Mass Transfer in Heterogeneous Catalysis

Mass Transfer in

Heterogeneous Catalysis Charles N. Satterfield

M.I.T. PRESS Cambridge, Massachusetts, and London, Eng lan d

Copyright © 1970 by The Massachusetts Institute of Te chn ology

Set in Monotype Times Roman. and bound in the United States of America by The Colon ia l Press Inc., Clinton, Massachusetts.

Printed

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262 19 062 1

(hard cover)

Library of Congress catalog card number: 70--87307

To Anne, Mark, and Joye

Contents Preface

xv

Diffusion 1.1 1 .2

1.3 1.4 1.5

I .6 1.7

Introduction 1.1.1 Reac t i on Regimes D iffu s i o n in Gases Diffusion Coefficients: Binary Gas Mixtures Diffusion in Liquids Solid Ca t alysts 1.5.1 Ca talys t Supports 1 . 5 . 2 Zeolites ("Molecular Sieves") I .5.3 Physical Characterization of Catalysts I .5.4 Bimodal Po re- Size Distribution I .5.5 Measurement of Diffusion in Porous Solids Bulk Diffusion in Porous Catalysts D i ffu si o n in Fine Pores 1.7.1 Knudsen Diffusion 1.7.2

1.7.3

I .8

1.9

1

1.7.4

The Transition Region Surface D i ffu sion

D i ffusion and Reaction in Zeolites(" Molecular Sieves")

Est i m at ion of Diffusion Coefficients in Porous S olid s I .8. I Parallel-Path Pore Model I .8.2 Anisotropic Structures I . 8 . 3 Other Mod els of Porous Structures Additional Comments

2 Mass Transfer to Catalyst Particles. 2. I Introduction vii

1

4 9 12

18 21 23 24 25 29 30 33 41 41 42

47 54 56 64 72 73

76

78 78

viii

CONTENTS

2.2

Fixed Beds 2.2.1 Mass Transfer

79 79

Heat Transfer

83 84

2.2.2

2.3

Fixed-Bed Reactor Performance with Mass Transfer Controlling 2.3.1 Effects of Space Velocity

2.4

2.3.2 Temperature Difference between Solid and Fluid Trickle Beds

2.5

Fluidized Beds 2.5. 1 2.5.2

2.6

3.1 3.2

Mass Transfer in Slurry Reactors

108

Mass Transfer to Bubbles and Suspended Solid Particles

Mass Transfer from Bubbles

Hydrogenation Capacity of Slurries

Diffusion and Reacti on in Porous Catalysts. Simple Treatment

I

Introduction Geometries Other than a Sphere Some Characteristics of Diffusion-Limited Reactions

3.4

Determination of Effectiveness Factor

3.4. I

141

Theoretical. The Modulus

be 0.27

Diffusion Coefficient i n a Liquid System

Esti ma te D 1 2 for thi o p h ene in dilute solution in hexane at 4 0 °C. Hexane is presumed not to associate, X i s t aken as 1 .0, Mz is 86 (hexane), and Vb was found in the previous numerical example to be 8 8 . 1 (thiophene). The viscosity of hexane at 40 °C is 0. 262 cP. Substituting in Equation 1 . 23, we o b ta in Du

3 1 3(86)1 1 2

=

74 . X w - 1 0 0 .0 0 2 62 (88 . 1 ) 0 . 6

=

5.6

X w - s cm l jsec.

S ince Vb is evidently much less than 0.27 (XMz)t . 87, apply.

Equation

1 .23 should

Perhaps the mos t significant feature of D1 2 for liquid syste ms is that ordinary temperat ures it is of the order of 1 0 4 times D1 2 for typical gas systems at atmospheric pressure, or one per cent of D1 2 for gas

at

-

1 00 atm. 1 .23 suggests that D1 2 is proportional to T/J1 and since viscosity d ecre ases with increase in temperature, D 1 2 increases

systems

at

Equati on liquid

rou ghly as the ·square of the absolute temperature. As for gases , the

activation e ne r gy

is

much smaller than for most chemical reacti ons. For

solutes in hexane, for

the range 0-70 °C

example, the apparent activation e nergy over is 2.23 kcal/g-mol . Data quoted by Maxwell [208]

show T/J1 for a 4 3 . 2 ° A .P.I. Pennsylvania crude fraction boi ling from

250-275°C

(482-52 7 ° F) to i ncrease as

the 4.4 power of T. A t 500°K of 4.3 kcalfg-mol.

(440° F) this corresponds to an activation energy

20

DIFFUSION

Table 1 . 5 i s an abbreviated list of experimental values of D� 2 for low sol u te concentrations in co mmon liquid systems . Himmelbla u [ 1 44] presents a d e t a i l ed review on di ffusion of di s s ol v ed gases in liquids. Diffusion i n general is treated i n the books by Sherwood and Pigford [3 1 9 ], by Bird , Stewart , and Lightfoot [32], and by Jost [ 1 63 ) and in a review b y Bird [3 1 ]. M ethods o f estimating diffus ion coefficients are summarized and evaluated in the book by Reid and Sherwood [264 ]. Crank [79 ] gives the mathematical solutions to the di ffusion equations for a variety of geometrical s hapes and boundary conditions. Table 1 . 5 Diffusion Coefficients for B i n a r y Liquid Systems at Low Solute Concentrations (Experimental values of D�2 (cm2/sec) x 1 0 �) Solute

In Water

Helium Hydrogen Oxygen Carbon dioxide Ammonia Chlorine Methane Propane Propylene Benzene Methanol Ethanol

298 298 298 298 285 298 293 277 333

298

i-Butanol i-Pentanol

298 288 283 288 288 288 288 293

Acetic acid Benzoic acid

293 293 298

n-Propanol n-Butanol

Ethylene glycol G l ycerol

Glycine Ethyl ace t a te A c e to ne F urfura l

Urea

Diethylamine An iline Acetonitrile Pyridine •

T(oK)

298 293 288 293 293 293 293 288 288

6.3 4.8

2.41 2.00 1 . 64 1 .25• 1 .49 0.55 2.7 1 1 .44 1 . 09 1 .26 0 . 84 0.87 0.77 0.77

0.69 1 .04 0.82 1.19 1 .2 1 1 . 05 1 .00 1 .22 1 .04 1 .20 0 .9 7 0.92 1 . 26

0.58

Equili brium mixture of hydrolyzed and unhydrolyzed chlorine.

1.5

SOLID CA TAL YSTS

21

Table 1 . 5-Continued

I n Benzene

Solute

reK>

D�z X 1 0 5

Acetic acid Carbon tetrachloride Ethylene chloride Ethanol Me th an o l Naphth ale n

298 298 28 1 288 298 281

2 .09 1 .92 1 .77 2.25 3 . 82 1.19

Acetic acid Benzoic acid

298 298

3.31 2.62

Carbon dioxide Pyridine Urea Water

290 293 285 298

3 . 20 1.10 0.54 1.13

Acetic acid Acetone Benzoic acid Ethanol

293 293 293 28 8

2.00 2.93 1 . 74

e

I n Acetone

In Ethanol

In

1.5

Toluene

3 . 00

Solid Catalysts

Industrial catalysts

comprise

a wid e variety

of

materials

and are

manufactured by a variety of methods. Many catalysts or poro us struc­

tures studied in the laboratory in fundamental investigations are chosen so as to have simple, uniform, or known structure rather than high ac tivity or good me c h an i ca l strength. Therefore, they are frequently of little immediate industrial interest. The commercially useful catalyst particle size is determined b y the process in which it is to be used. For fixed be d s particles ge ne r ally range from about -fer! in. in diameter. Diffusional resistance within the porous structure increases with partic le size and the large internal pore surface b ec o me s less "effective , " so particles larger than about ! in. are frequ e nt ly pierced with holes or formed as rings. Sizes much smaller than -h in. inay produce excessive pressure drop t hr ough the bed, be mechanically weak, or difficult to manufacture. In fluidized-bed reactors a substantial particle-size distri bution is desired for good fluidization ch aract er istics ; this is usually present in the catalyst powder supplied, or is pro d u ce d in the normal course of operation by the gradual attrition of t he catalyst in the highly turbulent environment. The particles present generally range fro m a bout 20-300 p. in d i a meter the mean being abo u t 50-75 Jl. Smaller particles are entrained from the reactor ; larger ,

22

DIFFUSION

p ar t icle s fluidize poorly. C a t al y s t s for slurry reactors are t yp i c all y 7 5-200 J1

in diamet e r . Finer powder s are d i fficu l t

to remove b y settling or filtra­ tion ; coarser powde rs may be m ore difficult to s u s pe n d and may be less effective per unit mass. The simplest me t ho d of cat al ys t manufacture an d one widely used is to impre gna te a c at aly st support with an appropr iat e solution, followed by d r ying and various treatments such as re d uc tio n or calc in ati o n , to pr o duce an active catalyst. The pore structure of the final catalyst is es s en t ial l y that of the s u pp or t (ca rr i e r) but the concentration of the active catalyst usually decreases toward the center of the particle. The degree of uni for mi ty varies with the adsorptive p r o per tie s of the carrier and the method of manufacture. Thus, the use of an a l c o h o l i c solution may pr o d u ce a substantially different co ncen t rat i o n distribution t han that obtained with an aqueous so l u t i o n . The drying fo l l o wi ng impregna­ tion by a solution u s ually t ake s pl ac e by e v ap o ration at the surface of the p el le t , to which the so l u ti o n is drawn by capillari ty . The so lute tends to be pr ec i p i t a t ed near the s u r fac e as t he s o l ve nt evaporates. If the active ingredient i s expensive, as with n o ble metals such as platinum or p all ad i um , it is s o m e ti m e s desired to support it in the form of a thin an n u l a r shell on the outside of the pellet if the reac t i on is so rapid that active material in the p ar ti cle interior would contribute little. With complex reactions, selectivity i s u su al l y diminished in the presence of s i gni fica n t concentration gradie nt s thr o u gh a porous ca t a l ys t (low effectiveness fact o r). Confining the active c at aly s t to a t h i n outside layer provides a method of eliminating this p r oble m while reta in ing a catalyst particle of a size e as y to work with. Alternatively, many industrial catalysts are prepa red by a process s t art i ng with precipitation fro m aqueous solution, sometimes in the presence of finely pow de red carriers, followed by dewatering and drying. The resulting solid p o wd er is then t y pi call y mixed w i th binders and lubricants, the mixture is pelletized or extruded, and the pe l lets or extrudates are subjected to a heat treatment or " activation " a t high temperatures, which ha s several purposes. It is usually desirable to decompose t he inorganic material present , such as a ni t rate or a car­ bonate, to form the c orres po nd i ng oxide. In most cases these reactions also cause fine p ores to be opened up in t he structure with a c orre s po n d ­ ing increase in total surface area. It is also desirable to eliminate com­ p one n t s undesired in the final co m p o s i t io n , such as organic bin d e rs and die ·l u bri cants, by burning them out with a i r . Someti mes a p arti a l sintering is requ i red to increase mechanical strength, although too much sintering as c au se d by exce s s i ve l y high te m p era tu re s may decrease t he

1 .5

SOLID CATAL YSTS

23

effective diffusivity in the structure (see, for examp l e , Table 1 . 1 1 ) . It is u s ua l ly desirable to h ea t the catalyst during m a n ufact u re to a tempera­ ture at l e a s t e q ual to t he highest reaction temperature to which it wil l be ultimately subjected , so that it wm be structurally stable in the reactor. This method of m anu fac ture u s ua ll y results in a uniform distribution of t he active catalyst throughout the pellet. Silica-alumina cracking catalysts are p rep a re d from a hydr o ge l , frequently incorporating fine crystals (about a micron i n .s ize) of a zeolite ( mol ec ul a r s i e ve). For use in moving-bed pr ocess e s , the hy d rogel is formed into " be ads " by d i s pe r sal a s dro ps of controlled size into h ot oil which co ag u l a t e s the gel. For use i n fluidized-bed reactors a " micro­ spheroidal " cat al yst is u sually u s e d , made by drying the h y d r o gel in a spray t o w e r .

1 . 5. 1

Catalyst Supports

The most ubiquitous catalyst carrier is al umina. It is i ner t to most reacting systems, structurally s t a b l e to relatively high t emper a t u re s and is available in a v a ri e t y of forms wi t h surface areas ranging from less t ha n 1 m2/g u p to about 300 m2/g. S i l i ca gel , either in the form of granu le s or powder, is also a u s e fu l carrier and is available with s urface areas up to about 800 m2/g. If the catalyst is to be composited , one may start with p o wde r ed silica ge l , with a hydrogel, with colloidal silica, or u t i l i ze a hydrogel fo r me d in situ by precipitation. Diatomaceous e a r t h (kieselguhr), a naturally o cc u rri ng form of s i l ica having surface areas in the nei ghb o r h oo d of 50 m2 /g, may be utilized as a p owd ere d carrier. Porous car b o n is thermally stable to temperat ures o f l 000°C or more under inert conditions and c e r t a i n forms have the highest kno w n surface 2 areas of any m at eri al , up to a b ou t 1 300 m /g. These " activated " carbons are c o mmo nly u s ed a s c at al ys t car rie r s for o r g ani c reactions, as in slurry reactors. For high-temperature re ac tio n s it may be n ece ssary to have a " refrac­ tory support " for mechanical stability. These are p ri ma ril y various forms of s i l i ca and alumina manufactured by high-temperature fusion (e.g. , 2000°C) in an e lec tri c furnace. The product is then crushed and sieved , formed int o i rreg ular granules, spheres , rings, cylinders, etc. , and fired in a kiln typi cal l y at about 1 400°C. The final products hav e surface areas of less than I m2 /g and p o re sizes typically in the r a nge of 20- 1 00 Jt. T he s e supports are also useful where high re act i on _ r at e per unit mass of catalyst is less i m p o r t a nt than other factors such as cat a ly s t cost, or where the desired product is a re ac ti o n intermediate and it is necessary to el i minat e fine p o re s in order to enhance re act i o n s ele ct i v i t y .

24

DIFFUSION

1 . 5 .2

Zeolites

("Molecular Sieves ")

A maj or cha nge in cr ac ki n g and hydrocracking processes in pe tr ol eu m been brought about in the last few years by the introd uction of zeolites as ca t a lys t s . Zeolites are highly crys tal line hyd r a t e d alumino­ si licates which upon deh yd ra t i o n develop in the i deal cr y s tal a udiform pore structure hav ing minimum channel diameters of fr o m 3- 1 0 A dep end ing upon the ty p e of zeolite and the n a tu re of the cati ons present. The s tr u c t ure consists of a three-dimensional framew ork of Si04 and AI04 te tra hed r a each of which c on t ai n s a silicon or aluminum atom in t h e center. The oxygen atoms are shared b e t ween adj oini ng tetrahedra. The t w o types of tetrahedra can be present in various r a t i o s a nd ar ran ge d in a v ari et y of ways. Zeolites have the general formula

refining ha s

,

Me21.0 :

Al 2 0 3 :

X Si02 :

Y H 20,

re prese nts a metal c at i on o f v a l e n ce n and X and Y, w hi ch w i th the type of zeol ite The metal c a t io n is present because for each alumina tetrahedron in the l att i ce there is a n over-all charge of - 1 . This req u i re s a cation to pr o d u ce electrical neutrality. Since the pore sizes of ze olite s are c l o s e l y comparable to the s i ze of t he molecules of ma n y substances of industrial interest, zeolites can ex hi b it a selective form of ad sorp ti o n based upon the exclusion from the po res of molecules whose size or shap e can not permit the m to be a cco mm oda t ed hence the term m o lec ul ar sieves. " They have been used fo r o ver a dec ad e in se p ar ati o n processes . To ac h i eve high cat al yt ic activity in aci d- catal y zed reactions such as cracking, the s o d i u m cations originally present in the ze o lite are replaced by other cations such as those of rare earths and by hydrogen to pro d uce acid si t e s . The zeolites of i n t ere s t in c atal ys i s are pri m ari l y tho se hav i ng relatively large p ores ; t he l arg es t pores are exhibited by faujasite-type zeolites, w hi c h are characterized by a three-dimensional s tr u ct u re of cages or cavi ti es inter­ connected by somewhat smaller portholes. Types 1 3X and l OX are, respectively, the sodium and calcium forms of a s y nt hetic zeolite having a structure identical to t h e natural mi ne ra l fauj asite and a Si/Al atomic r a t i o of ab ou t 1 t. Type Y has a s truc t ure similar to na t u ra l fauj asite, but an Si/Al atomic ratio of ab o u t 2!-. Bo t h X and Y can be pr epa red within certain ranges of Si/ AI rat i o s In the sodium form of Ty p e X or Y t he cages are abo ut 1 6 A in diameter and the interconnecting port­ h o l es about 1 0 A d i a me t e r For l OX the portholes are ab ou t 8 A in diamet er The next largest pores are found in mordenite, which has a t wo d i mens i o n a l pore structure cons i s t i ng of par a llel s l ig htl y e ll i p t i c a l s traight channels , h a v ing about 7 and 6 A major an d m i n o r di ame t er s, where Me

exceed unity, vary

.

,

"

.

.

.

-

1.5

SOLID CATAL YSTS

25

respectively, in the s odium form, the channels being effectively isolated one from another. The s m al l e r pore synthetic zeolites i nc l ude 4A and SA, whi ch are the sodium and calcium forms of Ty pe A, with po re diameters of about 4 and 5 A , respectively. The s tructures of zeolites in general are described in a book by H i rs c h [ 1 45], a review by Breck [45 ], and in a recent s y m po s i u m [ 4 5 ] . Synthetic fauj asi te-typ e zeolites are generally manufactured in the form of crystals abo u t 1 Jl in size. For use in cracking o r h y d r o cr ack i n g processes 5- 1 5 p er cent of the zeol it e is usually i ncorp o r a t ed in a s i l i ca­ alumina matrix, like t ha t comprising c o n ve n t io n al silica-alumina cata­ l ys t s , with powdered a-alumina s o me ti me s added for a ttr i ti on resistance, and the mix t ur e is then formed into beads, p o wders , etc. , of the de s ired size. 1 .5.3

Physical Characterization of Catalysts

The ultimate goal here is to be able t o p re di ct the effective diffusi v i ty o f react ant s an d p r o d u c t s in a porous cataly s t . The cl o s ely r elat e d p r o b l e m , that of pr e di c ti n g the permeability o f porous media to flow of fluids, ha s b ee n e x tens i vel y studied for many years [66]. Po ro u s structures are too c o m p li ca te d and too various to be cap ab l e of r ep re sent ati o n by a s ingle number or even one s i mp l e model . Prediction must be based on enlightened e mpi r ici s m , combining t he o retica l models and physical me as ure ments on ca t al y s ts wit h em p i r ical information on method of ma n ufac t u re and the p r o b a bl e effects of p he n o m e na o ccu rri ng d uring use in reactors. K now l e d ge of the total surface area of the ca t al y st is a basic require­ ment. This is ge n er a l ly obtained by the B . E .T. (Brunauer-Emmett­ Teller) method, i n which the effect o f the total pressure on the amount of a gas adsorbed on t he solid at constant temperature is mea s u r ed . T he e xp e ri me n t a l p roce d ure and the m e t h o d s of analys i s have been devel­ oped in great detail. For the most reliable measurements t he molecules of the gas chosen should be s mall, ap pr o x i ma tely sp he rical , inert so no che m i so rpti o n takes place, and the gas should be ea sy to hand le at the re q u i red te_mperature. The choice is usually n i tro gen ; measurements are required over a range of re l ative pressure P/P0 (ratio of gaseous pressure to vapor pressure over liquid p has e at the s a me te mper at ure) of a b o ut 0.05 to 0. 3 and , therefore, are m a d e at cr y o ge n i c t e mp e r a tures , u s u a lly u s i ng liqui d nitrogen as the co o la nt . The vari o u s methods of analyzing t he data are all di recte d to deter­ m i nat i o n of the quantity of adsorbate that corresponds to a molecular mon ol ayer on the sol id s u rface . C o mbi n ing this with the cross-sectional

DIFFUSION

26

are a o cc u pied per adsorbed m o le cu le, us u ally taken to be about

1 6. 2 A2

for ni tr oge n , gives the area of the catalyst. When the total area of a sample

is

sm al l ,

the am o u nt of nitrogen gas ad s orbe d becomes

s ma l l relative of measurement of higher b oil i ng p o i'b.t ,

to the total amount in the apparatus and the accuracy

becomes poor. By

using

as an ad so r bat e a vapor

measurements at liq u id ni t ro ge n temperature can be made at much lower pres s u re s to achieve the desired

·ran ge

of

P/P0 v al ues ,

so the amou n t

of gas adsorbed on the s o li d is now a much larger fractio n of that

present and can b e more preci se ly measured. The

gas

m o s t fre q uently

used for such lower pressure measurements is krypton, w hich has a vapor pressure of about

3 mm Hg

at the temperature of liquid nitrogen.

Pore-Size Distribution.

The most i mpo rt ant re qu irement for pr e di cti n g is a k no w led ge of the po re- si ze di stribution of the catalyst. In its absence, it is i m possi ble to make more than a gross g u e s s as to t h e effective diffusivity u nl e ss the d i ffu si o n is completely by the bulk mode, a co nd it i on encountered infrequently in gas -ph ase reactions. Much literature o n pore diffusion effects in catalysts, particularly the d iffu s ivity

earlier literature , provides i nsufficient informati on on the pore structure to permit i nterpreta ti on in

mo r e

than a

q ua li tative

manner.

The starting point is to assume that the complicated p o r e geometry

can be represented by

capillaries of

u ni fo rm

a simple model , usually an array o f cylindrical

but di fferent radii, rand o m l y oriented . The fact

that pore cross sec ti o n s are act u al l y highly irregular is not of maj o r consequence in the applicati on of theory. Th e

distribution of fine pores is us ual ly determined by a method de vel o ped by B arre t t , Joyner, and Halenda [2 1 ] and i mpr o v ed by Cranston and In kle y [80] . It app li e s the Kelvin equation, which relates vapor pressure above a l iq u i d in a c api l ­ lary tube to its s urface curvature. The increase in a mo u nt of vapor adsorbed onto a catalyst upon an i ncre me n tal increase in v ap o r pressure at constant temperature (usually ni tr oge n vapor at the temperature of liquid nitrogen) represents t he filling of cap i llar i e s of a size gi ven by the Kelvin eq u ati o n . This must be c orrected for an i ncrease in t hick ne ss of the adsorbed layer. The size of the largest pores that can be measured

is limited by the rapid change of meniscus radiu s with pressure as the

r e l ativ e pressure, PfP0 ,

300

A

nears unity.

This

is generally

taken

as about

in diameter, corresponding to a relati ve pressure of 0.93. The

smallest pore sizes that can be determined by this method are about 1 5-20

A

i n d i a met er . Although measurements may be re po r ted c or ­

respo nd i n g to smaller pore sizes, interpretati on of the results becomes

i n crea s i ng l y uncertai n . The method of analysis ass umes that the proper-

1 .5

21

SOLID CA TAL YSTS

ties of the condensed phase in the capillaries are the same as those of a bulk liquid, yet the concepts of surface tension or a curved surface must become increasingly unrealistic as pore size becomes of the order

of magnitude of the size of adsorbate molecules. A s omewhat different

relationship between amount of vapor sorbed and pressure is usually . obtained experimentally upon decreasing rather than increasing the pressure, and an extensive li terature exists attempting to relate these hysteresis effects to details of the pore structure or to explain them in terms of supersaturation effects, variatio n of contact angle, etc. These, however, are secondary matters which do not grea tly affect the use of pore-size-distribution measurements for our purpose. The desorption curve is preferred and good agreement has generally (although not

always) been found between the pore-size-distribution curve by nitrogen desorption and that determined by mercury porosimetry, described below. S tudies by J oyner, Barrett, and Skold [ 1 64] showed this to be true for a variety of charcoals ; indeed , a double peak in one sample having maxima at about 25 and

75 A

was traced out by both methods. The

surface area of the catalyst can also be determined from integration of all the pore volumes filled up as vapor is adsorbed and condensed, w hich provides an independent measurement of the s urface area from that determined by the usual B. E.T. method .

A simple application of these same principles was developed by Benesi, Bonnar, and Lee

[27].

They use a mixture of a nonvolatile and

a volatile liquid (e.g. , carbon tetrachloride and cetane

(n-C1 6H34))

at

room temperature to o btain the desired vapor pressure . A dried catalyst sample can be placed in a desiccator having the solution in the bottom and the gain in weight combined with the known density of carbon

tetrachloride gives the cumulative pore volume up to a critical diameter corresponding t o the relative pressure in the desiccator. By varying the ratio of carbon tetrachloride to cetane in

the liquid the partial pressure

of carbon tetrachloride can be varied , thus providing information on pore-size distribution. Mercury Porosimetry. Mercury does not wet most surfaces, s o an

external pressure is req uired to force it into a capillary . The relation­

ship i s

where

u

r

=

(- 2u c o s 0)/P,

( 1 . 25)

is the s urface tension of mercury, and (} is the contact angle

with a surface. Ritter and Drake

[270]

found that the contact angle

between mercury and a wide variety of materials such as charcoals and

28

DIFFUSION

metal oxides varied only between 1 35 ° and 1 42° and suggested that an average value of 1 40° could be used in general. Equation 1 .25 thus reduces to

r=

15 000/P,

( 1 .26)

where r is pore r adi u s (A) and P is in atmospheres. The increase in amount of mercury forced into ·the porous material up on an i ncre a se in pressure represents the filling of pores of a size given by Equation 1 .26. The smallest pore sizes that can be detected depend upon the pressure to which mercury can be subjected in a particular apparatus. Pore diameters qown to about 200 A can be determined with available commercial apparatus, and high-pressure porosimeters have been built to measure pores down to 30 A d i ameter Pores larger than about 7 . 5 J.l (75 000 A) will be filled at a tmos pheric pressure. Other investiga­ tors and laboratories h ave used somewhat different values of contact angle and surface tension, resulting in values of (r P) varying from 75 000 down to as low as 60 000, which can result in as much as a 25 per cent variation in repor ted pore size. The most commonly used value at present is about 62 000. .

·

Void Fraction. The total pore volume of a catalyst can be determined simply by measuring the increase in weight when the pores are filled with a liquid of known density. The liquid should preferably be of low molecular weight so that fine pores are filled ; water, hydrocarbons, or chlorinated hydrocarbons may be used satisfactorily. A s i mp le pro­ cedure is to boil a sample of dry pellets of known weight in distilled water for about 30 min, replace the hot water with cool water, transfer to a damp cloth, roll to remove excess water and reweigh. This will determine total pore volume between about 10 and 1 500 A. T h e accuracy, however, is limited by the difficulty of drying the external surface with­ out removing liquid from large pores , and some liquid tends to be held around the points of contact between particles. More accurate results are obtained by the so-called mercury-helium method. A container of known volume, V cm3 , is filled with a known weight of pellets or powder, W g. After evacuation helium is admitted, and from the gas laws is calculated the sum of the volume of the space between the pellets V' plus the void volume inside the pe l le t s V, The true den s ity of the solid is then .

p, =

V

-

w

(V'

+

V,) .

(1 .27)

1 .5

SOLID CATAL YSTS

29

The helium is then pumped out and the bulb filled with mercury at atmospheric pressure. Since the mercury does not penetrate the pores, its volume is that of the space between the pellets V'. The porosity or void fraction (} (cm3/cm3) is given by l

_

e

=

V - (V' + Vg) . V - V'

(1.28)

The density of t he pellets p, Wf( V - V') (g/cm3). The void fraction determined from the helium volume i s sometimes slightly greater than that determined by absorption of a liquid , since the smaller volume of the helium molecule permits it to penetrate into fine pores inaccessible to larger molecules. More details concerning methods for physical characterization of porous substances are given in the book by Gregg and Sing [ 1 24]. The proceedings of a symposium on structure and properties of porous materials a l so contain much useful information [ 1 02]. =

1 . 5 .4

Bimodal Pore-Size Distribution

Many catalysts or porous substances show a bimoda l pore-size dis­ tribution, sometimes termed a bidisperse, or macro-micro, distribution. This is the case, for example, of catalyst pellets prepared in the labora­ tory by compacting fine porous powders. One then deals with a fine pore structure within each of the particles of the o rig i na l powder, plus a coarser pore structure formed by the passageways around the com­ pacted particles . As compacting pressure i s increased the micropores remain unaffected unless or until the crushing strength of the solid is exceeded, but the macropores become successively reduced in size. As an examp l e , boehmite pellets pressed in the laboratory by Otani, Wakao, and Smith (2421 showed a micropore size distribution with a maximum at about 20 A radius , unaffected by degree of applied pres­ sure, and macr op o re size distribution maxima varying from 5000-500 A radius with increased pressure. A bimodal pore-size distribution is also found in most commercial forms of alumina. That used by Rothfeld [278] showed two pore-size-distribution peaks, one at 1 .25 Jl ( 1 2 500 A) an d one at 1 20 A (diameters). About 65 per cent of the total pore volume and 99 per cent of the surface was in the mi cr opores . A nickel oxide­ on-alumina catalyst prepared by Rao and Smi th (259] showed pore-size­ distribution peaks at about 30 and 1000 A radius. The fine particles themselves may also show a bimodal-type distribution. Thus, an exam­ ination by electron microscopy of a sample of commercial alumina by Bowen, Bowrey, and Malin [42] seemed to show the presence of three

30

DIFFUSION

kinds of pores : ( I ) l a rge irregular spaces between di scre t e particles, these particles being of the order of 2000 A d i a me ter, (2) smaller i rre g u l ar holes inside the particles, probably representing residual space between agglom­

erates which came together to form the particle, and (3) well-ordered rnic r opore s appr o x i mate l y cy l i nd ric al in cross section having a mean d i amete r of 2 7 ± 1 A. In this case, about 1 8 m2 /g of the t o t al area of about 275 m 2 /g was as s o ci ate d with the large macropores. M a ny com­ me rci al porous carbons also have a bi mo d al p o re-s i ze distri bu ti o n , a s d o catal y sts p r ep a red by i nc orp orati ng molecular sieves i n a gel matrix or pressing them into pellets. The p ore -size distribution into clearly defined macr opo re s and mi c r o pores is usually more ma r ked in laboratory pressed po w der s than in most c o m m e rci a l c a t al ys t s . Where a clear-cut division does not exist, the micropore region is frequently define d , s o me­

wh at a r bi tra ril y, a s c o mprisi n g pores under about 200-250 A di a me te r . Some industrial c at aly s t s are deli berately ma de to have a macro-micro­ pore system in order to mi nimize or eliminate diffusion limitations. An example is a recently pa t en te d catalyst for ammonia synthesis in w h i ch the catalytic ma ter i a l , a promoted iron oxide, is crushed i n t o a fine p ow d er and t he n formed into pellets. The catalyst must be reduced before use and the existence of a network of macropores a l l ow s t he w at e r vapor forme d by r e d uc t i o n to diffuse out of·catalyst pellets more rapidly. Studies by Nielsen and c o - w or ker s [Reference 227, pp. 97- 1 00 and p. 1 62] show that loss of surface area is t he r e by minimized and a ca t al y s t of higher i n t ri ns ic activity is obtained. An ammonia synthesis catalyst structure of this type is of more significance if the catalyst is to be reduced at relatively low pressures where diffusion is i n the Knudsen or t r an si t i on range where effective d i ffu s i t i vi t y is a fu n c t i o n of pore s ize ( S e c t i o n 1 .7) . 1 .5.5

Measurement

of Diffusion in Porous Solids

The most common method of m eas u ri ng counterdiffusion rates is by a s t e ad y - s t a t e technique app ar e n t ly first used by Wicke and Kal l e n ­ bach (387] and modified by Weisz [372, 377, 3 7 8 ] , and by Smith an d co-workers [ 1 40]. Two p u re gases are allowed to flow past opposite faces of one or more c y lind r ica l pellets affixed in pa r a l l e l i n a ti g h t­ fitting mount s u ch a s p l a s ti c tubing or a gum rubber disk. The fl uxes through the pellet are calculated fro m a kn o wled ge of the gas flow rates and c o mp o s i ti on leaving each side of the diffusion cell containing the pellets. No significant p re s s u r e gra d i e n t i s a l low ed to develop across the pellet. Typical gas pai rs are hydrogen-nitrogen, or helium-nitrogen. Si nce this method measures diffusion through the pe l le t it ignores d ead -

1.5

SOLID CA TAL YSTS

31

end pores, which may contribute to reaction and are included in a measured pore-size distribution. It may be used over a wide range of pressures but cannot be readily adapted to temperatures much exceeding ambient. It is primarily applicable to well-formed cylindrical pellets, although Weisz has used it for spheres by applying a geometrical correction factor. Steady-state flow under a pressure gradient gives results difficult to interpret unless all of the pores are sufficiently small that only Knudsen flow occurs. In the time-lag method of Barrer, one side of the pellet is first evacuated and then the increase in this downstream pressure is observed with time, the upstream pressure being held constant . The change in pressure drop across the pellet during the experiment is held to an insignificantly small value. There is a time lag before a steady­ state flux develops, and effective diffusion coefficients can be calculated from either the unsteady-state or the steady-state data. U nder unsteady­ state conditions correct analysis must allow for accumulation or depletion of material by adsorption, if this occurs, even if surface dif­ fusion is insignificant (see Equations 1 .46 and 1 .47). Several unsteady-state techniques have been developed for the measurement of pore diffusion. In one group of methods the solid is first saturated with one fluid and then the loss in weight is followed as a function of time after a vacuum is suddenly applied, or a second carrier gas is suddenly allowed to flow over the solid and the concentration of the leaving gas is fol lowed with time [ 1 22]. Most of the interchange in a spherical pellet occurs in a time t r 2/ I O D.rr where r is the pellet radius. For typical catalyst pellets and gases this time is so short as to make it difficult to obtain accurate results (e.g. , for r = 0.3 em and D. r r 1 0 - 3 cm2/sec, t 9 sec) and the method is primarily applicable to liquids, or very finely porous solids having low diffusivities such as crystals of molecular sieves. Deisler and Wilhelm [94] devised a sophisti­ cated frequency response techniq ue which, however, requires a com­ plicated test method and involved procedures to interpret the data. A promising unsteady-state method is based on gas chromatography, interpreting the broadening of an input p ulse. Only preliminary and scattered resuits have been reported and a detailed assessment of its degree of validity has not yet been made, but it provides a potential method of studying a variety of shapes and of making measurements under experimental conditions such as high temperatures that are not amenable to the Wicke- Kallenbach method. Surface diffusion, adsorp­ tion phenomena, and unsteady-state phenomena may make it difficult to interpret results (see Section 1 .7. 3), and the proper method of =

=

=

32

DIFFUSION

analyzing data from pellets having a wide pore-size distribution is not yet clear. Nevertheless, if an accurate effective diffusion coefficient can be obtained by thi s metho d , which relates to fl uxes in and out of the particle,

it

may be more representative of the situation occurring i n

reaction than steady-state diffusion measurements through a potous body, whenever the catalysts are highly anisotropic. The effective dif­

fu sivity may also be determined from measurements under diffu sion­ limited conditions of the rate of burnoff of carbo n from a uniformly coked catalyst (see Section

5 . 3 .2) .

Dead-end pores, if they are present, will contri bute to the flux in

unsteady-state methods, but not in steady-state measurements . Davis and Scott [9 1 ] used both methods t o study

diffusivities in spheres of After eliminating a

activated alumina and a Norton catalyst support.

skin containing fine pores, good agreement was found between results for the two methods , which i ndicates that dead-end p ores were un­ important for these catalyst supports . An effective di ffusion coefficient may also be determined from reac­ tion rate measurements on catalyst pellets of two or more sizes by the methods of Chapters

3

and 4, provided that the mathematical form of

the intrinsic kinetics of the reaction is well established. For

simple

power-law expressions it is not necessary to know the rate constant.

In any of these methods it is i mportant that the data be analyzed by the proper flux equation. This point may be illustrated with reference to the Wicke-Kallenbach method. The process of counterdiffusion changes as the diameter of the pores becomes progressively smaller. When the channels

are large (e.g. above about 10

Jl at 1 atm) equimolar counter­

diffusion occurs and D1 2 , e r r can be calculated di rectly from Equati on 1 . 1 4 if the flux is known . As the pore size is progressively reduced , the

relative rates of the counterditfusing gases gradually change, ultimately becoming i nversely proporti onal to the square root of the molecular weight ratio of the two gases. The proportionality of Nt fN2 to

J"M2 /M1

occurs well before the point at which the pore radius approaches the mean free path. This has been demonstrated both theoretical ly and experimentally by Scott and co-workers

[3 1 1 , 3 1 2].

In principle, the

relationship between flux ratio and molecular weight ratio can be employed to calculate effective diffusivities from measurements on only one of the two gas fluxes. In practice, however, measurements of both fluxes are highly desirable. Deviations fro m the t heoretical rati o may indicate experi mental difficulties such as leaks or the existence of surface diffusion.

1 .6

1.6

BULK DIFFUSION IN PORO US CATAL YSTS

33

Bulk Diffusion in Porous Catalysts

Pore diffusion may occur by one or more of three mechanisms : ordinary diffusion, Knudsen diffusion, and s urface diffusion. If the pores are large and the gas relatively dense (or if the pores are filled

with liquid) , the process i s that of bulk , or o rd i n ary diffusion, which has ,

been d i sc u ssed in preced i ng sections. If the pores are randomly oriented

the mean free cro s s section of the porous mass i s the same in any plane

and is identic a l with the volume fracti on voi d s 9. If the pores were

an array of cylinders para ll e l to the diffusion path, the d i ffusio n flux

per unit total cross sectio n of t he porous solid would be the fraction

() of the

fl ux under similar co ndit i o ns with no s olid present. However,

the length of the tortuous diffusion path in real pores is greater tha n the distance along a s trai ght line in t he mean direction of d i ffu sion More­ .

over , the channels through which diffusion occurs are of irregular shape and of v aryi ng cross section ; constrictions offer resistances that are not offset by the enlargements. Both of these factors cause the flux to be less than would be possible in a uniform pore of the same length and mean radius. We may thus express a bulk diffusion coe fficie n t per unit cross section of porous mass , D1 2 , e r r , as

D 1 2 0 - D 1 2 (} D 1 2 , e ff L'S'

-

where L' is a length or angle factor and

(1.29)

T

S'

a shap e factor, both be i ng

greater than unity, to allow for the two effects. There have been many attempts to develop theoretical expressions re l ati n g L' or

S'

to some

easily measu rable property such as porosity or particle size in a compact (see summary by M asam u n e and Smith

[200])

but none is of general

applicability, although L' is always pred i cte d to increase with decreasing

void fraction .

Theoretical models of p ores with constrictions to estimate the value of

S'

have been proposed by Petersen [248), Currie [83), and Michaels

[2 1 5 ]. Petersen represented the pores as a series of hyperbolas of revo­

l ution with constrictions at the v er ti ces

of the

hype r bo l a s Currie repre­ .

sented the p ores as tubes of sinusoidal form. Michaels represented the pore as a repetition of two cylind rical cap i lla r ies of different d ia m ete r s and lengths joined in series. In each case the rate of d i ffusio n is compared to that for a pore of uniform diameter and the same surface-to-volume ratio as that of the model. Figure graphi ng

1 /S'

1 . 3 compares the three models by

aga i ns t the ratio of maximum to minimum cros s s ection

.

34

DIFFUSION

1 .0 0.!) 0.8 0.7 0.0 � 0.5 �0.4 0.3 0.2 0. 1 0

Figure 1 . 3 . diffu sivity.

1

4

lll a x i m u m eross section M i n i m u m (·wss-sct·tiun

Theoretical effect of variation in p o re cross secti o n on effect ive

In Michaels' model the ratio of lengths of the two capillary c yli nders in series RL. is a fur t he r adj ustable parameter (or alternately, the ratio of their wid ths) and Fi gure 1 . 3 shows two extreme values . The val ue of S' is seen to de pend primarily o n the rati o of ma x i m um to minimuin pore area and is no t greatly affected by the details of the m o d el S' can reach values of 3 to 4 with moderate degrees of necking. For most rea l porous systems it is impossible to separate L' and S', .

1 .6

35

BULK DIFFUSION IN PORO US CATAL YSTS

are mu lt ipli e d together here a n d repre s e nt e d b y -r , a single fact o r al l o w for both va r y ing direction of diffusion and va rying pore cross section. For acc u r a t e w o rk r must be determined experimentally. It will be te r me d here the " tortuosity factor." In s o me literature L' i tself is te rmed a tortuosity factor. Petersen [25 1 ] terms 1 /S' a c o ns t ricti o n factor and his group ( r/a) is the same as r as u sed here . O ther literature refers to a l a byri nth factor," which is t he reciprocal of r in pre se n t nomenclature. Wheeler [382, 3 83 ] p r o p o sed a m o d e l in which the pores were visualized as cyl i n de r s of one fixed diameter which intersect a n y s o they to

"

= J2. D1 2, err re l a t e s to t he unit of face area of pores at any intersecting plane, which is less than t he flux per unit cross se c ti o n normal to the direction of dif-

plane at an av er age angle of 45°. Hence L'

flux per

l/J2

fusion in the cyl i ndri c a l pore, so another factor of is i nt r o d u ced . W = Consequently, in heeler's mod el -r 2. Weisz a nd Schwartz [378] ,

pro p o sed a model in w hich r Ji For diffusion through a randomly o ri ent ed system of long cylindrical pores the t o rt uos ity factor is 3 =

[1 6 1 ,

388].

Fi gure 1 .4 s ho ws the data of Currie [ 8 3 ] for diffusion in unconsol­ id a ted beds o f v ar i o u s powde r s a nd gran u la r materials, illustrating th e variation of D1 2 , e r r with porosity. For any given p o i n t on the fi g u re r is the ratio of abscissa to ordinate and it is seen t o vary up to a value of 2. H o o gsch a ge n [ 1 5 5 ] reported v a lu e s of -r of 1 .4 to 1 . 6 for loose po wd ers and beds of glass spheres having bed po ro sitie s of 0 3 5 to 0.43 . C at aly s t s are us u al l y consolidated p o r o u s media and somewhat larger values of r would seem to be required t h a n t h o s e for p acked beds . Figure 1 . 5 and Table 1 . 6 gather together d a t a reported on bulk diffusion in c a fa ly s t p e l le t s under a v ar i e t y of c ond i t io n s . Amberg and Echigoya [5] studied catalyst pellets prepared by pressing to various den s i t i e s po wd e rs of various sizes of a si l ve r alloy c ont ai ning 8 . 5 per cent calci um and the same material after " activation " in which most of the calcium w a s removed by ste am and acid treatment. (A catal yst of this co mp o s it i o n is reportedly used for the c o mme rc ia l oxidation of ethylene to ethylene o x i d e. ) The B.E.T. ( n i tr o ge n) surface areas varied from 2 0 . 22 to 1 . 50 m /g; a n d on the basis of fo rced fl o w experiments diffusion was believed to be al l in the bulk range. For the c a t a ly st prepared from acti vated p o w d er typical val ues of r w e r e 7 . 5 at 0 of abo ut 0. 6, 10 at (} of 0. 3 , and oo at (} of 0 . 1 . With po w d e r comprising the silver alloy be fore activation, stated to be more brittle, r w as smaller for a given void fr ac ti o n (e.g. , r = 6 at (} o f 0. 3) and d i ffu s i o n was not c o mp l e t e l y s t op p e d ( r --+ oo ) until the void fraction was reduced to 0 04 The void ,

.

-

.

.

DIFFUSION

36

1 .0 .------.--l---,l---.,--r--I---.I-"""TI-�I-r---11 f-

0. 8 f0.7

�

0.6

r-

0.5

r-

0.4

r-

o

lled of glass spheres

•

Carborundum

D

X

Sand

' Sod ium chloride 0 Soil crumbs

1 Pum i ce

ED Talc

fl Kaolin t Celite

X

Steel wool

'f/ Vermiculite V M ica

0.2

I-

v

-

'":---'..._ _., D e _ _ __.1-:----:� 1 _:'-:-J __.l-:----:"- l =--:":I 0 . 1 '-:--------:! C)

� Q

� � �

...... "'

40

DIFFUSION

mac ro ­ is used for the correlation in Figure 1 .5. Unre al i s t i c ­ ally high val u e s of -r (e.g. , 20 or hi gher) have been reported when i nvestigators have a s s u m e d diffusion to be com p l e t e ly in the bulk r ang e whereas in fact a considerable p ortion occ u r r ed in the transition region (see S e c t io n 1 . 7 ) . The data p o i nt s of Ma [29 7 ] and Pe lo s s o f [298 ] repr ese n t studies of the hyd rog e na t i o n of IX-methylstyrene t o cumene in t he l i q u i d phase on cat aly s t pe ll e t s of different sizes, from which an effective d i ffu si vi t y and t o r t u o s i t y can be calculated by the methods o utlined in Ch a p te r 3 . Pel ossof's cat a l y s t was palladium i mp regn ate d o n 0 . 8 25-cm-diameter alumi n a spheres .· Ma's catal y s t, t-in. by t-in. al u mi na pel l e t s , was made by a commercial catalyst manufacturer, the p o w d e r be i n g i m­ pr e gna te d with pal lad iu m befo re the pel lets were formed. Cadle [292] repo r te d studies on counterdiffusion of helium and n i t roge n at p r e ssu re s u p to 65 atm t h ro u gh five commercial p e ll e t ed catalysts. F r om the p o re ­ size distribution and the effect of pressure on the hel ium flux, it ap pe a r s that at 65 atm d iffus i o n w as e s s e ntial l y by the bul k mode. Commercial ca t al y s t s are ge n e ral l y calcined at high temperature for mechanical s tr e ngth , so i t is not surprising that their t o r tu o si ty factors are somewhat hi g he r than those for pressed powders, which in their mo re open forms more c l o se l y resemble beds of u nc o ns ol i d a t e d materi als. The va ri at i on between prereduced methanol s yn thesis ca tal ys t s from three manufac­ turers (all zinc promoted with chromium oxide ) is n ot e w or t hy, although the tortuosity factors fo r t he s eve n co mme r c ial c a t a l y s t s in Figure 1 . 5 all fall be tw ee n about 3 and 7 . 5 . Fi g u re 1 . 5 s ho w s that there is n o u n iq u e relationship b e t ween -r a nd 0 for porous cat aly s ts . Values of t o f 3 to 7 are re a d i ly reconciled with p hy s i cal reali ty, however, by multiplying together physically reasonable valu es of L' and S' . It is pe r haps remarkable t hat the range of va l ue s is this s mall when one c o n s i de r s t hat the t ortuosity factor i ncl ude s all t he deviations between t he real porous structure and the idealized m o d e l , i nc l udi n g dead-end p ores , skin effects, and various u nkn o w n degree s of an i sot r o py and inhomogeneity. In the above analysis , t is t ake n to b e characteristic of the porous structure, but not of the na t u re of t he diffusing molecules. When the size of t he di ffusing molecule b egin s to appr oac h that of the passageway, the w al l wo uld be exp ected to exert a r e ta rdi n g influence on the dif­ fu s i o n flux. As an example, Max t e d and Elkins [207 ) studied the h yd ro ­ ge na ti o n of c yc l ohe xe ne and of e thyl crotonate in alcoholic solution at 20°C and 1 atm on a p l at i n um c a ta l y s t s upp or te d on o n e of s ev e ra l tha t

o nl y

the macropo res contributed to d i fTusion and o n l y the

pore void fraction

DIFFUSION IN FINE PORES

1 .7

oxides . On rate of

an alumi na su pp o r t

with

average p o re

hydrogenation of ethyl cr o t o n at e

have b ee n expected fr o m

41

radius of 21 A t he lower than w o u ld

was much

results on supports

o f higher pore radi us.

(See also Section 1 .7.4, " Diffusi on in Zeolites . ")

1 .7 I .7. I

Diffusion in Fine Pores Knudsen Diffusion

If the

is low, or if the p ores are q uite small, or both, the collide with the pore wall much more frequently than w i t h each o ther. This i s known as " K n u d sen flow " o r " Knudsen diffusion." The molecules hitting the wall are momentarily adsorbed and then g i ve n off in random directio ns (diffusively reflected). The gas flux is reduced b y the wall " resistance " which causes a delay because of both the dif­ fuse reflection and the finite time the molecule is adsorbed. Knudsen diffusion is n o t observed in liquids. Kinetic theory provides the follow­ ing relations for Knudsen d i ffu si o n in ga s e s in a straight round pore : gas density

m o l e c ul e s

N

=

=

DK =

D K (ci x0 2r.

3RT

_

c2)

(BRT) nM

=

I ll

RT (P DK

I - P 2) x0

=

2r. ii (P I - p2) x0

3RT

(P I - P2) ,

9700r. JT{M.

( 1 . 30)

x0

( 1 . 3 1)

In Equation 1 . 3 1 r . is the pore radius in em, T the temperature in oK, and M the molecular w e i ght . The symbols refer to a s i ng l e component ;

since

mo l e c u lar

collisions are negligible, flow

and diffusi o n are synon­

ymous, and each component of a mixture behaves as though it alone

were present .

The internal geometries of consolidated porous solids are but poorly understood, and an empirical factor m u s t be i ntroduced to make the theory useful. The straight round pore of radius r. has a volume-to­ surface ratio of tr• . The p o r o u s material has a total surface S(J (cm2/g), and an average bulk or pellet den s i ty p11 (g/cm3). We may l o g ic a ll y define the mean p o r e radius as r. =

With t h i s

substi tution ,

solid becomes

2 ft;,

-

s(J

2()

= -- .

the K n u d s en

sg p,

diffusion coefficient for

( 1 . 32) a

porous

42

DIFFUSION

DK

•

eff

=

DK O

--

't"m

802 =

3 T:'" SII pP

J

2RT --

nM

=

19 400

02

---

't"m Sg pp

J

T

-. M

( 1 . 33)

As in E q u a ti o n 1 . 29, the void fraction 0 has been introduced so that the flux N given by D K , . r r will be based on the total cross section of porous solid, not just the pore cross section. As with bulk diffusion, the factor 't"m allows for both the tortuou� path and the effect of the varying cross section of individual pores. The subscript m reminds us that 't" m is the value of the tortuosity factor obtained when DK is calculated from a mean pore radi us, Equation 1 . 32. If a range of pore sizes exists, the proper average pore radius to use in E q u at io n 1 . 3 1 is given by Equation 1 . 50 (see Section 1 .8) rather than Equation 1 . 32, provided that the fl ux is completely in the Knudsen range.

I . 7.2 The Transition Region Bulk diffusion occurs when the collisions of molecules with the pore wall are u ni mp o r t ant compared to molecular collisions in the free space of the pore. Knudsen diffusion occurs when this condition is reversed. In a given pore there is a range of molecular concentra tions in which both types of collisions are i mportant. This is the " transition " region. As pressure is red uced the change from bulk to Knudsen diffusion, however, does not occur suddenly when the mean free path of the gas m ole cu le s becomes e q u a l to the pore radius. Until fairly recently only empirical o r c omp le x expressions of limited usefulness were available for use in the transition region between ordinary and Knudsen diffusion. Almost simultaneously two independent groups of workers [ 1 0 1 , 3 1 2] derived the same type of relation. For binary gas diffusion in a poro us solid at c ons t an t total pressure, this takes the form

N1

=

- (PfRT)(d Yi fdx) 1 - ( 1 + N 2/N 1) Y1 1 + DK l , eff D 1 2 , eff --

-------

=

- D .rr P d Y1 RT dx

=

-D

rr

dc 1 • dx '

( 1 . 34)

where D. r r is defined as the ratio of the flux N1 to the concentration gradient, whatever the transport mechanism. The coefficient D1 2 , e r r is the effective ordinary diffusion coefficient, as given by Equation 1 .29, and D K 1 , . r r i s the Knudsen coefficient for constituent 1 as given by Equation 1 . 3 3 . N2 / N1 is positive for cocurrent diffusion and negative for countercurrent diffusion.

1 .7

DIFFUSION IN FINE PORES

43

It is important to be clear on how a flux or a diffusion coefficient is defined , since in some cases the plane of no net molar flux is fixed with respect to fixed coordinates, but in other cases it moves. In studying diffusion in porous substances we generally measure N, not J, if the two are different. Furthermore, for most engineering purposes, including diffusion in porous catalysts, we are concerned with fluxes relative to the catalyst, i . e. , N. D. c r is defined by Equation 1 . 34 to give the flux relative to the experimental apparatus, not relative to the plane of no net molar flux. With nonequimolar counterdi ffusion in the transition region or in the bulk-type mode, N ¥ J and D.rr becomes a function of N1/N2 • The value of N1/N2 in turn depends on the physical situation. For bulk diffusion conditions, D1 2 , . r r � DK l , e ff • and Equation 1 .34 reduces to D 1 2 , er r

De rr _

1

(1

-

+

( 1 . 3 5)

NzfNt) Yt

Whether Knudsen or ordinary diffusion predominates depends on the ratio D1 2/ DK and rwt solely on the pore size or pressure. Note that D1 2 varies inversely with pressure, and does not depend on pore size ; DK is proportional to pore diameter and independent of pressure. If D1 2/ D K i s large, Equation 1 . 34 red uces to ( 1 . 36) ln the other limit, with

small and

D1 2/ DK

Nt

=

-

.

D 1 2 , crr P P2

N2

=

0, the relation becomes

dct

( 1 . 37)

dx '

where p2 is the partial pressure of constituent 2. For self-diffusion or equimolal counterdiffusion of gases, N1 - N2 at constant P, and Equation 1 . 34 reduces to =

N1

=

-

=

--

( l / D 1 2 , err)

I

+

dc1

-

(1 /DK t , err) dx

( 1 . 38)

and l

-

D .rc

1

Dx, err

1

+ -- .

D u , err

( 1 . 3 9)

This result was obtained earlier by Pollard and Present [255]. The form of Equation 1 .39 emphasizes that resistance to the motion of molecules

44

DIFFUSION

of type 1

is cau sed by collisions

collisions with

the

with

wall or both.

other gaseo us m ol ec ul es or by

For steady-state binary d i ffus i o n at c on sta nt P, integration tion 1 . 34 from ( Y1 ) 1 to ( Y1 )2 over the length x0 yields Nt

=

+

1

x

=

D u , err P + N2/N t)

R Tx0(1

-

n

[1 - (1 1 - (1

D .rr P ( Y2 R Tx0 --

+

+ -

N2/Nt ) ( Y1)2 N 2/N1)( Yd t

+

+

of Equa­

err)]

(D 1 2 ' err/DK t ' (D 1 2, errfDK t , err)

Y1 ) .

( 1 .40)

This reduces to E qu at io n 1 . 1 3 when o rd i nar y diffusion predominates (Du , eff � DKl , e rr) and to Equ ati o n 1 . 30 for Knudsen diffusion (with DK 1 , err replacing DK)· Equation 1 . 40 is confirmed by the data of Scott and Dullien [3 1 2] on binary diffusion in several porous solids over a 50-fold r an ge in total pressure. The second equality gives a means of defining the effective diffusion coefficient i n the transition zone when N2/N1 =F - 1 . D err

-

_

(1

x

The above Fick's first

+

In

D u , err N 2/N t)C Yt

Y.z)

[1 - ( 1 + N2/N 1 )( Yth 1 -

(1

+

-

N2 /Nt )( Yt ) t

+

+

D u , errfDK t , err . D u , err/DK t , err

J

(1 .4l)

exp re s s i on appears complicated because we are applying law to a region in which the diffusion coefficient is not really

constant. lf bulk diffusion predominates [i.e. , r e d uc e s to

(D1 2/ DK) --+ 0] , Equation L4r ( 1 .42)

Example 1.3

Estimation of D. rr for Gas Diffusion in a Porous Catalyst

Estimate D.rr for the diffusion of thiophene

in hydrogen

at 660°K and

30 atm in a catalyst having a B. E.T. surface of 1 80 m2/g, a void volume of 40 per cent, a pellet density of 1 . 40 g/cm3, and exhibiting a narrow pore­

size distribution.

1 .7

DIFFUSION IN FINE POR�S

For this system, Du was found to be 0.052

illustrated by Example

Du.

I.I.

cm 2/sec

4S

by the procedure

Substituting in Equation 1 .29, we obtain

D1 2 8

err = -- = T

0.052

X

'T

0.40

0.0208

= -'T

cm2/sec.

Substituting in Equation 1 . 33 , we have

D" · e rr = Taking

r.,

=

r,. X

1 9 400 x 0.42

1 800 000 X 1 .40

J

2, Du. err = 0.0 1 04 and D " ' ·

660

0.00344

84 = ---:;:::- cm 2 /sec. err

= 0.00 1 72 cm2 /sec. Knudsen

diffusion may be expected to predominate, since D u , than D " ' · e rr· As a good approximation, Equation 1

-

Deer

=

1

--

DK,

err

1

.rr

is so much larger

1 . 39 may be used to give

Den = 0.00 1 5 cm2/sec.

+ -- = 673 ' Du, err

The effect of t o tal pressure on diffusion of gases in pores obviously d epend s on t he relative importance of Kn u d sen and ordinary, or bulk, diffusion. The manner in which the diffusion flux varies with total press ure is illustrated in Figure 1 .6 . The curve shown is calculated for G' 0 ..... X .,

s

Io ���"�nr--,-.-rr��� 5

(,)

u

!."'

� '0 s

- i9 K

I'

.g

1 .0 0.5

d

.9 U1

16 A

0.1

L---.1---L.--L.-L...l....L.I...LJ._ .. --l_..L....J_,l-LJL.L.L..L...

0.1

0.5

1

Pressu re

5

(atm )

10

___.I_--l.-J..__.I_.L.L.L.l.J 50

1 00

Figure 1 . 6. Steady-state counterdiffusion of ethylene and hydrogen in the transition region calculated by the a ppro i ma te Equation 1 . 3 8 for a porous plug 1 .0 em thick, with pure ethylene at P on one side and pure hydrogen at P on the other. S, = 1 0 m 2/g ; T '=-' 298 ° K ; -r = 1 .0 ; (J = 0 .4 ; PP = 1 .4 g/cm 3 ; D1 2 P = 0. 602 cm2/sec ; r. = 570 A.

x

46

DIFFUSION

the case of a porous disk exposed to pure e t hyl e ne and pure hydrogen at the two faces, using the approximate Equation 1 . 3 8 . K n udsen diffusion predominates at low p r e s s u re , and the flux i ncre as e s with pressure because the concentration d iffe rence across the disk increases (DK is i ndependent of pressure). At h i gh pressure, the flux rate approaches the constant value for o rd i nary diffusion ; this is c o n s t a nt since the concentration difference i s di r ec tly , and D1 2 inversely, proportional to pressure. Figure 1 . 7 i l l u s t rates the effect of pore size a t fixed pressure on the

" &l d

., k d

., ...

0 ::0.

..... 0 .. = --':'._ 0 s

�

� ;:l '+=

:::: 0 ' iil

10- 5

10- 6

If A }0-7

���LllL---L-�JJ��--L-�LLLllLL--�

L-�

10

J 03

Pore radius ( A )

Figure 1 .7 . Counterdiffusing flux, N, of hydrogen a n d n itrogen as function of pore size. Total pre ss ure, 1 atm, T = 298°K. Saraf [290].

1 05

a

diffusion flux N for the binary system hydrogen-nitrogen . The ra t i o of the two species is i n v er sel y proport io nal to the square root of th e ratio of their molecular weights not onl y in the Knudsen and transition regime, but a l s o i nt o th e b u l k reg i me when the pores are of t he order of a micron in size. As the pores become of the order of 1 0 J1 or larger, the counterdiffusion molar fluxes a pp r o ach equality [3 1 1 , 3 1 2]. The range of p o re sizes for which diffusion w i l l occur in the transition region depends on t he r e l ati ve values of D K and D1 2 and moves in the the fluxes of

1 .7 .

DIFFUSION IN FINE PORES

47

direction of s m a ll er pores with incr eased press u re. To il l u s tr at e the effect of composition, Table l . 7 gives the transition

regio n limits for

Table 1 . 7 Transition Re g ion• for Selected Bi nary Gas Pairs ; 300°C, 1 atm

Limiting Pore Ra d ius (A) Lower Upper

245

H z-CO HrC6H6

H z-C , s H J o (pentadecene)

C , s H J o-Ct o H n (pentadecane) air-C,oHa (naphthalene)

1 37

19

8 76

26 300 1 8 1 00 3 1 80 535 6 350

• Diffusion flux 10 per cent or more below that predicted for Knudsen or bulk diffusion, respectively. Cadle [53].

some binary gas p ai rs at 300°C a nd 1 atm, arbitrarily defining the transi­ is 1 0 per cent or more below that predicted by the Knudsen or bulk diffusion equations,

tion region as that in which t he diffusion flux respectively [53].

The effect of temperature

is a ls o different for the two d i ffusion

mechanisms . As noted earlier, D1 2 for gases and liquids inc reas e s moderately with temperature, correspo nding activation energies bei ng

range 2 t o 5 kcal/g-mol. The Knudsen diffusion coefficient, how­ as the square root of the temperature. T he corresponding values of the ac tivati o n energy are 0.3 kcal at 20°C and 0.77 kcal at 500°C (when expressed in g-cal , the values are approximately eq ual t o the absolute t emper a t ure)

- in the

ever, i ncrea s e s

.

1 .7 . 3

Surface Diffusion

Molecules adsorbed on solid surfaces may e v i de nce considerable

mobility. Tr:ansport by movement of m o l ecu les over a surface is known as

"

s urface diffusion " and the direction is t hat of d ec re as ing s urface

concentration. Equi l ibrium ad s o rpt ion is

a func ti o n of the partial

pressure of the adsorbed constituent i n the gas adjacent to the surface,

and both tend to dec r eas e in the direction of diffusion. Thus, su rface

and gas-phase diffusion proceed in parallel. Surface diffusion has been

reviewed recently by Barrer [ 1 8], by D ace y and Weller [ 1 05].

[84], and by Field, Watts , Evidently, it canno t be s i gni fic an t unless appreciable

48

DIFFUSION

ad s o rp t i on occurs, yet

if ads orbe d molecules are held so s t r o ng ly as s u r face d i ffusion will be i n si g ni fi ca nt . I n s t e ad y - s tat e counterdiffusion t hr o u gh a finely poro us pe l l e t , the flux ratio N2/N1 s h o u l d be p r o p o r ti o n al to J MdM • If measuremenls 2 are made with helium and a seco nd gas as a counterdifTusing p air ; any excess flux of t he second gas o ver that calcul ated by this re la ti ons hi p of flux t o mol e c u l a r weight is attributed to surface diffusion. S u r face diffusion of hel i um has been believed to be i ns i g n i fican t at ambient temperature a nd pressure [ 1 8 , 105, 360], s i nce it is ad so r b ed so slightly, b u t a small amount of s u rface diffusion on p o r o u s Vycor has been r ep o rted by H w a n g and Kammermeyer [ 1 65]. An al ternate proced ure, appli ca bl e only in the completely Knudsen re g i o n , is t o measure the forced-flow flux of he li u m and of another gas singly t h r o u gh a particular porous solid. Agai n , any excess flux o ve r the theoretical is a tt rib u ted to surface d i ffu s i o n . There i s considerable con­ troversy o ve r how best to de s cri be surface mass transport, but e spe c i al ly for a m ou nts of adsorption corresponding to a fraction of a mo nolaye r , which occurs w he n the partial pressure of t he gas is substantially below the vapor press u re (e.g. , 0.05 or less) a Fi ck' s law t ype relati onship is usually ap p l i e d . The proporti01wli ty c o n s t an t Ds va r i e s somewh at with s ur face concentration, generally being greater at h i g h e r values of t he s urface concentration, but approaching s o me l i miting value as surface covera ge is d e c re ased . The s urface d i ffu s i o n flux per u n i t cross section area of porous catalyst J, may t hu s be expressed by to be essential ly i mmobile,

J. =

-D

where D. = D0

and

r,

accounts

for

.

� Pv S9

de.

(1 .43)

dx ,

e - E./R T

( 1 .44)

the tortu o us p at h of surface d i ffu s i o n ,

face concentration (molfcm2 ) , D, is

in cm2

c.

is the sur­

pellet c r o ss section per second , and the product p P S9 Sv is the surface area per u ni t volume 2 3 (cm /cm of pellet v o l ume) . There is no re a so n why r. s ho u ld e q u a l r a s used previ ously, although this as s u mp ti o n is s o met i me s made in t he absence of a ny other information . Surface diffusion will most l ike l y be s i gn ifica nt with high a r e a and , therefore, fine - p ore pe l l et s . If i t is assumed that t he bulk gas diffusion is all b y the K nu d s en mode, t he rat io of surface to K n ud s e n flux is then of

=

J.

JK

= rK D., Op P S9 T., D K

(de,) . dc9

( l .4S)

1 .7

DIFFUSION IN FINE PORES

49

At low surface coverages the adsorption is closely approximated by Henry's law and dc,fdc11 K8 , the absorption equilibrium constant (cm3 of gasfcm 2 of surface). Under unsteady-state conditions, in which diffusion occurs simul­ taneously with adsorption , the following relation hold s for flat-plate geometry : =

( 1 . 4 6)

=

where D •. e r r D.f'r: • . This assumes that adsorption equilibrium is established at all times and all positions. If adsorption is unimportant Equation 1 .46 reduces to

ik11

( DK) a 2 c9

at = t;

ax 2 '

( 1. 4 7)

where DK is based on the void portion of the pellet cross section. Equation 1 .46 has imp o rta nt implications for interpretation of diffusion measurements obtained by unsteady-state methods such as gas chro­ matography. Transient methods give the quantity in brackets in Equati on 1 .46 which may be either greater or less than DKfr:K . The term K11 D., e r r p P S11 reflects the contribution of surface diffusion to the flux and the term K8 pP S11 the effect on the flux caused by accumul a tion (or depletion) of adsorbed material. Even when surface mobility is unimportant, accumulation of ma t e r i a l by a d s o rp tio n may be of major significance, but this is sometimes overlooked in analysis of data. The term 0 will always be less than unity but K11 pP S9 can greatly exceed unity with high-area catalysts (e.g. 100 m2/g or higher) and adsorption corresponding to a substantial fraction of a monolayer (e. g. , 0. 1 or more). Values of D. at low surface coverages are typically in the range of w - 3 to w - s cm2/sec at ambient temperature for physically adso r be d molecules such as hydrogen, nitrogen, krypton, carbon dioxide, methane, ethane, propane, and butane on such surfaces as porous glass, carbon, silica gel, and typical commercial catalysts supported on porous oxides, such as alumina ( 1 8 , 1 1 9, 293, 308 , 323]. The activation energy E. in most cases is approximately one-half of the energy of p hy s ica l adsorption, but in a few cases closely approaches it in value [308]. J. sho uld then always decrease with increasing temperatu re since K11 and thus c. will decrease more rapidly than D. will increase.

50

DIFFUSION

[323 ] has r ecent l y de v el o p e d the gen era l correlation of s u rfac e s ho wn in Fi g u re 1 . 8, which brings together data on a wide va ri ety of systems , both of p hys i cal adsorption an d chemi­ sorption. Since s urface mobility s ho uld be re l ated to strength of adsorption, D. i s plotted ag ai ns t the parameter q/mRT, where q is \he heat of adsorption and m is an in t e ger having a value of 1 , 2, or 3 d e p en ding upon the na t ure of the su rface bond and of the s o li d , as shown in Table 1 . 8 . Sladek

diffusion coefficients

-2

·..

-4

..

I I I

� ... D"' I 6 "'

�\, l o

"•

:.;;, I

"Ca.rbola.c" I c&rbon : e He l - 6 A No 1

+ Hz

T SOz • Kr

·�A '•

'·

: :

1 Vycor · glaRS · SOz � + N2 I !J COt A I I-Pt l u NHa - Kr I CHt : 0 i-C.HJU • 0 -w -10 > C2Ho ! 0 C2H4 : T H-'� 1 • H-N1 Silica·alumina J + · Cs-W catalyst 1 :silica. 1 x Ba-W b. N2 1 powder I v A r-W - 1 2 [1 CHt 0 C2H 6 I X CF2Cho 0 Oz-0/ W : 0 COz-C02 fW v Ca Hs I · S 0 2 ::

-8

Vycor glaSS: Oz : 17

0

I

10

20

30

q mRT

•

.. •

40

60

50

Figure 1.8 . Correlation of surface diffusion coefficients. Sladek

[323].

For hydrogen and oxygen on metals, ads orpt i o n with dissociation was assu me d ; q was taken to be equal to one-half of the measured molecular heat of adsorption plus o ne- h alf of t he dissociation 'e ne rg y of t he diatomic gas. The assignment of values of m was made to achieve the best correlation of the data, and represents the e ase with which an adsorbed molecule can migrate r e l ative to the strength of b o nd i ng ; a larger value of m c or resp on d s to a larger value of D. , for a fixed value of q/RT.

Certain symbols are used more than once in Fi gure 1 . 8 ; three sets of

1 .7 V al u es of

Table 1 . 8

m

Van der Waals Polar Adsorbate N o n po lar Ad s o rb a te

Covalent

51

for Use in Figure 1 . 8

Bond

Ionic

DIFFUSION IN FINE PORES

Solid

Conductor

I ns u la t o r Cond uct o r Insulator Conductor Insulator Conductor Insulator

m

2 1

1 1

2 1 3 I

Examples of Available Surface Diffusion Data

S 0 2 -Carbon SOl , NH3-Giass A r-W, N 2-Carbon K r, C2H.vGlass Cs, Ba-W none H - Metals, 0-W none

legends are shown, each applying to a range of val ues of qfmRT. The data po i nt s shown for physically ad sorbe d s peci es are a representative selec tion from the much greater quantity of data used in the original correlation. The cor re l at io n is r em ar k a b le in that it rep resents eleven orders of magn i tude of D, to within about one and a half orders of m ag ni tu d e and over a qfmRT range of 60. Values of q from 0.3 to 200 and te mper atures of - 230 to + 600°C are represented as well as data obt a i ned by a va r i e t y of systems and expe ri men ta l techni ques. Kam­ mermeyer and Rutz [ 1 66) have published a useful correlation shown in Fi g u re 1 . 9, which also provides a simple visualization of the contribution of surface diffusion to the to tal flux for a number of g ase s and vapors . The product of the effective diffusivity at 25°C and the square root of ,

the molecular weight , D1 , e r r J M1 , is p l o tted as a fu nct i on of either the bo i li ng point or the cri tical temperature of the diffusing species. Taking the co ntribution of s u r face diffusion to the flux as b ei ng neg­ ligible for helium, the d i ffe re nce between the curve and the flat dashed line represents the contribution of the s u r face flux for other species. These data are for diffusion through porous V yco r of 200 m2/g adjusted in s ome cases where different samples were used and were obtained at u pstream pressures bel o w 3 atm, d ownstream pressure of 1 atm, under which co n d i tio ns the gas phase flux was c o m p l e te ly in the Knudsen r eg i o n On higher area substances surface diffusion will make a l arge r contri b u tion. For example, on " Carbolac " carbon plugs with an average p o re radius of 1 9 A , the surface diffusion flux of nitrogen, argon, o r krypton at am bi e nt conditions was fro m a third to the equal of the Knu d se n flux [ 1 8]. Schneider and Smith [308] used a chromatographic meth o d to measure surfa ce diffusion coefficients for ethane, propane, .

52

DIFFUSION

and n-butane on silica gel (832 m2/g ; re 1 1 A) over the t empe r at ure range of 50- 1 75°C and re v ie w ed all o the r available information on surface diffusivities of these hydrocarbons on v a rious substrates. The activation energies for the hydrocarbons were u s u a lly from about 3 to 5 kcal. A t 50°C surface d i ffu si on accounted for 64, 73, and 8., per cent of the transport for ethane, propane, and butane, respectively. At l 25°C, surface t r a nspo rt of n-butane represented 68 pe r cent of the total . =

so

Boiling point

70

curve

/.-- Critical

---./ f

I

60 -

� �

cs

temperature curve

50 40 30 20 10 0

•

T8

0

100

200

300

TB , boiling point or Tc , critical

400

temperature (°K)

500

550

Figure 1 . 9 . Correlation of the product (Dt • • rr)(M1 ) 1 1 2 with boiling point

and critical

temperature Tc

for

glass of 200 m 2/g. Measurements at Rutz [ 166].

a series of ga ses through p o r o us Vycor 25°C and 1 to 3 atm. Kammermeyer and

Studies by Cadle [293] of counterdiffusing he li u m and n itr o gen through five commercially pelleted c a talys t s (three methanol synthesis catalysts and t w o metal catalysts supported on oxides) at ambient temperatures showed that the surface diffusion of nitrogen made an increasing contribution to the total nitrogen flux with increase in p re s sure level . At the highest pressu r e 65 atm, the surface diffusion flux was of comparable magnitude to the volume d iffu s i o n flux. The adsorption equilibri u m constant was independent of pressure but the ga s phas e mode of diffusion varied from predominantly Knudsen at at mo s p he ri c pressure to predominantly bulk at 65 atm. Eq uat i on 1 .45 shows that the ratio o f surface diffusion flux to gas phase diffusion flux should indeed i ncrease w i t h pressure level , s ince the effective gas,

-

-

1 .7

DIFFUSION IN FINE PORES

53

phase diffusion coefficient wil l be gin t o dr op w i t h i nc re asin g pressure level. Wi th the wide po re-s i ze distribution t ypical o f most c o mmerc i al c a t alyst s , ri go r ous analysis of the pre ssu re effect is complex since the rel ati ve contributions to the total flux from pores of different sizes varies wit h press u re leve l . With chemisorbed species, Ds i s relatively small and the surface app ro ac hes saturation a t pressures that may be several orders of ma g­ nitude less than occurs fo r p hysical ad sorption of the same ga seous spec i es at the s ame temper at u re. Con se quently even on the highest surfac e area porous metal compacts available, s urface diffusion is a significant p o rti o n of the total flux only at very low pressures (e.g. , 40 JL Hg for hyd rogen on nickel, below about 1 mm Hg for hyd r og e n on platinum), where low s urface c o ve rage perm its a steep sl ope of the a d s orp ti o n isotherm (dcJdc11) in E qu ati o n 1 .45. The above discussion suggests that surface diffusion will make no

significant c o ntr ib u ti o n to

intraparticle flux in porous cata l y s ts at highl y is l i t t l e direct evidence one way o r the other. When the par t i al pressure of t he reacting sp ec i es is an ap pr ec iabl e fraction of the vapor pressure, surface diffusion may be of considerable i m po r t a nce , as has been suggested to exp l ai n studies - of the catalytic dehyd rat i o n of ethanol at 400-700°F a nd pressures up to 1 00 psi on a silica-alumina b ea d ca t alys t [ 1 77, 2 1 7]. S ince gaseous diffusivities in porous substances are usually measured at ambient or o nly s light ly elevated temperatures, results of such s tu die s , parti cu l a rl y on high - are a ma te rial s , must be a n a l yzed most c arefu l l y to be s u re that t he p o ssibl e contribution from surface flux has been isolated and taken

e l evate d te mp erature s , al th o ug h there

into account.

Somewhat more conclusive ev i de nce for sig n i fic a nt surface diffusion chemical reaction is presented by Bienert and Gelbin [29] who s t ud i ed the de hydra t io n of i s o pro pan ol at 200-250°C o n y-alumina in t he form of either powder or pellets compressed from the powder to vari o us densities. By co mpari s o n of the rates on the two d i ffere n t size s , effec tiveness fact ors o f the vari o us pellets were found in the range of 0.088-0.7 1 . Back-calculation (see Chapt er 3) gave values of the effective d i ffu sivity vary i ng from 0.0 1 5-0.06 c m 2 / sec, which for a gi ve n pellet reach e d a maximum value at a te m pe r a ture intermediate between the ex tremes of 200-250°C s tudied. A study by S terre tt and Brown [329] of the o-p h ydr ogen conversion a t 32 psia and 76°K gave values of the effective di ffusi v i ty back-calculated from reacti on rate data s ub s tan tially hi g he r than would be exp ect ed, indi cating that surface diffusion may h a ve b ee n s i gni fican t at this low reaction temperature, a conclusion in a

54

DIFFUSION

th at s eems pl a u s ible since the catalyst, a ferric o xi de ge l , contained pores that were all below about 30 A radius. Analysis of surface diffusion effects generally assumes that the thick­ ness of the adsorbed layer of ga s is not an appreciable fraction of the p o re cross section, i.e., that the adsorbed l ayer does not affect the e ffe c­ tive pore size for Knudsen diffusion. Evidently, this m u s t become invalid for s ufficien tly s mal l pores, and cases in wh ich this effect seems to be significant have been cited [9 1 , 1 06 , 292, 360]. As the size o f a m o l ec u l e becomes an appreciabl e fraction of the pore diameter, the m olec ul e comes u n d e r the influence of t h e wall d uri n g most of its movement an d representation of an y of t h e flux as Kn u d se n ­ type diffusion · must gradually bec o me invalid. Little is k n o wn about conditions under whi c h this occurs or the nature of d i ffu s io n under these c o nd i tio n s. ( S ee Section 1 .7.4 below and Secti on 3 . 5 . )

1 .7.4 Diffusion and Reaction in Zeolites ("Molecular Sieves ")

T he p hys i ca l structure of zeolites is su mm ari ze d in S e c t i o n 1 . 5 .2. Sorption and diffusion characteristics have bee n rece n t ly reviewed by Walker and co-workers [357]. M os t diffusion studies have been o n the s m al l -p o re zeolites (4-5 A diameter p o re s) in which, for nonpolar gases comprising small molecules such as nitrogen, methane, propane and n - b u ta ne, diffusion coefficients are in the range of 1 0 - 1 2- 1 0 - 1 4 cm 2 / s ec a t temperatures o f 20-200°C and the a p p ar e n t activation energies are i n the range of 3 to 1 1 kc al/ mol e. These d i ffu siviti e s are very small c o mpared t o typic al values of 1 0 - 3 to w - 4 cm 2 /sec found for the same gases in such high-area catalysts as conventional silica­ alumina. In s o d i u m mordenite (6 x 7 A diameter pores) d i ffu s io n coefficients at 25 ° C varied from about 33 x w - t o cm2/sec fo r methane t o 1 2 x 1 0 - 1 0 cm2/sec for n-C4H1 0 [295]. Fr o m cracking studies at 300-540°C w i th n-hexane on H - o ffre t i te (pores 3 . 7 by 4. 1 A) wh i c h were apparently highly diffusion limited, Mi a le, Chen, and Weisz [2 1 4] concl ud ed that the effective d i ffus i vi ty for this system was less than w - 9 cm 2 fsec. Little information is av ai l a ble on d i ffu s i on rates in the zeolites having l arge r pore sizes. From counterdiffusion s t u di e s of b enze ne and cumene at 2 5 ° C on catalyst Li nd e SK-500, a rare-earth exch a nge d form of Y ze ol i t e (about 10 A m i ni m u m diameter in the Na form), Katzer [1 68] c oncl u d ed that diffusion c o effici ent s were not less t han 1 0 - 1 1 cm 2 /sec, but they may h a ve been m uch l ar ger . On NaY, diffusion coefficients for c u me ne diffusing out into benzene we re 9 X w - 1 4 and 1 . 5 X 1 0 - 1 2 at 25° and 65°, respectively. On H -mord e n � te eq ui l i b ri u m could not be o b t ai n e d , but d i ffu s iv i ti e s of cumene u n de r

1 .7

DIFFUSION IN FINE PORES

55

desorption conditions were about 10- 1 3 (65°C} to 10- 14 ( 2 5 ° C) cm2 /sec. Evidently as molecular size and bulkiness of the reactant molecule is increased, a point will be reached at which diffusion of the reacting molecules into the pores of a particular zeolite will represent a limi tation on the observed rate of reaction, but little is known about conditions under which this occurs. Weisz and co-workers [373 ] compared the reaction rates of various compounds in binary mixtures of the straight chain and branched isomer forms, on the small-pore 4A and SA zeolites and also on l OX. As examples of their results, on SA zeolites at soooc (5 A diameter pores), cracking of 3-methyl pentane was essen­ tially prevented by diffusion limitation, whereas that of the n-hexene present in the mixture was no t , but no di ffusion limi tations were exhibited by either compound on the l OX sieve (about 8 A pore diam­ eter). 1-butanol was selectively dehydrated on the SA sieve but not the 2-methyl- 1 -propanol present. Catalysts with noble metals incor­ porated inside the cavities were shown to hydrogenate selectively n­ olefins in the presence of iso-olefins or to oxidize n-paraffins or n-olefins in the presence of the branched isomers. Further applications of shape­ selective catalysts have been described by Chen and Weisz [67 ]. On 5A sieve trans-butene-2 is hydrogenated from three to seven times faster than cis-butene-2, in mixtures of the two ; the diffusion of the trans­ isomer into the zeolite was also shown to be substantially greater than that of the cis isomer. On a sodium mordenite zeolite into which platinum had been incorporated , ethylene was selectively hydrogenated to ethane in the presence of propylene, at temperatures of 1 75-260°C, e.g. , in a 1 : 1 mole ratio in the presence of hydrogen at 260°C. A 28 per cent conversion of ethylene was obtained with less than 0. 1 per cent con­ version of propylene. Shape-selective catalysts may also be used as an internal method of generating or removing heat in a reactor. In a mixture of 2 per cent CO and 0 . 5 per cent n-butane in air at 427-540° C a platinum-loaded zeolite A was shown to cause 86 to 1 00 per cent reaction of the carbon monoxide and only 2.4 to 5 . 5 per cent reaction of the butane. Diffusion limitations can also develop by the buildup in the cavities of reaction products of higher molecular weight than that of reactants, as demonstrated for alkylation reactions by Venuto and Hamilton (347]. Norton [228] showed that for p r op y l ene polymerizati on, activity decreased in the order l OX > 1 3X > SA > 4A > 3A, that of 3A being zero. The data were interpreted in terms of differing intrinsic catalyst activities, but the results may have been markedly affected by the relative rates of diffusion of polymer product o ut of the pores.

56

DIFFUSION

Bryant and Kranich [49] studied the dehydration of ethanol and n-butanol over types X, A, and synthetic mordenite after the zeolites had been base-exchanged with various cations. The reaction rate and ratio of olefin to ether in the product were markedly affected by the type of molecular sieve and by the size and charge of the exchanged cation. Much more remains to be learned about diffusion in zeolites. The flux may not follow Fick's law closely and diffusion coefficients can be markedly affected by slight blockage of passageways. Furthermore, although pore structure is usually regarded as perfect and regular for purposes of analysis, zeolites are unquestionably polycrystalline and present a heter ogeneous set of passageways. In some cases the reaction may occur in part or exclusively on sites outside the fine intracrystalline pores. 1 .8

Estimation of Diffusion Coefficients in Porous Solids

Table 1 .9 and Figure 1 . 10 present the results of thirteen studies of diffusion or flow which illustrate some of the factors involved in pre­ dicting the effective diffusion coefficient. Table 1 . 10 presents additional data not shown in the figure. Since Knudsen flow predominated in all of the tests represented , flow and diffusion should be synonymous. The bulk-diffusion term has been ignored and Derr x 1 04 plotted as ordinate with the group [fJ2(T/M)1' 2 fS9 pp] x 1 08 as abscissa. The lines shown have been drawn with slopes of unity ; as read from the figure, an intercept with the ordinate (at a bsci ssa of unity) corresponds to 1 9 400 X 10 - 4/t, . Porous Vycor glass has been investigated the most frequently by far, at least nine studies being reported. For six, the results agree closely, the extreme values of t, being only 4. 7 and 6.6 with an average value of 5 .9. Three other studies report values df 3.2, 3 . 3 [ 1 4 1 , 360) and 1 0. 5 [3 10]. For the preparation of porous Vyco r a borosilicate glass is formed into the ultimately desired shape and is then annealed at a suitable temperature which allo ws the formation of two phases, but below that at which physical deformation occurs. One phase is rich in silica and the other rich in sodium oxide and borate. After cooling, the object is leached with dilute hydrochloric acid to leave behind a porous material co mpr i si ng 96 or more per cent silica. Both phases must be continuous, but the structure of the two phases, and hence that of the resulting pore structure after leaching, may vary substantially with the cooling procedure [275). It has also been suggested that the thickness

1.8

DIFFUSION COEFFICIENTS IN POR O US SOLIDS

/

Equation 1 . 33 �!might round tubes

(r = l )

/

/

/

/

/

/

/

"" " /

/

/

/

/

/

/

� �/ /

/

/

/

/

/

/ / /

, // // // // // • '/ ./ / / / /

57

1 /'

.-

/ / // // / � /" "/ � , /

/ / / /

/

1 �--�--U-����----�-L-L��LL�---L--L-�-LLLU 1 \()() 1 000

_8_2 {_'1' X SgPp '-J M __

J08

=

1�11 /1'_ X 1 0� 2 '-J M

Figure 1 . 10. Data on diffusion o f gases i n consolidated porous media compared with equation for K n u d sen diffusion in straight round pores. For key, see Table 1 .9. of the initial piece of Vycor may have a significant effect o n its diffusion characteristics since this may affect the cooling rate [ 1 4 1 ]. It is also evident that the leaching procedure may produce a somewhat different porou s structure for thin than for thick pieces of glass, since removal of material from the interior of a shape by solution becomes m ore d ifficult in thicker geometries. Three of the studies in Table 1 .9 were on silica-alumina commercial

bead cracking catalysts. In two of these the effectiveness factor of whole beads was determined from reaction studies by crushing the catalyst and

comparing the rate of reaction on whole beads with that on one or more sizes of the crushed cataly st. By the methods outlined in Chapter 3 ,

VI 00

Table

1 .9

Diffusion

and Flow in

Material

• Alumina plug com-

Finely Porous

Technique

Diffusion

Media

(Experimenta1

Gases N z , He, COz

data

reK) 303

shown in

r.

(A) 96

()

0.81 2

• r,.

0.85

pressed from p ow der

0

Figure 1 . 1 0) Reference Henry, Chennake-

sawan, and [140] Villet and Wilhelm [348 ] Smith

/:::, Fresh

and regenerated silica-alumina commercia! bead cracking catalyst X Vyco r glass

Flow

Hz , Nz

298

3 1 -50

0.464

2. 1

Flow

H , , He, Ar, N 2

298

30.6

0.3 1

5.9b

EB Water-gas shift

Diffusion

O z , N2

298

1 77

0.52

2.7

Diffusion

Oz , Nz

298

203

0.52

3.8

Bokhoven and van

He, Ne, Hz , Ar, N , , O , , Kr, CH4 ,

292,

0.298

5.9b

Barrer and Barrie

catalyst

D Ammonia catalyst

synthesis

0 Vycor glass

+ Vycor

glass

Flow

Flow

Gilliland, Baddour, and Russell [1 1 9] Bokhoven and van Raayen [39] Raayen [39]

30

[1 9 ]

294

CzH6 H z , He,

Ar, Nz

298

30

0.28

5.9b

Gilliland, Baddour, and Engel [1 18]

� 'ti � 0

:;;:

A

Silica-alumina commercial bead cracking catalysts Catalysts prepared from bead bydrogels

e

Vycor glass

\7 Silica-alumina cracking catalyst, plug compressed from powder T Silica-alumina cracking catalyst, plug compressed from powder 0 Vycor glass

0

Silica-alumina commercial bead cracking catalyst

• T is

b

Reaction

Gas oil cracking

3

3-10

Johnson, Kreger, and Erickson

5.9b 0 . 725

Whang [38 1 ]

16

0. 305 0.40

� � c

2733 23

24

0.53

0 2 85

Barrer and Gabor

He, CO :: , Nz , 0 2 , Ar Cumene cracking

298

44

5.9b

Kammermeyer and Rutz [1 66]

Hz , Nz

293

Gas oil cracking

755

Flow

Ar, Nz

298 273-

Flow

He, Ne, Ar, Nz

Flow

He, Ne,

Reaction Diffusion

-

Oa

Reaction

Flow

28

Johnson, Kreger, and Erickson

Nz

based on mean pore radius, Equations 1 . 32 and 1 .33. Average of five sets of data. m

755

323

693

5 1 -7 1 46

(24) (24)

[1 59] [1 59]

0.3 1 (0.50) (0.50)

.

5.6 23 .

Barrer and Gabor [20]

[20]

Weisz and Prater

[377]

1::1 ..,

�

8 �

�

10 ()()()

9 20

26 23 21

45 9

46

14 14

11 20

15 6

46

4

33

41

15 25 14

42

23

11

36

10

24

50 18

18 2

54 45

22 44

6 1

11 0

56

7 12

45 61

45

29

7 10

56

37

7

0.25

0

18

49

31

2

0

69

29

2

0

0

19

68

13

0

0

Topsee• •

Satterfield and Cadle [293].

b Diffusivity of hydrogen through nitrogen, measured at r o om temperature and atmospheric c -r, calculated from Equat ion 1 .48 . 4 Tm calculated from Equation 1 .33.

pressure,

average of 5 sets of samples.

Diffusivity of helium through nitrogen, multiplied by v 4/2, average of 2 sets of samples. f Prediction by parallel-path pore model, Equatio n 1 .48. " The predicted flux distribution for the Haldor--Topsee catalyst in the original p u bl ication has a typographical error. •

-·---- -- - -

Percentage of the Total Predictedf flux at Ambient Conditions from Pores in the Ranges

20

10

- ... ---· -·

Oo

I;:,

�

�

�

interchangeably, although in this book they have quite different meanings. =

3.3

Some Characteristics of Diffusion-Limited Reactions

If ¢1. is substantially less than about l ( ¢1 L < !), diffusion effects are relatively insignificant relative to kinetic effects, for first-, second-, or zero-order reactions, (but see Section 4.2 and Example 4.3). In this range the reactant concentration ·throughout the porous mass is essen­ tially the same as that at the outside surface, and 11 is near unity. At values of '1 less than about 0. 1 , where 11 approaches I I¢ L , substitution of the definition of the Thiele inodulus into the reaction-rate expression yields -

1 dn

- - =

Vc d

t

k'v Cm'1 •

=

c<m + 0' 2 • L

--

moles

JD• rr kv' sec · em 3 pellet volume ----:;--::---=--

(for 11


. = l/1;'1

(integer-power kinetics). (3. 33)

The relationship between ci>L and lflL or ci>. and l/1. is more complex for more complicated kinetic expressions. Equation 3.33 can be shown by formulating the rate of disappearance of reactant and eliminating k� by use of the definition of the Thiele modulus (e.g. , Equation 3.5). 1 dn _ k' m c. Derr .�.. C8 '1 - Ji2 'l's2 'l• 0 - Vc dt -

(3. 34)

_

If the order of the reaction is known and is of an integer power, then when '1 < "' 0. 1 , '1 = lflflL = l fci>L (for spheres, '1 3/l/J. = 9/ci>.). For higher values of '7, '1 may be determined by successive approximations from Figure 3.2 or a graph of '1 versus l/1 2 '7 ci> may be prepared to give '1 directly. Such a graph for first-order reaction in a sphere is given by the curve marked [3 0 in Figures 3 .4-3.7. (These figures are discussed in more detail in Chapter 4.). D er r can be estimated by the methods of Chapter 1 . In cases where the resistance to mass transfer fr. m fluid =

=

=

3.4

DETERMINA TION OF EFFECTI VENESS FA CTOR

1 43

Figure 3.4. Effectiveness factor 7J as a function of , . Example 3.1

Estimation of Effectiveness Factor First-Order Kinetics

rt :

Isothermal,

Archibald, May, and Greensfelder [9] studied the rate of catalytic cracking of a West Texas gas oil at 550° and 630°C and 1 atm pressure by passing the vaporized feed through a packed bed containing a silica-alumina cracking catalyst of each of several sizes ranging from 8 to 14 mesh to 35 to 48 mesh. They report that at 630°C the apparent catalyst activity was directly propor­ tional to the external surface for the three larger catalyst sizes studied. This implies that the catalyst is operating at a relatively low effectiveness factor. We can check this by suitable calculations based on their run on 8 to 14 mesh catalyst at 630°C. The average particle radius may be taken as 0.088 em. They report 50 per cent conversion at a liquid hourly space velocity (LHSV) of 60 cm3 of liquid per cm3 of reactor volume per hour. The liquid

·:..;_

1 44

....

.

-·,

DIFFUSION AND REA CTION IN POR O US CA TAL YSTS. I

,

y

Figure 3 . 5 . Effectiveness factor "' as a function of cl>, (Equation 3 . 3 1 ) for 20. First-order reaction in sphere. Weisz and Hicks [376].

=

density is 0.869, and its average molecular weight is 255. The effective density of the packed bed was about 0 . 7 g catalyst/cm3 reactor volume. The molec­ ular weight of the products was about 70. Let us take pore-structure characteristics for a commercial silicia-alumina cracking catalyst to be those reported by Johnson, Kreger, and Erickson [ 1 59], viz. , average pore radius = 28 A, catalyst particle density p, 0.95, 8 = 0.46, S, 338 m2/g, T is about 3.0 (see Table 1 . 1 1 ) . From Equation =

=

1 . 33,

Derr

=

82 1 9 400 -8

T

o PP

-

J

T

M

903 ( 1 9 400)(0.46) 2 - 8 · 0 x 1 0 - 4 crp. 2 /sec_. -_ - (3)(3 38 x 1 0 4)(0 .95) 255 J

3.4

DETERMINA TION OF EFFECTIVENESS FACTOR

1 45

Figure 3.6. Effectiveness factor TJ as a function of

Love and Emmett [1 92], 7J independent of particle size 0.8

0.05 0.36 0.78 -+ 1 0.40 0.89 0.95-1 .0 -+ 1 0.20

0.2�.33

0.80

0 . 1 2-0.23 1 .0

0.89-0.98 0. 1 5-0.40

.

Calculated by Wheeler [383 ], also 7J independent of particle size Bokhoven and van Raayen [39], by variation of particle size. Also calc'd from Derr measured at 1 atm, 20°C, with Nz - Oz. See also Peters and Krabetz [247] Data of Larson and Tour [1 87], 7J calcnlated by Bokhoven and van Raayen [39] D :ta of Nielsen [227], 7J calculated by Bokhoven and van Raayen [39]

C0 2 + H 2

1 42°C 300°C - 1 96°C, 1 00-400 psig

0.4

S, = 1 55 m 2 /g p, = 1 .9 1

0.4

d = 200 A S, = 1 6 m2 /g DK = 1 .77 X 1 0 - 2 cm 2/sec

film 2.2 x I Q - 4 em thick

S, = 91 m2 /g, . V, = 0.345 cm 3 /g d = 1 50 A

0.2-0.5

o H 2 -+ PH 2

19. Porous nickel film deposit

Hydrogenation of ethylene

23°C

20. Ni on aluminum oxide

Ethylene hydrogenation

80-l 40°C

2 1 . ZnO-Cr203

C0 + 2H 2 -+ CH30H

22. Dowex-50, cation exchange resin

Hydrolysis of ethyl formate

Assumed D.rr = 280 atm, 3 30-4WOC 8Df'"./2, 8 = 0.68 0-30 % conv. of CO. Gas comp'n = 1 2 % co, 80 % H 2 , 8 % inert 25°C, dilute solution

23. Amberlite IR- 1 20, cation exchange resin

Inversion of sucrose

50°C, 20 % solution

VI VI

-+ 1 . 0

S, = 1 72 m 2 /g

1 7 . Supported Ni catalyst 1 8 . 5 wt % Ni on y-alumina pelleted from powder

- 0. 1 0.43-0.45

-+ 1 for T > 1 25°C or thicker films, 1) < 1 < 0. 5 Calculated 'I) = 0.07-0. 1 9

0.5

X

1 .6

0.52-0.95

O.Q7

0.62

0.024

0.044

Vlasenko, Rusov, and Yuzefovich [350] Wakao, Selwood, and Smith [354], by variation of particle size Beeck, Smith, and Wheeler [24, 383]

Rozovskii, Shchekin, and Pokrovskaya [284 ], by effect of particle size ; calculated values of 'I) assume T= 1 Pasquon and Dente [246]

Smith and Amundsen [325], 'I) estimated by present author. See Section 3 . 5 Bodamer and Kunin [38]. See Section 3.5

Table 3 . 1-Continued

Catalyst 24. Amberlite IR-I 20, ion exchange resin 25. Commercial Co-Mo on Ah03

Vt 0'1

-

26. Dowex 50W x 8 cation exchange resin 27. Dowex 50W X 8

28. 0.5 wt % Pd on alumina, commercial-type pellets 29. I % Pd on alumina spheres

Reaction

Reaction Conditions

Catalyst Characteristics

Pellet Size (diam in em)

20-fold variation in degree of crosslinking of polymer d = so A

0.045 and somewhat greater 0.5 X 0.5

50-70°C

Dcrr = 27 X 1 0 - 8 cm 2/sec at 50°C

0.004 0.077

Hydration of isobutylene to t-butanol

74-97°C

D.rr = 1 .7

Hydrogenation of et-methylstyrene to cumene {liquid phase) Hydrogenation of et-methylstyrene to cumene (liquid phase)

70-1 1 5°C, I atm H 2 pressure

-r =

--

50°C, I atm H 2 pressure

-r =

--

Inversion of sucrose

50°C, 20 % solution

Hydrodesulfurization

350-375°C 50 atm liquid-phase trickle bed

Inversion of sucrose

X 10- 5 to 5.5 X 1 0 - s cm 2 / sec (varies inversely with temperature)

48-65 mesh X

0.32

Dtm 8 = 7.5 D.rr

0. 825

0.25 to 0.0009 � 0.36

-+ I

1 1 0-1 1 5 mesh mesh

Dtm 8 = 3 .9 Dcrr

Effectiveness Factor 7J

0.32

0. 1 3 0.25 0.97 0.60

(70°C) (50°C) (74°C) (94°C)

0.88 0.37 0. 1 1 0.07

(74°C) (94°C) (85°C) (I I 5°C)

0.006

Author and Method Bodamer and Kunin [38] . See Section 3.5 van Deemter [343], by variation of particle size. Reaction rate also increased with reduction in liquid viscosity O'Connell [230], by variation of particle size Gupta and Douglas [128], by variation of particle size and analysis Satterfield, Ma and Sherwood [297], by variation of particle size and analysis Satterfield, Pelosso( and Sherwood [298], by variation of par�icle size

3.6

VALIDA TION OF THEOR Y. ISOTHERMAL REACTION

1 57

of molecular sieves, etc. (Sections 1 . 5.2 and 1 .7.4), and those affecting the local intrinsic activity. The latter may develop from nonuniform distribution of catalyst on a carrier (Section 1 . 5. 1), the nonuniform buildup of carbonaceous deposits or other catalyst poisons (Section 5. 1), or by incorporation of molecular sieves into the structure. These problems must be handled on an individual basis, but if sufficient information is available they can frequently be analyzed satisfactorily. The principal problem in predicting the effective diffusivity given a reasonably isotropic pore structure, stems from the fact that reaction is usually at pressures and temperatures substantially different from those at which measurements have been made. The discussion in Chapter 1 gives guidance on methods of prediction and the following summarizes the conclusions. For gas-phase diffusion by the bulk mode the effects of pressure and temperature have been reasonably well developed. A tortuosity factor found under one set of conditions should apply with little error to other conditions provided that diffusion remains in the bulk mode. If no diffusion measurements are available on the catalyst structure of interest, assumption of a tortuosity factor in the range of 2 to 6 in Equation 1 .29 is recommended for the usual catalysts. If gas­ phase diffusion is by the Knudsen mode under both reaction conditions and those under which the effective diffusivity by itself has been meas­ ured, extrapolation by Equation 1 .33 assuming a constant tortuosity factor seems to be satisfactory. In the absence of a diffusion measure­ ment, the assumption of a tortuosity factor in the range of about 2 to 6 is recommended, provided that the pore-size distribution is narrow. Catalysts with a wide pore-size distribution such that much of the diffusion occurs in the transition range provide more of a problem since the relative contributions to the flux from pores of different sizes vary with pressure, temperature, and nature of the diffusing species. The best available procedure at present seems to be to use the parallel­ path pore model (Section 1 .8 . 1 ) for extrapolation, again applying a tortuosity factor in the range of 2 to 6 in the absence of other informa­ tion. Sections 1 .7.3 and 1 .7.4 give guidance as to how the possibility of surface diffusibn or molecular sieve-type diffusion may be examined. Chapter 1 showed that for bulk diffusion in small pores or transition­ range diffusion the effective diffusivity of one gas is a function of the opposing flux of a second gas. Under constant pressure conditions such as those provided in the Wicke-Kallenbach diffusion measurement method, the ratio of the fluxes is inversely proportional to the square root of the molecular weights of the gases. It is important to recognize that under reaction conditions the flux of reaction products out of a

1 58

DIFFUSION A ND REA CTION IN POR O US CA TAL YSTS. I

porous catalyst opposing the inward flux of reactant is determined by stoichiometry and not by molecular weight. For diffusion in the liquid

phase, extrapolation to higher temperatures is generally limited by the degree of knowledge concerning bulk diffusion for the syste!Jl of con­ cern rather than by the effect of the porous structure as such. Relatively little information is available on the effect of temperature on diffusion in the liquid phase and the diffusivity may vary substantially with concentration. Sufficient studies now exist to verify the close correspondence of theory and experiment when the above complications are not severe. Brief discussion of some of these will help clarify the state of knowledge. Wakao and Smith [356] have developed a method of predicting effective­ ness factors using their random-pore model for predicting the effective diffusivity and Smith and his co-workers have tested the theory in a number of experimental studies. The theory involves a microeffective­ ness factor relating to possible diffusion limitations through the in­ dividual powder particles comprising a catalyst pellet as well as a macroeffectiveness factor relating to the pellet as a whole. In all the studies reported, the microeffectiveness factor was essentially unity, and, indeed, this would be expected to be generally the case except for powder particles containing pores only slightly larger than the diffusing molecules (Section 4.6.2). Predictions of the effective diffusivity by the random-pore model have agreed fairly closely with experiment for pellets compressed from alumina boehmite powder. Two studies of the o-p hydrogen conversion on pellets of 7 per cent NiO on alumina [259] and on 25 per cent NiO on alumina [26 1 ] compressed to various densities gave agreement within 10 to 1 5 per cent between experimental and predicted effectiveness factors, which were in the range of 0. 1 5 to 0.30. For the first case about one-half of the diffusion flux was predicted to come via the macropores, and in the second case about 90 per cent. A study of the same reaction on .2 per cent NiO on porous Vycor [260] showed agreement within 10 per cent between experimental and predicted effectiveness factors of about 0.4, using experimentally measured values of the effective diffusivity. At low values of the effec­ tiveness factor, '1 is not highly sensitive to the effective diffusivity, being proportional to D �/l . In a study by Sterrett and Brown [329] of the same reaction on a ferric oxide gel catalyst of various sizes, '1 varied from 0. 78 to higher values. Theory and experiment could be brought together by a tor­ tuosity factor -rm = 1 .4 as calculated from Equation 1 . 33 or :t'p of 1 .7 as calculated by the parallel-path pore model. These unusually low

3.6

VALIDA TION OF THEOR Y. ISOTHERMAL REA CTION

1 59

values of the tortuosity factor suggest that a substantial fraction of the mass transport was by surface diffusion, which seems plausible since all the pores of the catalyst were below about 30 A radius. The ortho­ para hydrogen conversion at about - 1 96° C and I atm is a good model reaction since heat effects are negligibly small , the kinetics are relatively simple, and there is no change in number of moles on reaction. When diffusion is completely by the bulk mode, as in liquid systems, prediction of the effective diffusivity may be easier. Ma studied the hydrogenation of cx-methylstyrene to cumene on pellets of Pdfalumina catalyst at 70° to I I 5°C [297]. The effectiveness factors, found by varying particle size, were from 0.07 to 0. 1 3 on the large pellets and could be brought into agreement with theory using measured values of hydrogen diffusion in cx-methylstyrene and cumene and the reasonable tortuosity factor of 3.9. Pelossof [298] studied the same reaction at 50°C on alumina spheres impregnated with palladium and on powdered catalyst. The effectiveness factor was 0.0057 on the spheres. Bringing theory and experiment together required a relatively high tortuosity factor of 7.5. In an early investigation Johnson, Kreger, and Erickson [ 1 59] studied the cracking of gas oil at 480°C and I atm on silica-alumina catalysts treated in various ways. On an 0.44-cm-diameter 90 per cent silica-10 per cent alumina commercial bead catalyst, the effectiveness factor was found to be 0.55 by comparing the rates on the bead catalyst and powdered beads for the first 40 per cent of reaction. Theory and experiment could be brought together by a tortuosity factor of about 3 . 5 assuming the reaction was first order and a Knudsen diffusivity cal­ culated by Equation 1 . 33. This is remarkably close agreement between theory and experiment considering the uncertainties involved in repre­ senting this complex reaction of a wide mixture of substances as a simple first-order process. Diffusion measurements alone in homogene­ ous silica-alumina catalysts give a tortuosity factor of about 2 to 2.5 (Tables 1 .9, 1 . 1 1). Higher tortuosity factors were found from the reac­ tion studies on the catalysts treated in various ways, but no information is available on how treatment affected the diffusivity as such. Weisz and Prater [377] reported an early study of cumene cracking at 420°C and I atm pressure on a commercial silica-alumina bead catalyst in various sizes, in which '1 varied from unity down to 0. 1 5 . The effective diffusivity o f hydrogen i n the catalyst beads a t 27°C was measured and extrapolated to reaction conditions using the Knudsen relationship of Equation 1 .3 1 . Theory and observation could be brought

1 60

DIFFUSION AND REA CTION IN POROUS CA TAL YSTS. I

together by a tortuosity factor of 5 . 6, although the diffusion of hydrogen alone required a tortuosity factor of2. 3 . The biggest source of uncertainty here is probably associated with the complex intrinsic kinetics of the reaction, which was assumed to be simple first order in the allJllysis. A study by Weisz of burnoff in air of carbon from a silica-alumina cracking catalyst also showed reasonably good agreement between theory and experiment. The analysis is given in Example 4.7. Certain other experimental studies have shown greater deviation between theory and experiment. Otani and Smith [240] studied the rate of oxidation of carbon monoxide at 275° to 370°C on a 10 per cent nickel oxide-on-alumina catalyst in the form of both powder and i-in. spherical pell e ts compressed from the powder. The experimental effectiveness factors, determined by comparison of the rates on pellets and powder, varied from 0.37 to 0.64. Back-calculation gave values of the effective diffusivity 4 to 5 times lower than those that would be predicted by the Wakao and Smith random-pore model. The reason for the discrepancy is not clear. Although the random-pore model gave reasonably good predictions of the effective diffusivity for other pellets pressed from alumina, large spherical pellets such as these may be highly anisotropic [55, 300] or some of the micropores may have been sealed by the pelleting process. Finally, the rate of reaction is greatly inhibited by the product carbon dioxide and rather complex analysis was required to allow for this effect. 3. 7

Mass Transfer Limitations to and within Catalyst Particle

In Chapter 2 it was shown that, if the processes of mass transfer and reaction all occur in series, each of these steps may be regarded as a resistance and the observed rate may be set equal to the over-all con­ centration difference divided by the sum of the resistances. Equation 2. 34, for example, shows the formulation for a slurry reactor in which two mass transfer steps and a reaction at the outside surface of catalyst particles each contribute a resistance. As shown in Chapter 2, by proper variation of experimental conditions it is possible to separate the contributions of different steps. When reaction occurs simultane­ ously with pore diffusion, however, the process cannot be regarded as a case of resistances in series. If both mass transfer to the outside of a porous pellet and pore diffusion are significant resistances, bulk mass transfer coefficients can be determined by the methods of Chapter 2, but the concentration at the outside surface of the catalyst pellet, which is a boundary condition affecting the bulk mass transfer rate, is affected

3.8

DIFFUSION EFFECTS ON REA CTION INTERMEDIA TES

161

by the internal effectiveness factor. Consider, for example, a spherical pellet of radius R and equate the rate of first-order catalytic reaction within the pellet to the rate of mass transfer to the outside pellet surface : 3 1 dn - - - = k c 11 = kc(c 0 - c ) - . Vc dt v ·• s R Eliminating

c.

(3.42)

and rearranging, we obtain -

1 dn c0 . Vc dt = (R/3kc ) + (1 /k. 'l)

(3.43)

Although it would be possible from rate data alone to determine the value of the product of k. and '7, it is not possible to determine either separately without additional measurements, e.g. , of the effective diffusivity. With high-area porous catalysts of the sizes and activities typically u sed in chemical and petroleum processing in packed beds, with gas­ phase reaction at typical industrial mass flow rates, and with calculated effectiveness factors approaching unity, mass transfer to the outside surface of the pellet will normally represent an insignificant resistance. If the calculated effectiveness factor is a small fraction of unity, how­ ever, it would be wise to determine the possible contributions of bulk mass transfer and thence the proper value of c. to use in determining '1 · Conversely, if bulk mass transfer is found to be significant, the effectiveness factor of a high-area porous catalyst would be expected to be very low. These comments apply to systems in which the same phase exists in the pores as in the bulk. If the phases are different or a large distribution coefficient exists between bulk fluid and catalyst, the internal effectiveness factor may approach unity while bulk mass transfer constitutes a significant resistance. This is the case in some reactions catalyzed by ion-exchange resin particles. In laboratory work, mass velocities are frequently low and can lead to significant mass transfer resistance even at relatively low catalyst activities. The above genera1izations are intended to be but an approximate guide to expected behavior ; analysis of the effects of bulk mass transfer and pore diffusion for any particular case should be made by the quantitative methods described. 3.8

Diffusion Effects on Reaction Intermediates

Some catalysts provide more than one kind of site, and reaction proceeds by consecutive steps on different kinds of sites. These are

1 62

DIFFUSION AND REA CTION IN PORO US CA TA_L YSTS. I

termed " bifunctional " or " multifunctional " catalysts and are exem­ plified by the reforming catalysts used for converting petroleum stocks into high octane gasoline. They consist of a highly dispersed metal , usually platinum, on an acidic oxide support, usually an acidified form of alumina. A large variety of reactions occur industrially, but a model reaction which illustrates the key concepts is the isomerization of an n-paraffin to an iso-paraffin. A paraffin molecule is dehydrogenated to an olefin on a metal site a, the olefin diffuses to an acidic site b where it isomerizes, and the isomerized olefin diffuses to a metal site a where it undergoes hydrogenation. a

b

a

paraffin � olefin � iso-olefin � iso-paraffin.

(3.44)

Both kinds of sites are necessary for any appreciable conversion of paraffin to iso-paraffin. Weisz and Swegler have shown [369, 380] that a mechanical mixture of particles of platinum supported on carbon and particles of an acidic oxide will provide essentially the same yield of iso-paraffin as a dual function catalyst comprising platinum supported on silica-alumina, which is acidic, if the particles in the mechanical mixture are sufficiently fine, but neither type of particle alone forms appreciable product. As particle size of the mechanical mixture is increased, the yield diminishes. From this they conclude that the inter­ mediates are desorbable and are transported from site to site through the gas phase. This further indicates that the rate of the over-all reaction should be fundamentally affected by the thermodynamics of olefin formation since this sets an upper limit on the concentrations of olefins which can be attained. Weisz [368, 369, 37 1 ] has developed generalized criteria for proximity requirements for poly-step processes involving desorbable intermediates in terms of the free energy of formation of the intermediate, particle size, effective diffusivity within the catalyst par­ ticles, and other variables, and Gunn and Thomas [ 1 27] and Thomas and Thomas [334] have presented mathematical analyses of how the distribution of catalyst sites both within catalyst pores and on outer surfaces of catalyst particles may affect the outcome of the reaction. However, some reactions such as dehydrocyclization may involve sur­ face diffusion rather than gas-phase diffusion and the quantitative application of these theoretical considerations requires a fairly detailed knowledge of the true kinetics. Thus, if an olefin intermediate is not desorbed, the catalyst activity cannot be directly related to the olefin concentration in the gas phase. A good review of bifunctional cat�lysis has been recently published by Sinfelt [322]. ·

3.8

DIFFUSION EFFECTS ON REA CTION INTERMEDIA TES

163

Similar poly-step processes occur in noncatalytic reactions. For example, the decomposition of barium carbonate can be greatly en­ hanced by intimately mixing carbon particles with the powdered carbonate. The mechanism is presumably BaC0 3 .=: BaO + C02 , C0 2 + C .=: 2 CO.

(3.45)

The rate of reaction of carbon with C02 to form CO can frequently be the rate-limiting step, since it is found that the decomposition rate is greatly affected by the reactivi!Y of the carbon used [294].

4 Diffusion and Reaction in Porous Catalysts II. Complex Cases 4. 1

Temperature Gradients

The analyses of the effectiveness factor in Chapter 3 assume the porous structure to be isothermal. Wheeler and others have pointed out that substantial temperature gradients can sometimes occur in practice, and there have been many publications describing mathe­ matical analyses of this effect. A relationship between temperature and concentration which is useful for orientation may be simply derived by considering a boundary surface surrounding some portion of a porous structure. Under steady-state conditions, the diffusion flux of reactants across this boundary surface equals the rate of reaction within the surface. The heat released (or consumed) by the reaction must all be transferred across the same boundary. Hence

D.rr dxd e .

f:l.H

dT

= A dx '

(4. 1)

or, in integrated form, f:l. T

=

T - T•

=

(

-fl.H)(A D.rr) (c - c) •

'

(4.2)

where T is the temperature at any point within the pa rticle, c (mol/cm3). is the concentration at the same point, f:l. H (cal/m ol) is t he enthalp-y 1 64

4. 1

TEMPERA TURE GRA DIENTS

1 65

change on reaction, and A. (calf sec · em · °K) is the thermal conductivity of the porous solid ; T. and c. are boundary values, usually at the outer surface of the particle. This relation was first pointed out by Damkohler [87] and shown by Prater [256] to be valid for all kinetics and to apply to any particle geometry. To obtain temperature (or concentration) as a function of spatial coordinates, the reaction-rate constant is expressed as an Arrhenius function, and the differential equations for simultaneous mass diffusion and heat diffusion are solved, using Equation 4.2 to eliminate one variable. Analytical solution of the differential equations is quite difficult. Beek [25] and Schilson and Amundson [302] used linear approximations to obtain analytical solutions. Tinkler and Pigford [336] applied perturbation techniques to obtain approximate analytical expressions, a procedure that is applicable when temperature gradients are small but not negligible. Petersen [249] presents an asymptotic solution applicable at high values of the Thiele modulus. Other in­ vestigators [60, 335, 376] used machine computation methods. Tinkler and Metzner [335] and Weisz and Hicks [376] give families of curves correlating the effectiveness factor or the maximum tempera­ ture rise with other parameters. Tinkler and Metzner treat first- and second-order irreversible reactions in a spherical pellet and treat first­ order reaction in a semi-infinite flat plate. Carberry [60] treats first- and second-order reactions in a single pore, which is equivalent mathe­ matically to the semi-infinite plate treatment. Gunn [1 26] presents analytical expressions for concentration and temperature distributions inside the catalyst particle for first-order irreversible reaction in spherical or flat plate geometry. Hlavacek and Marek [ 1 49, 1 50, 1 5 1 , 1 98] treat zero-order reactions in various geometries. Weisz and Hicks treat irreversible first-order reaction in spherical geometry in terms of the w. parameter as well as the Thiele modulus, and their development will be summarized here. The effectiveness factor in all of these cases is defined as the ratio of the actual rate to that which would occur if the pellet interior were all exposed to reactant at the same concentration and temperature as that existing at the outside surface of the pellet. Two new independent parameters must now be introduced over the isothermal case ; one is the exponent in the Arrhenius reaction-rate expression, E/RT, arid the other a heat generation function c.

( - AH)De c rfA.T• .

Weisz and Hicks symbolize these by y and p, respectively. Graphs of the effectiveness factor '1 versus the Thiele modulus cp thus involve

1 66

DIFFUSION AND REA CTION IN PORO US CA TAL YSTS, II

families of families of curves. The parameter p represents the maxi­ mum temperature difference that can exist in the particle relative to the particle surface temperature, (T - T.)max/T. , under steady­ state conditions. This occurs when the concentration drops to essen­ tially zero within the pellet (Equation 4.2, when c 0). (Un der transient conditions the maximu!ll temperature difference may exceed that occurring under steady-state conditions [365] . For an exothermic reaction, p is positive. Four '1 versus ¢. graphs, for.first-order irreversible reaction in spheres, each presenting a family of curves for 0.8 5 p 5 - 0 ..8, are given by Weisz and Hicks [376]. These are for values of y of 1 0, 20, 30, and.40. Figure 4. 1 reproduces the graph for y = 20, which represents a fairly typical set of operating variables, e.g. , an activation energy of 24 000 cal/g-mol and reaction temperature of 600°K. The curve for p 0 represents the isothermal case. Figure 4. 1 shows that for exothermic reactions (p > 0), the effectiveness factor '1 may exceed unity, since under some sets of circumstances the increase in rate caused by the temperature rise toward the center of the particle more than offsets the decrease in rate caused by the drop in concentration. Thus, the over-all rate of reaction in the particle is greater than it would be if the interior were at the same concentration and temperature as that at the exterior surface. At high values of the Thiele modulus, '1 becomes inversely proportional to ¢, as in the isothermal case. Most of the reaction then occurs in a thin shell at the pellet surface, the interior being at a higher temperature than the. surface, but essentially isothermal. The shape of the curves on Figure 4. 1 shows that, for highly exothermic reactions at a low value of the Thiele modulus, the value of '1 is not uniquely defined by p, y , and ¢ • . In effect, for one value of ¢. there are three different possible values of '1 corresponding to three different sets of conditions under which the rate of heat production equals the rate of heat removal. The middle one can be shown to be metastable and not realizable in practice. Which of the two remaining sets of conditions will actually be obtained physically depends upon the direction from which the steady-state condition is approached. The fact that either of two rates of heat release can occur is analogous to the ignition situation with an exothermic surface reaction which occurs in the combustion of carbon and which has been observed in the heterogeneous decom­ position of hydrogen peroxide on a nonporous active catalyst [ 1 1 0, 299, 3 5 1 ]. In these cases, for example, a momentary disturbance around the metastable point can cause �he surface temperature either to rise to a stable level corresponding to a._process limited by mass transfer through the laminar layer outside the solid particle or to drop to another stabie =

=

4. 1

TEMPERA TURE GRA DIENTS

a

,., _

Cs( - AH) X7 '•

1 67

Dcff

4>.

Figure 4. 1 . Effectiveness factor 7J as a function of q,, = R V k./ Derr for 20. First-order reaction in sphere. Weisz and Hicks [376].

y=

level at which the rate of the surface reaction controls. Figures 3 .4-- 3 . 7, discussed below, make it evident, however, that for porous catalyst pellets the region of multiple solutions corresponds to combinations

1 68

DIFFUSION AND REA CTION IN PORO US CA TAL YSTS. 11

of large values of p and y seldom encountered in practice. The condi­ tions under which a unique steady state will occur have been the subject of various mathematical analyses [6, 1 1 7, I 85, I 96, 2 I 3, 364). The abscissa t/1. in Figure 4. I contains the intrinsic rate constant ku , which is frequently unknown. A more useful method of correlation proceeds by transforming the abscissa to the function w. used previously in the isothermal cases. Figures 3.4--3. 7 are graphs of t7 versus w. for values of y = I 0, 20, 30, and 40 [376] . The dashed portions of the curves . between arrows indicate the regions in which a unique solution does not exist, corresponding to the region of multiple solutions in the t7 versus t/1. curves. One striking conclusion from these graphs concerns highly exothermic reactions on a catalyst of low thermal conductivity. It is possible to have effectiveness factors greatly exceeding unity at relatively low observed reaction rates or at low values of the Thiele modulus, conditions under which analysis assuming isothermal operation (p = 0) would indicate an effectiveness factor of essentially unity. The shape of the curves also indicates that small changes in operating conditions or characteristics of the catalyst in this region could cause major changes in the observed reaction rate. Whereas under isothermal conditions the observed activation energy can be as low as one-half the true value when t7 is sufficiently low (Section 3 . 3) the opposite effect can occur when t7 > I , i.e., the measured o r apparent activation energy can exceed the true value [60). For kinetic expressions other than first order, such as those of the Langmuir-Hinshelwood type (Equation 4.4), nonisothermal behavior can cause the observed activation energy to even fall below one-half the true value, as shown by Schneider and Mitschka [307). (See also Section 4.2.2.) When q, L exceeds about 2 for a first-order reaction, the reactant concentration at the catalyst center approaches zero and the simpler asymptotic solution becomes valid. The number of new independent parameters required for handling nonisothermal conditions can then be reduced from two to one by use of a new parameter equal to P · y. This new parameter is termed !X by Carberry and e by Tinkler and Metzner, who present a family of curves for various values of !X (or e) on t7 versus q, coordinates for first- and second-order reactions. These show that heat effects becomes less important as reaction order in­ creases, i.e. , the family of curves of t7 versus q, for various values of (P · y) are more closely compressed for a second-order reaction than for one of first order. Weekman [362) has analyzed the combined effects of temperature

·

4.1

TEMPERATURE GRA DIENTS

1 69

gradients and volume change on reaction when bulk diffusion occurs. For endothermic reactions volume change has little effect on the above analyses. For exothermic reactions, volume contraction can cause a substantial increase in catalyst effectiveness factor and on internal temperatures over that otherwise calculated. Volume expansion works in the opposite direction. For high values of p and y the variation of P with temperature may be significant and Weekman has presented some analysis of this effect. (See also Section 4.2.2.) For endothermic reactions a good approximation for the effectiveness factoJO can be obtained by considering only the temperature gradients through the pellet and neglecting concentration gradients, even when these gradients are appreciable. Some analyses by this approach and comparison to the more rigorous method are given by Maymo, Cunningham, and Smith [21 0]. 4. 1 . 1 Thermal Conductivity of Porous Catalysts Whether or not thermal effects can be significant for a particular reaction on a particular catalyst is primarily governed by the value of p. This requires knowledge of the thermal conductivity of porous solids, but relatively little information has been published on the kinds of porous structures used in catalysis. The principal studies appear to be those of Sehr [3 1 3], of Mischke and Smith [220], and of Masamune and Smith [203]. The latter two have been further analyzed by Butt [5 1 ]. Table 4. 1 shows thermal conductivities for several porous oxide catalysts as reported by Sehr [3 1 3] at a mean temperature of about 90°C in air at atmospheric pressure. Mischke and Smith [220] measured the thermal conductivity of a series of alumina catalyst pellets in the presence of helium or air at pressures from vacuum to 1 atm. The catalysts were prepared by pelletization of alumina powder under different pressures. Masamune and Smith [203] reported similar studies in the presence of air, carbon dioxide, or helium on a series of metal catalysts prepared by compacting microporous silver powder. Representative results from these two studies are presented in Table 4.2, plus data reported by other workers, usually incidental to other studies. Oma c r o is the fraction of the porous pellet occupied by pores above about 1 00 or 1 20 A in radius. Omic r o is the fraction occupied by smaller pores. (I - Omac r o - Omic r o) is the solid fraction present. The spread of values of thermal conductivity is remarkably small for the group of nonmetallic substances and does not vary greatly with major differences in void fraction and pore-size distribution, brought

1 70

DIFFUSION AND REACTION IN POR O US CATAL YSTS. II

Table 4. 1 at 1 atm,

90

Thermal Conductivities of Some Porous Catalysts in Air C [3 1 3 ]

°

Catalyst Ni-W Co-Mo dehydrog"enation catalysts Chromia-alumina reforming . catalyst Silica-alumina cracking catalyst Pt-alumina reforming catalyst Activated carbon

Apartlcle •

(cal/sec em

q

·

.

o

1.12 x 10 - 3

{0..8358 0

X 10 - 3 X 10-3

Apowdcr8

(cal/sec .

0.73

o

q

·

em

X 10- 3

0. 5 1 X 1 0 - 3

0.33 x

to - 3

Density (g/cm 3) .

Particle 1 .8 3

Powder

1 .48

1 .63 b

1 . 56b

1 . 54
. for y 40 shows that, under highly diffusion-limiting cases, "' may be as small as one-quarter of that for isothermal conditions ; i.e., the observed rate could be as low as one-quarter of that obtainable under the same conditions if the same catalyst had a high thermal conductivity. The temperature in . the center of a catalyst pellet could be as much as (0. 1 8)(450 + 273) = l 30°C below that at the outside surface. In this particular reaction, the equilibrium composition shifts rapidly with temperature, and at 400°C instead of 450°C, for example, a subjltanti�l fraction of cyclohexane would remain at equilibrium. Figures 3.4 through 3.7 and 3 . 9 =

4. 1

TEMPERA TURE GRA DIENTS

1 75

assume reaction to be irreversible. Methods of an alyzi n g reversible reactions are discussed in Section 4.4.

4. 1 .2 Tests of Theory, Nonisothermal Reaction Two very similar studies, by Maymo and Smith [2 1 1 ] and by Miller and Deans [2 1 8], have provided a good test of the theory in the presence of temperature gradients within a catalyst particle. Both studied the reaction of oxygen in excess hydrogen on platinum-on-alumina catalysts of such large pore sizes that diffusion was essentially by the bulk mode. In the Maymo and Smith study, intrinsic kinetics were determined on catalyst powder ; the activity of a pellet compressed from the powder was varied by changing the ratio of catalyst powder to inert alumina in the mixture compressed to form a pellet. A si ngle 1 . 86-cm-diameter pellet was studied. With the most active pellet, temperature differences of as much as 1 l 5 °C occurred between bulk gas and outside pellet surface, and temperature differences between outside surface and pellet center were as much as 300°C. Indeed, the local surface temperature varied with position around the pellet (a sphere is not a " uniformly accessible " geometry unless it is surrounded by a stagant fluid, and the point heat transfer coefficient would be expected to vary with position). The intrinsic kinetic study showed the rate to be proportional to the 0.8 power of the oxygen and to have an apparent activation energy of 5.2 kcal. The, thermal conductivity of the pellets was measured, from . which the effective diffusivity was calculated, using Equation 4.2. Effectiveness factors were predicted by numerically integrating the mathematical expressions for simultaneous diffusion and reaction, using the intrinsic kinetic expression and the above effective diffusivities. Omitting the most rapid reaction studies, in which mass transfer to the outside surface of the pellet was highly important, the predicted effective­ ness factors, which varied from about 0.5 to I .4, averaged within about 7 per cent of those found experimentally. (Temperature differences between center and outside of the pellet surface as found experimentally varied from 8-1 02°C for this group of the studies.) }n the Miller and Deans study [2 1 8], commercial platinized alumina cylinders were used, the effective diffusivity and thermal conductivity were directly measured, and a rather complex procedure was used to isolate the intrinsic kinetics from data obtained on the cylinders. The final equation, representing the rate as proportional to the 0.8 power of the oxygen with an activation energy of 5.5 kcal, agreed closely with that found by Maymo and Smith for the same type of catalyst [2 1 1 ]. Their experimental effectiveness factors were obtained by a somewhat

·

·

1 76

DIFFUSION A ND REA CTION IN POROUS CATAL YSTS. II

indirect procedure and varied from 0.53 to 0.90. Those calculated assuming isothermal operation were from 10 to 40 per cent below these values. The experimental values of 11 were in turn 20 to 30 per cent below those predicted by the Weisz and Hicks method. Howevet, the kinetic expression, the geometry, and the nonequimolar counterdiffusion flux all differ from those used in the Weisz and Hicks [376] analysis. ' Observed temperature differences between center and outside pellet surface, from 60-30°C, agreed within an average of about 10 per cent with those calculated, and these in turn were from 60-90 per cent ofthe maximum temperature difference as calculated by an equivalent of Equatio_n 4.2 for the center concentration, c; 0. � study of the hydrogenation of ethylene (in a 1 7 per cent ethylene, 83 per cent hydrogen mixture) at 1 atm on a copper-magnesium oxide catalyst [8 1 1 on t-i n. spherical pellets at temperatures up to about 1 30°C showed temperature differences up to 20°C between ' pellet surface and pellet center. A study of the hydrogenation of benzene vapor on a single t-in. cylindrical pellet comprising 50 per cent nickel on kieselguhr at temperatures up to 1 33 °C [1 58] also showed substantial temperature gradients from the inlet to the exit of the reactor, from bulk gas to pellet surface, as well as through the pellet. Both reactions are highly exothermic and rapid, complex kinetically, and theoretical interpretation was uncertain. A theoretical analysis by Bischoff [35] gives the tempera­ ture distribution expected for diffusion-limited reaction in a sphere when the surface temperature varies linearly with position on the circum­ ference. =

4.2

4.2. 1

Complex Irreversible Kinetic Expressions

Isothermal Reaction

The mathematical analyses discussed in the previous sections of Chapter 3 apply to a single reactant, an irreversible reaction, and a simple power-law expression for the rate of reaction. We now consider methods of estimating the effectiveness factor for more complex kinetic expressions, as represented, for example, by a Langmuir-Hinshelwood (Hougen-Watson) type of rate equation. A first approach to developing a generalized method was published by Chu and Hougen, who were particularly concerned with the oxidation of NO to N 0 2 [72]. Several investigators have obtained closed form solutions by making various assumptions to simplify the mathematical treatment _ [3, ,Z84]. The specific case of cracking of cumene to benzene and propylene has been

4.2

COMPLEX IRREVERSIBLE KINEI'IC EXPRESSIONS

1 77

analyzed by Prater and Lago using numerical techniques [257]. A series of papers by Schneider and Mitschka [303-306] present the generalized results of numerical analyses for isothermal reaction in an infinite flat plate (infinite slab) for Langmuir-Hinshelwood type kinetics and for both reversible and irreversible reactions. The presentation that follows here is a generalized method developed by Roberts [272, 273] on the basis of numerical computations which has the practical advan­ tage of relating '1 to the modulus L , which contains only quantities that can be observed or predicted. The general chemical equation describing the reactions under con­ sideration is A + bB +

.

.

·

-+

xX + yY +

(4. 3)

· . . .

Two types of kinetic expressions will be considered. Type /

The rate equation is taken to be (4.4) where index i is used to denote any reaction product or reactant other than A. This expression includes reactions in which A decomposes or isomerizes by a first-order process, or reaction of A with B in which the concentration of B does not appear in the numerator, but may appear in the denominator. For the reaction of A with B, such an expression might result, for example, if adsorption of A on the catalyst is the rate-controlling process. It can be derived by assuming that the rate-limiting step is either the surface reaction or the adsorption of a single reactant without dissociation and allows for possible inhibition of the reaction rate by either reactants or products. Type II

The reaction rate is assumed to obey the equation

k 2 PA Pa /( 1 + KA PA + Ka Pa + Kx Px + · · ' ) 2 =

-

1 d nA V dt '

(4. 5)

When A is the only reactant, p8 is replaced by PA in Equation 4.5 and K8p8 disappears from the denominator. This expression thus includes the case of two or more molecules of A reacting with one another, in which case B = A.

1 78

DIFFUSION AND REA CTION IN POR O US CATAL YSTS. II

For both types of kinetic expressions it is assumed that the catalyst mass is infinite in two directions and of thickness L in the third, and is exposed to a reactant gas on one face and sealed on the other. It is further assumed that the pellet is isothermal, and that the ideal gas laws are applicable. Diffusion is assumed to obey Pick's Jaw and the diffusivities of all species are taken to be constant but not necessarily equal. Diffusion of each component is assumed to be unaffected by the flux of other species, so the total pressure in the catalyst interior may be substantially different than that atthe outside surface (see Section 4.5). For gases, the assumption of a constant effective diffusivity is justified for Knudsen diffusion, for binary, bulk equimolal counterdiffusion or if one component of the gas mixture is present in great excess. This major component might be a reactant, such as hydrogen in a hydro­ genation reaction or a " nonparticipating " component such as steam in a dehydrogenation reaction or an inert gas in the removal of carbon­ aceous deposits by oxidation. Equations 1 . 1 5 or 1 . 34 may be used to estimate whether a component is present in sufficient excess to justify the assumption of a constant effective diffusivity. For Type I kinetics, determination of the effectiveness factor requires the use of one additional modulus, KPA , s where P A , s is the partial pressure of reactant A at the outside surface of the pellet and K is given by

K and

=

[KA - DA L (K1 v;/D1)] /w i

(4.6)

W =

(4. 7 ) 1 + L K;[Pi , s + (PA , s V; DA /D;)] . i The stoichiometric coefficient v 1 includes all species other than A. It is

negative for a reactant other than A and positive for all products (i.e. , in Equation 4.3, b is negative and x and y are positive). The value of w will normally be positive, but in the case of a reaction having a reactant in addition to A, a negative value of w could result if the second reactant had a very large value of K1 and a very small value of Dp./ v. The method presented here cannot be used for negative values of w. Since w is dimensionless, K has the dimensions of an adsorption con­ stant. As the values of KA and the various K1 become so small that the reaction approaches simple first order, K approaches zero ; a negative value of K indicates that the sum of the groups KvDAI D 1 for the products is greater than that for the reactants. Qualitatively, a negative value of K indicates inhibition by reaction products.

4.2

COMPLEX IRREVERSIBLE KINETIC EXPRESSIONS

1 19

Figure 4.2 is a graph of '7 versus cl>L , for various values of KPA , s and for first-, second-, and zero-order reactions. The curve for zero-order reaction corresponds to a value of KPA , s approaching infinity and that for first-order, to a value approaching zero. That for a second-order expression, however, cuts across the family of curves. A negative value of KpA , • indicates significant product adsorption effects, as noted above. The minimum possible value of KpA, s is - 1 . Using this plot, '1 can be determined directly from the values of cl>L and KpA , s · The error involved in the use of an integer-power approximation can also be estimated for any case from the figure. The use of this plot is illustrated in Example 4.3. Estimation of Effectiveness Factor '1 ; Isothermal, Complex Irreversible Reaction

Example 4.3

The reaction of carbon dioxide with solid carbon is retarded by carbon monoxide, and a complex kinetic expression is required to represent the rate. Walker, Rusinko, and Austin [358] studied the reaction of carbon dioxide with spectroscopic carbon, a finely porous material, at temperatures ranging from 950-1 305°C, and at various carbon dioxide partial pressures. The mathematical relationships for this reaction are the same as those for decom­ position or isomerization of a single reactant on a porous catalyst, except that the porosity and, hence, the effective diffusivity will increase as reaction proceeds. The reaction-rate data of Walker et a/., however, are only for the first 1 1 per cent of reaction, so the change in diffusivity during a run is rela­ tively small. Starting with a single cylindrical shape weighing 8.8 g initially, at a carbon dioxide partial pressure of 0. 75 atm, and a temperature of 1 000°C, the rate of reaction was 0. 1 25 g of carbon per hour. Presumably, the partial pressure of carbon monoxide at the exterior of the carbon was zero during the run ; and it will be assumed that nitrogen, which was present in the feed stream, does not enter into the rate equation. The average rate of reaction during consumption of the first 1 1 per cent of the carbon was 0. 1 25/(8.8 x 0.945) = 0.0 1 5 g C/g C · hr. The average void fraction (cm3/cm3) o f this sample during the period of the burnoff was about 0.36. Taking 2.27 as the true density (p,) of carbon, the apparent density of the particle is then pP = (1

-

B)p, = (0.64)(2.27) = 1 .45 g/cm\

1 dfl (0.0 1 5) ( 1 .45) = = 5 .04 dt -u- 3600

- Vc

X

I0-7

mol/cm 3 · sec.

The effective diffusivity D.rr = 0.0 1 3 cm 2 /sec at normal temperature (294°K) and pressure. Diffusion apparently occurred in the transition region beween

I

I

I I

I

''I

I

I

I

I I I

q

� '!j S]

�

:to.

,. ... 0 .... 0 .,

�

gj

0.1

�

0.05

= Q) > ·..s 0

� t::l

0.5

.:!

-

� 9

� �

� 0.01

I

0.01

J

I

I

I

I

0.05

I

I I I

0.1

I

I

I

Figure 4.2. Effectiveness factor 7J as a function of 4.4). Roberts and Satterfield [272].

I

0.5

ci>L

I

I

I I

I

1 .0

�L

I

I

I

I

5.0

I

I

I I I

'\

10.0

!'\

I'

I

I

Y.bbN

50.0 •

100.0

(Equation 3.32). Type I kinetics in flat plate geometry (Equation

S] Q

r.i

�

�

�

:::::

4.2

COMPLEX IRREVERSIBLE KINETIC EXPRESSIONS

181

Knudsen and bulk diffusion. For carbon dioxide counterdiffusing through helium in a similar graphite electrode between 30° and about 400°C at a total pressure of 1 atm, Nichols [226] reported the temperature exponent to be about 0.98. Using this value, Derr at 1000°C = 0.01 3

( )

1 273 0.9& =0.0545 cm 2 /sec. 294

The external concentration of carbon dioxide is cA . • = 0.75/(82.06)(1 273) = 7. 1 6 x 1 0 - 6 mol/cm3 •

The dimension L, the ratio of volume to surface for the cylinder of carbon, was about 0.298 em. L2

L = DCA. •

(

L = 0. 1 08.

- 1 dn

V. dt

)

=

(0.298) 2 (5.04 x t o - ') (7.6 x w - 6)(0.0545) •

The authors did not determine a rate equation for the reaction, but several other investigators have reported that on each of several types of carbon it is of the form

r = kp co 2 /(1 + KcoPco + Kcoz Pco2 ) . Wu [393] reports values of the constants for electrode carbon over the temperature and pressure range of interest, from which we estimate that Kco 2 = 2.4 atm - 1 , and Kco = 63 atm - 1 • Assuming that these can be applied to the present case,

[

'

K = (KA - D A L.., K, v,j D,)jw = Kco 2 -

D co 2 D co

]

Kco Vco /w.

The diffusivity is approximately inversely proportional to the square root of molecular weight. ( D co2/Dco) =

.J 28/44 = 0.80,

K = (2.4 - 0.80 X 63 X 2)/w = - 99/w, w=1 +

=1+

2 K,[p,, , + (pA, . DA v,jD,)]

( ) D co2 D co

X

Pco2 , , Kco Vco = 1 + 0.80 X 0.75 X 63 X 2 = 77

(note that p,, , = P co , . = 0),

KpA , . = - (99/77) X 0.75 = - 0.965.

Interpolating between the curves for KpA , . - 0.95 and - 0.98 in Figure 4.2, the effectiveness factor TJ for thi'a run is about 0.5. Therefore, internal =

DIFFUSION AND REA CTION IN POR O US CATAL YSTS. II

1 82

diffusion effects are predicted to be significant. This conclusion is confirmed by the · nonuniform porosity profiles after reaction reported by Walker and co-workers. (A theoretical analysis of their data was subsequently published by Austin and Walker [14].) Since Kco increases with decreasing temperature more rapidly than does Kcoi , the retarding effect of CO would be expected to be even more significant at temperatures below 1 000°C. From the intrinsic reaction rate data on electrode carbon published by Reif [265] and taking pA, . = 0.75, the value of KpA , . = - 0.970, which is very close to that calculated from Wu's data. Actually, Wu's correlation for coal coke gives at this temperature a value of KpA , • = - 0.965, identical to the value for electrode carbon. For the same observed rate of reaction, the effectiveness factors for these two types of carbon would, therefore, be very close to the value calculated above. More details of this calculation have been published [272]. Note that, if the reaction were assumed to be of simple first order, the effectiveness factor would erroneously be calculated to be nearly unity - i.e., diffusional effects would be thought to be insignificant. Even if a simple second-order reaction were assumed, the effectiveness factor would be taken to be about 0.92. The use of charts for «L instead of j«l>, makes little difference (see Section 4.3). The same reaction was studied earlier by Wicke and Hedden [386] at tem­ peratures of 1 000-1 1 00°C and carbon dioxide partial pressures of about 0. 1 -0.7 atm. They demonstrated that pore diffusion was highly significant for their system by two kinds of results : ( 1 ) At .a fixed temperature of 1 074°C and fixed partial pressure of carbon dioxide of 1 24 Torr the rate of reaction in the presence of carbon dioxide alone exceeded that of a mixture of carbon dioxide and helium, which exceeded that of a mixture of carbon dioxide and nitrogen. (2) The apparent activation energy was about 43 kcal/g-mol versus values of about 85 reported by other investigators.

For TypeJI kinetics, an additional parameter E is required in addi­ tion to KPA , . and L . K and ro are defined as for Type I kinetics, where i is any species other than A and the stoichiometric coefficient is taken to be negative for a reactant and positive for a product ; i.e. , i n Equation 4 . 3 , b is negative and x and y are positive. E i s defined by

v

I

E

s

( - Ds Ps, s) _ Vs DA PA ,s

1.

(4. 8)

For any real system, A is chosen to designate that reactant which per­ mits E to be zero or positive. It is seen that E will have a large value when B is present in considerable excess over A and/or has a substan­ tially higher effective diffusivity than A. It may be thought of as a " modified stoichiometric excess. "

4.2

COMPLEX IRRE VERSIBLE KINETIC EXPRESSIONS

1 83

Figures 4.3 and 4.4 are plots of 17 versus $L for selected values of For Figure 4.3, E = 0 ; for Figure 4.4, E = 1 0. A plot for E = I has been published [273] as well as plots of 17 versus a modified Thiele modulus for values of E = 0, I , and 10. For orientation, the zero-order curve is shown in Figures 4.3 and 4.4 and the second-order curve for E = 0 i n Figure 4.3. Two characteristics of the curves in Figure 4.4 deserve comment. In the first place, effectiveness factors greater than unity result over a range of values of $ L when KpA , s = 1 0 or 1 00. This is a consequence of the fact that the rate equation, Equation 4.5, possesses a maximum under certai n conditions. In the Langmuir-Hinshelwood model, the reaction rate is proportional to the product of the concentrations of adsorbed A and adsorbed B, but the two reactants compete for sites on the catalyst surface. If A is strongly adsorbed relative to B, an i ncrease i n P A at constant p 8 will displace B from catalytic sites and this can cause the reaction rate to decrease. It can be shown that effectiveness factors greater than unity will result when KPA , s is greater than (E + 2)/E. Secondly, for such combinations as E = 1 0, KP A , s = 10 or 100, and for E = I , KPA , . = I 00, a range exists for which 17 is a multiple-valued function of the Thiele modulus, analogous to that found for noniso­ thermal systems obeying simple kinetics discussed in Section 3.7. Here such effects are also predicted to be possible for isothermal reactions and complex kinetics. These regions, represented by dashed lines on the figures, represent unstable operation ; the steady-state reaction rate cannot be uniquely determined by specifying the conditions outside the catalyst pellets. The direction from which steady state is approached may i nstead determine which effectiveness factor is eventually realized. Luss and Amundson [ 1 97] have investigated the kinds of kinetic expres­ sions for which a multiple state solution is impossible. The effect of geometry on predicted effectiveness factor is probably also much greater in this region than in most others. The 17 - $ L curve for Type II kinetics always lies below the curve for Type I kinetics, if KpA , s is the same in both cases and is negative. This is illustrated by the dashed curve in Figure 4.3 for Type I kinetics, Kp A , s = - 0.90. Figure 4.4 also shows that when KPA , s is large and E is greater than zero, the 17 $ L curve lies above that for a zero-order reaction. How­ ever, in these cases, the diffusional retardation persists to lower values of $ L than it does for a zero-order reaction. Figure 4.5 is a crossplot that illustrates the effect of the modified stoichiometric excess E on the effectiveness factor at constant values KPA , s ·

-

1 .00

f

___.. _

I

� - -I

I �------� ------

T7T

::t::t:: I

I

I

I

C:f::::��

I

I

1

1

�

I

I I

00

""

0.5

l::l

2

"' c.S

"'

J(po\, ,, -+ 0 ( second 0, 1

� Ji

�

:'

n

o n

Type I kinetics, KpA, s

� �

=

I

-------=

�

"'

,

" I " " '

- 0.90

� L (Equation 3.32) for reversible first-order reaction. C = PA /PA

B defined by Equation 4. 1 2. Kao and Satterfield [ 1 67].

• •

• •

!\.! N

10.0

=

0.9.

�

� � �

"' -.I

l

.O

��

0.5

;:-

....

.E 0 � ., "' "' = "'

> :;l 0

B = 0.98, C = O.O B = 0.98, C = 0. 1 B = 0.98, C = O. S B = 0.98, C = 0.5, B = 0.98, C = 0.7 B = 0.98, C = 0.9

0.1

�

� 0.05

�r ����� �I

j_l l __

!

I I I I I�

i t;:,

;;

� � � )...

· B = 50.0, C = O.O B = 50.0, C = 0. 1 > ?'51.. > .c B = 50.0, c; = 0.3 � B = 50.0, C = 0.5 B = 50.0, C = 0.7 B = 50.0, C = 0.9

� >

.c

'-..

"-.. '-.. �

, ,/),

3).

(5.5)

A plot of f8 versus fA from Equation 5 . 3 compared to a plot of f8 versus fA from Equation 5 . 5 shows the effect of a low effectiveness factor in altering the amounts of B formed. Figure 5 . 3 shows such a comparison for an intrinsic selectivity factor kt fk 2 of 4.0. The maximum yield of B obtainable when a porous catalyst is operating at a low effectiveness factor is only about one-half that obtainable if the effective­ ness factor approaches unity. Whenever the ratio of rate constants (intrinsic selectivity factor S) exceeds 1, the effectiveness factor with respect to A will always be less than that with respect to B, assuming the differences between the effective diffusivities to be negligible. Hence, at low effectiveness factors for both, the ratio of rate of reaction of B in a catalyst pellet to that of A will be greater than it would be on a plane surface, and thus a poorer yield of B is obtained. For intrinsic selectivity factors varying up to 1 0, the maximum yield loss due to a low effective­ ness factor is reported to be about 50 per cent [382].

POISONING, SELECTIVITY, GA SIFICA TION O F GOKE

216

Although the effectiveness factor continues to drop as the Thiele modulus increases, the observed selectivity becomes independent of the modulus at values of ¢L exceeding about 3 (1'/ < 0.3). The decrease in selectivity all occurs at effectiveness factors between unity and llbout 0. 3 . If a series reaction is being carried out on catalyst pellets of such size that the effectiveness facto r is substantially less than unity but above . l OO r----,--�--�

I=Q 0

+'

�

.s QJ

"' ...

�

0

+> � QJ QJ

"' ...

P-t

20

Percent conversion (total) of

A

Figure 5.3. Effect of effectiveness factor on catalyst selectivity. Reaction of kt �2 the type A --+ B --+ C. Calculated values for kdk 2 = 4.0. Wheeler [382, 383].

about 0.3, it should be possible to increase the yield of a desired inter­ mediate substantially by subdividing the catalyst or altering the pore structure so as to increase the effective diffusivity. If, however, the effectiveness factor is well below 0.3, a large reduction in pellet size or large increase in effective diffusivity is required to achieve significant improvement in selectivity. Other forms of selectivity may also be considered. For the case of two independent simultaneous reactions, termed Type I by Wheeler, k,

k2

A -� products an d B -� pro d ucts

(Type I selectivity).

(5.6)

5.2

SELECTI VITY IN POROUS CA TAL YSTS

217

The selectivity, for any value of 17 , is given by -

At low effectiveness factors, tion 5.7 becomes

d eB d cA 11

=

CB k l i1 B CA k 1 11A .

(5.7)

is inversely proportional to