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Effect of sintering and porosity changes on rates of gas—solid reactions
The Chemical Engineering Journal, 14 (1977) 137 - 146 0 Elsevier Sequoia S.A., Lausanne -Printed in the Netherlands
Effect of Sintering and Porosity Changes on Rates of Gas-Solid Reactions
P. A. RAMACHANDRAN Department
(Received
of Chemical
and J. M. SMITH Engineering,
University
of California,
Davis, Calif. 95616
(U.S.A.)
8 March 1977; in final form 17 June 1977)
Abstract
sintering. Szekely and Evans [2] and Szekely et al. [3] proposed an exponential decay in diffusivity with reaction but made no attempt to relate the diffusivity to the changes in pore structure. Rehmat and Saxena [4] studied the effects of density differences on the conversion-time relationship in terms of the sharp interface (shrinking core) model; Ramachandran and Smith [5] have analyzed the effects of density differences by focusing attention on the changes in dimensions of a single pore. For analyzing the combined effects of density differences and sir&ring, the particle pellet [6] or grain [7] concept provides an attractive basis. Structural changes can be expressed in simple geometric terms and nonisothermal cases are easily formulated. This paper employs such a model as a basis for establishing conversion and intrapellet temperature profiles as a function of reaction time. Costa and Smith [8] measured both temperature profiles and conversion for the hydrofluorination of uranium dioxide pellets, and these data are used to explore the suitability of the model.
During the course of most gas-solid non-catalytic reactions the solid undergoes structural changes caused primarily by swelling or shrinkage, due to the chemical reaction, or by sin tering. In this paper a theory is developed to account for the effects of such structural changes on the conversion-time relationship and on temperature profiles in the pellet. The theory is based on the particle pellet concept and accounts for differences in density between reactant and product solids and for changes of porosity and pore interconnections due to sintering. The rate of sintering is assumed to obey an Arrhenius-type equation. Predicted conversion-time and tempemturetime-position curves based upon the theory are compared with experimental results for the hydrofluorination of pellets of uranium dioxide.
1. INTRODUCTION
Many non-catalytic reactions occur under conditions such that the solid undergoes various structural changes during the course of the reaction. Such changes may be due to density differences caused by chemical reaction, which results in swelling or shrinkage of the individual particles that comprise the pellet, or to the effects of temperature which cause sintering. Only a few reports in the literature refer to the effects of structural changes, and particularly limited are papers that consider sintering. Calvelo and Cunningham [l] assumed that the changes in the effective diffusivity due to structural changes were a function of porosity. This procedure may account for the effects of density differences but is inadequate to account for the effects of
2. BASIS OF THE MODEL
The particle pellet model refers to an assembly of spherical non-porous particles (of uniform size at the beginning of the reaction) compressed into a spherical pellet. The reaction occurs by diffusion of gaseous reactant into the pellet to the reactive surface of a particle. Suppose the reaction is of the form A(g) + bR(s) --, gG(s) + P(g) As time progresses a layer of porous G builds up around the particles. Owing to the differences in stoichiometric coefficients and in the molal density of the product and the reactant, 137
138
the particles expand or shrink, resulting in a change in radius. These changes affect the macroporosity and hence the effective diffusivity. The additional structural changes which can occur are due mainly to the effects of temperature, namely sintering. Sintering can cause considerable reduction in the effective diffusivity due to removal of pore interconnections. An approximate predictive equation to account for this effect has been derived [9] and use of this will be made in the present work. Various assumptions are made in this model in order to keep the mathematics tractable. These are as follows. (1) The radius of the pellet does not change. (2) The particle shape is spherical initially and the particles retain their original shape during the course of the reaction. (3) Sintering affects the diffusivity in the macropores but not the diffusivity through the product layer formed around the particles. (4) Only the solid product (the product layer) is assumed to sinter. The original solid reactant does not sinter. (5) The effective thermal conductivity of the pellet is independent of the extent of reaction . (6) The analysis is restricted to first-order irreversible reactions with respect to the gaseous reactant, and to first-order reactions with respect to the reactive surface, according to the rate equation
- -dn, = kCA&
(1)
dt
Equation (1) implies that the reaction occurs at a sharp interface between the non-porous reactant and the porous product layer in the particles. With respect to the pellet radius reaction does not occur at a sharp boundary. (7) The temperature gradient within individual particles is neglected, but not the intrapellet gradients. The geometry of the model is shown in Fig. 1. 3. MATHEMATICAL PROBLEM
FORMULATION
OF THE
In order to develop equations for the diffusion and reaction of gas A in the pellet, it is first necessary to obtain the rate of reaction per unit volume of the pellet. The rate per particle
PARTICLE
PELLEl
Product Layer
Reactant
0
lc
‘. -tR
Fig. 1. Geometry of the pellet and particle: - - -, particle radius t-0 at t = 0 for y > 1; - -, particle radiusroatt=Ofory< 1.
can be obtained by considering the consecutive processes of diffusion within the product layer and the reaction at the sharp interface of area 47rrz. These individual rates are, for diffusion, --= dnA dt
4nrgrC DG rg -r,
(CA
-
CAi)
and, for chemical reaction,
-- ‘hi dt
= 47V:kCA,
(3)
Eliminating the interfacial Concentration CAi
where rC is the radius of the reaction interface and rg the radius of a particle at any time. The number of spherical particles per unit volume at time zero is 3(1 - eo)/4m$, where r. is the initial radius of the particles, and this number is assumed to be constant during the reaction. Hence, the rate of disappearance of A per unit volume of pellet is
In terms of eqn. (5) the mass conservation expression for A within the macropores of the pellet is
139
Here, pseudo-steady state and equimolal counter-diffusion have been assumed. Also D, is indicated to be a function of the radial position R in the pellet. Note, however, that a more general formulation would include the term {dD,(R)/dR }dCJdR. An order of magnitude analysis indicates that this term is small with respect to the other terms in eqn. (6). The rate of disappearance of B can be expressed in terms of the rate of change of r, with time and the stoichiometricahy equivalent amount of gas A that has reacted, i.e.
From the last two parts of eqn. (7) the expression for the rate of change of the reaction interface is
(8)
The radius rg will be a function of t and R for those cases where there is shrinkage or swelling of individual particles. This radius can be expressed in terms of the densities of the reactant and product solids by a mass balance for the product layer: i
(moles of G formed)
= i (moles of B reacted)
or 3 fg -
rc3 = y(r$
-
rf)
where y is the molal volume ratio defined
(9) as
The parameter y determines the change in particle dimensions: if y < 1 the particles shrink during reaction; if y > 1 swelling occurs; for y = 1 no change in particle radius occurs. For non-isothermal conditions it has been assumed that the particles are small enough that temperature gradients in them can be neglected. However, both temperature and
concentration gradients can exist in the pellet. Under these conditions eqns. (6) and (8) are applicable provided that k is taken to be an Arrhenius function of temperature : lz=Aexp(-&)
(11)
The energy balance in the pellet, assuming pseudo-steady state heat transfer, a constant heat of reaction AH and a constant effective thermal conductivity k,, is k,
There is some justification for taking k, as independent of R while D, is considered to be a function of R. This is that the range of values for k, is rather limited, perhaps because the solid phase contributes to 12, but not to LX,. When the effective diffusivity is constant, it is possible to eliminate the energy balance (eqn. (12)) by using the Damkoehler relationship, which expresses temperature in terms of the concentration. However, for the general problem considered here such a simplification is not possible, and both mass and energy conservation equations must be solved simultaneously. 3.1. The effect of structural changes on the diffusivity The effect of structural changes on the effective diffusivity will be developed using the results of Kim and Smith [9] who studied the effect of sintering on the effective diffusivity in non-reactive systems. The starting point is the random pore model (RPM) [lo] which is based upon a probability approach to pore interconnections. According to the RPM, the effective diffusivity for a monodisperse unsintered porous pellet containing only macropores is given by ~,=!z=De2 Tf
(13)
since 71 = l/E
(14) and D is a composite diffusivity accounting for Knudsen and bulk diffusion. Equation (13)
140
may be used to correlate changes in D, due to changes in e caused by density differences (shrinkage or swelling). However, it is not adequate to account for the effects of sintering. As shown experimentally [ 91, sintering causes a much larger increase in the tortuosity factor than can be accounted for by changes in porosity alone. As an explanation it was postulated that sintering causes not only a decrease in porosity but also removal of some pore interconnections. The fractional increase g(@) in rf due to sintering was correlated as a function of the fraction I$of pores removed :
(@) Tf (0) Tf
(15)
&@I = -
where Tf($I) is the tortuosity factor when a fraction $Jof the pores are removed and ~~(0) is the tortuosity factor for $ = 0. The correlation for g(@) as a function of $Ias developed previously [9] is shown graphically in Fig. 2. A more complicated model for this effect of sintering has been developed [ 111, but the additional accuracy is perhaps not warranted for reacting systems. The changes in porosity due to changes in the particle radii rg can be expressed as l---f I
I..
t
-=-
‘g
\3
( 1
l--e0
(16)
r.
where E+is the hypothetical porosity which would have resulted if there were no sintering. The actual porosity due to both chemical reaction and sintering is E = E+(l -- $J) =
i
1-(1--e+-
(
3 (l-4) it
(17)
where the second equality follows from the RPM and eqn. (16). Then the tortuosity factor is given by Tf
=
g(41 f+(l- @)
(18)
Thus from eqns. (13), (17) and (18) the effective diffusivity is D
D, = g(rg)
(I-
(P)2 (19)
Equation (19) predicts how the effective diffusivity is affected by structural changes, during reaction, due both to density variations
FRACTION OF PORES REMOVEO, $
Fig. 2. The effect of pore interconnections tortuosity factors.
on the
and sintering. The only additional information required is the variation of I#Iwith time. This is modelled in this work as a first-order rate process governed by an Arrhenius-type equation
& ~=(l-$)A,
exp
1
(20)
where Es is the activation energy for sir-&ring and T, is a characteristic temperature corresponding to the onset of sintering. Generally T, will correspond to the Tamman temperature, which is approximately half the melting point of the solid. The model parameters introduced to account for the effect of sintering are A,, Es and T,. These can be most accurately estimated by independent sintering experiments on the product ‘solid, i.e. by measuring the porosity and the effective diffusivity of species G for various sintering times. 3.2. Method of solution The mass and energy conservation equations (eqns. (6) and (12)) have been solved numerically at a given time, using eqn. (19) for D,, to obtain temperature and concentration profiles. The variation of rc and Q with time was then obtained by solution of eqns. (8) and (20). This process was repeated for successive increments of time. The average conversion of reactant B for the whole pellet, as a function of time, can now be obtained by integration, i.e.
141
3 R2dR The boundary and initial conditions sary for these calculations are
dctt
= k&A,
De---
dR
-CA)
atR=R,
(21) neces-
(3) Activation
energy for sintering:
Es
(22)
and
Ar, = RsTo
(34)
(9) Dimensionless
Tamman
temperature
tic = 5
keg =hf (To - n
atR=R,
dC, -=dR
atR=O
(24)
r, = r-0
att=O
(25)
&J=O
att=O
(26)
dT
=o
dR
(23)
(1) Thiele-type
I
for sir&ring:
4w0
4. RESULTS
parameter: (28)
To&
(3) Arrhenius
parameter:
E Ar= R,To
(29)
(4) Biot number
for the reaction:
DG
(30)
roW’o) (5) Modified k,R, Sh = D% (6) Modified
The conversion versus time relationship was obtained for chosen sets of these parameters. The main study was directed to the role of the parameters accounting for structural changes, i.e. the molal volume ratio and the rate constant A, of the sir&ring process. The results are presented as plots of conversion versus dimensionless time, defined as (37)
(27)
I
Q-0
(-AfWe,,C~,
Nu=-
rate constant
t* = tMBCAobk(To)/Ptro
u2
(2) Heat of reaction
Bi =
(10) Dimensionless
modulus :
3k(l -co)
h=R,
(35)
To
a=
:
MBC.@dTo)
For convenience, the equations were cast in dimensionless form. The dimensionless parameters affecting the conversion-time relationship are then as follows.
P=
(33)
Sherwood
number: (31)
Nusselt number:
WL ke
(32)
The additional parameters needed to account for the effect of structural changes are as follows. (7) Molal volume ratio:
AND DISCUSSION
4.1. Iso thermal case For isothermal conditions the parameters B, Ar and Nu are not involved so that the conversion depends on h, Bi, Sh, y and the rate constant for sintering. The influence of h and Bi has been studied by Calvelo and Smith [6], neglecting structural changes. In this paper, we are interested primarily in the effects of 7 and the sintering parameters. The influence of y was evaluated over the range 0.5 - 2.5 for the specific case of h = 6.3 and Bi = 1.0. These values were chosen in the range of practical interest. The results, shown in Fig. 3, indicate that the time required to achieve a given conversion is reduced as y is reduced; i.e. the time required is less for a gas-solid system with a smaller numerical value of y. This is due to an increase in porosity and in effective diffusivity as the particles shrink (7 decreases). For the high y value of 2.5, the conversion of the solid approaches a maximum asymptotic value equal to 22.5%. This maximum occurs because the particles near the surface expand to such an extent that the macroporosity approaches zero, resulting
142
0.6 ,m g
0.5
& E B
0.4
0.1
0.1
I
0
0
0.1
0.2
0.3
0.4
OIUENSICNLESS
0.5 TIME
0.6
0.7
0.8
,
0.9
, t’
0 0
0.2
0.4
06
0.8
DIMENSIONLESS
1.0 TIME.
1.2
1.4
1.6
1.8
1’
Fig. 3. The effect of density changes on conversion for h = 6.3, Bi = 1.0, ~0 = 0.5, no sintering.
Fig. 4. The effect of sintering on conversion for (Y= 4.8 x log, Ar, = 2.3, 0, = 0.9 and 7 = 0.5, h = 6.3, Bi = 1.0, EO = 0.5.
in pore closure. Such asymptotic conversions can result for systems for which the molal volume of the product formed is much greater than that of the reactant. This phenomenon has been experimentally observed for the sulfation of CaO [12] and the hydrofluorination ofUOs[8]. The effect of sintering on the conversiontime behavior for an isothermal system is shown in Fig. 4 for y = 0.5. The sintering parameters were chosen as (Y= 4.8 X log, Ar, = 2.3 and Bc = 0.9. These values represent a case where 50% of the pores are removed for a dimensionless time of unity (t* = 1.0). The increase in relative tortuosity factor for this extent of sintering would be about 2.5, as is seen from Fig. 2. The curves in Fig. 4 show that sintering causes a large decrease in conversion. Since y is less than unity, the effects of density changes and sintering are in the opposite direction; density changes cause the porosity to increase, while sintering decreases E. If the effect of sintering dominates, the conversion can approach asymptotic values of less than 100% even though y is less than unity.
intermediate values for the activation energy and heat of reaction. The Thiele modulus h = 11.7 ensures that significant concentration and temperature gradients exist within the pellet at zero time. The effect of y alone (no sintering) is illustrated in Fig. 5 for the following practical values of the other parameters: h = 11.7, e. = 0.70, Bi = 1.6, Nu = 12.2 and Sh = 640. The temperature at the center of the pellet for these conditions is shown in Fig. 6. The conversion at any time is reduced as y is increased, and the center temperature decreases with reaction time, the decrease being particularly sharp for large values of 7. This decrease in temperature occurs after the initial temperature transients disappear. The initial conditions (t =, 0) are that the pellet and gas temperatures are equal. Hence for an exothermic reaction the center temperature would rise rapidly from T/T,, = 1.0 to a maximum value. After a short time period, the temperature would fall according to the curves shown in Fig. 6. The initial rise is not seen in Figs. 6 and 8 (or in Fig. 11) because the transient terms have been neglected in formulating the energy balance (eqn. (12)), i.e. the pseudosteady state assumption has been used. The effect of sinking is illustrated for values of 7 = 0.5,l.O and 1.5 in Fig. 7. Two values of the sintering parameter c1(4000 and 8000) are considered. The corresponding temperature profiles are displayed in Fig. 8.
4.2. Non-iso thermal case The effects of y and the sintering parameters were studied for a non-isothermal system for which the thermal reaction parameters are 0 = 0.11 and Ar = 8.6. These values correspond to
0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
DIMENSIONLESSTIME, 1'
Fig. 5. The effect of density changes on conversion for a non-isothermal system: h = 11.7, Bi = 1.6, Ar = 8.6, fl= 0.11, Nu = 12.2 and Sh = 640.
Cl
0.2
0.4
0.6
0,s
1.0
1.2
1.4
1.6
1.8
DlYEWStONLESSTIME, t'
Fig. 7. The effect of sintering and density changes on conversion for a non-isothermal case (h = 6.3, Bi = 1.6, p = 0.11, Ar = 8.6, Nu = 12.2, Sh = 640) for sintering parameter values Ar, = 1.58 and Be = 0.9.
1.32
1.18 1.24 1.16 1.22
1.20
1.18
1.16
1.14
1.12
1.10 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
OtYENSIONLESSTINE. t*
Fig. 6. The effect of density changes on temperature at the center of the pellet when no sintering occurs (parameter values as in Fig. 5).
The conversion- time-temperature relationship is affected in a more complex manner than can be attributed to the effects of diffusivity alone. The variation in diffusivity causes changes in the temperature profile in the pellet, as can be seen from Fig. 8. For large values of y the temperature drops significantly with time. This in turn results in a reduced rate of sintering. The effect is more pronounced at the higher sinteringrate (e = 8000), and for this case the conversion for y = 1.5 is
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
DIYENSIONLESSTINE, t*
Fig. 8. The effect of sintering and density changes on temperature at the center of the pellet (parameter values as in Fig. 7).
higher than that for y = 0.5 or 7 = 1.0. Figure 9 shows the fraction of the pores removed owing to sinking as a function of radial position in the pellet for various reaction times. Since only the solid product is assumed to sink, the fraction removed is a maximum in the region between the center and the outer surface of the pellet.
144 0.7
0.7
0.4
0.3
Exwrinwntal
l
/ 0.2
-
Predicted Y: 17.
’
Dab
(8)
Curr# IDI Bi =Z7, h = 7.1
0 0
0.2
0.4 RADIAL
0.6 POSITION,
0.8
1.0
Fig. 9. The fraction @ of pores removed owing to sintering vs. radial position in the pellet (parameter values as in Fig. 7).
5. APPLICATION
0.1
R / R, nn
V.”
0
400
16W
12QQ
600 TIME,
t, set
Fig. 10. Conversions for the hydrofluorination single pellets of UO2 at a bulk gas temperature 640 K.
of of
OF THE MODEL
The only system for which complete experimental temperature and conversion data are available, to our knowledge, is [ 81
575
I
I
I
I
UO,(s) + 4HF(g) -+ UF4(s) + 2H,O(g) Calvelo and Smith [6] analyzed the data on the basis of the particle pellet model, ignoring the effects of structural changes. However, the molal volume ratio of UFI to UOs is 1.7 (y = 1.7) so that significant changes in porosity would be expected as a result of reaction. In fact, the experimental data indicated that the conversion approaches a maximum asymptotic value of 63% for a reaction temperature To of 640 K. The maximum temperature attained in the pellet was 800 K and the average surface temperature was about 700 K, while the melting point of the product UF4 is 1233 K. Since the melting point is high, the effect of sintering is not expected to be significant for this system. Therefore, as an approximation, structural changes can be attributed solely to density changes. Results predicted on this basis are compared with the experimental data in Figs. 10 and 11. The parameter values for which predicted and experimental results most closely agree are h = 7.1, Bi = 7.7 and r. = 0.2 X 10m2cm, which correspond to h
= 65 X 1O-3 cm s-l
DG = 10e3cm2 s-l The corresponding values obtained for the best fit by Calvelo and Smith were k = 84.6 X 10e3 cm s-l, r. = 0.5 X 10m2cm and Do =
Q
200
4W
600
MI0 TINE,
l.wO
1. SIC
Fig. 11. Predicted and experimental temperature profiles for the hydrofluorination of UO2.
lo-* cm2 s-l. Figure 10 shows that the experimental conversion-time data are in good agreement with the theory. In particular the asymptotic conversion of 63% agrees well with the experimental results. There are some differences between predicted and experimental temperatures (Fig. 11). Predicted results are 20 to 40 “C lower than the experimental temperatures. In view of the temper-
145
ature level and the sensitivity of temperature to the model assumptions, these deviations are not unexpected. The higher Do obtained with our present model can be explained by the structural changes in the solid, i.e. the decrease in porosity (y > 1) which is accounted for in our model. In order to obtain the same rate of reaction at a lower porosity, an increased effective diffusivity through the product layer has to be used in the model. This larger Do suggests that the product layer is somewhat permeable, with a porosity of about 6%. It is noteworthy also that the present model gives temperature profiles which agree much more closely with the experimental results than the profiles predicted without considering structural changes.
Bi cA
‘Ai
C‘40
D D,
De0 DG
E
ES g
g(41 6. CONCLUSIONS
A model has been developed to account for the combined effects of sintering and density changes during the course of a non-catalytic gas-solid reaction. The effects of density changes are expressed in terms of an additional parameter y, the molal volume ratio of product to reactant solid. Sintering effects are interpreted in terms of removal of pore interconnections, and the rate of sintering is supposed to be an activated first-order process. Application to the hydrofluorination of uranium dioxide shows that the model more accurately predicts temperature profiles and conversions than a model which does not consider the effect of structural changes during reaction. In particular the proposed model can predict the asymptotic conversion observed for the system.
h
hf AH k
ke
kg MB MG
-dnA/d t
Nu R RS R, r0 rc
rg rA
NOMENCLATURE rB
A 4 Ar Ar, b
frequency factor in the Arrhenius equation frequency factor (rate constant) in the rate equation for sintering E/RgTo, Arrhenius number E,/R,To, Arrhenius number for sintering stoichiometric coefficient of the solid reactant B
SB
Sh t t*
T To
Da /kr& Biot number in the particle concentration of gas A at any point in the pellet concentration of gas A at the reaction interface within a particle concentration of A in the bulk gas molecular diffusion coefficient of A effective diffusivity of A in the pellet effective diffusivity at time t = 0 diffusivity of A through the product layer around the particles activation energy for the main reaction activation energy for the sintering process stoichiometric factor of the product G factor by which the tortuosity is increased owing to sir&ring R, Pk(l - eN%,ro) 1’2, Thieletype modulus heat transfer coefficient in the gas film heat of reaction per mole of A first-order reaction rate constant, cm s-l effective thermal conductivity of the solid gasfilm mass transfer coefficient molecular weight of B molecular weight of G rate of disappearance of A per particle hfR,/k,, modified Nusselt number radial distance in the pellet radius of the pellet gas constant initial radius of the particle radius of the unreacted core in the particle radius of a particle at time t rate of disappearance of A per unit volume of pellet rate of disappearance of B per unit volume of pellet reactive surface area per particle kgR,/De , modified Sherwood number time tM&Aobk(To)/ptro, dimensionless time temperature temperature of the bulk gas
146 TC
XB
Tamman temperature, corresponding to the onset of sintering conversion of the solid reactant B in the pellet
Greek symbols a! AOp,r,/MBC,Obk(T,),
constant P Y CO
E E+
fG @ ec Pt
PG Tf
dimensionless
REFERENCES
rate
for sintering
(-AH)D,OC,O/T,k,, heat of reaction parameter gpJ4, /bMBpc( 1 - EG), molal volume ratio initial porosity porosity at time t hypothetical porosity corresponding to no sintering porosity of the product layer G fraction of pores removed owing to sintering T iTo, dimensionless Tamman tempera .re d lsity of the solid,reactant B dt nsity of the solid product G to:tuosity factor
6 7 8 9 10 11 12
A. Calvelo and R. E. Cunningham, J. Catal., 17 (1970) 1. J. Szekely and J. W. Evans, Metall. Trans., 2 (1971) 1699. J. Szekely, W. H. Ray and T. K. Chuang, Chem. Eng. Sci., 28 (1973) 683. A. Rehmat and S. C. Saxena, Ind. Eng. Chem., Process Des. Dev., 15 (1976) 2. P. A. Ramachandran and J. M. Smith, Paper presented at the 83rd Nat1 Meeting, A.Z.Ch.E., Houston, March 1977. A. Calvelo and J. M. Smith, Proc. Chemeca’ 70, Butterworths, Australia, 1971, Paper 3.1. J. Szekely, J. W. Evans and H. T. Sohn, Gas-Solid Reactions, Academic Press, New York, 1976. E. C. Costa and J. M. Smith, A.Z.Ch.E. J., 17 (1971) 947. K. K. Kim and J. M. Smith, A.Z.Ch.E. J., 20 (1974) 670. N. Wakao and J. M. Smith, Chem. Eng. Sci., 17 (1962) 825. S. F. Chan and J. M. Smith, Indian Chem. Eng., 18 (1976) 42. M. Hartman and R. W. Coughlin, Znd. Eng. Chem., Process Des. Dew, 13 (1974) 248.
Effect of Sintering and Porosity Changes on Rates of Gas-Solid Reactions
P. A. RAMACHANDRAN Department
(Received
of Chemical
and J. M. SMITH Engineering,
University
of California,
Davis, Calif. 95616
(U.S.A.)
8 March 1977; in final form 17 June 1977)
Abstract
sintering. Szekely and Evans [2] and Szekely et al. [3] proposed an exponential decay in diffusivity with reaction but made no attempt to relate the diffusivity to the changes in pore structure. Rehmat and Saxena [4] studied the effects of density differences on the conversion-time relationship in terms of the sharp interface (shrinking core) model; Ramachandran and Smith [5] have analyzed the effects of density differences by focusing attention on the changes in dimensions of a single pore. For analyzing the combined effects of density differences and sir&ring, the particle pellet [6] or grain [7] concept provides an attractive basis. Structural changes can be expressed in simple geometric terms and nonisothermal cases are easily formulated. This paper employs such a model as a basis for establishing conversion and intrapellet temperature profiles as a function of reaction time. Costa and Smith [8] measured both temperature profiles and conversion for the hydrofluorination of uranium dioxide pellets, and these data are used to explore the suitability of the model.
During the course of most gas-solid non-catalytic reactions the solid undergoes structural changes caused primarily by swelling or shrinkage, due to the chemical reaction, or by sin tering. In this paper a theory is developed to account for the effects of such structural changes on the conversion-time relationship and on temperature profiles in the pellet. The theory is based on the particle pellet concept and accounts for differences in density between reactant and product solids and for changes of porosity and pore interconnections due to sintering. The rate of sintering is assumed to obey an Arrhenius-type equation. Predicted conversion-time and tempemturetime-position curves based upon the theory are compared with experimental results for the hydrofluorination of pellets of uranium dioxide.
1. INTRODUCTION
Many non-catalytic reactions occur under conditions such that the solid undergoes various structural changes during the course of the reaction. Such changes may be due to density differences caused by chemical reaction, which results in swelling or shrinkage of the individual particles that comprise the pellet, or to the effects of temperature which cause sintering. Only a few reports in the literature refer to the effects of structural changes, and particularly limited are papers that consider sintering. Calvelo and Cunningham [l] assumed that the changes in the effective diffusivity due to structural changes were a function of porosity. This procedure may account for the effects of density differences but is inadequate to account for the effects of
2. BASIS OF THE MODEL
The particle pellet model refers to an assembly of spherical non-porous particles (of uniform size at the beginning of the reaction) compressed into a spherical pellet. The reaction occurs by diffusion of gaseous reactant into the pellet to the reactive surface of a particle. Suppose the reaction is of the form A(g) + bR(s) --, gG(s) + P(g) As time progresses a layer of porous G builds up around the particles. Owing to the differences in stoichiometric coefficients and in the molal density of the product and the reactant, 137
138
the particles expand or shrink, resulting in a change in radius. These changes affect the macroporosity and hence the effective diffusivity. The additional structural changes which can occur are due mainly to the effects of temperature, namely sintering. Sintering can cause considerable reduction in the effective diffusivity due to removal of pore interconnections. An approximate predictive equation to account for this effect has been derived [9] and use of this will be made in the present work. Various assumptions are made in this model in order to keep the mathematics tractable. These are as follows. (1) The radius of the pellet does not change. (2) The particle shape is spherical initially and the particles retain their original shape during the course of the reaction. (3) Sintering affects the diffusivity in the macropores but not the diffusivity through the product layer formed around the particles. (4) Only the solid product (the product layer) is assumed to sinter. The original solid reactant does not sinter. (5) The effective thermal conductivity of the pellet is independent of the extent of reaction . (6) The analysis is restricted to first-order irreversible reactions with respect to the gaseous reactant, and to first-order reactions with respect to the reactive surface, according to the rate equation
- -dn, = kCA&
(1)
dt
Equation (1) implies that the reaction occurs at a sharp interface between the non-porous reactant and the porous product layer in the particles. With respect to the pellet radius reaction does not occur at a sharp boundary. (7) The temperature gradient within individual particles is neglected, but not the intrapellet gradients. The geometry of the model is shown in Fig. 1. 3. MATHEMATICAL PROBLEM
FORMULATION
OF THE
In order to develop equations for the diffusion and reaction of gas A in the pellet, it is first necessary to obtain the rate of reaction per unit volume of the pellet. The rate per particle
PARTICLE
PELLEl
Product Layer
Reactant
0
lc
‘. -tR
Fig. 1. Geometry of the pellet and particle: - - -, particle radius t-0 at t = 0 for y > 1; - -, particle radiusroatt=Ofory< 1.
can be obtained by considering the consecutive processes of diffusion within the product layer and the reaction at the sharp interface of area 47rrz. These individual rates are, for diffusion, --= dnA dt
4nrgrC DG rg -r,
(CA
-
CAi)
and, for chemical reaction,
-- ‘hi dt
= 47V:kCA,
(3)
Eliminating the interfacial Concentration CAi
where rC is the radius of the reaction interface and rg the radius of a particle at any time. The number of spherical particles per unit volume at time zero is 3(1 - eo)/4m$, where r. is the initial radius of the particles, and this number is assumed to be constant during the reaction. Hence, the rate of disappearance of A per unit volume of pellet is
In terms of eqn. (5) the mass conservation expression for A within the macropores of the pellet is
139
Here, pseudo-steady state and equimolal counter-diffusion have been assumed. Also D, is indicated to be a function of the radial position R in the pellet. Note, however, that a more general formulation would include the term {dD,(R)/dR }dCJdR. An order of magnitude analysis indicates that this term is small with respect to the other terms in eqn. (6). The rate of disappearance of B can be expressed in terms of the rate of change of r, with time and the stoichiometricahy equivalent amount of gas A that has reacted, i.e.
From the last two parts of eqn. (7) the expression for the rate of change of the reaction interface is
(8)
The radius rg will be a function of t and R for those cases where there is shrinkage or swelling of individual particles. This radius can be expressed in terms of the densities of the reactant and product solids by a mass balance for the product layer: i
(moles of G formed)
= i (moles of B reacted)
or 3 fg -
rc3 = y(r$
-
rf)
where y is the molal volume ratio defined
(9) as
The parameter y determines the change in particle dimensions: if y < 1 the particles shrink during reaction; if y > 1 swelling occurs; for y = 1 no change in particle radius occurs. For non-isothermal conditions it has been assumed that the particles are small enough that temperature gradients in them can be neglected. However, both temperature and
concentration gradients can exist in the pellet. Under these conditions eqns. (6) and (8) are applicable provided that k is taken to be an Arrhenius function of temperature : lz=Aexp(-&)
(11)
The energy balance in the pellet, assuming pseudo-steady state heat transfer, a constant heat of reaction AH and a constant effective thermal conductivity k,, is k,
There is some justification for taking k, as independent of R while D, is considered to be a function of R. This is that the range of values for k, is rather limited, perhaps because the solid phase contributes to 12, but not to LX,. When the effective diffusivity is constant, it is possible to eliminate the energy balance (eqn. (12)) by using the Damkoehler relationship, which expresses temperature in terms of the concentration. However, for the general problem considered here such a simplification is not possible, and both mass and energy conservation equations must be solved simultaneously. 3.1. The effect of structural changes on the diffusivity The effect of structural changes on the effective diffusivity will be developed using the results of Kim and Smith [9] who studied the effect of sintering on the effective diffusivity in non-reactive systems. The starting point is the random pore model (RPM) [lo] which is based upon a probability approach to pore interconnections. According to the RPM, the effective diffusivity for a monodisperse unsintered porous pellet containing only macropores is given by ~,=!z=De2 Tf
(13)
since 71 = l/E
(14) and D is a composite diffusivity accounting for Knudsen and bulk diffusion. Equation (13)
140
may be used to correlate changes in D, due to changes in e caused by density differences (shrinkage or swelling). However, it is not adequate to account for the effects of sintering. As shown experimentally [ 91, sintering causes a much larger increase in the tortuosity factor than can be accounted for by changes in porosity alone. As an explanation it was postulated that sintering causes not only a decrease in porosity but also removal of some pore interconnections. The fractional increase g(@) in rf due to sintering was correlated as a function of the fraction I$of pores removed :
(@) Tf (0) Tf
(15)
&@I = -
where Tf($I) is the tortuosity factor when a fraction $Jof the pores are removed and ~~(0) is the tortuosity factor for $ = 0. The correlation for g(@) as a function of $Ias developed previously [9] is shown graphically in Fig. 2. A more complicated model for this effect of sintering has been developed [ 111, but the additional accuracy is perhaps not warranted for reacting systems. The changes in porosity due to changes in the particle radii rg can be expressed as l---f I
I..
t
-=-
‘g
\3
( 1
l--e0
(16)
r.
where E+is the hypothetical porosity which would have resulted if there were no sintering. The actual porosity due to both chemical reaction and sintering is E = E+(l -- $J) =
i
1-(1--e+-
(
3 (l-4) it
(17)
where the second equality follows from the RPM and eqn. (16). Then the tortuosity factor is given by Tf
=
g(41 f+(l- @)
(18)
Thus from eqns. (13), (17) and (18) the effective diffusivity is D
D, = g(rg)
(I-
(P)2 (19)
Equation (19) predicts how the effective diffusivity is affected by structural changes, during reaction, due both to density variations
FRACTION OF PORES REMOVEO, $
Fig. 2. The effect of pore interconnections tortuosity factors.
on the
and sintering. The only additional information required is the variation of I#Iwith time. This is modelled in this work as a first-order rate process governed by an Arrhenius-type equation
& ~=(l-$)A,
exp
1
(20)
where Es is the activation energy for sir-&ring and T, is a characteristic temperature corresponding to the onset of sintering. Generally T, will correspond to the Tamman temperature, which is approximately half the melting point of the solid. The model parameters introduced to account for the effect of sintering are A,, Es and T,. These can be most accurately estimated by independent sintering experiments on the product ‘solid, i.e. by measuring the porosity and the effective diffusivity of species G for various sintering times. 3.2. Method of solution The mass and energy conservation equations (eqns. (6) and (12)) have been solved numerically at a given time, using eqn. (19) for D,, to obtain temperature and concentration profiles. The variation of rc and Q with time was then obtained by solution of eqns. (8) and (20). This process was repeated for successive increments of time. The average conversion of reactant B for the whole pellet, as a function of time, can now be obtained by integration, i.e.
141
3 R2dR The boundary and initial conditions sary for these calculations are
dctt
= k&A,
De---
dR
-CA)
atR=R,
(21) neces-
(3) Activation
energy for sintering:
Es
(22)
and
Ar, = RsTo
(34)
(9) Dimensionless
Tamman
temperature
tic = 5
keg =hf (To - n
atR=R,
dC, -=dR
atR=O
(24)
r, = r-0
att=O
(25)
&J=O
att=O
(26)
dT
=o
dR
(23)
(1) Thiele-type
I
for sir&ring:
4w0
4. RESULTS
parameter: (28)
To&
(3) Arrhenius
parameter:
E Ar= R,To
(29)
(4) Biot number
for the reaction:
DG
(30)
roW’o) (5) Modified k,R, Sh = D% (6) Modified
The conversion versus time relationship was obtained for chosen sets of these parameters. The main study was directed to the role of the parameters accounting for structural changes, i.e. the molal volume ratio and the rate constant A, of the sir&ring process. The results are presented as plots of conversion versus dimensionless time, defined as (37)
(27)
I
Q-0
(-AfWe,,C~,
Nu=-
rate constant
t* = tMBCAobk(To)/Ptro
u2
(2) Heat of reaction
Bi =
(10) Dimensionless
modulus :
3k(l -co)
h=R,
(35)
To
a=
:
MBC.@dTo)
For convenience, the equations were cast in dimensionless form. The dimensionless parameters affecting the conversion-time relationship are then as follows.
P=
(33)
Sherwood
number: (31)
Nusselt number:
WL ke
(32)
The additional parameters needed to account for the effect of structural changes are as follows. (7) Molal volume ratio:
AND DISCUSSION
4.1. Iso thermal case For isothermal conditions the parameters B, Ar and Nu are not involved so that the conversion depends on h, Bi, Sh, y and the rate constant for sintering. The influence of h and Bi has been studied by Calvelo and Smith [6], neglecting structural changes. In this paper, we are interested primarily in the effects of 7 and the sintering parameters. The influence of y was evaluated over the range 0.5 - 2.5 for the specific case of h = 6.3 and Bi = 1.0. These values were chosen in the range of practical interest. The results, shown in Fig. 3, indicate that the time required to achieve a given conversion is reduced as y is reduced; i.e. the time required is less for a gas-solid system with a smaller numerical value of y. This is due to an increase in porosity and in effective diffusivity as the particles shrink (7 decreases). For the high y value of 2.5, the conversion of the solid approaches a maximum asymptotic value equal to 22.5%. This maximum occurs because the particles near the surface expand to such an extent that the macroporosity approaches zero, resulting
142
0.6 ,m g
0.5
& E B
0.4
0.1
0.1
I
0
0
0.1
0.2
0.3
0.4
OIUENSICNLESS
0.5 TIME
0.6
0.7
0.8
,
0.9
, t’
0 0
0.2
0.4
06
0.8
DIMENSIONLESS
1.0 TIME.
1.2
1.4
1.6
1.8
1’
Fig. 3. The effect of density changes on conversion for h = 6.3, Bi = 1.0, ~0 = 0.5, no sintering.
Fig. 4. The effect of sintering on conversion for (Y= 4.8 x log, Ar, = 2.3, 0, = 0.9 and 7 = 0.5, h = 6.3, Bi = 1.0, EO = 0.5.
in pore closure. Such asymptotic conversions can result for systems for which the molal volume of the product formed is much greater than that of the reactant. This phenomenon has been experimentally observed for the sulfation of CaO [12] and the hydrofluorination ofUOs[8]. The effect of sintering on the conversiontime behavior for an isothermal system is shown in Fig. 4 for y = 0.5. The sintering parameters were chosen as (Y= 4.8 X log, Ar, = 2.3 and Bc = 0.9. These values represent a case where 50% of the pores are removed for a dimensionless time of unity (t* = 1.0). The increase in relative tortuosity factor for this extent of sintering would be about 2.5, as is seen from Fig. 2. The curves in Fig. 4 show that sintering causes a large decrease in conversion. Since y is less than unity, the effects of density changes and sintering are in the opposite direction; density changes cause the porosity to increase, while sintering decreases E. If the effect of sintering dominates, the conversion can approach asymptotic values of less than 100% even though y is less than unity.
intermediate values for the activation energy and heat of reaction. The Thiele modulus h = 11.7 ensures that significant concentration and temperature gradients exist within the pellet at zero time. The effect of y alone (no sintering) is illustrated in Fig. 5 for the following practical values of the other parameters: h = 11.7, e. = 0.70, Bi = 1.6, Nu = 12.2 and Sh = 640. The temperature at the center of the pellet for these conditions is shown in Fig. 6. The conversion at any time is reduced as y is increased, and the center temperature decreases with reaction time, the decrease being particularly sharp for large values of 7. This decrease in temperature occurs after the initial temperature transients disappear. The initial conditions (t =, 0) are that the pellet and gas temperatures are equal. Hence for an exothermic reaction the center temperature would rise rapidly from T/T,, = 1.0 to a maximum value. After a short time period, the temperature would fall according to the curves shown in Fig. 6. The initial rise is not seen in Figs. 6 and 8 (or in Fig. 11) because the transient terms have been neglected in formulating the energy balance (eqn. (12)), i.e. the pseudosteady state assumption has been used. The effect of sinking is illustrated for values of 7 = 0.5,l.O and 1.5 in Fig. 7. Two values of the sintering parameter c1(4000 and 8000) are considered. The corresponding temperature profiles are displayed in Fig. 8.
4.2. Non-iso thermal case The effects of y and the sintering parameters were studied for a non-isothermal system for which the thermal reaction parameters are 0 = 0.11 and Ar = 8.6. These values correspond to
0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
DIMENSIONLESSTIME, 1'
Fig. 5. The effect of density changes on conversion for a non-isothermal system: h = 11.7, Bi = 1.6, Ar = 8.6, fl= 0.11, Nu = 12.2 and Sh = 640.
Cl
0.2
0.4
0.6
0,s
1.0
1.2
1.4
1.6
1.8
DlYEWStONLESSTIME, t'
Fig. 7. The effect of sintering and density changes on conversion for a non-isothermal case (h = 6.3, Bi = 1.6, p = 0.11, Ar = 8.6, Nu = 12.2, Sh = 640) for sintering parameter values Ar, = 1.58 and Be = 0.9.
1.32
1.18 1.24 1.16 1.22
1.20
1.18
1.16
1.14
1.12
1.10 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
OtYENSIONLESSTINE. t*
Fig. 6. The effect of density changes on temperature at the center of the pellet when no sintering occurs (parameter values as in Fig. 5).
The conversion- time-temperature relationship is affected in a more complex manner than can be attributed to the effects of diffusivity alone. The variation in diffusivity causes changes in the temperature profile in the pellet, as can be seen from Fig. 8. For large values of y the temperature drops significantly with time. This in turn results in a reduced rate of sintering. The effect is more pronounced at the higher sinteringrate (e = 8000), and for this case the conversion for y = 1.5 is
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
DIYENSIONLESSTINE, t*
Fig. 8. The effect of sintering and density changes on temperature at the center of the pellet (parameter values as in Fig. 7).
higher than that for y = 0.5 or 7 = 1.0. Figure 9 shows the fraction of the pores removed owing to sinking as a function of radial position in the pellet for various reaction times. Since only the solid product is assumed to sink, the fraction removed is a maximum in the region between the center and the outer surface of the pellet.
144 0.7
0.7
0.4
0.3
Exwrinwntal
l
/ 0.2
-
Predicted Y: 17.
’
Dab
(8)
Curr# IDI Bi =Z7, h = 7.1
0 0
0.2
0.4 RADIAL
0.6 POSITION,
0.8
1.0
Fig. 9. The fraction @ of pores removed owing to sintering vs. radial position in the pellet (parameter values as in Fig. 7).
5. APPLICATION
0.1
R / R, nn
V.”
0
400
16W
12QQ
600 TIME,
t, set
Fig. 10. Conversions for the hydrofluorination single pellets of UO2 at a bulk gas temperature 640 K.
of of
OF THE MODEL
The only system for which complete experimental temperature and conversion data are available, to our knowledge, is [ 81
575
I
I
I
I
UO,(s) + 4HF(g) -+ UF4(s) + 2H,O(g) Calvelo and Smith [6] analyzed the data on the basis of the particle pellet model, ignoring the effects of structural changes. However, the molal volume ratio of UFI to UOs is 1.7 (y = 1.7) so that significant changes in porosity would be expected as a result of reaction. In fact, the experimental data indicated that the conversion approaches a maximum asymptotic value of 63% for a reaction temperature To of 640 K. The maximum temperature attained in the pellet was 800 K and the average surface temperature was about 700 K, while the melting point of the product UF4 is 1233 K. Since the melting point is high, the effect of sintering is not expected to be significant for this system. Therefore, as an approximation, structural changes can be attributed solely to density changes. Results predicted on this basis are compared with the experimental data in Figs. 10 and 11. The parameter values for which predicted and experimental results most closely agree are h = 7.1, Bi = 7.7 and r. = 0.2 X 10m2cm, which correspond to h
= 65 X 1O-3 cm s-l
DG = 10e3cm2 s-l The corresponding values obtained for the best fit by Calvelo and Smith were k = 84.6 X 10e3 cm s-l, r. = 0.5 X 10m2cm and Do =
Q
200
4W
600
MI0 TINE,
l.wO
1. SIC
Fig. 11. Predicted and experimental temperature profiles for the hydrofluorination of UO2.
lo-* cm2 s-l. Figure 10 shows that the experimental conversion-time data are in good agreement with the theory. In particular the asymptotic conversion of 63% agrees well with the experimental results. There are some differences between predicted and experimental temperatures (Fig. 11). Predicted results are 20 to 40 “C lower than the experimental temperatures. In view of the temper-
145
ature level and the sensitivity of temperature to the model assumptions, these deviations are not unexpected. The higher Do obtained with our present model can be explained by the structural changes in the solid, i.e. the decrease in porosity (y > 1) which is accounted for in our model. In order to obtain the same rate of reaction at a lower porosity, an increased effective diffusivity through the product layer has to be used in the model. This larger Do suggests that the product layer is somewhat permeable, with a porosity of about 6%. It is noteworthy also that the present model gives temperature profiles which agree much more closely with the experimental results than the profiles predicted without considering structural changes.
Bi cA
‘Ai
C‘40
D D,
De0 DG
E
ES g
g(41 6. CONCLUSIONS
A model has been developed to account for the combined effects of sintering and density changes during the course of a non-catalytic gas-solid reaction. The effects of density changes are expressed in terms of an additional parameter y, the molal volume ratio of product to reactant solid. Sintering effects are interpreted in terms of removal of pore interconnections, and the rate of sintering is supposed to be an activated first-order process. Application to the hydrofluorination of uranium dioxide shows that the model more accurately predicts temperature profiles and conversions than a model which does not consider the effect of structural changes during reaction. In particular the proposed model can predict the asymptotic conversion observed for the system.
h
hf AH k
ke
kg MB MG
-dnA/d t
Nu R RS R, r0 rc
rg rA
NOMENCLATURE rB
A 4 Ar Ar, b
frequency factor in the Arrhenius equation frequency factor (rate constant) in the rate equation for sintering E/RgTo, Arrhenius number E,/R,To, Arrhenius number for sintering stoichiometric coefficient of the solid reactant B
SB
Sh t t*
T To
Da /kr& Biot number in the particle concentration of gas A at any point in the pellet concentration of gas A at the reaction interface within a particle concentration of A in the bulk gas molecular diffusion coefficient of A effective diffusivity of A in the pellet effective diffusivity at time t = 0 diffusivity of A through the product layer around the particles activation energy for the main reaction activation energy for the sintering process stoichiometric factor of the product G factor by which the tortuosity is increased owing to sir&ring R, Pk(l - eN%,ro) 1’2, Thieletype modulus heat transfer coefficient in the gas film heat of reaction per mole of A first-order reaction rate constant, cm s-l effective thermal conductivity of the solid gasfilm mass transfer coefficient molecular weight of B molecular weight of G rate of disappearance of A per particle hfR,/k,, modified Nusselt number radial distance in the pellet radius of the pellet gas constant initial radius of the particle radius of the unreacted core in the particle radius of a particle at time t rate of disappearance of A per unit volume of pellet rate of disappearance of B per unit volume of pellet reactive surface area per particle kgR,/De , modified Sherwood number time tM&Aobk(To)/ptro, dimensionless time temperature temperature of the bulk gas
146 TC
XB
Tamman temperature, corresponding to the onset of sintering conversion of the solid reactant B in the pellet
Greek symbols a! AOp,r,/MBC,Obk(T,),
constant P Y CO
E E+
fG @ ec Pt
PG Tf
dimensionless
REFERENCES
rate
for sintering
(-AH)D,OC,O/T,k,, heat of reaction parameter gpJ4, /bMBpc( 1 - EG), molal volume ratio initial porosity porosity at time t hypothetical porosity corresponding to no sintering porosity of the product layer G fraction of pores removed owing to sintering T iTo, dimensionless Tamman tempera .re d lsity of the solid,reactant B dt nsity of the solid product G to:tuosity factor
6 7 8 9 10 11 12
A. Calvelo and R. E. Cunningham, J. Catal., 17 (1970) 1. J. Szekely and J. W. Evans, Metall. Trans., 2 (1971) 1699. J. Szekely, W. H. Ray and T. K. Chuang, Chem. Eng. Sci., 28 (1973) 683. A. Rehmat and S. C. Saxena, Ind. Eng. Chem., Process Des. Dev., 15 (1976) 2. P. A. Ramachandran and J. M. Smith, Paper presented at the 83rd Nat1 Meeting, A.Z.Ch.E., Houston, March 1977. A. Calvelo and J. M. Smith, Proc. Chemeca’ 70, Butterworths, Australia, 1971, Paper 3.1. J. Szekely, J. W. Evans and H. T. Sohn, Gas-Solid Reactions, Academic Press, New York, 1976. E. C. Costa and J. M. Smith, A.Z.Ch.E. J., 17 (1971) 947. K. K. Kim and J. M. Smith, A.Z.Ch.E. J., 20 (1974) 670. N. Wakao and J. M. Smith, Chem. Eng. Sci., 17 (1962) 825. S. F. Chan and J. M. Smith, Indian Chem. Eng., 18 (1976) 42. M. Hartman and R. W. Coughlin, Znd. Eng. Chem., Process Des. Dew, 13 (1974) 248.