NEOH, K. G., HOWARD, J. B., AND SAROFIM, A. F.: Particulate Carbon: Formation ... G. R.: AIChE J. 26, 577 (1980). 24. SIMONS, G. A.: Fuel 59, 143 (1980). 25.
Twentieth Symposium (International) on Combustion/The Combustion Institute, 1984/pp. 941-949
FRAGMENTATION
DURING AND
CARBON
CONVERSION:
PREDICTIONS
MEASUREMENTS
ALAN R. KERSTEIN aND STEPHEN NIKSA Sandia National Laboratories Livermore, California 94550
The first quantitative predictions of fragmentation phenomena during carbon conversion are presented. Predictions of the porosity at which fragmentation occurs based on deterministic and statistical models are compared. Analysis of the statistical models is based on percolation theory; which is the geometrical theory of the connectedness of irregular objects. Percolation theory predicts fragmentation at a porosity of about 0.7 for homogeneous samples. Corrections for some aspects of heterogeneity are provided by the theory. In addition, percolation theory predicts that the mass distribution of fragments will exhibit a power-law range f ( m ) ~ m -z15. The distribution of fragment radii is predicted to exhibit a power-law range g(r) ~ r 386. The fragments are predicted to be highly irregular, with surface roughness characteristics different than those of the unreacted sample. These predictions are qualitatively consistent with previous observations. New measurements of the porosity at fragmentation for six carbon composite materials are presented. These measurements indicate that the porosity at fragmentation is a reproducible quantity for a given material, and that it depends in a complex way on material properties. These theoretical and experimental results indicate the likelihood that fragmentation contributes significantly to weight loss during conversion. In addition, fragmentation may influence the size distribution of particulate effluents produced during conversion.
1. Introduction Numerous observations of the formation of fragments during the combustion, gasification, or pyrolysis of porous carbon substances have been reported in the literature.l-7 A variety of mechanisms for fragment formation have been identified, the operative mechanism i n any instance depending upon such factors as material composition, pore morphology, thermochemical environment, and mechanical stress. An aspect of fragmentation which has received little attention is the underlying geometrical criterion for the transition from a single connected object to a highly disconnected, or fragmented, state. Though this criterion may seem intuitively obvious, it is shown here that significant quantitative predictions are obtained from purely geometrical considerations, independent of details of material composition or morphology, or the physical mechanism of fragmentation. In fact, the predictions are not limited to the conversion of carbonaceous materials, although they are presented in this context because char conversion is the principal technological application of the theory. In the literature, fragmentation has generally been treated as an incidental occurrence particular to the system under study. Hence, the subject has been 941
approached from disparate viewpoints, and therefore a review of the experimental literature is best postponed until Sec. 4, after presentation of the concepts, models, and predictions of the present analysis. With regard to modeling, several references are noteworthy. Gavalass incorporates into his char combustion model a criterion for regression of the perimeter of a char particle based on an assumed critical value, ~*, of the porosity at which pore mouth coalescence occurs. He takes the critical value to be e* = 0.8 based on observations of char particle disintegration by Dutta, Wen and Belt. 3 Gavalas does not attempt to predict or interpret either this value or any other aspect of the fragmentation process. In Beshty's9 analysis of char combustion, he notes the possibility of partial collapse of the particle due to pore enlargement and merging. However, he does not address the crucial distinction between connectedness of th e pores and connectedness of the solid matter. In the present study, a statistical approach has been adopted for the analysis of fragmentation. The analysis is based on the principles and results of percolation theory, which is the theory of the connectedness of irregular objects. Percolation theory predicts fragmentation at a critical porosity of about 0.7 for pure substances with homogeneous pore morphology. Corrections for heterogeneity are pro-
942
COMBUSTION-GENERATED PARTICULATES
vided by the theory. In addition, percolation theory predicts power-law functional forms for the mass and size distributions of the fragments. A characterization of the surface roughness of fragments is also provided. Following the discussion of models and predictions, previous experimental work providing evidence of fragmentation is reviewed, and new measurements of the porosity at fragmentation are presented. Finally, some technologically important consequences of fragmentation are identified.
2. Homogeneous Fragmentation 2.1 Porosity at Fragmentation
TABLE I Predictions of the critical porosity for fragmentation Pore Network Structure square/cubic close-packed random network random network and random pore size
Cylinder
Pore Shape Complement Sphere of Sphere
0.785 0.907
0.965 0.960
0.476 0.260
0.675
0.968
O.705
0.682
(n.a.)
0.697
Several geometrical models, deterministic as well as statistical, of pore structure at the onset of fragmentation are now considered. These models are pores. Using only models for which published renot intended to be accurate representations of the sults are available, we consider two types of models. pore structure of real substances, but rather are used The third row of Table I gives the critical porosity to examine the sensitivity of the critical porosity for for models in which the pores are identical in size fragmentation to modeling assumptions. The models and shape but their spatial locations are random. are homogeneous in the sense that the local pore (However, in the case of cylindrical pores, the cylmorphology for a given model is the same through- inders are still assumed to be parallel.) The fourth out the material. Consideration of the evolution of row of Table I gives results for models in which the the pore structure prior to fragmentation, as influ- size of pores is also random, with a uniform distrienced by reaction rates, transport, etc., is deferred bution of pore radius from zero to a maximum value. until Sec. 2.4. Effects of heterogeneity are exam- Details of the computations leading to estimates of ined in Sec. 3. the critical porosity for these models are provided In the deterministic models, the pores are as- by Pike and Seager,10 who obtained all the quoted sumed to be identical in sh~/pe and to form a reg- results for the statistical models except the result ular lattice. Therefore, the pore structure is spec- for spherical pores, which was obtained by Elam, flied by the pore shape and the network structure Kerstein, and Rehr.ll of the lattice. In each model, the porosity depends Before drawing inferences from the numerical reon one parameter governing pore size (e.g., the ra- sults, it is useful to consider the nature of the fragtio of pore diameter to lattice spacing). The critical mentation mechanism in a substance whose strucporosity for fragmentation corresponds to the value ture is irregular. Since the deterministic models are of the pore size parameter at which the pore walls spatially periodic structures, the criterion for conin adjacent lattice cells cease to be connected. The nectedness of the solid matter is simply that each results for several deterministic models, obtained cell of the lattice be connected to its neighbors. In exactly by straightforward application of this geo- irregular substances, the relationship between the metrical criterion, appear in the first two rows of fine-scale structure and the criterion for overall Table I. connectedness is less obvious, since a certain fracIn the first column of Table I, the pores are as- tion of the pore walls may be disconnected locally sumed to be parallel cylinders, forming a lattice without interrupting the global connectedness of the which is two-dimensional in the plane perpendic- solid matter. In fact, it is not self-evident that the ular to the axes of the cylinders. The lattice is either transition from a connected to a fragmented strucsquare or close-packed (triangular). In the second ture is sharp, nor that the critical porosity is recolumn, the pores are spheres and the (three-di- producible from sample to sample. mensional) lattice is either cubic or hexagonal-closeThe study of the relationship between overall packed. In the third column, the solid matter forms connectedness and the fine-scale structure of irrega lattice (cubic or hexagonal-close-packed) of spheres ular objects is known as percolation theory. (Ziman12 so the pores comprise the region complementary to provides a lucid introduction to percolation theory the spheres. and its applications. Several combustion Statistical versions of these models may be ob- applications13 15 have previously been identified.) tained by introducing random variations of the size, Extensive theoretical and experimental investigashape, orientation and/or network structure of the tions of percolation phenomena have addressed the
FRAGMENTATION DURING CARBON CONVERSION questions raised above. It has been found that the transition from a connected to a fragmented structure is indeed sharp, and the critical porosity, or more properly, its analogs in the physical problems studied thus far, is remarkably reproducible both in Monte Carlo computations and in experiments. A further result of percolation theory is that the parameter or parameters governing the transition from a connected to a disconnected structure are often insensitive to structural details. In the present context, this result implies that the critical porosity may vary only slightly from one substance to another. This insensitivity, called "dimensional invariance, ''1~ is violated in certain situations, for instance if the substance consists of randomly oriented plates or whiskers, or if it is heterogeneous. For the moment, we examine predictions based on homogeneous, irregular pore morphologies. In Table I, we note that with one exception, the estimates of the critical porosity based on the random models are quite close to each other, clustering about a value of 0.7. This is in marked contrast to the estimates obtained using the deterministic models, which vary widely. 2.2 Fragment Mass Distribution The contrast between deterministic and statistical models is even greater with respect to the predicted mass distribution of the fragments. As noted above, deterministic models involve regular arrays of identical cells. In this context, fragmentation involves the disconnection of each cell from its (identical) neighbors, so all fragments have identical mass. This conclusion applies to all spatially periodic deterministic models. (A more general class of deterministic models based on the "Koch construction''16 can give nondegenerate mass distributions, but these models are not considered here.) The fragment mass distribution predicted by percolation theory is based on a concept known as scaling. 17 Scaling is best understood by considering the evolution of a porous material as the porosity increases from just below to just above the critical porosity. Just below the critical porosity, sufficient pore coalescence has occurred so that the material is barely intact. Within the network of connected material, local regions of fragmentation are found, although the fragments cannot yet escape because they are caged by the connected network. In the language of percolation theory, the connected network is a fragment of infinite extent, while the abovementioned local regions contain fragments of finite extent. Precisely at the critical porosity, the integrity of the infinite fragment is lost and the fragments can escape and separate, hence the association of the mathematical concept of percolation with the physical phenomenon of fragmentation. At porosities below the critical porosity, the
943
presence of the infinite fragment causes the massweighted fragment mass distribution to exhibit a singularity at infinite mass. At the critical porosity, the singularity is replaced by a distribution which vanishes in the infinite limit but extends to arbitrarily large mass values. At large mass values, the fragments are so much larger than the typical pore diameter or pore wall thickness that it is unlikely that the mass distribution of these fragments is significantly influenced by pore morphology. The scaling concept is based on the postulate that, for large mass values, the fragment mass distribution is a universal curve. Statistical analysis based on this postulate 17 gives the result that the fragment mass distribution is a power law. The exponent, denoted T, has been estimated by means of Monte Carlo calculations and other methods, giving the result 17
f(m) = Am -2"15,
(1)
where f(m) is the probability density function for fragment mass. The prefactor A is non-universal. It is not determined by the normalization condition on f because Eq. (1) is applicable only to a limited range of m. A lower bound on this range is implicit in the abovementioned requirement that the fragment size be much larger than the local pore structure. An upper bound follows from the observation that reasoning based on an infinite range of fragment masses must break down when applied to physical samples which are necessarily finite in extent. Therefore, for purposes of comparison to measured mass distributions, Eq. (1) is applicable only for m much smaller than the mass of a single piece of the unreacted sample (i.e., for particulate samples, the mass of a single particle). 2.3 Size Distribution and Surface Roughness One would suppose that the fragment size distribution follows from the mass distribution in a straightforward manner. However, Mandelbrot 16 has pointed out that fragment mass does not vary with size in the usual fashion (assuming, again, that the scaling concept is applicable). One would ordinarily expect that m - r D, where D = 3. However, the scaling concept leads to the conclusion that fragments are highly irregular shapes whose mass-size relationship is of the power-law form, but with D not necessarily equal to 3. These highly irregular shapes, which have been dubbed "fractals, ''16 are characterized by their "fractal dimension," D -< 3. Statistical analysis gives the result D = 2.49 for fragments, is Based on the fractal dimension concept, we generalize the usual relation between size and mass distributions to obtain g(r) = Br D-D'-I = Br -a'86,
(2)
944
COMBUSTION-GENERATED PARTICULATES
where g(r) is the probability density function for fragment radius and B is a nonuniversal prefactor. (For an irregular particle, r is defined to be the radius of gryation.) The range of validity of this prediction is analogous to that of the mass distribution. The irregularity of individual fragments, in addition to influencing the mass-size relationship, implies a high degree of surface roughness. This conclusion, again, is a consequence of the fragmentation process and does not depend on details of pore morphology or the surface roughness of the unreacted sample. Since surface roughness can influence the gas-phase flowfield, heat and mass transfer to and from the particle can be affected. 2.4
Fragmentation Dynamics
In Sec. 2.2, a qualitative account was given of the evolution of a porous solid as the porosity approaches and finally reaches the critical porosity. Implicit in the discussion was the assumption that the porosity at any instant is uniform throughout the sample. Under this assumption, the sequence of events outlined in Sec. 2.2 corresponds to the intuitive notion of disintegration. A qualitatively different process occurs if the rate of pore surface regression is a slowly varying function of location in the solid. For instance, in a particle undergoing Zone II (pore-diffusion-limited) or Zone III (film-diffusion-limited) combustion or gasification, the rate of pore growth is greatest at the perimeter of the particle. Therefore the local porosity at any instant is greatest at the perimeter, so the perimeter reaches the critical porosity before the interior of the particle. This results in perimeter fragmentation in which the exterior boundary of the particle shrinks as pieces of solid matter at the perimeter become disconnected from the particle. Provided that the spatial variation of the porosity is small over a distance corresponding to the typical pore diameter or pore wall thickness, the critical porosity for perimeter fragmentation is the same as the critical porosity for disintegration. However, the fragment mass and size distributions are likely to differ from those predicted for disintegration, because the porosity gradient introduces an additional length scale into the system, possibly invalidating in this instance the postulate of a universal mass distribution. Nevertheless, the mass distribution should be broad because the qualitative reasoning of Sec. 2.2 should still be applicable. Thus, fragmentation is anticipated for all regimes of carbon conversion. The rate-limiting process for a given regime will determine whether disintegration or perimeter fragmentation occurs. 3. Heterogeneous Fragmentation
The analysis thus far has focused on pure substances with homogeneous pore morphology. Next
we consider the possible effects of heterogeneity: Recognizing the idealizations inherent in any characterization of a complex material, for present purposes we model heterogeneity by viewing the material as a two-phase composite, consisting of a continuum phase and a second phase in the form of isolated inclusions. At any instant, the porosity in a given phase is spatially uniform, but the porosity values in the two phases differ due to differences in initial porosity, chemical reactivity, or diffusional resistance. We denote the porosities of the continuum phase and of inclusions as cc and ei respectively, and we denote the volume fraction of the continuum phase as q. This model might be appropriate, for instance, for estimating the impact of mineral matter inclusions on fragmentation. If we adopt the simple assumption that the presence of inclusions does not affect quantitative predictions for fragmentation of the continuum phase, then the criterion for fragmentation of the composite material is ec = ~*, where ~* is the critical porosity predicted by percolation theory for the continuum phase. Since the volume-averaged porosity is e = qec + (1 - q)ei, the composite material fragments at a volume-averaged porosity of e* = qe* + (1 - q)~i.
(3)
If we take ~* = 0.7, then et is greater or less than 0.7 depending on whether ~i is greater or less than 0.7 when fragmentation occurs. Thus, if the inclusions are more reactive than the continuum phase, the volume-averaged porosity at fragmentation tends to increase, but if the inclusions are less reactive or even inert, as may be the case for mineral matter inclusions, the volume-averaged porosity at fragmentation is reduced. The assumption that fragmentation of the continuum phase is unaffected by the inclusions can be justified based on percolation theory. Specifically, Turban 19 has estimated the effect of the inclusions on the numerical value of the critical porosity ~* appearing in Eq. (3). If q is close to unity (that is, the continuum phase is the principal constituent), then ~* is unchanged from its homogeneous value provided that the size of a typical inclusion is much greater than the typical pore size. Thus, the assumption is justified for certain forms of heterogeneity but not for all forms. A form of heterogeneity of possible importance in carbon conversion is the presence of "active sites" within the solid at which the reactivity is higher than the bulk average. If pore wall regression occurs predominantly in the vicinity of active sites, then the spatial distribution of active sites may influence the pore structure at high conversion. For instance, if the active sites are isolated points, then spherical pore structures may develop, while sites lined up along crystalline interfaces may lead to cylindrical or planar voids. To deal with these com-
FRAGMENTATION DURING CARBON CONVERSION plex situations, extensions of percolation theory will be needed. 4. Measurements
945
agreement of this result with the value 0.7 predicted by percolation theory for homogeneous fragmentation is encouraging. 4.2 Kinetically-Controlled Conversion of Carbon
Composites 4.1 Previous Observations Most observations of fragmentation reported to date were incidental to the main objectives of the investigations, so the discussions are brief and the techniques and results are difficult to evaluate. Fragmentation of graphite has been observed at 0.5 conversion, 2 char disintegration has been observed at 0.8 conversion, 3 soot breakup has been observed at 0.8 burnout, 5 and the appearance of fragments during lignite pyrolysis has been reported. 4 Fragment formation during carbon-composite heat-shield ablation is a well-documented occurrence of recognized practical importance. 7 However, it has been attributed exclusively to spallation, or pressure-induced pore wall fracture, and the possibility of alternative mechanisms not involving fracture has not been recognized in the ablation literature. Chemical mechanisms for carbon fragmentation have been discussed in the catalysis literature~ but they are not yet understood qualitatively or quantitatively. 6 The only previous measurement for which porosity, rather than conversion or burnout, is reported involved Zone II gasification of rods of spectroscopic carbon. 1 Measured porosity profiles of partially gasified rods were extrapolated to the perimeter, giving an estimated range of 0.7-0.8 for the critical porosity for perimeter fragmentation. Though hardly a basis for definitive conclusions, the
The objectives of our measurements were first, to establish a valid and systematic procedure for determining the critical porosity for fragmentation, and second, to investigate the dependence of the critical porosity on the composition and morphology of various carbon-composite materials. Regarding the first objective, it is important from a theoretical as well as an experimental viewpoint to establish that, for a given sample material and conversion environment, the critical porosity for fragmentation is a reproducible quantity. We performed a series of measurements in which round disks (51 mm dia.; 1.6 mm thickness) of various carbons were supported at four points on wire stands and reacted isothermally in pure oxygen. As shown below, the temperature was low enough that diffusional resistances did not alter the reaction rate so reaction occurred uniformly throughout each disk (Zone I). Fragmentation was deemed to have occurred when the disk crumbled under its weight. We reacted two samples of each of six different carbons, with the characteristics and physical properties shown in Table II. These carbons represent broad ranges of bulk density, initial porosity, grain size, pore size, and pore size distribution. Coal chars were avoided at this stage because substantial amounts of minerals and heteroatoms are present and the morphology is not amenable to simple clas-
TABLE II Sample characteristics
Sample
Typea
Bulk Density g/cc
580d 890Sa 7716a H451 e BCe D857 f
EG EG AC RG AC AC
1.74 1.66 1.39 1.74 1.63 1.84
Maximum Ash ppm
Total Porosity %
Specific BET Area m2/g
Pore Size~' Mac/Trn microns
Generale Form
1500 5000 5000 1000 na 1500
14 21 31 14 17 14
0.30 1.15 0.52 0.73 0.52 0.37
4.66/. 027 2.50/. 030 4.66/-3.65/. 015 3.65/-0.51/--
BM BM MV BM MV MV
~EG and RG denote electro- and reactor graphites, respectively. Both were extruded then carbonized at 2800 K. AC denotes amorphous carbon which was baked at 1200 K in a mold. bPore sizes are the maxima in incremental volume vs. size from Hg porosimetry. For bimodal systems, both macropore (Mac) and transitional (Trn) pore sizes are given. ~ pore systems (BM) which have significant porosity in pores smaller than 0.1 micron are distinguished from macropore systems (MV) which do not. dSupplied by Airco Carbon, St. Marys, PA. eSupplied by Great Lakes Carbon, Niagara Falls, NY. rSupplied by Duramic Products, Inc., Palisades Park, NJ.
946
COMBUSTION-GENERATED PARTICULATES
sification. Morphology of the six carbons is also complex and varied, but the morphologies can be classified at least qualitatively based on the methods by which the materials were prepared, providing a suitable test of the range of variation of the critical porosity. All six carbons consist of a particle phase (petroleum coke, carbon black, graphite flowers, needle coke) and binder (coal tar, pitch). The amorphous carbons, which were baked during manufacture at 1200-1300 ~ K, contain up to twenty percent binder phase, while the carbon graphites, which were carbonized at 2800-3000~ K, contain about five percent carbonized binder. For the graphites, the similarity of the two phases should mitigate the impact of composition on the measured critical porosity. For the amorphous carbons, the impact of compositional heterogeneity is likely to be significant. Samples were dried in vacuum at 383~ K. The initial total porosity was determined by measuring the apparent densities in helium and mercury. Pore size distributions and specific surface areas were determined from Hg porosimetry and N2 BET, respectively. Preliminary trials established the reaction temperatures at which the fragmentation porosity would be reached within 150 to 250 hours. The experiments were conducted in a preheated muffle furnace purged continuously with dry oxygen. Each sample was removed and weighed twice daily. The porosities were calculated from the initial total porosity and the normalized weight loss. The critical porosity was assigned from the weight of the sample when mechanical failure was observed. The low ash levels in these samples mitigate concerns that catalytic sites are accumulating with conversion. At these ash levels, catalytic effects are insignificant. (Even at higher impurity levels, the determination of the critical porosity is insensitive to catalysis provided that the catalytic species are uniformly dispersed throughout the solid.) Very low values for the Thiele Moduli corroborate the absence of diffusional limitations. The Thiele Moduli are less than 5 x 10-2, computed on the following basis: effective diffusivities based on Knudsen and bulk coefficients; an effective pore size equal to the mean macropore size or from the N2 BET surface area and total porosity; a characteristic diffusion path equal to one-half the disk thickness; and reaction rates from the measured burnoff profiles. The smaller pores in the distribution were characterized with N2 rather than CO2 because the total surface area from CO2 adsorption is not a relevant reactivity parameter for carbon gasification. 2~ When the reactive sites are not accumulating and diffusional limitations are absent, a characteristic form is expected for the plot of fractional burnoff, X = (Wo - W(t))/Wo, versus the dimensionless time scale, t/to.s, which, by definition, equals unity at
X = 0.5. This universal burnout profile is interpreted solely in terms of the pore system. 22-24 Ini~ tially (X < 0.15) the rate increases as narrow apertures burn away to expose additional pore surface, but during the later stages (X > 0.65) the rate decreases as surface is lost due to pore coalescence. Compared to the profile for coal char, 2z the profile from this study, Fig. 1, shows a stronger initial curvature due to pore enlargement. Nevertheless, we neglected the dead volume and used the initial porosities to relate the weight loss to changes in porosity because most of the porosity is associated with very large pores. A comparison of the cumulative pore volume from Hg porosimetry and the total pore volume from the densities in helium and mercury indicates that, for the amorphous carbons, pores larger than 0.1 micron comprise all the porosity and, for the graphites, pores of this size contribute more than sixty percent of the total porosity. Except for the reactor graphite (GLC H451), the samples disintegrated into very fine fragments at the porosities and fractional conversions listed in Table III. The results of replicate runs for each material agree to within two percent; indeed, fragmentation was a distinct .transition. The large pieces from the initial fracturing of the disks disintegrated upon handling into a powder with fragments as large as several microns. Both the disk size and the observed surface texture were unaffected until fragmentation occurred. At porosities within 2-3 percent of the critical value the disks became pliable and sagged. For the reactor graphite, however, large pits formed and developed into holes. These holes were the same size as the particle grains visible initially (ca. 1-2 mm), suggesting that the particle phase was more reactive than the binder phase. Beyond forty percent burnoff the binder phase formed a strong 0.9 0.8 0
0.7
Z
0.6
,.n _J z~
0.5-
o
F-
ii
0.4
o.30.2
2
0.1 0 0
0.4
0)8
1.2
1.;
NON-DIMENSIONALTIME, t/t0.s FIG. 1. Burnout profile for five different carbons (A 580; O 890S; 9 H451; [] BC; + D857) compared to that for coal char (solid line) from reference 22.
FRAGMENTATION DURING CARBON CONVERSION TABLE III Conversion indices at the fragmentation threshold
Sample
Critical Porosity %
Fractional Burnoff
Time to X = 0.5 hr
Reaction Temperature K
580 890S 7716 H451 BC D857
63 77 50 none 58 85
.57 .71 .28 none .49 .83
188 141 na 131 100 103.5
801 801 684 800 684 750
lacy skeleton which grew thinner as the' carbon burned away but retained its structural integrity. No consistent trend relating the measured critical porosity to the sample type or any physical property has been identified. The range of variability of the critical porosity is greater than that predicted by the homogeneous random models, and less than that predicted by the deterministic models. We expect that heterogeneous random models will prove to be the most promising route to the development of a predictive capability. An obstacle to model development is the lack of a demonstrated capability to experimentally characterize the morphology and reactivity of the constituent phases, which typically are polycrystalline with wide pore size distributions. Investigation of alternative experimental criteria for fragmentation is desirable in order to verify the insensitivity of the critical porosity to sample shape and method of support. 5. Discussion
5.1 Consequences of Fragmentation Two consequences of fragmentation which are potentially of practical importance for carbon conversion are enhanced weight loss and enhanced production of particulate effluents. To estimate the impact on weight loss, we consider Zone II combustion with perimeter fragmentation at a critical porosity of 0.7. If the initial porosity is 0.1 (a typical value) and if fragments escape at 0.7 porosity, then the porosity increase from initial to fragmentation value is 0.6, with the remaining weight loss (equivalent to an additional 0.3 porosity increase) attributable to fragmentation. In this example, therefore, 1/3 of the weight loss is attributable to fragmentation, a rather substantial fraction in view of previous inattention to this weight loss mechanism.
947
With regard to effluents, fragmentation may influence the size distribution of flyash produced during char burnout. Small flyash particles are of particular concern because they are difficult to remove from the exhaust stream. On a mass-weighted basis, most of the small mineral matter inclusions in char are believed to fuse and coalesce during burnout, 25 forming relatively small numbers of large flyash particles. However, if substantial fragmentation occurs prior to mineral matter coalescence, small inclusions within fragments are likely to maintain their integrity and therefore may contribute substantially to the total number of flyash particles. Any methods which are proposed for suppressing the formation of small flyash particles should take into account the possibility of this hitherto unrecognized source of such particles. 5.2 Summary Although fragmentation of carbons and other porous solids has been observed and reported in various contexts, a systematic, quantitative investigation of this phenomenon and its consequences has not previously been undertaken. We have found that careful examination of the geometrical criterion for fragmentation reveals a commonality previously obscured by the diverse physical mechanisms involved in specific cases. Most importantly, fragmentation is predicted to be a distinct and reproducible transition. Percolation theory provides quantitative predictions concerning the critical porosity for fragmentation, the mass distribution of fragments, and related quantities. Our experimental studies of six carbons verify that fragmentation is indeed a distinct transition, occurring at a porosity which is reproducible for a given material, though it depends in a complex way on material properties. Additional theoretical studies in the framework of percolation theory, as well as experimental studies, are needed to interpret this dependence more clearly. The principal technological application of these results is to char conversion. We have identified two previously unrecognized consequences of fragmentation, enhanced weight loss and enhanced production of fine flyash. The quantitative predictions should aid in identifying these effects in combustors and reactors.
Acknowledgments The authors would like to thank D. J. Holve and R. E. Mitchell for stimulating and helpful discussions, and A. Salmi for assistance with the experiments. This research was supported by the Offices of Basic Energy Sciences and Fossil Energy, U.S. Department of Energy.
948
COMBUSTION-GENERATED PARTICULATES REFERENCES
1. WALKER, P. L., JR., RUSINKO, F., AND AUSTIN, L. G.: Adv. Catal. 11, 133 (1959). 2. LANG, F. M. AND MAGNIER, P.: Chemistry and Physics of Carbon (P. L. Walker, Jr., Ed.) Vol. 3, Marcel Dekker, 1968. 3. Du'rrA, S., WEN, C. Y., AND BELT, R. J.: Ind. Eng. Chem., Proc. Des. Dev. 16, 20 (1977). 4. SUUBERG, E. M., PETERS, W. A., AND HOWARD, J. B.: Ind. Eng. Chem., Proc. Des. Dev. 17, 37 (1978). 5. NEOH, K. G., HOWARD, J. B., AND SAROFIM, A. F.: Particulate Carbon: Formation During Combustion (D. C. Siegla and G. W. Smith, Eds.), p. 261, Plenum, 1981. 6. GUINOT, J., AUDIER, M., COULON, M., AND BONNETAIN, L.: Carbon 19, 95 (1981). 7. PARK, C.: AIAA J. 21, 1588 (1983). 8. GAVALAS, G. R.: Combost. Sci. Tech. 24, 197 (1981). 9. BESHTY, B. S.: Combust. Flame 32, 295 (i978). 10. PIKE, G. E. AND SEAGER, C. H.: Phys. Rev. B 10, 1421 (1974). 11. ELAM, W. T., KERSTEIN, A. R., AND REttR, J,
J.: Phys. Rev. Lett. 52, 1516 (1984). 12. ZIMAN, J. M.: Models of Disorder, Cambridge Univ. Press, 1979. 13. PETERS, N.: Combust. Sci. Tech. 30, 1 (1983). 14. KERSTE1N, A. R. AND LAW, C. K.: Nineteenth Symposium (International) on Combustion, p. 961, The Combustion Institute, 1982. 15. KERSTEIN, A. R.: Combust. Sci. Tech. 37, 47 (1984). 16. MANDELRROT, B. B.: The Fractal Geometry of Nature, Freeman, 1982. 17. STAUFr'ER, D.: Phys. Rep. 54, 1 (1979). 18. GAUNT, D. S. AND SYKES, M. F.: J. Phys. A 16, 783 (1983). 19. TURBAN, L.: J. Phys. C 12, L191 (1979). 20. MAttAJAN, O. P., KOMATSU, M., AND WALKER, P. L., JR.: Fuel 59, 3 (1980). 21. RADOVI(~, L. R., WALKER, P. L., JR., AND JENKINS, R. G.: Fuel 62, 849 (1983). 22. MAHAJAN, O. P., YARZAB, R., AND WALKER, P. L., JR.: Fuel 57, 643 (1978). 23. GAVALAS, G. R.: AIChE J. 26, 577 (1980). 24. SIMONS, G. A.: Fuel 59, 143 (1980). 25. SMITII, R. D.: Prog. Energy Combust. Sci. 6, 53 (1980).
COMMENTS B. Gerhold, Phillips Petroleum Co., USA. The critical porosity for homogeneous solid fracture was shown to be about 0.70. In coal char, the ash can be nonreactive inclusions in a reaction carbon matrix or for some ash species, the ash can be catalytic. These two possibilities are inconsistent with an assumed equal probability of reacting anywhere in the solid. Also, in some applications, the c0al char experiences high temperature where the ash inclusions can melt and flow locally thereby healing potential fractures. Can you modify the theory to account for these effects? Perhaps a Monte Carlo calculation with random ash inclusions could better represent coal. Variation of the critical porosity with the precent ash and ash inclusion size may then explain the different friability of different ranks and deposits of coal. Authors' Reply. Catalysis of gasification reactinns by mineral species is not necessarily inconsistent with the assumption of uniform reactivity throughout the solid. If the catalytic mineral forms are well dispersed, so that their influence is essentially uniform on the size scale of fragments, this assumption may be justified. In many coals, various catalytic mineral forms are uniformly dispersed on the scale of a few hundred Angstroms, and in low-rank coal
chars, alkaline earth cations are bonded into the organic matrix as chelates. Since the degree of dispersion influences the effectiveness of a catalyst, the effect of the dispersed minerals may dominate over any catalysis by the larger, discrete inclusions. To the extent that large (micron-size) mineral inclusions influence the conversion process, whether because they are catalytic or nonreactive, they should be modeled as a separate phase. Extension of the percolation mechanism to encompass nonreactive inclusions is discussed in Sec. 3, and will be addressed in greater detail in future work. The bonding of fragments by molten ash requires a more complex model because, presumably, various mineral inclusions must coalesce before the melt could heal potential fractures. Data on the friability of coals under various conditions and at various ash levels would help to identify the most sensitive parameters and to determine which phenomena should be incorporated into a Monte Carlo simulation model.
J. Lahaye, C.N.R.S., France. The percolation threshold theory is quite useful to determine the limit above which the solid particle collapses. Don't
FRAGMENTATION DURING CARBON CONVERSION you believe that in practical conditions, because of fracture propagation, the fracture occurs for pore volumes much lower than the threshhold as determined by the percolation theory?
Authors" Reply. Some reported observations of fragmentation during devolatilization are indeed indicative of a pressure-induced shattering process. However, conversion rates during char gasification are generally too slow to induce mechanical failure.
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In some fluidized or entrained flows, abrasion of the char may be an important fragmentation mechanism. However, in slow conversion processes in fixed beds, the percolation mechanism should predominate. Unambiguous verification of the percolation mechanism may be achieved by experimental confirmation of the predicted fragment size and mass distributions. Efforts to achieve such confirmation are in progress.
