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Energy & Fuels 1996, 10, 1091-1098
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Modeling of Ignition and CO Oxidation in the Boundary Layer of a Single Char Particle Shakti Goel, Chun Hyuk Lee, John P. Longwell, and Adel F. Sarofim* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received November 22, 1995X
A model is developed for CO oxidation in the boundary layer of a single char particle. The model includes char oxidation and a 56 reaction gas phase kinetic scheme which is coupled with the diffusive properties of the 12 species involved in the CO oxidation. The model is compared with the experimental data of Tognotti and co-workers. The temperature reached on ignition, the resulting CO2/CO ratio, and the effect of changing water vapor concentrations are well described. Studies on the effect of water concentration show that significant CO oxidation at low temperatures requires a high surrounding water concentration. The presence of water vapor or hydrogen is found to be necessary to obtain a high degree of CO oxidation at low temperatures; however, there is a minimum temperature below which significant CO oxidation in the boundary layer does not occur irrespective of how high the water concentration is. In addition, CO oxidation is negligible even at a surface temperature as high as 2500 K when water and hydrogen are absent. The model has also been applied to predict the effects of changing parameters. Catalytic acceleration of the rate of carbon oxidation, for example by the addition of calcium, leads to both a high particle temperature overshoot and a significant increase in CO oxidation over the particle surface.
Introduction The temperature history of a burning char particle is needed to predict the carbon consumption rate, the ratio of CO2 to CO, the burning time, and the rate of vaporization of mineral constituents. Earlier experimental studies include the measurement of particle temperature history in an electrodynamic balance1,2 and the CO2/CO ratio as a function of particle temperature.2-4 Particle temperature histories have been modeled frequently by assuming pure heterogeneous reaction5 or by using global kinetics for the oxidation of CO to CO2. During char combustion the two major products are CO and CO2. CO formed by surface reactions is further oxidized to CO2 either in the pores (macro and meso) of the particle or in the gas phase, affecting the energy balance in and around the particle. Modeling of CO oxidation around the particle is complicated due to the high temperatures involved, the complex coupling between the partial differential equations governing mass and energy conservation, and the stiff gas phase chemistry. Most of the modeling studies to date have either assumed quasi-steady state, constant gas thermodynamic and transport properties, or global reaction kinetics. Amundson and co-workers,6 using global kinetics and average gas and solid properties, estimated * Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, June 1, 1996. (1) Bar-Ziv, E.; Jones, D. B.; Spjut, R. E.; Dudek, D.; Sarofim, A.; Longwell, J. P. Combust. Flame 1989, 75, 81. (2) Tognotti, L.; Longwell, J. P.; Sarofim, A. F. Twenty-Third Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1990; p 1207. (3) Otterbain, M.; Bonnetain, L. Carbon 1968, 6, 877. (4) Phillips, R.; Vastoal, F.; Walker, P. Carbon 1970, 8, 205. (5) Jones, D. B. Carbon oxidation in an electrodynamic balance. Ph.D. Thesis, Department of Mechanical Engineering, MIT, Cambridge, MA, 1989. (6) Sunderasen, S.; Amundson, N. R. AIChE J. 1981, 27, 679.
CO oxidation as a function of temperature, particle size, and moisture content. They concluded that CO oxidation is not accelerated in the presence of water for small particles (0.1 was found to be necessary. If one considers the Spherocarb as a conglomerate of small micrograins26 with no oxygen diffusion limitations to the micrograins, then η ) 1 can be obtained. Alternatively, the phenomenon of only macropores participating in the reaction can also lead to a large effectiveness factor. Since the macropore surface area is about 1% of the total surface area and the macropore volume is about 0.4, one can get an η in the range of 0.1-1, sufficient to ignite the particle. This treatment of pore diffusion is clearly very simplistic.
Np Np
b X k+1 k - Yi,j ) Yi,j l)1 q)1 l,q,p i,j (Yi,j 2 ) (R∞- Rp) + + ∆tDim,j Xi,j h2
2(R∞ - Rp)
Sg ) (920.9089 + 2661.0706Xc - 27273.1303X2c +
{
1. at nodes within elements
N p Np
∑ ∑ al,q,pXi,j l)1 q)1 2 hpXi,j
(R∞- Rp)
∑ ∑ al,q,pYi,j l)1 q)1 N p Np
∑ ∑ al,q,pYi,j l)1 q)1
∑ ∑ al,q,pXi,j l)1 q)1 -Yi,j
hp
+
hp
N p Np
T Di,j
∑ ∑ bl,q,pTj l)1 q)1
FjDim,jTj
hp2
[
N p Np
+
T Di,j
∑ ∑ al,q,pTj l)1 q)1
2(R∞ - Rp)
FjDim,jTj
hp2
r
N p Np
Np Np
∑ ∑ al,q,pTj l)1 q)1
)
+ hp
N p Np
Xi,j
-
∑ ∑ l)1 q)1
(
T al,q,pDi,j )
N p Np
}
+ hpTj
]
∑ ∑ al,q,pTj) l)1 q)1
(
h2pTjFjDim,j
+
k+1
ωi,j
(R∞- Rp)2 Dim,jFj
(13)
2. continuity of flux and concentration profiles maintained at the boundaries of the elements N p Np
k+1 ) Si,j
|
∑ ∑ al,q,p-1Yi,j l)1 q)1 hp-1
N p Np
p-1
|
al,q,pYi,j k+1 ∑ ∑ l)1 q)1
k+1
hp
p
(14)
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1094 Energy & Fuels, Vol. 10, No. 5, 1996
Goel et al.
3. boundary conditions
[
surface
k+1 Si,1
)
Np
∑ a1,qY1,j
-Ur,1Yi,1(R∞-Rp)
q)1
+
+ h1
D1,1
Np
T D1,1
ωc(R∞-Rp)
∑ a1,qTj
q)1
+ D1,1F1
h1T1F1
]
k+1
(15)
at ∞ k+1 Si,N ) Yi,Nt - Yi,∞ t
(16)
Following a similar approach, the energy conservation equation can also be discretized. Solution of the differential equations using the Newton-Raphson method requires the evaluation of the Jacobian. Since such calculations can be numerically expensive, first-order derivatives should be calculated with respect to those variables that affect the dependent variable the most. The numerical scheme employed here is given by
F ) Frxn + Ftransp
(17)
where F is the function for which the first-order derivative needs to be estimated. The function is broken down into two terms representing the reaction kinetics (Frxn) and the transport (Ftransp) terms. Reaction rates are a strong function of both the concentration of the species and the gas temperature. Since the calculation of reaction rates is not computationally expensive, the calculation of derivatives is also fast. The calculation of transport properties such as diffusivity coefficients and thermal conductivity requires expensive matrix operations. The derivative of Ftransp, therefore, was calculated only with respect to temperature as perturbations in temperature affect the transport properties the most. This approach gives the matrix approximate of the Jacobian. Having developed the single-particle model and discussed its solution strategy, our next step was to compare its predictions with the experimental data. The next section describes the model calibration results and the practical ramifications of this study. Results and Discussion Comparison with Experimental Data. Tognotti et al.2 obtained the particle temperature history profile as a function of combustion time and the CO2/CO ratio as a function of particle temperature for oxygen concentrations of 100% and 21% both in the absence and in the presence of water. The ambient gas temperature was maintained at 300 K for their experiments. For temperatures below 1300 K and at 100% oxygen, the CO2/ CO ratio could be attributed to a certain known particle temperature. Since the particle had not ignited for these conditions, it was possible to obtain time-resolved data for CO and CO2. At temperatures above 1300 K when the ignition occurred, the temperature rise was so steep that the CO2/CO ratio reported could only be
Figure 1. Comparison of the experimentally measured temperature profile with the model predictions using a fully transient model accounting for all of the gas phase reactions. The Spherocarb was burned in 100% oxygen and in the absence of any external water vapor. The rates of Spherocarb combustion (×104 ) and carbon conversion (×103 ) as a function of time are also presented.
an average over a temperature range of 1500-2500 K (the peak surface temperature was around 2500 K). However, it was possible to calculate the instantaneous CO2/CO ratio as a function of time for the transient model presented in this paper by using eq 18. At 21% oxygen the combustion was not rapid and averaging the CO2/CO ratio obtained experimentally over a temperature range could be avoided.
|
[CO2] [CO]
∫rR ∫rR
∞
[CO2]tr2 dr
∞
[CO]tr2 dr
cf
) t
cf
(18)
The primary of aim this paper is to validate the model by comparing the modeling results with the experimental data and to explain the following characteristics: (1) the steep rise in the temperature profile above 1300 K and the combustion time of 0.06-0.07 s; (2) the decrease in the CO2/CO ratio with increases in temperature in the absence of ambient water; (3) the upward curvature in the CO2/CO ratio profile at 100% oxygen and at very high temperatures in the absence of ambient water vapor; and (4) the ignition at lower temperatures both for 100% and for 21% oxygen in the presence of water vapor. Figure 1 shows the comparison between the modelpredicted particle temperature profile obtained by performing transient analysis for a burning Spherocarb with regressing particle radius and the experimentally observed profile in an electrodynamic balance. The predictions are very close to the experimental observations both in terms of the combustion time and the particle temperature overshoot. The figure also shows that the rate of char gasification as a function of combustion time continuously increases with increases in conversion and then falls off rapidly as the complete conversion approaches. To test the model further, our next step was to explain the observations 2-4 on mass conservation listed above. When hydrogen is absent, oxidation of CO in the gas phase does not occur except at extreme conditions, and the presence of water is normally required for its
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Energy & Fuels, Vol. 10, No. 5, 1996 1095
conversion to CO2.27 The H-containing radical known to control CO oxidation is primarily OH:
CO + OH ) CO2 + H The H radical generated may react with O2 and H2O to generate more OH radicals. CO oxidation may be further augmented by the HO2 radicals as given by the following reaction:
CO + HO2 ) CO2 + OH A source of hydrogen for the OH and HO2 radicals is its emanation from the surface of the burning Spherocarb (the Spherocarb particle under investigation contained 0.74% hydrogen by weight). Therefore, the task of model comparisons for the CO2/CO ratio was undertaken for the following scenarios: (1) absence of any water and hydrogen-containing radicals in the surroundings; (2) instantaneous release of all hydrogen present in the Spherocarb, assuming fast pyrolysis (this phenomenon will lead to a water vapor concentration of a few ppm in the surroundings); (3) supplying water vapor externally at a concentration of up to a few percent; (4) absence of ambient water vapor but release of H radicals from the particle at a rate proportional to carbon consumption; and (5) same as (4) but in the presence of added water vapor. Model simulations and comparisons with the experimental data for cases 1-3 are presented in Figures 2a and 3a and for cases 4 and 5 in Figures 2b and 3b. For 100% oxygen the particle combustion is at quasi-steady state for temperatures below 1300 K. Above 1300 K the particle temperature rises rapidly, suggesting ignition, and the particle combustion then reaches unsteady state conditions. The modeling results presented in Figure 2 are for unsteady state conditions above 1300 K. Parts a and b of Figure 2 show that the model predictions for all five possibilities discussed above are close to the experimental data and any deviations are within the bounds of experimental error. The model predicts the increases in the CO2/CO ratio at very high temperatures (>1600 K) or in the presence of moisture in the surroundings above the values expected for the surface ratios (eq 10). Figure 2a shows that there is no deviation from the empirical correlation at 100% oxygen even at a surface temperature of 2500 K when water is not available in the surroundings. This suggests that the presence of H-containing radicals is necessary to ignite CO in the boundary layer. Since water acts as a source of H-containing radicals, its presence is very important. In its absence, ignition does not occur, and the CO2/CO ratio continuously decreases with increases in temperature. On the other hand, one needs only a few ppm of water to ignite CO at high temperatures and high oxygen concentrations. The modeling results for 800 ppm ambient water vapor were found to match the experimental data more closely. This small amount of water concentration may be achievable if the Spherocarb undergoes rapid pyrolysis, releasing all of the bound hydrogen at once. Alternatively, if water is supplied externally (3.5%), the concentration of OH radicals increases. This concentration is high enough such that CO can now be ignited in the gas phase even (27) Adomeit, G.; Mohiuddin, G.; Peters, N. Sixteenth Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1977; p 731.
Figure 2. Comparison of the model predictions with the experimental data for the CO2/CO ratios for 100% oxygen in the ambient when (a) hydrogen flux from the particle is absent and (b) hydrogen flux from the particle at a rate proportional to carbon consumption is present.
at low temperatures (∼1250 K). This is manifested by the upward curvature in the calculated CO2/CO profile at 1250 K. Figure 2a also shows that by changing the water concentration, one can control the ignition temperature of the particle. There is, however, a lower limit on surface temperature below which ignition does not occur even on further increase of the water concentration. This limit is set due to a reduction in the dissociation of H2O. In the absence of any radicals, the only possible way of H2O dissociation is its reaction with a third body and O2 and CO molecules. The H, OH, and HO2 radicals generated from the dissociation of H2O are not high enough in concentration at low temperatures to further dissociate water molecules. Also, the concentration of OH radicals is not high enough to compensate for the decrease in the rate constant for the CO + OH reaction. A similar argument holds true for the HO2 radicals, which also control CO oxidation. Figure 2b shows the effect of release of hydrogen radicals from the particle surface on CO ignition. The hydrogen radicals emanating from the particle surface at a rate proportional to carbon consumption help in igniting CO. If ambient water vapor is also present, the ignition temperature is further lowered. Moreover, if only ambient water vapor is present (Figure 2a), the ignition temperature is higher as compared to when
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1096 Energy & Fuels, Vol. 10, No. 5, 1996
Figure 3. Comparison of the model predictions with the experimental data for the CO2/CO ratios for 21% oxygen in the ambient when (a) hydrogen flux from the particle is absent and (b) hydrogen flux from the particle at a rate proportional to carbon consumption is present.
both the water vapor and hydrogen flux are present (Figure 2b). Hydrogen radicals lead to the generation of OH and HO2 radicals, which in turn lead to ignition. If water vapor is absent in the ambient, then the concentrations of the OH and HO2 radicals generated from the hydrogen flux are not high enough to lower the ignition temperature. The modeling study discussed above shows that the CO2/CO ratio is highest when both external water vapor is present in the surroundings and hydrogen also emanates from the particle. In this case the CO2/CO ratios are greater as compared to when external water in the ambient is present but the hydrogen flux is absent. The difference, however, is not significantly large. However, the CO2/CO ratios are much lower when external water is absent but hydrogen continuously evolves from the Spherocarb during combustion or when the hydrogen devolatilization is instantaneous, leading to a few ppm of water in the surroundings. In the absence of any hydrogen-containing radicals, CO oxidation in the gas phase is negligible. In summary, the CO2/CO ratio decreases in the following order: case 5 > case 3 > cases 4 and 2 > case 1. Parts a and b of Figure 3 show similar results but for an ambient oxygen environment of 21%. At an oxygen concentration of 21% the particle is not found to ignite in the absence of
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Goel et al.
external water vapor or hydrogen emanation from the particle surface when the bulk gas temperature is 300 K. The model is once again able to predict the experimentally observed levels of CO2/CO ratios. The CO2/ CO ratios are found to be the largest when external water is available in the surroundings and lowest when it is absent. Figure 3a exhibits deviation in the CO2/ CO profile at 1200-1300 K when 3.5% water is fed to the electrodynamic balance chamber. These deviations are a consequence of the significant CO oxidation in the boundary layer. The ignition temperature is the lowest (of all the five cases described above) when both the external water vapor and hydrogen emanation from the particle surface are present (Figure 3b). A higher ignition temperature is required when only 200 ppm of water vapor is available (Figure 3a) or when hydrogen flux is the only source of hydrogen-containing radicals (Figure 3b). Both the modeling and the experimental studies show that CO oxidation in the boundary layer can be significant even at a low ambient oxygen concentration of 21% when hydrogen-containing radicals are present in the surroundings. Figure 4 presents the OH, HO2, and O profiles in the boundary layer at particle temperatures of 800 and 1250 K for the cases (a) when there is only hydrogen flux from the particle surface and (b) when external water is also present in the surroundings. There is a significant difference in the concentration levels of the radicals for the two cases at 1250 K. Figure 4a shows that the OH mass fractions are higher at 1250 K and 3.5% water as compared to when external water is absent (1250 K) or when water is present but the temperature is 800 K. This suggests that the OH concentration is sufficiently high at 1250 K and 3.5% water to ignite CO in the boundary layer. Similar trends are observed for the HO2 profile (Figure 4b), and now both the OH and the HO2 radicals control CO oxidation. At 800 K the OH concentration is more than 1000 times lower than that at 1250 K even in presence of moisture. If one further augments the water concentration in the surroundings at 800 K, CO oxidation still cannot occur. It is interesting to note that there is a reversal in trends in the O radical profiles. The concentration of O radicals is higher when external water is absent in the ambient. This may be due to their consumption by the O + H2O ) 2OH reaction. During the numerical simulations we observed that there could be at least two solutions to the system of PDEs being solved when water was present in the surroundings. One form of the solution which suggests that significant CO oxidation occurs in the boundary layer in the presence of water has been shown earlier in Figures 2 and 3. The other solution shows that the system is immune to the ambient water vapor and even at 3.5% (molar basis) water CO ignition does not occur, irrespective of the oxygen concentration. One cannot possibly discern the validity of the two solutions in the absence of any experimental data. Since the work of Tognotti and co-workers presents evidence that water augments CO oxidation, we considered the former solution (one in which CO oxidation is significant) to be the valid one. The two numerical solutions found here correspond to the extremes of the ignited and nonignited cases. Possibly, more than two numerical solutions exist for these nonlinear coupled differential equations.
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Ignition and CO Oxidation
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Energy & Fuels, Vol. 10, No. 5, 1996 1097
Effect of Mineral Matter. Under naturally occurring conditions coal is a complex agglomerate of mineral matter, carbon, hydrogen, oxygen, and other elements. In other words, its composition is significantly different from that of a Spherocarb. Earlier work in this area has aimed at categorizing the catalytic activity of different metal cations28-30 on carbon oxidation. For example, the CO2/CO ratio for the combustion of lowrank coal chars has been found to be high, and this has been attributed to the catalytic effect of mineral matter21 such as ion exchange calcium.31 These high values may be either due to an increase in the number of sites favoring CO2 production or due to a decrease in the activation energy barrier for the surface intermediates. Another possible explanation for the effect of mineral matter on the CO2/CO ratio is that it can affect the particle temperature overshoot and, therefore, the CO2 formation reactions. This hypothesis will be tested here. Several earlier experimental studies have shown that ion exchange calcium acts as a catalyst in surface oxidation of carbon, preferentially accelerating the channel producing CO2 over that producing CO.32 Du et al.33 have provided the following empirical equation for the higher conversion of carbon to CO in the presence of ion-exchanged calcium: 0.21 2575/Ts fCO ) 1/(1 + 3.57XO2 e )
(19)
This equation gives higher CO2/CO ratios than eq 10 for the uncatalyzed Spherocarb. For the present study the kinetic data for char containing ion-exchanged calcium was taken from the work of Floess et al.32 The kinetic rate of char consumption used was
rc ) 1.2 × 1012e-14897/TsMCoXO2,∞ (g of C/s)
Figure 4. (a) OH radical profiles in the boundary layer for different conditions of particle surface temperature (0, 1250 K; O, 800 K) and surrounding water vapor concentrations. The concentrations are multiplied by a factor of 103 for 800 K. CO oxidation is not significant at 800 K due to low levels of OH concentration. (b) HO2 radical profiles in the boundary layer for different conditions of particle surface temperature (0, 1250 K; O, 800 K) and surrounding water vapor concentrations. The concentrations are multiplied by a factor of 10 for 800 K. Low HO2 concentrations do not support CO oxidation even at 3.5% external water vapor concentrations. (c) O radical profiles in the boundary layer for different conditions of particle surface temperature (0, 1250 K; O, 800 K) and surrounding water vapor concentrations. The concentrations are multiplied by a factor of 104 for 800 K.
The next step was to conduct model simulations to predict the particle temperature overshoot and compare the temperature overshoot and the CO2/CO ratios for the Spherocarb, both in the presence and in the absence of ion-exchanged calcium. Figure 5 shows the particle temperature profiles as a function of combustion time for different oxygen concentrations. It can be seen that the particle temperature is much higher in the presence of calcium and the combustion time lower as compared to the case when calcium is not present in char for the same ambient oxygen concentrations. Ignition occurs even for an oxygen concentration level as low as 21%, and the particle undergoes complete combustion. Ignition did not occur at 21% when calcium was absent (this is true when the surrounding gas temperature is 300 K; at higher ambient temperatures ignition may occur even in absence of calcium), and the particle temperature dropped over time to ambient levels. For these increased temperatures, eq 19 predicts that CO2 will be the dominant product in the presence of mineral matter. The carbon conversion to CO2 in the presence of catalyst was greater than 97% compared to levels of less than 50% for uncatalyzed Spherocarb (see Figures (28) Amariglio, H.; Duval, X. Carbon 1966, 4, 233. (29) McKee, D. W. In Chemistry and Physics of Carbon; Walker, P. L., Thrower, P. A., Eds.; Dekker: New York, 1981; Vol. 16, p 1. (30) Walker, P. L.; Mahajan, O. P.; Komatsu, M. Prepr. Pap.sAm. Chem. Soc., Div. Fuel Chem. 1979, 24, 10. (31) Hurt, R. H.; Longwell, J. P.; Sarofim, A. F. Fuel 1986, 65, 451. (32) Floess, J. K.; Longwell, J. P.; Sarofim, A. F. Energy Fuels 1988, 2, 756. (33) Du, Z.; Sarofim, A.; Longwell, J. Energy Fuels 1991, 5, 214.
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1098 Energy & Fuels, Vol. 10, No. 5, 1996
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matter is particularly important for practical combustion conditions at which the ambient oxygen concentration is very low. Nomenclature
Figure 5. Effect of ion-exchanged calcium present in char on the particle temperature history.
2 and 3). Increases in char consumption rates due to catalysis, therefore, provide an alternative explanation for the significant CO2 formation, due to the impact of catalysts and temperature on the surface oxidation channels rather than any gas phase oxidation of surfacegenerated CO. This example shows the importance of both the surface and gas phase chemistries in determining the CO2/CO ratio. Summary and Conclusions A single char particle model for CO oxidation in the boundary layer characterized by 12 species and 56 reactions has been developed. The model was tested by comparison with the experimental data of Tognotti and co-workers for the CO2/CO ratios as a function of particle temperature for different levels of water vapor. At least two solutions were found to exist for the set of PDEs solved in this paper, and experimental data were used to identify the valid numerical solution. An important parameter is the evolution of hydrogen from char. Two models were considered, one of instantaneous release at the beginning of oxidation and another in which the hydrogen was released at the same rate as carbon to limit the cases of practical interest. Modeling results show that hydrogen from the char or atmospheric water vapor is critical in determining ignition temperatures. However, at very low surface temperatures CO ignition does not occur even at high water concentrations. The modeling studies also show that in the presence of external water CO oxidation can occur in the boundary layer at 21% oxygen concentration in the ambient. Of the several cases investigated, the lowest temperature at which ignition of CO in the boundary layer occurred was in the presence of ambient water vapor and hydrogen emanation from the particle. The ignition temperature increased with decreases in the ambient water vapor levels. The presence of mineral matter in char can also lead to significant amounts of CO2 production at the particle surface, in turn affecting the energy balance around the particle. In addition, the particle temperature overshoot is considerably higher and combustion time lower than when calcium is not present. The presence of mineral
a, b ) coefficients for the derivatives used in the collocation method Cp ) specific heat capacity D ) diffusivity fCO ) conversion of char carbon to CO hp ) length of the differential element p in spatial discretization H ) heat of formation kg ) external mass transfer coefficient ks ) rate constant for the surface reaction of oxygen with char Mc ) weight of the Spherocarb Ne ) number of elements Np ) number of nodes in an element including the end points Nsp ) total number of species Nt ) total number of nodes in the numerical domain ) Ne(Np - 1) + 1 r ) radial position rc ) kinetic rate of char consumption rcf ) radius after which the fluxes of CO and CO2 are constant Rp ) particle radius Rs ) rate of surface reaction R∞ ) radial position at the end of numerical domain Sg ) surface area of Spherocarb Sj ) residual of the differential equations obtained after spatial discretization at node j t ) time T ) temperature U ) velocity wc ) overall char consumption rate X ) mole fraction Xc ) carbon conversion Y ) mass fraction Greek/Other Symbols and Sub/superscripts [ ] ) concentration R, β ) constants ) emissivity σ ) Stefan-Boltzmann constant F ) mass density ω ) reaction rate λ ) thermal conductivity η ) effectiveness factor ∞ ) refers to bulk conditions c ) char or Spherocarb g ) gas phase i ) species index j ) node index ) (p - 1)(Np - 1) + q (1 . . . Nt) k ) time step index l, q ) index for the node in a given element j (1 . . . Np) m ) mixture 0 ) initial conditions p ) element index or particle r ) radial position s ) surface
Acknowledgment. We are grateful to the U.S. Department of Energy for funding this research through the Fossil Energy Program (Solicitation No. DE-PS2294PC94200 and DE-PC89774-3). S.G. thanks the PEEER foundation at MIT for the Martin Environmental Fellowship award. EF9502416
