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Reaction of microporous solids: The discrete random pore model
Carbon Vol. 34, No. 11, pp. 1383-1391, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 000%6223/96 $15.00 + 0.00
Pergamon PII: SOOOS-6223(96)00080-2
REACTION OF MICROPOROUS SOLIDS: THE DISCRETE RANDOM PORE MODEL S. Department
of Chemical
K.
Engineering,
BHATIA*
Indian
B. J. VARTAK
and
Institute of Technology, India
Bombay
Powai, Bombay
400 076,
(Received 1 May 1996; accepted in revised form 20 May 1996)
Abstract-The universally used assumption of proportionality between reaction rate and pore surfacearea is modified for application to solids dominated by micropores, where it is shown that the discreteness of the solid phase necessitates some corrections. A new discrete random pore model, which modifies the earlier random pore model results, is proposed for such solids. An exact method and two other approximate methods for predicting the evolution of pore volume distribution are discussed in light of the discrete random pore model. It is observed that the structural parameter characterizing reactivity is effectively reduced in value due to the discreteness of the solid. The new results are used to interpret the difference between reaction rate and structurally determined values of this parameter reported in the literature for the char steam reaction. The importance and need for the development of better molecular models for such systems is stressed. Copyright 0 1996 Elsevier Science Ltd Key Words-Gas-solid
reactions,
gasification,
microporous
SE’
NOMENCLATURE
A
r” g k
ks
L 1 LCI LE L EO LG I6 P” r r0 rAV
S s S(X) So SE S*(x) S EO
carbons,
s
normalization constant Lennard-Jones molecular size parameter pore volume size distribution (PVSD) fin terms of I., 8 rate constant of reaction pore-size independent growth rate total length of pore axes per unit volume pore length total initial pore segment length per unit volume total length of pore axes of non-overlapped system per unit volume L, at t=O total initial pore segment length per unit mass number of layers reacted approximate continuous probability function exact discrete probability function radius of the pore Y at t=O mean pore size of the distribution surface area per unit volume standard deviation used for log normal distribution surface area per unit volume at conversion X initial total surface area per unit volume total surface area of non-overlapped system per unit volume
S;
*Corresponding author. Present address: Department of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia. 1383
corrected surface area corresponding intermediate contour at (r + a/2) SE’ at t=O
to the
SE/SE0 total initial surface area per unit mass time volume of pores per unit volume V volume of pores per unit volume v total volume non-overlapped system per unit VE volume VEO VE at t=O VE(>P) VE for all pores with size greater than p instantaneous mass of solid reactant W initial mass of solid reactant K X conversion Greek Symbols discreteness parameter (eqn (13)) dimensionless length of pores per unit volume ; PO(P) initial pore length distribution /I in terms of A, e Bl porosity E E at t=O CO dimensionless pore size (eqn (46)) I e dimensionless time (eqn (47)) dimensionless pore size, r/g P p at t=O PO parameter for the Rayleigh distribution PO0 mean p of the distribution PAV true density PT radial interlayer spacing between atoms of osolid phase dimensionless time t structural parameter, (eqn (5)) effective structural parameter (eqn (14))
SG t
S(X)/So S, at t=O
’
pore structure.
S. K. BHATIA and B. J. VARTAK
1384 1. INTRODUCTION
Since its development more than 15 years ago, the random pore model [ 1,2] has found extensive application in the interpretation of reaction-rate data for gas-solid reactions, as well as in theoretical analyses. Brief accounts of many of the applications have appeared in recent reviews [3,4]. Arguably its most important, and certainly the widest, application has been in the correlation of reactivity data for chars and carbonaceous materials. These applications have been prompted by the ability of the model to represent the often-observed rate maximum with increase in conversion, without the use of arbitrary adjustable additional parameters. This is principally due to the competitive mechanisms of surface-area increase and loss related to pore growth and intersection respectively, as reaction occurs, which are inherent to the overlapping pore concept assumed in the model. Considering the usual assumption that the reaction rate in any pore is proportional to its associated surface area, i.e: dv - =Sd’ =Sk, dt dt where k, is a pore-size independent random pore yields the expressions,
SW
~
(1) growth [ 11:
rate, the
[7,8] have also reported the values of $ from structural characterization for a large number of chars, their interest was in studying the use of $ as a characterization parameter rather than in testing the model. Consequently, the studies of Perlmutter and coworkers, who compared the values of $ estimated for different chars at various temperatures for the char-O, reaction, [S] and the char-steam reaction, [6] with those from structural characterization, perhaps constitute the most definitive tests of the model. However, while both studies showed no systematic variation of the reactivity-based value of + with temperature, the latter investigation showed the average value of this parameter to be about 35-40% lower than that evaluated from structural characterization. Table 1 presents the structural parameters and the two independent estimates of $ for the two chars, reported by Chi and Perlmutter [6], illustrating this difference. On the other hand, no such significant difference was reported by Su and Perlmutter [S] for the earlier study of the char-O, reaction. It may be mentioned here that the structurally determined ci/ values reported in Table 1 were calculated by Chi and Perlmutter [6] based on the relation: $=
=(l-X)dl-$ln(l-X)
4xL, ~
(7)
%T
(2)
S0 dX -= dt
k,S, -(l-X)fi-$ln(l-X) (1 --E0) X(r)= I -exp[
in which the parameter defined as: *=
(3)
-(T + $t2/4)] cl, is a structural
47cLE0 7
(4) constant,
(5)
Ai0
Here L,, is the total length of the pore axes per unit volume, and S,, a hypothetical total initial surface area per unit volume, if the pores were not overlapping, given by:
se,=
so
7
(1 -%I)
While the list of published applications of the above to chars and carbonaceous materials is rather extensive, it is pertinent to mention here that most of the studies have involved verification of the ability of eqn (3) to correlate reaction rate data, and thereby empirical estimation of the structural parameter $. More definitive tests of the model would also involve independent evaluation of the parameter $ from pore structure characterization data, and only a few comprehensive studies of this kind have been reported [ 5,6]. Such studies are particularly important in view of interest [ 7, S] in considering $ as a characterization parameter for char reactivity. While Miura et a/.
and the reaction rate-determined values of $ are those reported from their nonlinear fit of eqn (3) to rate data, as the linear fit using the transformation of this equation was sensitive to scatter at high conversions and reportedly less reliable. One possible source of error in applying the random pore model eqns (2)-( 6) to chars is that these are highly microporous materials, with pore sizes typically only a few multiples of the spacing between the reacting sites in the char. For example, the modal pore radius of the chars used by Chi and Perlmutter [6] was reported to be about 6 8, which is comparable to the inter-layer spacing of about 3.35 A in graphite [9]. Under these circumstances the concept of continuous motion of the interface is clearly inappropriate, and the molecularity or discreteness of the reacting species must be considered.
Table 1. Char
structural parameters reported Perlmutter [6]
by Chi and
Coal char Parameter Specific surface area, SG (m’g-‘) Total specific pore length, L(mg-‘) I++(pore structure determined) $ (reaction rate determined)
PSOC-3-10
PSOC-3-20
157.1
174.8
3.69 x IO”
4.18 x 1O’O
12.94 8.52
12.54 7.06
Reaction
of microporous
2.1 Reaction in a cylindrical micropore the gasification
reaction:
A(s)+bB(g)-+products(g) occurring in a cylindrical pore of radius r and length 1, in a solid A comprised of spherical molecules, as depicted in Fig. 1. We further assume that the radial inter-layer spacing in the solid is cr, and that reaction in the pore effectively occurs by the layerwise radial erosion of the solid phase. Upon consumption of a single layer the change in the volume of the pore is given by: Au = nl [(r + a)” - r’] = 2&(r
1385
2.2 Reaction in an overlapping pore structure 2.2.1 Overall behaviour. Here we modify
2. THEORY
Consider
solids
+ a/2) = 2rcI(r + 5/2)Ar
(8) where Ar=a is the corresponding change in pore radius. In the continuum limit, we write k,=dr/dt as the growth rate, and eqn (8) yields:
eqns (l)-(6) in light of eqn (9). Since it is assumed that reaction occurs by radial erosion only, the total length of the pore axes LE is unchanged and is given by: (10)
0
The surface area of the system SE is given by:
non-overlapped
pore
SE = 27~7 r( t,ro)L(ro) dr,
(11)
0
We recognize that for eqns (2) and (5) to be applicable to the discrete solid, S(X) and So must correspond to the intermediate contour in Fig. 1. In light of eqn (9) the non-overlapped area corresponding to this intermediate contour is given by: S,‘=2n%
do - =2nl(r+o/2)k,
= 7 I@,,) dr,
L,=L,,
[r(t,r,)fc/2]
L(r,)dr,
(12)
dt
= s, + asa, Comparison with eqn (1) shows that the area S associated with the pore reaction rate is not the true pore surface area 2nrl, but in fact that corresponding to the intermediate contour at (r+o/2). This latter surface will, in general, not pass through the molecular centers, since the inter-layer spacing c~ may be different from the molecular size &, depending upon the packing geometry. For example, for tetrahedral packing, 0 = ad,. It is also evident that the correction a/2, having magnitude of about 2 A, is not negligible in comparison to the pore radius r (which, for micropores, is also in the region of a few A).
where c( is a discreteness
parameter,
defined as:
RoLEO -
c(=
(13)
S EO We may now define an effective structural ter I+&using eqns (12) and (13), as: 47&o l//E= s EO
= -!L ( 1 + a)’
parame-
(14)
where $ is the measured structural parameter defined by eqn (5). The factor LYaccounts for the finiteness of the size of the reacting solid units, and I,&-+$ as a-+0. Following the basic relationship of the random pore model [l] we now write, for any overlapped surface: S=S,(l-c)
(15)
We define conversion by X =( Wo- W)/W, which, in term of porosities, is represented as: X=
E--E0 ~ (l-co)
(16)
This expression for the conversion is clearly approximate for a discrete solid. More exact expressions for such a case may also be considered, but it is readily established that eqn (16) should be accurate for most practical purposes. For the original random pore model eqns (2) (6) (15) and (16) readily yield: SE = SE0 (VI-*ln(l-X)
Fig. 1. A pore of radius I surrounded of a solid. The interlayer
by spherical molecules spacing is 6.
>
(17)
which must actually be applied to the non-overlapping surface area corresponding to the intermediate contour in Fig. 1. Therefore the expression for the
S. K. BHATIA and B. J. VARTAK
1386
discrete
random
pore model is:
SE’= S,,’ (\i’l-$Eln(l-X)
(18) 1
Equations
(12), ( 13) and ( 18) combine
We characterize pore structure by the length distribution of the pore axes, and the pore volume size distribution. We can define the instantaneous length distribution of the pore axes as: mt(0) c P,(Wo(P
to give &(r,p) =
S,=SEO
(l+r)ql-$,ln(l-X)-x
(19) 1
1 which, following
eqns (15) and (16), yields
S(X)=S,(l-Xx)
(l+a)ql-$,ln(l-X)-a 1
1
where L,(p) is the initial length distribution. It should be noted that as n can assume only integral values. the upper limit of the summation is the integer part of p. We define a dimensionless length distribution /j(~,p) given by: B(?P)
(20) Also, eqn (3) must be rewritten
following
as:
(28)
- n)
I!=0
(29)
= na3L(?P)
which eqns (27) and (28) combine
to yield:
dX = k,S,,‘( 1 -X) dt which combines dX -= dt
with eqns (12) and (15) to yield:
k,S, -----(l-x)(1+a) (1-G)
cdl-&ln(l-X)
(22) Equations (20) and (22) represent a more general result replacing the original random pore model equations (eqns (2) and (3)). They may also be derived independently of the latter from first principles, as in the earlier work [ 11. Comparison with the original model equations shows that while the functional dependence of the reaction rate on conversion is unchanged, albeit with the effective structural parameter tiE, the form for the pore surface area variation is modified. 2.2.2 Evolution of pore structure. Following Fig. 1 we first consider a single pore of initial radius r,,, in which n layers have reacted, so that: r=r,+ncr which yields, in dimensionless
(23) form: (24)
P=Po+n
Upon defining P,(t) as the probability that n layers have reacted in time, t, we can write the equation governing its evolution:
df’n
-
dt
==kP,_,-kP,
where k is a suitable rate constant for the layerwise reaction. In dimensionless form, eqn (25) yields
This expression can be used to obtain the length distribution of the pores after any time 7, given an initial distribution /IO(p).We can then quantify the evolution of pore volume size distribution (PVSD). Upon defining the non-overlapped volume corresponding to pores larger than p by:
VE(>,>,=% P' WZ>P')dp’ we obtain
the corresponding
(31)
pore volume as:
V l>p)= 1 -ev
(- vE(,p,)
(32)
following the random pore model [Z]. Equations (3 1) and (32) provide the dimensionless PVSD .f(p) = P’B(~P)
exp (-
V,, > piI
(33)
which holds at any time T, given an initial distribution Be(P). 2.2.3 Approximations. In the foregoing analysis, we have made the simplifying assumptions of negligible diffusional resistance and a uniform time-independent rate. When such assumptions are not applicable, the analytical result in eqn (27) does not hold and approximations to the exact model need to be worked out. One of the approximations considered here hinges upon the representation of the discrete probability function by a continuous one. This approximation can be called the “continuous approximation to the exact model” (CAEM), and is obtained by rewriting eqn (27) in a different form to make it continuous, as: ~“-“~exp(-z) P(GPlPo)
(26) following which is readily solved to yield the Poisson tion: P”(T) = since Pn(0)=
6,,.
rnexp
(- z)
?I!
Vn20
=
P,(t) =
(34)
UP-Po+l)
which eqn (30) is rewritten
as:
distribuB(LP)
(27)
=
j: P(T>P/P,Mo(P,) 0
dpo
(35)
upon approximating the summation by the integral. Use of eqn (34) in eqn (35) would give the length distribution of the pores using CAEM. The PVSD
Reaction
of microporous
would still be predicted by eqn (33) except for the fact that now b(7,p) would have to be evaluated using eqn (35) instead of eqn (30). The second approximation addressed in this article uses a partial differential equation (PDE) to evaluate the probability function, and is called the partial differential equation model (PDEM). In this approach we rewrite eqn (26) as:
1387
as the initial length distribution of the pores. In eqns (42) and (43), A is a normalization constant satisfying: $ AP$(PO)
dpo= Go=
-ln(l
-co)
P(z,p-
=
l/PO)-
P(74dPO)
dt
-ln(l-Eo)
(45)
s P;P(Po) dpo
(36)
0
Equation (36) can be converted to a PDE in P(7, p/p& by using a second-order Taylor series approximation for P(7,p - l/p,). The resulting PDE is: ap(7dPO)
W7dPO) =-
+
!
azp(7dPO)
2
ap
3%
Once the initial distributions have been specified, the models can be used to obtain the length distribution B(7, p) and the PVSD f(z,p) of the pores after any time 7. However, for purposes of presentation it is instructive to express the results in terms of the dimensionless variables:
(37) The conditions
for the solution
P
r
i=-_-
of eqn (37) are:
rAV
P(O,PlPo) = 0
VP ’ PO
(38)
vz
(39)
P(z,co/p,)=0
(46)
PAV
and: ,=k”=i
7 P(~,plpo)dp= 1 V7 (40) PO Solving eqn (37) using initial condition eqn (38) and boundary conditions in eqns (39) and (40) we obtain:
exp(P -
P(GPlP0) =
2n
po) s[
(P - POS ~ -1-(P-PO) t
0
x exp
(P-Po)2 ~ 2t
[
t - 2
1.
t3P
clt
1 (41)
Use of eqn (41) in eqn (35) yields the length distribution of the pores using PDEM. As before, eqn (33) will predict the PVSD using the PDEM if P(z,p/p,,) is evaluated using eqn (41). 3. RESULTS
3.1 Evolution
(44)
yielding: A=m
dP(wltd
a7
solids
(47)
rAV
leading tion:
to a modified
PAV
dimensionless
k(W)
length
= Pi”B(7,P)
distribu-
(48)
and pore volume distribution: g(kn)=
(49)
pA"f(7d
Figure 2 compares the results for Pi(n) from the exact method with those from the proposed approximations, using a Rayleigh distribution. The CAEM proves to be a better approximation, with the results more closely matching the exact result. A similar
0.6
AND DISCUSSION
ofpore
structure
The models proposed require an initial distribution of the pores to be specified. The results presented here assume either a Rayleigh distribution:
Po(po)=A
(p-poo)exp{-~(~~)
(Pa” - Poo)2
or a log-normal
distribution:
(42)
-
[In(poJpA~)]
PEPS)=
stspo
exp
2s2
i
1 (43)
1.0
Dimensionless
2.0
3.0 pore size (h)
Fig. 2. Comparison of results for PI(n) from the exact method with those from the CAEM and the PDEM. A Rayleigh initial distribution is assumed, with parameters poO= 1, p*“=2, Eg=0.2S, 8=1.
S. K. BHATIA and B. J. VARTAK
1388
trend can be observed for the PVSD in Fig. 3. In what follows we report results from computations using the exact method. The earlier continuum random pore model assumed c to be 0 (no discreteness) which is equivalent to the condition pAv+ co. Figure 4 (log-normal initial distribution) and Fig. 5 (Rayleigh initial distribution) depict the effect of the discreteness parameter pAv on the dimensionless length distribution PI(i), with the dashed curve representing the continuum model (P~~-‘co). It is evident that at a sufficiently high pAv (> 50) the earlier random pore model is accurate enough. However, at smaller pAv the discrete random pore model must be applied for more reliable results. In the case of the PVSD results also, as shown in
__ _ cl
Random Pore Model
0.8
Dimensionless
0.8 /\
------- CAEM --_ PDEM
0.6
2.0
1.0
0.0
__
Exact method
1.0
0.0
i 1 / ,
2.0
Dimensionless
3.0
--_
~,=0.25,
4.0
Fig. 5. Effect of pAVon b,(A). The earlier random pore model prediction is shown as a dashed curve for comparison. A Rayleigh initial distribution is assumed, with parameters poO= 1, ~~=0.25, O= 1.
1 \ \
4.0
5.0
pore size (h)
Fig. 3. Comparison of PVSD results from the exact method with those from the CAEM and the PDEM. A Rayleigh initial distribution is assumed, with uarameters P,,“= “” 1. pAV=2,
3.0 pore size (1)
o= i.
Fig. 6 (log-normal initial distribution), the convergence of the curves to the dashed curve is visible as pAV increases. Figures 4 and 5 reveal another interesting feature. At smaller pAV, the PI(i) curves are not smooth and kinks can be observed. These kinks are smoothened out as pAv increases. This is a consequence of the finite thickness layerwise reaction concept assumed in the discrete random pore model. These kinks are not very prominent in the PVSD plots due to the smoothening effect of the integral involved in the computations in eqn (31) and eqn (33). The effect of 0 on fil(l) is depicted in Fig. 7 for a Rayleigh initial length distribution with pAv = 2. As before. the random pore model results are shown as 0.4 -
Random Pore Model
Random Pore Model
0.3
G 2
0.2
L 0. I
1.0
2.0
Dimensionless
3.0
4.0
pore size (L)
Fig. 4. Effect of pAv on ,0,(,?).The earlier random prediction log-normal
0.0
pore model is shown as a dashed curve for comparison. A initial distribution is assumed, with parameters s=OS, ~,=0.25, B=l.
1.0
2.0
Dimensionless
3.0
4.0
pore size (h)
Fig. 6. Effect of pAV on PVSD profiles. The PVSD profile predicted by the earlier random pore model is shown as a dashed curve for comparison. A log-normal initial distribution is assumed,
with parameters
s = 0.5, q, = 0.25, 0 = 1.
Reaction 1.2
of microporous
-
Random Pore Model
-
1.0
2.0
Dimensionless
3.0 pore
size
(h)
Fig. 7. Effect of 8 on PI(I). The results predicted by the earlier random pore model are shown as dashed curves for comparison. A Rayleigh initial distribution is assumed, with parameters paa = 1, pAv = 2, fzO= 0.25.
dashed profiles. It is obvious that the random pore results deviate from the exact discrete pore model results as B increases, indicative of the importance of a discrete analysis for microporous solids having a small pAV. The latter is also evident in Figs 5-7. Figure 8 shows the PVSD results as 0 increases, for a log-normal initial length distribution, again clearly indicating that the more accurate discrete random pore model should be applied for microporous solids. 3.2 SigniJicance of mod$ications The essential feature of the difference between the discrete random pore model (eqns (13) (14), (20) and and the earlier random pore model (21)) (eqns (2)-(6)) lies in the introduction of the new 0.6 - -
solids
1389
parameter c(. For cc-+0 the two models become identical. From the definition of a in eqn (13) and considering SE,, as 2nr,,L,, where rAV is a characteristic mean initial pore radius, one estimates that a = a/2r,,. Thus, the parameter a may be construed as representing the relative importance of the discreteness of the microporous solid. As an order of magnitude estimate, if we consider the reasonable values of 0 cr=4A, SE,=250 m2cme3 and $=lO, eqn(13) provides a=O.25. This would seem to be a significant correction in eqns (20) and (22), since a is not negligible in comparison to unity in the terms involving the factor (1 +a). For these parameter values we also estimate that I,& =6.4, which is considerably smaller than the value Ic,= 10 based on measured pore structure characteristics, underscoring the relevance of the proposed correction. For highly microporous chars, as well as activated carbons, SE,, will often exceed 1000 m2 cm-’ and the value of a may approach or even exceed unity, making the present modification even more significant. On the other hand, for relatively macroporous materials, typified by surface areas of the order of 10 m2 cmm3, the present corrections would be insignificant. 3.3 Model behaviour Figure 9 depicts the variation of the dimensionless pore surface area with conversion, at $= 10, for various values of a, as predicted by eqns (14) and (20). The result for a= 0 corresponds to the earlier random pore model, and with increase in a the value of the surface area is reduced. At the same time the position of the surface area maximum is shifted to a lower conversion, an effect which is even more readily evident for the lower value of 11/= 3, as seen in Fig. 10. In this case the surface area is monotonic for a= 1, with no initial increase. Figure 11 shows a comparison of the dimensionless surface area and the reaction rate variation with conversion, for $ = 10 and a =0.5,
Random Pore Model
a=0
1.0
2.0
Dimensionless
3.0
0.50
-
0.25
-
4.0
pore size (h)
Fig. 8. Effect of B on PVSD profiles. The PVSD profiles predicted by the earlier random pore model are shown as dashed curves for comparison. A log-normal initial distribution is assumed, with parameters s = 0.5, E,, = 0.25, pAv = 2.
I 0.00
I
I
I
I
\I
0.2
0.4
0.6
0.8
1.0
Conversion Fig. 9. Effect of u on the 5(X)/S(O) *=10.
(X) profiles,
with measured
1390
S. K. BHA~IAand B. J. VARTAK Table 2. Estimated values of i and (r Coal char Parameter
PSOC-3-10
PSOC-3-20
0.232 3.15
0.333 4.43
i. a(A) o^ 0.75 ;; A x, * 0.50
forward to estimate c( using eqn (14), and subsequently 0 using the S, and L, values in Table 1 along with eqn (7) and the modified form of eqn (13):
0.25
0.00
I
I
0.2
0.4
I 0.6
Conversion
I
0.8
1.0
(X)
Fig. 10. Effect of 61on the S(X)/S(O) profiles, with measured *=3.
. /_--~ 4 \ 1.50
__
,.25
S(X)/S(O)
/T
rate ‘;;i;
(O)
1
/’
‘,
1.00
,
\
s
z
= x 0.75 VI.
\
\
\\
\
\
\
\
0.50
\
\\
Table 2 lists the values of c( and u thus estimated for the two chars. The values of 0 are remarkably close to the 3.35 A inter-layer spacing of graphite, or the Lennard-Jones size parameter for the C-C interaction, indicative of the viability of the present approach. However, it should be emphasized that our findings are based on a small data set and much more carefully obtained experimental data needs to be analyzed. In this regard, it may be mentioned that the results of Su and Perlmutter [5] do not show similar differences in $, and are therefore not subject to the current interpretation. While this may suggest that CI<<1 for the C-O, reaction with the chars used by them, another complication arises from the fact that in both of these studies, and most others concerning the random pore model, the initial structural parameters are estimated using the expressions:
0.25
t
\ I
0.00
0.2
I 0.4 Conversion
I 0.6
I
0.8
1.o
L,=L,(
1 -Ed)=
7 &(r) 0 ?l?
(X)
Fig. 11. Comparison of S(X)/S(O) and rate(X)/rate(O) curves, with measured $ = 10 and a = 0.5
following eqns (20) and (22), respectively, as well as that predicted by the earlier random pore model. With the discrete random pore model, the predicted surface area and reaction rate variation are no longer identical, with the latter curve being lower. Careful experimentation is necessary to verify these effects. As such, the reaction-rate variation still has the same form as the original model, but corresponds to that for the lower value $E.
(52)
which ignore intersections. While they were indeed originally suggested by Bhatia and Perlmutter [ 11, alternate expressions based on application of the random pore model to the initial structure as well have been derived by Gavalas [2] and should be used. These provide: ru
s
S,=(l-Eo)
0
r
2f(4 -
1
7I
dr
f(s) ds
e
fIrI L,=
3.4 Application to experimental data As a direct application of the new results we may attempt to reconcile the difference between the structurally determined and reaction rate determined values of $ reported by Chi and Perlmutter [6], for the char-steam reaction, and presented here in Table 1. Considering the reaction rate determined values instead as I/&, following eqn (22), it is straight-
dr
1
1
(53)
dr
(54)
0
I and future applications of the model, particularly the discrete random pore model, need to be conducted using these relations. This comment also holds for the published S(X) data, which is invariably based on eqn (51) instead of eqn (53), and therefore not analyzed here.
1391
Reaction of microporous solids
3.5 Further directions While we have accounted for the discreteness of the solid phase in the framework of the random pore model, it is clear that this approach is equally applicable to other models such as that due to Petersen [lo]. In all of these it would suffice to consider the reaction area as that corresponding to the intermediate contour at (r + a/2) in Fig. 1, in place of the actual pore surface area at Y. Thus, application of the approach using other structural models is an open area for study. It is also clear that molecular models of discreteness must recognize the possibility of anisotropic reactivities, a well-known feature of graphite which is also related to the frequently observed appearance of etch pits [ 111. However, it is not clear as to what extent these findings for graphite apply to microporous chars and other carbonaceous solids, since the structure of the latter and its relation to reactivity is currently subject to much debate [ 12,131. Even the discrete random pore model utilizes the discreteness parameter 0 which may also vary with the nature of the reactant gas and the reaction mechanism. Thus, r~ may vary for carbons and also differ for gasification by oxygen, steam or CO,, leading to small differences in the normalized rate curves. Such differences, if determined by careful and accurate experiments, may also help in understanding the reaction chemistry.
Acknowledgements-Part of the work reported here, and the ideas discussed, originated from research conducted by one of the authors (S.K.B.) while he held a visiting appointment at the Institute for Kemiteknik, Technical University of Denmark. during 1994. The supDort of his stay by ’ the Danish ReseaFch Council i’sA gratefully acknowledged.
REFERENCES 1. S. K. Bhatia and D. D. Perlmutter, AIChE J. 26, 379 (1980). 2. G. R. Gavalas, AZChE J. 26, 577 (1980). 3. M. Sahimi, G. R. Gavalas and T. T. Tsotsis, Chem. Eng. Sci. 45, 1443 (1990). 4. S. K. Bhatia and .I. S. Gupta, Rev. Chem. Eng. 8, 177 (1992). 5. J. L. Su and D. D. Perlmutter, AZChE J. 31, 973 (1985). AZChE J. 35, 1791 6. W. K. Chi and D. D. Perlmutter, (1989). I. K. Miura, K. Hashimoto and P. L. Silveston, Fuel 68, 1461 (1989). 8. K. Miura, M. Makino and P. L. Silveston, Fuel 69, 580 (1990). 9. Steele, W. A., Properties of Interfaces, In The International Encyclopedia of Physical Chemistry and Chemical Physics (Edited by D. H. Everett), Pergamon, Oxford (1974). 10. E. E. Petersen, AIChE J. 3, 443 (1957). 11. R. T. Yang and R. Z. Duan, Carbon 23, 325 (1985). 12. R. I. Ben-Aim, Znt. Chem. Eng. 27, 70 (1987). 13. J. G. Bailey, A. Tate, C. F. K. Diessel and T. F. Wall, Fuel 69, 225 (1990).
Pergamon PII: SOOOS-6223(96)00080-2
REACTION OF MICROPOROUS SOLIDS: THE DISCRETE RANDOM PORE MODEL S. Department
of Chemical
K.
Engineering,
BHATIA*
Indian
B. J. VARTAK
and
Institute of Technology, India
Bombay
Powai, Bombay
400 076,
(Received 1 May 1996; accepted in revised form 20 May 1996)
Abstract-The universally used assumption of proportionality between reaction rate and pore surfacearea is modified for application to solids dominated by micropores, where it is shown that the discreteness of the solid phase necessitates some corrections. A new discrete random pore model, which modifies the earlier random pore model results, is proposed for such solids. An exact method and two other approximate methods for predicting the evolution of pore volume distribution are discussed in light of the discrete random pore model. It is observed that the structural parameter characterizing reactivity is effectively reduced in value due to the discreteness of the solid. The new results are used to interpret the difference between reaction rate and structurally determined values of this parameter reported in the literature for the char steam reaction. The importance and need for the development of better molecular models for such systems is stressed. Copyright 0 1996 Elsevier Science Ltd Key Words-Gas-solid
reactions,
gasification,
microporous
SE’
NOMENCLATURE
A
r” g k
ks
L 1 LCI LE L EO LG I6 P” r r0 rAV
S s S(X) So SE S*(x) S EO
carbons,
s
normalization constant Lennard-Jones molecular size parameter pore volume size distribution (PVSD) fin terms of I., 8 rate constant of reaction pore-size independent growth rate total length of pore axes per unit volume pore length total initial pore segment length per unit volume total length of pore axes of non-overlapped system per unit volume L, at t=O total initial pore segment length per unit mass number of layers reacted approximate continuous probability function exact discrete probability function radius of the pore Y at t=O mean pore size of the distribution surface area per unit volume standard deviation used for log normal distribution surface area per unit volume at conversion X initial total surface area per unit volume total surface area of non-overlapped system per unit volume
S;
*Corresponding author. Present address: Department of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia. 1383
corrected surface area corresponding intermediate contour at (r + a/2) SE’ at t=O
to the
SE/SE0 total initial surface area per unit mass time volume of pores per unit volume V volume of pores per unit volume v total volume non-overlapped system per unit VE volume VEO VE at t=O VE(>P) VE for all pores with size greater than p instantaneous mass of solid reactant W initial mass of solid reactant K X conversion Greek Symbols discreteness parameter (eqn (13)) dimensionless length of pores per unit volume ; PO(P) initial pore length distribution /I in terms of A, e Bl porosity E E at t=O CO dimensionless pore size (eqn (46)) I e dimensionless time (eqn (47)) dimensionless pore size, r/g P p at t=O PO parameter for the Rayleigh distribution PO0 mean p of the distribution PAV true density PT radial interlayer spacing between atoms of osolid phase dimensionless time t structural parameter, (eqn (5)) effective structural parameter (eqn (14))
SG t
S(X)/So S, at t=O
’
pore structure.
S. K. BHATIA and B. J. VARTAK
1384 1. INTRODUCTION
Since its development more than 15 years ago, the random pore model [ 1,2] has found extensive application in the interpretation of reaction-rate data for gas-solid reactions, as well as in theoretical analyses. Brief accounts of many of the applications have appeared in recent reviews [3,4]. Arguably its most important, and certainly the widest, application has been in the correlation of reactivity data for chars and carbonaceous materials. These applications have been prompted by the ability of the model to represent the often-observed rate maximum with increase in conversion, without the use of arbitrary adjustable additional parameters. This is principally due to the competitive mechanisms of surface-area increase and loss related to pore growth and intersection respectively, as reaction occurs, which are inherent to the overlapping pore concept assumed in the model. Considering the usual assumption that the reaction rate in any pore is proportional to its associated surface area, i.e: dv - =Sd’ =Sk, dt dt where k, is a pore-size independent random pore yields the expressions,
SW
~
(1) growth [ 11:
rate, the
[7,8] have also reported the values of $ from structural characterization for a large number of chars, their interest was in studying the use of $ as a characterization parameter rather than in testing the model. Consequently, the studies of Perlmutter and coworkers, who compared the values of $ estimated for different chars at various temperatures for the char-O, reaction, [S] and the char-steam reaction, [6] with those from structural characterization, perhaps constitute the most definitive tests of the model. However, while both studies showed no systematic variation of the reactivity-based value of + with temperature, the latter investigation showed the average value of this parameter to be about 35-40% lower than that evaluated from structural characterization. Table 1 presents the structural parameters and the two independent estimates of $ for the two chars, reported by Chi and Perlmutter [6], illustrating this difference. On the other hand, no such significant difference was reported by Su and Perlmutter [S] for the earlier study of the char-O, reaction. It may be mentioned here that the structurally determined ci/ values reported in Table 1 were calculated by Chi and Perlmutter [6] based on the relation: $=
=(l-X)dl-$ln(l-X)
4xL, ~
(7)
%T
(2)
S0 dX -= dt
k,S, -(l-X)fi-$ln(l-X) (1 --E0) X(r)= I -exp[
in which the parameter defined as: *=
(3)
-(T + $t2/4)] cl, is a structural
47cLE0 7
(4) constant,
(5)
Ai0
Here L,, is the total length of the pore axes per unit volume, and S,, a hypothetical total initial surface area per unit volume, if the pores were not overlapping, given by:
se,=
so
7
(1 -%I)
While the list of published applications of the above to chars and carbonaceous materials is rather extensive, it is pertinent to mention here that most of the studies have involved verification of the ability of eqn (3) to correlate reaction rate data, and thereby empirical estimation of the structural parameter $. More definitive tests of the model would also involve independent evaluation of the parameter $ from pore structure characterization data, and only a few comprehensive studies of this kind have been reported [ 5,6]. Such studies are particularly important in view of interest [ 7, S] in considering $ as a characterization parameter for char reactivity. While Miura et a/.
and the reaction rate-determined values of $ are those reported from their nonlinear fit of eqn (3) to rate data, as the linear fit using the transformation of this equation was sensitive to scatter at high conversions and reportedly less reliable. One possible source of error in applying the random pore model eqns (2)-( 6) to chars is that these are highly microporous materials, with pore sizes typically only a few multiples of the spacing between the reacting sites in the char. For example, the modal pore radius of the chars used by Chi and Perlmutter [6] was reported to be about 6 8, which is comparable to the inter-layer spacing of about 3.35 A in graphite [9]. Under these circumstances the concept of continuous motion of the interface is clearly inappropriate, and the molecularity or discreteness of the reacting species must be considered.
Table 1. Char
structural parameters reported Perlmutter [6]
by Chi and
Coal char Parameter Specific surface area, SG (m’g-‘) Total specific pore length, L(mg-‘) I++(pore structure determined) $ (reaction rate determined)
PSOC-3-10
PSOC-3-20
157.1
174.8
3.69 x IO”
4.18 x 1O’O
12.94 8.52
12.54 7.06
Reaction
of microporous
2.1 Reaction in a cylindrical micropore the gasification
reaction:
A(s)+bB(g)-+products(g) occurring in a cylindrical pore of radius r and length 1, in a solid A comprised of spherical molecules, as depicted in Fig. 1. We further assume that the radial inter-layer spacing in the solid is cr, and that reaction in the pore effectively occurs by the layerwise radial erosion of the solid phase. Upon consumption of a single layer the change in the volume of the pore is given by: Au = nl [(r + a)” - r’] = 2&(r
1385
2.2 Reaction in an overlapping pore structure 2.2.1 Overall behaviour. Here we modify
2. THEORY
Consider
solids
+ a/2) = 2rcI(r + 5/2)Ar
(8) where Ar=a is the corresponding change in pore radius. In the continuum limit, we write k,=dr/dt as the growth rate, and eqn (8) yields:
eqns (l)-(6) in light of eqn (9). Since it is assumed that reaction occurs by radial erosion only, the total length of the pore axes LE is unchanged and is given by: (10)
0
The surface area of the system SE is given by:
non-overlapped
pore
SE = 27~7 r( t,ro)L(ro) dr,
(11)
0
We recognize that for eqns (2) and (5) to be applicable to the discrete solid, S(X) and So must correspond to the intermediate contour in Fig. 1. In light of eqn (9) the non-overlapped area corresponding to this intermediate contour is given by: S,‘=2n%
do - =2nl(r+o/2)k,
= 7 I@,,) dr,
L,=L,,
[r(t,r,)fc/2]
L(r,)dr,
(12)
dt
= s, + asa, Comparison with eqn (1) shows that the area S associated with the pore reaction rate is not the true pore surface area 2nrl, but in fact that corresponding to the intermediate contour at (r+o/2). This latter surface will, in general, not pass through the molecular centers, since the inter-layer spacing c~ may be different from the molecular size &, depending upon the packing geometry. For example, for tetrahedral packing, 0 = ad,. It is also evident that the correction a/2, having magnitude of about 2 A, is not negligible in comparison to the pore radius r (which, for micropores, is also in the region of a few A).
where c( is a discreteness
parameter,
defined as:
RoLEO -
c(=
(13)
S EO We may now define an effective structural ter I+&using eqns (12) and (13), as: 47&o l//E= s EO
= -!L ( 1 + a)’
parame-
(14)
where $ is the measured structural parameter defined by eqn (5). The factor LYaccounts for the finiteness of the size of the reacting solid units, and I,&-+$ as a-+0. Following the basic relationship of the random pore model [l] we now write, for any overlapped surface: S=S,(l-c)
(15)
We define conversion by X =( Wo- W)/W, which, in term of porosities, is represented as: X=
E--E0 ~ (l-co)
(16)
This expression for the conversion is clearly approximate for a discrete solid. More exact expressions for such a case may also be considered, but it is readily established that eqn (16) should be accurate for most practical purposes. For the original random pore model eqns (2) (6) (15) and (16) readily yield: SE = SE0 (VI-*ln(l-X)
Fig. 1. A pore of radius I surrounded of a solid. The interlayer
by spherical molecules spacing is 6.
>
(17)
which must actually be applied to the non-overlapping surface area corresponding to the intermediate contour in Fig. 1. Therefore the expression for the
S. K. BHATIA and B. J. VARTAK
1386
discrete
random
pore model is:
SE’= S,,’ (\i’l-$Eln(l-X)
(18) 1
Equations
(12), ( 13) and ( 18) combine
We characterize pore structure by the length distribution of the pore axes, and the pore volume size distribution. We can define the instantaneous length distribution of the pore axes as: mt(0) c P,(Wo(P
to give &(r,p) =
S,=SEO
(l+r)ql-$,ln(l-X)-x
(19) 1
1 which, following
eqns (15) and (16), yields
S(X)=S,(l-Xx)
(l+a)ql-$,ln(l-X)-a 1
1
where L,(p) is the initial length distribution. It should be noted that as n can assume only integral values. the upper limit of the summation is the integer part of p. We define a dimensionless length distribution /j(~,p) given by: B(?P)
(20) Also, eqn (3) must be rewritten
following
as:
(28)
- n)
I!=0
(29)
= na3L(?P)
which eqns (27) and (28) combine
to yield:
dX = k,S,,‘( 1 -X) dt which combines dX -= dt
with eqns (12) and (15) to yield:
k,S, -----(l-x)(1+a) (1-G)
cdl-&ln(l-X)
(22) Equations (20) and (22) represent a more general result replacing the original random pore model equations (eqns (2) and (3)). They may also be derived independently of the latter from first principles, as in the earlier work [ 11. Comparison with the original model equations shows that while the functional dependence of the reaction rate on conversion is unchanged, albeit with the effective structural parameter tiE, the form for the pore surface area variation is modified. 2.2.2 Evolution of pore structure. Following Fig. 1 we first consider a single pore of initial radius r,,, in which n layers have reacted, so that: r=r,+ncr which yields, in dimensionless
(23) form: (24)
P=Po+n
Upon defining P,(t) as the probability that n layers have reacted in time, t, we can write the equation governing its evolution:
df’n
-
dt
==kP,_,-kP,
where k is a suitable rate constant for the layerwise reaction. In dimensionless form, eqn (25) yields
This expression can be used to obtain the length distribution of the pores after any time 7, given an initial distribution /IO(p).We can then quantify the evolution of pore volume size distribution (PVSD). Upon defining the non-overlapped volume corresponding to pores larger than p by:
VE(>,>,=% P' WZ>P')dp’ we obtain
the corresponding
(31)
pore volume as:
V l>p)= 1 -ev
(- vE(,p,)
(32)
following the random pore model [Z]. Equations (3 1) and (32) provide the dimensionless PVSD .f(p) = P’B(~P)
exp (-
V,, > piI
(33)
which holds at any time T, given an initial distribution Be(P). 2.2.3 Approximations. In the foregoing analysis, we have made the simplifying assumptions of negligible diffusional resistance and a uniform time-independent rate. When such assumptions are not applicable, the analytical result in eqn (27) does not hold and approximations to the exact model need to be worked out. One of the approximations considered here hinges upon the representation of the discrete probability function by a continuous one. This approximation can be called the “continuous approximation to the exact model” (CAEM), and is obtained by rewriting eqn (27) in a different form to make it continuous, as: ~“-“~exp(-z) P(GPlPo)
(26) following which is readily solved to yield the Poisson tion: P”(T) = since Pn(0)=
6,,.
rnexp
(- z)
?I!
Vn20
=
P,(t) =
(34)
UP-Po+l)
which eqn (30) is rewritten
as:
distribuB(LP)
(27)
=
j: P(T>P/P,Mo(P,) 0
dpo
(35)
upon approximating the summation by the integral. Use of eqn (34) in eqn (35) would give the length distribution of the pores using CAEM. The PVSD
Reaction
of microporous
would still be predicted by eqn (33) except for the fact that now b(7,p) would have to be evaluated using eqn (35) instead of eqn (30). The second approximation addressed in this article uses a partial differential equation (PDE) to evaluate the probability function, and is called the partial differential equation model (PDEM). In this approach we rewrite eqn (26) as:
1387
as the initial length distribution of the pores. In eqns (42) and (43), A is a normalization constant satisfying: $ AP$(PO)
dpo= Go=
-ln(l
-co)
P(z,p-
=
l/PO)-
P(74dPO)
dt
-ln(l-Eo)
(45)
s P;P(Po) dpo
(36)
0
Equation (36) can be converted to a PDE in P(7, p/p& by using a second-order Taylor series approximation for P(7,p - l/p,). The resulting PDE is: ap(7dPO)
W7dPO) =-
+
!
azp(7dPO)
2
ap
3%
Once the initial distributions have been specified, the models can be used to obtain the length distribution B(7, p) and the PVSD f(z,p) of the pores after any time 7. However, for purposes of presentation it is instructive to express the results in terms of the dimensionless variables:
(37) The conditions
for the solution
P
r
i=-_-
of eqn (37) are:
rAV
P(O,PlPo) = 0
VP ’ PO
(38)
vz
(39)
P(z,co/p,)=0
(46)
PAV
and: ,=k”=i
7 P(~,plpo)dp= 1 V7 (40) PO Solving eqn (37) using initial condition eqn (38) and boundary conditions in eqns (39) and (40) we obtain:
exp(P -
P(GPlP0) =
2n
po) s[
(P - POS ~ -1-(P-PO) t
0
x exp
(P-Po)2 ~ 2t
[
t - 2
1.
t3P
clt
1 (41)
Use of eqn (41) in eqn (35) yields the length distribution of the pores using PDEM. As before, eqn (33) will predict the PVSD using the PDEM if P(z,p/p,,) is evaluated using eqn (41). 3. RESULTS
3.1 Evolution
(44)
yielding: A=m
dP(wltd
a7
solids
(47)
rAV
leading tion:
to a modified
PAV
dimensionless
k(W)
length
= Pi”B(7,P)
distribu-
(48)
and pore volume distribution: g(kn)=
(49)
pA"f(7d
Figure 2 compares the results for Pi(n) from the exact method with those from the proposed approximations, using a Rayleigh distribution. The CAEM proves to be a better approximation, with the results more closely matching the exact result. A similar
0.6
AND DISCUSSION
ofpore
structure
The models proposed require an initial distribution of the pores to be specified. The results presented here assume either a Rayleigh distribution:
Po(po)=A
(p-poo)exp{-~(~~)
(Pa” - Poo)2
or a log-normal
distribution:
(42)
-
[In(poJpA~)]
PEPS)=
stspo
exp
2s2
i
1 (43)
1.0
Dimensionless
2.0
3.0 pore size (h)
Fig. 2. Comparison of results for PI(n) from the exact method with those from the CAEM and the PDEM. A Rayleigh initial distribution is assumed, with parameters poO= 1, p*“=2, Eg=0.2S, 8=1.
S. K. BHATIA and B. J. VARTAK
1388
trend can be observed for the PVSD in Fig. 3. In what follows we report results from computations using the exact method. The earlier continuum random pore model assumed c to be 0 (no discreteness) which is equivalent to the condition pAv+ co. Figure 4 (log-normal initial distribution) and Fig. 5 (Rayleigh initial distribution) depict the effect of the discreteness parameter pAv on the dimensionless length distribution PI(i), with the dashed curve representing the continuum model (P~~-‘co). It is evident that at a sufficiently high pAv (> 50) the earlier random pore model is accurate enough. However, at smaller pAv the discrete random pore model must be applied for more reliable results. In the case of the PVSD results also, as shown in
__ _ cl
Random Pore Model
0.8
Dimensionless
0.8 /\
------- CAEM --_ PDEM
0.6
2.0
1.0
0.0
__
Exact method
1.0
0.0
i 1 / ,
2.0
Dimensionless
3.0
--_
~,=0.25,
4.0
Fig. 5. Effect of pAVon b,(A). The earlier random pore model prediction is shown as a dashed curve for comparison. A Rayleigh initial distribution is assumed, with parameters poO= 1, ~~=0.25, O= 1.
1 \ \
4.0
5.0
pore size (h)
Fig. 3. Comparison of PVSD results from the exact method with those from the CAEM and the PDEM. A Rayleigh initial distribution is assumed, with uarameters P,,“= “” 1. pAV=2,
3.0 pore size (1)
o= i.
Fig. 6 (log-normal initial distribution), the convergence of the curves to the dashed curve is visible as pAV increases. Figures 4 and 5 reveal another interesting feature. At smaller pAV, the PI(i) curves are not smooth and kinks can be observed. These kinks are smoothened out as pAv increases. This is a consequence of the finite thickness layerwise reaction concept assumed in the discrete random pore model. These kinks are not very prominent in the PVSD plots due to the smoothening effect of the integral involved in the computations in eqn (31) and eqn (33). The effect of 0 on fil(l) is depicted in Fig. 7 for a Rayleigh initial length distribution with pAv = 2. As before. the random pore model results are shown as 0.4 -
Random Pore Model
Random Pore Model
0.3
G 2
0.2
L 0. I
1.0
2.0
Dimensionless
3.0
4.0
pore size (L)
Fig. 4. Effect of pAv on ,0,(,?).The earlier random prediction log-normal
0.0
pore model is shown as a dashed curve for comparison. A initial distribution is assumed, with parameters s=OS, ~,=0.25, B=l.
1.0
2.0
Dimensionless
3.0
4.0
pore size (h)
Fig. 6. Effect of pAV on PVSD profiles. The PVSD profile predicted by the earlier random pore model is shown as a dashed curve for comparison. A log-normal initial distribution is assumed,
with parameters
s = 0.5, q, = 0.25, 0 = 1.
Reaction 1.2
of microporous
-
Random Pore Model
-
1.0
2.0
Dimensionless
3.0 pore
size
(h)
Fig. 7. Effect of 8 on PI(I). The results predicted by the earlier random pore model are shown as dashed curves for comparison. A Rayleigh initial distribution is assumed, with parameters paa = 1, pAv = 2, fzO= 0.25.
dashed profiles. It is obvious that the random pore results deviate from the exact discrete pore model results as B increases, indicative of the importance of a discrete analysis for microporous solids having a small pAV. The latter is also evident in Figs 5-7. Figure 8 shows the PVSD results as 0 increases, for a log-normal initial length distribution, again clearly indicating that the more accurate discrete random pore model should be applied for microporous solids. 3.2 SigniJicance of mod$ications The essential feature of the difference between the discrete random pore model (eqns (13) (14), (20) and and the earlier random pore model (21)) (eqns (2)-(6)) lies in the introduction of the new 0.6 - -
solids
1389
parameter c(. For cc-+0 the two models become identical. From the definition of a in eqn (13) and considering SE,, as 2nr,,L,, where rAV is a characteristic mean initial pore radius, one estimates that a = a/2r,,. Thus, the parameter a may be construed as representing the relative importance of the discreteness of the microporous solid. As an order of magnitude estimate, if we consider the reasonable values of 0 cr=4A, SE,=250 m2cme3 and $=lO, eqn(13) provides a=O.25. This would seem to be a significant correction in eqns (20) and (22), since a is not negligible in comparison to unity in the terms involving the factor (1 +a). For these parameter values we also estimate that I,& =6.4, which is considerably smaller than the value Ic,= 10 based on measured pore structure characteristics, underscoring the relevance of the proposed correction. For highly microporous chars, as well as activated carbons, SE,, will often exceed 1000 m2 cm-’ and the value of a may approach or even exceed unity, making the present modification even more significant. On the other hand, for relatively macroporous materials, typified by surface areas of the order of 10 m2 cmm3, the present corrections would be insignificant. 3.3 Model behaviour Figure 9 depicts the variation of the dimensionless pore surface area with conversion, at $= 10, for various values of a, as predicted by eqns (14) and (20). The result for a= 0 corresponds to the earlier random pore model, and with increase in a the value of the surface area is reduced. At the same time the position of the surface area maximum is shifted to a lower conversion, an effect which is even more readily evident for the lower value of 11/= 3, as seen in Fig. 10. In this case the surface area is monotonic for a= 1, with no initial increase. Figure 11 shows a comparison of the dimensionless surface area and the reaction rate variation with conversion, for $ = 10 and a =0.5,
Random Pore Model
a=0
1.0
2.0
Dimensionless
3.0
0.50
-
0.25
-
4.0
pore size (h)
Fig. 8. Effect of B on PVSD profiles. The PVSD profiles predicted by the earlier random pore model are shown as dashed curves for comparison. A log-normal initial distribution is assumed, with parameters s = 0.5, E,, = 0.25, pAv = 2.
I 0.00
I
I
I
I
\I
0.2
0.4
0.6
0.8
1.0
Conversion Fig. 9. Effect of u on the 5(X)/S(O) *=10.
(X) profiles,
with measured
1390
S. K. BHA~IAand B. J. VARTAK Table 2. Estimated values of i and (r Coal char Parameter
PSOC-3-10
PSOC-3-20
0.232 3.15
0.333 4.43
i. a(A) o^ 0.75 ;; A x, * 0.50
forward to estimate c( using eqn (14), and subsequently 0 using the S, and L, values in Table 1 along with eqn (7) and the modified form of eqn (13):
0.25
0.00
I
I
0.2
0.4
I 0.6
Conversion
I
0.8
1.0
(X)
Fig. 10. Effect of 61on the S(X)/S(O) profiles, with measured *=3.
. /_--~ 4 \ 1.50
__
,.25
S(X)/S(O)
/T
rate ‘;;i;
(O)
1
/’
‘,
1.00
,
\
s
z
= x 0.75 VI.
\
\
\\
\
\
\
\
0.50
\
\\
Table 2 lists the values of c( and u thus estimated for the two chars. The values of 0 are remarkably close to the 3.35 A inter-layer spacing of graphite, or the Lennard-Jones size parameter for the C-C interaction, indicative of the viability of the present approach. However, it should be emphasized that our findings are based on a small data set and much more carefully obtained experimental data needs to be analyzed. In this regard, it may be mentioned that the results of Su and Perlmutter [5] do not show similar differences in $, and are therefore not subject to the current interpretation. While this may suggest that CI<<1 for the C-O, reaction with the chars used by them, another complication arises from the fact that in both of these studies, and most others concerning the random pore model, the initial structural parameters are estimated using the expressions:
0.25
t
\ I
0.00
0.2
I 0.4 Conversion
I 0.6
I
0.8
1.o
L,=L,(
1 -Ed)=
7 &(r) 0 ?l?
(X)
Fig. 11. Comparison of S(X)/S(O) and rate(X)/rate(O) curves, with measured $ = 10 and a = 0.5
following eqns (20) and (22), respectively, as well as that predicted by the earlier random pore model. With the discrete random pore model, the predicted surface area and reaction rate variation are no longer identical, with the latter curve being lower. Careful experimentation is necessary to verify these effects. As such, the reaction-rate variation still has the same form as the original model, but corresponds to that for the lower value $E.
(52)
which ignore intersections. While they were indeed originally suggested by Bhatia and Perlmutter [ 11, alternate expressions based on application of the random pore model to the initial structure as well have been derived by Gavalas [2] and should be used. These provide: ru
s
S,=(l-Eo)
0
r
2f(4 -
1
7I
dr
f(s) ds
e
fIrI L,=
3.4 Application to experimental data As a direct application of the new results we may attempt to reconcile the difference between the structurally determined and reaction rate determined values of $ reported by Chi and Perlmutter [6], for the char-steam reaction, and presented here in Table 1. Considering the reaction rate determined values instead as I/&, following eqn (22), it is straight-
dr
1
1
(53)
dr
(54)
0
I and future applications of the model, particularly the discrete random pore model, need to be conducted using these relations. This comment also holds for the published S(X) data, which is invariably based on eqn (51) instead of eqn (53), and therefore not analyzed here.
1391
Reaction of microporous solids
3.5 Further directions While we have accounted for the discreteness of the solid phase in the framework of the random pore model, it is clear that this approach is equally applicable to other models such as that due to Petersen [lo]. In all of these it would suffice to consider the reaction area as that corresponding to the intermediate contour at (r + a/2) in Fig. 1, in place of the actual pore surface area at Y. Thus, application of the approach using other structural models is an open area for study. It is also clear that molecular models of discreteness must recognize the possibility of anisotropic reactivities, a well-known feature of graphite which is also related to the frequently observed appearance of etch pits [ 111. However, it is not clear as to what extent these findings for graphite apply to microporous chars and other carbonaceous solids, since the structure of the latter and its relation to reactivity is currently subject to much debate [ 12,131. Even the discrete random pore model utilizes the discreteness parameter 0 which may also vary with the nature of the reactant gas and the reaction mechanism. Thus, r~ may vary for carbons and also differ for gasification by oxygen, steam or CO,, leading to small differences in the normalized rate curves. Such differences, if determined by careful and accurate experiments, may also help in understanding the reaction chemistry.
Acknowledgements-Part of the work reported here, and the ideas discussed, originated from research conducted by one of the authors (S.K.B.) while he held a visiting appointment at the Institute for Kemiteknik, Technical University of Denmark. during 1994. The supDort of his stay by ’ the Danish ReseaFch Council i’sA gratefully acknowledged.
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