Effective diffusivities for catalyst pellets under reactive conditions

Chemrcni Engmeenng Scaence Vol 35, pp 1CL- 16 Pergamon Press Ltd , 1980, Pnnted m Great Bntaln

2

EFFECTIVE

DIFFUSIVITIES REACTIVE

D.

Ryan,

R.G.

FOR CATALYST CONDITIONS

Carbonell

and

PEZLETS

S.

UNDER

Whitaker

Department of Chemical Engineerzng University of California Davis, Californza 95616 U.S.A.

ABSTRACT A theory is developed that allows the calculatron of the components of the effective diffusivity tensor under reaction conditions In spatially periodic porous media. Under normal conditions, it can be shown that the effective diffusivity should be independent of the rate of the chemical reaction. Preliminary calculations are provided illustratlng the effect of the particle void fraction on the effective diffusivlty.

KEYWORDS Effective diffusivities; porous me&a.

catalyst

pellets,

reaction

conditions;

spatially

periodic

INTRODUCTION lateness of using effective Many investigators have raised the questlon of the appro y&ve conditions m the design diffusivlties for catalyst pellets measured under non-r *ed Naruse (1974) used asimRecently, Wakao (1974) and Wa 3 of packed bed reactors. ple grid of macro and micro pores to show that effective diffuslvitles under passive conditions would be larger than those under reaction conditions, but that this difference would decrease conslderably in the presence of a small percentage of dead-end pores. These results are in agreement with earlier studzes by Wakao, Kimura and Shibata (1969) who found experimentally that effective diffusivitles under non-reactive conditions were larger by a factor of 3 or 4 than the reactive effective diffusivities in

the

para-ortho

H2

later results of Toei catalyst. Balder and fective diffusivities cal.

conversion

reaction.

and co-workers Petersen (1968) under reactive

This

also

(1973) on the on the othet and non-reactive

seems

to

hydrogenation found hand, conditions

be

in

agreement

with

the

of ethylene on a Ni experimentally that efare essentially identi-

used the Foster and Butt (1966) convergent-divergent Similarly, Steisel and Butt (1967) pore model and the pore size distribution reported by Otanl and Smith (1966) to show that the reactive and non-reactive effective diffusivitles for the oxidation of CO there seems to be some conflicting information as to came out to be the same. Thus, the effect of reaction rate on effective diffusivitles. we present the results of an analysis of combined molecular diffusion In this paper, This model offers some with chemical reaction in a spatially periodic porous medium. in catalyst pellets [Foster and distinct advantages over previous models of the pores Johnson and Stewart (1965): Wakao and Smith <1962), Brown and Haynes Butt (1966); Since It can Haynes and Manogue (1969); Youngquist (197O)l. (1969, 1971:; Brown, it yields all of the components of the effective diffusihandle anisotropic systems, vity tensor, and can be used to study the effect of packing and particle shape on the In the sections that follow we present the theoretical developeffective diffuslvity. and a comparison with experimental ment, the results of calculations on a model system, data on effective diffusivities for unconsolidated porous media, Theoretical

Development 10

Effectivedlffumvlhes for catalyst pellets under reactiveconditions

A-2

Consxder

the

particle

catalyst

to

Fzgure medxum is the fluid phase being transported in the fluid phase

be

1

while the o by molecular is governed

a

porous

medium

AveragIng

as

shown

V rn

volume

phase 1s the solid diffusion alone so by the equation,

a

in

11

Fig.

The

1.

phase

a

porous

phase. Withln that the local

a solute the pores, or point concentration

o=w2c

1s

(1)

where we have assumed the molecular diffusivity to a steady-state At the fluid-solid em. is taking place Jo that the boundary condition -DsEc=kc

on

a unit normal !z sents the fluid-solid of the intrlnsxc rate to the solid phase.

restrxct chemical

ourselves reaction

(2)

ou

pointing

with

We are interested txon in the fluid

A

is a constant and we interface a first order for Eq. (1) becomes,

from

the

fluid

into

the

solid

interfacial area. The reaction rate constant and an equilxbrium constant

in obtaining an phase [Wbxtaker

equation (1970)],

for

the

phase

constant for the

intrinsic

phase

and

Aoo

repre-

k is the product adsorption of solute

average

concentra-

(3)

where

V

(x

particles concentration plying the

is as

the

volume

shown

in

of

Fig.

locally within spatial averaging



= v-cc> +

the 1.

CL This

phase is

within

an

averaging

a measure

of

the

the particle. We theorem FWhitaker

can obtain (1970)],

volume

average an

V

containing

interstitial

equation

for

many

solute ccBa

by

ap-

1

(4)

dV

(5)

where c

12

A-2

1s

the

phase-average

concentrations with

c

=

concentration 01

cc> =

and

xc>

and

are

Aoo

related

is

by

the

the

interfacial

void

fraction

area

V.

within

The

E,

a

ECC>

(6)

Vu/V_

Takxng the phase tion Eq. (2) one

average quickly

of Eq. arrives

(1) at

and the

making result,

~v-

0 -

use

of

Eq.

(4)

and

the

boundary

condi-

-b;[cdA

(7)

CUJ By

writing

the c

and

makxng

point

concentration

c

in

terms

of

use

(9)

of

the

a

Note

that

ferential by taking substituting

V

if

and

a

deviation

z

[Gray

(1975)],

Oc+ ;

=

Eq.

(8) (6),

we

can

transform

Eq. ka - 2 D

In Eq. pellet,

Cc>u

quantity

=

Am/v

we

is

av

equal

to

the

(7)



to

the

form

CL

surface

(9)

area

per

unit

volume

of

catalyst

.

could

(10) express

equation for the the point equation Eq. (8),

z

In

terms

of

interstitial for c, Eq.

<,>a,

Eq.

concentration (l), and the

(9)

would

~. boundary

yield

a

This can be condition,

governing

dxf-

accomplished Eq. (2), and

(11) -Dn-VT -1 If the
-

field The

k"c =

ko

+

were available, "c solution will

Dn*_

Vu

on

(12)

AW

Eqs. (11) and (12) undergo significant

would result changes over

in a description of distances Y on the a order of the radius of a pore between particles, while average concentration varies over distances L on the order of the catalyst particle radius, Y/L << 1. From an order of magnitude estimate of the terms in the boundary condition Eq. (12) it is quickly concluded that the ratio, (13) for all normal reaction rates encountered in practice. Similarly, it ing the theory of Green's functions [Arfken <1970)] that the solution be closely approximated by the solution to the homogeneous problem,

v%= whenever y/L z simplifies -U In

order

to

where

L

and

-

obtain

. s

uscan

(14)

no 0;

Z=f

shown (11)

0

<< 1. to,

c

can be to Eq.

Using

result

Eq.

=

k~

a

solution

lv

..?-

are

the

a

+

s

a vector

to

Eq.

(13)

on

A

(14).

we

in

Eq.

<12),

the

boundary

condition

for

(15)

aa represent

the

function

E

as

a (16) and

a

scalar

function

of

position.

Substituting

Eq.

(16)

13

Effectivedlffusnrrnesfor catalyst pelletsunder reactivecon&uons

A-2

Into Eqs. (14) and (15) we can develop boundary conditions at the fluid-solid reaction rate and is governed by,

equations interphase.

for

both The g

r and field

s with appropriate does not depend on

the

(17)

- n-Vf

=

Y-

The s shown

field to be

is

on

reaction

v2s = -

n

(18)

Aoo

rate

dependent

and

its

governing

differential

equation

may

be

(19)

0

=

n-Vs _I

k/D

on

(20)

Aoo

f and s equations as given are still formidable, since the The solutions to both the However, if we 1 is extremely complex. geometry of a porous medium as shown in Fig. consider the special case of a spatially periodic porous medium, as described in detail one can show that the f and s fields should be spatially periodic. by Brenner <1979), 2 we show a typical two dimensional spatially periodic array of particles. In Fig.

Two dlmenslonal Frgure 2 rlodlc porous medrum with Ice vectors Rland IL.2 vectors

The and

fl

entire as z2

array shown.

spatially unit cell

may be generated from a The spatial periodicity

pelatt-

single of

f

unit cell and s

with lattice implies that,

f (2)

= f(f

+ _el)

1 =

1,2

(21)

s(5)

=

+ _e,)

i =

1,2

(22)

S(f

One can easily extend this idea to a threeIS a spatial position vector. where r The spatially periodic porous medium does not have to dimensi%al array of particles. Since the and the unit cell can contain more than one type of particle. be isotropic, (17) and (19) need only be solved within a unit cell, f and s fields are periodic, Eqs. znd this greatly simplifies the computational effort required to obtain a solution for z. Once Eq. tam,

the (9).

f functions the governizg

use

dzffusivity

have for

,eff:~~~c>a - ka/c>o

O=D where

and s equation

has

been

tensor,

made

of

defined

Eq. by,

(13).

been the

+

one can substitute Eq. determined, intrinsic phase average concentration

DE

~-Vo _-

In

Eq.

(23)

(16) to

into ob-

(23) the

quantity

,Deff

is

the

effective

14

Knwtzcsand

sDeff -= where

U .z

is

the

vector

unit

that

zeff

5

is

J A

A-2

TJ

(24)

?z

tensor

and

T r

r =a!--- 1 vo v

Note

D Et&+

E

Cataiysrs

is

the

tortuosity

tensor

fWhitaker

(1967)]

(25)

n f dA _CLCS

is

independent

defined

of

reaction

rate

since

f

is

independent

of

k.

The

by,

(26)

and it appears in a convective term in Eq. (23). The vector 6 depends on rate through the parameter f3 = kY/D. This parameter appears Then Eq. (20) mensionless using a mean pore radius, Y as the characteristic length. It ficult to determine from order of magnitude estimates for real systems that -4

normally less than 10 , and that as a essentially equal to zero. This makes and results in, 0 as the section

=

CL

gef f :_o~

governing equation that follows, we

-

for the consider

isotropic two dimensional the results to experimental

Results

and

As a simple sional unit

spatially cell shown

with

s

and term

5 should be very small, x;Eq. (23) insignificant

(27)

kava

versely compare

Comparison

result both the convective

average interstitial solute some preliminary calculations spatially periodic observations.

Experimental

periodic in Fig.

r eat t ion is made diis not difB << 1,

model

of

concentration. In the of for a transQeff a catalyst particle, and

Data

model of the The cell 3.

catalyst pellet, we consider is identically square with

Figure 3 Two dxmenslonal spatially rlodlc model for a catalyst pellet blocks are cubes of length IL The of the unzt cell have a length of The void fractxon E = 1 - ab/R2

peAll srdes

I

the two dimenside length e,

A-2

1s

Effective&ffumvltles for catalystpelletsunder reactivecondltlons

The value of the void fracand gives rise to transversely isotropic porous medium. Equation (17), subject to can be varied by changing the values of a and b. tion E and the spatial periodicity constraint Eq. (21). was boundary condition Eq. (181, and f The problem was namely, fx solved for both components of the vector ,fa Ysolved numerically using a Gauss-Seidel successive over-relaxation scheme [Roache The relaxaAx = Ay = l/60. The unit cell was divided into steps of size (197611. and the calculations took approximately 5000 Ittlon parameter used was equal to 1.4, were calculated directly by the The components of the tortuosity tensor 5 erations. area Integral in Eq. (25). Since the system 1s transversely isotropic then =-C and T =-c =O. This me&s there is only one independent non--zero comT YY Yx xY ponent of the effective diffuslvity tensor, and It is given by D eff -

D

according In

Fig.

to 4 we

=

[l +

E

Eq.

(28)

Txx3

(24).

show

theoretical

values

Deff /U

of

for

various

values

of

E

for

the

unit

0 SOI 1 crumbs

05

A Pumice -Talc . Kaolxn ce1rte

04

r\

1

2 w=-

I

l

I

rVermrculltc vM1.X i

a 011 02

01

I

I

03

04

LIIIf 05nbO708

10

E

Comparison of theoretlcal Figure 4 with expertpredlctlons Eq (281, mental data of Currle [ '601 on the drffuslon of hydrogen In air through beds of unconsolidated sollds (After SatterEffect of the porosity field 1'701 ) E: on the ratlo of the effective dlffuslvlty to the molecular dlffuslvlty cell of Fig. 3. These are compared to the experimental results of Currie (1960) for diffusion in beds of various unconsolidated porous media as presented by Satterfield (1970). When a = b, the agreement between theory and experiment 1s excellent, except for materials such as vermiculite and mica that exhibit a more layered structure. When the ratio a/b = 3, we find that the slope of the theory line is much steeper as the experimental results for these materials seem to indicate,

NOMENCLATURE a A

Surface

V

cc0

C

CC>

z u

zeff

Fluid-solid Point Phase

cc> a

area

per

unit

volume

interfacial

of

catalyst

area

concentration average concentration

Intrinsic

phase

Deviation Molecular Effective

of c from diffusivity diffusivity

average

concentration

XC>

a

tensor

Kmetzcs and Catalysm

16

A-2

Vector function of position Reaction rate constant Lattice vector Length of cube in two-dxmensional array Characteristic catalyst pellet dimension Unit normal from the fluid to solid on

AaB

Characteristx pore dimension Spatial position vector Scalar function of position Unit tensor Volume of the averagxng volume phase with V Volume of a Greek

Symbols

a 5 : F r r

Fluid phase Solxd phase Dimensionless reaction Rate dependent vector Voxd fraction Tortuosity tensor

rate

parameter

Methods

for

REFERENCES Arfken .A,;.

(1970).

Mathematical

Physicist,

2nd

Ed.,

Academic

Press,

New

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