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Modelling diffusion-limited gasification of carbons by branching pore models
THE CHEMICAL EN$M&JG ELSEVIER
The ChemicalEngineering Journal 64 ( 1996) 77-84
Modelling diffusion-limited gasification of carbons by branching pore models Salvatore Borrelli a, Michele Giordano b, Piero Salatino a** aDipartimento di Ingegneria Chimica. Llniversitb degli Studi di Napoli Federicu II, P. le Tecchio, 80125 Napoli, Italy ’ Dipartimento di Ingegneria dei Maten’ali e della Produzione. Universitci degli Studi di Napoli Federico II, P. Ie Tecchio, 80125 Nap&. Italy
Abstract A kinetic model of gasification of porous carbon particles characterized by broad pore size distributions is developed. The interaction between the intrinsic kinetics of surface oxidation and intraparticle diffusion of reactants is modelled, taking into account the wide variation of local diffusivities within the pore space as the length scale of pores changes. In particular, the model considers the configurational diffusion mechanism, which dominates transport in pores whose size is of the order of the diffusing molecule size. The model is based on the iterated application of Thiele analysis to branching pore sequences of different topology. Computations are directed to investigate the sensitivity of the model to the pore space topology, to the parameters of the pore structure and to variables determining the rate of surface reaction and of intraparticle diffusion. Keywords:
Diffusion-limited
gasification;
Branching
pore model; Kinetic model; Porous carbon particles
1. Introduction In most practical contexts, modelling gasification of single particles of porous carbons implies considering the interaction between the intrinsic kinetics of surface oxidation and kinetic resistances arising from intraparticle diffusion of gaseous reactants. In the framework of the theory of heterogeneous catalysts, this problem is dealt with by recourse to the classical Thiele analysis by the evaluation of effectiveness factors from the generalized solution of balance equations on diffusing and reacting species. Unfortunately, the straightforward application of this well-established approach to carbon gasification reaction is usually prevented by the following aspects: 1. Carbons are frequently characterized by very broad ranges of pore size distributions, typically spanning over 4 or 5 orders of magnitude of pore radii. 2. Diffusion coefficients in pores change depending on the size of cavities within which the diffusing molecule is constrained. Accordingly, molecular, Knudsen or configurational diffusion mechanisms may prevail, depending on the relative magnitude of the pore size, of the mean free path of the molecule and on the size of the molecule itself. * Corresponding author. Tel.: + 39 81768 2258; fax: + 39 81 593 6936; email: [email protected] 0923~0467/96/$15.00 Q 1996 Elsevier Science S.A. All rights reserved PUSO923-0467(96)03106-5
The combination of these factors makes the issue of modelling combined diffusion and chemical reaction in a broadly distributed pore space a rather formidable problem. Paths followed in the past to accomplish this task differ considerably from each other. Some authors mapped the actual pore space onto an equivalent regular network of pores, by analogy with the flux of electrical current in random resistor networks. Paths eventually diverged either to the formulation of Effective Medium Approximation (EMA) procedures [ l-31, or to the development of tools based on percolation concepts on regular lattices [ 4-61. The final goal was in any case the determination of effective transport properties to be used in a continuous formulation of balance equations. A different approach is that of lumping pores into a finite number of classes arranged in a hierarchical sequence. The basic idea is the iteration of the Thiele analysis to the entire cascade of pore sets in order to calculate the overall effectiveness factor of the internal surface area. Starting from the simplest micro-macro models [7,8] this concept has been extended to finite sequences of more than two pore classes r91. The idea of an arrangement of pores in a branching sequence has been strongly supported by Simons et al. in a number of papers [ 10-131. Based on purely statistical arguments, Simons et al. concluded that a broadly distributed pore
78
S. Borrelli et al. /The Chemical Engineering Journal 64 (1996) 77-84
space with a random isotropic dispersion and orientation of cylindrical pores naturally leads to branching patterns. Following this argument, these authors developed models of diffusion-limited gasification of coals based on the pore tree concept. Further simplifying assumptions implied the possibility of formulating the diffusion-reaction problem as a ldimensional problem along a curvilinear coordinate in the pore tree. The governing equations were eventually solved under limiting cases of kinetically and diffusion-controlled gasification of coal. The concept of pore branching naturally leads to recent views based on the description of the pore space of carbons as surface fractals 1141. Friesen and Mikula [ 151, Salatino et al. [ 161, Jaroniec [ 171, Foster and Jensen [ 181, to cite a few, give the evidence of fractal-like structures of carbons from the analysis of porosimetric data obtained with different techniques. The coincidence of fractal exponents obtained with various experimental methods (Hg intrusion, gas adsorption, small-angle X-ray scattering) reinforces the soundness of the use of the fractal geometry to describe the pore space of coal chars. The fractal-like structure of bituminous coals has also been simulated by numerical computation [ 191. In the present paper a model for diffusion-limited heterogeneous reaction of a porous solid characterized by a broad pore size distribution is presented. The main concern is the gasification of coal chars, whose pore spaces are amenable to a fractal description. The model is based on the recursive application of Thiele analysis to branching sequences of pores. The changes of the effective diffusivity of the gaseous reactants with the pore radius is taken into account by considering molecular, Knudsen and configurational mechanisms. Two topological models of the pore space are adopted: the first is based on a classical prototype of the fractal structures, i.e. the Menger sponge [ 201; the second is a branching model constructed as indicated by Sheintuch and Brandon [ 211. The latter model turns out not to be a fractal in a strict sense, but exhibits fractal behaviour in the asymptote of infinite number of pore classes. Computations are directed to shed light on the relevance of the topology and of the diffusional parameters to the model results.
Intraparticle diffusion of the reactant results from different mechanisms, namely molecular, Knudsen or configurational. Depending on the relevant mechanism, the dependence of the diffusivity on the pore radius changes in the following way: Molecular diffusion. Under the hypothesis of negligible Stefan flow it is: D=D, Knudsen diffusion. The diffusivity the pore lateral size. Accordingly: D=k,r
r=k,c
(1)
where r is expressed as moles of gaseous reactant consumed per unit time and unit exposed surface, and c is the reactant concentration in moles per unit volume. It is assumed that the solid is characterized by uniform reactivity, that is k, does not depend on the position on the solid surface.
changes linearly with (3)
Configurational difision. Different relationships have been reported in the literature to express configurational diffusivity in microporous solids. They are reviewed and compared by Froment and Bischoff [ 71, who noted that, in spite of the apparent differences among functional relationships expressing the influence of pore size on D, the various expressions yield results practically coincident as regards the hindrance effect within micropores. With reference to diffusion in carbonaceous micropore structures, Floess and VanLishout [ 221 suggest that configurational diffusivity is an activated process, the activation energy Ed being related to the adsorption potential of the diffusing molecule within the micropore. On the other hand Dubinin [ 231 first showed that the adsorption potential in microporous adsorbents is inversely proportional to the pore width. A combination of these findings reinforces the soundness of the empirical relationship proposed by Satterfield et al. [ 241: D=k,exp(
-a/r)
(4)
where CTis a temperature-dependent parameter defined by the equation u/r = E,IRT. As discussed by Salatino and Zimbardi [ 251 it is further assumed that k, = kg. It follows: D=kgexp(
-a/r)
(5)
Following Satterfield [ 261, the expression for the intrapore diffusivity valid over the whole range of pore sizes is obtained by combining the mechanisms considered above: 1 -=-+ D(r)
2. Theory The proposed model is based on the following assumptions: The gasification reaction obeys a first-order kinetic rate equation of the type:
(2)
1
1 D,
(6) k,rex
P(
_a
r1
2.1. Topological and geometric representation of the pore space The pore space has been modelled by using branching selfsimilar pore networks. Pores are hierarchically arranged in a sequence of levels i 2 0. Each pore is characterised by lateral size 2ri. The sequence of pore levels is constructed so as to keep constant the reduction ratio: ‘I+‘=,<1 ri
(7)
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84
For fixed pore geometrical properties, two different topological arrangements of pores have been considered: 1. Pores of ith level may branch out from any larger pore of index j < i (Fig. 1 (a) ) . The topology which arises from this arrangement is, for instance, that of the Menger sponge (Fig. 2). 2. Pores of ith level branch out only from pores belonging to the (i - 1) th level (Fig. 1 (b) ) . This arrangement is similar to that described by Sheintuch and Brandon [ 211. Whatever the pore network topology, the geometrical characteristics of the pores are assumed to be the same. Each pore is characterised by a square cross-section and by aspect ratio A = 1,/2r,,where Zjis the pore length. The square cross-section of pores has been assumed in the development by analogy with the Menger sponge. This assumption, like others relative to the shape of pores, has a minor impact on the model results when compared with the influence of the topological properties of the network (connectivity, fractal dimension, fractal range). In the construction procedure of both types of networks, ui is the surface of pores of level i prior to the introduction of pores of level j > i, 4‘” is the surface of the pores of level i as
/( a
19
b
a
Fig. 3. Definition of terms related to pore branching.
it is left over after introducing pores of level j > i; m is the number of pores of the i+ 1 level per pore belonging to the ith class. The following coefficients can be defined:
(8)
With reference to Fig. 3 we have:
p252a+b and
p&L
2a+b
From this point on the description of the pore space topologies will follow different paths for the Type 1 Menger sponge (MS) and for the Type 2 branching pore tree (BPT) .
08
(a)
Fig.
1.Schematic representation
of pore space topologies.
2.1.1. MS The construction rules of a triadic Menger sponge are reported elsewhere [ 15,201. The triadic sponge is a surface fractal with dimension D, = 2.2619. Non-triadic sponges of 2.2619
A,=uw+mi
(10)
2.1.2. BPT Each pore of level i branches into m smaller pores of level i + 1 of decreasing linear dimension with similarity ratio (Y. Coefficients j3 and y are related to the other parameters by the relationships: Table 1 TopologicaI
Fig. 2. The Menger sponge.
and geometrical
parameters
of the Menger sponges
‘heavy’
‘light’
DS
2.2619
(L m A P Y
I/3 12 1 819 112
2.3717 l/7 101 1 36149 65136
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84
80
p+!g
(11)
(1%
kn-,=ksP(l+mJ
where the contribution of pores of level n is given &a,,weighted by the effectiveness factor q,,. (n - 2) th level: proceeding in a similar manner, it is:
Py= mcy*
This arrangement
i,
of pores yields: (20)
k,-2=k,P2(l+y77n)(l+y~“-,)
Iterating the procedure, the general term of the succession of k,,_ i is given by In comparing the two pore structures, it is worth noting that the Menger sponge is a surface fractal in the strict sense. This means that the relationship holds:
=ln(Ai+lIAi)
2_D
5
Wff)
(14)
with D, constant for all values of i. The BPT is not a fractal. However it obeys a relationship of the type in Eq. ( 14) in the limit i + to. The fractal nature of the two pore structures can be made to coincide in this limit (i.e. lim DS,BpT= Ds,,.& if I and only if: PA
1+ x44s) = PEPT’YWT
The relationship
(15)
holds:
i-l (21)
k-i=kPifl(l+YTn-j) j=O
which, for the external yields: k,=k#“fi(l+
surface of the particle
lim~~)=~(PnW!;_l+l)
(22)
YVi)
BPT Model: For the BPT model, the derivation of the effective reaction rate at each pore level is formally similar to that for the MS, accounting for the different way in which the pore sequence is arranged: nth level:
(n-
(23)
1)th level:
1+ Prrl,l
The Menger sponge is a fractal object characterized by an upper cutoff of the fractal range coincident with the size of the object itself. For the sake of simplicity, and without loss of generality, the assumption that the upper limit of the hierarchical sequence of pores is coincident with the size of the object, i.e.
k” -
r, = rP
The general term of the succession
(17)
will be made even with reference to the BPT model. The theory can be easily modified in order to account for cases in which the upper cutoff r,, is smaller than the object size r,,. 2.2. Kinetic model 2.2.1. Iterative application of the Thiele analysis The iterative application of the Thiele analysis is based on the definition of an effective reaction rate constant, ki, per unit surface of the pores belonging to the ith level. It is calculated as the sum of contributions to the reaction rate from pores smaller than ri, divided by the oxygen concentration at the ith level and by the surface area ai of pores of size r,. In the following, the derivation Of ki for the two topological models will be demonstrated with the aid of the bidimensional simple branching pattern of Fig. 3. MS Model: Starting from the nth level it is: nth level: k, = k,; (n-
1)th level:
(18)
0)
i=l
k,=k,; (16)
(Level
1=
k,P + UWr-/n
(n-2)th
=W
(24)
level:
k,-,=k,P+k,P’yPrln-,+k,[Py12P?7n-,71n =UUl
k,+=k,p
+Pyr)n+
l+ ‘&j [
j=l
(25)
[Prl*rl,-,71nl
“E’
is:
1
71~
k=n-i+l
(26)
which, for n - i = 0, yields:
b=k,P[ kP’YifITj] j=O
(27)
i=o
2.2.2. Evaluation of effectiveness factors Calculation of k, -i and of b from, respectively, Eq. (21) and Eq. (22) for the MS and Eq. (26) and Eq. (27) for the BF’T requires the knowledge of the effectiveness factors 77i for each class of pores. This is accomplished by relating vi to Thiele moduli pi of the pores of level i:
(28) For the sake of simplicity the expression for the effectiveness factor of a dead-end pore of cylindrical shape is adopted:
S. Borrelli et ul. /The Chemical Engineering Journal 64 (19%) 77-84
However, this expression has been modified to satisfactorily comply with the case in which the penetration depth of the reactant within the pore becomes comparable to or smaller than the pore radius. This takes place as 40,+ CCand in this limit Eq. (29) predicts vi + 0. This limit is physically sound only for pores having infinite aspect ratio, i.e. the ratio between the pore length and the lateral size. For pores of finite length a minimum non-zero value of the pore effectiveness factor exists even in the limit of large Thiele modulus, that is for infinitely fast reaction’. This lower bound is given by the ratio of the pore cross-sectional area over its internal surface area. In terms of the already defined geometrical parameters:
vy=- 1-P Pr An approximate expression of vi which preserves behaviour of Eq. (29) (as cpr= 1) and of Eq. (30) cp,-+ + ~0) is simply given by:
1-p +
%’ PY
(30) the (as
(31)
and will be adopted in the computations. In either the MS or BPT versions of the model, the computation procedure consists of iteratively evaluating Eq. (21) (or Eq. (26) ) starting from the nth level, up to the 0th level. At each iteration step calculation of k,,_i is preceded by the evaluation of qn _ i + 1 by means of Eq. (31), with ‘pn_i+l given by Eq. (28)) and once k, _, + 1 is known from the previous step. The overall reaction rate 8 is given by: ~=JWkoAocS
(32)
The choice of the parameter IZin Eq. (22) and Eq. (27) deserves some discussion. Its value can be fixed if there is evidence of the existence of a lower bound to length scales associated with the pore space. Otherwise, it is easily recognized that the successions:
P’I-p+Yrlj)
81
values of rj (of the order of the parameter a), of diffusional resistances stronger than those determined by the Knudsen mechanism. One important consequence is that the choice of n, in this case, becomes merely a problem of numerical convergence of the series in Eq. (22) and Eq. (27). This is the way it has been treated in the computations.
3. Results and discussion The results of the computations are presented and discussed with the aim of outlining the influence of a number of variables either related to the topology and the geometry of the pore space or to the kinetics of the reaction/diffusion processes on the apparent gasification rate of the solid. 3.1. The influence of the pore space topology Fig. 4 compares the cumulative surface areas of the pore spaces for different choices of the topological parameters. In particular, curves for D,=2.2619 and D,=2.3717 are presentedforeithertheMS(Eq. (lO))andtheBPT(Eq. (13)) pore structures. These choices of D, correspond to the limits within which Menger sponges, ranging from ‘heavy’ to ‘light’ ones, can be constructed. For a given D,, the reduction ratio (Y,the pore aspect ratio h and the multiplication factor m have been kept at the same values regardless of the topology. For D, > 2.3717 only the BPT structure applies. Since equal values of A0 have been assumed for the two topologies, Eq. (16) yields: +1
li
(33)
The non-fractal nature of the BPT structure, as opposed to the fractal character of MS, is evident from the deviation from linearity of the curves for BET at large values of rr. In this respect, it is interesting to note that several authors found similar ‘bending’ of plots like those of Fig. 4 determined from
j=l
t
for the MS, and:
-
:----.MSD,-2.3717
AI*,
-------BPT D,=2.3717
-
-MS D,-2.2619 BPT D,=2.2619
P'Y'nTj j=O
ld
for the BPT, appearing in Eq. (21) and Eq. (26), converge respectively to 1 and to 0 as i + to. This property is a direct consequence of the following conditions: ( 1) the succession of ‘pi is strictly increasing in the neighbourhood ofj + @J;(2) a minimum value of rli, given by Eq. (30)) exists as uj + CO. In particular, Condition 1 is fulfilled due to the onset, at small ’ This apparent inadequacy of the Thiele analysis arises from the limitations intrinsic to the application of a 1-dimensional approach to the problem of diffusion and reaction in a pore of a finite length. Multidimensional approaches, or models based on integral approaches, e.g. those developed by Chambri? [27] and Verhoff and Strieder [28], would overcome these limitations.
1 ot
d
1 00 lo-5
lir’
10-l
l@ r/r,
Fig. 4. Dimensionless cumulative surface areas versus pore space size fat various pore topologies and D..
82
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84 -
South
African
A/A,
100 Lo/b 10
1 O1 0.001
Marine South African
2.6 2.9
0. 0.9
1.9 1.35
I
2
1 rfr0 Fig. 5. Dimensionless cumulative surface. areas for samples of 2 coal chars. Curves are best-fit lines based on the BPT pore model.
kO/lu
0.1
10
1 0.001
0.01
0.1
Fig. 6. Dimensionless reaction L&=2.2619; r”= 1 X lo-“
t
rate constant
0.1
Fig. 7. Dimensionless reaction 0.=2.3717; rp= 1 X lo-“
I
11 0.01
0.01
10
versus
100
Thiele
lCO0
modulus.
porosimetric characterization of coal and chars at large values of pore radius [ 15,161. Further evidence is given in Fig. 5, where cumulative surface area distributions obtained by mercury porosimetry of samples of chars from two medium-rank coals are compared with curves obtained by the application of the BPT pore mode1 with suitably defined geometrical and topological parameters. In both cases, values of D, = 2.6-2.7 are asymptotically observed at small values of the pore radius, but significant deviations are observed at r = 1 pm. Fig. 6 and Fig. 7 report on the influence of the pore space topology on the gasification rate. Results are presented in terms of the ratio between the apparent reaction rate constant k. referred to the external surface A, divided by the intrinsic reaction rate constant k,. k,lk, values are plotted as functions of a particle Thiele modulus of the sample, defined as follows:
1 + rate constant
10 versus
100 Thiele
loo0 modulus.
external surface area. This asymptote is usually neglected in rj versus cpplots from classical Thiele analysis as it is implicitly assumed that the external surface area is negligible with respect to the internal. At low values of cpanother asymptote is attained. The ordinates of the curves as cp+ 0 substantially reflect the values of A,IA, where i is the pore level at which significant diffusional resistances associated with the configurational mechanism arise. Differences between the asymptotic values of k,lk, for the two topologies as cp+ 0 are related to the different values of A,/A,, expressed by Eq. (33). The decay of the apparent reaction rate as rp increases is steeper for the BPT pore topology than for the MS. The maximum slopes of plots in Fig. 6 and Fig. 7, corresponding to the inflection point of the curves, are about - 1.1 (D,=2.2619) and -0.95 (0,=2.3717) for the BPT. The maximum slopes for the MS are of the order -0.6 and - 0.65. This behaviour is easily interpreted in the light of the following argument. As the global Thiele modulus increases, that is, as the importance of intraparticle diffusional resistances is raised, the fall of the effectiveness factor takes place earlier for coarse pores, i.e., those in which the dominant diffusion mechanism is molecular (let us refer to these pores as M pores), than for pores where the Knudsen mechanism prevails (K pores). In the MS arrangement a significant fraction of K pores branches out directly from the external surface, whereas in the BPT arrangement the same pores can only be reached through M pores. For both topologies, the transitional range between the asymptotes broadens as D, increases. On the whole, the results corresponding to the two topologies, all other parameters remaining the same, get closer to each other as the fractal exponent D, gets larger.
8h2rPk, ‘= C--D(Q,,,>
3.2. The influence of the particle size
Parameters of the curves which have been kept constant in the two figures are the particle size (rt, = 1.10e4 m), the parameter kd of the expression for Knudsen diffusivity (k, = 500 m s-l), and the parameter of the expression of configurational diffusivity ((T = 2.10p8 m). Notably, two asymptotes are recognized in the plots. As cp+= + 00 all curves tend to the value b/k, = 1, which expresses the fact that under strong intraparticle diffusional limitations the reaction rate is proportional, through k,, to the
Further analysis has been only carried out with reference to the BPT pore structure, using D, = 2.7. Geometrical and topological parameters adopted in the computations are: m = 20, (Y= l/3, A = 1. Fig. 8 compares curves of b/k, versus Q or different particle radii rP. The effect of increasing r,, is that of increasing the apparent reaction rate throughout the range of Q. In fact, the effect of rP on the overall reaction rate is twofold. On the one hand, increasing rP increases A,,, and
S. Borrelli et al. /The Chemical Engineering Journul64
(19%) 77-84
83
loww 10000 kO/ks
low 100
10 10 I 0.001
0.01
0.1
10
loo
low
Fig. 8. Dimensionless reaction rate constant versus Thiele modulus. BPT: D,==2.727; 0=2X lo-’
this, in turn, increases the reaction rate. On the other hand, increasing rP shifts the upper cutoff of the pore sequence towards larger values and widens the range of pore intervals in which the more efficient molecular and Knudsen diffusion mechanisms dominate transport. As a consequence, both the extent and the effectiveness of the internal surface area are increased. It is worth noting that the asymptotic value of kJ k, at low 50 does not increase in proportion to the particle radius. This implies that reaction rates when expressed per unit volume of the particle, rather than per unit external surface area, decrease with particle radius.
3.3. The influence of the difSusion parameters
The values of the molecular diffusivity D, and of the constant kd, which determines the value of the Knudsen contribution to the overall diffusivity, are well established. In particular the values of D, = 5. 10m5 m2 se1 and of kd = 500 m s-’ used throughout this work correspond to oxygen molecules diffusing at 500 “C. The relevance of configurational diffusion to the pore hindrance in a fractal pore space has been addressed by Coppens and Froment [29]. These authors based their development on the assumption that a sharp boundary can be established between accessible and non-accessible pores. In the present analysis the role of configurational diffusion is assessed by checking the sensitivity of the model to the parameter u appearing in Eq. (6). The value of (r used in base case computations is that suggested by Salatino and Zimbardi [25] for diffusion of oxygen at 500 “C, though, as the authors point out, this figure was guessed in an indirect way. Fig. 9 compares values of k,lk, as a function of rp for two values of u. The patterns previously found can still be observed in this figure. At large rp the curves converge towards k,/k, = 1. At low rp, two asymptotes are observed, corresponding to different extents of the internal surface area effectively accessed by the reactant. It is interesting to note that contributions to the gasification rate from pores smaller than about a/ 10 are negligible since the effectiveness of such pores is virtually zero.
0.001
0.01
0.1
10
100
loo0
Fig. 9. Dimensionless reaction rate constant versus Thiele modulus. BFT: D,=2.727; r,,= 1 X 1OA4
4. Conclusions A model has been developed to simulate gasification of porous carbon particles characterized by broad distribution of pore sizes in the diffusion-limited regime. On the basis of previous work and on the available experimental evidence, two topological models of the pore space have been adopted. The first is based on the representation of the particle as a surface fractal, the Menger sponge. The second is a strictly sequential pore tree. It is not a fractal, but exhibits fractal behaviour in the limit of a large number of pore levels. The kinetic model is based on the iterative application of the Thiele analysis, under the hypothesis of linear kinetics of surface oxidation and using a constitutive relationship for pore diffusivity which embodies terms relative to molecular, Knudsen and configurational diffusion mechanisms. An approximated relationship has been adopted to express the effectiveness factor in pores having low aspect ratio. Computations show that differences between model results based on the two topologies are smaller, the larger the fractal exponent of the structure. The BF’T structure is characterized by a steeper decrease of the apparent reaction rate due to a build-up of intraparticle diffusional restrictions as surface oxidation gets faster. Forfractal exponents larger than 2.3717, that is closer to those actually found in pore size distributions of a wide class of carbons, only the branching pore tree applies. Computations using the latter model have been directed to shed light on the influence of the particle size on the gasification rate and to establishing the sensitivity of the model to the value of U. the key parameter in the expression of configurational diffusion coefficients.
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The ChemicalEngineering Journal 64 ( 1996) 77-84
Modelling diffusion-limited gasification of carbons by branching pore models Salvatore Borrelli a, Michele Giordano b, Piero Salatino a** aDipartimento di Ingegneria Chimica. Llniversitb degli Studi di Napoli Federicu II, P. le Tecchio, 80125 Napoli, Italy ’ Dipartimento di Ingegneria dei Maten’ali e della Produzione. Universitci degli Studi di Napoli Federico II, P. Ie Tecchio, 80125 Nap&. Italy
Abstract A kinetic model of gasification of porous carbon particles characterized by broad pore size distributions is developed. The interaction between the intrinsic kinetics of surface oxidation and intraparticle diffusion of reactants is modelled, taking into account the wide variation of local diffusivities within the pore space as the length scale of pores changes. In particular, the model considers the configurational diffusion mechanism, which dominates transport in pores whose size is of the order of the diffusing molecule size. The model is based on the iterated application of Thiele analysis to branching pore sequences of different topology. Computations are directed to investigate the sensitivity of the model to the pore space topology, to the parameters of the pore structure and to variables determining the rate of surface reaction and of intraparticle diffusion. Keywords:
Diffusion-limited
gasification;
Branching
pore model; Kinetic model; Porous carbon particles
1. Introduction In most practical contexts, modelling gasification of single particles of porous carbons implies considering the interaction between the intrinsic kinetics of surface oxidation and kinetic resistances arising from intraparticle diffusion of gaseous reactants. In the framework of the theory of heterogeneous catalysts, this problem is dealt with by recourse to the classical Thiele analysis by the evaluation of effectiveness factors from the generalized solution of balance equations on diffusing and reacting species. Unfortunately, the straightforward application of this well-established approach to carbon gasification reaction is usually prevented by the following aspects: 1. Carbons are frequently characterized by very broad ranges of pore size distributions, typically spanning over 4 or 5 orders of magnitude of pore radii. 2. Diffusion coefficients in pores change depending on the size of cavities within which the diffusing molecule is constrained. Accordingly, molecular, Knudsen or configurational diffusion mechanisms may prevail, depending on the relative magnitude of the pore size, of the mean free path of the molecule and on the size of the molecule itself. * Corresponding author. Tel.: + 39 81768 2258; fax: + 39 81 593 6936; email: [email protected] 0923~0467/96/$15.00 Q 1996 Elsevier Science S.A. All rights reserved PUSO923-0467(96)03106-5
The combination of these factors makes the issue of modelling combined diffusion and chemical reaction in a broadly distributed pore space a rather formidable problem. Paths followed in the past to accomplish this task differ considerably from each other. Some authors mapped the actual pore space onto an equivalent regular network of pores, by analogy with the flux of electrical current in random resistor networks. Paths eventually diverged either to the formulation of Effective Medium Approximation (EMA) procedures [ l-31, or to the development of tools based on percolation concepts on regular lattices [ 4-61. The final goal was in any case the determination of effective transport properties to be used in a continuous formulation of balance equations. A different approach is that of lumping pores into a finite number of classes arranged in a hierarchical sequence. The basic idea is the iteration of the Thiele analysis to the entire cascade of pore sets in order to calculate the overall effectiveness factor of the internal surface area. Starting from the simplest micro-macro models [7,8] this concept has been extended to finite sequences of more than two pore classes r91. The idea of an arrangement of pores in a branching sequence has been strongly supported by Simons et al. in a number of papers [ 10-131. Based on purely statistical arguments, Simons et al. concluded that a broadly distributed pore
78
S. Borrelli et al. /The Chemical Engineering Journal 64 (1996) 77-84
space with a random isotropic dispersion and orientation of cylindrical pores naturally leads to branching patterns. Following this argument, these authors developed models of diffusion-limited gasification of coals based on the pore tree concept. Further simplifying assumptions implied the possibility of formulating the diffusion-reaction problem as a ldimensional problem along a curvilinear coordinate in the pore tree. The governing equations were eventually solved under limiting cases of kinetically and diffusion-controlled gasification of coal. The concept of pore branching naturally leads to recent views based on the description of the pore space of carbons as surface fractals 1141. Friesen and Mikula [ 151, Salatino et al. [ 161, Jaroniec [ 171, Foster and Jensen [ 181, to cite a few, give the evidence of fractal-like structures of carbons from the analysis of porosimetric data obtained with different techniques. The coincidence of fractal exponents obtained with various experimental methods (Hg intrusion, gas adsorption, small-angle X-ray scattering) reinforces the soundness of the use of the fractal geometry to describe the pore space of coal chars. The fractal-like structure of bituminous coals has also been simulated by numerical computation [ 191. In the present paper a model for diffusion-limited heterogeneous reaction of a porous solid characterized by a broad pore size distribution is presented. The main concern is the gasification of coal chars, whose pore spaces are amenable to a fractal description. The model is based on the recursive application of Thiele analysis to branching sequences of pores. The changes of the effective diffusivity of the gaseous reactants with the pore radius is taken into account by considering molecular, Knudsen and configurational mechanisms. Two topological models of the pore space are adopted: the first is based on a classical prototype of the fractal structures, i.e. the Menger sponge [ 201; the second is a branching model constructed as indicated by Sheintuch and Brandon [ 211. The latter model turns out not to be a fractal in a strict sense, but exhibits fractal behaviour in the asymptote of infinite number of pore classes. Computations are directed to shed light on the relevance of the topology and of the diffusional parameters to the model results.
Intraparticle diffusion of the reactant results from different mechanisms, namely molecular, Knudsen or configurational. Depending on the relevant mechanism, the dependence of the diffusivity on the pore radius changes in the following way: Molecular diffusion. Under the hypothesis of negligible Stefan flow it is: D=D, Knudsen diffusion. The diffusivity the pore lateral size. Accordingly: D=k,r
r=k,c
(1)
where r is expressed as moles of gaseous reactant consumed per unit time and unit exposed surface, and c is the reactant concentration in moles per unit volume. It is assumed that the solid is characterized by uniform reactivity, that is k, does not depend on the position on the solid surface.
changes linearly with (3)
Configurational difision. Different relationships have been reported in the literature to express configurational diffusivity in microporous solids. They are reviewed and compared by Froment and Bischoff [ 71, who noted that, in spite of the apparent differences among functional relationships expressing the influence of pore size on D, the various expressions yield results practically coincident as regards the hindrance effect within micropores. With reference to diffusion in carbonaceous micropore structures, Floess and VanLishout [ 221 suggest that configurational diffusivity is an activated process, the activation energy Ed being related to the adsorption potential of the diffusing molecule within the micropore. On the other hand Dubinin [ 231 first showed that the adsorption potential in microporous adsorbents is inversely proportional to the pore width. A combination of these findings reinforces the soundness of the empirical relationship proposed by Satterfield et al. [ 241: D=k,exp(
-a/r)
(4)
where CTis a temperature-dependent parameter defined by the equation u/r = E,IRT. As discussed by Salatino and Zimbardi [ 251 it is further assumed that k, = kg. It follows: D=kgexp(
-a/r)
(5)
Following Satterfield [ 261, the expression for the intrapore diffusivity valid over the whole range of pore sizes is obtained by combining the mechanisms considered above: 1 -=-+ D(r)
2. Theory The proposed model is based on the following assumptions: The gasification reaction obeys a first-order kinetic rate equation of the type:
(2)
1
1 D,
(6) k,rex
P(
_a
r1
2.1. Topological and geometric representation of the pore space The pore space has been modelled by using branching selfsimilar pore networks. Pores are hierarchically arranged in a sequence of levels i 2 0. Each pore is characterised by lateral size 2ri. The sequence of pore levels is constructed so as to keep constant the reduction ratio: ‘I+‘=,<1 ri
(7)
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84
For fixed pore geometrical properties, two different topological arrangements of pores have been considered: 1. Pores of ith level may branch out from any larger pore of index j < i (Fig. 1 (a) ) . The topology which arises from this arrangement is, for instance, that of the Menger sponge (Fig. 2). 2. Pores of ith level branch out only from pores belonging to the (i - 1) th level (Fig. 1 (b) ) . This arrangement is similar to that described by Sheintuch and Brandon [ 211. Whatever the pore network topology, the geometrical characteristics of the pores are assumed to be the same. Each pore is characterised by a square cross-section and by aspect ratio A = 1,/2r,,where Zjis the pore length. The square cross-section of pores has been assumed in the development by analogy with the Menger sponge. This assumption, like others relative to the shape of pores, has a minor impact on the model results when compared with the influence of the topological properties of the network (connectivity, fractal dimension, fractal range). In the construction procedure of both types of networks, ui is the surface of pores of level i prior to the introduction of pores of level j > i, 4‘” is the surface of the pores of level i as
/( a
19
b
a
Fig. 3. Definition of terms related to pore branching.
it is left over after introducing pores of level j > i; m is the number of pores of the i+ 1 level per pore belonging to the ith class. The following coefficients can be defined:
(8)
With reference to Fig. 3 we have:
p252a+b and
p&L
2a+b
From this point on the description of the pore space topologies will follow different paths for the Type 1 Menger sponge (MS) and for the Type 2 branching pore tree (BPT) .
08
(a)
Fig.
1.Schematic representation
of pore space topologies.
2.1.1. MS The construction rules of a triadic Menger sponge are reported elsewhere [ 15,201. The triadic sponge is a surface fractal with dimension D, = 2.2619. Non-triadic sponges of 2.2619
A,=uw+mi
(10)
2.1.2. BPT Each pore of level i branches into m smaller pores of level i + 1 of decreasing linear dimension with similarity ratio (Y. Coefficients j3 and y are related to the other parameters by the relationships: Table 1 TopologicaI
Fig. 2. The Menger sponge.
and geometrical
parameters
of the Menger sponges
‘heavy’
‘light’
DS
2.2619
(L m A P Y
I/3 12 1 819 112
2.3717 l/7 101 1 36149 65136
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84
80
p+!g
(11)
(1%
kn-,=ksP(l+mJ
where the contribution of pores of level n is given &a,,weighted by the effectiveness factor q,,. (n - 2) th level: proceeding in a similar manner, it is:
Py= mcy*
This arrangement
i,
of pores yields: (20)
k,-2=k,P2(l+y77n)(l+y~“-,)
Iterating the procedure, the general term of the succession of k,,_ i is given by In comparing the two pore structures, it is worth noting that the Menger sponge is a surface fractal in the strict sense. This means that the relationship holds:
=ln(Ai+lIAi)
2_D
5
Wff)
(14)
with D, constant for all values of i. The BPT is not a fractal. However it obeys a relationship of the type in Eq. ( 14) in the limit i + to. The fractal nature of the two pore structures can be made to coincide in this limit (i.e. lim DS,BpT= Ds,,.& if I and only if: PA
1+ x44s) = PEPT’YWT
The relationship
(15)
holds:
i-l (21)
k-i=kPifl(l+YTn-j) j=O
which, for the external yields: k,=k#“fi(l+
surface of the particle
lim~~)=~(PnW!;_l+l)
(22)
YVi)
BPT Model: For the BPT model, the derivation of the effective reaction rate at each pore level is formally similar to that for the MS, accounting for the different way in which the pore sequence is arranged: nth level:
(n-
(23)
1)th level:
1+ Prrl,l
The Menger sponge is a fractal object characterized by an upper cutoff of the fractal range coincident with the size of the object itself. For the sake of simplicity, and without loss of generality, the assumption that the upper limit of the hierarchical sequence of pores is coincident with the size of the object, i.e.
k” -
r, = rP
The general term of the succession
(17)
will be made even with reference to the BPT model. The theory can be easily modified in order to account for cases in which the upper cutoff r,, is smaller than the object size r,,. 2.2. Kinetic model 2.2.1. Iterative application of the Thiele analysis The iterative application of the Thiele analysis is based on the definition of an effective reaction rate constant, ki, per unit surface of the pores belonging to the ith level. It is calculated as the sum of contributions to the reaction rate from pores smaller than ri, divided by the oxygen concentration at the ith level and by the surface area ai of pores of size r,. In the following, the derivation Of ki for the two topological models will be demonstrated with the aid of the bidimensional simple branching pattern of Fig. 3. MS Model: Starting from the nth level it is: nth level: k, = k,; (n-
1)th level:
(18)
0)
i=l
k,=k,; (16)
(Level
1=
k,P + UWr-/n
(n-2)th
=W
(24)
level:
k,-,=k,P+k,P’yPrln-,+k,[Py12P?7n-,71n =UUl
k,+=k,p
+Pyr)n+
l+ ‘&j [
j=l
(25)
[Prl*rl,-,71nl
“E’
is:
1
71~
k=n-i+l
(26)
which, for n - i = 0, yields:
b=k,P[ kP’YifITj] j=O
(27)
i=o
2.2.2. Evaluation of effectiveness factors Calculation of k, -i and of b from, respectively, Eq. (21) and Eq. (22) for the MS and Eq. (26) and Eq. (27) for the BF’T requires the knowledge of the effectiveness factors 77i for each class of pores. This is accomplished by relating vi to Thiele moduli pi of the pores of level i:
(28) For the sake of simplicity the expression for the effectiveness factor of a dead-end pore of cylindrical shape is adopted:
S. Borrelli et ul. /The Chemical Engineering Journal 64 (19%) 77-84
However, this expression has been modified to satisfactorily comply with the case in which the penetration depth of the reactant within the pore becomes comparable to or smaller than the pore radius. This takes place as 40,+ CCand in this limit Eq. (29) predicts vi + 0. This limit is physically sound only for pores having infinite aspect ratio, i.e. the ratio between the pore length and the lateral size. For pores of finite length a minimum non-zero value of the pore effectiveness factor exists even in the limit of large Thiele modulus, that is for infinitely fast reaction’. This lower bound is given by the ratio of the pore cross-sectional area over its internal surface area. In terms of the already defined geometrical parameters:
vy=- 1-P Pr An approximate expression of vi which preserves behaviour of Eq. (29) (as cpr= 1) and of Eq. (30) cp,-+ + ~0) is simply given by:
1-p +
%’ PY
(30) the (as
(31)
and will be adopted in the computations. In either the MS or BPT versions of the model, the computation procedure consists of iteratively evaluating Eq. (21) (or Eq. (26) ) starting from the nth level, up to the 0th level. At each iteration step calculation of k,,_i is preceded by the evaluation of qn _ i + 1 by means of Eq. (31), with ‘pn_i+l given by Eq. (28)) and once k, _, + 1 is known from the previous step. The overall reaction rate 8 is given by: ~=JWkoAocS
(32)
The choice of the parameter IZin Eq. (22) and Eq. (27) deserves some discussion. Its value can be fixed if there is evidence of the existence of a lower bound to length scales associated with the pore space. Otherwise, it is easily recognized that the successions:
P’I-p+Yrlj)
81
values of rj (of the order of the parameter a), of diffusional resistances stronger than those determined by the Knudsen mechanism. One important consequence is that the choice of n, in this case, becomes merely a problem of numerical convergence of the series in Eq. (22) and Eq. (27). This is the way it has been treated in the computations.
3. Results and discussion The results of the computations are presented and discussed with the aim of outlining the influence of a number of variables either related to the topology and the geometry of the pore space or to the kinetics of the reaction/diffusion processes on the apparent gasification rate of the solid. 3.1. The influence of the pore space topology Fig. 4 compares the cumulative surface areas of the pore spaces for different choices of the topological parameters. In particular, curves for D,=2.2619 and D,=2.3717 are presentedforeithertheMS(Eq. (lO))andtheBPT(Eq. (13)) pore structures. These choices of D, correspond to the limits within which Menger sponges, ranging from ‘heavy’ to ‘light’ ones, can be constructed. For a given D,, the reduction ratio (Y,the pore aspect ratio h and the multiplication factor m have been kept at the same values regardless of the topology. For D, > 2.3717 only the BPT structure applies. Since equal values of A0 have been assumed for the two topologies, Eq. (16) yields: +1
li
(33)
The non-fractal nature of the BPT structure, as opposed to the fractal character of MS, is evident from the deviation from linearity of the curves for BET at large values of rr. In this respect, it is interesting to note that several authors found similar ‘bending’ of plots like those of Fig. 4 determined from
j=l
t
for the MS, and:
-
:----.MSD,-2.3717
AI*,
-------BPT D,=2.3717
-
-MS D,-2.2619 BPT D,=2.2619
P'Y'nTj j=O
ld
for the BPT, appearing in Eq. (21) and Eq. (26), converge respectively to 1 and to 0 as i + to. This property is a direct consequence of the following conditions: ( 1) the succession of ‘pi is strictly increasing in the neighbourhood ofj + @J;(2) a minimum value of rli, given by Eq. (30)) exists as uj + CO. In particular, Condition 1 is fulfilled due to the onset, at small ’ This apparent inadequacy of the Thiele analysis arises from the limitations intrinsic to the application of a 1-dimensional approach to the problem of diffusion and reaction in a pore of a finite length. Multidimensional approaches, or models based on integral approaches, e.g. those developed by Chambri? [27] and Verhoff and Strieder [28], would overcome these limitations.
1 ot
d
1 00 lo-5
lir’
10-l
l@ r/r,
Fig. 4. Dimensionless cumulative surface areas versus pore space size fat various pore topologies and D..
82
S. Borrelli et al. /The Chemical Engineering Journal 64 (19%) 77-84 -
South
African
A/A,
100 Lo/b 10
1 O1 0.001
Marine South African
2.6 2.9
0. 0.9
1.9 1.35
I
2
1 rfr0 Fig. 5. Dimensionless cumulative surface. areas for samples of 2 coal chars. Curves are best-fit lines based on the BPT pore model.
kO/lu
0.1
10
1 0.001
0.01
0.1
Fig. 6. Dimensionless reaction L&=2.2619; r”= 1 X lo-“
t
rate constant
0.1
Fig. 7. Dimensionless reaction 0.=2.3717; rp= 1 X lo-“
I
11 0.01
0.01
10
versus
100
Thiele
lCO0
modulus.
porosimetric characterization of coal and chars at large values of pore radius [ 15,161. Further evidence is given in Fig. 5, where cumulative surface area distributions obtained by mercury porosimetry of samples of chars from two medium-rank coals are compared with curves obtained by the application of the BPT pore mode1 with suitably defined geometrical and topological parameters. In both cases, values of D, = 2.6-2.7 are asymptotically observed at small values of the pore radius, but significant deviations are observed at r = 1 pm. Fig. 6 and Fig. 7 report on the influence of the pore space topology on the gasification rate. Results are presented in terms of the ratio between the apparent reaction rate constant k. referred to the external surface A, divided by the intrinsic reaction rate constant k,. k,lk, values are plotted as functions of a particle Thiele modulus of the sample, defined as follows:
1 + rate constant
10 versus
100 Thiele
loo0 modulus.
external surface area. This asymptote is usually neglected in rj versus cpplots from classical Thiele analysis as it is implicitly assumed that the external surface area is negligible with respect to the internal. At low values of cpanother asymptote is attained. The ordinates of the curves as cp+ 0 substantially reflect the values of A,IA, where i is the pore level at which significant diffusional resistances associated with the configurational mechanism arise. Differences between the asymptotic values of k,lk, for the two topologies as cp+ 0 are related to the different values of A,/A,, expressed by Eq. (33). The decay of the apparent reaction rate as rp increases is steeper for the BPT pore topology than for the MS. The maximum slopes of plots in Fig. 6 and Fig. 7, corresponding to the inflection point of the curves, are about - 1.1 (D,=2.2619) and -0.95 (0,=2.3717) for the BPT. The maximum slopes for the MS are of the order -0.6 and - 0.65. This behaviour is easily interpreted in the light of the following argument. As the global Thiele modulus increases, that is, as the importance of intraparticle diffusional resistances is raised, the fall of the effectiveness factor takes place earlier for coarse pores, i.e., those in which the dominant diffusion mechanism is molecular (let us refer to these pores as M pores), than for pores where the Knudsen mechanism prevails (K pores). In the MS arrangement a significant fraction of K pores branches out directly from the external surface, whereas in the BPT arrangement the same pores can only be reached through M pores. For both topologies, the transitional range between the asymptotes broadens as D, increases. On the whole, the results corresponding to the two topologies, all other parameters remaining the same, get closer to each other as the fractal exponent D, gets larger.
8h2rPk, ‘= C--D(Q,,,>
3.2. The influence of the particle size
Parameters of the curves which have been kept constant in the two figures are the particle size (rt, = 1.10e4 m), the parameter kd of the expression for Knudsen diffusivity (k, = 500 m s-l), and the parameter of the expression of configurational diffusivity ((T = 2.10p8 m). Notably, two asymptotes are recognized in the plots. As cp+= + 00 all curves tend to the value b/k, = 1, which expresses the fact that under strong intraparticle diffusional limitations the reaction rate is proportional, through k,, to the
Further analysis has been only carried out with reference to the BPT pore structure, using D, = 2.7. Geometrical and topological parameters adopted in the computations are: m = 20, (Y= l/3, A = 1. Fig. 8 compares curves of b/k, versus Q or different particle radii rP. The effect of increasing r,, is that of increasing the apparent reaction rate throughout the range of Q. In fact, the effect of rP on the overall reaction rate is twofold. On the one hand, increasing rP increases A,,, and
S. Borrelli et al. /The Chemical Engineering Journul64
(19%) 77-84
83
loww 10000 kO/ks
low 100
10 10 I 0.001
0.01
0.1
10
loo
low
Fig. 8. Dimensionless reaction rate constant versus Thiele modulus. BPT: D,==2.727; 0=2X lo-’
this, in turn, increases the reaction rate. On the other hand, increasing rP shifts the upper cutoff of the pore sequence towards larger values and widens the range of pore intervals in which the more efficient molecular and Knudsen diffusion mechanisms dominate transport. As a consequence, both the extent and the effectiveness of the internal surface area are increased. It is worth noting that the asymptotic value of kJ k, at low 50 does not increase in proportion to the particle radius. This implies that reaction rates when expressed per unit volume of the particle, rather than per unit external surface area, decrease with particle radius.
3.3. The influence of the difSusion parameters
The values of the molecular diffusivity D, and of the constant kd, which determines the value of the Knudsen contribution to the overall diffusivity, are well established. In particular the values of D, = 5. 10m5 m2 se1 and of kd = 500 m s-’ used throughout this work correspond to oxygen molecules diffusing at 500 “C. The relevance of configurational diffusion to the pore hindrance in a fractal pore space has been addressed by Coppens and Froment [29]. These authors based their development on the assumption that a sharp boundary can be established between accessible and non-accessible pores. In the present analysis the role of configurational diffusion is assessed by checking the sensitivity of the model to the parameter u appearing in Eq. (6). The value of (r used in base case computations is that suggested by Salatino and Zimbardi [25] for diffusion of oxygen at 500 “C, though, as the authors point out, this figure was guessed in an indirect way. Fig. 9 compares values of k,lk, as a function of rp for two values of u. The patterns previously found can still be observed in this figure. At large rp the curves converge towards k,/k, = 1. At low rp, two asymptotes are observed, corresponding to different extents of the internal surface area effectively accessed by the reactant. It is interesting to note that contributions to the gasification rate from pores smaller than about a/ 10 are negligible since the effectiveness of such pores is virtually zero.
0.001
0.01
0.1
10
100
loo0
Fig. 9. Dimensionless reaction rate constant versus Thiele modulus. BFT: D,=2.727; r,,= 1 X 1OA4
4. Conclusions A model has been developed to simulate gasification of porous carbon particles characterized by broad distribution of pore sizes in the diffusion-limited regime. On the basis of previous work and on the available experimental evidence, two topological models of the pore space have been adopted. The first is based on the representation of the particle as a surface fractal, the Menger sponge. The second is a strictly sequential pore tree. It is not a fractal, but exhibits fractal behaviour in the limit of a large number of pore levels. The kinetic model is based on the iterative application of the Thiele analysis, under the hypothesis of linear kinetics of surface oxidation and using a constitutive relationship for pore diffusivity which embodies terms relative to molecular, Knudsen and configurational diffusion mechanisms. An approximated relationship has been adopted to express the effectiveness factor in pores having low aspect ratio. Computations show that differences between model results based on the two topologies are smaller, the larger the fractal exponent of the structure. The BF’T structure is characterized by a steeper decrease of the apparent reaction rate due to a build-up of intraparticle diffusional restrictions as surface oxidation gets faster. Forfractal exponents larger than 2.3717, that is closer to those actually found in pore size distributions of a wide class of carbons, only the branching pore tree applies. Computations using the latter model have been directed to shed light on the influence of the particle size on the gasification rate and to establishing the sensitivity of the model to the value of U. the key parameter in the expression of configurational diffusion coefficients.
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