Kinetic combustion parameters for chars using to IFRF solid fuel data base

In this study, kinetic combustion parameters are determined for chars in the IFRF solid fuel data base (SFDB) in order to model the combustion history of the chars. Experimental burn-out data are used to determine the kinetic parameters that best fit the experimental data according to two standard char combustion models. The SFDB contains burn-out data, for more than 130 different chars of which more than 100 are coal chars, obtained from drop tube experiments. In previous IFRF-reports, kinetic parameters have been determined for several of the chars. The parameters are, however, model specific and only valid for the model that has been used when determining the parameters. In this study, two standard particle models available in Fluent have been used: the kinetic/diffusion surface reaction rate model and the intrinsic char combustion model. The intrinsic model is more detailed and requires a larger amount of fuel specific information. In order to simplify the use of the intrinsic model, some properties of the fuel can be assumed, which will affect the accuracy of the modeling results. The main purpose of the study is to compare which of the models that is most suitable for modeling the combustion history of char particles, while choosing some fuel properties to be constant for different coals in the intrinsic model.

K in etic co m b u stio n p a ra m eters for ch ars u sin g th e I F R F solid fuel d a ta b a se Oskar Karlstroma, Anders Brinka, Jaroslaw Hercogb, Mikko Hupaa, Leonardo Tognottic aAbo Akademi University, Faculty of inorganic chemistry bInstitute of Power Engineering, cIFRF Abstract In this study, kinetic combustion parameters are determined for chars in the IFRF solid fuel data base (SFDB) in order to model the combustion history of the chars. Experimental burn-out data are used to determine the kinetic parameters that best fit the experimental data according to two standard char combustion models. The SFDB contains burn-out data, for more than 130 different chars of which more than 100 are coal chars, obtained from drop tube experiments. In previous IFRF-reports, kinetic parameters have been determined for several of the chars. The parameters are, however, model specific and only valid for the model that has been used when determining the parameters. In this study, two standard particle models available in Fluent have been used: the kinetic/diffusion surface reaction rate model and the intrinsic char combustion model. The intrinsic model is more detailed and requires a larger amount of fuel specific information. In order to simplify the use of the intrinsic model, some properties of the fuel can be assumed, which will affect the accuracy of the modeling results. The main purpose of the study is to compare which of the models that is most suitable for modeling the combustion history of char particles, while choosing some fuel properties to be constant for different coals in the intrinsic model. Keywords: kinetic parameters, char combustion, modeling 1 INTRODUCTION Today, several different types of coals are normally used in a coal-fired utility boiler. Therefore, predicting the behavior of coal fired utility boilers for various types of coals is essential. Computational fluid dynamics (CFD) tools are important for this purpose. 1,2,3One of the main limitations in CFD modeling of coal combustion is the use of simplified sub-models for char combustion. 1,4 Char can be defined as the porous solid residue remaining from the thermal decomposition (pyrolysis, devolatilization) occurring during the heating of the particle. In coal combustion, the char combustion takes much longer than the devolatilization. Thus, the residence time needed to achieve complete burn-out of the particles is mostly affected by the char combustion. Consequently, the char combustion determines whether the residence times in the furnace are sufficient. Char combustion is controlled by a complex coupling between oxygen transport to the reacting char surface and chemistry between the reactant species. The oxygen transport can be divided into oxygen diffusion to the external surface of the particle and intra-particle pore diffusion. Factors affecting the heterogeneous process are among others: particle size, reactant - and product diffusion, chemical reactions, pore structure, diffusion in pores, pore size distribution, changes in pore sizes (and the internal surface area) during the combustion, mineral content and fracturing of the char5. Several types of models have been presented in literature.6,7,8,9 In spite of the complexity, strongly simplified char combustion models have successfully been used, questioning the necessity of detailed, complex models. Ballester and Santiago9 (2005) pointed out that the effects of property heterogeneities and deactivation phenomena play minor role if the particle size distribution is taken into account when modeling pulverized coal combustion, even in the case of narrow size distribution. Taking the particle size distribution, obtained from ground sieving, into account and using a simplified char combustion model, they managed to accurately model the complete combustion history of char particles. In order to use conventional combustion models for a single char particle, kinetic parameters pre-exponential factor and activation energy - need to be determined. Kinetic char combustion parameters are fuel specific and must be determined from experimental combustion data9,10. A suitable experimental facility for performing char particle combustion experiments in order to derive kinetic parameters is a drop tube /entrained flow reactor. 11 Traditionally, kinetic char combustion parameters have been determined from Arrhenius plots.5,10 In reality, the "lines" in Arrhenius plots will never completely go through all experimental points obtained from drop tube experiments.9 As alternatives to Arrhenius plots, kinetic char combustion parameters have been determined from fitting modeled particle combustion history to experimentally observed particle combustion history 9 and by using neural networks.1,12 In this study, experimental burn-out data are used to determine kinetic parameters that best fit experimental data according to two standard char combustion models: the kinetic/diffusion surface reaction rate model6,13 and the intrinsic char combustion model10. The intrinsic model is more detailed and requires more fuel specific information than the kinetic/diffusion model. In order to simplify the use of the intrinsic model, some of the fuel properties can be assumed, which will affect the accuracy of the modeling results. The main purpose of the study is to compare which of the models that is most suitable for modeling the combustion history of char particles, while choosing several fuel properties to be constant in the intrinsic model. Both models have frequently been used and evaluated in literature, but often the models have only been tested for a few fuels. The IFRF solid fuel data base contains burn-out data, of more than 130 different chars from which 1 0 0 are coal chars, obtained from drop tube tests in realistic combustion conditions. All fuels are similar-sized, providing excellent comparison possibilities for the modeling. 2 EXPERIMENTS - IFRF SOLIF FUEL DATA BASE The IFRF solid fuel database contains proximate- and ultimate analyzes of more than 130 fuels and of 130 chars derived from the fuels. Additionally, ash fraction analyses, true density-, apparent density- and internal specific area measurements have been performed on several of the fuels. The degrees of burnout of the fuels and the chars have been measured in a drop tube reactor at different residence times. The data base contains burnout- versus time data of the fuels and of the chars derived from the fuels. 2.1 Experimental setup and tests Devolatilization and combustion experiments have been performed in two IFRF drop tube reactors. The first one was used in the period 1985-1993 and the second has been used since 1994. The principles of the two reactors are similar. The major difference is the height: the height of the older reactor was 2 meter and of the new reactor is 4 meter. Most of the experiments have been performed in the newer drop tube reactor. A detailed description of the setups can be found in a previous work14. Here, a brief description of the newer reactor will be given. The drop tube reactor consists of a 4 m long tube, where the combustion takes place. The tube consists of eight modules which can be controlled and heated independently. A K-tron sample feeder provides a continuous mass flow of pulverised fuel. At the outlet of the K-tron, the pulverised fuel is mixed with preheated gases in order to transport the particles to the furnace. A fuel feeder injects the fuel-gas mixture into the tube-reactor. The furnace is perforated with sampling points at different levels. From the holes in the tube, partially burned particles can be rapidly quenched and burn-out fractions at different residence times can be obtained using an ash tracer. In the experiments, fuels were first devolatalized in an oxygen free gas at heating rates typical for real combustion systems. The remaining chars were collected for combustion tests. Most of the chars were combusted at three different temperatures in the range 1000-1500 °C and at three different oxygen fractions below 20 molar %. The fuel rate for the char combustion tests was set to achieve a maximum relative oxygen drop of 10% over the reactor length. For each experimental condition, seven fuel samples were taken. Table 1 Properties of two coals. (%daf) refers to mass fraction of dry- and ash free fuel. Property Coal/char Columbian Coal South African Coal CC - char Moisture (%) 3.20 2.50 0.50 Ash (%) 11.30 11.10 17.30 Fixed C (%) 51.90 56.40 76.60 Volatiles (%) 36.80 32.50 6 .10 Volatiles -droptube (%daf) 39.10 47.80 C (%daf) 77.75 77.39 89.90 H (%daf) 5.46 4.76 1.80 O (%daf) 14.30 14.93 5.80 N (%daf) 1 .6 8 2.07 1.90 S (%daf) 0.82 0.85 0.60 Mass mean diameter (|jm) 48.85 36.80 44.18 1498 True density (kg/m3) 1407 840 1746 Apparent density (kg/m3) 1691 477 Mean particle mass (ng) 0.086 0.039 0.038 SAC -char 0.30 19.30 77.70 3.00 89.72 1.42 6 .02 2.04 0.80 41.70 844 544 0.032 2.2 Experimental results Tab. 1 and Fig. 1 exemplify char data available in the SFDB for two coal chars. Fig. 1 shows burn-out curves of the two coal chars. Fig. 2 compares burn-out curves of the two chars. Tab. 1 shows fuel properties of the chars and of the initial fuels (i.e. before char preparation). Tab. 1 shows that the coals and their coal chars are very similar, but Fig. 2 shows that the char conversion rates of the chars differ significantly. In order to model the burn-out curves of the two chars, it is, therefore, evident that the same models or model input parameters can not be used when modeling the two chars. Fig. 1. Burn-out (U) as function of time of Columbian (left) and South African (right) coal t (m s) Fig. 2. Comparison of Burn-outs (U) as function of time of Columbian and South African coal 3 COMBUSTION MODELS When choosing a char combustion model, the regime under which the combustion takes place must be considered. Normally, three char combustion regimes (or zones) are defined. Regime I means that the chemistry is slow compared to the reactant diffusion (internal- and external-) and, as a result, the gas species penetrates fully inside the char particle during the combustion ( Th << 1). Combustion under regime III conditions occurs when the chemistry is fast compared to the reactant diffusion and, consequently, the combustion takes place close to the surface of the char particle ( Th >> 1). Combustion under regime II takes place when the reactant species penetrate partially inside the particle. Analogously, regime I can be referred as chemical kinetic control, regime III can be referred as mass transfer control and regime II then is a mixing of chemical kinetic- and mass transfer control. Generally, if coal char particles are less than 100 ^m and the combustion temperature are below 1800 °C, chemistry controls the conversion partially or fully15. Thus, the combustion in the IFRF drop tube tests can be considered to be in regime I - or in regime II conditions, since the particle diameters of the chars are in the range of 50 ^m and the reactor temperatures are between 1000-1500 °C. In this study, two standard char particle combustion models for pulverized fuels, suitable for regime II conditions, available in Fluent have been used: the kinetic/diffusion limited surface rate model6 and the intrinsic char combustion model10. The models do not take intra particle temperature gradients into account which is a realistic assumption for the chars in the SFDB (d ~ 50 ^m, Bi << 0.1). The intrinsic model is more detailed and more realistic than the kinetic/diffusion limited model. Thus, the intrinsic model requires more fuel specific input information than the kinetic/diffusion model. In this study, some of the fuel properties are assumed in the intrinsic model. The purpose is to determine which of the models - the kinetics/diffusion model or the intrinsic model with some of the fuel properties chosen constant - that is most suitable for modeling the combustion history of char particles. The two standard models are described briefly here with the same equations as are presented in the Fluent 6.3 manual. More detailed descriptions of the models can be found in Fluent manuals or in studies by the model developers6 10. 3.1 Kinetic/diffusion limited surface rate model In Fluent 6.3, the char particle temperature is obtained from an approximate analytical solution to the following heat balance of the particle: '" ' C ' d i ' = hAp ( - T )- - Tp ) °) where m p is the mass of the particle, Ap is the outer surface area of the particle, Tp is the temperature of the particle, Tx is the temperature of the gas surrounding the particle, f h is the fraction of heat that the particle absorbs from the heat released by the surface reaction H reac, s p is the emissivity of the particle surface and 0R is the radiation temperature. The char combustion rate in the kinetic/diffusion limited surface reaction rate model is governed by the following equation: D 0^ dm (2) ~ t where p ox is the partial pressure of oxidant species in the gas surrounding the particle. The term D 0^ /( ( 0 + is the overall reaction rate coefficient. The diffusion rate coefficient D 0 and the kinetic rate ^ are calculated using the expressions D 0 = C, [ M l (3) dp and a -(Ekd/ RTp ) = A KDe yKD p) (4) where C is a diffusion coefficient, d p is the diameter of the particle, A KD is a pre-exponential factor, E KD is an activation energy and R is the ideal gas constant. In this model, the kinetic rate, * , takes the effect of chemical reactions on the internal surface of the char particle as well as the effect of pore diffusion into account. 3.2 Intrinsic char combustion model The intrinsic model also uses Eq. 1 for solving the particle temperature and Eq. 2 for calculating the char combustion rate. In this model, the chemical rate depends both on intrinsic chemical- and pore diffusion rates: di * i = n P p As k, (5) 6 n is the ratio of the real combustion rate to the rate achievable if no pore diffusion existed16: 3 n = - 2- ( c o t h ^ - 1 ) <P (6 ) where <j> is the Thiele modulus: 1/2 S bPpA gkiPox '= d t (7) D ePox p p is the apparent density of the char particle, Ag is the mean value of the internal surface area during the char conversion and p ox is the density of the oxidant in the bulk gas. Sb is a mass stoichiometric coefficient of the char combustion reaction: Char(s) + Sb ox(g) ^ Products(g) (8 ) The intrinsic reactivity ki and the effective diffusion coefficient D e are calculated as follows: kt = Ae -(e, /RTp) (9) and 1 D 1 ' H---D (1 0 ) where Ai is a pre-exponential factor and E i is the intrinsic activation energy and T is the tortuosity of the pores. 0 is the porosity of the char particle as calculated from the true density and apparent density of the char particle: 0 = 1- P P pt (11) D is the bulk molecular diffusion coefficient and the Knudsen diffusion coefficient D Kn : T D Kn = 97rp P M w,ox (1 2 ) rp is the mean pore radius and M wox is the molar mass of the oxidant in the gas. It is important to note that using a constant value for pore radius is a strong simplification because of the pore size distribution characterizing coal chars. In this study, the mean pore radius is calculated according to17 20T rv = - - AgPp (13) The outer surface area, A p , in Eq. 1, is constant in the kinetic/diffusion model. In the intrinsic model Ap and, hence, d p is related to the fractional degree of burnout U : -i- = d p,0 (1 - U )a (14) (15) U = m p0af For spherical particles a = 0 corresponds to a constant particle size with decreasing density during burnout, which is typical for regime I conditions. a = 1/3 corresponds to a decreasing particle size with a constant density occurring during regime III conditions. Combustion under regime II conditions means that both the particle diameter and density decreases during the conversion, and, consequently a should be between 0 and 1/3. 4 DETERMINATION OF KINETIC PARAMETERS AND RESULTS Eq. 1-14 are used to model the combustion of the pulverized char particles along the IFRF drop tube reactor, based on the experimental conditions and the fuel properties. The fuel properties that are taken from the SFDB are: ash-, volatiles- and char content, true and apparent density and mean mass diameter. Remaining properties, except the kinetic parameters, are chosen to be the same for all coals (see Table 2). The internal specific surface areas of coals can vary between ~11000 m2/g. 10 For many coal chars the internal specific surface area is typically in the range of 100 m2/g . 18 During the combustion of a char particle, the internal specific surface area changes and therefore it might be inappropriate to use a constant value for the area as required in the intrinsic model. In this work, the internal specific surface area has been chosen to 300 m2/g for all coal chars. Nevertheless, if an inappropriate value of the internal specific surface area is used in the modeling, the pre-exponential factor in the Arrhenius reactivity expressions can be modified in order to improve the accuracy of the model. For calculating the diameter evolution, and thus the change in outer surface area and density, a = 0.25 has been used. Ballester and Santiago9 (2005) pointed out that using a higher value of a , than the real value of a , can improve the modeling of coal combustion in drop tube tests if the particles are treated as mono-sized in the modeling. In this study, the particle size distribution is neglected and, therefore, it might be appropriate to use a = 0.25 even if the real value of a is far below 0.25. Table 2 . Parameter values used for all coals in the modeling. = 2300 J/kgK Ag = 300000 m2/kg cp C h = 5*10 -12 s/K 0 75 = Nu / km d p W /m2K D H fh = M wox = 0.032 kg/mol Sb = 0.75 a = 0.25 sp = 0.9 T = 1.4142 1 = 0.24* 104 * (T» /298)175 m2/s reac = 9.583*106 J/kg 4.1 Calculation procedure In order to determine the kinetic parameters for combustion modeling of the char particles in the SFDB, a parameter determination method used in this study is similar to the one used by Ballester and Jimenez9. The calculation of kinetic parameters follows the approach described below: • Values of kinetic parameters - pre-exponential factor and activation energy - are discretized for a wide range of values e.g. E KD = 30000, 31000, 32000...200000 J/mol. • For each kinetic parameters pair, the burn-out curve U M (t ) is modeled: d U M = _ l dm dt m c0 d t (16) • An object function is defined using the equation . = Jf min j k max '- Jmax k ‘'- j ‘''Jk (17) where f ,t = U m - Ue) (18) where U E is the experimentally measured burn-out fraction, j equals to the j th sampled point in the experiments for the k th experimental test condition. • The object function is calculated for each kinetic parameters pair. Based on the optimal values the kinetic parameters are discretized into a new smaller range of values. Then, the procedure is repeated as described above as long as the value of the object function is decreasing. The reason to using this brute-force reminding optimization approach is that the optimization problem is nonconvex and, therefore, it is inappropriate to use conventional optimization methods like the Simplex method19 for non-linear optimization problems. Fig. 3. Level plot diagram of pre-exponential factor and activation energy of a Columbian coal (see table 1). The levels represent the object function in Eq. 17 that compares the difference of modeled char combustion history data to experimental observations. The graph to the right is a more detailed description of the graph to the left around the area of the optimum point marked with a black sphere. The optimum values are •sf min ■ = 0.069,■ 'A.i = 5.523*107kg/m2sPa and E.i = 71107 J/mol x 10 A i (kg/m2sP a ) x 10 Fig. 4. Level plot diagram of pre-exponential factor and activation energy of Columbian coal. The values of the levels are included and represents the object function in Eq. 17 that compares the difference of modeled char combustion history data to experimental observations 4.2 Results Fig. 3 shows the level plot diagram of the optimized parameters A i and E i of the Columbian coal (see Tab. 1). The minimum object function value f min = 0.069 is obtained for Ai = 5.52*106 kg/m2sPa and E i = 71100 J/mol. Fig. 4 includes the values of the levels of the optimized kinetic parameters of the Columbian coal. The levels shows that there is a large range of kinetic parameter values giving approximately equal object function values compared to the optimum kinetic parameters. The levels demonstrate the sensitivity of the object function against the change in kinetic parameters: if e.g. the optimum activation energy is fixed, the pre­ exponential factor can be varied to some extent without significantly decreasing the accuracy of the modeling results. C o l u m b i a n C oa l kinetics/diffusion m o d e l 1 0 1 0.8 7 ....... 0.6 0.4 0.2 C o l u m b i a n Coa lintrinsic m o d e l A expl ■ exp2 O exp3 A exp4 ------ modi -------- mod2 -------- mod3 .......... mod4 0.0 0 500 1000 1500 t (m s) t (m s) S o u t h A fric an Coal intrinsic m o d e l S o u t h A fric an Coa l c h a r kinetics/diffusion m o d e l 1.0 expl exp2 O exp3 exp4 ▲ -------- modi -------- mod2 -------- mod3 .......... mod4 0.8 0.6 0.4 0.2 0.0 0 500 1000 t (m s) 1500 t (m s) Fig. 5. Modeled and experimental burn-out fraction U v.s. time for Columbian and South African Coal. "1": 1223 °K 4% O 2, "2": 1223 °K 12% O 2,"3": 1423 °K 4% O 2,"4": 1623 °K 4% O2. f minlCC = 0.069, A lCC = 5.523*107 kg/m2sPa and E iCC = 71107 J/mol. fminiSAC = 0067, AjSAC = 9.514*107 kg/m2sPa and E iSAC = 70215 J/mol. (The given numbers are for the intrinsic model). Fig. 5 compares modeling results, of the two different models using optimized kinetic parameters, with experimental results in Fig 1. The figures show that the combustion rates of the diffusion/kinetics model are constant. In the intrinsic model, the diameter is allowed to decrease and, consequently, the combustion behavior of the fuels is better modeled. The intrinsic model reproduces the experimental burnout curves accurately, except in the latest stages of the burnout in some of the curves. A possible reason is that the particles are considered mono-sized in the model and, thus, the particle size distribution is not taken into account, which could explain the deviations9. The determined intrinsic activation energies of the two coal chars are similar, but the pre-exponential factors, and thus the reactivity, differ by a factor of 2 , which is interesting since the fuels are very similar (see experimental results section). The deviation could be explained by a difference in the internal surface areas of the two coal chars, since the same internal specific surface areas have been used when determining the kinetic parameters for both of the fuels. 1.E+01 £ tz I Q l (A C <U"C §f a i ?gu ^ 1.E-01 .♦ 1.E-03 - * ♦ 1.E-05 1.E-07 1.E-09 0 50000 100000 150000 200000 Activ a tio n e n e r g y E (J/m ol) Fig. 6 . Optimized pre-exponential factors versus calculated intrinsic activation energies for 60 coal chars. Each point represent a different coal char. fmin intrinsic model Fig. 7. Minimized object function values in Eq. 17 that compares the difference of modeled char combustion history data to experimental observations of 60 different coal chars. Each point represents a different coal char. X-axis: minimized object function values for the intrinsic model. Y-axis: minimized object function values for the Kinetic/diffusion limited model. The grey line distinguishes where the object function values are equal for the two models. Fig. 6 shows optimized pre-exponential factors versus optimized activation energies for 60 coal chars. The greater part of the activation energies are in the range 50000-130000 J/mol, which can be considered as typical values510. The lowest and highest activation energy of the chars in Fig. 6 are 30504 and 187253 J/mol. These values might be unrealistically low and high respectively, suggesting that there might be some uncertainties either in experimental data or in the fuel properties that was used in the intrinsic model. Fig. 7 compares the accuracy of the two models for 60 coal chars. For all the chars, the object function values for the intrinsic model are close to or below 0.10. The object function values for the kinetic/diffusion model are for many of the coals close to 0.15. Thus, the intrinsic model was more flexible than kinetic diffusion model, even though some fuel properties were chosen constant for all coals using the intrinsic model. E-02 n • E-03 1000 C Kinetic/diffusion model A 1300C "i? Q_ E-04 *t uj * < 8 E-05 * A & E-06 AA* A 9 E-07 E-08 1 0 10 20 30 40 50 60 Voldaf % Fig. 8 . Kinetic rate versus dry and ash-free (daf) volatile content in initial fuel sample (before char preparation) at 1000 °C and 1300 °C for 60 coal chars. The kinetic rate is calculated with the kinetic parameters obtained for the kinetic/diffusion limited model. Each point of the same symbol represents a different coal char. Fig. 8 shows the kinetic rate for 60 different coal chars at 1000 and at 1300 °C. The kinetic rate is here calculated with the optimized parameters for the kinetic/diffusion model. The figure shows that the scatter of the kinetic rate is large, but that there generally is an increasing trend between the kinetic rate and the coal rank 5 CONCLUSIONS AND FURTHER WORK Determining of kinetic combustion parameters for coal chars in order to model the combustion history of the chars has been demonstrated. Kinetic parameters were determined from fitting modeled particle combustion history data to experimental particle combustion history data, obtained from drop tube experiments. Optimized kinetic parameters for 60 coal chars were used in two standard char particle models: the kinetic/diffusion limited surface rate model and the intrinsic char combustion model. The intrinsic model is more detailed and requires more fuel specific input data. The study shows that the intrinsic model was more flexible than the kinetic/diffusion model for 60 coal chars, even though several fuel properties were chosen to be constant for all coals in the intrinsic model. In further studies, the burning mode a and the internal specific surface area A will be chosen based on values typical for different kinds of coals and on combustion conditions. Also, the two models will be compared while allowing the diameter to decrease in the kinetic/diffusion model. specific internal surface area Ag (m2/kg) pre-exponential factor in intr. model A (s/m) pre-exponential factor in kin/diff A kd model (s/m) surface area of particle (m2) Ap heat capacity of the particle (J/kgK) s coefficient for calc. diffusion rate c, (s/K0 75) bulk molecular diffusion coefficient D (m2/s) diffusion rate coefficient (s/m) D0 D ee effective diffusion coefficient in g /s) c res po D Kn Knudsen diffusion coefficient (m /s) dp diameter of particle (m) d p,0 initial particle diameter (m) Ei activation energy in intr. model (J/mol) EKD activation energy in kin/diff model (J/mol) . object function to be minimized (-) Jf min convective heat transfer coefficient h (W/m2K) H reac heat released by surface reaction (J/kg) intrinsic reactivity (s/m) ki thermal conductivity of the k« continuous phase (W/mK) particle absorbed heat fraction (-) A mp mass of particle (kg) m paf mass o f ash free particle (kg) m p0af initial mass o f ash free particle (kg) m c0 initial mass of char particle (kg) M w.ox molar mass of oxidant species (kg/mol) Nu Nusselt number (-) partial pressure of oxidant species pox (Pa) mean pore radius (m) rp % ideal gas constant (J/molK) kinetic rate in kin/diff model (s/m) intrinsic chemical rate (s/m) Sb stoichiometric coefficient (-) Tp temperature of the particle (K) Ta, U temperature of continuous phase (K) fractional degree of burn-out (-) experimentally calculated U (-) R UE UM modeled U (-) burning mode for the diameter evolution (-) emissivity of particle surface (-) F a efficiency factor (-) n G porosity of the char (-) Or radiation temperature (K) P'ox density of the oxidant species (kg/m2) Pp density of the particle (kg/m3) P pa apparent density of the particle (kg/m3) 1Pvt Fl C true density of the particle (kg/m3) Stefan-Boltzmann constant (5.67 x 10-8 W/m2K4) T tortuosity of the pores (-) Thiele modulus (-) $ 1 Korytnyi E, Saveliev R, Perelman M, Chudnovsky B, Bar-Ziv E. 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