studies in gas solid reactions

8/10/2019 Studies in Gas Solid Reactions

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Studies in Gas Solid Reactions: Part I. A Structural

Model for the Reaction of Porous Oxides with a

Reducing Gas

J. SZEKELY AND J. W. EVANS

A str uc tur al model is pr esented for desc ribin g the rea ction of a porous m etal oxide pellet

with a reducing gas . It is suggested that the p ellet is made up of a large n um ber of gra ins

and the ov eral l rate of reaction is computed by summing the contribu tions of all these in-

dividual grain s. The model thus incorporates stru ctu ral param eter s, such as grain size,

porosity pore size distribution) and allows a qu antitative ass ess me nt of the role played

by these quan tities in determining the rate of progr ess of the reactio n.

T H

eac t ion of porous meta l oxide aggregates wi th a

reducing gas i s of considerable technologica l impor-

taslce arid has consequently received a great deal of

a t tention. A la rge proport io n of these s tudies was con-

cerned wi th i ron oxide reduct ion but many other sys-

tems have a l so been extensive ly s tudied.

Most of the in vest iga to rs to da ta have interpre t ed

the i r resul t s by us ing a sui table var iant of the shr ink -

ing core mod el . 1 8 This model , ske tched sc hemat ic-

al ly in Fig. 1 is based on the assu mpti on that after

some reac t ion had occurred the sol id phase consis t s of

an unreac ted core , s urrounded by a reac ted she l l .

These two zones a re separa t ed by a sharp phase bound-

ary where the chemica l reac t ion takes place . As the

reac t ion proceeds , the reac ted she l l expands and di ffu-

sion of the reactan ts and products through this region

may become one of the ra te l imi t ing fac tors .

The shrinking cor e model {model based on top o-

chemi ca l r eac t i ons ) ha s been remarkabl y success fu l

for the interpre ta t ion of experimen ta l res ul t s ; in gen-

era l , the equat ions based on thi s representa t ion de-

scr ibed the overa l l ra te in te r ms of a 'dr iving for ce

and three se t s of res i s t anc es , namely:

i) Gas phase mass t r ans fer .

i f) Diffusion of reac tan ts and products through the

reac ted she l l .

i i i ) Chemica l reac t ion occurr ing a t the interface

separa t ing the reac ted and the unreac ted regions .

Of these quant i t i es , the gas phase mas s t rans fer

coeffi cient and the rat e of por e diffusion may be pre -

d i c ted , o r a t le a s t es t i ma t ed . The pa rame t e r s cha rac -

te r iz ing the chemi ca l kine t ics , however , have to be

measu red exper imenta l ly. Indeed, the form of the ra te

equation and the numer ica l va lues of the ra te constant

se rved a s t he pr i nc i pa l ad j us t ab l e pa rame t e r s whi ch

i l lowed the matching of exper imenta l and the ore t ic a l

re su l t s .

While the shrinking core model has been very widely

ased, i t suffers f rom two major shortcomings:

a) The important postula te of topo chemt ca l rea c-

t io n , i . e . , the exis tence of a sharp boundary be tween

J. SZEKELY is Professor of Chemic-,d Engineering and Director,

Center for Process Metallurgy at the State University at Buffalo, BuD

falo, N.Y. J. W. EVANS, formerly Graduate Student in Chemical Engi-

neering at the State University of New York at Buffalo, is now with the

Ethyl Corp. Baton Rouge, La.

Manuscript submitted July 24 1970.

the reac ted and unreac ted zones i s not universa l l y

supported by experim enta l evidence . In the i r s tudy of

hemat i te redu ctio n, G ray and Henderson ~ found that the

reduced and unreac ted sec t ions were sep ara ted by a

t ransi t ion zone , which conta ined both tota lly reduced

and only par t ia l ly reduced gra ins . Simi lar f indings

wer e rep ort ed by Weisz and Goodwin t~ in thei r inve s-

t iga t ion concerned wi th the combust ion of carbon de-

posi t s in porous ca ta lys t s .

b) Perha ps the most ser iou s drawback of the shr ink-

Lag cor e mo del is the fact that st ruct ura l effec ts, such

as porosi ty, gra in s ize , and so for th, a re impl ic i t ly

incorporated in the chemical rate constant and do not

appear expl ic i t ly . I t fol lows tha t any measurement

made of reduct ion kine t ics i s necessar i ly spec i f ic to

a given mater i a l o r even to a given ba tch of mater i a l ,

and no unique reac t ion parameters can be ass igned to

a given ore or s inter . The shr inking core model pro-

vides l i t t le guidance on the role played by structural

pa ram e t e r s i n de t e rmi n i ng t he ove ra l l r e act i on ra t e .

INRE CTED CORE

/ - / I/C.EM,CA REACT,ON AT

/ / T.E P.ASE .OONDARY

/ REAC TED ZONE

- P O R E D I F F U S I O N

GAS PHASE MASS TRANSFER

OF REACTANTS AND PRODUCTS

Fig. t--Schematic represen~tion oftheshrinkingcore model.

METALLURGICALTRANSACTIONS VOLUME 2, JUNE 1971-1691


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These apparent shortcomings of the topochemical as-

sumption prompted a number of recent studies,~x-x4 of

which the work report ed by Ishida and Wen is prob-

ably the most noteworthy. The se authors regard ed the

porous solid matrix as a homogeneous medium and de-

scr ibed the gas-sol id react ion in ter ms of a homo-

geneous ra te constant . This repres enta t i on led to a

diffuse reaction front but still would not allow the a

pr io r i predic t ion of the react ion ra tes in ter ms of

s t ruc tu ra l cons ide rat ions .

Fina lly, in a recent paper the authors proposed a

str uct ura l model ts for the reacti on between a s emi -

infinite porous medi um and a gaseous react ant.

Through this model it was po ssible to identify some

of the s ignif icant s t ruct ura l p ara met ers , but the geom-

etry chosen precluded d irec t comparison with meas-

u r e m e n t s .

The work reported in the pres ent s eri es of papers

was undertak en with a view of developing a st ruc tur al

model which can be compared d irec t ly with exper i men-

ta l me asur emen ts . The u l t imate objec t ive of the inves-

tigation is to define the

p h y s i c a l c r i t e r i a

that affect the

reactivity of porous solids, and hence to define the

optimal condit ions for effec t ing these react ions .

In Par t I we shal l present a mathemat ica l model ,

which is a further development of that desc ribed in the

ear l ier publ ica t ion , and in the subsequent Par t I I the

predictions based on the model will be compared with

experimenta l measurements obta ined, us ing the sys-

tem NiO-H2.

FORMULATION

Let us consider a spherical pellet of the solid re-

ac tant , made up of a la rge numbe r of spher ica l gra ins

of uniform radius ,

r s,

Let the porosity of the sample

be P, and the distance from the center (macroscopic

radi al coordinate) be designated by R. The sampl e is

brought into contact with a gas, A, with which it reacts

to form a solid product and a gaseous product, B.

The following majo r assu mpti ons are made:

i) The initial physical stru ctu re is maintained

throughout the reaction, and

ii) the react ion of each grain p roceeds from the

outside toward the cent er, so that the position of the

react ion front w i t h i n e a c h g r a i n exhibi ts spher ica l

symmetry-- th is behavior may be descr ibed as the mi-

croscop ic shrink ing core. Thus the rate at which each

grai n reacts is proportional to the surface area of

the react ion front at any given tim e. The radius of

th is mi croscopic react ion front is des ignated by r ,

which is a funct ion of time and

o f R i . e . ,

radia l posi-

tion within the sample).

The model described above is sketched in Fig. 2

where i t is seen tha t the reactant gas is t ransferred

from the bulk gas st rea m, diffuses between the grains

and then through a solid product layer within each

gra in and reacts a t the spher ica l react ion in terface .

The product gas thus generated diffuses back through

the solid product layer and between the grains before

undergoing a mass tra nsf er step into the bulk gas

s t r e a m .

At this stage it may be worthwhile to comm ent

br ief ly on the appropria teness of these assumptions .

Assumption i ) is res tr ic t ive , nonetheless i t is thought

to be valid for a range of practical conditions. It is

agreed that s in ter i ng or agglomerat ion of the reduced

phase will occur in many ins tanc es, hut it is suggested

that these more complex s i tua t ions are best descr ibed

in ter ms of an appropria te ly modif ied model .

Assumpti on ii ) is thought to repr esent a reasonabl e

approximation to rea l i ty , a t leas t in a macorscopic

sense .

Let us now proceed by stating the appropriate con-

servat ion equat ions for the gaseous reactants and

products .

A mas s balance on component A yields the following

differential equation describing diffusion of A between

the gr ains :

D_s a (R aC A/ - (1 -~ O) p, . ~ = 0 il l

R ~ a--R- _.z ~R

where

P m=

t rue molar densi ty of reactant so l id .

= rate of disappearance of gaseous reactant per

mole of initial solid reactant.

DJi = the effect ive diffusivi ty of the gase ous rea cta nt

between the grains of the porous solid.

CA = molar concentra t ion of the gaseous reactant in

the in ters t ices between the gra ins .

A similar equation may be written for the gaseous

product:

R a 8R 8R

where n s is the mole s of B for med by the react ion of

one mole of A. Subsequently,

n s

will be taken as unity,

Bu lk g as s ream

Fig. 2--Schematic representation of the structural model.

1692-VOLUME2, JUNE t971 METALLURGICAL TRANSACTIONS


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but the modification to account for other values of

n s

is s t ra ightforward .

Eqs. [ ] and [2] are valid for eq uimo lar co unter dif-

fusion (the present case) or where diff usion between

the grai ns is pur ely of the Knudsen type. It is to be

noted that no accumu lati on ter ms appea r in the two

equat ions, thus the 'quasi -s tead y s ta te ' assumpti on is

implic i t in th is formula t ion . The quasi -s te ady s ta te

assumption is readi ly jus t i f ied on physica l grounds

because the amount of gaseous reactant within the

pores is negligible, compar ed with the net input and

the rate of reaction.

Before we can proceed fur ther , the react ion term,

has to be relat ed to the conce ntra tio n of the re ac-

tants and products and also the rate of progress of the

react ion front , with in the gra ins a t a pa r t icu l ar loca-

tio n, R 9

We shall make the following further assumptions:

lit) The reaction between the gas and the solid is of

f i rs t ord er in both the forward and rev ers e d i rec t io ns ,

and

iv) there is negligible resistance due to diffusion

through the product layer w i t h i n t h e g r a i n s ; thus the

concen trat ion at the reac tion su rface within a grain is

identical with that in the inte rst ice s at the sa me radi al

coordinate .

The assumption regarding f i rs t order k inet ics is

res t r ic t ive , nonetheless i t wil l apply to many sy stems ,

in par t i cu lar i t wil l hold as a reasonable app roxima-

tion in the vicinity of equilibrium,

i . e . , i n

the mixed

contro l regimes. Nonlinear k inet ic express i ons could

be readi ly accomm odated but in view of the much

larger demand on computer t ime th is poss ib le ref ine-

ment was not incorpo rated at this stage.

Assu mpti on iv) is thought to be reas onabl e for sm all

grai ns as the diffusion path through the (porous} grai ns

is very much smal le r than the diffusion path through

the bulk of the porous solid.

With these assumptions, il l) and iv), the advance-

meat of the reaction front within a grain is written as:

d t P rn A - K E

where k is the chemical reaction rate constant and

K E

is the equi l ibr ium constant .

g, the reaction ter m, may now be readily r elated to

the advanceme nt of the react ion front within a grai n,

as follows:

For the ease where one mole of A re acts with one

mole of solid,* we have:

R e a d y a l l o w a n c e c o u l d b e m a d e f o r o t h e r s t o i d a i o m e t r i e s t h r o u g h t h e u s e o f

a a a p p r o p r i a t e c o n s t a n t o f p r o p o r t i o n a l i t y .

4 ~ r z " d r ( C _ C _~ ) [4]

- ~ P m = 4 n r 2 k A K E

Thus

. . . . 3

A [ 5 ]

P m r s K E

On substi tuti ng for 9 fro m Eq. [5] into Eq. [1] and [2]

we obtain:

OR 3 R / r s K E

R ~ a ~ a R J ~ 0 1 7 ]

Here menti on may be made of a more genera l case

where the grains are not of uniform size. Let

f ( r s ) 9 d r s be the weight fraction of grains with radii

between

r s

and

~r s + d r s )

and assume thatf is inde-

penden t of R. Now

prn r~ \ C A I-~E

and Eq. [3] becomes:

d r ( r s , R ) _ k ( C A - K ~ E ) [9]

d t Pm

Clea rly, this additional sophi sticat ion would be appro-

pr ia te only if accura te information were avai lable on

the gra in s ize d is t r ibut ion .

I N I T I A L A N D B O U N D A R Y C O N D I T I O N S

T h e i n i t i a l a n d b o u n d a r y c o n d i t i o n s f o r t h e g o v e r n i n g

E q s . [ 3 ] , [ 6 ] , a n d [ 7 ] h a v e t o e x p r e s s t h e f o l l o w i n g

physica l constra in ts :

a) the initial position of the reaction front within

each gra in ,

i . e . ,

that no reaction had taken place be-

fore t = 0

b) the fact that the concent ratio n profiles are sym-

metrical about the center of the pellet, and

c) the continuity of the m olar fluxes at the outer

surface of the pellet,

i . e . ,

diffusive flux across the

outer surface of the pellet = couvective flux through

the "gas f i l m" surrounding the pel le t .

These boundary conditions are given in Eqs. [10]

through [14]:

r = r s

for at lR at t = [10]

~ C A = 0 a t R = 0 [11]

OR

~ z

= o a i r

o [ 12 ]

~R

D~4 ~ = h ( CA o CA )

at R =Ro [13]

~R

D[3 ~ = h ( CB o - CB )

at R =Ro [14]

3R

where Ro is the radius of the spherical sample, h is

the mass transfer coefficient from the bulk gas stream

to the solid sample, CAo and CBo are the concentrations

of the gaseous rea cta nt and product in the bulk gas

s t r e a m .

C O R R E C T I O N F O R N O N I ~ K T H E R M A L B E H A V I O R

I n g e n e r a l , t h e t e m p e r a t u r e o f t h e p e l l e t w i l l d i f f e r

f r o m t h a t o f t h e g a s s t r e a m , b e c a u s e o f t h e h e a t g e n -

e r a t e d o r a b s o r b e d b y t h e c h e m i c a l r e a c t i o n . F o r

r e l a t i v e l y s m a l l h e a t s o f r e a c t i o n r e a d y a l l o w a n c e

m a y b e m a d e f o r n o n - i s o t h e r m a l i t y , b y a s s u m i n g t h a t

t h e p e l l e t i s a t u n i f o r m t e m p e r a t u r e a t a n y g i v e n

t i m e ; ~ 6 t h e n t h e u n s t e a d y s t a t e h e a t b a l a n c e y i e l d s t h e

following:

M E T A L L U R G I C A L T R A N S A C T I O N S V O L U M E 2 , J U N E 1 9 7 1 - - t6 9 3


4/8

d T H d e 3

d t C p d t

Ro 1 -

P ) p m C p

[ h t T - T E ) + e a T ~ - T~)] [15]

w ith

T = T E

at t = 0 [16]

where T and T E are the pel le t and environment tem-

pera tur es in absolute units, H is the heat of react ion,

and the other qu antiti es are defined in the list of

symbols .

Fro m the knowledge of d ~ / d t the pel le t temperature

may thus be computed and the property values may

then be evaluated a t th is correc t temper ature . For the

practical syste m to be consid ered in Par t H, this cor-

rect ion amount ed only to a ma xi mum of about 20 ~ to

30~ so that the princ ipa l effect was conf ined to the

react ion ra te constant .

Eqs. [3], [6], [7], and [15], together with the bound-

ary condit ions contained in Eqs. [10] through [14]

repres ent a complete s ta tement of the problem. The

subsequent m anipulat ion of these equations and the

numer ica l technique used for the ir so lu tion is d is-

cussed in the subsequent section.

THE TECHNIQUE OF SOLUTION

Rearrangement of the Equations

The governing equations may be rearranged on noting

that in the absence of coupled fluxes Eqs. [6] and [7]

are not independentand one of them may be eliminated

by the following procedure:

Stoichiometry dictates that:

fur the rmor e on combinin g Eqs. [6] and [7] we have:

On integrating Eq. [18] once, with respec t to R, and

using Eq. [17 ] yields after a furt her integrat ion:

D~4 Constant - C a ) = D ~ C B [19]

which is valid for all value s of R.

CB

may now be expressed with the aid of Eqs. [17]

and [19] to obtain:

( D A

- 1 ) 9 , [ 2 0 ]

C B = C Ao + \ D ~ C A R o - - C A D ~D B

Thus on substi tution Eqs. [3] and [6] to [7] are t ran s-

f o r m e d t o E q s . [ 2 0 ] a n d [ 2 1] r e s pe c t i ve l y .

d r k {C A l+ ~ 1

R aR

[ 2 2 1

In generating a solution, we seek values of C and

also CB) and r, for all values of R and t. However, in

order to compare the computed results with experi-

mental data, it is convenient o define a quantity,

termed the extent of reactionwhich bears a direct

relationship to experimentally measured weight change

of the specimen.

The overall extent of reaction e may be expressed

as follows :

R z

Ro~ r s

- r /rs)] may be consider ed a

l o c a l e x -

he quan tit y [1 3

t e n t o f r e a c t i o n and will be desig nated ~. Thus the

right side of Eq. [21] is the weighted average value of

O/

c is not a funct ion of R, but is, of cou rse , a funct ion

of t.

The Numerical Solution

In order to put the governing equations in a finite

d ifference form, le t us es tabl ish a two-dimensional

grid in R and t, designa ting the mes h spaci ng in R by

AR and the time step by At.

The values of R at the vari ous grid points wilt be

designated by subst i tu t ing appropria te numerica l

values into the index i of the qua nti ty Ri , which now

denotes discrete values of the spatial coordinate. The

index i is so chosen that:

i = 1 at R = 0

i = n

ai r =Ro [24]

The index of any mes h valu e of R di ffers from its

neighbors by unity.

Eq. [21] may now be replaced by n equations of the

type

d t P m D B K E ] K E

where the subscripts i denote values of the appropriate

var iables a t radius Ri .

If the values of C A i wer e known as a function of

time, the set of equations represented by Eq. [251

could be readily integrated, e.g. by the Runge-Kutta

method.

In order to obtain C A i , let us rec all Eq. [22], which

may be writ ten as follows:

[ R 2 OR R=R rs

The left hand side of Eq. [26] may be expr ess ed in a

f in i te d ifference form as:

0R ~ R ] -~ -~ + 2R ~R ~ AR2

i

[CAi__1 - 2 C A i + C a i+ l ]

L 2 , J

1694-VOLUME2, JUNE 1971 MFTALL:URCICAL TRANSACTIONS


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6/8

C H E M I C A L R E A C T I O N

Extent

I.o

Radial Variation

of Extent of

Reaction ( ~ ) :

Concentration

Profile :

Occurs :

C O N T R O L

0 I

CONG.

Co

0 Re

Distance from Centre

0 L t

0 Ro


Low Temperatures

High Porosities

Small Spheres

Fig. 4--Typical plot of the local extent of reaction and of the

concentrati on profile for chemical control.

The molecular diffusion coefficient may be est imated

from the mol ecul ar theory of gase s, and the Knudsen

diffusion coefficient may also be est imated if the pore

s ize d i s t r ibut i on (gra in s ize d is t r ibut ion) and the por -

osity are known.

Alternat ively , D.~ may be measur ed exper imenta l l y

through both the reacted and unreacted matrices.

F ina l ly , r s the gra in s ize may be e i ther measured

direc t ly by e lec tron microscopy, or X-ray d iff rac t ion

techniques; a l ternat ivel y , a va lue for

r s

may be de-

duced from pore s ize d is t r ibut ion meas urem ents .

It is str ess ed that the effective diffusion coefficient

and the grain size may be strongly interrelated if the

component due to Knudsen diffusion is s ignifican t.

A fur ther , more deta iled d iscuss io n wil l be presented

in Part ]I of the paper regarding the techniques avail-

able for es t i mating and measur ing D~, k and r s.

In view of the very large number of parameters that

have to enter the computati on, any set of computed

res ult s will be specific to a part icu lar , given applica -

tion. At this stage it is thought desirable, to defer the

pres enta tion of detailed computed curves to Part II of

the paper and to confine ours elv es to a disc ussi on of

the general features of the solution.

Figs . 4 to 6 show sketches of the concent ratio n pro-

files and of the spatial dist ribu tio n of a, the extent of

react ion , a t some in ter media te t ime s, corresponding

to various types of asymptotic behavior.

The profiles of a and of CA sketched in Fig. 4, cor-

respond to systems where the overa l l ra te is contro l led

by chemical k inet ics ; in prac t ica l s i tua t ions chemical

kinetics tend to control at low temperatures, at high

poro siti es, and in the case of sma ll pellet s. Inspection

of Fig. 4 shows that both the concen trat ion and a pro-

f i les are uniform.

Fig. 5 shows syst ems wher e inte rnal diffusion is the

rate control ling step, which is realiz ed at high tern-

I N T E R N A L D I F F U S I O N

Extent

I O

Radial Variation

of Extent of

React ion ( = ) :

C O N T R O L

Cone

Co

0 I-

0 Ro


Concentration

Profil e : o

o R=

Distance

r o m

Centre

Occurs : High Temper atures

Low Porosities

Large Spheres

Towards End of Reaction


concentrati on profile for pore diffusion control.

MI XE D (chemical reaction 4- internal diffusion) CON TR OL

Extent

I.o

Radial Variation /

of Extent of

Reaction ( ~ ) : o -

b

0 Ro

Distance

from Centre

Conc. T

Prof ile : o ' , 9

O Re



concentration profile for mixed control.

pera tures , low porosi t ies , and for la rge pel le ts . I t is

seen that the profi le of c~ und ergo es a s tep change from

unity to zero at s ome int erme diat e value of R, which of

course , corresponds to a sharp react ion boundary . In-

spection of the concentr ation profile shows that gra -

dients in concent ratio n are confined to a region between

Ro and the reaction front, i . e . to the reacted shell.

Fina lly, the behavior of a re gime of mixed control

(chemical rea ction + interna l diffusion) is sketched in

Fig. 6. It is seen that the profi le of a is a smooth

curve , a significant portion of which falls between 0

and 1; it follows that this situat ion corres pond s to a

1696-VOLUblE 2, JUNE 1971 METAL[.URGICAL TRANSACTIONS


7/8

o V ; ,b ,~ ~o ' ,% ~o' ,2o

TIME Minutes)

Fig. 7--Plot of the overall extent of reaction against time for

nickel oxide reduction with hydrogen. The numbers on the

curves correspond to:

No. Temp, ~ R0, cm P, Initi al

299 0.770 0.729

2 301 1.178 0.734

3 303 0.800 0.384

4 293 1.128 0.384

dLffuse react ion zone. The concentrat ion profi le shown

also ref lec t s thi s behavior .

Inspection of Figs. 4 to 6 shows that a sharp react ion

front exis t s only in one par t icular case , when the proc-

ess i s cont rol led ent i re ly by interna l di f fus ion; in a l l

other case s the r eact io n zone wil l consi st of a diffuse

regio n, the width of which Ls det ermi ned by the r ela t iv e

magni tude of the var ious parameters .

Since Figs . 4 to 6, l ike a l l preceding i l lus t ra t ions ,

were ske tches of computed behavior , i t may be of in-

ter est to show actual computed resu lts. This is done

in Fig. 7 on a plot of the overal l extent of react ion

agains t t ime. The par t icular sys tem considered i s in

the intermedia te regime where both di ffus ion and

chemica l kine t ics a re s igni f icant . The four curves

shown indicate that both porosi ty and pel let size have

a s igni f icant e ffec t on the over a l l reac t ion ra te .

DISCUSSION

In the paper an a l te rn a t ive i s proposed to the con-

vent ional shr in king cor e model , extensive ly used

for descr ibing the reduct ion kine t ics of sol id, porous

meta l oxides .

The model presented in the paper descr ibes the

over a l l ra te of the reac t ion by summing the cont r ibu-

t ions from the individual grains that make up the por-

ous sol id mat r ix.

On assu ming that the sol id struc tur e is not modified

in the cour se of the reac t ion, the sys tem i s repr e-

sented by two se t s of s imul taneous di ffe rent ia l equa-

t ions , one se t descr ibing the reac t ion of individual

layers of sol id gra ins , and the other descr ibing the

diffusion of reac tant s and produ cts within the porous

ma t r i x .

A sa t i s fac tory numerica l scheme was developed for

the solut ion of these equat ions , and thus cur ves w ere

genera ted giving the appro pria te t rans ient co ncent ra-

t ion profi les and the extent of react ion with t ime.

The potent ia l a t t rac t iveness of the present model i s

twof o ld :

a) The profi les generated for the local extent of re-

ac t ion are in qual i ta t ive agreement wi th experiment a l

f indings; a cer ta in combinat ion of c i rc ums tan ces can

lead to di ffuse reac t ion zones , whereas there a re con-

di t ions (when pore di ffus ion cont rols) where the re -

ac ted and unreac ted zones a re separa ted by a sharp

reac t ion boundary.

b) A mor e significant fe ature of the mode l is that i t

i ncorpora t e s s t ruc t ura l pa rame t e r s , such a s pore

di ffus ion coeff ic ient , gra in s ize , porosi ty, and the l ike ,

into the ove ra l l reac t ion scheme. These quant i t i es

may be meas ure d in dependently, and if adequate infor-

mation is avai lable on the chemica l kin et ic s, i t should

be poss ible to predic t the behavior of cer ta in sys tems

f rom pure l y phys i ca l measurement s .

A be t te r unders tanding of the role played by s t ruc-

t u ra l pa rame t e r s i n t he ove ra l l r e ac t i on scheme could

lead us to specia l techni ques, or to the modifica t ion of

exis t ing techniques of sol ids prepara t ion wi th a view

of achieving opt imal performance of the reac tor uni t

i n whi ch t hey a re p r ocessed . Ul t i ma t el y , t a i l o r -made

st ru c tur es may be evolved to suit a given, par t i cular

gas - so l i d reac t i on .

It is noted, that in i ts present state the model is

res t r ic t ive , because of the assumpt ion made for the

re t a i nment o f t he or i g i na l s t ruc t ure .

Wi thin the f ramework of the model , a l lowances could

be made for s t ruc tura l changes occurr ing in the course

of the reac t ion. The incorpora t ion of these fac to rs ,

through addi t ional equat ions , i s qui te s t ra ight forward

in concept but would, of course , increase the complexi ty

of computa t ion. At present ther e i s not enough inf orma-

t ion ava i lable on s inter ing kine t ics for meta l -meta l

oxide sys tem s of inte r es t to make such an effor t worth -

whi le . Nonet he less , future work both in the author s '

l abora tory and e l sewhere , could tend to the const ruc t ion

of more gene ra l mode l s .

ACKNOWLEDGMENTS

The authors wish to thank the New York State Science

and Technology Foundation for part ial support of this

invest iga t ion. Thanks are a l so due to the t rus tees of

the C. C. Fu rnas Fel low ship for the support of J. W. E.

during the academic year 1969-70.

CA CB

CA~ CBo

CARo

Cp

D~4 D~

e

9

LIST OF SYMBO LS

Gaseous reac tant and product molar

concent ra t ions wi thin pores

Gas concent ra t ion s in bulk gas s t r eam

Reactant gas concent ra t ion a t pe l le t

surface

Molar specific heat of pel let

Effec t ive dt ffus ivi t ies of gaseous rea c-

tant and product within porous pel let

Tot a l hemi sphe r i ca l emi ss i v i t y o f

pe l le t

Gra in s ize di s t r ibut ion funct ion

Ra te of d i sappea rance of ga seous reac -

tant per mole of ini t ial sol id reactant

METALLURGICAL TRANSACTIONS VOLUME 2, JUNE 1971-1697


8/8

H

h

ht

k

K E

rl s

P

R

o

rs

T

T

H e a t o f r e a c t i o n

M a s s t r a n s f e r c o e f f i c i e n t

H e a t t r a n s f e r c o e f f i c i e n t

C h e m i c a l r e a c t i o n r a t e c o n s t a n t

E q u i l i b r i u m c o n s t a n t

S t o i c h i o m e t r y c o e f f i c i en t

P o r o s i t y

R a d i a l c o o r d i n a t e w i t h i n sp h e r i c a l

pellet

Radius of pellet

Radius of reaction front within grain

Radius of grain

Pellet temperature

nvironment temperature

Time

G r e e k L e t t e r s

L o c a l e x t e n t o f r e a c t i o n

E

P D1

O v e r a l l e x t e n t o f r e a c t i o n

T r u e m o l a r d e n s i t y o f s o l i d r e a c t a n t

S t e f a n - B o l t z m a n n c o n s t a n t

R E F E R E N E S

1. S . Yagi and D. Kun i i : P roc . 5 th ln te rm . Syrup . on Com bust io n 1955 p . 231 .

2. W. M. McKewan:

? rang TMS-AIME,

1962 vol . 224 p . 2 .

3. H. W. St. Clair : Trans. TMS-AIME, 1965 vol . 233 p . 1145 .

4 . W. K. Lu and G. B i ts ianes : Trahx TMS-AIME, 1966 vot . 236 p . 531 .

5 . R . H. Spi tze r F . S. Manning and W. O. Phi lbrook: Trans. TMS-ADtE, 1966

vol . 236 p . 1715 .

6 . J . Shen and J . M. Smith : Ind. Eng. Chem. FundaTnentals, 1965 vol . 4 p . 293 .

7 . M. l sh ida an d C . Y. Wen: Chem. Eng. ScL, 1968 vol . 23 p . 125 .

8 . G . S . G . Beve t idge and P . J . Gold ie : Chem. Eng. Sci., 1968. vo t. 23 p . 913 .

9 . N . B . Gray and J . Hende r son: Trans. TMS~41ME, 1966 vol . 236 p . 1213 .

10 . P . B . We isz and R . D. Good win : J . Catalysis, 1963 voL 2 p . 397 .

1 1 . J . M . A m m a n a n d C . C . W a t s o n : Chem. Eng. Sci, 1962 vol . 17 p . 323 .

12 . D . T . Lacey J . H . Bowen and K. S . Basden: Ind. Eng. Chem. Fundamentals,

1965 vol . 4 p . 275 .

13 . A . K. Lah i r i and V. Seshadr i : J . Iron Steel Inst., 1968 vol . 206 p . 1118 .

14 . M. I sh ida and C . Y. Wen:

At~ Inst. Chem. Eng.

J . 1968 vo i . 14 p . 311 .

15 . J . Szeke ly an d J . W. Evans : Chem. Eng. ScL, 1970 vol . 25 pp . 1095-1107 .

1 6 . D . L u s s a n d N . R . A m u n d s o n : Ant Inst. Chem. Eng. J., 1969 vol. 15. p. 194.

17 . J . W. Evans : Ph .D. D isse r ta t ion S ta te Un ive r s i ty of New Y ork a t Buf fa lo

1970.

1 6 9 8 - V O L U M E 2 J U N E 1 9 71 M E T A I _ I . U R G I C A L T R A N S A C TI O X ~S

studies in gas solid reactions

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