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The pore tree structure of porous char
Nineteenth Symposium (International) on Combustion/The Combustion Institute, 1982/pp. 1067-1076
THE PORE TREE STRUCTURE OF POROUS CHAR GIRARD A. SIMONS
Physical Sciences Inc. Andover, MA 01810 U.S.A. A theory is presented which describes the tree-like pore structure of carbon char. Each pore that reaches the exterior surface of a char particle is depicted as the trunk of a tree. A statistically derived and empirically verified pore distribution function specifies the size distribution of the tree trunks on the particle surface. A continuous branching description of the pore tree predicts the number and size of the pores as a function of the skewed distance into the pore tree. The pore structure is completely specified through measured values of the porosity 0, internal surface area sp, and the particle radius a. The oxidation rate of carbon char is shown to be sensitive to the structure of the pore tree and the present theory is shown to accurately model the oxidation data base. 1. Introduction A theory describing the structure of porous carbon char has been reported by Simons and Finson [1] and by Simons [2]. The unique feature of the theory is that it possesses cylindrical pores whose branching sequence is depicted as an ordinary tree or river system. The concept of the pore tree was developed from a statistically derived and empirically verified pore distribution function. This distribution function quantitatively describes the number of pores with radius between rp_ and rp_ + drp and is continuous over several decades in the pore dimensions. The primary significance of the pore tree is the influence of the pore branching sequence on the gasification process. The reactant species must diffuse through the tree trunk and the larger branches in order to reach the total surface area St contained in the fine pores. The net reactivity of each pore tree may be limited by the kinetic rate acting at the surface St, or by diffusion through the trunk and branches of the tree. The smaller trees tend to be kinetically limited while the larger trees are diffusion limited. This conclusion is supported by Walker [3] through independent derivation and is just the reverse from that which is concluded [4-9] for single, isolated pores. Hence, the concept of pore branching is fundamental to the construction of a char gasification model. Using the pore tree theory of pore structure, a transport theory has been developed [10] and demonstrated to accurately predict the rates of char oxidation [11] and gasification [12]. The reactivity of a pore tree is assumed to be either diffusion limited in the tree trunk or kinetically limited on the surface St. The reactivity of each tree is integrated
over the entire size distribution of trees to determine the net oxidation/gasification rate of the char. In developing the transport theory [10], the details of the branching of the pore tree were not developed and the continuous evolution of the reactivity of the tree from kinetic to diffusion control was not demonstrated. It is the purpose of this paper to develop the details of pore branching and to demonstrate that the limiting cases considered in the transport theory [10] are sufficiently accurate to describe char reactivity. In addition, the current description of the pore tree provides a framework within which the role of species transport during pyrolysis may ultimately be assessed. 2. Pore Structure Theory Before developing a detailed description of the pore tree; the pore structure theory [1, 2] is briefly reviewed. Consider a spherical char particle of radius "a" that contains randomly distributed cylindrical pores of radius r~ and length/?~. The number of pores within the'bulk volume ~whose pore radius is between rp and rp + drp is denoted by Vf(rp)drp. The pore volume is expressed as the 'trr z ep moment of f(rp) and the internal surface area is the 2~rrp{?p moment of f(rp). It is convenient to express the pore structure in terms of a second pore distribution function. The quantity $(r_)A drp denotes the number of pores within the }~ulk cross-sectional area A whose pore radius lies between r P and r P + dr_~' The pore distribution function ~(rp) represents an average over all inclination angle between the axis of the pore and the normal to the plane containing the area A. Due to the random orientation of the pores, the
1067
1068
COAL COMBUSTION KINETICS AND MECHANISMS
intersection of a circular cylinder with a plane is an ellipse of average area 21rrZ_. Hence the porosRy is the 2Wrp moment of g(rp) and the internal surface area is the 47rrp moment of ~(rp). The pore distribution functions are clearly not independent. Simons and Finson [1] have related ~(rp) to f(rp) by 1
g(%) = 2 tpf (rp).
(1)
Equation (1) simply states that the probable number of pores intersecting an arbitrary plane increases with the length of the pores and with the total number of pores. The length of a pore is determined by an arbitrary intersection with another pore and is expressed [1] as a collision integral over the pore distribution functions. The analysis suggests that f~, 9 3 ~' g(r,) and f(r,) are proportional to rp, 1/rp and r 4 ,, . 1/rp, respectwely. The constants of proportmnahty are obtained from integral constraints, i.e. the total porosity and internal surface area contained in the pore structure9 The chars contain a continuous distribution of pore sizes from rmin to rm,x. The pore distribution function ~(rp) is given by 0 ~(rp) - 2,rr[3rp3,
(2)
where 0 is the total porosity and
~ = ,n(rmax I \ rmin
]
The expression for f(rp) is given by 04/3
f (%) = 1ff3Korp4,
(3)
a significant fraction of the particle radius. The radius of the smallest pore is determined from %, the specific internal surface area (typically several hundred m2/g), and is given by r~in = eO /f~pssp
(5)
where Ps is the density of the non-porous solid (2.21 g/cc). The radius of the largest pore is determined under the assumption that there are no internal voids in the particle. The largest pore must terminate at the exterior surface of the particle. This constraint leads to 2a01/3 rm~ -
3Ko
,
(6)
where the constant Ko reflects the pore aspect ratio (pore length to radius) and cannot be determined theoretically. An empirical measure of Ko is obtained [1] by comparing the predicted values of rmax to the data of Mackowsky and Wolf [15]. The results are not unique, but the data suggest a value of Ko near 5. This value of Ko then suggests that the pore aspect ratio is of the order of ten. Consider a small pore located in the interior of the char particle9 If the aspect ratio of this pore is of the order of ten, it cannot reach the particle exterior directly. It must branch from a larger pore which, in turn, branches from a still larger pore, etc. This picture resembles that of a tree or river system. Random pore models proposed by Gavalas [6,7] and by Bhatia and Perlmutter [8,9], consider isolated pores whose length is equal to the particle radius. These models allow all pores to reach the particle exterior directly and branching is not required. Hence, the pore tree model is quite distinct from other [6-9] random pore models. While the concept of the pore tree was previously introduced [1], a detailed description of the tree had not been generated. Such a description is developed in the following sections.
where Ko is an arbitrary constant of integration which relates Cp to rp [1]
~p = Ko rJO I/3.
3. Statistical Description of the Pore Tree (4)
and it has been assumed that pyrolysis and gasification increase r_ at constant e_. This insures that the pore distribution functionseare invariant in a Lagrangian frame moving with rp in time. These expressions for f(rp) and ~(rp) have been validated [1] through extensive comparisons of the predicted volume and surface area distributions with the data of Berger et al. [13] and with that of Stacy and Walker [14]. The pore dimensions range from a microscale, of the order of angstroms, to a macroscale which is
A spatially dependent description of the pore tree, i.e., a mathematical description of the pore tree depicting the number and size of the pores as a fimction of distance into the pore tree, cannot be calculated without first extending the statistical description of the pore tree. The concept of the pore tree depicts that every pore that reaches the exterior surface of the char particle is the trunk of a tree. Let Nt be defined as the branch distribution function where N t drp is the number of pores of radius rp (within size range drp) in a tree whose trunk radius is r t. Since
PORE TREE STRUCTURE 4/3 1ra~f drp represents the total number of pores of radius rn (within size range drp) in a sphere of radius a, the distribution function N t can be related to f by expressing 4/3 ~ra~f drp as the sum of all pores of radius rp contained within every tree in the char sample, plus all pores of radius rp that are themselves the trunk of a tree. Hence,
While a statistical description of the pore tree has depicted the number of pores of various sizes within the tree, an additional hypothesis is necessary in order to depict the branching sequence. This hypothesis is developed in the following section.
~prnax
~ ~a3f =
N t ~(rt) 4~a2drt + 4"tra2~(rp),
Nt = /'t//rp. 3 4
(8)
The branch distribution function Nt completely characterizes the pore tree. The internal surface area and pore volume associated with each pore tree are denoted by St(rt) and Vt(rt), respectively, and are expressed as the sum of the contributions from the trunk and that from the branches.
St(rt) = 2~rrte t +
~rrt
2~rrpepNtdr p
(9)
~rr~tpNtdr p
(10)
rain
Vt(rt) = ,r163 t +
~rrt
4. Continuous Branching Model
(7)
where ~(rt) is the number of tree trunks per unit external area of the char sample and only those trees whose trunk radius is greater than rp may contain a pore of radius rp. Using 8, f and rm~ as given by Eqs. (2), (3) and (6), respectively, Eq. (7) is identically satisfied by
1069
Each pore that reaches the exterior surface of the char particle has been depicted as the trunk of a tree. To describe the sequence of the branching within a pore tree, it is necessary to develop a basic mathematical rule of branching that is consistent with the statistical description of the pore tree. Consider a pore tree which branches in the manner illustrated in Fig. 1. A branching structure with a continuously decreasing pore radius is depicted as being attached to a trunk of radius r t. After a statistical length Ct, the trunk branches into smaller pores of radius rp which, after statistical length ep, themselves branch into even smaller pores, etc. Any distance x measured along the pores corresponds statistically to a particular pore radius rp. Hence, both the radius rp and number of pores of radius r~ is a unique single valued function of the skewed ~tistance x into the pore tree. Let n(x) represent the number of pores of radius rp at location x in a tree of trunk radius r t. The internal surface area and pore volume in the pore tree are expressed as
min
St(rt) =
Using Eq. (8) for Nt, St and Vt become
St(rt)=2"irrt~t(rr-~in) ( 1 - 0 )
(11)
2"rrrp n(x)dx + 27rrte t
t
(13)
and
~t
xt
Vt(rt) =
and
Vt(rt) = ~rr2tet (1 + ln(rt/rmin) ),
(12)
where the (1 - 0) in St has been included to account for pore combination [2]. The above expression for St(rt) is identical to that previously derived [1] from an integral constraint. The surface area associated with the pore tree may be several orders of magnitude greater than the surface area of the trunk. However, the volume of the pore tree may, at most, be one order of magnitude greater than that of the trunk. It should also be noted that the above expressions for St and Vt reduce to those appropriate to a single cylindrical pore in the limit of r t ---> rmin (the leaf of the tree). Furthermore, the integrals of Wt(rt) and St(l't) o v e r all ~(rt) recover the total pore volume and surface area, respectively.
~rr~ n(x)dx +
~rr~et,
(14)
! rp Appropriate
{ \ \ \ \ \ \ \ \ \ ~\ \ l~\ ~i ~
x
Mean Radius at x
t
I=\\l~\\\\\\\~\\\\\\~Particle Surface
FIG. 1. Continuous branching description of the pore tree.
1070
COAL COMBUSTION KINETICS AND MECHANISMS
respectively, where x t is the skewed height of the pore tree measured from the edge of the char partjcle. Comparison of Eqs. (13) and (14) to Eqs. (9) and (10) indicates that n(x) is given by n(x) = - fpNt ~ ,
(15)
ax
where s and N t are given by Eqs. (4) and (8), respectively, and the negative sign appears because the limits of integration are such that x = x t corresponds to rp = rrnln and x = ~t corresponds to rp = r t. Equation (15) is a direct consequence of the definitions of n(x) and Nt(rp). The number of pores of radius r_ and length ~_ in a tree of trunk radius r t is denoted by Ntdr p ~vhereas n(x) represents the number of pores of radius rp and length dx in the same tree. It therefore follows that n(x) is ep/dx times greater than Ntdrr, and this statement is expressed by Eq. (15). Equation (15) is an expression relating two unknowns: n(x) and drp/dx. Since all previously determined pore distribution functions have been shown to scale as the pore radius to a constant power, it is logical to assume that n(rp) also scales in the same manner. It is therefore assumed that drp/dx is proportional to r ; where to is a constant. The constant of proportionality is set by the requirement that each tree possesses one trunk (n = 1 at rp = rt). Hence drp/dx and n(x) are expressed as dr--e-P=-rt(rp) ~ dx et \ r t /
(16)
made by examining the skewed height of the pore tree, x t. The value of x t is expressed as the integral of dx/drp over all rp between r t and rmin. It suffices to say that only the value of to identically equal to unity allows the pore tree to degenerate into a single pore in the limit of rt --+ rmin. Thus, the physical description of the pore tree is dependent upon the value of to and a realistic description requires that to be identically equal to unity. While the continuous branching model of the pore tree has been shown to be quite plausible, the true test of the model is its ability to accurately describe the transport properties of the pore tree. The oxidation of the pore tree is examined in the following sections. 5. Oxidation Within the Pore Tree When porous char is placed in an oxygen environment, Oz will diffuse into the pore tree and oxidize the carbon which constitutes the walls of the pores. The diffusion of 02 through n pores of radius rv is balanced by the reaction of O~ at the walls of the pore. This is expressed as d I"
~ dc\
~npcDTr~ ~ )
= n2~rr~,k
(18)
where c denotes the gaseous mass fraction of oxygen, Pc is the gas density within the pore, D is the self-diffusion coefficient of oxygen in an arbitrary gaseous environment and k is the kinetic rate with which oxygen reacts with carbon. The total oxidation rate of the pore tree, Ivlt, is related to the gradient of c at x = 0 by
and n ~
(
r_,/3-~
\ rp/
(17)
respectively. The continuous branching description of the pore tree has related physical space (x) to the pore radius (r_). It therefore follows that some constraint in physical space must be used to determine to. The expression for the volume of the pore tree, Eq. (14), is rewritten to include only that pore volume between r t and rp, i.e., between x = 0 and x. For to = 1,
V,(x) = ~ x which implies that pore volume is homogeneously distributed in physical space. No other value of to satisfies this basic requirement of the porous media. An independent choice for the value of to is
Ir t = - p c D ~ r ~
aX
(19) O'
However, to obtain the value of dc/dx at x = 0, Eq. (18) must be integrated subject to the boundary conditions that c = c o at x = 0 (rp = rt) and dc/dx = 0 at x = x t (rp = rmin). It is assumed that the oxygen concentrations are sufficiently low that the kinetic rate k is adsorption limited, i.e., k is proportional to c k = k a pG c,
where Pc is the gas pressure and the adsorption rate constant ka is a function only of the char temperature. The diffusion of oxygen within the pore is assumed to be dominated by gas-wall collisions. The corresponding diffusion coefficient D is given by D = 2Vrp/3,
(20)
PORE TREE STRUCTURE 10
where ~z is the mean thermal speed of an O2 molecule at the char temperature and 2rp is the mean free path between collisions with the walls. Within these restrictions, the total oxidation rate of the pore tree is obtained from Eq. (19) and the solution for the oxygen concentration profile, In the limit of rt >> rmin, Mt is expressed as ( 4kaPGPC~1/2 r e2~ " 1 1
= ,etCo \-
)
1071 1
|
I
l.-
KINETICALLY LIMITED r
/
1
_ D,
oS.ON L'M'TEO
~ ~ S O L U T I O N
..=. ,,m 10_ 1 =.
(21)
=,
=,
where
I lO2
t lo
10-2
I lO3
lO4
rt / rmin -- RADIUS OF TREETRUNK/MINIMUM PORE RADIUS
(3ka'Z~Pc~]'2(~mi~): rt
FIC. 2. Transition from kinetic to diffusion control. Equation (21) clearly demonstrates two limits: for K > 1, the Thiele [4] diffusion solution is recovered whereas for K < 1, the oxidation rate is limited by the kinetic rate k acting on the total surface of the pore tree, St(rt). This demonstrates that large trees (large K) are diffusion limited whereas small trees (small K) are kinetically limited. The transition from kinetically limited to diffusion limited oxidation is illustrated in Fig. 2. The total oxidation rate of the pore tree is normalized by the kinetic rate and plotted as a functign of trunk radius. For all practical purposes, Mt may be approximated by the kinetic solution for K < 1 and by the diffusion solution for K > 1. This is precisely the approach used in developing the char transport [10] and char oxidation models [11]. Validation [11] of the char oxidation model offers indirect verification of the detailed shape of the pore tree.
consuming oxygen faster than it can diffuse through a non-reacting trunk of length f r This intermediate mode does not occur for the case of o~ equal to unity. The char transport model [10] has been modified to accommodate the case of to = 0. The predictions for the oxidation rate of carbon char are illustrated in Fig. 3, The oxidation rate is determined in grams of carbon removed per cm2 o f external surface area per s per arm 02. The theory is compared to the data of Field [16]. The choice of to = 0 is clearly wrong whereas that of to = 1 is consistent with the data. Hence, the transport processes are sensitive to the structure of the pore tree and the details of the pore tree derived herein have been validated with respect to char oxidation.
I
6. Sensitivity of the Oxidation Rate to the Shape of the Pore Tree Since the transport processes provide the most sensitive tests for the structure of the pore tree, a demonstration of the sensitivity of the char oxidation rate to the constant to will further validate the choice of to equal to unity. The sensitivity of the char oxidation rate to the value of to is readily demonstrated by determining Mt for oJ = 0. Equation (18) again recovers the Thiele [4] solution for large trees and the kinetic solution for small trees. However, there is also an intermediate solution which is diffusion limited, but quite distinct from the Thiele diffusion solution. The Thiete diffusion solution represents a diffusion process whereby the gradient in oxygen concentration is established by oxygen consumption on the walls of the pores. The intermediate solution represents the limit in which the leaves and branches of the tree are capable of
I
i
I
I
9 FIELD [16] 38//. OIAMETER PARTICLE
8to) = o.8o 10% 02 (0.1 atm 02 )
E= I
//,HEORy,,O.',,
10--1
2
I/
10--2 1000
1200
I 1400
I 1600
I 1800
I 2000
TEMPERATURE- K
F]C. 3. Oxidation of low rank char.
2200
1072
COAL COMBUSTION KINETICS AND MECHANISMS 7. Summary and Conclusions
A continuous branching description of the pore tree has been developed. The pores branch in a continuous downsizing manner. Both the radius and the number of pores is a single valued function of the skewed distance x into the pore tree. The number of pores of radius rp is given by
n(x)- ~(x)' and the coordinate x is related to rp by
dry_
rp
dx
et '
The size distribution of tree trunks on the exterior surface of the particle is given by ~(rt). Hence, the entire pore structure is completely specified by measured values of the char porosity, internal surface area, and particle radius. The features of the pore tree are best verified through its transport properties. Char oxidation is shown to be sensitive to the structure of the pore tree. The continuous branching model developed herein accurately reproduces the kinetic and diffusion solutions which were utilized in developing the transport model [10]. The validation of the transport model in describing char oxidation [11] indirectly verifies the continuous branching description of the pore tree. The current description of the pore tree provides a realistic pore structure in which pore evolution and species transport during pyrolysis may be assessed. It is anticipated that such a theory may be used to assess the secondary chemistry which occurs during pyrolysis. These important potential functions of the pore tree model are under current investigation [17, 18].
ep ft Ivlt n(x) Nt PG rn rmln rm~ rt St sn V Vt x xt
Greek Symbols 13 0 K PG p~ oJ
Work supported by U. S. Department of Energy, Pittsburg Energy Technology Center, under contract DE-AC22-80PC-30293.
REFERENCES 1. SIMONS, G.
2. a C
Co
D
f k
ko
Ko
radius of char particle cross sectional area of sample mass fraction of oxygen in pore mass fraction of oxygen at surface of char particle oxygen self diffusion coefficient pore distribution function per sample volume pore distribution function per sample area kinetic rate: mass of reactant gas/area-time adsorption rate constant constant ~ ratio of pore length to diameter =5
In (rm~xlrrnin) porosity constant, Eq. (22) gas density density of non-porous material constant, Eq. (16)
Acknowledgment
Nomenclature A
length of pore of radius ro length of the pore that represents the trunk of a tree oxidation rate per pore tree number of pores of radius rp at distance x into pore tree branch distribution function gas pressure radius of pore radius of smallest pore radius of largest pore radius of the pore that represents the trunk of a tree surface area of a pore tree specific internal surface area (area/mass) mean thermal speed of a molecule volume of sample volume of a pore tree skewed distance into pore tree skewed height of pore tree
3. 4. 5. 6.
A. AND FINSON, M. L., "The Structure of Coal Char: Part I. Pore Branching," Comb. Sci. Tech,, 19, 5 & 6, 217 (1979). SIMONS, G. A., "The Structure of Coal Char: Part II. Pore Combination," Comb. Sci. Tech., 19, 5 & 6, 227 (1979). WALI~ER,Jl~., P. L., "Pore System In Coal Chars. Implications for Diffusion Parameters and Gasification," Fuel, 59, 809 (1980). THIELE, E. W., "Relation between Catalytic Activity and Size of Particle," Ind. Eng. Chem., 31, 916 (1939). WHEELER, A., "Reaction Rates and Selectivity in Catalyst Pores," in Advances in Catalysis, 3, 249 (1951). GAVALAS,G. R., "A Random Capillary Model with Application to Char Gasification at Chemically Controlled Rates," AIChE Journal, 26, 577 (1980).
PORE TREE STRUCTURE 7. GAVALnS, G. R., "Analysis of Char Combustion Including the Effect of Pore E n l a r g e m e n t , " Combustion Science and Technology, 24, 197 (1981). 8. BHATIA, S. K., AND PERLMUTfER, D. D., "A Random Pore Model for Fluid-Solid Reactions: I. Isothermal, Kinetic Control," AIChE Journal, 26, 379 (1980). 9. BHATIA, S. K., AND PERLMUTrER, D. D., "A Random Pore Model for Fluid-Solid Reactions: II. Diffusion and Transport Effects," AIChE Journal, 27, 247 (1981). 10. SIMONS, G. A., " C h a r Gasification: Part I. Transport Model," Comb. Sci. Tech., 20, 3 & 4, 107 (1979). 11. LEWIS, P. F. AND SIMONS, G. A., "Char Gasification: Part II. Oxidation Results," Comb. Sci. Tech., 20, 3 & 4, 117 (1979). 12. SIMO~S, G. A., "The Unified Coal-Char Reaction," Fuel, 59, 143 (1980). 13. BERGER, J., SIEMIENIEWSKA, T. AND TOMKOW,
14.
15.
16.
17. 18.
1073
K., "Development of Porosity in Brown Coal Chars on Activation with Carbon Dioxide," Fuel, 55, 9 (1976). STACY, W. O. AND WALKER, JR., P. L., "Structure and Properties of Various Coal Chars," Coal Research Section, College of Earth and Mineral Sciences, The Pennsylvania State University, September 1972. MACKOWSKY, M. T. AND WOLF, E. M., "Microscopic Investigations of Pore Formation during Coking," in Coal Science, Advances in Chemistry Series 55, Am. Chem. Society, ed. R. F. Gould (1966) p. 527. FIELD, M. A., "Rate of Combustion of Size Graded Fractions of Char from a Low-rank Coal Between 1200 K and 2000 K," Comb. and Flame, 13, 237 (1969). SIMONS, G. A., "Pyrolysis I. Evolution of the Pore Tree," in preparation. SIMONS, G. A., "Pyrolysis II. Species Transport through the Pore Tree," in preparation.
COMMENTS Stephen Niksa, Sandia National Laboratories, Livermore, USA. The transport equation relating the flux of oxygen through arbitrary passages in the particle to the net consumption by surface reaction reduces the pore tree to a one-dimensional tube with diminishing cross section. As the pore radius decreases the mass transport mechanism shifts from bulk flow to Knudsen flow to, perhaps, activated diffusion. Have you a c c o u n t e d for the various mechanisms and, are the predictions from the model sensitive to the values of the diffusivity or the points in the pore at which these transitions occur?
Author's Reply. Both Knudsen and continuum diffusion are considered in the transport model [10]. The diffusion coefficient (D) is the smaller of either the Knudsen or continuum value. There is an "'order one" error in D at the pore radius where transition occurs. Upon integrating over all pores, this error becomes minimal. Activated diffusion is not explicitly included. This concept is replaced with that of kinetic control on the internal surface area. This area is a function of burnoff but not diffusion residence time. The continuous branching description of the pore tree does not allow small branches to emerge from large trunks. In this respect, the pore tree fails to duplicate the common shade tree. While the similarities between the common shade tree and the pore tree far outnumber the differences, one still cannot use this analogy to prove the concept.
Eric Suuberg, Brown University, USA. How was the diffusion coefficient determined? Does it include any correction for bulk flow, which your transport model did not explicitly allow for? This bulk flow correction would probably be small under combustion conditions, but u n d e r pyrolysis conditions during which internal pressures were predicted to be of order 1000 atm this may not be true. Author's Reply. Within any given pore, the diffusion coefficient is the smaller of either the Knudsen or continuum diffusion values [10]. The generation or depletion of moles (one Oe ~ two CO) induces bulk flow within the pore. During char oxidation, this p h e n o m e n o n may be described through a moderate correction to the diffusion coefficient. I do not include it. During pyrolysis, bulk flow is the dominant transport phenomenon and is included in that model [17, 18].
Nahum Gat, TRW-Space & Technology Group, USA. As combustion proceeds inside the pores and branches, the walls between them disappear. Then initially the surface area increases but as the walls burn out the surface area drops significantly. Is that the correct physical process and does the model account for this phenomenon?
1074
COAL COMBUSTION KINETICS AND MECHANISMS
Author's Reply. This exact process i s described by the present model [2] and the results have been used to successfully describe [12] the universal burnoff profile in the limit of kinetic control.
C. J. Lawn, CEGB, Southampton, U.K. Do I understand correctly that because your pore "trees' are all 'rooted' on the outer surface of the particle, the pore size distribution in your model becomes finer as one moves radially inward? If so, is this realistic? Author's Reply. As one moves into a single tree, the pore size becomes finer. However, trees are rooted from all sides of a porous sample and the largest trees are as large as the sample. Hence, as one moves radially inward, one may move either into or out of a tree. In this manner, a homogeneous, isotropic pore structure is retained.
j. Lahaye, CNRS, France. In your model you have taken into consideration only macropores and mesopores larger than 10 nm diameter. In char, small mesopores and micropores are expected. From the point of view of reactivity at high temperature they are probably unimportant at low burnoff. W h e n oxidation proceeds and the activation of micropores occurs, the contribution of pores with initial diameters below 10 nm may become important in the oxidation process. Did you check, by adsorption and desorption of CO2, e.g,, for the presence or the absence of this population of pores? Author's Reply. The 1/r~ pore size distribution predicts that a significant fraction of the porosity is contained in the micropores. This is validated by CO2 adsorption. The pore tree model includes pores as small as a few angstroms. They exist on the extremities of the pore tree (i.e., the leaves). At elevated temperatures, the oxygen is consumed before it can reach this pore size. The relative importance of the micropore increases with increased burnoff and with decreased temperature.
Alan Kerstein, Sandia National Laboratories, Livermore, USA. Could the prediction of high internal pressures during gasification be a consequence of neglect of the molecular sieve effect, namely, the existence of large pores in the char particle interior which are accessible from the exterior only through smaller pores?
Author's Reply. There is no pressure rise during gasification. The high pressures were predicted to occur during pyrolysis [17, 18] and are a consequence of the bulk flow through the network of fine pores which is modeled as the pore tree. Molecular sieve effects into internal voids would tend to slightly reduce this pressure.
Philip R. Westmoreland, MIT, USA. Driving pyrolysis volatiles or natural moisture from the coal pores have analogous effects on internal pressures. Internal temperatures have been measured during the heating at 3 K/rain of 150-ram-diameter blocks of Wilcox (Texas) lignite. 1 This lignite contained 37.5 wt % moisture, its natural moisture content. The heating rate and diameter are comparable by a conduction model to 1-mm-diameter particles heated at 1000 K/s. At the time when the drying front reached the center of the block, the boiling temperature had reached 107~ although the block surface was at 1 atm. This temperature corresponds to an internal pressure less than 2 atm. Please comment on the difference between this pressure and the pyrolysiscaused internal pressure of 1000 atm which you have calculated. REFERENCE 1. P. R. WESTMORELAND, R. C. FORRESTER, AND J. B, GIBSON, Pyrolysis and Physical Properties of Coal Blocks, Oak Ridge (TN, USA) National Laboratory, report ORNL/TM-7313, 1980.
Author's Reply. The fluid transport associated with pyrolysis and the boiling of natural moisture are identical. The internal gas pressures which occur during pyrolysis are a function of the pyrolysis rate and the geometric location in the pore tree [17, 18]. The 1000 atmospheres I quoted will occur in the smallest branches of the largest trees. Submicron size pores will experience an over pressure of the order of a few atmospheres. Average pressures, as inferred in the manner your described, are difficult for me to interpret with this model. However, I would assert that these predictions are not inconsistent with your observations.
R. H. Essenhigh, Ohio State University, USA. I have two questions. (1) I am still bothered by the assumption of reverse adsorption of 02 and the impTication of molecular adsorption. All evidence now sup-
PORE TREE STRUCTURE ports dissociative adsorption of atoms with no evidence for reverse desorption back to molecular oxygen. The fact that it seems to work, i.e. to give agreement with experiment, does suggest the structure needed in the mathematics; but can you use this to justify the inclusion of evidently incorrect physical/chemical assumptions? (2) The overall development and results of the model are impressive and valuable. Would you agree, however, that there is still a problem of uniqueness; and, ff so, do you have plans to do sensitivity studies and evaluation of alternative assumptions to reduce ambiguities?
Author's Reply. (1) The fluid mechanical description of gas diffusion through the pores with a chemical reaction occurring on the walls of the pores requires a global description of the C plus 02 kinetics. To this end, the concept of molecular adsorption yields a carbon consumption rate whose oxygen partial pressure dependence and temperature dependence agree with data. While the mechanism of dissociative adsorption is certainly a specific step in the detailed kinetic process, we have been unable to develop an accurate global rate using dissociative adsorption as a rate limiting process. I hope that your comment will inspire further work in this area. (2) There is a problem of uniqueness. At the very minimum, pore trees are not isolated. The leaves or fine branches must overlap to insure permeability. Percolation theory has the potential of developing entire classes of pore structures that obey the 1/r3p distribution. We have considered time dependent mercury porosimetry measurements that would test specific details of the pore tree. No practical experiment has yet been designed.
1075
incorporated as corrections to the pore structure and kinetic rates, respectively. Smith [Fuel, 57, 409 (1978)] used the Thiele model to reduce a vast amount of oxidation data to infer the intrinsic rates. He reported that the intrinsic rates varied four orders of magnitude with coal type. To attribute this variation entirely to catalysis or graphitization is not completely correct. For example, Smith found that petroleum coke was 1000 times more reactive than brown coal at 1250 K and ten times more reactive at 700 K. Lewis and Simons [11], using Smith's data, found that petroleum coke was a factor of six more reactive and independent of temperature and particle size. It is my opinion that the Thiele model is too crude to accurately achieve the goal that Smith sought. First, the Thiele model does not describe the change in reactivity with burnoff. This effect may influence particle reactivity by a factor of ten and intrinsic reactivity by a factor of 100. Second, the Thiele model describes one pore size and one oxidation mode. The pore tree model describes three simultaneously occurring oxidation modes in three pore size ranges. All modes scale differently with reaction order, temperature and oxygen pressure. Third, Smith correlated data from 10 -~ to 1 arm of oxygen using an empirically derived reaction order m. Lewis and Simons showed that at 10 -2 atm, the reactivity is adsorption limited (m = 1) but desorption limited (m = 0) at 1 arm. Hence, m itself is an unknown variable in this pressure range and data reduction is even more difficult and inaccurate. In the final analysis, I believe an accurate pore structure/transport model will reduce the large variations in the inferred intrinsic rates. I agree that catalysis does enhance char reactivity and must ultimately be incorporated into the model. I only disagree in the degree to which naturally occurring agents may catalyze the oxidation reaction. III
A. F. Sarofim, MIT, USA. This is an elegant contribution to the literature on char burning models. You should, however, be cautious in implying universal rate constants for chemical reactivity. It is known that carbon reactivity decreases with extent of graphitization or increases by catalysis. The data on intrinsic kinetics derived by Smith in his plenary lecture show evidence of both trends with carbons that had been graphitized by heat-treatment at high temperatures showing low reactivity and petroleum cokes with a high vanadium content showing high reactivity.
Author's Reply. The universal rate constants apply only to uncatalyzed carbon surfaces. The effects of graphitization and catalysis may be modeled and
William Bartok, Exxon Research & Engineering, USA. Your pore tree structure model for porous char has been developed for the description of the mechanism of char oxidation by 02 . You mentioned an extension of this model to char gasification conditions, where the dominant oxidizing species are H20 and COy Can the kinetics of char oxidation by the latter species be readily adapted to your model? A second question is to what extent have you taken into account changes in pore structure (pore diameter, surface area) that occur during reaction? Is there a model treatment similar to that of Gayalas?
Author's Reply. The pore structure and transport models have been extended to gasification condi-
1076
COAL COMBUSTION KINETICS AND MECHANISMS
tions [Finson et al., Physical Sciences Inc., PSI TR136, 1978]. This formulation accounts for char conversion by HzO and CO~, as well as water gas shift reactions occurring on the walls of the pores. For many gasifier conditions, the heterogeneous shift reactions can be quite important. Approximate integral methods are used to solve the coupled diffusion equations, and the time integrations are completed numerically.
The pore structure does evolve with b u r n o u t [10]. The pore radius and porosity increase, particle size decreases and the surface area passes through a maximum. Although Gavalas treats surface recession in greater detail than do I, the models give similar results [Comb. Sci. Tech., 24, 211 (1981)].
THE PORE TREE STRUCTURE OF POROUS CHAR GIRARD A. SIMONS
Physical Sciences Inc. Andover, MA 01810 U.S.A. A theory is presented which describes the tree-like pore structure of carbon char. Each pore that reaches the exterior surface of a char particle is depicted as the trunk of a tree. A statistically derived and empirically verified pore distribution function specifies the size distribution of the tree trunks on the particle surface. A continuous branching description of the pore tree predicts the number and size of the pores as a function of the skewed distance into the pore tree. The pore structure is completely specified through measured values of the porosity 0, internal surface area sp, and the particle radius a. The oxidation rate of carbon char is shown to be sensitive to the structure of the pore tree and the present theory is shown to accurately model the oxidation data base. 1. Introduction A theory describing the structure of porous carbon char has been reported by Simons and Finson [1] and by Simons [2]. The unique feature of the theory is that it possesses cylindrical pores whose branching sequence is depicted as an ordinary tree or river system. The concept of the pore tree was developed from a statistically derived and empirically verified pore distribution function. This distribution function quantitatively describes the number of pores with radius between rp_ and rp_ + drp and is continuous over several decades in the pore dimensions. The primary significance of the pore tree is the influence of the pore branching sequence on the gasification process. The reactant species must diffuse through the tree trunk and the larger branches in order to reach the total surface area St contained in the fine pores. The net reactivity of each pore tree may be limited by the kinetic rate acting at the surface St, or by diffusion through the trunk and branches of the tree. The smaller trees tend to be kinetically limited while the larger trees are diffusion limited. This conclusion is supported by Walker [3] through independent derivation and is just the reverse from that which is concluded [4-9] for single, isolated pores. Hence, the concept of pore branching is fundamental to the construction of a char gasification model. Using the pore tree theory of pore structure, a transport theory has been developed [10] and demonstrated to accurately predict the rates of char oxidation [11] and gasification [12]. The reactivity of a pore tree is assumed to be either diffusion limited in the tree trunk or kinetically limited on the surface St. The reactivity of each tree is integrated
over the entire size distribution of trees to determine the net oxidation/gasification rate of the char. In developing the transport theory [10], the details of the branching of the pore tree were not developed and the continuous evolution of the reactivity of the tree from kinetic to diffusion control was not demonstrated. It is the purpose of this paper to develop the details of pore branching and to demonstrate that the limiting cases considered in the transport theory [10] are sufficiently accurate to describe char reactivity. In addition, the current description of the pore tree provides a framework within which the role of species transport during pyrolysis may ultimately be assessed. 2. Pore Structure Theory Before developing a detailed description of the pore tree; the pore structure theory [1, 2] is briefly reviewed. Consider a spherical char particle of radius "a" that contains randomly distributed cylindrical pores of radius r~ and length/?~. The number of pores within the'bulk volume ~whose pore radius is between rp and rp + drp is denoted by Vf(rp)drp. The pore volume is expressed as the 'trr z ep moment of f(rp) and the internal surface area is the 2~rrp{?p moment of f(rp). It is convenient to express the pore structure in terms of a second pore distribution function. The quantity $(r_)A drp denotes the number of pores within the }~ulk cross-sectional area A whose pore radius lies between r P and r P + dr_~' The pore distribution function ~(rp) represents an average over all inclination angle between the axis of the pore and the normal to the plane containing the area A. Due to the random orientation of the pores, the
1067
1068
COAL COMBUSTION KINETICS AND MECHANISMS
intersection of a circular cylinder with a plane is an ellipse of average area 21rrZ_. Hence the porosRy is the 2Wrp moment of g(rp) and the internal surface area is the 47rrp moment of ~(rp). The pore distribution functions are clearly not independent. Simons and Finson [1] have related ~(rp) to f(rp) by 1
g(%) = 2 tpf (rp).
(1)
Equation (1) simply states that the probable number of pores intersecting an arbitrary plane increases with the length of the pores and with the total number of pores. The length of a pore is determined by an arbitrary intersection with another pore and is expressed [1] as a collision integral over the pore distribution functions. The analysis suggests that f~, 9 3 ~' g(r,) and f(r,) are proportional to rp, 1/rp and r 4 ,, . 1/rp, respectwely. The constants of proportmnahty are obtained from integral constraints, i.e. the total porosity and internal surface area contained in the pore structure9 The chars contain a continuous distribution of pore sizes from rmin to rm,x. The pore distribution function ~(rp) is given by 0 ~(rp) - 2,rr[3rp3,
(2)
where 0 is the total porosity and
~ = ,n(rmax I \ rmin
]
The expression for f(rp) is given by 04/3
f (%) = 1ff3Korp4,
(3)
a significant fraction of the particle radius. The radius of the smallest pore is determined from %, the specific internal surface area (typically several hundred m2/g), and is given by r~in = eO /f~pssp
(5)
where Ps is the density of the non-porous solid (2.21 g/cc). The radius of the largest pore is determined under the assumption that there are no internal voids in the particle. The largest pore must terminate at the exterior surface of the particle. This constraint leads to 2a01/3 rm~ -
3Ko
,
(6)
where the constant Ko reflects the pore aspect ratio (pore length to radius) and cannot be determined theoretically. An empirical measure of Ko is obtained [1] by comparing the predicted values of rmax to the data of Mackowsky and Wolf [15]. The results are not unique, but the data suggest a value of Ko near 5. This value of Ko then suggests that the pore aspect ratio is of the order of ten. Consider a small pore located in the interior of the char particle9 If the aspect ratio of this pore is of the order of ten, it cannot reach the particle exterior directly. It must branch from a larger pore which, in turn, branches from a still larger pore, etc. This picture resembles that of a tree or river system. Random pore models proposed by Gavalas [6,7] and by Bhatia and Perlmutter [8,9], consider isolated pores whose length is equal to the particle radius. These models allow all pores to reach the particle exterior directly and branching is not required. Hence, the pore tree model is quite distinct from other [6-9] random pore models. While the concept of the pore tree was previously introduced [1], a detailed description of the tree had not been generated. Such a description is developed in the following sections.
where Ko is an arbitrary constant of integration which relates Cp to rp [1]
~p = Ko rJO I/3.
3. Statistical Description of the Pore Tree (4)
and it has been assumed that pyrolysis and gasification increase r_ at constant e_. This insures that the pore distribution functionseare invariant in a Lagrangian frame moving with rp in time. These expressions for f(rp) and ~(rp) have been validated [1] through extensive comparisons of the predicted volume and surface area distributions with the data of Berger et al. [13] and with that of Stacy and Walker [14]. The pore dimensions range from a microscale, of the order of angstroms, to a macroscale which is
A spatially dependent description of the pore tree, i.e., a mathematical description of the pore tree depicting the number and size of the pores as a fimction of distance into the pore tree, cannot be calculated without first extending the statistical description of the pore tree. The concept of the pore tree depicts that every pore that reaches the exterior surface of the char particle is the trunk of a tree. Let Nt be defined as the branch distribution function where N t drp is the number of pores of radius rp (within size range drp) in a tree whose trunk radius is r t. Since
PORE TREE STRUCTURE 4/3 1ra~f drp represents the total number of pores of radius rn (within size range drp) in a sphere of radius a, the distribution function N t can be related to f by expressing 4/3 ~ra~f drp as the sum of all pores of radius rp contained within every tree in the char sample, plus all pores of radius rp that are themselves the trunk of a tree. Hence,
While a statistical description of the pore tree has depicted the number of pores of various sizes within the tree, an additional hypothesis is necessary in order to depict the branching sequence. This hypothesis is developed in the following section.
~prnax
~ ~a3f =
N t ~(rt) 4~a2drt + 4"tra2~(rp),
Nt = /'t//rp. 3 4
(8)
The branch distribution function Nt completely characterizes the pore tree. The internal surface area and pore volume associated with each pore tree are denoted by St(rt) and Vt(rt), respectively, and are expressed as the sum of the contributions from the trunk and that from the branches.
St(rt) = 2~rrte t +
~rrt
2~rrpepNtdr p
(9)
~rr~tpNtdr p
(10)
rain
Vt(rt) = ,r163 t +
~rrt
4. Continuous Branching Model
(7)
where ~(rt) is the number of tree trunks per unit external area of the char sample and only those trees whose trunk radius is greater than rp may contain a pore of radius rp. Using 8, f and rm~ as given by Eqs. (2), (3) and (6), respectively, Eq. (7) is identically satisfied by
1069
Each pore that reaches the exterior surface of the char particle has been depicted as the trunk of a tree. To describe the sequence of the branching within a pore tree, it is necessary to develop a basic mathematical rule of branching that is consistent with the statistical description of the pore tree. Consider a pore tree which branches in the manner illustrated in Fig. 1. A branching structure with a continuously decreasing pore radius is depicted as being attached to a trunk of radius r t. After a statistical length Ct, the trunk branches into smaller pores of radius rp which, after statistical length ep, themselves branch into even smaller pores, etc. Any distance x measured along the pores corresponds statistically to a particular pore radius rp. Hence, both the radius rp and number of pores of radius r~ is a unique single valued function of the skewed ~tistance x into the pore tree. Let n(x) represent the number of pores of radius rp at location x in a tree of trunk radius r t. The internal surface area and pore volume in the pore tree are expressed as
min
St(rt) =
Using Eq. (8) for Nt, St and Vt become
St(rt)=2"irrt~t(rr-~in) ( 1 - 0 )
(11)
2"rrrp n(x)dx + 27rrte t
t
(13)
and
~t
xt
Vt(rt) =
and
Vt(rt) = ~rr2tet (1 + ln(rt/rmin) ),
(12)
where the (1 - 0) in St has been included to account for pore combination [2]. The above expression for St(rt) is identical to that previously derived [1] from an integral constraint. The surface area associated with the pore tree may be several orders of magnitude greater than the surface area of the trunk. However, the volume of the pore tree may, at most, be one order of magnitude greater than that of the trunk. It should also be noted that the above expressions for St and Vt reduce to those appropriate to a single cylindrical pore in the limit of r t ---> rmin (the leaf of the tree). Furthermore, the integrals of Wt(rt) and St(l't) o v e r all ~(rt) recover the total pore volume and surface area, respectively.
~rr~ n(x)dx +
~rr~et,
(14)
! rp Appropriate
{ \ \ \ \ \ \ \ \ \ ~\ \ l~\ ~i ~
x
Mean Radius at x
t
I=\\l~\\\\\\\~\\\\\\~Particle Surface
FIG. 1. Continuous branching description of the pore tree.
1070
COAL COMBUSTION KINETICS AND MECHANISMS
respectively, where x t is the skewed height of the pore tree measured from the edge of the char partjcle. Comparison of Eqs. (13) and (14) to Eqs. (9) and (10) indicates that n(x) is given by n(x) = - fpNt ~ ,
(15)
ax
where s and N t are given by Eqs. (4) and (8), respectively, and the negative sign appears because the limits of integration are such that x = x t corresponds to rp = rrnln and x = ~t corresponds to rp = r t. Equation (15) is a direct consequence of the definitions of n(x) and Nt(rp). The number of pores of radius r_ and length ~_ in a tree of trunk radius r t is denoted by Ntdr p ~vhereas n(x) represents the number of pores of radius rp and length dx in the same tree. It therefore follows that n(x) is ep/dx times greater than Ntdrr, and this statement is expressed by Eq. (15). Equation (15) is an expression relating two unknowns: n(x) and drp/dx. Since all previously determined pore distribution functions have been shown to scale as the pore radius to a constant power, it is logical to assume that n(rp) also scales in the same manner. It is therefore assumed that drp/dx is proportional to r ; where to is a constant. The constant of proportionality is set by the requirement that each tree possesses one trunk (n = 1 at rp = rt). Hence drp/dx and n(x) are expressed as dr--e-P=-rt(rp) ~ dx et \ r t /
(16)
made by examining the skewed height of the pore tree, x t. The value of x t is expressed as the integral of dx/drp over all rp between r t and rmin. It suffices to say that only the value of to identically equal to unity allows the pore tree to degenerate into a single pore in the limit of rt --+ rmin. Thus, the physical description of the pore tree is dependent upon the value of to and a realistic description requires that to be identically equal to unity. While the continuous branching model of the pore tree has been shown to be quite plausible, the true test of the model is its ability to accurately describe the transport properties of the pore tree. The oxidation of the pore tree is examined in the following sections. 5. Oxidation Within the Pore Tree When porous char is placed in an oxygen environment, Oz will diffuse into the pore tree and oxidize the carbon which constitutes the walls of the pores. The diffusion of 02 through n pores of radius rv is balanced by the reaction of O~ at the walls of the pore. This is expressed as d I"
~ dc\
~npcDTr~ ~ )
= n2~rr~,k
(18)
where c denotes the gaseous mass fraction of oxygen, Pc is the gas density within the pore, D is the self-diffusion coefficient of oxygen in an arbitrary gaseous environment and k is the kinetic rate with which oxygen reacts with carbon. The total oxidation rate of the pore tree, Ivlt, is related to the gradient of c at x = 0 by
and n ~
(
r_,/3-~
\ rp/
(17)
respectively. The continuous branching description of the pore tree has related physical space (x) to the pore radius (r_). It therefore follows that some constraint in physical space must be used to determine to. The expression for the volume of the pore tree, Eq. (14), is rewritten to include only that pore volume between r t and rp, i.e., between x = 0 and x. For to = 1,
V,(x) = ~ x which implies that pore volume is homogeneously distributed in physical space. No other value of to satisfies this basic requirement of the porous media. An independent choice for the value of to is
Ir t = - p c D ~ r ~
aX
(19) O'
However, to obtain the value of dc/dx at x = 0, Eq. (18) must be integrated subject to the boundary conditions that c = c o at x = 0 (rp = rt) and dc/dx = 0 at x = x t (rp = rmin). It is assumed that the oxygen concentrations are sufficiently low that the kinetic rate k is adsorption limited, i.e., k is proportional to c k = k a pG c,
where Pc is the gas pressure and the adsorption rate constant ka is a function only of the char temperature. The diffusion of oxygen within the pore is assumed to be dominated by gas-wall collisions. The corresponding diffusion coefficient D is given by D = 2Vrp/3,
(20)
PORE TREE STRUCTURE 10
where ~z is the mean thermal speed of an O2 molecule at the char temperature and 2rp is the mean free path between collisions with the walls. Within these restrictions, the total oxidation rate of the pore tree is obtained from Eq. (19) and the solution for the oxygen concentration profile, In the limit of rt >> rmin, Mt is expressed as ( 4kaPGPC~1/2 r e2~ " 1 1
= ,etCo \-
)
1071 1
|
I
l.-
KINETICALLY LIMITED r
/
1
_ D,
oS.ON L'M'TEO
~ ~ S O L U T I O N
..=. ,,m 10_ 1 =.
(21)
=,
=,
where
I lO2
t lo
10-2
I lO3
lO4
rt / rmin -- RADIUS OF TREETRUNK/MINIMUM PORE RADIUS
(3ka'Z~Pc~]'2(~mi~): rt
FIC. 2. Transition from kinetic to diffusion control. Equation (21) clearly demonstrates two limits: for K > 1, the Thiele [4] diffusion solution is recovered whereas for K < 1, the oxidation rate is limited by the kinetic rate k acting on the total surface of the pore tree, St(rt). This demonstrates that large trees (large K) are diffusion limited whereas small trees (small K) are kinetically limited. The transition from kinetically limited to diffusion limited oxidation is illustrated in Fig. 2. The total oxidation rate of the pore tree is normalized by the kinetic rate and plotted as a functign of trunk radius. For all practical purposes, Mt may be approximated by the kinetic solution for K < 1 and by the diffusion solution for K > 1. This is precisely the approach used in developing the char transport [10] and char oxidation models [11]. Validation [11] of the char oxidation model offers indirect verification of the detailed shape of the pore tree.
consuming oxygen faster than it can diffuse through a non-reacting trunk of length f r This intermediate mode does not occur for the case of o~ equal to unity. The char transport model [10] has been modified to accommodate the case of to = 0. The predictions for the oxidation rate of carbon char are illustrated in Fig. 3, The oxidation rate is determined in grams of carbon removed per cm2 o f external surface area per s per arm 02. The theory is compared to the data of Field [16]. The choice of to = 0 is clearly wrong whereas that of to = 1 is consistent with the data. Hence, the transport processes are sensitive to the structure of the pore tree and the details of the pore tree derived herein have been validated with respect to char oxidation.
I
6. Sensitivity of the Oxidation Rate to the Shape of the Pore Tree Since the transport processes provide the most sensitive tests for the structure of the pore tree, a demonstration of the sensitivity of the char oxidation rate to the constant to will further validate the choice of to equal to unity. The sensitivity of the char oxidation rate to the value of to is readily demonstrated by determining Mt for oJ = 0. Equation (18) again recovers the Thiele [4] solution for large trees and the kinetic solution for small trees. However, there is also an intermediate solution which is diffusion limited, but quite distinct from the Thiele diffusion solution. The Thiete diffusion solution represents a diffusion process whereby the gradient in oxygen concentration is established by oxygen consumption on the walls of the pores. The intermediate solution represents the limit in which the leaves and branches of the tree are capable of
I
i
I
I
9 FIELD [16] 38//. OIAMETER PARTICLE
8to) = o.8o 10% 02 (0.1 atm 02 )
E= I
//,HEORy,,O.',,
10--1
2
I/
10--2 1000
1200
I 1400
I 1600
I 1800
I 2000
TEMPERATURE- K
F]C. 3. Oxidation of low rank char.
2200
1072
COAL COMBUSTION KINETICS AND MECHANISMS 7. Summary and Conclusions
A continuous branching description of the pore tree has been developed. The pores branch in a continuous downsizing manner. Both the radius and the number of pores is a single valued function of the skewed distance x into the pore tree. The number of pores of radius rp is given by
n(x)- ~(x)' and the coordinate x is related to rp by
dry_
rp
dx
et '
The size distribution of tree trunks on the exterior surface of the particle is given by ~(rt). Hence, the entire pore structure is completely specified by measured values of the char porosity, internal surface area, and particle radius. The features of the pore tree are best verified through its transport properties. Char oxidation is shown to be sensitive to the structure of the pore tree. The continuous branching model developed herein accurately reproduces the kinetic and diffusion solutions which were utilized in developing the transport model [10]. The validation of the transport model in describing char oxidation [11] indirectly verifies the continuous branching description of the pore tree. The current description of the pore tree provides a realistic pore structure in which pore evolution and species transport during pyrolysis may be assessed. It is anticipated that such a theory may be used to assess the secondary chemistry which occurs during pyrolysis. These important potential functions of the pore tree model are under current investigation [17, 18].
ep ft Ivlt n(x) Nt PG rn rmln rm~ rt St sn V Vt x xt
Greek Symbols 13 0 K PG p~ oJ
Work supported by U. S. Department of Energy, Pittsburg Energy Technology Center, under contract DE-AC22-80PC-30293.
REFERENCES 1. SIMONS, G.
2. a C
Co
D
f k
ko
Ko
radius of char particle cross sectional area of sample mass fraction of oxygen in pore mass fraction of oxygen at surface of char particle oxygen self diffusion coefficient pore distribution function per sample volume pore distribution function per sample area kinetic rate: mass of reactant gas/area-time adsorption rate constant constant ~ ratio of pore length to diameter =5
In (rm~xlrrnin) porosity constant, Eq. (22) gas density density of non-porous material constant, Eq. (16)
Acknowledgment
Nomenclature A
length of pore of radius ro length of the pore that represents the trunk of a tree oxidation rate per pore tree number of pores of radius rp at distance x into pore tree branch distribution function gas pressure radius of pore radius of smallest pore radius of largest pore radius of the pore that represents the trunk of a tree surface area of a pore tree specific internal surface area (area/mass) mean thermal speed of a molecule volume of sample volume of a pore tree skewed distance into pore tree skewed height of pore tree
3. 4. 5. 6.
A. AND FINSON, M. L., "The Structure of Coal Char: Part I. Pore Branching," Comb. Sci. Tech,, 19, 5 & 6, 217 (1979). SIMONS, G. A., "The Structure of Coal Char: Part II. Pore Combination," Comb. Sci. Tech., 19, 5 & 6, 227 (1979). WALI~ER,Jl~., P. L., "Pore System In Coal Chars. Implications for Diffusion Parameters and Gasification," Fuel, 59, 809 (1980). THIELE, E. W., "Relation between Catalytic Activity and Size of Particle," Ind. Eng. Chem., 31, 916 (1939). WHEELER, A., "Reaction Rates and Selectivity in Catalyst Pores," in Advances in Catalysis, 3, 249 (1951). GAVALAS,G. R., "A Random Capillary Model with Application to Char Gasification at Chemically Controlled Rates," AIChE Journal, 26, 577 (1980).
PORE TREE STRUCTURE 7. GAVALnS, G. R., "Analysis of Char Combustion Including the Effect of Pore E n l a r g e m e n t , " Combustion Science and Technology, 24, 197 (1981). 8. BHATIA, S. K., AND PERLMUTfER, D. D., "A Random Pore Model for Fluid-Solid Reactions: I. Isothermal, Kinetic Control," AIChE Journal, 26, 379 (1980). 9. BHATIA, S. K., AND PERLMUTrER, D. D., "A Random Pore Model for Fluid-Solid Reactions: II. Diffusion and Transport Effects," AIChE Journal, 27, 247 (1981). 10. SIMONS, G. A., " C h a r Gasification: Part I. Transport Model," Comb. Sci. Tech., 20, 3 & 4, 107 (1979). 11. LEWIS, P. F. AND SIMONS, G. A., "Char Gasification: Part II. Oxidation Results," Comb. Sci. Tech., 20, 3 & 4, 117 (1979). 12. SIMO~S, G. A., "The Unified Coal-Char Reaction," Fuel, 59, 143 (1980). 13. BERGER, J., SIEMIENIEWSKA, T. AND TOMKOW,
14.
15.
16.
17. 18.
1073
K., "Development of Porosity in Brown Coal Chars on Activation with Carbon Dioxide," Fuel, 55, 9 (1976). STACY, W. O. AND WALKER, JR., P. L., "Structure and Properties of Various Coal Chars," Coal Research Section, College of Earth and Mineral Sciences, The Pennsylvania State University, September 1972. MACKOWSKY, M. T. AND WOLF, E. M., "Microscopic Investigations of Pore Formation during Coking," in Coal Science, Advances in Chemistry Series 55, Am. Chem. Society, ed. R. F. Gould (1966) p. 527. FIELD, M. A., "Rate of Combustion of Size Graded Fractions of Char from a Low-rank Coal Between 1200 K and 2000 K," Comb. and Flame, 13, 237 (1969). SIMONS, G. A., "Pyrolysis I. Evolution of the Pore Tree," in preparation. SIMONS, G. A., "Pyrolysis II. Species Transport through the Pore Tree," in preparation.
COMMENTS Stephen Niksa, Sandia National Laboratories, Livermore, USA. The transport equation relating the flux of oxygen through arbitrary passages in the particle to the net consumption by surface reaction reduces the pore tree to a one-dimensional tube with diminishing cross section. As the pore radius decreases the mass transport mechanism shifts from bulk flow to Knudsen flow to, perhaps, activated diffusion. Have you a c c o u n t e d for the various mechanisms and, are the predictions from the model sensitive to the values of the diffusivity or the points in the pore at which these transitions occur?
Author's Reply. Both Knudsen and continuum diffusion are considered in the transport model [10]. The diffusion coefficient (D) is the smaller of either the Knudsen or continuum value. There is an "'order one" error in D at the pore radius where transition occurs. Upon integrating over all pores, this error becomes minimal. Activated diffusion is not explicitly included. This concept is replaced with that of kinetic control on the internal surface area. This area is a function of burnoff but not diffusion residence time. The continuous branching description of the pore tree does not allow small branches to emerge from large trunks. In this respect, the pore tree fails to duplicate the common shade tree. While the similarities between the common shade tree and the pore tree far outnumber the differences, one still cannot use this analogy to prove the concept.
Eric Suuberg, Brown University, USA. How was the diffusion coefficient determined? Does it include any correction for bulk flow, which your transport model did not explicitly allow for? This bulk flow correction would probably be small under combustion conditions, but u n d e r pyrolysis conditions during which internal pressures were predicted to be of order 1000 atm this may not be true. Author's Reply. Within any given pore, the diffusion coefficient is the smaller of either the Knudsen or continuum diffusion values [10]. The generation or depletion of moles (one Oe ~ two CO) induces bulk flow within the pore. During char oxidation, this p h e n o m e n o n may be described through a moderate correction to the diffusion coefficient. I do not include it. During pyrolysis, bulk flow is the dominant transport phenomenon and is included in that model [17, 18].
Nahum Gat, TRW-Space & Technology Group, USA. As combustion proceeds inside the pores and branches, the walls between them disappear. Then initially the surface area increases but as the walls burn out the surface area drops significantly. Is that the correct physical process and does the model account for this phenomenon?
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Author's Reply. This exact process i s described by the present model [2] and the results have been used to successfully describe [12] the universal burnoff profile in the limit of kinetic control.
C. J. Lawn, CEGB, Southampton, U.K. Do I understand correctly that because your pore "trees' are all 'rooted' on the outer surface of the particle, the pore size distribution in your model becomes finer as one moves radially inward? If so, is this realistic? Author's Reply. As one moves into a single tree, the pore size becomes finer. However, trees are rooted from all sides of a porous sample and the largest trees are as large as the sample. Hence, as one moves radially inward, one may move either into or out of a tree. In this manner, a homogeneous, isotropic pore structure is retained.
j. Lahaye, CNRS, France. In your model you have taken into consideration only macropores and mesopores larger than 10 nm diameter. In char, small mesopores and micropores are expected. From the point of view of reactivity at high temperature they are probably unimportant at low burnoff. W h e n oxidation proceeds and the activation of micropores occurs, the contribution of pores with initial diameters below 10 nm may become important in the oxidation process. Did you check, by adsorption and desorption of CO2, e.g,, for the presence or the absence of this population of pores? Author's Reply. The 1/r~ pore size distribution predicts that a significant fraction of the porosity is contained in the micropores. This is validated by CO2 adsorption. The pore tree model includes pores as small as a few angstroms. They exist on the extremities of the pore tree (i.e., the leaves). At elevated temperatures, the oxygen is consumed before it can reach this pore size. The relative importance of the micropore increases with increased burnoff and with decreased temperature.
Alan Kerstein, Sandia National Laboratories, Livermore, USA. Could the prediction of high internal pressures during gasification be a consequence of neglect of the molecular sieve effect, namely, the existence of large pores in the char particle interior which are accessible from the exterior only through smaller pores?
Author's Reply. There is no pressure rise during gasification. The high pressures were predicted to occur during pyrolysis [17, 18] and are a consequence of the bulk flow through the network of fine pores which is modeled as the pore tree. Molecular sieve effects into internal voids would tend to slightly reduce this pressure.
Philip R. Westmoreland, MIT, USA. Driving pyrolysis volatiles or natural moisture from the coal pores have analogous effects on internal pressures. Internal temperatures have been measured during the heating at 3 K/rain of 150-ram-diameter blocks of Wilcox (Texas) lignite. 1 This lignite contained 37.5 wt % moisture, its natural moisture content. The heating rate and diameter are comparable by a conduction model to 1-mm-diameter particles heated at 1000 K/s. At the time when the drying front reached the center of the block, the boiling temperature had reached 107~ although the block surface was at 1 atm. This temperature corresponds to an internal pressure less than 2 atm. Please comment on the difference between this pressure and the pyrolysiscaused internal pressure of 1000 atm which you have calculated. REFERENCE 1. P. R. WESTMORELAND, R. C. FORRESTER, AND J. B, GIBSON, Pyrolysis and Physical Properties of Coal Blocks, Oak Ridge (TN, USA) National Laboratory, report ORNL/TM-7313, 1980.
Author's Reply. The fluid transport associated with pyrolysis and the boiling of natural moisture are identical. The internal gas pressures which occur during pyrolysis are a function of the pyrolysis rate and the geometric location in the pore tree [17, 18]. The 1000 atmospheres I quoted will occur in the smallest branches of the largest trees. Submicron size pores will experience an over pressure of the order of a few atmospheres. Average pressures, as inferred in the manner your described, are difficult for me to interpret with this model. However, I would assert that these predictions are not inconsistent with your observations.
R. H. Essenhigh, Ohio State University, USA. I have two questions. (1) I am still bothered by the assumption of reverse adsorption of 02 and the impTication of molecular adsorption. All evidence now sup-
PORE TREE STRUCTURE ports dissociative adsorption of atoms with no evidence for reverse desorption back to molecular oxygen. The fact that it seems to work, i.e. to give agreement with experiment, does suggest the structure needed in the mathematics; but can you use this to justify the inclusion of evidently incorrect physical/chemical assumptions? (2) The overall development and results of the model are impressive and valuable. Would you agree, however, that there is still a problem of uniqueness; and, ff so, do you have plans to do sensitivity studies and evaluation of alternative assumptions to reduce ambiguities?
Author's Reply. (1) The fluid mechanical description of gas diffusion through the pores with a chemical reaction occurring on the walls of the pores requires a global description of the C plus 02 kinetics. To this end, the concept of molecular adsorption yields a carbon consumption rate whose oxygen partial pressure dependence and temperature dependence agree with data. While the mechanism of dissociative adsorption is certainly a specific step in the detailed kinetic process, we have been unable to develop an accurate global rate using dissociative adsorption as a rate limiting process. I hope that your comment will inspire further work in this area. (2) There is a problem of uniqueness. At the very minimum, pore trees are not isolated. The leaves or fine branches must overlap to insure permeability. Percolation theory has the potential of developing entire classes of pore structures that obey the 1/r3p distribution. We have considered time dependent mercury porosimetry measurements that would test specific details of the pore tree. No practical experiment has yet been designed.
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incorporated as corrections to the pore structure and kinetic rates, respectively. Smith [Fuel, 57, 409 (1978)] used the Thiele model to reduce a vast amount of oxidation data to infer the intrinsic rates. He reported that the intrinsic rates varied four orders of magnitude with coal type. To attribute this variation entirely to catalysis or graphitization is not completely correct. For example, Smith found that petroleum coke was 1000 times more reactive than brown coal at 1250 K and ten times more reactive at 700 K. Lewis and Simons [11], using Smith's data, found that petroleum coke was a factor of six more reactive and independent of temperature and particle size. It is my opinion that the Thiele model is too crude to accurately achieve the goal that Smith sought. First, the Thiele model does not describe the change in reactivity with burnoff. This effect may influence particle reactivity by a factor of ten and intrinsic reactivity by a factor of 100. Second, the Thiele model describes one pore size and one oxidation mode. The pore tree model describes three simultaneously occurring oxidation modes in three pore size ranges. All modes scale differently with reaction order, temperature and oxygen pressure. Third, Smith correlated data from 10 -~ to 1 arm of oxygen using an empirically derived reaction order m. Lewis and Simons showed that at 10 -2 atm, the reactivity is adsorption limited (m = 1) but desorption limited (m = 0) at 1 arm. Hence, m itself is an unknown variable in this pressure range and data reduction is even more difficult and inaccurate. In the final analysis, I believe an accurate pore structure/transport model will reduce the large variations in the inferred intrinsic rates. I agree that catalysis does enhance char reactivity and must ultimately be incorporated into the model. I only disagree in the degree to which naturally occurring agents may catalyze the oxidation reaction. III
A. F. Sarofim, MIT, USA. This is an elegant contribution to the literature on char burning models. You should, however, be cautious in implying universal rate constants for chemical reactivity. It is known that carbon reactivity decreases with extent of graphitization or increases by catalysis. The data on intrinsic kinetics derived by Smith in his plenary lecture show evidence of both trends with carbons that had been graphitized by heat-treatment at high temperatures showing low reactivity and petroleum cokes with a high vanadium content showing high reactivity.
Author's Reply. The universal rate constants apply only to uncatalyzed carbon surfaces. The effects of graphitization and catalysis may be modeled and
William Bartok, Exxon Research & Engineering, USA. Your pore tree structure model for porous char has been developed for the description of the mechanism of char oxidation by 02 . You mentioned an extension of this model to char gasification conditions, where the dominant oxidizing species are H20 and COy Can the kinetics of char oxidation by the latter species be readily adapted to your model? A second question is to what extent have you taken into account changes in pore structure (pore diameter, surface area) that occur during reaction? Is there a model treatment similar to that of Gayalas?
Author's Reply. The pore structure and transport models have been extended to gasification condi-
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tions [Finson et al., Physical Sciences Inc., PSI TR136, 1978]. This formulation accounts for char conversion by HzO and CO~, as well as water gas shift reactions occurring on the walls of the pores. For many gasifier conditions, the heterogeneous shift reactions can be quite important. Approximate integral methods are used to solve the coupled diffusion equations, and the time integrations are completed numerically.
The pore structure does evolve with b u r n o u t [10]. The pore radius and porosity increase, particle size decreases and the surface area passes through a maximum. Although Gavalas treats surface recession in greater detail than do I, the models give similar results [Comb. Sci. Tech., 24, 211 (1981)].