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Energy & Fuels 1999, 13, 826-831
Combustion Characteristics of Carbon: Dependence of the Zone I-Zone II Transition Temperature (Tc) on Particle Radius Robert H. Essenhigh* and Heather E. Klimesh Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210
Dieter Fo¨rtsch Institute of Process Engineering and Power Plant Technology (IVD), University of Stuttgart, Stuttgart, Germany Received November 3, 1998
Reformulation of the standard equations governing internal reaction of carbon, used to predict the Zone I-Zone II transition temperature, Tc, shows that Tc is, primarily, an inverse function of particle radius (a). The expression obtained has the form: log(a) ∝ (1/Tc). This result is supported by the (limited) experimental data available. The numerical values also show that, for particles of pulverized coal size, the values of Tc are in the flame temperature range, thus possibly allowing a discontinuous drop in reaction rate toward the end of a flame; this has relevance to increased LOI due to staging for NOX control. The results also permit evaluation of the internal diffusion coefficient indicating that internal reaction most probably involves only the macropores, and total internal surface area measurements are probably irrelevant in determining reaction rates.
Introduction In the combustion of porous carbon particles it is wellknown from the standard extensions1-7 of the original Thiele analysis8 that, with rising temperature, there is a Zone I/Zone II transition in the reaction mechanism at a critical temperature, Tc, with reaction fully penetrating the particle in Zone I, when T < Tc, but with only partial penetration in Zone II when T > Tc. The transition is associated with a change in effective activation energy, falling from the true value of EI in Zone I to an effective value EII, in Zone II, where EII ) 1/2EI.1-7 The transition point is most commonly defined by setting the Thiele Modulus,2,3 φ, to unity; and using this criterion we now show in this Communication that the temperature, Tc, at which the Zone I/Zone II * Corresponding author. Department of Mechanical Engineering, The Ohio State University, 206 West 18th Avenue, Columbus, OH 43210. Telephone: (614) 292-0403. Fax: (614) 292-3163. E-mail:
[email protected]. (1) Wicke, E.; Rossberg, M. Z. Elektrochem. 1953, 57, 641-645. Wicke, E. Proceedings of the 5th Symposium (Int.) on Combustion; Reinhold Publ. Co.: New York, 1955; pp 245-252. Wicke, E.; Wurtzbacher, G. Int. J. Heat Trans. 1962, 5, 277-289. (2) Walker, P. L.; Rusinko, F.; Austin, L. G. Adv. Catal. 1959, 11, 133-221. (3) Peterson, E. E. Chemical Reaction Analysis; Prentice-Hall: New York; 1965. (4) Thomas, J. M.; Thomas, W. J. Principles and Practice of Heterogeneous Catalysis; VCH: Weinheim Germany, 1997. (5) Aris, R. Ind. Eng. Chem. Fundam. 1967, 6, 316-318. (6) Mulcahy, M. F. R.; Smith, I. Rev. Pure Appl. Chem. 1969, 19, 81-108. (7) Essenhigh, R. H. Fundamentals of Coal Combustion. In Chemistry of Coal Utilization: Second Supplementary Volume; Ellior, M. A., Ed.; John Wiley and Sons: New York, 1981; Chapter 19, pp 11531312. (8) Thiele, E. W. Ind. Eng. Chem. 1939, 31, 916-920.
transition takes place, is an inverse function of the particle radius, a. The significance of this transition is that, for particles of sizes used in pulverized coal (pc) flames, the values of Tc are in the flame temperature range. The implication of this dependence, as a driver for this study, is the possibility of an almost discontinuous change in the reaction rate of a given particle if the particle cools through its transition temperature as it moves toward the end of the flame. Under certain conditions of firing coal in pc boilers, such as staging for NOx control, this transition behavior, we find, could be responsible for the now well-recognized phenomenon of increased carbon in the fly ash, commonly referred to as the loss on ignition (LOI). This problem of increased carbon in the ash is currently a matter of substantial economic concern on account of the reduced salability of the ash, used in manufacture of cement, due to the higher carbon levels. Thus, the engineering problem generated by the economic concerns of higher carbon in the ash (LOI) is evidently dependent for solution on precision of knowledge at the fundamental level. It is for this reason that we are examining the relationship between transition temperature and particle radius We also identify further outcomes, including definition of a number of critical problems requiring clarification or solution and relating particularly to the influence of the density-diameter index, R, and the internal surface, Sv, on the reaction process.
10.1021/ef980241g CCC: $18.00 © 1999 American Chemical Society Published on Web 06/30/1999
Zone I-Zone II Transition Temperature and Particle Radius
Development General Formulation of Equations. What is original in this treatment is the development of the Tc-a relationship from standard expressions in the literature,1-10 as we now show. The development also provides additional results not previously described. Using Peterson’s3 general form for the Thiele Modulus we have
1 n + 1 1/2 (n-1)/2 φ ) λa ys 3 2
(
)
(1)
or
Energy & Fuels, Vol. 13, No. 4, 1999 827
where F is a numerical multiplier of order unity, with a half order dependence on the oxygen concentration, ys, for n ) 0. The effective activation energy, E, takes the value EI or EII on either side of the transition temperature but, at the transition point between the two zones, since the transition is not, in fact, discontinuous3 (though commonly shown as such), it commonly takes the average value Eav ) (EI + EII)/2. At the intersection of the two zones, at T ) Tc, we then have
( )
kc ) koI exp -
( )
EI EII ) koII exp ) RTc RTc
( )
ko exp 1 φ ) λa 3
(for n ) 1)
(1a)
where
λ2 )
Svk FDe
(2)
λ)
3Sv R
(3)
Eliminating the internal surface area term, Sv, from eq 2 by dividing by eq 3, gives
λ)
Rk 3FDe
(4)
and using this result to eliminate λ in eq 1, by substitution, yields
φ)
(
)( )
1 Rk n + 1 a 3 3FDe 2
1/2
y(n-1)/2 s
(5)
noting that, in this form of the equation, φ now does not explicitly depend on the internal surface, Sv. This extension and conclusion is believed to be original here. Equations at Transition. The Zone I/Zone II interface, when T ) Tc, is commonly defined as already noted by setting the value of φ to unity.2,3 For the values of the other parameters: the (true) reaction order, n, has bounding values of 0 and 1; and, in Zone II, the experimental values of R at flame temperatures have been shown mostly to range from 1 to 310 (although R can be orders of magnitude larger at ignition temperatures, in Zone I, governed by the Arrhenius group parameter10); experimentally, R has also been shown to be essentially constant through the combustion lifetime of a single particle.9,10 Using these values, eq 5 can then be rearranged to relate the operational reaction velocity constant, kc, at the transition temperature, Tc, to the particle radius, taking the form
(
kc ) k0 exp -
)
E 1 ) F(FDe) RTc a
As an alternative approximation reduction we note that if we use eq 3 to eliminate λ in the standard reduced form (eq 1a) of the Thiele Modulus, i.e., φ ≈ (λa), then at φ ≈ 1, we get
Sv ≈
where k ) k° exp(-E/RT) For the Zone II reaction we also have9,10
(6)
(9) Essenhigh, R. H. Proceedings of the 22nd Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1988; pp 8996. (10) Essenhigh, R. H. Combust. Flame 1994, 99, 269-279.
Eav (7) RTc
R 3a
(8)
For the set of conditions that R ≈ 3,10 this reduces to Sv ≈ 1/a. This result also is believed to be original here; its significance and potential for interpretation is discussed briefly below. Reduced Equation: Tc ) f(a). In eq 6, the product term (FDe) is temperature dependent and in standard form can be written as: (FDe)o (T/To)m, with 0.2 < (1 + m) < 1.7 where the lower and upper bounds on (1 + m) represent, respectively, limiting control either by Knudsen diffusion (lower value) or by molecular diffusion (upper value).4,7,11 The effective (stp) diffusion coefficient (Deo) is jointly dependent on the (reciprocal) sum of the (stp) molecular (DMo) diffusion and the Knudsen (DKo) diffusion terms (see Nomenclature), and iteration is required to determine the appropriate value of m and thence of Deo in given cases.4 With the given temperature correction included and incorporated in the radius term, eq 6 can be rewritten in the (log) form
[( )] (
log a
To Tc
m
) log
)
F (FDe)o ko
+
E 1 2.3R Tc
(9)
This is the inverse functional relation between radius, a, and transition temperature, Tc, identified in the opening paragraph as the principal result of this Communication, and numerical validation is provided, as we show next. Validation is limited in range at this time due to the relative paucity of known experimental results; within that limitation, however, eq 9 is experimentally supported, as we show. We note in advance that the available data nevertheless range in radius from roughly 10 µm to 1 cm, a range of 3 orders of magnitude; within that range, the temperature correction term (To/Tc)m is a second-order perturbation so that the dominant variable parameter groups, for approximately constant R, are log(a) and (1/Tc). Thus, the dominant result of the theoretical development in this Communication is the inverse dependence (11) Valix, M.; Harris, D. J.; Smith, I. W.; Trimm, D. L. Proceedings of the 24th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1992; pp 1217-1223.
828 Energy & Fuels, Vol. 13, No. 4, 1999
Figure 1. Variation of transition temperature (Tc) with particle size (a). Data sources: refs 12-18. Values in the 1 mm (1000 µm) range are inaccessible due to boundary layer control (see text).
between log(a) and Tc. The two most important supplementary results are (1) demonstration in eq 5 of the elimination of the explicit dependence of φ on the internal surface, Sv, and (2) demonstration in eq 8 that the value of Sv at transition is of the order of the inverse value of the particle radius, a. Numerical Evaluation Data Sources. Although the Zone I/Zone II transition has been identified and discussed extensively in the past four decades,1-7 usable joint data on values of Tc and a for validating eq 9 are unexpectedly sparse, with only eight data sets identified at this time.12-18 These are illustrated in Figure 1, as plot of Tc vs a, with the theoretical line, extrapolated to the range 1 µm-10 cm, calculated from eq 9 using parameter values determined as given below. Usable data for this application are limited to singleparticle experiments that also exhibit the Zone transition. This excludes, for example, the transitions obtained in solid bed studies, or in the Smith et al.11,19 solid-bed/ single-particle comparisons; these applications both require separate evaluation. In addition, data are available only for either large particles of order 1 cm (diameter) where the boundary layer control is vitiated by natural or forced convection; or for small particles, (12) Tu, C. M.; Davis, H.; Hottel, H. C. Ind. Eng. Chem. 1934, 26, 749. (13) Froberg, R. W.; Essenhigh, R. H. Proceedings of the 17th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1979; pp 179-187. (14) Lester, T. W.; Seeker, W. R.; Kerklin, J. G. Proceedings of the 18th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1981; pp 1257-1265. (15) Hurt, R. H.; Mitchell, R. E. Proceedings of the 24th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1992; pp 1233-1241. (16) Monson, C. R.; Germane, G. J.; Blackham, A. U.; Smoot, L. D. Combust. Flame 1995, 100, 669-683. (17) Mitchell, R. E.; McLean, W. J. Proceedings of the 19th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1982; pp 1113-1122. (18) Shaddix, C. H.; Johnson, E. T.; Walsh, P. M. Proceedings of the 1st Joint Meeting of the U.S. Sections of the Combustion Institute; Washington, DC, March 14-17, 1999; pp 833-836. (19) Smith, I. W.; Tyler, R. J. Comb. Sci. Technol. 1974, 9, 87-94.
Essenhigh et al.
Figure 2. Variation of log[af(Tc)] with 1/Tc,: Test of eq 9.
of the order of or less than 100 µm (diameter), where the boundary layer is no longer controlling or dominant. Particles of about 1 mm are typically under boundary layer control,20,21 and the Zone transition is not seen (is suppressed) or does not occur. Data Plots Figure 2 is the relevant re-plot of the Figure 1 data12-18 in the format of eq 9, to show, for constant ko, the dependence of log[af(Tc)] on 1/Tc. The linear fit that is set on the graph of Figure 2 is in accordance with (though, for paucity of data, only marginally tests) the linearity predicted by eq 9; nevertheless, this linearity is well-suppported by numerical evaluation (below) of the relevant constants (slope and intercept). One of the data sources for Figure 2 is Hurt and Mitchell15 who presented their rate data, for Pocahontas coal char, with a linear fit on an Arrhenius plot. Their data are reproduced in Figure 3, with fitted lines at slopes corresponding to activation energies of EI and EII () 1/2EI) to show the clear break in the curve that we identify here as the Zone I/Zone II transition. The nonrandom variation from the author’s original linear fit (dashed line) is strongly indicative of more complex behavior identified here. This type of re-plot is the procedure adopted for the other sources12,14,16,18 where no transition break had previously been identified and the reported plots were averaged by a single line. Parameter Evaluations (a) Activation Energy. In Figure 2 the slope takes the value according to eq 9 of: (E/2.3‚R); yielding a value for E of 30 180 cal/mole. This, of course, is Eav as indicated in the third term of eq 7. This is almost exactly the value originally reported by Hurt and Mitchell, of 30 000 cal/mole, for their linear fit to their data (dash line in Figure 3). It also closely corresponds to the average values obtained for the other cases listed. These are reexamined in Figure 4, below. (20) Beeston, G.; Essenhigh, R. H. J. Phys. Chem. 1963, 67, 13491355. Essenhigh, R. H. J. Engrg. for Power 1965, 85, 183-188. Essenhigh, R. H.; Yorke, G. C. Fuel 1965, 44, 177-186. (21) Nettleton, M. A. Ind. Eng. Chem. Fundam. 1967, 6, 20-25.
Zone I-Zone II Transition Temperature and Particle Radius
Figure 3. Re-plot of Hurt and Mitchell15 Arrhenius’ graph for Pocahontas Coal: illustrating Zone I/Zone II transition. (Dash-dot line: original linear fit.)
Figure 4. Auto-correlation plot of log(ko) with activation energy, E. Zone I: (diamonds) 40-50 kcal/mol; Zone II: (diamonds) 20-25 kcal/mol; Mean: (squares) 30-37.5 kcal/ mol.
(b) Diffusion Parameter Values. The iteration to obtain the appropriate value of (1 + m) resulted in a value of 0.8. The magnitude of Deo was then extracted from the intercept value (I) in Figure 2 that, from eq 8, is given by I ) log(FFoDeo/ko). From the data plots, the value of ko was obtained as 240 g/cm2 s, giving a value for FFoDeo of 8.8 × 10-5 g cm-1 s-1. Correcting for stp air density (Fo ) 1.3 × 10-3 g/cm3) and taking F ≈ 1 gives Deo ) 0.068 cm2/s. Using DMo ) 0.207 cm2/s,5 this gave DKo ) 0.10 cm2/s. The ratio of the two diffusion coefficients (DKo/DMo) ≈ 0.5, and using the standard result that this is approximately the ratio of the (average) pore diameter to the molecular mean free path, this gives a value for the diameters of the pores involved in the reaction of approximately 0.05 µm (i.e., half the magnitude of, or the order of the molecular mean free path at stp22). (c) Internal Surface Area (Sv). This is obtained by eliminating λ between eqs 3 and 4, so that Sv ≈ k/(FDe), (22) deBoer, J. H. The Dynamical Character of Adsorption; Oxford University Press: New York, 1953.
Energy & Fuels, Vol. 13, No. 4, 1999 829
and then calculating Sv by appropriate substitution of (back-calculated) values of k, F, and De, taken at temperature Tc. The values of Sv in the different cases are startlingly differentsabout 200 cm2/cm3 for the smaller diameter (≈100 µm) char particles,14-18 and 0.3 cm2/cm3 for the larger diameter (1 to 2 cm) manufactured carbons,12,13 a ratio of about 600. This result is, nevertheless, entirely in accordance with eq 8, with Sv decreasing with increasing a (at constant R). Correspondingly, at the Zone transition, the product, aSv ≈ φ ≈ 1, for all the particles, as required by postulation. This implies that the magnitude of the true internal surface (measured, for example, by N2 or CO2) may have minimal bearing on the rate of the reaction process. With a potential perturbation under limited circumstances from a variability in R, this conclusion is foreshadowed by the demonstration in eq 5, jointly, of the independence of φ on Sv and of the inverse relation between Sv and a, and this has some original consequences as we outline below. (d) Data Summary. The additional key results obtained from the data analysis are the values of the activation energies (E) and preexponential factors (ko), and these are presented and compared in the Figure 4 auto-correlation plot23 of log(ko) vs E. The graph shows a band, with three groups of data points representing, respectively, the Zone I values (highest) with EI in the range 40-50 kcal/mol, the Zone II (lowest) values (EII) at half that, in the range 20-25 kcal/mol, and with the middle group (30-37.5 kcal/mol) representing the average for values determined across the transition zone. This differentiation helps to clarify the problem of the “true” activation energy of the carbon oxidation. The very wide range of values reported in the literature over the last half century (see Table 19.6 in ref 7) show values broadly in the range 20-50 kcal/mol. Figure 4 supports the view that: the higher range is obtained when the reaction is (predominantly) in Zone I; the lower range when the reaction is (predominantly) in Zone II; and the middle range when the reaction is crossing the Zone I/Zone II transition. The band in Figure 4 represents the bounds (corrected for oxygen concentration) of the values compiled by the Sandia National Laboratory24 reported in ref 23. The majority of the points are inside the data band or within a common range. Two “exceptions” are seen as two points on the high side in the Zone I and Zone II groups that are nearly 2 orders of magnitude higher than the rest of the data. Excluding those two exceptions, as special cases as identified below, the common autocorrelation carries or implies the probability of essentially common baseline kinetics rates for all the different samples, in support of the common-kinetics view for coal chars advanced, notably, by Fu and associates.25 The two exception points are the Zone I and Zone II parameter values (ko and E) for measurements, on coal, (23) Essenhigh, R. H.; Misra, M. K. Energy Fuels 1990, 4, 171177. (24) Hardesty, D. R. Quarterly Reports of the Sandia National Laboratories Combustion Research Facility, July 1987-June 1988. Mitchell, R. E.; Hurt, R. H.; Baxter, L. L.; Hardesty, E. R. Compilation of Sandia Coal Char Combustion Data and Kinetic Analyses, Sandia report SAND 92-8208, 1992. (25) Fu, W. B.; Zhang, E. Z. Combust. Flame 1992, 90, 103-113. Fu, W. B.; Zhang, B. L.; Zheng, S. M. Combust. Flame 1997, 109, 587598.
830 Energy & Fuels, Vol. 13, No. 4, 1999
carried out in a shock tube14 so that heating rates are in the range 107 K/s. The parameter values show the following characteristicssthat the EI and EII values are in the same group range as for the other data, but that the frequency factor values are high. Since the fuel used was coal, not pre-carbonized char, the char reaction includes a prior devolatilization. At very high heating rates (> 106 K/s) it is known from other (high-intensity) flame studies24 that the char formed under very fast pyrolysis is about an order of magnitude higher in reactivity than for chars formed at rates less than 105 K/s. The ko values obtained from ref 14 are thus in accordance with the results from the high-intensity flame measurements (noting also that the Tc value for this fuel, shown as the lowest points on Figures 1 and 2, are nevertheless in agreement with the predicting equation, eq 9). Commentary The most significant result of this development is the demonstration of the inverse dependence of the transition temperature on the particle radius (when R is approximately a constant) combined with the notation that the temperatures are in the flame temperature range for pc particles. The most significant further developments are (1) the demonstration of evident macropore dominance in the internal reaction, and (2) the independence of φ on Sv. (1) The significance of the first result is that if the internal reaction is assumed, ab initio, to be taking place only in the macropores, of diameter about 0.1 µm, then the whole of the argument developed above can be reversed to predict the line of Figure 2 as a theoretical expectation, and the experimental values then become direct experimental support and verification of the model predictions. This thus identifies a deniable (i.e., testable) postulate of the development presented here. To falsify the initial postulate requires demonstration of significant reaction in the transition and micropores in normal combustion. (2) Significant reaction in the transition and micropores has often or commonly been assumed, but the direct evidence is believed at this time to be inconclusive. The assumption has generally been based on measurements showing internal surface areas as high as 100-500 m2/g, with as much as 85% associated with micropores, and with the further assumption that this micropore surface area participates in the reaction. Accessibility to that large (micropore) surface, however, during fast reaction becomes problematical. The effective accessible pore area per molecule in the pore can be shown in a standard analysis to be approximately inversely proportional to the pore diametersthis has a structural correspondence in eq 8. For the micropores, this reduces the accessible area by about 2 orders of magnitude compared with the macropores. This evaluation is not definitive but it is consistent with the finding above that the relevant pore diameter was that of the macropores. (26) Goldberg, P. M.; Essenhigh, R. H. Proceedings of the 17th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA., 1979; pp 145-154. Farzan, H.; Essenhigh, R. H. Proceedings of the 19th Symposium (Int.) on Combustion; The Combustion Institute: Pittsburgh, PA, 1982; pp 1105-1111.
Essenhigh et al.
(3) This position is further supported by the earlier finding10 that the Zone II penetration factors, characterized by the density-diameter index, R, are mostly of order unity (range 1 to 3) rather than 10 or higher. For deep penetration and/or for extensive utilization of the internal surface, the magnitude can be 1000 or higher (as reported in one case at ignition10). The position is also consistent with the theoretical result of eq 5 showing the lack of dependence of the Thiele Modulus, φ, on the internal surface parameter, Sv. (4) This evaluation suggests that the past emphasis on relevance and measurement of the internal surface, though clearly reasonable on its face, may have been a misdirection. The focus should perhaps be on the accessibility of the internal surface under (fast) reaction conditions, and with more emphasis on the fraction of the accessible area that is actually utilized. This is already characterized in the Thiele analysis8 by the effectiveness factor, η, defined as the ratio of the actual (internal) reaction rate (Ri) to the maximum (internal) reaction rate, Rimax:η ) Ri/Rimax, with the further relation to the Thiele Modulus: φ ≈ 1/η under Zone II conditions. (5) The further characteristic of interest with regard to the internal surface is the inverse relation shown by eq 8: Sv ≈ 1/a at transition; with the supporting calculations given above (and with the proviso regarding the value range of R). A mechanistic relationship between an independently measurable (e.g., by N2 or CO2) internal surface area and the particle radius clearly makes no sense: the possibility is denied by considering a large particle that is broken into pieces; clearly, there is no increase in actual internal area on that account. What Sv may represent, however, is the effectively accessible area involved in reaction, corresponding to an “inverse” diffusion boundary layer penetrating into the particle and determined by the radius of curvature. The meaning of Sv clearly requires more analysis; comparisons of predictions by pore tree, or random pore, or by fractal models would be particularly valuable for this. (6) Separately, the result shown by Figure 4 of the three groupings of data, at approximately 20+, 30+, and 40+ kcal/mol, provides a basis to resolve the question regarding the variability or otherwise of carbon reactivity from one sample to another. The wide range reported in the standard literature, mostly from 20 to 50 kcal/ mol, has commonly been used to support the proposition of variable reactivity (and the need, therefore, for variable kinetics constants parameters). The structure of Figure 4 with the interpretation in terms of Zone I, Zone II, and transition values indicates that if reactivity variability does exist it has, nevertheless, very much smaller range than is commonly assumed or proposed. The real reactivity variations reported can be due to measurement in the different zones or to measurement across the zones. In support of this position, as noted earlier, there is also now the independent set of results from Fu et al.25 in support of this proposition. Summary. The principal result, that the critical temperature for transition from Zone I to Zone II, Tc, depends on a determined inverse function of particle radius, a, is supported, with the quantitative further result that values of Tc are in the range found in
Zone I-Zone II Transition Temperature and Particle Radius
pulverized coal flames, for particles of relevant size. The further outcomes of this examination, that the participating pores in the internal reaction are the macropores, and that the magnitude of the internal surface is not, apparently, critically significant (for reasons yet to be determined), are presented as deniable (i.e., testable) postulates and, thus, as targets for critical test in future studies. Nomenclature a ) particle radius. De ) effective diffusion coefficient inside porous particle; ) 1/[(1/DK) + (1/DM)] DK ) Knudsen diffusion coefficient DM ) molecular diffusion coefficient E ) activation energy F ) numerical multiplier; ≈ 1 k ) kinetics velocity constant; ) ko exp(-E/RT) kc ) velocity constant at Tc ko ) frequency factor
Energy & Fuels, Vol. 13, No. 4, 1999 831 n ) effective reaction order at the solid surface Sv ) internal surface area (area/unit volume) Tc ) Zone I/Zone II transition temperature yox ) oxygen mole fraction concentration R ) power index in density/radius relation: σ/σo ) (a/ao)R φ ) Thiele Modulus; ) 1/3 λa [(n + 1)/2]1/2ys(n-1)/2 λ ) Thiele modulus parameter; ) [(Svk)/(FDe)]1/2 F ) gas density σ ) particle density subscript ) I, II: Zone I and Zone II values superscript ) o: base values (e.g., at stp)
Acknowledgment. We appreciatively acknowledge support for this work provided by the Department of Energy under DOE Project No. DE-FG22-96PC06269 (Project Monitor P.L. Goldberg) in association with Brown University (Lead: Principal Investigator, Dr. R. Hurt). EF980241G
