diffusion in catalyst pellets

7/24/2019 Diffusion in Catalyst Pellets

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Chemical Engineering Science, 1962, Vol. 17, pp. 825-834. Pergamon Press Ltd., London. Printed in Great Britain.

Difhsion in catalyst pellets

N. WAKAO

and

J. M.

SMITH

University of California, Davis, California

(Received 10 April 1962)

Abstract-A theory is proposed for predicting diffusion rates at constant pressure through bi-disperse

porous media. The total rate is the

sum of separate contributions for diffusion through macropores,

micropores and a series path. To apply the theory requires a knowledge of the pore volume-pore

radius distribution for the porous material.

Experimental diffusion measurements are reported for five high-area alumina pellets of different

densities made from the same Boehmite powder. The results show that for the least dense pellets

macropore diffusion is dominant. In contrast the micropore contribution controls the diffusion rate

in the most dense pellet. The theory predicts rates in good agreement with the data over the pressure

range investigated, 1-12 atm. Comparison with other diffusion data is hindered by the lack of pore-

volume-distribution information. However, such data are available for a silver catalyst and for a low-

area alumina. For these different materials the theory also predicts reliable diffusion rates.

DIFFUSIONN CATALYST ELLETS

THE

importance of pore diffusion resistance in

some gas-solid catalytic reactions (for example

cumene cracking [l] and o&o-hydrogen conversion

[2]) has resulted in several studies of pore diffusion

rates [3-71. Because of the range of pore sizes that

exist in many solid catalysts, both Knudsen and bulk

diffusion processes may be significant. The inter-

pretation of diffusion data in terms of the appro-

priate diffusivities in such cases lead to some

confusion in the early work. However, correct

and equivalent interpretations have been develop-

ed recently by

SCOTT

and

DULLIEN [8], MASON

et

al [9] and ROTHFELD and WATSON lo].

For optimum rates of reaction the pore-size

distribution should be such that the total of

diffusion and reaction resistances are a minimum.

One can anticipate the time when catalysts can be

prepared specifically to give a pore-size and area

relationship for optimum performance. Before

this can be done, it is necessary to develop a means

of predicting diffusion rates as a function of the

physical properties of the porous material. The

objective of the present study is to propose a method

for correlating and predicting diffusion rates in

terms of the porosity and pore size-distribution

data. The work is restricted to pelleted type

materials prepared by compressing particles of

catalyst powder, which themselves are porous.

Thus the resulting pellet has two pore systems:

micropores within the powder particles and macro-

pore space between the powder particles. To

evaluate the proposed method experimental dif-

fusion measurements were made on high-surface

alumina pellets over a range of pressure and

macropore sizes. These data constitute the only

results that cover a broad enough range of variables

and include sufficient pore size-distribution informa-

tion to evaluate the method critically. However,

partial comparisons can be made with published

data on silver [5] and low-surface alumina [IO]

catalysts.

MECHANISMOF DIFFUSION IN CATALYSTPELLETS

In a bi-disperse porous system at constant pressure

the mass transport will be a combination of diffusion

through (1) macro- and (2) micropores, the re-

lationship between the two depending upon the

bulk density (macropore volume). Fig. 1 is proposed

to represent the system as far as diffusion is con-

cerned. The dotted squares identify the powder

particles containing micropores and the spaces

between them represent the macropores. It will

be assumed that the area void fractions are the

same as the volume void fractions. Then the

probability that a given location (on a plane

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FIG. 1. Diffusion mechanisms in b&disperse porous

materials.

perpendicular to the direction of diffusion) will

be in a void space is E. If E,,, si, and E, represent the

volume fractions of macropores, micropores and

solid,

E, + Ei +

Es = 1

(1)

Suppose the sample is cut at the plane and the two

surfaces are rejoined. Then the void area per unit

total area on the rejoined plane will be the proba-

bility of two successive events, or (E)(E) = E.

The diffusion through the plane can be divided

into three additive parts (parallel mechanisms):

(1) Through the

macro-pores

with an area of E:

and an average pore radius Z,,.

(2) Through

the particles

having an area (1 - 8,)

and an average pore radius L&. The effective void

area of micropores per unit area in the particles is

2

( 2 Or

2

6

(1 - &,)2

(3) Through the macropores and micropores in

series. The area for this contribution is 1 - E: -

(1 - &a)2or 2~,( 1 - EJ.

The general equation for the rate of diffusion of

A in a circular capillary per unit area at constant

pressure in a binary system of gaseous A and B

[8-lo] is

NAYP

1

RT (1 - uyJID AB + l/D,

(2)

N. WAKAO and J. M. SMITH

This expression accounts for bulk diffusion accord-

ing to the molecular diffusivity

D,,

and Knudsen

diffusion with a diffusivity

D,

determined by the

pore radius a

D, = 2ijA a/3

(3)

Equation (2) is now applied to the model of Fig. 1,

adding the contributions to mass transfer of A for

each of the three mechanisms. The resulting

diffusion rate per unit cross-sectional area of the

pellet is

N RT

T-

- -s,z ,

2 - (1 - Q2Di 2 -

(Mechanism 1) (Mechanism 2)

2

- 2E(1 - s) (l/D,) +

( l /Di)

Ax

*A (4)

(Mechanism 3)

In formulating this expression the unit cell that is

repeated to make up the pellet has a length Ax,

from the centre of one powder particle to that of its

neighbour (Fig. 1). The composition change across

the element is AyA.

The diffusivities

Da

and

D,

apply for the macro- and micro-regions re-

spectively, and are given by the equations

1

Da = (1 - ay,+)/DA, + I/&,,

I(1 - &,)2

Di = (1 -

~YAYDAB + (l/h,)

(6)

Here Bkti and Dk, are the Knudsen diffusivities for

component A in the macro- and micropore regions.

Since the pores cover a range of sizes, mean values

of the diffusivity based upon mean macro- and

micropore sizes should be used.

Equation (4) applies to the unit cell of Fig. 1.

If it is summed to include the whole pellet of thick-

ness

L,

and if equations (5) and (6) are introduced

for

D,

and

Di,

the diffusion flux NA can be expressed

in terms of

L

and the end compositions yA, and

yA,.

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Diffusion in catalyst pellets

1-aYA2 +(DA~/Dki)

1 - aYAl +(DAB/%,>

+

4sLl(I - e3

+ 1 + [(i - &,)2/E;]

1 _ ayA

2

I +I [tl - Ea)2/si21@k,/k.)

Dki

1 + [(l -

qJ2/ J

1 _ ayA, : + cc1 - %~)~/~i Zl + @k k.)

Dk,

1 + [(l - qJ2jEiZ1

(7)

If either the macro- or micropores do not exist,

i.e. E =

0 or ~~ = 0, equation (7) reduces to the

pressures on the two ends of the pellet, contained

simple integrated form of equation (2);

in a stainless steel chamber [7], were maintained by

slowly bubbling streams of each gas (helium and

N RTLa

=E21n 1 - aYA2 +cDAdBk)

A PDAB

1 - YA, +

(D.& k)

@)

In most bi-disperse porous catalysts the macro-

pore radius Za is much larger than the micropore

value, &. Then Dki/Ljko is small with respect to

(1 &a)

2

- /&i .

Also the ratio DAB/nki iS large enough

for the arithmetic mean to approach closely the

logarithmic mean. Hence, equation (7) is closely

represented by a simpler result:

nitrogen) into a surge tank [3] containing water.

The lines containing the two gases extended to

the same depth in the surge tank. The pure gases

(nitrogen 99.97 per cent pure and helium 99.99

per cent) were directed tangentially into the chamber

near the end surfaces of the pellet. The exit streams

from the chamber were analysed in calibrated,

thermal conductivity cells (Gow-Mac Co., Type

30s) immersed in a constant temperature bath

maintained at 25C. The entire flow rates of each

1 - aYAz + @?& /p~k.)

a(yA, -

YAz)

1 - aYAl +

< /pDk ,> 1 -

d(YA, + YA,)/21 + ( /p~k,)

4 0 - 80)

+

1 + [(l - &J2/&l

.

l -a[(YA~ +

YA2)/21 + 1 +z$(

)2]

(9)

i -a

a(YAt - J )AI)

-

Here D.& ,

= PD,, and is independent of pressure.

Equation (9) predicts the effect of macro- and

microporosities and mean pore sizes (through

9, and 9, and equation 3) on the diffusion rate.

The result is based upon the model of Fig. 1 which

includes the assumption that the three contributions

are independent of each other. Of particular

significance is the simple result that in a random

distribution of pores the effect of porosity is pro-

portional to s2.

EXPERIMENTAL ORK AND RESULTS

FIG. 2. Experimental apparatus.

1,2 He, Na cylinders

3 Surgetank

4 Manometer

5 Pressure gauge

6 Porous pellet

7 Pellet container

8 He, NZ reference gases

9 Thermal conductivity cell

10 Soap-film meter.

The diffusion apparatus shown schematically in

Fig. 2 is similar to that used and described by

HENRY et al. [3]. It is based upon the original pro-

cedure by WICKE [ 111, for measuring counter-current

diffusion rates through porous materials under

constant pressure conditions. Equal and constant

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N.

WAKAO

nd J. M. SMITH

stream were passed through the cells. Soap film

sented elsewhere [13]. The distribution curves for

meters were employed to measure the rates which the micropores were essentially the same for all the

varied from O-5 to 2 cma/min (S.T.P.). pellets with a narrow peak at about 20 A. This result

Data were obtained for the nitrogen (component indicates that the pelleting process did not destroy

A) and helium (component B) system, at pressures the powder particles. The minimum in the volume

from 1 to 12 atm and at 24C. distribution vs. pore-radius curves occurred at

The pellets consisted of cylindrical disks of about 100 A, and this was taken as the boundary

Al,OZ - lHzO (Boehmite) 1 in. in diameter and between micro- and macropores for determining

about 4 in. in thickness (see Table 1). They were void fractions. The densities of the six pellets were

Table 1. Physi cal Properti es of Boehmi t e pel lets

Pell et *

Total Density Total M icropores Macropores

Thi ckness pore volume Total void

(mm.)

Vt cm3/g)t

Wm3)

fr action Voidfraction -

.%

1

Voidfraction -

ea

&

A

12.6 2.14 0.488 O-80 0.17 25 0.63 15,000

B

12.6

l-39

0.679 O-72 0.27 24 0.45 2300

C 13.0 1.10 0.800 0.67 0.33 23 0.34 1100

D 12.7 0.728 1.041 0.57 0.39 23 0.18 460

E 12.6 O-560 l-206 0.51 0.42 23 0.09 370

Pellets are disks with a diameter of 1 in.

t Pore volume is per gramme of AlaOs; i.e. per gramme of pellet weighed after ignition.

made by compressing the Boehmite powder in a

Carver Model-B Press without adding lubricant or

other contaminant. The powder was contained in

a stainless steel ring, 1 in. I.D., and the ring and

powder together placed in the press to form the

pellet. The ring and pellet as a unit were then

installed in the stainless steel chamber using

flanges with O-rings to obtain a tight seal. This

arrangement allowed operation up to 15 atm

pressure without leakage. The powder used was

spray dried alumina from the American Cyanamid

Company and had an average particle size of about

90 F. Pellets (A to E) of five different densities,

and hence different macropore sizes and void

fractions, were obtained by using different masses

of alumina in the press.

The total pore-volume data and volume distribu-

tion vs. pore-radius calculations were obtained by

Mr. Marvin F. L. Johnson of the Sinclair Research

Laboratories, Harvey, Illinois, by nitrogen adsorp-

tion and mercury penetration measurements.

Measurements were made on six pellets, each of a

different density but all made from the same

Boehmite powder. Examples of the volume

distribution-pore-radius curves have been pre-

in the same range as that of pellets A-D studied

in this investigation. Hence, the void volume for

samples A-D were obtained by interpolation and E

by extrapolation, The resulting physical properties

for the five samples are summarized in Table 1.

The measured diffusion rates, related data and

calculated results are given in Table 2. It has been

shown (9) that the ratio - Na/N, should be inversely

proportional to the square roots of the molecular-

weight ratio, regardless of type of diffusion. For

helium and nitrogen this result is -N,,/N, = 2.64.

The measured rates given in Table 2 indicate ratios

close to this value.

ANALYSIS OF DATA AND THEORY

1.

Evaluat ion of mean pore r adii

In using equation (9) to calculate the diffusion rate

the mean micro- and macropore radii are necessary

in order to evaluate i&, and &,. This requires

data on the distribution of micro- and macropore

sizes. The distribution curves [13] for the micro-

pores are narrow with a very sharp peak at 20 A,

so that cj will be close to this value. Since the mean

radii are used to evaluate the Knudsen diffusion

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Table 2. Experi ment al data and t he compari son w i t h t heoret i cal val ues.

Dif fusion r ate

mole raction of

mol/sec)

RTLNA RTLNA Calculated rom Eq. [9]

Run Pres- He in

Na in Helium Nitr ogen

NB

Contribution of

no. sure Na-side

He-side-SNB S.NA --

JJA, - YA#%B @A, - YA,)~~B

observed macro- micro-

series

P

1 - YAI YAz

x 10-4 x 10-4

NA

A 1 1GO 0.353 0.127 0.333 0.125 2.66 o-209 O-228 0.226 o+MO293 0~00200

2

1.92

0402 0.133 0.302

0.119 2.54 0.222 0.248 0.244 oGoO555 0.00377

3 3.25 0.273 0.124 0.410 0,162 2.53 0.234 0.247 0.240 O+MlO938 0.00620

4 4.81 0.326 0.0915 0.397 0.169 2.35 0.252 0.271 0.261 oMI134 oGO9Oo

B 1 1.26 0.358 0.115 0.1264 0.0485 260 0.0800 OG90 0.086 o+loO929 0GO301

2 5.05 0.391 0.106 0.1622 0.0655 2.54 0.114 0.133 0.118 0.00351 0.0113

3 12.0 0.372 0.109 0.2198 0.0845 260 0.142 0.155 0.124 0.00754 0.0237

c 1 1.92 0.154 0*0500 0.1044 OWO8 2-56 OGKO 0.0584 0.0428 OW132 0.00425

2 4.23 0.240 0.0620 0.1297 0.0498 2.61 0.0641 0.0697 0.0553 0.00443 0.00896

3 7.80 0.321 0.0931 0.1258 0.0484 2.60 0.0740 0.0862 0.0629 o+IO777 0.0155

4 12.4 0.320 OWI80 0.1507 0.0584 2.58 0.0900 0.101 0.0663 0.0115 0.0228

D 1 1.05 0.0387 0.0203 0.0281 0.0107 2.63 0.00996 0.00889 OGO571 0+0170 OGO148

2 2.70 0.0560 0.0298 0.0473 0.0184 2.57 0.0176 0.0182 0.0103 OGO424 0.00367

3 3.28 0.147 0.0335 0.0523 0.0200 2.62 0.0214 0.0210 0.0115 o+IO511 oGo441

4 5.18 0.180 0.0483 0.0648 0.0249 260 0.0282 0.0285 0.0139 0.00785 OGO671

5 6.58 0.202 0.0640 0.0604 0.0241 2.51 0.0287 0.0335 0.0154 0.00975 0.00832

6 11.5 0.197 0.0641 0.0872 0.0352 2.48 0.0418 0+473 0.0180 0.0159 0.0134

E 1 1.68 0.0451 0.0127 0.0184 0.00714 2.58 O+IO659 003566 OGO154 0.00296 O+KU16

2 3.56 0.0582 0.0149 0.0321 0.0122 2.63 0.0114 0.0108 0.00239 OTlO605 0.00236

3 6.25 0.140 0.0890 0.0390 0.0147 2.65 0.0166 0.0171 0GO303 0.0102 oGO393

4 9.30 0.0853 0.0375 0.0630 0.0237 2*66 0.0235 0.0233 ow343 0.0144 oGO553

5 10.86 0.147 0.0522 0.0564 0.0233 2.42 0.0253 0.0267 0.00377 0.0166 0.00637

All runs are at

24C,

A = Nitrogen, B = Helium, S = Cross-sectional area of end of pellets = 5.07 cm2.

contributions in equation (9), the averaging pro-

cedure should be for this type of mass transfer.

The total rate of Knudsen diffusion through the

micropores, using equation (3) is

QA = - & + 2

s

Ob (xa2)n(a)da

(10)

0

The function

n(a)&

represents the number of pores

with a radius between

a

and

a + da,

and

a,

the

upper limit of the microradius, in this case 100 A.

In terms of the mean diffusivity &, the total

diffusion rate is

QA = ; i& 2

s

*kz2n(a)da

0

P

2vA by, _

=-

---a,

s

ab

RT3Ax o

na2n(a)da

(11)

Combining equations (10) and (11) and solving

for pi gives

- s

6%

axa2n(a)da

0

ai =

ali

s

12

7ca2n(a)da

0

The quantity

za2n(a)da

is the volume of micropores

between

a

and

a + da,

per unit length of pore.

Hence, equation (12) can be written in terms of the

known pore volume per gramme of alumina.

- s

Vi

adVi

0

ai=vi

(13)

This

expression was used to calculate ai from the

pore-volume-pore-radius distribution curves and

found to be 23-25 A. The integral is applied over

the microvolume region, 0 to

Vi.

The result

B

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N. WAKAO and J. M. SMITH

depends upon the somewhat arbitrary separation

of the distribution data into separate curves for

the micro- and macroregions. Since the micropore

distribution was the same for all the pellets, this

value of pi is applicable for pellets A-E.

The diffusion data for the most dense pellets

also can be used to obtain a value of Zi. Thus as

s,, + 0, equation (9) reduces to the micropore term

only. Under these conditions,

NA

RTL

P(YAi - YA&%S

=P{l -a[(yA, + yAz)/21) + @%/~d

(14)

Furthermore, at low pressures (l-3 atm), the term

involving the pressure in the denominator of the

right-hand side of equation (14) becomes small with

respect to

D; i B/ bk i .

ig. 3(a) is a plot of

N

RTL

(YAl

- YA,P&

vs. E,.

Equation (13) can also be used to evaluate Z,,

for the macropores by changing the limits of

The intercept, obtained by extrapolation to s,, = 0,

should be equal to

EF&/ D~~.

rom Fig. 3(a) this is

EiD

2 =

O+M20atm-

D

( 1 5 )

AR

The value of the micropore void fraction at .sll= 0

can be estimated by extrapolating the sl data given

in Table 1. The result, illustrated by the graph in

Fig. 3(b), is

E~)_~ =

O-44. Then from equation

(15)

&, = 0~0020(0~697)~(044)2

= 0.0072 cm/sec

(16)

where Di B = 0.697 atm(cm)/sec is the value of

DA,

btained from the Chapman-Euskog expression

[12] for the N,-He system at 1 atm pressure and

24C. Using this result for & in equation (3)

gives Z, = 23 A assuming circular pores. The

excellent agreement between the two results is

probably due to the narrow distribution of micro-

pore radii.

00

8

6

4

2

0.001

0

0.2

0.4

6

FIG. 3. Estimation of

c

0.4

i

.;o

8

0.3

it

0

9

0.2

e

.o

r

0.1

0

0.2 0.4

c-6

Macro void fractlan , r,

Knudsen diffusion in micropores.

830


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IO'

o

6

r

IO'

0

I 2

Total pore volume, V

cdh

FIG. 4. Mean macropore radii for Boehmite pellets.

integration to cover the macropore region V, =

V, - Vi. The values of (z,

SO

calculated for the

original six pellets are plotted vs. total pore volume

V, in Fig. 4. Values of Za for pellets A-E were

obtained by using the V, data in Table 1 and Fig. 4.

The values of (5, so evaluated are given in the last

column of Table 1. These were used to determine

the corresponding mean Knudsen diffusivities

Dk,.

Total pressure, P. atm

P,

atm

P,

atm

o Exp data of pellet B

o Pellet D o Pellet E

2.

Comparison with data for high-area alumina

Knowing i& (OGO720 cm2/sec) and &._ equa-

tion (9) can be used to predict diffusion rates through

the pellets as a function of density and gas pressure.

The porosities a. and ai are available from the pore-

volume measurements. The observed and predicted

total rates are compared in Fig. 5 for three of the

pellets by plotting N,R7X/(yA, - vA,)D& vs.

pressure.

Experimental and calculated results for

all the pellets are given in Table 2. Also included

in Fig. 5 are curves for the separate contributions

to the total rate, corresponding to the separate terms

in equation (9). For the low-density pellets, illustra-

ted by the results for pellet B in Fig. 5, diffusion

through the macropores is dominant. The con-

tributions of the micro- and series mechanisms are

10 per cent or less of the total. For the most dense

pellet E, this is reversed and the contribution of the

macropore diffusion is least signitkant. The total

rate decreases sharply as the density increases.

The agreement between the predicted and experi-

mental result lends confidence to the proposed

theory, particularly since thereare no arbitrary con-

stants involved in equation (9).

Evaluation of equation (9) requires data (a) for

pellets of different densities to test the validity of

the porosity functions in each term, and (b) for

FIG. 5.

Comparison of experimental data for high-area alumina with theory.

1: Macro contribution. 2: Micro contribution. 3 : Series contribution.

831


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N. WAKAO nd J. M. SMITH

different pressures to test the validity of the func-

tions giving the relative significance of bulk and

Knudsen diffusion. Complete data of this nature

do not appear to be available except for that pre-

sented here. However, some information has been

published which permits partial evaluation of

equation (9).

3. Comparison w it h sil ver-pell et data

MASAMUNE and SMWH [5] measured diffusion

rates through silver catalyst pellets of different

densities at 1 atm pressure. Void-fraction data

were obtained but not volume pore size-distribution

information. Hence, the appropriate values of

D,. and Dkki re not available. However, for low-

density pellets under conditions where Knudsen

diffusion is negligible, such information is not

needed to apply equation (9).

The results of most prior investigators have been

reported in terms of an effective diffusivity, D,,

defined by the expression

dYA

e

1 - ay, dx

(17)

If this is integrated at constant pressure conditions,

there results

1 -

aYA

N,RTLu = D,P

In

l- MyA,

(18)

Equation (18) can be combined with Equation (9)

to give an expression for D, in terms of the proper-

ties of the porous material. Expressed as the

diffusibility, De/DAB, the result is

I.0 _

8

I II

i IL

n

6

FIG. 6. Comparison of silver and alumina data with

theory.

restricted conditions, equation (19) reduces to the

simple form

D

e=E

D ,

AB

(20)

The experimental results for silver-catalyst pellets

are shown in Fig. 6 plotted as D,/DAB

VS. ~2.

The dotted line on the Fig. represents equation (20).

The results for the least-dense pellet (si = O-57)

agree reasonably well with Equation (20). The

data for the higher-density pellets show large

deviations for equation (20). This is expected, from

D

e=

1

D

B ln[(l - ayAz)/(l - ayAl)l

1 -RYAN +(D~B/P&.)

dyA, - YAz)

+ 1 - @[(YA, + YAJ PI + (KBIP~,,) +

4 (1 - )

+ 1 + [(l - &,)/S] *

dYA, - YAz)

1 - a[(YA, +

YAz)/~~ + 1

+; fe

121

I cl

(19)

For low-density pellets only the macropore con-

tribution should be signmcant (see Fig. 5). Also if equation (19), because of the increasing importance

the macropores are large enough, or the pressure

of Knudsen diffusion in the macropores and

high enough, the Knudsen part of the diffusion in

increasing importance of the micropore and series

the macropores should be negligible. Under these contributions.

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4. Compari son w it h ow -area alumi na

ROTHFELD

and

WATSON

[lo] measured diffusion

rates through Q in. cylindrical pellets of alumina

(Type Al-0104-T, Harshaw Chemical Co.) and

presented data for the isobutane (A)-helium (B)

system at pressures of

200

to

700

mm Hg. This

material corresponded approximately to the more

dense pellets D or E in Table 1 with respect to void

fractions. The values of E, and Ei were 0.18 and

0.34 respectively. However, the micro- and macro-

pores were much larger and the surface area much

smaller (76 m2/g vs. 350 m2/g for pellets A-E).

The pore volume vs. pore-radius distribution data

did not show a sharp minimum between macro- and

microregions in contrast to the high-area alumina.

There was a signi6cant volume in pores of inter-

mediate radii. Therefore, taking the minimum in

the distribution curve as the boundary between

micro- and macropores introduces errors in calcu-

lating ii and Cr,. This difficulty was circumvented by

extrapolating the micro- and macrosections to give

separate distribution curves. From these curves,

using equation (13), the mean pore radii were found

to be Zi = 84 A and & = 4800 A. Composition

information was not available so that yA1 and Ye,

were unknown. This prevents an exact comparison

of theory and experimental data. However, most

diffusion measurements of this type are made with

nearly pure gases on the opposite faces of the pellet

so that yA, = 1 and yA2

= 0, is probably a good

approximation. Using these results diffusion rates

were calculated from equation (9). The experi-

FIG

7. Theoretical and experimental results for low-

area lumina [lo].

mental and calculated

values are shown in Fig. 7

based upon the same co-ordinates as Fig. 5. The

agreement suggests that the theory is also applicable

to this different type of porous material.

In summary a model has been proposed for

diffusion in bi-disperse porous catalysts. The

development gives a relationship between diffusion

rate, porosity and pore-size characteristics, that

agrees well with available data. To predict diffusion

rates from the equations it is necessary to know

the pore-volume-pore-radii distribution in the

porous material.

Acknowledgement-This

study was sponsored by the

United States Army Research Office (Durham) through

Grani

DA-ARO(@-31-124-Gl91.

a

li,

CZb

tit

DAB

D'AB

De

DC2

NOTATION

Pore radius cm

Mean radius for diffusion in the macropores

cm

Upper limit of radius of micropores

cm

Mean radius for diffusion in the micropores

cm

Binary bulk diffusivity for system A-B

cm2/sec

DABP atm

cn+/sec

Effective diffusivity defined by equation (17) cm2/sec

Composite diffusivity, defined by equation (5)

Composite diffusivity, defined by Equation (6)

Mean Knudsen diffusivity of gas A in the micropores,

defined by equation (11)

cm2/sec

Mean Knudsen diffusivity of gas A in the macropores

Geometrical length of porous pellet cm

Molecular weight

g/g mole

Diffusion rate per unit area of gas A (nitrogen)

T

V

BA

X

YA

Number distribution of pores

cm-

Total pressure

atm

Diffusion rate

g mole/set

Gas constant cm3(atm)/(K)g mole

Cross-sectional area of pellet

cm2

SNA = total diffusion rate through pellet

Temperature

K

Pore volume per g (ignited,

cm3/g

Vt = total pore volume,

Vt = micropore volume

Average molecular velocity of gas A

Cm/SeC

Distance in the direction of diffusion cm

Mole fraction of gas A in the system;

y_Q = mole fraction nitrogen on nitrogen side of

pellet

;

yA2 =

mole fraction nitrogen on helium side of

pellet

OL

E

Ratio of diffusion rates 1 + NBINA

Void fraction

cm2/sec

CllG/StZC

cm2/sec

g mole/(cm2)sec

Subscripts

a, Macro- and micropores, respectively

A, B Nitrogen and helium, respectively

1,2 Nitrogen and helium sides of pellet

833


10/10

N. WAKAOand J. M. Surrn

RFFERENCES

[l] PRAYER . D. and LAGOR. M.,

Advances in Catal ysis

Vol. VIII, p. 294. Academic Press, New York 1956.

[2] WAKAON., SELWOOD

. W.

and SETH 5. M.,

Amer. I nst. Chem. Engr s. .I .

To be published.

[3] HENRY . P., CHENNAKF..SAVAN. and SMITH . M.,

Amer. Inst. Chem. Engr s. J. 1961 7 10.

[4]

HOOGSCHAGEN.,

I ndustr. Engng. Chem. 1955 47 906.

[5]

MASAMUNE. and %rrrr J. M.,

Amer. I nst. Chem. Engr s. J. 1962 8 217.

[6] Scorr D. S. and Cox K. E., J. Chim. Phys. 1960 57 1010.

[7] WEISZ P. B.,Z. Phys. Chem. 1957 11 1.

[8] Scorr D. S. and DULLIEN . A. L., Amer. inst. Chem. Engr s. J. 1962 8 293.

[9] EVANSR. B., WATSONG. M. and MACON . A., Gaseous Di ff usion n Porous Media at Uni form Pressure, IMP-AEC-15.

Institute for Molecular Physics, University of Maryland, 1 June 1961.

[IO] ROTHFELD. B. and WATSONC. C., Gaseous Counter Diffusion in Catalyst Pellets. Paper presented at the 54th Ann.

Meeting Amer. I nst. Chem. Engrs., New York, 3-7 December 1961.

[ll] WICKEE. and RALLENBACH., KoNoidZ. 1941 97 135.

[12] HIRXHFELDER. O., CURT~SS . F. and BIRD R. B., Molecular Theory of Gases andliqui ds, Chap. 8. John Wiley, New

York 19.54.

[13] MISCHKE . A. and Sm J. M., Thermal Conductivity of Alumina Catalyst Pellets. Submitted for publication to

I nd. & Eng. Chem., Fundamentals Quarterl y.

R&urn&La

auteurs proposent une theorie pour ltvaluation des vitesses de diffusion a pression

constante a travers un milieu poreux ;i double dispersion.

La vitesse totale est la rtkultante des vitesses de diffusion a travers les macro et micropores et

diverses voies. Lapplication de la theorie nkcessite la connaissance de la repartition des volumes

et rayons des pores du mat&au.

Des mesures experimentales de diffusion sont effectukes pour cinq sortes de granules dalumine

a grande surface active de differentes densites, obtenues a partir de la m&me poudre de Boehmite. Les

r&hats montrent que la diffusion par les macropores est predominante pour les plus faibles densites.

Inversement, la diffusion par les micropores correspond aux plus grandes densitb. Les vitesses

prevues theoriquement dans lintervalle 1-12 atm correspondent bien aux r&hats experimentaux.

Le manque dinformation sur les distributions des volumes des pores concemant dautres rksultats

de diffusion empkhe toute comparaison.

Cependant ces donnks sont valables pour un catalyseur dargent et pour une alumine a faible

surface spkcifique.

La theorie donne aussi des evaluations de vitesses de diffusion correctes pour ces differents

mattriaux.

834

diffusion in catalyst pellets

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