à à à È(pk)R È(pk)R − (p¯ k)R É − (pk)R p É N−1 p N (A11) The comparison of Eq. (A10) and Eq. (A11) with Eq. (22) and Eq. (24), respectively, leads to the results L= RpMF − 1

(A12)

C= − HF − 1

(A13)

Eq. (A12) and Eq. (A13) were used to generate the values given in Table 3 for a quartic polynomial approximation for different values of Rl/Rp. Since in that case, F is a 2× 2 matrix, it is relatively easy to derive analytical expressions for the elements of L and C (see Ref. [29]). The procedure that is followed for the computation of the coefficients for piecewise continuous polynomial representation of the partial pressure profiles is basically the same as that presented above. The only difference is that the continuity conditions at the breakpoints are used to eliminate some of the ai,k coefficients, leaving only N of them for N subdomains.

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[20] A. Serbezov, S.V. Sotirchos, Ind. Eng. Chem. Res. 36 (1997) 3002. [21] A. Serbezov, S.V. Sotirchos, Adsorption 4 (1998) 93. [22] R.M. Marutovsky, M. Bu¨ low, Gas Sep. Purif. 1 (1987) 66. [23] X. Hu, G.N. Rao, D.D. Do, Gas Sep. Purif. 7 (1993) 39. [24] A.M.M. Mendes, C.A.V. Costa, A.E. Rodrigues, Gas Sep. Purif. 10 (1996) 141. [25] A. Serbezov, S.V. Sotirchos, Chem. Eng. Sci. 54 (1999) 5647. [26] R. Jackson, Transport in Porous Catalysts, Elsevier, New York, 1977. [27] E.A. Mason, A.P. Malinauskas, Gas Transport in Porous Media: The Dusty-Gas Model, Elsevier, New York, 1983. [28] S.V. Sotirchos, AIChE J. 35 (1989) 1953. [29] A. Serbezov, Ph.D. Thesis, University of Rochester, Rochester, NY, 1997. [30] S. Farooq, M.N. Rathor, K. Hidajat, Chem. Eng. Sci. 48 (1993) 4129. [31] M.M. Tomadakis, S.V. Sotirchos, Chem. Eng. Sci. 48 (1993) 3323. [32] W.G. Pollard, R.D. Present, Phys. Rev. 73 (1948) 762. [33] M. Asaeda, J. Watanabe, M. Kitamoto, J. Chem. Eng. Jpn. 14 (1981) 129. [34] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. [35] R.D. Present, The Kinetic Theory of Gases, McGrawHill, New York, 1958.

References [1] E. Glueckauf and J. I. Coates, J. Chem. Soc. 41 (1947) 1315. [2] E. Glueckauf, Trans. Faraday Soc. 51 (1955) 1540. [3] S. Nakao, M. Suzuki, J. Chem. Eng. Jpn. 16 (1983) 114. [4] E. Alpay, D.M. Scott, Chem. Eng. Sci. 47 (1992) 499. [5] G. Carta, Chem. Eng. Sci. 48 (1993) 622. [6] D.H. Kim, Chem. Eng. Sci. 51 (1996) 4137. [7] P. Sheng, C.A.V. Costa, Chem. Eng. Sci. 52 (1997) 1493. [8] B.A. Finlayson, Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York, 1980. [9] C.H. Liaw, J.S.P. Wang, R.A. Greenkorn, K.C. Chao, AIChE J. 25 (1979) 376. [10] D.D. Do, R.G. Rice, AIChE J. 32 (1986) 149. [11] C. Yao, C. Tien, Chem. Eng. Sci. 47 (1992) 457. [12] C. Yao, C. Tien, Chem. Eng. Sci. 47 (1992) 465. [13] C. Yao, C. Tien, Chem. Eng. Sci. 53 (1998) 3763. [14] E.S. Kikkinides, R.T. Yang, Chem. Eng. Sci. 48 (1993) 1169. [15] G.G. Botte, R. Zhang, J.A. Ritter, Chem. Eng. Sci. 53 (1998) 4135. [16] G. Carta, A. Cincotti, Chem. Eng. Sci. 53 (1998) 3483. [17] T.S.Y. Choong, D.M. Scott, Chem. Eng. Sci. 53 (1998) 847. [18] J. Lee, D.H. Kim, Chem. Eng. Sci. 53 (1998) 1209. [19] S. Sircar, J.R. Hufton, Adsorption 6 (2000) 137.

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