Analysis of the frequency response method for sorption kinetics in bidispersed structured sorbents

Cbmlcal Engineering Scknce, Vd. F-rintcdinGmt Britain.

4CNo.6,pp. 1103-1130.1993.

CCC&-2549/93 Sfi.M+O.Kl Q 1993l'qamon F'mm Ltd

ANALYSIS OF THE FREQUENCY RESPONSE METHOD FOR SORPTION KINETICS IN BIDISPERSED STRUCTURED SORBENTS ROBIN Department

G. JORDI and DUONG

D. DO’

of Chemical Engineering, University of Queensland, St. Lucia, Qld. 4072, Australia (First received

7 January

1992; accepted

in revised form 18 May

1992)

Abstract-A model for the sorption of gasesonlo bidispersedstructured sorbcnt material in a batch system,

subjected to small periodic perturbations of the system volume is developed. The model takes account of external film mass transfer resistance, coupled macropore and micropore resistance, and the surface barrier. The general model is linearised. and analytical solutions for the characteristic functions for slab, cylindrical and spherical macroparticle and microparticle geometries are determined, for both the general model and degenerate models which are applicable when one or more of the mass transfer resistances are negligible. The characteristic functions are examined to determine the effects of macroparticle and microparticle geometry and the rate controlling mechanisms on the frequency response. spectrum. The limitations of the batch frequency response technique for investigation with sorbatesorbent systems exhibiting H2- or LA-type isotherms are b&fly considered.

INTRODUCTION

The adsorption of gases in bidispersc structured sorbents such as zeolites, activated carbon and carbon molecular sieve is an area of considerable importance in chemical engineering. The adsorption rate in such systems may be dependent on an interplay of processes such as film mass transfer, macropore and micropore diffusion and the microparticle surface resistance, or surface barrier. The relative contribution of each dynamic process to the overall mass transfer resistance depends on their relative rates and the distribution of the sorbate capacity in the macropore void and microparticles. The frequency response method is a relaxation technique in which the equilibrium state of the system is perturbed periodically in order to measure the response of the state variables under periodic steadystate conditions. The response of the state variables depends on the time scales of the dynamic processes affecting the state variables relative to the time scale of the perturbation, the type of perturbation, the sorption mechanisms, and physical characteristics of the system. Measurement of the response of the system over the perturbational frequency spectrum facilitates the identification of the controlling sorption mechanisms and the determination of the associated dynamic parameters. Frequency response methods have wide applicability in process control theory and have been used in stability analysis and process identification procedures (Stephanopoulous, 1984). Yasuda (1982) proposed a modified batch frequency response procedure for the determination of the micropore diffusion coefficients of sorbed species in zeolite adsorbents. In this technique, the sorbate

‘Author to whom correspondence should be addressed.

equilibrium is perturbed by the application of small sinusoidal variations of the batch system volume, typically having amplitudes of only 1 or 2%. The pressure change induced by the volume perturbations is monitored using a sensitive pressure transducer, to determine the amplitude ratio and phase lag over a range of perturbational frequencies. The in-phase and out-of-phase characteristic functions, or components of the pressure response in-phase and out-ofphase with the applied volume perturbation, may then be determined and compared with standard solutions. Yasuda applied the batch frequency response technique to measure diffusion rates of krypton in Na-mordenite. Although pellet&d sorbent material having a bidispersed structure was used, the results were analysed using a single-particle mode1 which assumed that micropore diffusion was the sole rate controlling process. Furthermore, they foufid that the in-phase and out-of-phase characteristic functions were not mutually asymptotic at high perturbational frequencies, as predicted by the theory, and suggested the existence of a second rapid kinetic process. The utility of the frequency response method for the determination of sorption kinetics and mechanisms is illustrated by the work of Yasuda and Yamamoto (1985) who applied the batch frequency response technique in their study of diffusion of ethane and propane in zeolite 5A. In both cases, they found two peaks in the out-of-phase characteristic function which is indicative of the presence of at least two kinds of sorbed species with different mobilities. They modelled the system in terms of two Fickian diffusion processes occurring in parallel. In their experiment with propane on Linde 5A at an temperature of 366 K and absolute pressure of 8.9 torr, they found that the in-phase and out-of-phase characteristic functions crossed one another, which was interpreted in terms

1103

1104

ROBING. JORDIand DUONGD. Do

of an inadequacy of Fick’s law, however, this kind of behaviour may be indicative of a surface barrier, (Yasuda, 1991), or in terms of a reversible intracrystalline mass exchange process (Yasuda et al., 1991). Later, Billow et al. (1986), van-den-Begin and Rees (1989), and van-den-Begin et al. (1989) utilised the batch frequency response method to investigate the diffusion of ethane, propane and n-butane in ZSMS-silicalite systems. Their experimental technique involved the application of a small square wave perturbation of the system volume. They analysed their results using a similar approach as that of Yasuda. They found that the depth of the zeolite crystal bed influenced their experimental frequency response measurements, and could lead to intra-crystalline diffusivities appearing to be two orders of magnitude lower than those determined by using the NMR pulsed field gradient and the NMR fast tracer desorption techniques. They concluded that for rapidly diffusing species, the adsorption and desorption rates could be influenced by heat release effects. The frequency response method has also been applied to study heat and mass transfer in flow systems. Boniface and Ruthven (1985) determined axial dispersion coefficients, micropore diffusion coefficients, and adsorption equilibrium constants for argon, nitrogen and oxygen on 4A, Na-X and Na-mordenite zeolites. They imposed a sinusoidal perturbation on the inlet concentration of a chromatographic column and monitored the periodic steady-state response of the outlet concentration. The parameter estimation was performed by matching the response to the model prediction in the frequency domain, allowing for macropore and micropore diffusion and axial dispersion mechanisms. Huber and Jones (1988) determined interphase heat transfer coefficients and effective axial conductivities for air and carbon dioxide in a packed bed of uniform alumina spheres over a temperature range of 375-1300 K. They induced a small sinusoidal disturbance of the inlet gas-phase temperature, and monitored the gas-phase temperature response at two locations on the axis of the bed. The parameters were estimated by matching the response to the model prediction in the frequency domain. Li et al. (1989) determined the adsorption and desorption rate constants of carbon dioxide over a supported metal catalyst. They imposed a sinusoidal perturbation on the inlet concentration and monitored the response of the surface concentration over a range of perturbational frequencies. The adsorption and desorption rate constants were estimated by matching the response to the model prediction in the frequency domain. Perhaps the first modelling work dealing with adsorption in bimodal solids was that of Ruckenstein et al. (1971) who presented an analysis of the transient sorption of a single component. They assumed a pore diffusion mechanism in the macropore and micropore, and included adsorptive capacity in both pore

systems. They applied the theory to the adsorption of carbon dioxide, sulphur dioxide and ammonia on an ion exchange resin. Subsequent workers proposed a different type of model for adsorption in bidispersed structured sorbents. The essential features include pore diffusion with no adsorption in the macropore, rapid adsorption at the micropore mouth and activated diffusion in the micropores. This type of model was used by Kawazoe et al. (1974) in a chromatographic study of the sorption dynamics and equilibrium of nitrogen in carbon molecular sieve adsorbent, by Shah and Ruthven (1977) for their study on the sorption of light hydrocarbons on 5A zeolite, and by Andrieu and Smith (1980, 1981) for the determination of sorption rate and equilibrium parameters of carbon dioxide, sulphur dioxide and hydrogen sulphide on activated carbon by means of a chromatographic technique. Although past research work on adsorption in bimodal solids has frequently ignored the presence of a surface barrier, apparent intra-crystalline diffusivities determined by NMR pulsed field gradient and NMR fast tracer desorption techniques have indicated the presence of a surface barrier for some systems (KBrger and Caro, 1977; Kiirger and Ruthven, 1989). Later, Yasuda (1991) identified the presence of a surface resistance using the frequency response technique in experiments with methane on Linde SA. This article attempts to address the need for a more systematic approach to the modelling and analysis of the batch frequency response technique. We develop a general model describing the frequency response of a batch system containing bidispersed structured sot-bent material, subjected to small harmonic perturbations of the system volume, and allow for film mass transfer, macropore and micropore diffusion, and surface barrier resistances. The general model, and degenerate models identified by Do (1990), are linearised, and solved using the Laplace transform technique for slab, cylindrical and spherical macroparticle and microparticle geometries, in order to determine the characteristic functions from the periodic steady-state solutions. MATHEMATICAL FORMULATION We consider a batch system containing bidispersed structured sorbent material subjected to a small sinusoidal perturbation of the system volume about the equilibrium state. We make the following assumptions in the analysis: (i) The system is isothermal. (ii) The diffusional processes under consideration are Fickian. (iii) The diffusion coefficients are time and position invariant, and are constant over the induced concentration range. (iv) The adsorption at the pore mouth of the micropore follows the Langmuirian kinetics

Analysis

of the frequency response method

with constant adsorption and desorption rate constants. (4 Adsorption on the microparticle exterior surface is negligible. 64 Only one adsorbable component is present. (vii) Bach sorbent particle is exposed to the same bulk-phase environment. (viii) The sorbent particles are identical to one another. Assumption (iv) implies that the microparticle uptake follows the Langrnuir isotherm under equilibrium conditions. This assumption is not critical, since as the batch frequency response technique is a differential technique, the intensity of the frequency spectrum under set experimental conditions is dependent on the local gradient of the isotherm, and not on the global nature of the isotherm. Assumptions (vii) and (viii) reduce the sorption mass balance equations to the single-particle analysis. The following mass transfer processes are considered in this analysis: (i) External diffusion through the film surrounding the sarbent particle. (ii) Macropore diffusion through the large pores. (iii) Adsorption at the pore mouth of the micropartitle. (iv) Activated diffusion in the micropores. Subject to the above assumptions, the mass balance equations describing the sorbate concentration distribution within the microporous particle become Macropore

for sorption

kinetics

3

at

1105

= DyIV2CP.

(W

The boundary conditions imposed at the micropartitle surface and at the microparticle centre are

Uf)

ac,

ar, ,, =0

0.

(W

=

The RHS of eq. (If) is the net local adsorption rate per unit exterior surface area of the microparticle. Note that the rate is assumed to follow the Langmuirian kinetics. Initially, the sorbate in the micropores is in equilibrium with the sorbate in the surrounding macropore void: C,ult=o = c,..

(lh)

Batch system mass balance:

d

&

[

W v(t) Z@T

1 +

e = dt

o

’

(10

The batch system volume is perturbed harmonically about the mean volume, thus:

v(t) = K[l - u&y

)01< 1.

(lj)

Initially, the bulk-phase pressure is given by P(t)l,=o

= p,.

(lk)

The total amount of sorbate in the sorbent material, B (mol) is given by

mass balance:

x jrr;C,,dr,]dr.

(11)

(la) where C, is the sorbate concentration in the microsphere (mol/cm3 of microsphere), and C is the sorbate concentration in the particle (mof/cm” of macropore void). Note that we have taken the mass transfer driving force to be proportional to the sorbate concentration gradient. We impose the film mass transfer boundary condition at the particle exterior and the symmetry boundary condition at the particle centre:

Using the nondimensional parameters defined in Table 1 eqs (la)-(11) may be nondimensionalised as follows: Macropore mass balance:

aA

ax dA 0 iG X=0=

ac dr

r=O=

Cl,=0 = c,. Micropore mass balance:

Al,=0 = 0.

0.

Initially, the sorbate in the macropore librium with the hulk gas phase:

(24 W)

Micropore mass balance: void is in equi(Id)

1

w

ROBIN G. JORDI

1106

aA,

Ix,=~=0

PI3

ax,

A,I,=o

0.

=

(W

Batch system mass balance: &C(a.+l)(Y*+I)l+K’~=O v*c7)

=

-

(2i)

&m*r

Id< 1

(29

AsI,=,, = 0

CW

LI* = (SM + 1)

c 1

1

Q-A,, dx, dx. J JO Equations (2a)-(21) are the general model equations describing the response of the batch system to sinusoidal perturbations of the batch system volume. The equations consist of a linear parabolic partial differential equation (2a), with boundary conditions (2b) and (2~) and initial condition (2d), coupled via the microparticle boundary condition (2f), to the linear parabolic partial differential equation (2e), with boundary conditions (2f) and (2g) and initial condition (2h), and the non-linear ordinary differential equation (2i), with initial condition (2j). X

SIGNIFICANCE

OF THE NON-DIMENSIONAL PARAMETERS

The parameters e1 and u2 represent the fractional equilibrium hold-up in the macropore void and the

Table

1.

Non-dimensionalvariables and parameters

Variables G - G. A, = CBe B*=-

B - Be B,

A _ C - GJc C8e v-v V’ = 2

C, - C,, A, = c L= P - P, P* =P

K

e

h4.t

,=L R

T = CG, + (1 - 8,~) C,,/GlR=

@* = wR’CQ4 + (1 - %)C”./C,.l xp = -5 EM4 4 Capacity parameters

and DUONGD. Do microparticles, respectively. For most commercial sorbents, ur is close to zero, and ur is close to unity. The parameter I is a measure of the approach to saturation of the Langmuir isotherm. Small values of A indicate an equilibrium state in the linear region of the Langmuir isotherm, while saturation is approached as R increases. The parameter K* is the ratio of sorbate adsorbed by the adsorbent material to that in the gas phase at equilibrium. The parameter z%Iis the ratio of the adsorption rate to the micropore diffusion rate. When & is large, equilibrium at the pore mouth of the micropore is rapidly attained. The parameter y is the ratio of the micropore diffusion rate to the macropore diffusion rate. The parameter Bi is the Biot number, which is a measure of the resistance through the external fluid film surrounding the particle, small Biot numbers being a characteristic of large film resistance. DEGENERATE

MODELS

The relative importance of the three internal rate processes are dictated by the magnitude and relative magnitude of the two dynamic parameters 1 and y. Do (1990) identified seven independent degenerate models in which one or two of the internal rate processes control the adsorption uptake rate, as well as their criteria of validity, by means of a perturbation technique. The models may be designated as follows: Model 1: macroporemicropore diffusion and adsorption model (general model). diffusion model. Model 2: macropore-micropore Model 3: macropore diffusion model. Model 4: micropore diffusion model. diffusion and adsorption Model 5: micropore model. Model 6: adsorption model. diffusion and adsorption Model 7: macropore model. Model 8: external film mass transfer model. The degenerate models and the criteria for their validity are given in Appendix A. The parametric domain of validity of the parent and degenerate models is illustrated in Fig. 1. APPROXIMATION

In order to determine the frequency response, the periodic steady state of the system has to be determined. Since the model equations are generally nonlinear, numerical techniques or approximate analytical methods must be used to obtain solutions. The approach taken here involves two stages of approximation, namely: (i) The linearisation of the coupling equation (If) or (2f) about the equilibrium state using a Taylor series approximation, neglecting terms of O( u’) and higher,

Analysis of the frequency response method for sorption kinetics

1

I

I

I

I

41 Model

Model

21

I

I -------+-------I I I

1

+BB

Model

Model

3

I I

5IModel

11

I I

I

Fig. 1. The parametric domain of validity of model 1 and its degenerate models.

giving

- kd(l + d)(C,, - C,,) + O(u’)

(3a)

When 1 is very small, the adsorption isotherm is linear, and eqs (3a) and (3b) become exact expressions. (ii) The assumption that the bulk-phase pressure and concentration response may be taken as harmonic in the periodic steady state, i.e P(t) = pC[l + pe’(ror+O)]

WI

1107

a harmonic perturbation of the external fluid concentration. These! equations are, thus, susceptible to linear integral transform techniques, such as the Laplace transform. The method of solution is illustrated in Appendix B for the case of coupled macropore and micropore diffusion and adsorption mechanisms (model 1) in a bidisperscd structured sorbent particle with spherical macroparticle and microparticle geometry. The frequency domain solutions of the in-phase and out-of-phase characteristic functions associated with the parent and all degenerate models for slab, cylindrical and spherical macroparticle and micropartitle geometries are presented in Tables 2-4 respectively, whilst the corresponding Laplace domain solutions are given in Appendix C. The evaluation of the characteristic functions at a given dimensionless angular frequency w*, for the macroparticle geometry, microparticle geometry and mode1 of interest, may be affected using either the frequency or the Laplace domain solutions; however, in the latter case, complex arithmetic is required. Evaluation of the modified Bessel functions of the first kind of orders zero and one for complex arguments, required in the case of cylindrical macroparticle geometry, may be affected using the algorithm presented by Amos (1986), whilst the evaluation of the Kelvin functions and their derivatives may be affected using the algorithms presented by Burgogne (1963) and Abramowitz and Stegun (1965). Typically, the evaluation of the characteristic functions over the relevant portion of the frequency spectrum required about 2 CPU seconds on an IBM 308 1 mainframe computer. The time-domain periodic steady-state solution for the sorbent uptake B* may be. expressed in the form: B* = pe i(m*r++’ (da,.l - i a,,,)

IPI G IUI & = p&m-r+*)

(W

IPI

Q

(5)

IU

with

IPI =s lvl. Implicit in this approach is the assumption that the higher-order harmonics in the pressure response may be neglected. This assumption greatty simplifies the analysis. Although the choice of functional form is arbitrary, for harmonic perturbation of the batch system volume, experimental studies have confirmed that eqs (4a) and (4b) provide a consistent representation of the experimental data (Yasuda, 1982; Yasuda and Sugasawa, 1984; Yasuda and Yamamoto, 1985). It may be noted that once the system of equations has been linearised, the model equations may be formulated in terms of a linear Volterra integral equation of the third kind. SOLUTION OF THE MODEL

EQUATIONS

Equations (2a)-(2e) and (2g) and (2h) together with eqs (3b) and (4b) form a linear system of coupled parabolic partial differential equations, which describe the sorbate concentration distribution within a bidispersed structured sorbent particle subjected to

where Q is the phase shift corrected for apparatus effects determined from a blank run, and dRcll and 6 hs are the overall in-phase and out-of-phase characteristic functions, respectively (Yasuda, 1982), and are of the form

(64 6 Imrs = Aswhs

(

+ &,a,.,

~1 + (1 “: I) &+..I -(1 y k) &=*n

> (6b)

where AsyPerl and A+,,,_, are the macroparticle inphase and out-of-phase characteristic functions for macroparticle shape factor sM, and A.“_, and

ROBIN G. JORDI and DUONG D. Do

1108 Table 2. In-phase

and out-of-phase

characteristic functions geometry Model 1 characteristic

Macroparticle Case 1. Film mass transfer resistance Bi

( Bi &

611 = r

( 2Bi fi

611 = r

cos (O/2)] + sin (O/Z) sin [2$

cos

{COS

(e/2) sinh [2+

cos

sin (O/2)]} >

sin (O/2)]} +

cos (O/2)] - cos (e/2) sin [2fi

(e/z)] (r + Bi’) +

sin (e/2)]} >

sin (O/2)] (Bi2 - r) +

(8/Z)] - cos [2$

cos

{sin (O/2) sinh [2&

and microparticle

+

{cos (O/2) sinh [2J r ~0s (O/Z)] - sin (e/2) sin [2J;

cash [2& ( 2 Ei &

sin (e/2)]}

cos (O/2)] (r + BP) + cos [2fi

r sin (0) {cash [2$ ( Bi fi

macroparticle

functions

cos (O/2)] - cos [24

{cos (O/2) sinh [2J

cash [2fi

Bi

Case 2. Negligible

r cos (0) {cash [2&

for slab

[2,1?

sin (O/2)] 1 >

sin (e/2) J (Bi* - r) +

cos (e/2)] - sin (O/2) sin [2&

sin (O/Z)]} >

film mass transfer resistance d

6

_ cos (e/2) sinh [2J r 1Rfi {cash [2& 1R

cos

(e/2)]

cos

+ sin (e/2) sin [2J; + cos [2&

(e/2)]

= sin (O/2) sinh [2,/ z co6 (e/2)] - COB(e/2) sin [2J; fi

{cash [2&

cos (O/2)] + cos [2&

sin (e/z)]

sin (O/Z)]} sin (e/2)]

sin (O/2)]}

where )I.,, = re” u2a?y {q,’ [cash (q,,) - ~0s (v,Jl+ ‘I~@Csinh (a) - sin(qJ1) qM= (1 + I){cosh(q,)(q: + 28*) - cos(tl.)(q: - 2~37 + 2W,,Csinh(rl,) - sin(q

+ (1 + 1){Gosh (+)(v;

iu#y v,, [sinh (a,) + sin (rl,Al + 287 - co8 (rl,)(rt: - ZW + 29%Csinh

Microparticle characteristic As for mode1 5 Macroparticle As for model

(%I -

sin(%)I}

+ ia10*

functions

Mode1 2 characteristic

functions

1 with qM given by:

u2yqr[sinh(q,) - sin( + in w+ + io2yv,Csinh(v,,) + sin( G = 2(1 + L)[cosh (q,) + cos (qr)] ’ 2U + 4 [a-h (rl,A+ ~0s(q,Jl

Microparticle characteristic As for model 4 Macroparticle Case 1. Film mass transfer resistance

bl = qH {cash 611 =

functions

Model 3 characteristic

functions

BP [sinh (qnr) + sin (v~)] (qH)(q:/2

i BP)

-

cos(vM)(vb/2

-

Bi2) + rlHBi Csinh (tld

-

sin (bdl

1

vu Bi [cash (~3 - cos (q~)] + Bi’ [sinh (q~) - sin (qM)] qna {Gosh (x&&/2

Case 2. Negligible

+ BP) - cos(rl~s)(&/2

- W

+ rtMBi [sinh (rl& - sin

film mass transfer resistance 6

=

1s

-

Csinh(VW)+ sinhM)l vnsCcosh (srr) + cos h)l rsinh (vu) - sin ftrnr)l

1,

rlrs

Ccoshh)

+

~0s

h1)l

where: qw = ,/ZW*[Q~ + u2/(1 + A)] Microparticle

characteristic

s,, =

1

61, = 0

functions

(rid I

1109

Analysis of the frequency response method for sorption kinetics Table 2. (conrd.) Model 4 Macroparticle characteristic functions &a = 1 s,, = 0 Microparticle characteristic functions a,R

=

d

=

I’

Csinh(rl,) + sinhAI vpC=-h (vp) + ~0s WI CsinhhJ - sin(tlr)l ‘I,,Ccosh(G) + cos (rlr)l ‘Ill= J=G

Model 5 Macroparticle characteristic functions As for model 4 Microparticle characteristic functions &a =

Bz Csinh(v,) + sin (tf,,)] rlr {cash (rlJW2

+ @)

- cos(~,)(t13/2 - 8’)

+ rfBaBCsinh(q,) - sin(rt,)lI

Macroparticle characteristic functions As for model 4 Microparticle characteristic functions

Model 7 Macroparticle characteristic functions As for model 1 with Q, given by

Microparticle characteristic functions As for model 6

A..+.s are the microparticle in-phase and out-ofphase characteristic functions for microparticle shape factor sr. Substitution of eqs (4b) and (5) into eq. (2i) together with eq. (2j) gives on differentiation with respect to time and division by eiwar: e4[1

- Zue’“” + K*&.,

Equating Re:

- ifSI,,)]

the real and imaginary

cos (4) - 2uco.s (~9 + K* [cos (4) c&,

- ;=

0.

(7)

parts to zero gives

+ #) + sin (c$) Sr,,J

= i

(8a)

Im:

sin(+) - 2usin(w*r

+ 4)

+ K* [sin (4) &,.I - cos(#

6,-J

= 0.

(8b)

Neglecting the terms of O(u), and solving for &.., and &n** gives 0

;

cos(#

0 5

- 1 = IC*r&.,

Pa)

sin (4) = K+&,.

Alternatively, the terms involving c.r*r in eqs (8) may be eliminated giving the following identity linking the characteristic functions to experimentally de-

ROBIN G. JORDI and DUONG

1110 and out-of-phase

Table 3. In-phase

characteristic

functions

for cylindrical

Model 1 characteristic

Macroparticle

D. Do macroparticle

and microparticle

functions

Case 1. Film mass transfer resistance 2Bi

[Br cos (e/2) + I, sin (O/2)] {J;

[cos (O/2)% - sin (8/2)f,]

( + [I, cos (f@) - 9P, sin (O/2)] {J &=

J;{J; 2Bi

[cos(8/2)fSr

- sin(8/2)&]

+ BiiRc}*

[aI ~0s (e/2) + I1 sin (O/2)] {J;

+ {J;

[cds (e/2)1,

( - [I,00s (e/2) - 53, sin (e/2)] {J;

[cos(8/2)1,

Case 2. Negligible

cc0s(e/2)aI

- sin(W2)IJ

+ Bi a,}s

+ sin(O/2)*,]

+ Bi I,]}’

+ sin (@/2)Sr, 1 + Bi IO}

[COS@/2)st,

- sin (e/2)1,]

621 = fi{J;

+ Bi 9ZO}

[cos (O/Z) 1, + sin (O/2) 9?J + Bi I,} >

+ Bi 9?,} >

[cos(e/2)r, + sin (e/2)a,]

+ {J;

+ Bi r,l}l

film mass transfer resistance s

_ 2 cc0s (e/2) (ra, 4 2s -

+ IO1, ) + sin (e/2) (9, II - r, aI ) 1 J;

r, =

- 2 $ k-1

(&

+ 1;)

( - iP rzt-, cJ; cos (e/2)1 .h- I CJ sin (e/z)] % =

ReCk(

w, = g ( - Ilkr,, - 1[JF ~0s (e/2)1 {J,, CJ; sin (em3 - Jzk- z Cd sin wi k-i

1

I1 = Kmcr, (&)I 4 =$y

‘+ 1h cd

f r,IJ;

cos W2)l {J2,-I C& sinW2)l - Jzk+ 1 CJ sin W)l 1

co8 (e/2)1 4 CJ; sin (em and qnr = re*

rl,=&% Microparticle characteristic As for model 5

functions

Model 2 Macroparticle characteristic functions As for model 1 with nrr given by

%=Jollr. Microparticle characteristic As for model 4

Macroparticle Case 1. Film mass transfer resistance

Model 3 characteristic

functions

functions

geometry

Analysis of the frequency response method for sorption kinetics Table 3. (contd.)

Case 2. Negligible film mass transfer resistance

where qM =

o*[q

+ a,/(1 + A)]

Microparticle characteristic functions IS*, = 1 b*, = 0 Model 4 Macroparticle characteristic functions s 2R = 1 62, = 0 Microparticle characteristic functions

Model 5 Macroparticle characteristic functions As for model 4 Microparticle characteristic functions

Model 6 Macroparticle characteristic functions As for model 4 Microparticle characteristic functions 6

rR

=-

1 (1 + $9 &I

--

821 - (1 + %=

v:,

CO* ( 21y >

Model 7 Macroparticle characteristic functions As for model 1 with qM given by ~+~*%I ‘lnr

=

(1

+

A)(1

+

q:,

UP %I= 28y ( > +

iw

[ O1

+

(1

+

02 A)(1

+

Microparticle characteristic functions As for model 6

s:,

1

1112

ROBIN G. JORDI and DUONG

Table 4. In-phase and out-of-phase

characteristic

functions

for spherical

Model 1 characteristic

Macroparticle

D. Do

macroparticle

and microparticle

geometry

functions

Case 1. Film mass transfer resistance cos (0) {cash [2,,‘7 coo (O/2)] (r - Bi + 1) + cos [2& 3Bi

i + St,,& sin (t?){sin (e/2) sinh [2&

&I =

l)&

(g/2)] -

+ (Bi - 2)&

631 =

cos

sin (O/2)] [r - (Bi -

ces (O/2)] (r - Et + 1) + cos [2.,6

[cos (g/2)] sinh [2$

sin (O/2)] (r + Bi - 1) sin (g/2)]}

cos @J/Z)][r + (Bi - l)*] + cos [2fi (O/2) sinh [2fi

sin (e/2)]} >

sin (e/2)]}

cash [2&

1

1)2]

(e/z) sin [2$

cos (6J){sin (e/2) sinh [2J r co9 (e/z)] - co9

{cos

(e/2)]}

cos (g/2)] + sin (g/2) sin‘[2fi

( - Blti

’ ( + Z(Ei - 1)fi

sin (8/Z)]}

(g/2) sin [2,,&sin

{cos (e/2) sinh [2J T cos (e/2)] + sin (ep) sin [2J

sin (8) {cash [2$ 3Bi

Case 2. Negligible

cos

cash [2,,& cos (g/2)] [r + (Bi - l)s] + cos [24 r ( + 2(Bi -

sin (O/2)] (r + Bi - 1)

[cm (IT/~)] sinh [2 ,/? cos (O/2)] + sin (e/2) sin 124

+ (Bi - 2).$

sin (O/2)] [r - (Bi -

cos (O/2)] + sin (O/2) sin [2$

1)2]

sin (O/2)]} >

film mass transfer resistance co* (e/2)] - sin (O/2) sin [2J; 4 6

_ 3 SR -

{cash [2J;

cos (O/2) sin [t& fi

c0s (e/2)] -

cos

[2J:

sin (O/2)] + sin (e/2) sinh [2&

{cash C2&

cos (g/2)] -

cos

sin (e/2)]

--

sin (e/2)]}

[2$

cos (e/2)]

coa (0) r

--

sin (e/2)}

>

sin (0) r

>

where qkl = reJe 30~9~ {cos,h(qJi~~ - 2W - 01 + co~(~,,)id + W3 - 111 + r1,W- 2)isinhhJ+ ~inolr)l~ ‘M =(l + 1){cosh(qJ[qf + 2(ia - l)‘] + cos (q,,) [q: - 2(S9 - I)‘] + 2(9 - l)q& [sinh (t&J + sin (qJ]} i3cs@yq,, + (1 + Wash

(UCq:

[sinh (qJ - sin (q,)]

+ 2W - l)*] -+ COS(~~)[~; - 2(zIf - l)‘]

+ 2(1-

l)rl,[sinh(qJ

+ sin(%)])

+ f”rru*

‘Ir =&S Microparticle characteristic As for model 5

functions

Model 2 Macroparticle characteristic functions As for model 1 with qM given by 30s~

(s) +. sin hJ1 - 21 i3wv,,Csinh hJ - sinh,Jl + ialo+ + 2(1 + .I)[cosh(q,,) - cos(~,)J boshh,) - ~0shJ1

Wsinh

‘IM= 2(1

‘184=

&m

Microparticle characteristic As for model 4

Case 1. Film mass transfer resistance

Macroparticle

functions

Model 3 characteristic

functions

3Bi* qH [sinh (qM) - sin (l]~)] 63R = t&{cosh(q,)[?&/2 631 =

3Bi{cosh(q,)[q& q&{cosh(~,)[t&/2

Case 2. Negligible

+ (w - l)a] + cos(qH)i&/2 - 2(Bi - I)] +

cos(q~)Cq

-Pi $

l)‘l + cosh)i&/2

+ (Bi -

+

-

1Yl + hnr(Bi - Uisinhhd + ~idfdl~

2(Bi - l)] + qM(Bi - 2)[sinh(~,) + sin (qr,r)]} - @i - I)‘1 + m(Bi - Uisinhh) + sinhdl}

film mass transfer resistance

d

3,

=

3 isinh 0)~) + sinbdl rlnr Cashh) - ~0sh)l

where qrr = J2o*

[at + q/(1

_- 6 s: $ A)]

Analysis of the frequency response method for sorption

kinetics

1113

Table 4. (coned.) Microparticle

characteristic

functions

.&a = 1 ssr= Macroparticle

0

Model 4 characteristic

functions

63* = I 63, = 0 Microparticle 6 d

3R

_

”

characterisfic

= 3 [sinh

functions

(~1 - sin (~13 WI

q, Cash (a) - cos

3[sinh(q,) + sin( 6 Q,[cash (v,,) - ~0s@I,,)] -7 Vr

Model 5 Macroparticle characteristic functions As for model 4 Microparticle characteristic functions 3@5

“,

= vi Wh

(v,,,)Clr,2/2+ W - 0’1

3ra{~sh(q,,)[‘1;

Csinh (v,) - sin (%,)I

+ cos 07,AC’1:/2 - 9

- 2t.68 - 01 + cos(tl,)Cv:

“I = q:{cosh(q,,)[t&2

+ (9 - 1j21 + cos(rt,)Er&2

- 1)‘l + %W - Nsinh h) + sin hJ1)

+ 2W - 1)l + ?I# - (9 - 0’1

Model 6 Macroparticle characteristic As for model 4

- 2)Csi~hJ + sinh,hl~ + %A@ - 1)CsinhOrJ + sin(%

functions

Microparticle characteristicfunctions

Model 7

Macroparticle characteristicfunctions As for model 1 with Q., given by

Microparticle characteristicfunctions As for model 6 tennined variables: 01 4-L + 2; ccos (d)(l P3> ( + sin(t$)K*L,,.,l - (K*&,,J2

= 0.

+ K* L.1)

- (1 + K* SRe.d2 (10)

It is advantageous to define corrected overall in-phase

and out-of-phase characteristic Functions, based on the normalisation of the overall in-phase character-

istic function, &,,, to unit magnitude at very low angular Frequency. We shall denote these normal&d characteristic functions by ARcal and AhI, respectively. The upper and lower limits for v/p derived in Appendix D, are given by V

-=l+K* ’

[ai +&I (limit for very small u*)

(lla)

1114

ROBING. JORDIand DUONGD. Do 1 .oo

V

- = I (limit for very large w*). P

(ll’4

The variable 4 has a value of zero for very small and for very large values of o*. Thus, the corrected in-phase and out-of-phase characteristic functions are defined by

0.75

% 0.50

--

v:

0.06

--

V!

0.16

v: 0.26 -

APproxim&ion.

4 0.25

*r”“‘=[O1

(12b)

Z1),

0.001

The characteristic functions 811ca1 and Blm.l may be determined from the relative amplitude of the volume and pressure variations, v and p, respectively, and the corrected phase angle 4. Since K* [or + uZ/(1 + A)] is directly proportional to the gradient of the equilibrium isotherm, the intensity of the frequency spectrum is proportional to the gradient of the adsorption isotherm at equilibrium (Yasuda, 1976a).

COMPARISON OF

APPROXIMATE

EXACT

SOLUTIONS

WITH

SOLUTIONS

In order to quantify the nature and extent of the errors introduced by the approximation technique we have used, a comparison of the approximate and exact solutions is presented. The exact solutions were determined numerically by a combination of the orthogonal collocation method (Villadsen and Michelsen, 1978) for the space variable discretization and integration of the resulting differential algebraic system using a stiff differential algebraic equation solver which implements the backward difference formulae (Petzold, 1982). In summary, two types of approximation have been used, namely: (i) Linearisation of the governing equations (2) using a Taylor series approximation of O(v”). (ii) An assumption that the response of the batch system pressure may be taken as harmonic in the periodic steady state. It is expected that the accuracy of the approximations will degrade for large fractional changes of the batch system volume, and under conditions where the isotherm exhibits appreciable local non-linearity. We shall consider the case of the micropore diffusion model (model 4). The exact and approximate normalised overall inphase and out-of-phase characteristic functions for the micropore diffusion model (model 4) are illustrated in Fig. 2 with v, the maximum fractional batch system volume change, as parameter. Typical values of the parameter v taken from the literature range from 0.01 to 0.05. It is clear from Fig. 2 that, in the

0.01

1

0.1

10

100

1000

+ 0

Fig. 2. The effect of the magnitudeof the volume perturba-

tion on the exact and approximate normalised overall characteristic functions, AR and A,, under micropore diffusion control (model 4), for spherical microparticle geometry. Note

that the approximateand exact out-of-phasecharacteristic functionsare superimposed.

linear region of the isotherm, the parameter v has little effect on the out-of-phase characteristic function, since the approximate and exact functions are almost coincident. In the case of the in-phase characteristic function,, the approximation deviates from the exact function for large v particularly at low and high angular frequencies. This is presumed to be mainly due to the error incurred by approximation (ii). The exact and approximate overall in-phase and out-of-phase characteristic functions for the micropore diffusion model (model 4) are illustrated in Fig. 3 with 2, a measure of the approach to saturation of the Langmuir isotherm, as parameter. In this case, the approximate in-phase characteristic function is larger than the exact in-phase characteristic function over the whole range investigated. It should be noted that the relative error of the approximation increases as A increases, The approximate and exact out-of-phase characteristic functions are almost coincident over the whole range investigated. After extensive computation we have found that for maximum fractional changes in the batch system volume of less than 0.05, the approximate in-phase and out-of-phase characteristic functions for the micropore diffusion control model, are in very good agreement with the exact characteristic functions, for an equilibrium uptake of up to 70% of saturation. Furthermore, the general trends in the exact characteristic functions are matched by the approximate functions. We have found that the exact and approximate out-of-phase characteristic functions are almost coincident over a wide range of the parameter space, except at very high angular frequency. MODEL

SIMULATIONS

The influence of a particular step in the sorption mechanism on the frequency response depends on the

Analysisof the frequencyresponsemethod for sorption kinetics

% 0.25

0.00 0.001

0.01

0.1

1

10

100

1000

+ 0

Fig. 3. The effect of the equilibrium uptake on the exact and approximate overall characteristic functions, 6, and S,, under micropore diffusion control (model 4). for spherical microparticle geometry. Note that the approximate and exact out of phase characteristic functions are almost coincident.

time scale for that step relative to the time scale of the perturbing processes. If the time scale of a process is much larger than that of the perturbing processes, the process is uninfluenced by the perturbations. Conversely, if the time scale of a process is much shorter than that of the perturbing processes, the process approaches equilibrium with regard to the perturbations. When the time scales of the process and perturbing processes are comparable, the process variables oscillate at the imposed frequency, but remain out-of-phase with the perturbations. When the sorption mechanism involves several steps in series, the influence of the successive processes may only be observed if the time scales of the proceeding processes are shorter than the time scale of the process in question. In this case we would expect to observe peaks in the out-of-phase characteristic function at the angular frequencies corresponding to the time scales of the slowest process and preceding processes, while their relative magnitudes would reflect the fractional sorptive capacity associated with each process. For sorption mechanisms involving several steps in parallel, the influence of all processes will be observed, since in this case, all processes are subjected to the same perturbation. In this case we would expect to observe distinct peaks in the out-of-phase characteristic functions at the angular frequencies corresponding to the time scales of the individual processes, while their relative magnitudes would reflect the fractional sorption capacity associated with each process. In the case of a sorption process involving film mass transfer, macropore diffusion, the surface barrier and micropore diffusion, macropore diffusion acts in series with film mass transfer and in a series/parallel manner with the microparticle sorption mechanisms, thus there can be no simple statement of the overall effect. However, simple limiting cases may be identified. If

1115

film mass transfer is not rate-limiting, and if the time scale for micropore diffusion is significantly shbrter than the time scales for the surface barrier or macropore diffusion, we would expect the response spectrum to reflect the contribution of micropore capacity but to contain little information about micropore diffusion dynamics. Similarly, if the time scale of the surface barrier is much shorter than the time scale for macropore diffusion, we would expect the response spectrum to be uninfluenced by the surface barrier. The resonant frequency corresponding to the time scale of each single resistance model, has been determined using the approximate analytical solutions and the resulting relations are presented in Table 6. Once the resonant frequency corresponding to each sorption mechanism has been determined, the dominant sorption mechanisms may then be predicted. We present the salient features of the single resistance models before considering coupled sorption mechanisms. Model 4: micropore diffusion model (B P 1 and y Q 1) This type of mode1 was used by Billow et al. (1986). van-den-Begin et al. (1989) and van-den-Begin and Rees (1989) in their studies of diffusion of light linear hydrocarbons in silicalite/ZSM-5 zeolite, and by Yasuda (1982) and Yasuda and Sugasawa (1984) in their study of diffusion of krypton in Na-mordenite and krypton and xenon in 5A zeolite. The overall norrnalised in-phase and out-of-phase characteristic functions for slab, cylindrical and spherical microparticle geometry, under micropore diffusion control are presented in Fig. 4. The normal&d in-phase characteristic function is asymptotic to unity for w+ (L,,) =$0.1, decreases rapidly until o+(L,,) x 10 and thereafter decreases asymptotically to zero as tanh (m)/(m)

[for n1 Q u2/(l + A)].

1.00

0.75

% 0.50

--.

Sphere Cylinder Slab

4 0.25

0.00 n

Fig. 4. The effect microparticle shape on the normal&d overall characteristic functions, An and A,, for slab, cylindrical and spherical microparticle geometry, under micropore diffusion control (model 4).

ROBING. JORDIand DUONGD. Do

1116

The magnitude of the normalised in-phase characteristic function. A(s,&,,, decreases as the microparticle shape factor sp increases, i.e. A(2) ~ca,< A(1) lien,< A(O)...,.

ml:

(13)

The location of the maximum in the out-of-phase characteristic functions expressed in terms of the dimensionless angular frequency based on the micropore diffusion time scale, w+(L,, s,,), are offset as follows (cf. Table 6):

z 2.541

(14a)

0.00 : 0.001

or in the ratio: w+(L,,2):w+(L,,

l):o+(L,,O)

(14b)

A(2),,,,

z 0.355:A(l),,,,

0.01

x 1.00:1.23:1.97.

The maximum magnitudes of the out-of-phase characteristic functions are weakly dependent on the macropore capacity as indicated in eqs (6b) and (12b), however, for (or + u2/(1 + A), the maxima are given approximately by z 0.377:A(O)r,,, z 0.417. (15) It is interesting to note that at high angular frequency, the in-phase and out-of-phase characteristic functions for slab, cylindrical and spherical microparticle geometry are mutually asymptotic, when displayed in terms of the non-dimensional angular frequency w+(L,,). This is expected, since as the frequency of the perturbation increases, the amplitude of the concentration waves propagating into the microparticle decrease increasingly more strongly with increasing distance from the particle surface; thus, at very high frequency, only a thin shell near the surface of the microparticle experiences appreciable fluctuations in concentration. This assertion is verified for the case of a semi-infinite slab exposed to harmonic oscillations in the surface concentration, which is considered in Appendix E. Model 6: adsorption model [s.+?< 1 and y -z O(l)] This type of model was used by Yasuda (1976bl and Yasuda and Saeki (1978) in their study of the sorption kinetics of ethylene over zinc oxide, and by Goodwin et al. (1985) in their study of the role of the support on hydrogen chemisorption on supported rhodium catalysts. The overall normalised in-phase and out-of-phase characteristic functions for slab, cylindrical and spherical microparticle geometry, under adsorption control are presented in Fig. 5. The characteristic functions are identical for the various microparticle geometries when expressed in terms of the dimensionless angular frequency based on the adsorption time scale, w- (Lfl). The normalised in-phase characteristic function is asymptotic to unity for w- (L,) S 0.1, and asymptotes to zero for o-(L,) 3 10 when oi < 02/(1 + A). The

K-: 1.00

\

40.251 /\\

w+(L,, 2) C 1.29O:w+(L,, 1) c 1.581:wf(L,,,0)

\

I

0.01

vs: 0.88 *

0.1

1

10

1 I

1 loo

1000

0

Fig. 5. Approximate and exact normalisedoverall characteristicfunctions,An and A,. for slab.cylindricaland spherical microparticle geometry, under adsorption control (model 6), illustrating the independence of microparticle shape.

out-of-phase characteristic function is symmetric about w-(L,) = 1 when displayed on a log axis. The location and magnitude of the maximum of the out-of-phase characteristic function expressed in terms of the non-dimensional angular frequency, o-&), are given by (cf. Table 6) CL-(&) = 1

A

R.2.l - Inns= 1 1 -A

(16)

=z [ O1 + 2(1 + A) (X0.5 for small ai). er

(17)

Model 5: micropore d&%&ion/adsorption model [Go E O(1) and y a 11 The salient features of the overall normal&d inphase and out-of-phase characteristic functions for spherical microparticle geometry under mixed micropore diffusion/adsorption control are presented in Fig. 6, with 1 as parameter. For I r 10 the model degenerates into the micropore diffusion model (model 4), while for W < 0.1 the model degenerate into the adsorption model (model 6). The frequency response technique is sensitive to the presence of a surface barrier for B < 10 since the resulting characteristic functions are significantly different from those resulting from micropore diffusion control alone. Importantly, the in-phase characteristic function crosses over the out-of-phase characteristic function at a moderately large angular frequency and remains a noticeable effect up to values of L%? z 10. This is not unexpected, since at high angular frequency, the microparticle properties near the particle surface influence the frequency response to a much larger extent than the nature of the microparticle interior, as indicated in Appendix E. This feature may be used to test for the presence of a surface barrier

Analysis of the frequency response method for sorption kinetics

%

1117

--

Sphars.

0.50

4 0.25

: 0.00 = 0.001

0.01

0.1

l+

10

100

1000

w

Fig. 6. The effect of the relative magnitude of the adsorption rate to the micropore diffusion rate on the approximate normal&d overall characteristic functions, Aa and A,, for spherical microparticle geometry, under micropore diffusion and adsorption control (model 5). illustrating the limiting bchaviour for small and large values of 1. and, hence, to distinguish between the micropore diffusion and micropore diffusion/adsorption mechanisms, without performing experiments with different crystal sizes, if macropore diffusional resistance may be shown to be negligible. The location and magnitude of the maximum of the out-of-phase characteristic functions for slab, cylindrical and spherical microparticle geometry under mixed micropore diffusion/adsorption control are presented in Fig. 7, with 93 as parameter. This allows the determination of the micropore diffusion and adsorption time scales from experimental measurement of the magnitude of the peak maximum in the outof-phase characteristic function. The sensitivity of this method is greatest for spherical microparticle geometry, since the difference between the maximum magnitudes of the out-of-phase characteristic functions under adsorption control and under micropore diffusion control is greatest.

Fig. 7. The location and magnitude of the maximum of the normal&d overall out of phase characteristic function, A,, for slab, cylindrical and spherical microparticle geometry under mixed micropore diffusion and adsorption control (model S), with L&? as the parameter.

0.25

-

0.001

Model Y s.

3: macropore

diffusion

model [s%?> O(1)

11

This mode1 incorporates film mass transfer and macropore diffusion mechanisms. Film mass transfer controls the sorption process for small Biot numbers, and macropore diffusion controls the sorption process for large Biot numbers. The overall normalised in-phase and out-of-phase characteristic functions for slab, cylindrical and spherical macroparticle geometry, under macropore diffusion control are presented in Fig. 8. The normalised in-phase characteristic function is asymptotic to unity for uP(L) + g&l + A)] B 0.1, decreases rapidly until w*(L)[oi + az/(l + A)] x 10 and thereafter decreases to zero asymptotically as

tanh

{,bw*WC~t + e/(1 + All>/ J2o*(L)Ca,

0.01

0.1

1

10

100

1

90

and

+ u2/(1 + A)].

Fig. 8. The effect macroparticle shape on the normalised overall characteristic functions, As and A,, for slab, cylindrical and spherical macroparticle geometry, under macropore diffusion control (model 3).

The magnitude of the normalised in-phase character-

istic function, A?I(s~M)~~.~, decreases as the macropartitle shape factor sM increases, i.e. A(2) ~e.1G AU) RC=IG A(O),,,. The location characteristic dimensionless pore diffusion

(18)

of the maximum in the out-of-phase functions expressed in terms of the angular frequency based on the macrotime scale, w*(L, sM), are offset as fol-

1118

ROBING.JORDI

lows (cf. Table 6): w*(L, 2) x o*(L,

1) x

w*(L, 0) ;tr

1.290 1.581 + 1):

W(J-)[a,

2.541 e1 + %/(I

(194

+ 1)

or in the ratio

w*(L, 2):w*(L,

l):w*(L,

0) w 1.00: 1.23: 1.97.

(19b)

The maximum magnitudes of the out-of-phase characteristic functions are independent of the distribution of sorption capacity between macropore and microparticle phases, as implied by eqs (6b) and (12b). The maxima for the three macroparticle geometries are given by

~(~hma,x 0.355:A(l),,,,

z 0.377:A(O),,,,

z 0.417. (20)

At high angular frequency the characteristic functions are independent of macroparticle geometry when displayed in terms of the non-dimensional angular frequency w*(L)[ar + ez2/(1 + A)]. Under film mass transfer control, the characteristic functions are not dependent on the macroparticle geometry when expressed in terms of the dimensionless angular frequency based on the film mass transfer time scale, c+(L)[a, + oJ(l + ,I)]. The overall normalised characteristic functions under film mass transfer control are presented in Fig. 9. For wr(L)[r~r + aJ(1 + A)] d 0.1, the normalised totic

in-phase characteristic function is asympto unity, and asymptotes to zero for

wr(L)[ar

+ q/(1

D.Do

acteristic function is symmetric about o,(L)[a, + uz/ (1 + ,I)] = 1 when displayed on a log axis. The location and magnitude of the maximum of the out-of-phase characteristic function expressed in terms of the non-dimensional angular frequency, wr(L.)[er f us/(1 + A)], are given by (cf. Table 6)

01 + Q/(1 + A): er + $/(l

and DUONG

+ ,I)] > 10. The out-of-phase char-

+ %/(I

A Rell -A -

+

41 = 1

(21)

rmns= l/2.

(22)

The overall normal&d in-phase and out-of-phase characteristic functions for spherical macropore geometry, under mixed macropore diffusion/film mass transfer control are presented in Fig. 10, with Bi as parameter. For Bi > 10 the model degenerates into the macropore diffusion model (model 3: case 2), while for Bi G 0.1 the model degenerate into the film mass transfer model (model 8). This is consistent with Do and Rice (19PO), who formulated a model of the sorbate uptake into a single particle having a pseudohomogeneous structure, incorporating pore and surface diffusion and assuming a linear equilibrium isotherm. By comparing the model prediction with that predicted in the case of external film mass transfer control, and by considering the magnitude of gasphase molecular diffusivities, they concluded that for systems where surface diffusion provides a negligible contribution to the mass transfer rate, external film mass transfer is never rate-controlling. They showed that an approximate lower limit of the Biot number for gas-phase systems with negligible surface diffusion is Bi 2 10. The location and magnitude of the maximum of the out-of-phase characteristic functions for slab, cylindrical and spherical macroparticle geometry under mixed macropore diffusion/film mass transfer control are presented in Fig. 11, with Bi as parameter.

--

-

Bi:

10.0

Bi:

1.00

w: 0.10

-

0.75 --

v: 0.06

4 0.25 -

0.001

0.001

0.01

0.1

1

10

100

1000

Fig. 9. Approximate and exact normalised overall characteristic functions, An and A,, for slab, cylindrical and spherical microparticle geometry, under film mass transfer control (model B), illustrating the independence of macroparticle shape.

0.01

0.1

1

10

100

1000

ol(Q1+uz/( 1+A)) Fig. 10. The effect of the relative magnitude of the film mass transfer rate to the macropore diffusion rate on the approximate norraalised overall characteristic functions, AI and A,, for spherical microparticle geometry, under macropore diffusion and film mass transfer control (model 3), illustrating the limiting behaviour for small and large values of Bi.

Analysis

of the frequency

responsemethod for sorption

4&X

--.-

0.4

t ’

0.3 0.001

1119

kinetics

‘-k;..

Sphere. Cylinder. Slab.

‘\‘\ \

'.. 'L-_ '--_ i

I

0.01

0.1

B'i

10

100

I

1000

Fig. 11. The location and magnitude of the maximum of the normal&d overall out of phase characteristic function, A,, for slab, cylindrical and sphericalmacroparticlegeometryundermixed macroporediffusion and i&n mass transfer control (model 3), with Bi as the parameter.

Table 5. Definition of non-dimensional angular frequency based on macropore diffusion, micropore diffusion and adsorption rate time scales Non-dimensional

angular frequency based on characteristic length Basis. Macroparticle time scale &@)

_

cd’ Cenr + (1 -

hf) C,JC,l

s&

Basis. Adsorption

time scale

time scale

Characteristic L=

particle half-width

Particle External

Particle geometry Infinite slab Infinite cylinder Spherical

d and y (Bi B 1) The parametric domain (9, y) may be subdivided into regions reflecting the dominance of a singlesorption mechanism on the overall sorption process, by means of the relations presented in Tables 5 and 6. The domain for 6, = 0.01, CT,= 0.99 and d = 1.00 is illustrated in Fig. 12. The criteria for the subdivision and the dominant and the secondary sorption mechanisms are presented in Table 7 in the case of no significant film mass transfer resistance, and in the case of macropore diffusion and film mass transfer resistances. Since macropore diffusion acts in a series/parallel manner with the microparticle sorption mechanisms this subdivision can only be regarded as approximate. General

(cf. Table 1).

Basis. Film mass transfer time scale

Basis. Microparticle

macroparticle geometry, since the difference between the maximum magnitudes of the out-of-phase characteristic functions under film mass transfer control and under macropore diffusion control is greatest.

volume

area available

for diffusion

Characteristic length L=R L, = R, L = R/2 L, = RJ2 L = R/3 L, = R-13

This facilitates the determination of the macropore diffusion and film mass transfer time scales from experimental measurement of the magnitude of the peak maximum in the out-of-phase characteristic function. The sensitivity of this method is greatest for spherical

Case 1. The micropore diffusion rate slower than the adsorption rate. For D x 10.0, the time scale for adsorption is about an order of magnitude less than the time scale for micropore diffusion. The characteristic functions for ~8 = 10.0, (rl = 0.01, gz = 0.99 and d = 1.00, for spherical macroparticle and micropartitle geometry, are illustrated in Fig. 13 with y as parameter. These conditions are represented by the circles in Fig. 12. For y < 0.01, the characteristic functions, AR and A,, exhibit features typical of micropore diffusion control (model 4), as the micropore diffusion time scale is

1120

ROBIN G. JORDI and DUONG D. Do

to peak function

Table 6. Angular frequencies corresponding maxima in-phase the out-of-phase characteristic

66

Model 3. Macropore diffusion model Macroparticle geometry Peak maximum Infinite slab

2.54064 w*(L)*

---_.

[u, + cr*/(l + A)]

cl

J

Infinite cylinder Spherical

6

w’(L) x

1.28986 C% + %/U

4

+ a1

I

0.00

0.25

0.00

0.50 -Y_-

1+7 Fig. 12. The parametric domain (9&y) with approximate subdivisions based on the relative magnitudes of the intraparticle mztss transfer mechanisms, for spherical macropartitle and microparticles, with or = 0.01, tra = 0.99 and 1 = 1.05. The circles,squaresand diamonds representrel-

Model 6. Adsorption model Microparticle geometry Peak maximum Infinite slab fC(L,) = 1 Infinite cylinder u-&J = 1 Spherical w- (L,) = 1 Model 8. Film mass transfer model Macroparticle geometry Peak maximum

V

V

V

Model 4. Micropore diffusion model Microparticle geometry Peak maximum Infinite slab w+ (L&J = 2.54064 Infinite cylinder o+(L,) iz 1.58130 Spherical a+ W,) x 1.28986

ative magnitudesof the adsorption rate to the micropore diffusionrate (Eo)of 10.0,1.00and 0.10,respectively.

.

Infinite slab 1 .oo

Infinite cylinder Spherical

0.75

4 0.50

about two orders of magnitude larger than the macropore diffusion time scale, and about an order of magnitude larger than the adsorption time scale. At high angular frequency, the in-phase and out-of-phase characteristic functions show a small degree of overlap, and asymptote to a low plateau at high angular frequency, reflecting the small contribution of macropore capacity to the overall capacity (ai d u2), For y = 0.10s the adsorption time scale and the macropore diffusion time scales are of similar order, while the micropore diffusion time scale is an order of magnitude larger; thus, we expect micropore diffusion to be the dominant rate-controlling mechanism. Interestingly, the characteristic functions AR and A,, remain almost identical to those for y = 0.01, except for a minor increase in the extent of overlap between the two functions at high angular frequency. For y = 1.00, the micropore diffusion rate is of similar order to the macropore diffusion rate, and the characteristic function A,, and AI. overlap closer to the maximum in the out-of-phase characteristic function. Furthermore, the maximum magnitude of the out-of-phase characteristic function is larger than would be expected for either micropore diffusion control (model 4) or macropore diffusion control (model 3). Interestingly, the maximum magnitude of the out-of-phase function occurs when the ratio of the

4 0.25

0.00 k 0.001

0.01

Oil

1

l

10

loo

1000

w

Fig. 13. The effect of the relative magnitude of the micropore dilfusion rate to the macropore diffusion rate on the approximate normal&d overall characteristic functions, Aa and Ar, for spherical microparticle geometry, under mixed macropore and micropore diffusion and adsorption control (model I), for negligible film mass transfer resistance, when the micropore diffusion rate is significantly slower than the adsorption rate (a cz 10.0).

macrogore diffusion time scale to the micropore diffusion time scale just exceeds unity. For y 2 10.0,the macropore diffusion rate is slower than the micropore diffusion rate by more than an order of magnitude; therefore, we would expect the characteristic functions, AR and A,, to converge to those for macropore diffusion control (model 3), in the limit as y +co, as illustrated in Fig. 13. Case 2. The micropore diffusion rate comparable with the adsorption rate. For 9 rc 1.00,the time scales

Analysis of the frequency response method for sorption kinetics

1121

Table 7. Criteria for subdivision of the parametric domain (a, y) Case 1. Iii(L)

5~ C(s,)

Region

Dominant mechanism

Secondary mechanism

1 2 3 4

Micropore diffusion Micropore diffusion Surface resistance Surface resistance Macropore diffusion Macropore diKusion

Surf&. resistance Macropore diffusion Micropore diffusion Macropore diffusion Surface resistance Micropore diffusion

:

c(0) = 2.541

c(l) = 1.581

c(2) = 1.290

Criteria

Region

WL,) ) CD,) and RW,)~(L

y(L LJc(sJ

LJ

<

C(SM) Cl + s/(1

GM) and %L,,)T(L, < Q1 + aJ(l + 1)

+ -9 -%) >

C(%) QI + @i/(1 + 1)

3

6

WL,,)

z- C&J and Y(L, LJ CW

>

C(sbr)

01 + Ml

+ 1)

Case 2. Bi(L) = 0 [C(s,)] Criteria

Dominant mechanism

Secondary mechanism

Bi(L.) -z C(S,)

Film mass transfer Macropore diffusion

Macropore diffision Film mass transfer

SW)

> C(G)

Case 3. The micropore diffusion rate faster than and micropore diffusion are of the % 0.10, the time scale for same order. The characteristic functions for 1= 1.00, the adsorption rate. For LX? oi = 0.01, + = 0.99 and 1= 1.00, for spherical adsorption is about an order of magnitude larger than macroparticle and microparticle geometry, are illus- the micropore diffusion time scale. The characteristic functions for sZI= 0.10, o1 = 0.01, a, = 0.99 and trated in Fig. 14 with y as parameter. These conditions ,I = 1.00, for spherical macroparticle and microparare represented by the squares in Fig. 12. For y d 0.10, the macropore diffusion rate is at title geometry, are illustrated in Fig. 15 with y as parameter. These conditions are represented by the least an order of magnitude faster than the micropore diffusion rate, and the characteristic functions. AR and diamonds in Fig. 12. For y d 10.0, the macropore diffusion rate is at AIj exhibit features typical of micropore diffusion/ adsorption control (model 5) for comparable rates of least an order of magnitude faster than adsorption adsorption and micropore diffusion. At high angular rate, and the characteristic functions, AR and A,, corfrequency, the in-phase and out-of-phase character- respond to those for the adsorption model (model 6). istic functions asymptote to a low plateau, reflecting The limit of macropore diffusion control occurs only the small contribution of macropore capacity to the for very large values of y. overall capacity (aI Q Q). For y = 1.00,the adsorption time scale is about two LIMITATIONS OF THE FREQUENCY RESPONSE times larger than the macropore diffusion time scale, TECHNIQUE and the characteristic functions, Aa and AI, remain It is important to recognize the limitations of the almost the same as those for y = 0.10. For values of y > 1.00, the system changes from frequency response technique. Broadly, the main limitations relate to the limited range of angular freadsorption control (model 6) to macropore diffusion control (model 3), and the characteristic functions, AR quencies experimentally obtainable, the accuracy with which measurements may be obtained, and the inacand AI, converge to those for macropore diffusion curacy in correcting for apparatus effects. Also incontrol (model 3), in the limit as y 4 co, as illustrated herent in the technique is the requirement to perform in Fig. 14. for adsorption

ROBIN G. JORDT and

1122

DUONG D. Do

1 .oo

0.75

*I3 0.50 *r 0.25

0.00 0.t 30

1 0.01

0.1

* J

1 w

10

100 1000

W

Fig. 14. The effect of the relative magnitude of the micropore. diffusion rate to the macropore diffusion rate on the approximate normal&d overall characteristic functions, Aa and AI, for spherical microparticle geometry, under mixed macropore and micropore diffusion and adsorption control (model 1), for negligible film mass transfer resistance, when the micropore diffusion rate and the adsorption rate arc comparable (6’ z 1.00).

Fig. 15. The effect of the relative magnitude of the micropore diffusion rate to the macropore diffusion rate on the approximate normalised overall characteristic functions, AR and A,. for spherical microparticle geometry, under mixed macropore and micropore diffusion and adsorption contiol (model l), for negligible film mass transfer resistance, when the micropore diffusion rate is significantly faster than the adsorption rate (1 w 0.10).

measurements

important for very large and very small angular frequencies where the two phase angles are similar in magnitude. Interestingly, no batch system frequency response experiments have been reported for activated carbon or carbon molecular sieve material. This is surprising, as these sorbents frequently exhibit isotherms with appreciable gradients over a much larger range of pressure than do zeolite adsorbents. The gas phase pressures reported in the literature range from about 1 to 10 torr, necessitating the use of very accurate pressure transducers.

under periodic

steady-state

conditions

over a range of angular frequencies, which may be very time consuming for systems with slow dynamic characteristics. Furthermore, only two parameters, namely, the phase angle and amplitude ratio, are extracted from a large body of measured data. Clearly, in order to discern all of the dynamic process active in a given system, one needs to perform measurements covering the complete range of dynamic time scales. The range of angular frequency covered in the reported literature ranges from 0.005 to 60 rad/s, corresponding to a dynamic process timescale range of about 0.015-1500 s. Another important limitation of the frequency response method is the limited pressure range that may bc covered before the accuracy of the experimental data is significantly degraded. Since the intensity of the frequency spectrum is proportional to the gradient of the adsorption isotherm at equilibrium, systems with isotherms exhibiting a plateau or asymptotic features may only be investigated over a limited range of pressures. Such features are found in the L4 system of o-xylene on silicalite at intermediate and high equilibrium uptake, and in the H2-type isotherms, typical of zeolite systems, at high equilibrium uptake. Investigation of eqs (9) and (11) reveals that the relative accuracy of the experimentally determined phase angle is greatest at low angular frequency, whilst the relative accuracy of the experimentally determined amplitude ratio is greatest at high angular frequency, since the pressure oscillation is greatest at high angular frequency. Finally, since the determination of the corrected phase angle necessitates the subtraction of the phase angles determined from a blank experimental run, and an experimental run with adsorbent material, it is to be expected that errors will be more

CONCLIYSIONS

In this paper, we have formulated a model describing the pressure response of a batch system containing bidispersed sorbent material, subjected to a small periodic volume perturbation. This model and associated degenerate models were linearised and solved for slab, cylindrical and spherical macroparticle and microparticle geometries, for the case of small harmonic perturbations of the system volume, under the assumption that the pressure oscillates harmonically in a periodic steady state. The features of the characteristic functions associated with the parent and degenerate models have been illustrated. We have shown that the out-of-phase characteristic function is most faithfully represented by the approximation technique over a large range of angular frequency, when the volume perturbations are small. Further, we have quantified the effect of particle geometry on the characteristic functions, and have found that differences in particle shape offset the angular frequency at which the maximum in the outof-phase characteristic function occurs, by at most

Analysis

of the frequency response method for sorption

a factor of two, when the length scales are expressed in terms of characteristic length. The discrimination between mass transfer rate limiting resistances for cases involving coupled macropore and micropore resistances hinges on rather subtle differences between the characteristic functions associated with the dominant mass transfer resistances. For cases where the dominant mass transfer resistance lies in the microparticle phase, the batch frequency response method is a powerful technique capable of discerning the active mass transfer rate limiting mechanisms over a wide range of time scales, and offers the exciting possibility of investigating sorption dynamics with response times in the intermediate range between those accessible to NMR techniques and those accessible to classical macroscopic sorption techniques. Since the batch frequency response technique involves only small perturbations about the equilibrium, non-isothermal effects which have plagued classical macroscopic experiments may be markedly reduced. Important disadvantages of the batch frequency response method include the specialized nature of the equipment, the need for very accurate pressure measurements, and the limited range experimentally accessible angular frequencies. I Acknowledgelnents-Support from the Australian Research Council is gratefully acknowledged. Support from the Department of Chemical Engineering and the University of Queensland in the form of a University of Queensland Postgraduate Research Scholarship awarded to the first author is gratefully acknowledged.

NOTATION

A

AB

4 b B B*

a? Bi C C, C Be C, Cl4 C cc as 48:6-H

non-dimensional sorbate concentration in macropore void defined in Table 1 non-dimensional bulk sorbate concentration defined in Table 1 non-dimensional sorbate concentration in microparticle defined in Table 1 Langmuir constant (=k./k,), cm3 mole1 total amount of sorbate adsorbed on sorbent material, mol non-dimensional sorbate uptake defined in Table 1 non-dimensional parameter defined in Table 1 Biot number for mass transfer defined in Table 1 sorbate concentration in the macropore void, mol cme3 bulk sorbate concentration, mol crK3 bulk sorbate concentration under equilibrium conditions, mol cm- 3 sorbate concentration in the macropore void in equilibrium with Ce., mol cmm3 sorbate concentration in the micropartitle, mol cm-’ sorbate concentration in the micropartitle in equilibrium with C,,, mol cm-”

ct4 4 k,

kinetics

1123

sorbate concentration in the micropartitle at saturation, mol cm - ’ macropore diffusivity based on void area, cm2 s-1 micropore diffusivity, cm2 s- ’ rate constant for adsorption, cm4 molt 1 s-1

kd

kf KH K*

L

Me %f

rate constant for desorption, cm s-l external film mass transfer coefficient, cm s-’ Henry’s law constant, dimensionless ratio sorbate adsorbed on sorbent to sorbate remaining in bulk phase at equilibrium defined in Table 1, mol/mol characteristic macroparticle half-width based on particle volume per unit area defined in Table 5, cm characteristic microparticle half-width based on particle volume per unit area defined in Table 5, cm mass of sorbent material, g macroparticle shape factor, dimensionless microparticle shape factor, dimensionless maximum fractional batch pressure deviation from equilibrium pressure, dimensionless absolute bulk fluid pressure in batch system, Pa radial coordinate in macroparticle, cm radial coordinate in microparticle, cm macroparticle half-width, cm universal gas constant, J mol- 1 R- ’ microparticle half-width, cm real time, s absolute temperature, K batch system volume, cm3 equilibrium batch system volume, cm3 non-dimensional batch system volume defined in Table 1 non-dimensional radial coordinate in macroparticle defined in Table 1 non-dimensional radial coordinate in microparticle defined in Table 1 gas compressibility factor, dimensionless

Greek letters non-dimensional parameter defined in Y Table 1 6 IIn89 overall out-of-phase characteristic function, dimensionless A SYInuE macroparticle out-of-phase characteristic function, dimensionless A Q~C microparticle out-of-phase characteristic function, dimensionless 6 Real overall in-phase characteristic function, dimensionless A .Irle*l macroparticle in-phase characteristic function, dimensionless A srla.1 microparticle in-phase characteristic function, dimensionless

ROBIN

G.

JORDI

macropore void fraction, dimensionless non-dimensional Langmuir isotherm favourability parameter defined in Table 1 maximum fractional batch volume deviation from equilibrium volume, dimensionless sorbent particle density g cmm3 fractional capacity of macropore defined in Table 1 fractional capacity of micropore defined in Table 1 non-dimensional time based on macropore diffusion time scale defined in Table I phase angle corrected for apparatus effects, rad phase angle measured from blank experiment performed without sorbent, rad experimental phase angle, rad angular frequency, tad s- 1 non-dimensional angular frequency based on macropore diffusion time scale, defmed in Table 5 non-dimensional angular frequency based on micropore diffusion time scale, defined in Table 5 non-dimensional angular frequency based on adsorption time scale, defined in Table 5 Mathematical

i Im

0 Re s Bessel

Bei, Bei; Bet-, Ber; &I J”

svmbols

imaginary number, i = J--i imaginary part of a complex number order real part of a complex number Laplace transform variable and associated

mathtmaticalfinctions Kelvin function of order n (-Im [J. x (xe 3X1/4)] ) derivative of Kelvin function Bei. with respect to the argument Kelvin function of order n { =Re[.J, x (xe3xf’4)]} derivative of Kelvin function Ber,, with respect to the argument modified Bessel function of the first kind of order n Bessel function of the first kind of order n

REFERENCES

Abramowitz, M. and Stegun, I. A., 1965, Handbook ofMathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York. Amos, D. II., 1986, Algorithm 644: a portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Sojt. 12, 265-273. Andrien, J. and Smith, J. M., 1980, Rata parameters for adsorption of CO in beds of carbon paticks. A.1.Ch.E. J. 26. 944-948.

Bmd

DUONG D. Do

Andrieu, J. and Smith, J. M., 1981, Adsorption rate for sulphur dioxide and hydrogen sulphide in beds of activated carbon. A.I.Ch.E. J. 27, 840-842. Boniface, H. A. and Ruthven, D. M., 1985, Chromatographic adsorption with sinusoidal input. Chem. Engng Sci. 40, 2053-2061. Bglow, M., Schlodder, H., Rees. L. V. C. and Richards, R. E., 1986, Molecular mobility of hydrocarbon ZSMS/silicalite systems studied by sorption uptake and frequency rcsponse methods, in New Developments in Zeolite Science and Technoloav. ___ Proceedinas of the 7th International Zeolife Con@ence (Edited by Y. &m&ami, A. Iijima, and

I. W. Ward) Tokyo, August 17-22, uu. __ 579486. Elsevier, Amsterdam. Burgogne, F. D., 1963, Approximations to Kelvin functions. Math. Comput. 83, 295-298. Carslaw, H. S. and Jaeger, I. C.. 1959, Conduction ofHeat In Solids. Oxford University Press, Oxford, London. Do, D. D.. 1990, Hierarchy of rate models for adsorption and desorption in bidispersed structured sorbcnts. Chem. Enanc Sci. 45. 1373-1381. Do, D. b. and Rice, R. G., 1990, Applicability of the extemaldiffusion model in adsorption studies. Chem. Engng Sci. 45, 1419~1421. Goodwin, J. G.. Jr., Lester, J. E., Marcellin, G. and Mitchell, S. F., 1985, Frequency response chemisorntion studies of carbon monoxide hydrogenation cataly&s. ACS Symp. Ser. 288.67-78. Hubsr, M. L. and Jones, M. C., 1988, A frequency response. study of packed bed heat transfer at elevated temperatures. Int. J. Heat Mass Transfa 31, 843-853. Kiirger, J. and Caro, J., 1977. Interpretation and correlation of zeolite diffusivities obtained from nuclear magnetic resonance and sorption experiments. J. Chem. Sot. Faraday Trans. I 73, 1363-1376. Kiirger, J. and Ruthven, D. M.. 1989, On the comparison between macroscopic and N.M.R., measurements of intracrystalline diffusion in zeolites. Zeofltes 9, 267-281. Kawazoe, K., Suzuki, M. and Chihara, K., 1974, Chromatographic study of diffusion in molecular sieving carbon. J. Chem. Engng Japan 7(3), 151-157. Li, Y.-E., Willcox, D. and GonaaleG R. D., 1989, Deterrnination of rate constants by the frequency rcsponsc method: CO on Pt/SiO,. A.I.Ch.E. J. 3S, 423-428. Petzold. L. R.. 1982. A descrintion of DASSL: a differentialjaigebraih equation system solver. Sandia Technical Report SAND82-8637, Livermore, CA. Ruckenstein, E., Vaidyanathan, A. S. and Youagquist, G. R., 1971. Sorotion bv solids with bidisnore structures. Cherk En& Scf:Za, 1305-1318. 1 X Shah, D. B. and Ruthvcn, D. M., 1977, Measurement of xeolitic diffusivities and equilibrium isotherms by chromatography. A.1.Ch.E. J. 23,804-809. Stephanopoulous, G., 1984, Chemical Process Control: an Introduction to Theory and Practice. Prentice-Hall, Englewood Cliffs, NJ. van-den-Begin, N. G. and Rees, L. V. C.. 1989, Diffusion of hvdrocsrbons in silicalite usinn a frwuencv-resnonse method, in Zeolites: Facts, Fig&, Future, Pr&eed&s of the 8th Internatfonal Conference on Zeolftes (Edited by P. A. Jacobs and R. A. van Santen), pp. 915-924. Elsevier, Amsterdam. van-den-Begin, N. G., Rees, L. V. C., Care, J., Biilow, M., Hunger, M. and KHrger, J., 1989, DilTusion of ethane in silicalite-1 by frequency response, sorption uptake and nuclear magnetic resonance techniques. J. Chem. Sot. Faraday Trans. I 85, 1501-1509. Villadsen, J. and Michelsen, M. L., 1978, Solution of Difirential Equations by Polynomial Approximation. PrenticeHall. Ennlewood Cliffs. NJ. Yasuda, Y_r1976a, Frequency response method for study of the kinetic behaviour of a gas-surface system: 1. Theoretical treatment. J. phys. Chem. 80, 1867-1869. Yasuda, Y., i976b, Frequency response method for study of

Analysis

of the frequency response method

the kinetic behaviour of a gas-surface system: 2. An ethylene-on-zinc oxide system. J. phys. C&m. So, 1870-1875. Yasuda. Y. and Saeki, M., 1978, Kinetic details of a gas-surface system by the frequency response method. J. phys. Chem. 82, 74-80. Yasuda, Y.. 1982, Determination of vapor diffusion cam& dents in zeolite by the frequency response method. J. phys. Ck %6, 1913-1917. Yasuda, Y. and Sugasawa, G., 1984, A frequency response technique to study zeolitic diffusion of gases. J. Catal. 88, 530-534. Yasuda, Y. and Yamamoto, A., 1985, Zeolitic diffusivities of hydrocarbons by the frequency response method. J. Carol. 93, 176-181. Yasuda, Y., 1991, Detection of surface resistance in a gas/porous-adsorbent system by frequency response method. Bull. them. Sot. Japan 64,954-961. Yaauda, Y., Suzuki, Y. and Fukada. H., 1991, Kinetic details of gas/porous adsorbent system by frequency response method. J. phys. Chem. 95.2486-2492.

APPENDIX

A; DEGENERATE

MODELS

OF MODEL

2 (macropore-micropore d@iion model) The criteria for validity of this model arc

are the same as eq. (2). after eq. (2f) is A

%I,_~

WI

1125

=p[Am*-AU].

(A4a)

The non-dimensional time scale has been chosen micropore dition time scale:

as the

(3 >

Yr=

RZ

NW

t

L

Model 6 (adsorption model) The criteria for validity of this model are B < 1 and y < O(1). The model equations

become

(ASa) together with eqs (Zi)-(2k). The time scale for this model is

Byr=($g)t. This time scale is independent istics.

I %=1 and y - O(1).

4 xr _, = [l + 41 + A)]-

kinetics

with

1

Model

The model equations replaced by

for sorption

Wb)

of the macropore

character-

Model 7 (macropore-diffusion and mocropore-adsorption model) The criteria for validity of this model are

y % 1 and yl The model equations

I O(1).

become

Model 3 (macropore digrSion model) The criteria for validity of this model are

1

P > O(1) and y 9 1. The model equations

together with cqs (2b)-(2d),

become

dA

u,~+~~=V’A

aA,

(AW

Model 8 (externalfilm mass tranqfer model) The criteria for validity of this model are Bi 4 1. r0 > O(1) and y & 1.

A

AE= [l together with eqs (2b)-(2d)

The model equations

+ A(1 + A)]

4 (micropore di@usion model) Tlie criteria for validity of this model

become

(1 +a

and (2i)-(2k).

Mode[

Qi)-(2k) and eq. (A2a).

Q’+QZ[1+il(l+A)]2

= (s,, + l)Bi(A,

-

A)

together with eqs (A2b), (2d) and (2i)-(2k).

are

Iglandy41. The model equations

become APPENDIX

$

T

= V’A,

WW together with eqs (Zgj-[2k). It should be noted that the time scale for this model is Yr

=

t. (A.5 %i>

643~)

This time scale is dependent only on micropore characteristics which is expected when micropore dillbsion controls the sorption rate. Model 5 (micropore di&ion and adsorption model) The criteria for validity of this model are

I The model equations

1~ O(1) and y 4 1. are eq. (A3a) and eqs (2g)-(2k) together

B: METHOD OF SOLUTION FOR MODEL

1

Here we consider the case of spherical macroparticle and spherical microparticle geometry: Taking the Laplace transform of eq. (2e) with respect to z gives

SJ$(S, x, x,) - A,s(O, x, x,) =

-$* $ c

After application of the initial condition, solution of (Bl) is given by &(s, x. x,) = 2

cash (fi

x,) + 2

(JW

eq. (2h), the general

sinh (fi

xJ

(B2)

where C1 and Cz arc arbitrary to be determined by application of the boundary conditions (2f) and (2g). Taking the Laplace transform of eq. (2g) with respect to zr diaerentiation of eq. (B2) with respect to x, and substituting gives c, =o

(B3a)

1126

ROBIN G. JORDI

or & (s, x, x,) = 2

sinh (fi

x,).

(B3b)

Taking the Laplace transform of eq. (2f) with respect to r, differentiation of eq. (B3b) with respect to r,,, gives on substitution

(1 +

Here Cs and C4 are arbitrary constants to be determined by application of the boundary conditions (2b) and (2c). Taking the Laplace transform of eq. (2~) with respect to 5, differentiation of eq. (B6a) with respect to x, gives on substitution c,

= 0

@7a)

Or

4ad(s, x)

c3 =

and DUONG D. Do

N,,&cosh(,h% + W

- l)sinh(&ll Pa)

I%@, x) = $

sinh (6

x).

(B7b)

or Taking the Laplace transform of eqs (Zb) and (3b) with respect to 5, differentiation of equation (B7b) with respect to x, gives after substitution

J$T,(s,& x,1 BX(s, = (1 + A)x, [fi

x) sinh (&

cash (fi)

x,)

+ (9 - 1)sinh (.&)I.

p ei* Ei (B4b)

Taking the Laplace transform

c4 =

(s-ii~*)[~cosh(J;;;;)+(Bi-

l)sinh(&)]

of eq. (2a) with respect to

(B8a)

p ei* Bi sinh (fi

WI After application of the initial condition, eq. (2d), the general solution of eq. (BS) is given by

= (s - iw*) x [Gcosh

(6)

x)

+ (Bi -

1)sinh (A)]‘ (B8b)

Taking the Laplace transform of equation (2k) with respect to r, substitution of eqs (B4b) and (B8b) gives on integration 3 Bi C&

cash (&)

cash (6) 3&@ Cfi

cash (&I

(1 + A)(s/Y) Cfi

x(s, x) = 2

+

3~3J

(1 + 4 Cfi

C.,&

cash (Ax)

cash (,/&I

+ 2

sinh (6

(B6a)

x)

- sinh (J&f1

@W

cod (d&I + (a - 1)hh (,/%)I

where eq. (J34b) has been used.

cash (&)

+ Bi& cash [2&

cos{B/2) sinh [2J

characteristic

+ (Bi - 2)J; - Bi&

sin (O/2)] (r + Bi -

r co8 (O/2)] + sin (O/2) sin [2 J;

(~0s (e/2) sinh [2&

~0s (O/2)] + sin (O/2) sin [2&

(O/2) sinh [2&

l)&

{COS

(8/2) sinh [2J

sin (U/2)]}

cos (O/2)] - cos (O/Z)sin [2fi

1)

sin (e/2)]} sin (O/2)]}

r sin (O/2)3 Cr - (Bi - l)‘]

cos (e/2)] + sin (O/2) sin [2J

(Bl la)

sin (O/2)1) >

(O/2)] (r + Bi -

* cos (e/2)1 + sin (O/2) sin [2J;

r cos (e/2)] [r + (Bi - 1)2] + cos [2

1)

sin (e/z)]}

r sin (S/2)] [r - (Ei - l)‘]

cos (O/2)] (r - Bi + 1) + cos [2,,&sin

co9 (e/2) sinh [2J

cos (0) {sin

. (B9)

functions:

sin (0) {sin (O/2) sinh [2< r cos (O/2)] - cos (6/2) sin [2J;

sin (0) {cash [2&

(&$I

out-of-phase characteristic functions, respectively, and 6 J,W and +,~s are the microparticle in-phase and outof-phase characteristic functions, respectively, and am given by

cos (O/2)] [r + (Bi - l)*] + cos [2

+ 2(B1 - l)&

1) sinh

wherebMsoa~ and bylm., are. the macroparticle in-phase and

cos (0) {cash [2,,& cos (O/2)] (r - Bi + 1) + cos [2&

&a -

- sinh (&)I + (a -

Applying the inverse Lgplace transform using the residue theorem for the pole s = iw*, corresponding to the periodic steady state, gives

Macroparticle

+ (BI - 2)&

- sinh C&j]

+ (Bi - 1) sinh (&)I

sin (O/2)]) >

(Bllb)

1127

Analysis of the frequency response method for sorption kinetics where ?jM= lx?‘@ 3a#y

{cash (rl,,)Ctl: - 2W - 1) + cos (+)Cq:

vM= (1 + 1) {cash (q,,) [q: + 2(8 - lf’]

+ 2W - l)l + q,,(a - 2)Csinh (QJ + sin (~JII

+ cos(tl,) [vi - 2(68 - l)‘]

+ 2(a - l)rt, [sinh(Q

(Bllc)

+ sin @+)I}

i3uzgp2y qfl[sinh (q,,) - sin (a)] + 2(6@- I)‘] + cos(vwt,)[s: - 2(4 - 1)1] + 2(# - l)q* [sinh (q,,) + sin (q,,)]} + Iuiw*

+ (1 + Q{cosh(~,,)[~:

Macroparticle characteristic functions: 63, =

S: {cash (~[$/a

+ (I

- WI

39%Csinh(rlr) - sinhA + cos(rl,)Caf/2 -(a - 11’3 + rl,(-@ - 1)Csinh (%J + sin(s)])

(Bl2a)

- 2(9# - l)] + cos(rl,)Clt: + 2@ - 111 + rt,M - 2)CsinhOtJ + sin(&

3a{cosh(~J[~;

“’ = ~;{cosh(~,)C~:/2

+ (ro - lY1 + coshJCt@

(B12b)

- W - U”l + tt&U- UCsinhhJ + sin(rlrHI

where

Essentially, the same procedure is followed to determine the characteristic functions corresponding to any macropartitle or microparticle geometry for any of the other parent or degenerate models. Alternatively, they may be derived from the corresponding solution to model 1. for the geometries of interest, using the criteria for validity of the desired degenerate model, determined by Do (1990). using L’ Hospital’s theorem. APPENDIX

C: IN-PHASE AND

CHARACTERISTIC

FUNCTION

Model

3

Macroparticle characteristic functions As for model with 1 flM given by

OUT-OF-PHASE

DOMAIN

LAPLACE

SOLUTIONS, FOR SLAB, CYLINDRICAL MACROPARTICLE

Microparticle characteristic functions As for model 4.

Microparticle characteristic functions

AND SPHERICAL

AND MICROPARTICLE

GEOMETRY

Note: The substitution s = io* into the relevant model below gives the in-phase and out-of-phase characteristic functions of the chosen model at that dimensionless angular frequency w+.

6,. = 1 61, = 0. Model 4

Macroparticle characteristic functions s,l=

(I) Slab genmtry Model

61, = 0.

1

Macroparticle characteristic functions Case 1 (film mass transfer resistance.):

Microparticle characteristic functions 6,.

Bi sinh (&) ‘lR = Re

1

+ ~i~sh(&)l

,/&C&sinh(,/&)

S,r = -1m

Bi sinh (A)

61, = -Im

&CA

= Repz)]

sinh (J;;;;)

+ Bi cash (&,I

Case 2 (negligible film mass transfer resistance):

Model

[fag)].

5

Macroparticle characteristic functions As for model 4. Microparticle characteristic functions B sinh (,/‘&I

S,, = Re fi where qN

=

UlS

wafi

+

(1 + 4

C&

= -1m

sinh Cfi)

sinh Q’&)

+1

co& Cfi)l’

Microparticle characteristic functions As for model 5. Model

2

Macroparticle characteristic functions As for model 1 with Q, given by

Model

C,.&

sinh (a)+

I cash C &?$)I

0 sinh (a) J’%[.f%sinh(fi)

+ ~cosh(,&)l

6

Macroparticle characteristic functions As for model 4. Microparticle characteristic functions

’

ROBIN 0. JORDI and DUONG D. Do

1128 -1m

S,,=

+J c s+ly

>

Model 5

.

Macroparticle characteristic functions As for model 4. Microparticle characteristic functions

Model 7

Macroparticle characteristic functions As for model 1 with qM given by

Il” = OiS+ (1 +

w-vs A)@ + Lay)’

Microparticle characteristic functions As for model 6. (ll~~d~~

Model 6

geometry

Macroparticle characteristic functions As for model 4.

Macroparticle characteristic.functions Case 1 (film mass transfer resistance):

Microparticle characteristic functions

. Case 2 (negligible film mass transfer resistance): s

_

*a -

Re

2MJi) [ &M&)

S2, = -1m

21, (&) [ $L

L&/G)

where

Model 7

Macroparticle characteristic functions As for model 1 with qs, given by

1

2u+i?ys

1

Microparticle characteristic functions As for model 5. Model 2

Macroparticle characteristic functions As for model 1 with q~ given by

tl” = uls + (1 + A)(s + 2Sy)’ Microparticle characteristic functions As for model 6. (III) Spherical geometry Model 1 Macroparticle characteristic functions Case 1 (film mass transfer resistance) 3 WC&

6 1R= Re

dJI = Re

Macroparticle characteristic functions As for model 1 with Q, given by

Model 4

3 Q’&

Y sinh (,/&)I

(&)

+ (a-

lrnr

1

3 C,/‘& mth (6)

11

-

rlrr

>

where VM

=

cls

+

302N Cfi

cash (A)

(1 + n)C&cosh(fi)

- sinh (fl)J + (9 -

l)sinh(fiIl’

Macroparticle characteristic functions &R = 1 61, = 0. Microparticle characteristic functions

}

1) sinh (&)I

- 11

coth (&)

{

as1= -1m

fh=s[u, +&j. slr = 0.

=h

cash (6)

>

Case 2 (negligible Elm mass transfer resistance):

Model 3

&fi = 1

(,/&I + 03 - 1) sinh (&)I

3 IBiC& VMC&

Microparticle characteristic functions

- sinh (&)I)

cash (&)

VUC&co&

6,, = - lm Microparticle characteristic functions As for model 4.

.

Microparticle characteristic functions As for model 5. Model 2

Macroparticle characteristic functions As for model 1 with qM given by

q&# = u,s +’

302~ Cfi

coth (fi) (l+L)

-

11 .

Microparticle characteristic functions As for model 4.

>.

1129

Analysis of the frequency response method for sorption kinetics

When the time scale of the volume perturbation is much larger than the time scales of the dynamic sorption processes, the concentration gradients in the macropore void and in the microparticles approach zero, and sorbate uptake ap proaches that corresponding to equilibrium with the bulk phase. Under these conditions, equation (Dl) becomes

Model 3

Macroparticle characteristic functions As for model 1 with qM given by

Microparticle characteristic functions

NT=

6sa = 1

Macroparticle characteristic functions

The relative pressure amplitude is given by

&I = 0.

P-P.

Microparticle characteristic functions

3C,/&otht@i) -

1

PI

-

11

&Clk

For small volume prrturbations comes

(DW

.

G - G,

Ap=P=-. .

(S/Y)

. (D3)

Z 2 Z, and eq. (D3) be-

P - P.

- 11

3C../%coth(&)

1

zc,-z&t.

Ap=P= c

(S/Y)

Ss, = -1m

C bC s&Z,. i! (1 - Ed) = 1 + bC,.

NT = KC,. + 5

8,s 2 1

bsR = Re

l+bC,,

The overall mass balance on .the adsorbable component under equilibrium conditions prevailing at the start of the experiment is given by

6,, = 0. Model 4

C&G

VCB+z s.,,C~)+ti-e,)-

C’W

C,

The relative volume amplitude is given by Model 5

Macroparticle characteristic functions As for model 4.

Au=-.

0

6,s = Re

S/Y C&

dsr = - Im

cash (,j’%

3 (4 C& S/YCfi

- sinh (&)I +

(1 -

wsh L.6)

cash (fi)

1) sinh

(D5)

K

The amplitude ratio becomes

Microparticle characteristic functions 3 {a C,/‘% cash (a)

V-v,

-I_-= P

1

AV

--- CB= AV K AC,

AP

C&%1 >

- sit& (&)I

1

+ (9 - 1) sinh C&h1 > .

cm.Nr

(D6)

For Nr = Nr( t: Cs) we may write the differential form: dNT(Y,Cs)=(~~*VdCs+(~)=‘dk’.

Model 6

Macroparticle characteristic functions As for model 4.

(D7)

Dividing by dC, and taking limits as dNr + 0 gives 0 = (!%y*”

Microparticle characteristic functions

+ (!!v+c’(&>“‘~r

(Dg)

Differentiation of eq. (D2) gives = V+ z

C& EM+ (1 - EM) (1 + b&,1’

Model i’

1(W (DW

Macroparticle characteristic functions As for model 1 with Q,, given by

From eqs (D8)-(DlO)

we obtain

3a#?oys qH = U’s + (1 + 1)(s + 3&?y)’ Microparticle characteristic functions As for model 6. APPENDIX

D: LIMITING ANGULAR

VALUE

enr+tl -hdtl

OF u/p FOR SMALL

FREQUENCY

Cd +

bC_)’

Ill

C,,. (Dll)

For small perturbations of the batch system volume C, 1: Cs,. Substitution of eq. (Dll) into eq. (D6) gives

An overall mass balance on the adsorbable component g&S Nr=

VCa+~ce,c+(L

-&&C&J

(DU

where C is the volumetric mean sorbate concentration in the macropore void and Cp is the volumetric mean sorbate concentration in the microparticles.

PW Writing in non-dimensional

:=l+K* P

terms (D13)

ROBIN G. JORDI and

1130 APPENDIX

E: FREQUENCY A SEMI-INFINITE

RESPONSE

OF

DUONG

D. Do

The solution

to eq. (El) is given by

SLAB

Here we consider the case of a semi-infinite stab exposed to a harmonic oscillation of the surface concentration. The. foLIowing discussion parallels that of Carslaw and Jaeger (1959), who considered the response of a semi-infinite slab to harmonic oscillations in surface temperature.. The equations describing the response of a semi-infinite homogeneous macroporous slab to harmonic oscillations in surface concentration, expressed in dimensionless form are given by

+isin(w*r-xJZjS++)]

WI Macropore

mass balance: 8A

d2A

a7

ax2

_=-

Al,=,

= AB for all z 20.

A is finite for all x and T. AIrso

= 0 for all x 2 0.

(El@ @lb) (EW (Eld)

Bulk phase: AD = pe+r*++l.

We)

The integral expression represents the transient behaviour, and decays to zero as r increases. The periodic solution has the following properties: (1) The amplitude of the concentration nentially with distance as

wave decays

IAl cc e-” (2) There is a progressive lag of magnitude phase of the oscillation. where k = m number.

expo(E3)

(kx) in the is the wave