Measurement of effective diffusivities using a spinning basket reactor

Chemxal Engmeering Science. Vol. 44. No. Printed in Great Bntain

12, pp 2843

2852,

1989.

DCKX-25D9/89 $3.DO+D.DD p 1989 Pergamon Press plc

MEASUREMENT OF EFFECTIVE DIFFUSIVITIES A SPINNING BASKET REACTOR

School (Firsr

S. T. KOLACZKOWSKI’ and U. ULLAH of Chemical Engineering, University of Bath, Bath BA2 received

8 January

1989; accepred

in reuised_form

USING

7AY,

10 March

U.K.

1989)

Abstract-An experimental technique is developed using a spinning basket reactor to determine effective diffusivities under reaction conditions for both products and reactants. This technique overcomes the extreme difficulties of determining effective diffusivities of the products under reaction conditions in pellet diffusion cells. The system studied was the dehydration of ethanol on a 13X zeolite catalyst at 623 K and 101 kPa. Equations are derived relating the rate and equilibrium parameters for adsorption and reaction in a spherical pellet to measurable properties acquired in pulse response experiments. To overcome the problem of multiple solutions, a simplification is developed for the numerical technique employed to solve the series of differential equations. These values of effective diffusivity are then compared with those determined in pellet diffusion cells under reaction and inert conditions. It is shown that for the system studied, the effective diffusivity of a component is of the same order of magnitude irrespective of the presence or absence of chemical reaction. The validity of the mean transport pore model as a technique for determining effective diffusivities from permeability and diffusion experiments is also tested and shown to predict comparable values.

INTRODUCTION

evaluating the performance of a porous heterogeneous catalyst, intraparticle diffusion is a significant and often rate controlling step. In order to account for this mechanism it is necessary to determine experimentally the effective diffusivity in the catalyst pellet. These diffusivities are often measured under inert conditions, using well established experimental techniques (Smith, 1982; Eberly, 1969; Ma and Mantel, 1972). Although determined under non-reaction conditions, they have provided useful estimates for the purpose of reactor design. The determination of effective diffusivities under reaction conditions, although desirable, is not always practicable. Several experimental and theoretical techniques have been developed to determine the effective diffusivity under reaction conditions. These include use of the Wicke and Kallenbach (1941) type of diffusion cell (Dogu and Smith, 1976; Burghardt and Smith, 1979) and fixed bed chromatographic techniques (Suzuki and Smith, 1971). Although pellet diffusion cells have been used successfully to determine effective diffusivities of reactants, it is difficult to determine values for the products of the reaction since they may diffuse in either direction across the cell. The use of packed columns has been considered to be less accurate than single pellet experiments (Burghardt and Smith, 1979), since axial dispersion and fluid-particle mass transfer effects must be accounted for, before intraparticle parameters may be evaluated. In order to overcome some of these difficulties, Park and Kim (1984b) adopted the use of a recycle loop in their experiments. In

‘Author to whom correspondenceshould be addressed. CES

44:12-E

2843

From a recent literature review (Park and Kim, 1984a), it is evident that there are many conflicting views as to the comparative nature of effective diffusivities measured under reaction and inert conditions. These differences are to some extent not surprising since it is well recognised (McGreavy and Siddiqi, 1980; Baiker et al., 1982) that each method of measuring effective diffusivities has its own characteristic. Therefore values determined by different methods and a variety of investigators are likely to differ especially when the additional complexity of a reaction system is included in these studies. In an endeavour to resolve these differences of opinion, Park and Kim (1984b) studied the isomerization of cyclopropane to propylene on three types of zeolite catalysts, LiY, NaY (both active) and KY (inactive) in a recycle packed bed reactor. They reported that for the active catalyst NaY and LiY, the diffusivities measured under reaction conditions were one order of magnitude smaller than those measured under inert conditions. In order to overcome some of the difficulties of measuring effective diffusivities under operating conditions, Fott and Petrini (1982) proposed an interesting approach known as the mean transport pore model (MTPM). In this model the effective binary diffusivity is determined from: (a) theoretically calculated values of Knudsen diffusivity and binary bulk diffusion coefficient, and (b) experimentally determined transport parameters. These parameters are only dependent on the characteristics of the porous medium and independent of experimental conditions (e.g. temperature, pressure, gas composition). The authors proposed that these parameters, once evaluated for the porous catalyst, may be used to predict mass transport under different situations and condi-

and U.

S. T. KOLACZKOWSKI

2844

tions. Schneider and Gelbin (1985) also proposed a mean transport pore model making similar claims. In a recent parallel study of the effect of intraparticle diffusion on the catalytic selectivity of ethanol dehydration, Thomas and Ullah (1988), reported that similar order of magnitude values were determined for the effective diffusivity of the reactant ethanol under reaction and inert conditions. They performed these experiments in a single pellet diffusion cell applying both steady state and dynamic techniques based on models developed by Dogu and Smith (1976). In this paper, an experimental technique is developed using a spinning basket reactor to determine effective diffusivities under reaction conditions for both products and reactants. The spinning basket reactor was selected for the following three main reasons:

(a) external mass transfer effects may be eliminated/

ULLAH

Equations are derived relating the rate and equilibrium parameters for adsorption and reaction in a spherical Pellet to measurable properties acquired in pulse response experiments. To overcome the problem of multiple solutions, a simplification is developed for the numerical technique employed to solve the series of differential equations. Values of effective diffusivities are then compared with those determined in pellet diffusion cells under reaction and inert conditions; thus providing additional evidence in the vexed debate of whether diffnsivities determined from experiments under inert conditions compare with those determined under reaction conditions. Applying the mean transport pore model, effective diffusivities are also determined from permeability and diffusion experiments, and compared with the values determined in the spinning basket reactor.

minimized;

THEORETICAL

(b) unlike the chromatographic (cl

method, this technique does not suffer from potential problems of axial dispersion; effective diffusivities under reaction conditions may be determined for both reactants and products. In a single pellet diffusion cell it is difficult to measure the effective diffusivities of the products since they diffuse in both directions appearing in the lower as well as the upper chambers of the cells.

aci 5-=at

1: C,H,+

2C,H,OH and involves

H,O

1: C,H,OCzH,

(1.1) + H,O

(1.2)

molar expansion.

_.---

‘i,s

\

l-

l(a).

Pulse

Input

‘Spherical

F

S

/

pellets

i

To anolytlcol section

(b)

(a) Fig.

(2.1)

Fs t-

ijuu

i a(t-2Ni)

-- r2 ~-Nip, ar

where ci is the concentration of the diffusing component, a, is the porosity, Ni is the molar Aux in the porous medium, N; is the molar flux from the catalyst pores to the catalyst surface and pp is the apparent pellet density. At high temperatures (i.e. substantially above the boiling point of the species considered), it is reasonable to assume that since the thickness of any adsorbed layer is small, surface flux is minimal in comparison with the flux through the gas phase (Schneider and Smith, 1968). Surface diffusion effects are therefore neglected in this analysis. In addition, when using a

The reaction studied was the dehydration of ethanol on a 13X zcolite catalyst at 623 K and 101 kPa. The reaction stoichiometry is represented by the following: &H,OH

DEVELOPMENT

Introducing a pulse of ethanol into an inert stream of nitrogen llowing into a spinning basket reactor, the differential mass balance for the transport of a reactant or product into a spherical catalyst pellet [see Fig. l(a)], is

A cross-sectional

view

of spherical

pellets

in a spinning

spinning basket reactor.

basket reactor. (b) A schematic of the

Measurement of effective diffusivities pulse response technique, since the concentrations of both the products and the reactants are low, then surface concentrations would also be low, so the accumulation of species on the surface is neglected. The molar flux, Ni, is given by

(2.2)

(2.3)

Ni=N,+N2+N,+N,+N,

i i=l

where subscripts 1, 2, 3, 4 and 5 are assigned to represent ethylene, diethyl ether, ethanol, water and nitrogen respectively. From stoichiometry and for a spherical pellet, the following relationships apply: N,=

-_(N,+2N,)

(2.4)

+N,

(2.5)

N,=N, and

(2.6) -r2

N;=r, N&=

+2r,

(2.9)

-_(r,+r,)

k,K,,,c, (I+ K,, lC3Y

(3.1)

where k,, k, are the respective reaction rate constants and K,,,, K,., are the adsorption equilibrium constants for reactions (1.1) and (1.2) respectively. For the reactor assembly shown in Fig. l(b), upon the introduction of the pulse input the differential mass balance is represented by:

v,zac; =F,Mis(t)-F,ci.B+NiI,=~~ s

at

3w

(4.1)

RPP

where sb is the bed voidage of the reactor, V, is the volume of the cell, ci,S is the concentration of species i in the cell, w is the mass ofcatalyst, F, is the volumetric flowrate and M, is the strength of the input pulse [represented by s(t)]. The following initial and boundary conditions apply, for i= 1, 4 for

ci = 0, hi

when

may be evaluated METHOD

t>O

(5.3)

from eq. (4.1).

OF SOLUTION

The developed model which accounts for molar expansion and hence non-equimolar counter diffusion, results in a complex system of coupled differential equations. The solution of these would require making estimates of a range of Di values and subsequent solution by trial and error. This would be complex and generate multiple solutions which would be impracticable to solve. To overcome this constraint the following technique was developed which includes approximations which simplify the problem avoiding the occurrence of multiple solutions. Considering the diffusion of ethylene and starting from eqs (2.1) and (2.2), the following apply:

cpp= --+I

at

1

r’

2rN,

+r2-

8N, ar

,c 1 I[ k,K +

(I+K~~~c~)~

”

when

O
=0 _ ar ,=o

and for this system since

when

k

(2.8)

(3.2)

&*

and Nil,=,

(2.7)

where rl and r2 represent the adsorption of ethanol followed by surface reaction for stoichiometric eqs (1.1) and (1.2), respectively. From a recent study of the catalytic dehydration of ethanol (Birk et al., 1985), the reaction rates rI and r2 may be represented by: rl =

cil,=,=ci,Jt)

(6.1)

N;=-r, N;=

2845

t30

t=O

(5.1) (5.2)

N,=N,+N,

(6.2)

i=l

then N, = -&2+y,(N,

+N,).

(6.3)

As the concentration of the pulse injected was small the molar flowrate of nitrogen, N,, was assumed to be constant and known. In the first time differential step, neglecting the dehydration reaction that forms diethyl ether [eq. (1.2)], uncouples eqs (6.1) and (6.3) from the other species equations. This enables eqs (6.1) and (6.3) to be solved by a finite forward difference numerical technique, solving for D, by trial and error. For diethyl ether:

kzW,.zc,)2 I[ +

(1 +K,.,c,)=

1 ”

(7.1)

and N,=

-D2g+y2(NI

+N,).

(7.2)

In this next numerical step, although the dehydration reaction to form diethyl ether [eq. (1.2)] was primarily considered, the influence of the concentrations evaluated in the first step [from eqs (6.1) and (6.3)] were taken into account when developing the finite difference network for the first time step. In this step D, was evaluated. Having evaluated the values of N, and N, in the first time step of the finite difference network, N, and N, may be determined from eqs (2.4) and (2.5) at each point on the grid. The diffusivities of the remaining

2846

S.T. KOLACZKOWSKI

species may then be determined N,=

-D,~+y,(N,+iVs)

N,=

--D,2fy,(N,

from: (8.1)

and +N,).

(8.2)

A trial and error method was adopted estimating D, in eq. (8.1) until the boundary condition (5.2) was satisfied and then likewise for D, in eq. (8.2). In this manner effective diffusivities were evaluated for each of the four species for the first time step. The procedure was then repeated for another 20 to 30 steps (depending on the experiment) and a mean value of Di was calculated. EXPERIMENTAL In both the spinning

basket reactor and the multipellet diffusion cell the required shape of pellets was formed from 13X zeolite pellets originally supplied by Laporte. These were initially crushed in an agate mortar, screened to 200 mesh and then compressed to the desired shape. Spherical pellets were formed in a mould for the spinning basket reactor experiment. For the multi-pellet diffusion cell, the pellets were formed by compressing the material directly into the cylindrical holes in the cell. Regular checks were made to ensure that a leak-tight seal was achieved and maintained between the cylindrical walls of the pellets and those of the cell. These were performed by creating a differential pressure across the cell and monitoring the change in pressure as a function of time. A rapid change would have indicated the presence of a leak. The physical properties of the pellets are shown in Table 1. The porosity of the pellets was experimentally determined by a combination of the mercury penetration and the nitrogen desorption method.

and U.ULLAH

Inconel 600 and type 321 stainless steel were the primary materials of construction for the reactor. The nitrogen supply was split into two lines, passed through a bed of silica gel which served as a drying agent, and was then carefully metered. One of the nitrogen feeds was directed to the reactor, while the other feed passed through a temperature controlled saturator containing ethanol, providing the reactant for the pulse input. The concentration of ethanol vapour flowing from the saturator was calibrated as a function of temperature and flowrate using a gas chromatograph. The composition of the effluent stream from the reactor was analysed chromatographically using a thermal conductivity cell and a Porapak Q 8&100 mesh packed glass column. Both the cell and the column were maintained at 523 K. The retention times of ethylene, water, ethanol and diethyl ether were 50 s, 80 s, 240 s and 600 s respectively. As the retention time of diethyl ether was very high. It was not possible to obtain several experimental points during the course of one experiment. Therefore samples were analysed from several identical experiments at different times. The experiments were conducted at 623 K and 101 kPa. The 13X zeolite pellets were placed in a stainless steel gauze cylindrical shaped basket. The spherical pellets were 8 mm in diameter and the basket rotation speed was 2000 revolutions per minute. This speed of rotation was selected after experiments had been performed at steady-state conditions to identify the speed of rotation at which further increases did not affect conversion, and hence external mass transfer was not the rate limiting step.

Table 1. Physical properties of (a) spherical pellets used in spinning basket reactor, and (b) cylindrical

(a) Spinning basket reactor The experiments were performed in the apparatus illustrated in Fig. 2, using a spinning basket reactor.

Apparent Measured

r--

_ -

-

pellets used in the multi-pellet fusion cell

density (g cm-a) porosity, a,

_ -

-

- -

___a

dif-

(a)

(b)

1.23 0.37

0.87 0.52

1

Fig. 2. Schematic diagram of the apparatus used for dynamic pulse response experiments in a spinning basket reactor: (1) nitrogen feed, dry and metered, (2) ethanol saturator, (3) sampling valve and loop, (4) spinning basket reactor, (5) thermostatted furnace, (6) chromatograph.

Measurement of effectivediffusivities

2847

(1982) equations are derived for flow in a circular capillary and applied to a porous medium. In assuming that only some of the pores, called transport pores, contribute to the transport of mass, these pores are characterized in terms of two transport parameters. These are the mean transport pore radius, r, and the geometric constant IJ% (the ratio of the porosity to tortuosity of the transport pores). Using MTPM the effective bulk and Knudsen diffusion coefficients can be expressed as

Df = r@Dt.

/

‘\

/

Upper

chamber

Lower

VIEW

chamber

A-A

Fig. 3. Schematic of multi-pellet diffusion cell.

(b) Multi-pellet difSusion cell Permeability and binary diffusion experiments were performed in a specially designed multi-pellet diffusion cell (see Fig. 3). The cell was constructed from brass and contained five cylindrical holes, 2.07 cm in diameter and 1.45 cm in length. The pellets were formed by compressing the material directly into the cylindrical holes to form the required pellet. Each face of the pellet was in flush with the surface of the cell. The volume of the upper chamber (V,) was equal to 25.8 cm3 and was of the same volume as the lower chamber ( V,). The nitrogen and ethylene used in the experiments were 99.9 and 99.98% pure respectively. Ethanol and diethyl ether were of an analar grade. Permeability experiments. In the mean transport pore model (MTPM) proposed by Fott and Pet&i

(9.2)

In order to determine the transport parameters rt&and + in the mean transport pore model, permeability experiments were performed in the multi pellet diffusion cell and associated apparatus illustrated in Fig. 4. Initially both sides of the chamber were evacuated using a vacuum pump and then the cell was isolated. Nitrogen was then introduced into the upper chamber and the pressure differential between the two cells as the nitrogen diffused through the pellets as a function of time was recorded. In this manner permeability experiments were performed at 291 and 323 K for a range of initial differential pressures (6.6 to 13.3 kPa) across the cell. From the relationship (Fott and Petrini, 1982): ln($J=rg)t

(9.3)

where 2V”VL v=----. vu+ VL By plotting

In

vs t, the value of the permeability

B may be determined

from the slope where

gradient =From

(9.4)

2AB LV

(9.5)

_

the relationship: B=D:(l+wK-‘)@+p

r2tiP 8~

Fig. 4. Schematic of apparatus for permeability studies: (1)differential pressure cell, (2) multi-pellet diffusion cell, (3) vacuum pump, (4) manometer, (5) thermostatted furnace.

2848

S.T.

a=

1

KOLACZKOWSKI

(9.7)

l+K-’

in the Knudsen region the value of Df may be determined since as K-l -0, then B= Df. By performing a binary effective diffusivity experiment on this system at the same temperature, using any pair of gases under non-reaction conditions, the effective diffusivity Di,j may be experimentally determined. For equimolar counter diffusion the effective diffusivity may be represented as

Substituting

eqs (9.1) and (9.2) into (9.8) 1 ~=___

1

DLmj r$Df

1 +l/lot-

Once the theoretical values of 0: and Df,j have been calculated, the parameters rl(l and + may then be evaluated for the system. These transport parameters may then be used in eq. (9.9) to predict a theoretical value of the binary effective diffusivity for any pair of species in the reaction scheme studied. Non-reaction, difision experiments. A steady state method was employed using well established experimental techniques. Each of the components in the reaction scheme was studied in turn to determine its effective binary diffusion coefficient in the remaining species. In this series of experiments one of the components flowed through the upper chamber, while the other flowed through the lower chamber. The concentration of the diffusing component was then monitored as a function of time by taking a number of discrete batch samples and analysing them chromatographically. In the series of experiments where ethanol was one of the components studied, the temperature was maintained below 448 K to prevent the dehydration reaction from occurring. Binary diffusion coefficients were then extrapolated to 623 K applying the Chapman-Enskog formula (Hirschfelder er al., 1954) where at moderate temperatures and pressures:

D.I., or T’.5.

(10.1)

RESULTS

(a) Spinning basket reactor After the introduction of a pulse of ethanol into an inert N, stream which flows into the reactor (operating at 623 K and 101 kPa), the response was analysed in the output from the cell and is illustrated in Figs 5(a) and (b). From this response it was evident that as the ethanol was dehydrated so diethyl ether, water and ethylene were formed. The sharp increase in ethanol concentration in the output at the instant the reactant was injected indicates that the reactor was well mixed. Measurements of the gradient of the curve at various time intervals enables NilrcR to be determined from eq. (4.1) while the boundary condition

and U. ULLAH (5.3) was directly determined from the concentration time plot. Prior to implementing the numerical technique proposed, it would have been necessary to evaluate the reaction rate and equilibrium adsorption constants for eqs (3.1) and (3.2). Both k, and k, have already been evaluated for the system studied (Birk et al., 1985). Likewise the relationships for K,, 1 and K,,, as a function of temperature for this 13X zeolite have been presented (Thomas and Ullah, 1988) in equation form according to the van? Hoff relation. Values for these constants are presented in Table 2. Applying the developed numerical technique for the operating conditions described in Table 3, multicomponent effective diffusivities for each of the species were determined and are presented in Table 4. (b) Permeability experiments The results of plotting the experimental data in accordance with the relationship expressed by eq. (9.3) are illustrated in Fig. 6. For the system studied the permeability, B, was determined to be 0.07 1 cm2 s- r. From a series of binary effective diffusivity experiments at 291 and 373 K, using ethylene and nitrogen as the binary system the transport parameters for the pellet were evaluated (r&=0.011, r1(1= 5.5 nm). These values were then used applying eq. (9.9) to determine effective binary diffusivities for the reaction scheme studied. These are presented in Table 5. (c) Non-reaction, diffusion experiments Values of effective binary diffusion coefficients determined with all of the five components, including nitrogen, are presented in Table 6.

DISCUSSION

Comparing Tables 5 and 6, it is evident that similar values of effective diffusivity are obtained from both steady state diffusion and permeability experiments (applying MTPM). These results provide additional supporting evidence as to the value of this interesting approach where, once the transport parameters have been characterized, they can be used to calculate the effective diffusivity for a specific binary pair. However, when applying the MTPM in reactor design, one should be cautious in applying the results from permeability experiments determined in pellet diffusion cells where transfer through the side wall is excluded. Although comparative values have been determined in this study, transfer through the side walls was excluded in both of the experiments. Effective diffusivities measured for a completely accessible cylinder have been shown (Waldram, 1976) to be on average about one half that of the same particle with sealed side walls. In an analogous fashion, similar observations could be expected in the application of MTPM. In comparing the results of values of effective diffusivities determined in the presence and absence of chemical reaction, binary effective diffusivities deter-

Measurement

of effective

I

2849

diffusivities

I

r

0 (a)

\

800

0

zz 5

600

z-.

3

z L

0 .g IE 2 5

400

V

200

L

0

Time

Time Fig.

1200

800

1200

(~1

400

0

800

400

1600

(s)

5. (a) Concentration change with time at the exit of the reactor for ethanol. (b) Concentration with time at the exist of the reactor for (0) ethylene, ( x ) water and (0) diethyl ether.

Table

2. Values

of reaction

rate and equi-

librium adsorption constants 10-6molg-1s-’ 1.35 x lop6 molg-‘s-l

k,=8.415x

k,=

K,, 1 = 1.26 x lo4 cm3 molY 1 K 3.2 =4.411

x IO5 cm’mol-’

mined in the absence aged according to:

of chemical

Di=i(i$,

change

reaction

were aver-

Di.i>j+i

for each of the key components. These are presented in Table 7 and compared

(10.2)

with

S. T. KOLACZKOWSKI and U. ULLAH

2850

Table 3. Operating conditions and experimental parameters for spinning basket reactor

Mass of catalyst in reactor, w (g) Dimension of cylindrical reactor cell: diameter (cm) length (cm) Volumetric flowrate, F. (cm3 s- ’ ) Temperature (K) Pressure (kPa) Mass injected in input pulse, G (mol) Bed voidage, E+_

Table 5. Calculated effective binary diffusivities at 623 K, 101 kPa, from experimental data applying the mean transport pore model (r$ = 5.5 nm. $ =O.Oll)

15.65 Components in binary system

5.0 1.5 1.67 623 101 2.88 x 10-a 0.38

C,H,, C,H,OC,H, C,H,, C,H,OH C,H,, H,O C,H,, N, C,H,OC,H,, C,H,OH C,H,OC,H,. H,O C,H,OC,H, >N, C,H,OH, H,O C,H,OH, N, H,O, N,

Table 4. Multicomponent effective diffusivities determined under reaction conditions in a spinning basket reactor at 623 K, 101 kPa

ElTective binary diffusivity (cm: cm-’ s-‘) 4.58 3.86 6.25 6.51 1.98 3.11 4.15 4.09 5.01 8.76

x x x x x x x x x x

lo-3 10-a 10-a lO-3 lO-3 lo-3 10-S lo-3 10-a lO-3

Multicomponent effective diffusivity

(cm:cm-‘s-l)

Component

1.74 2.17 0.53 3.97

Ethylene Diethyl ether Ethanol Water

I

I

I

x x x x

Table 6. Values of effective binary diffusivities in the absence of chemical reaction determined in the multi-pellet diffusion cell (623 K, 101 kPa)

10-J lo-’ 10-a 10-j

I

Components in binary system C,H,, C,H,OC,H, C,H,, C,H,OH C,H,, H,O C,H,. N, C,H,OC,H,, C,H,OH C,H,OC,‘H,, H,O CIH,OC,H,, N, C,H,OH, I-I,0 C,H,OH, N, HA N,

I

0

0

100 -

/ 0

/ 0

0

20

40

60

Time

Fig. 6. Determination

00

100

120

(s)

of permeability from eq. (9.3).

the values of multicomponent effective diffusivities determined under reaction conditions. The value for ethanol under reaction conditions in a single pellet diffusion cell (at 623 K, 101 kPa) determined by

Effective binary diffusivity (cm:cm-‘s-1) 2.42 2.56 4.68 6.30 1.41 1.77 3.57 3.41 4.69 8.62

x x x x x x x x x x

10m3 10m3 lO-3 lO-3 1o-3 10-a lo-3 10-j 1O-3 1O-3

Thomas and Ullah (1988) is also presented in this table. In their work, the material from which their pellet was made was the same as in this study, and therefore comparative observations are valid. When comparing the values determined in this work (see Table 7), although close comparison is obtained for the products, the diffusivity of the reactant ethanol differs by a factor of about five. This may be due to the effective diffusivity of the reactant being influenced by the conditions of chemical reaction due to the effects of adsorption etc., whereas the diffusivities of the products remain practically independent of reaction conditions, as may be expected for the product species once they have been desorbed. If the alternative view is considered, that the effective diffusivity is independent of chemical reaction but is a function of pore characteristics, then perhaps under reaction conditions changes took place in the pores which affected their characteristics. It is difficult to draw firm conclusions on this vexed subject, and it is likely to remain a topic in future debates. Nevertheless, it is interesting to note that similar order of magnitude values are obtained irrespective of the presence or absence of chemical reaction. In general, values for cylindrical pellets are higher than those for spherical pellets. This observation is in line

Measurement Table 7. Comparison

of effective diffusivities

of the effective diffusivity of a component absence of chemical reaction Multicomponent effective diffusivity (cm:cm-‘s-l) spinning single pellet basket cell (Thomas reactor and Ullah, 1988)

Key component

1.74 2.17 0.53 3.97

Ethylene Diethyl ether Ethanol Water

x 10-J x lO-3 x 10-a x10-3

14.5 7.5

0.37

0.42

Reaction conditions

yes

yes

Sealed area

none

Porosity, ep

c--

with the effect of sealed side walls previously discussed (Waldram, 1976). In addition, when correcting for the effect of particle porosity, according to the relationship that (10.3)

Di cc iczp

a closer comparison is obtained. In addition, when comparing results for ethanol with Thomas and Ullah (1988), the diffusivity value under reaction conditions has a substantially lower value than that reported earlier for a single pellet cell. In their analysis molar expansion effects had been ignored. If these are combined with the effects of porosity and sealed side walls then this may account for the lower value determined. Binary diffusion coefficients for ethanol and nitrogen were also observed to differ. In their paper they reported a value of 0.671 x lo- 3 compared with the present determination of ) at 623 K and 101 kPa. In 4.69 x low3 (cm:cm-‘s-l retrospect, this discrepancy is believed to have occurred as a result of not allowing their system sufficient time to reach equilibrium during the course of the experiment. This aspect is particularly important when dealing with an adsorbing species. In this study, after gaining previous experience with this system, extra care was taken to ensure steady state was reached. CONCLUSION It is possible both tions

products

to determine and

reactants

effective under

diffusivities reaction

condi-

Average effective binary diffusivity (cm:cm-‘s-l) multi-pellet diffusion MTPM c‘.?ll x x x x

10-a 1O-3 lo-3 10-j

3.99 2.29 3.01 4.62

cylindrical 20.7 14.5

x x x x

10-a 10m3 10-J lO-3

20.7 14.5

0.52

0.52

no

no

cylindrical walls -

Alternatively, discrete samples may need to be taken as a function of time for detailed analysis. For the system studied, the effective diffusivity of a component is shown to be of the same order of magnitude irrespective of the presence or absence of chemical reaction. A closer comparison was obtained for the products, indicating that the diffusivity of the reactant may have been influenced by reaction conditions. The mean transport pore model was applied and shown to predict comparable values with diffusion experiments performed in the same experimental cell as that in which permeability experiments were conducted. Although the pellets in this comparative study were constructed from the same material, differences in diffusivity values would have occurred as a result of the effects of varying porosity, pellet geometry and accessible surface area for mass transfer. NOTATION

A B

ci ci.s

total cross-sectional area of pellet(s), cm2 effective permeability in a porous

medium 2 cm’ s _ 1 concentration of diffusing component the pore volume species i, mol cm; 3 concentration of diffusing component total volume, species i, mol cmw3

in

species i,

effective intraparticle cm:cm-‘s-l

0:

effective Knudsen diffusivity, species i, cm: cm- 1s- ’ Knudsen diffusivity in a capillary of unit radius, cm: cm- I s- 1 effective intraparticle diffusivity for a binary mixture, cm: cm-’ s-l bulk diffusion coefficient in a binary mixture, cmfcm-‘s-l volumetric flowrate, cm3 s- ’ mass injected in the input pulse, mol

0: Di.j Rj

FS G

diffusivity,

in

Di for

in a spinning basket reactor. Experimental investigators already using the spinning basket reactor (to determine intrinsic rate data) should consider this technique as a means of evaluating Di, thereby eliminating some of the errors caused by using different experimental systems. Analysis of the dynamic response of the required species may not always be practicable and multiple analytical techniques may need to be employed to give on-line measurement.

determined in the presence and

5.3 3.46 3.73 5.55

3.30 x 10-J

spherical 8.0

Pellet shape diameter (mm) length (mm)

2851

2852 k,,

S. T. KOLACZKOWSKI and U. ULLAH reaction rate constants for eqs (1.1) and (1.2) respectively, mol g- ’ s- ’ Knudsen number adsorption equilibrium constants for eqs (I. 1) and (1.2) respectively, cm3 mol - ’ length of pellet, cm input pulse = G/F,, strength of molscm-3 molar flux of species i in a porous medium based on total cross-sectional area, mol s-l cm-’ rate of mass transfer of species i from the fluid in the pores to the catalyst surface, mol g -Ls-l pressure, kPa mean transport pore radius or radial distance, cm rate of reaction per unit stoichiometric coefficient for eqs (1.1) and (1.2) respectively, molg-‘s-l radius of spherical particle time, s temperature, K parameter defined by eq. (9.4), cm3 volume of reactor cell, cm3 volume of upper and lower chambers respectively, in the multi-pellet diffusion cell, cm3 mass of catalyst in reactor, g mole fraction of component i

k,

K K

3.19 K3,2

L M

Nf

P r rlr

r2

R t T V v, V”,

V,

W

Yi

Greek 6(t) AP

Ap” &b

%

w

letters

Dirac delta function, s-l pressure difference across the cell, kPa pressure difference across the cell when t=O, kPa bed voidage internal void fraction or porosity, cm: cmp3 viscosity, g cm ’ s 1 apparent density of pellet, g cm - 3 parameter defined by eq. (9.7) geometric constant (ratio of porosity to tortuosity of the transport pores) slip constant in eq. (9.6)

Subscript

f

fluid (used to indicate that volume based on pore or fluid volume)

is

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reaction on effective diffusivities within biporous catalysts-l. Chem. Engng Sci. 39, 523-531.

Park, S. H. and Kim, Y. G., 1984b, The effect of chemical reactions on effective diffusivities within biporous catalysts-11. Chem. Engng Sci. 39, 533-549. Schneider, P. and Gelbin, D., 1985, Direct transport parameters measurement versus their estimation from mercury penetration in porous solids. Chem. Engng Sci. 40,

1093-1099.

Schneider, P. and Smith, J. M., 1968, Chromatographic study of surface diffusion. A.1.Ch.E. J. 14, 88&X95. Smith, J. M., 1982, Chemical E‘ngineering Kinetics (3rd edn), pp. 45-73. McGraw-Hill. Suzuki, M. and Smith, J. M., 1971, Kinetic studies by chromatography. Chem. Engng Sci. 26, 221-235. Thomas, W. J. and Ullah, U., 1988, Effect of intraparticle diffusion on the catalytic selectivity of ethanol dehydration. Chent. Engng Res. Des. 66, 138-146. Wicke, E. and Kallenbach, R., 1941, Kolloid-2 17, 135. Waldram, S. P., 1976, Unsteady-state diffusion in porous solids. PhD Thesis. University College London.