at Elevated Temperatures with a Modified Random Pore - American

Energy & Fuels 2003, 17, 961-970

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Unification of Gasification Kinetics of Char in CO2 at Elevated Temperatures with a Modified Random Pore Model Hao Liu,†,‡ Chunhua Luo,‡ Masahiro Kaneko,† Shigeru Kato,† and Toshinori Kojima*,† Department of Applied Chemistry, Seikei University, 3-3-1 Kichijojikita-machi, Musashino-shi, Tokyo 180-8633, Japan Received October 11, 2002. Revised Manuscript Received April 14, 2003

The gasification of a char has, so far, been mainly studied at low temperatures and low heating rates, using thermogravimetric analysis. Studies on the gasification of a char in CO2 at elevated temperatures and high heating rates are necessary to develop integrated coal gasification combined-cycle (IGCC) technology. Coal gasification is a very complicated phenomenon, which makes a generalized understanding and description of gasification reactivity of a char very important. Using a unique fluidized bed, the reactivity of a char pyrolyzed at elevated temperatures and high heating rates was investigated. A generalized expression of the gasification reactivity of a char under various conditions was approached with a modified random pore model proposed in this work. It was clarified that a shrinking core model was not appropriate to describe char gasification in CO2. On the basis of our results and those in the literature, it was identified that the char reactivity could be unified with a master curve not only for chars pyrolyzed at low temperatures and low heating rates, but also for chars pyrolyzed at high temperatures and high heating rates. This master curve was independent of coal type, temperature, and the pyrolysis conditions under which a char was derived, despite the fact that these factors had complicated influences on the reaction rates. Our model can also predict the data in the literature very well. The model proposed in this work gave better predictions than the random pore model, up to high carbon conversions. It was recommended as a submodel for the comprehensive simulation of coal gasifiers.

Introduction As a key component included in integrated coal gasification combined-cycle (IGCC) technology, entrained flow gasifiers have been highlighted in recent years, because of their high gasification efficiency and smooth discharge of molten ash. However, the high temperature in an entrained flow gasifier tends to cause different gasification behavior. Accordingly, the clarification of char gasification characteristics under conditions similar to those of entrained flow gasification becomes very important. Coal gasification, which is influenced by many factors (such as temperature, coal property, pyrolysis time, etc.), is a very complicated phenomenon, particularly at elevated temperatures. A generalized understanding and description of gasification reactivity of a char are very important for the development of entrained flow gasification and IGCC technology. In recent years, Japan has developed a 200 ton/day entrained flow IGCC gasifier and a 50 ton/day hydrogen-from-coal (HYCOL) gasifier. One of the problems encountered, particularly for simulation and scaling up of gasifiers, is how to * Author to whom correspondence should be addressed. E-mail: [email protected]. † Seikei University. ‡ Research Fellow, New Energy and Industrial Development Organization (NEDO), Japan.

choose an appropriate model to describe, simply and satisfactorily, the char reactivity and its variation during gasification. The random pore model has been developed by Bhatia and Perlmutter1-3 to account for the relationship between the surface area and the carbon conversion during the char gasification. According to the unification approach proposed by Mahajan et al.,4 char gasification curves in the form of carbon conversion X versus gasification time t for different temperatures, pressures, gasifing agents, and chars could be approximately reduced to a single curve, when X was plotted against the dimensionless time τ (the ratio of reaction time to the half-life, with half-life being the time required for 50% carbon conversion). Moreover, with this approach, a unified expression of experimental gasification data was also successfully obtained by Kasaoka et al.5 and Peng et al.6 From the experimental data of a lignite derived with thermogravimetric analysis (TGA) in the temperature range of 1023-1473 K, Raghunathan and (1) Bhatia, S. K.; Perlmutter, D. D. AICHE J. 1980, 26, 379-386. (2) Bhatia, S. K.; Perlmutter, D. D. AICHE J. 1981, 27, 247. (3) Bhatia, S. K.; Perlmutter, D. D. AICHE J. 1981, 27, 226-234. (4) Mahajan, O. P.; Yarzab, R. Y.; Walker, P. L., Jr. Fuel 1978, 57, 643-646. (5) Kasaoka, S.; Sakata, Y.; Tong, C. Int. Chem. Eng. 1985, 25, 160175. (6) Peng, F. F.; Lee, I. C.; Yang, R. Y. K. Proc.sAnnu. Pittsburgh Coal Conf., 1986, 3rd, 730-744.

10.1021/ef020231m CCC: $25.00 © 2003 American Chemical Society Published on Web 05/20/2003

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Figure 1. Reaction rate of gasification versus carbon conversion at various temperatures for Senakin char (pyrolysis time, 30 s; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values. Table 1. Properties of Senakin Coal Proximate Analysis (as received, wt %) moisture 4.60

ash

VM

Ultimate Analysis (dry, ash free, wt %) FC

10.60 41.60 43.20

C

H

N

O

S

79.42 6.09 1.34 12.36 0.80

Table 2. Experimental Conditions property static bed height (mm) alumina particles diameter (mm) density (g cm-3) diameter of coal particles (mm) bed temperature (K) minimum fluidization velocity at bed temperature (m s-1) for alumina particles for coal particles terminal velocity (m s-1) (at bed temperature) for alumina particles for coal particles superficial gas velocity at bed temperature (m s-1)

value 100 0.119 (0.075-0.149) 3.0 0.194 (0.177-0.210) 1273-1873 0.0083-0.0061 0.012-0.0092 0.429-0.328 0.420-0.333 0.080-0.120

Yang7 found that the conversion-time behavior predicted by the model was in agreement with the experimental data, up to a carbon conversion of ∼70%. Ochoa et al.8 found that the discrete random pore model also (7) Raghunathan, K.; Yang, R. Y. K. Ind. Eng. Chem. Res. 1989, 28, 518-523. (8) Ochoa, J.; Cassanello, M. C.; Bonelli, P. R.; Cukierman, A. L. Fuel Process. Technol. 2001, 74, 161-176.

enabled the experimental results to be described satisfactorily with effective structural parameters. Truls and Krister developed a semiempirical gasification kinetic model and reviewed the most common rate models.9 In addition, a quantitative model for gas-solid reactions has been used by Marban and Fuertes10 to analyze the effect of percolation on the evolution of the overall particle properties (particle size, overall porosity, and reactive surface area) during gasification and combustion reactions. However, many problems remain to be clarified. Most of the previous experiments were conducted at a much lower temperature than that in an entrained flow gasifier, where the temperature can reach 1773 K or even higher.11,12 Many of the previous studies about char gasification were performed via TGA,13-15 which has many limitations, because of its low gas throughput, low heating rate, etc. A fixed bed usually also suffers from the same shortcomings as TGA. A drop tube furnace facilitates mass transfer and can be used for hightemperature experiments; however, it has several draw(9) Truls, L.; Krister, S. Fuel 1997, 76, 29-37. (10) Marban, G.; Fuertes, A. B. Chem. Eng. Sci. 1997, 52, 1-11. (11) Jahnke, F. C. Proceedings of the Advanced Clean Coal Technology International Symposium ’97, Tokyo, Japan, 1997; pp 111-120. (12) Chen, C.; Miyoshi, T.; Kamiya, H.; Horio, M.; Kojima, T. Can. J. Chem. Eng. 1999, 77, 745-750. (13) Russell, N. V.; Beeley, T. J.; Gibbins, C. K.; Man, J. R.; Williamson, J. Fuel Process. Technol. 1998, 57, 113-130. (14) Beamish, B. B.; Shaw, K. J.; Rodgers, K. A.; Newman, J. Fuel Process. Technol. 1998, 53, 243-253. (15) Devi, T. G.; Kannan, M. P. Energy Fuels 2000, 14, 127-130.

Unification of Gasification Kinetics of Char in CO2

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Figure 2. Dimensionless reaction rate of gasification versus carbon conversion at various temperatures for Senakin char (pyrolysis time, 30 s; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

backs, such as the difficulty to separate pyrolysis and gasification processes, nonuniform temperature distribution, and uncertainty regarding reaction temperature and residence time, as well as short reaction time. All these factors mean that more research is necessary on the gasification of char pyrolyzed at a high heating rate and an elevated temperature with an adequate experimental facility. Furthermore, it is very important to clarify the char gasification reactivity, not only at low conversions, but also at high ones, because the conversion of a char in an entrained flow gasifier can attain values of >90%. A model that can describe char gasification behavior up to high conversions is required, whereas the random pore model tends to overestimate the experimental results at conversions of >70%. Considering the previously detailed drawbacks and the need for more research, we studied the gasification reactivity of a char directly from the change in CO concentration in a unique fluidized bed that could be operated at temperatures up to 1873 K. This technique provided more-reliable results, because of its advantages of uniform temperature distribution, stable temperature, changeable pyrolysis time, both low and high heating rates during pyrolysis, in-situ and direct measurements, etc. The main objective of this work was, as fundamental research for IGCC technology development, to obtain the gasification kinetics of a char under conditions similar to those in an entrained flow gasifier, and to achieve a generalized understanding of char gasification, through a unification approach of the obtained kinetics. We proposed our own modified random pore model. On the basis of our own experimental data

derived from chars pyrolyzed at high heating rates and elevated temperatures in a unique reactor, we unified the gasification reactivity up to a high carbon conversion. Moreover, the unification of gasification reactivity of a char derived from various pyrolysis times was also approached. Experimental Section A unique fluidized bed, with an inner diameter of 35 mm, was used as the reactor. The bed, including the delicately fabricated distributor, was made of alumina and heated by siliconit electric heaters, so that it could be operated at high temperatures. Bed particles were inert alumina. Fluidizing gas (20% CO2, with N2 as the balance) was supplied from cylinders. The flow rates of N2 and CO2 into the bed were controlled by needle valves and measured by orifice flow meters. The tested coal was screened to a narrow size range of 0.177-0.210 mm (with a mean diameter of 0.194 mm). First, the bed was heated from room temperature to the desired temperature in a N2 atmosphere. Coal particles then were pneumatically transported into the reactor by a nitrogen carrier gas through an alumina tube with an inner diameter of 5 mm. The char was kept for the desired time (residence time). During this process, the char particles were heated rapidly, i.e., at a high heating rate, to the bed temperature. This process was nonisothermal and was called pyrolysis. In this paper, heating rate refers to the heating rate of the pyrolysis prior to gasification. After pyrolysis, the gas line was immediately switched to a CO2-N2 mixture so that gasification of the coal char could begin. The gasification process is isothermal, because the bed temperature was kept constant. The gasification rate was derived from the CO concentration in the exhausted gas at the exit of the reactor, which was analyzed using gas chromatographs. The composition of the pyrolysis product, i.e., the volatile matter released from pyrolysis, was measured as a background in a nitrogen

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Figure 3. Dimensionless reaction-rate-carbon-conversion curves described by various models (Φ ) 1).

Figure 4. Master curve predicted by various models with different parameters. atmosphere through separate experiments. The production rate of CO from pyrolysis was subtracted from the total production rate of CO when the gasification reaction rates were calculated, to avoid the influence of CO that was produced from pyrolysis.

An Indonesia coal (Senakin) was adopted in the present work. Its proximate and ultimate analysis results are listed in Table 1. Experimental conditions are listed in Table 2. The details about the experiment were described in our previous article.16



Figure 5. Carbon-conversion-dimensionless-time master curve at various temperatures for Senakin char (pyrolysis time, 30 s; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

Modeling Random Pore Model. For each gasification run, researchers usually fit rate expressions of the form

r)

dX ) r0F(X) dt

p) (1)

where X refers to the carbon conversion and t refers to the time of the gasification reaction; r0 refers to the initial reaction rate of gasification (i.e., dX/dt at the beginning of gasification); and r refers to the instantaneous reaction rate (i.e., dX/dt at time t). F(X) is a ratio of the instantaneous char reactivity to the initial reactivity. The random pore model (RPM), developed by Bhatia and Perlmutter1-3 to account for the relationship between the surface area and the carbon burnoff during the char gasification, has been used extensively in recent years.1-3,5,7,8,10 This model gives

dX ) r0(1 - X)[1 - f ln(1 - X)]1/2 dt

(2)

where

f)

4πLE0 S2E0

[ (

(3)

)]

pfτ2 4

(4)

2[(1 + f ln 2)1/2 - 1] f

(5)

t t0.5

(6)

X ) 1 - exp - p τ +

τ)

where f is a structural factor. LE0 is the total length of the pore per unit volume, and SE0 is a hypothetical total initial surface area per unit volume. τ is the dimensionless time and t0.5 is the half-life, i.e., the time required for 50% carbon conversion. These expressions were tested with the experimental data in the literature.1,7,8,17 The conversion-time behavior predicted by these equations was in agreement with the experimental data, up to ∼70% conversion. Extension of Unification Approach: A Modified Random Pore Model. As mentioned previously, the RPM could predict the conversion-time curve up to ∼70% conversion but overestimated the conversion, to some degree, at higher conversions. Thus, we modified the RPM using a structural factor that varied with the reaction:

Integrating eq 2 gives (16) Luo, C.; Watanabe, T.; Nakamura, M.; Uemiya, S.; Kojima, T. Fuel 2000, 80, 233-243.

f* ) f exp(-Φτ)

(7)

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Figure 6. Reaction rate of gasification versus carbon conversion at 1473 K for Senakin char (pyrolysis time, 0 s; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

where f* is a structural factor used in the modified random pore model (MRPM) and Φ is a coefficient, whereas τ is the dimensionless reaction time, as defined previously. This empirical modification was derived from experimental data with a trial-and-error procedure. The appropriate value of factor f was determined from the reaction-rate-carbon-conversion curve when X < 0.5, and the value of Φ was determined from the master curve when X > 0.5. It was identified that the MRPM could appropriately describe the reactivity of various chars under different conditions up to high carbon conversions. The details are described below. Results Reactivity at Various Temperatures. To clarify the temperature dependence of gasification reactivity, experiments were made on Senakin coal at various temperatures (pyrolysis time of 30 s). In this article, RPM refers to the random pore model and MRPM refers to the modified random pore model proposed in this work. Moreover, a value of Φ ) 1 was adopted for the predictions by the MRPM in all the figures except for Figure 4, which shows the dependence of the predicted master curve on Φ. In Figure 1, the experimental results (17) Liu, G. S.; Tate, A. G.; Rezaei, H. R.; Beath, A. C.; Wall, T. F. Dev. Chem. Eng. Miner. Process. 1999, 7, 525-536.

of reaction rates versus carbon conversion are compared with the predictions by the MRPM. The shape of the predicted curves in Figure 1 was in good agreement with that of the experimental results. Figure 2 shows the dimensionless reaction rates of gasification versus carbon conversion at various temperatures for Senakin char (pyrolysis time of 30 s). The dimensionless reaction rates r/r0 at various temperatures could be unified into a single curve with the MRPM, although the reaction rates at different temperatures were significantly different. Moreover, the predictions by the MRPM agreed better with the experimental results than those by the RPM. The correlation between experimental data and predictions by models were examined through a correlation index β, which is defined as

{ [ n

β)1-

abs ∑ i)1

]}

(r/r0)exp - (r/r0)pred (r/r0)pred n

(8)

where abs denotes the absolute value and n is the total number of experimental data sets. The correlation indexes, which are calculated based on experimental data shown in Figure 2, were listed in Table 3. Obviously, the prediction by the MRPM with f ) 2.5 has the best correlation with the experimental data.



Figure 7. Carbon-conversion-dimensionless-time master curve at 1473 K for Senakin char (0 s pyrolysis time; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

Figure 3 shows the dimensionless reaction rate (r/r0)conversion curves, as described by various models, including the MRPM. Comparison between Figures 2 and 3 suggests that among these models, the MRPM proposed by us could describe the dimensionless reaction-rate-conversion behavior the best, whereas a shrinking core model was not appropriate to describe char gasification in CO2. The dependence of the predicted master curve on Φ was shown in Figure 4. It was revealed that Φ mainly influenced the predicted master curve at X > 0.5. However, when Φ is greater than 1, the influence of Φ on the master curve was limited. Figure 5 shows the carbon conversions at various dimensionless times for Senakin char. This curve was also called the master curve in the literature.7,8 Obviously, both the RPM and the MRPM could satisfactorily predict the carbon conversions up to 70%. However, above ∼70% carbon conversion, the RPM overestimated the carbon conversion, with the MRPM giving much better predictions. Despite the fact that the experimental data in Figure 5 covered a wide temperature range, satisfactory unification of the reactivity was achieved by the MRPM. Reactivity of Chars Derived from Various Pyrolysis Times. We are also interested to know if the reactivity of chars derived from different pyrolysis times can be unified by a master curve. Therefore, the reactivity of Senakin chars derived from pyrolysis times of 0 s and 10 min was also examined at 1473 K (Figures 6 through 10). The main data were also listed in Table 4. It can be seen that the predictions by the MRPM agreed much better with the experimental results than those

by the RPM. The overestimation by the RPM in the high conversion range, as shown in Figures 7 and 9, was improved by our model (MRPM). This finding is highly significant, particularly for entrained flow gasifiers, where the carbon conversion can be very high. Figure 10 summarizes the unification of the reactivity of Senakin chars derived from pyrolysis times of 0 s, 30 s, and 10 min. The experimental data were compared with the predictions by different models or parameters. It can be seen that the experimental data were successfully unified with a master curve that is described by the MRPM with f ) 2.5, despite the complicated influence of pyrolysis time on the reaction rates. Comparison between the Predictions and the Data from Literature. Figure 11 shows the experimental data from literature4-6,18 versus the predictions by the MRPM and the RPM. Obviously, the predictions by the MRPM agreed better with the experimental data than the predictions by the RPM, despite the fact that the experimental data from literature were obtained at various conditions and with different particle sizes. Discussion On the basis of this work and those in the literature,4,7,8 it was clarified that the char reactivity could be unified with a master curve, despite the fact that factors such as coal type, temperature, and pyrolysis condition had complicated influences on the reaction rates. It seems that the reactivity of chars derived from (18) Schmal, M.; Monteiro, J. L. F.; Castellan, J. L. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 256-266.

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Figure 8. Reaction rate of gasification versus carbon conversion at 1473 K for Senakin char (pyrolysis time, 10 min; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

various coals under various pyrolysis conditions, at different gasification temperatures, differed mainly in initial reaction rates, whereas the conversion-dimensionless-time behavior followed approximately the same pattern (master curve) from the beginning to the end of the gasification process. Combining the results in the literature and this work, it was identified that this pattern, which is described by the master curve, was independent of coal type, temperature, and pyrolysis conditions under which a char was derived. The RPM is correct as a mathematical model. It is a simple model, using only one parameter. However, it overestimated the experimental data at higher conversions, possibly because of its oversimplicity and its assumption that a particle is composed of only carbon, i.e., ash-free. After being modified with a structural factor f* decreasing with carbon conversion, the predictions by the MRPM agreed much better with experimental results than the predictions by the RPM. However, the reason to account for this improvement was not clear. The possible interpretations of this improvement included the following: (i) as the reaction continued, the reaction regime shifted from chemical control to both chemical and diffusion control, because of the formation of ash layer (the enrichment of ash in a particle increased the diffusion resistance, particularly at high carbon conversions); (ii) the influence of crystal change or mineral reaction; (iii) the ash enrichment at

Figure 9. Carbon-conversion-dimensionless-time master curve at 1473 K for Senakin char (pyrolysis time, 10 min; Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.



Figure 10. Dimensionless reaction rate of gasification versus carbon conversion at 1473 K for Senakin chars derived from various pyrolysis times (Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values. Table 3. Correlation Index model

RPM, f ) 2.5

f ) 2.5

MRPM f ) 1.0

f ) 5.0

correlation index

0.65

0.86

0.79

0.77

Table 4. Experimental Data and Model Predictionsa Pyrolysis Time ) 0 s t/t0.5 0.11204 0.19608 0.28011 0.36415 0.5042 0.64426 0.78431 1.0644 1.3445 1.6246 2.0448 2.465 2.8852 3.3053 4.1457 4.986 6.6667 8.3473 10.028

Xexp

Pyrolysis Time ) 10 min t/t0.5

0.056 0.33333 0.101 0.44017 0.146 0.65385 0.194 0.86752 0.273 1.0812 0.348 1.4017 0.414 1.7222 0.526 2.0427 0.617 2.3632 0.688 3.0043 0.767 3.6453 0.823 4.9274 0.864 6.2094 0.895 7.4915 0.937 8.7735 0.962 10.056 0.987 11.338 0.995 0.998

Xpred t/t0.5 (for Xexp prediction) by RPM by MRPM

0.165 0.226 0.341 0.445 0.534 0.64 0.722 0.786 0.831 0.893 0.928 0.963 0.978 0.985 0.99 0.994 0.996

0.01 0.21 0.405 0.605 0.805 1.005 1.205 1.405 1.605 1.805 2.005 2.205 2.405 2.605 2.805 3.005 3.205 3.405 3.605 3.805 3.955 5.5 7

0.005229 0.11064 0.21306 0.31516 0.4121 0.50216 0.58413 0.65732 0.72146 0.77666 0.82335 0.86218 0.89392 0.91947 0.93969 0.95544 0.96753 0.97666 0.98345 0.98842 0.99122

0.005239 0.11373 0.22045 0.32432 0.4187 0.50194 0.5739 0.63531 0.68735 0.73131 0.76844 0.79988 0.82659 0.84937 0.86889 0.88569 0.90021 0.9128 0.92375 0.93329 0.93965 0.97856 0.99228

a X refers to carbon conversion, RPM refers to the random pore model, and MRPM refers to the modified random pore model. Values of f ) 2.5 and Φ ) 1 were used for all the model predictions.

high conversions caused changes in pore structure in a way different from the assumption of the RPM that the value of f, i.e., 4πLE0/S2E0, was constant. Chi and Perlmutter noted that, when pores overlapped, they evidently maintained a fairly constant value of f as they

lost both pore surface area and pore length.19 However, it seems that this argument only holds true at carbon conversions up to 70%, but not over 70%. Nevertheless, further detailed study is required to find direct evidences of structure change, which we will study next. Moreover, if the aforementioned interpretation holds true, an appropriate correlation between the value of Φ and coal property (e.g., ash composition) may achieve even better unification. The MRPM modified the RPM by simply introducing a function exp(-Φτ), and could give much better predictions than the RPM. It is convenient, as a submodel, for the comprehensive simulation of gasifiers. Conclusions The gasification reactivity of an Indonesia char in CO2 was investigated through experiments at chars pyrolyzed at elevated temperatures and high heating rates in a unique fluidized bed. With a modified random pore model (MRPM) that was proposed in this work, a unification approach of char reactivity obtained from our experiments was performed. The following conclusions were reached: (1) A shrinking core model was not appropriate to describe char gasification in CO2. (2) Our model can satisfactorily predict both our own experimental data and the data in the literature. It was clarified that the char reactivity could be unified with a master curve not only for chars pyrolyzed at low temperatures and low heating rates, but also for chars pyrolyzed at high temperatures and high heating rates. This master curve was independent of coal type, temperature, and pyrolysis conditions under which a char (19) Chi, W. K.; Perlmutter, D. D. AIChE J. 1989, 35 (11), 17911802.

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Figure 11. Predictions by the MRPM and the RPM versus the data from literature (Φ ) 1). Symbols represent experimental data, whereas curves represent predicted values.

was derived, despite the fact that these factors had complicated influences on the reaction rates. (3) In addition to the conversion-dimensionless-time master curve, the dimensionless reaction-rate-conversion behavior could also be satisfactorily reduced to a single curve with the MRPM proposed in this work. The model proposed in this work gave better predictions up to high carbon conversions than the random pore model. It was recommended as a convenient submodel for comprehensive simulation of char gasification, particularly for entrained flow gasifiers, where the carbon conversion can be very high. Acknowledgment. The authors would like to thank NEDO/CCUJ for financial support of this work, under the BRAIN-C program. Nomenclature f ) structural factor used in the random pore model f* ) structural factor used in the modified random pore model

LE0 ) total length of the pore per unit volume m ) grain shape factor for grain model (Figure 4) n ) total number of experimental data sets r0 ) initial reaction rate of gasification (s-1) r ) instantaneous reaction rate of gasification (s-1) SE0 ) hypothetical total initial surface area per unit volume t ) reaction time (s) t0.5 ) half-life, defined as the time required for 50% carbon conversion (s) X ) carbon conversion Greek Letters β ) correlation index τ ) dimensionless time (reaction time/half-life) Φ ) coefficient used in the modified random pore model Subscript exp ) experimental pred ) predicted EF020231M

at Elevated Temperatures with a Modified Random Pore - American

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