Energy Fuels 2010, 24, 871–879 Published on Web 12/21/2009
: DOI:10.1021/ef900994k
Extended Application of the Moving Flame Front Model for Combustion of a Carbon Particle with a Finite-Rate Homogenous Reaction Jian Zhang, Mingchuan Zhang,* and Juan Yu Institute of Thermal Energy Engineering, School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China Received September 10, 2009. Revised Manuscript Received November 17, 2009
The moving flame front (MFF) model with the assumption of an infinitely fast homogeneous reaction was successfully extended to use in the finite-rate cases, applying the concept of “characteristic combustion rate of CO relative to its generation” and a universal “effectiveness-transforming formula”. The predictions by the amended MFF model agree well with those by the rigid continuous-film model, no matter what kinetics of the homogeneous reaction are taken and no matter what the particle diameters are. In addition, the prediction of the particle ignition temperature is accurate too. Under certain conditions, the CO flame may yet appear in the boundary layer of a small carbon particle less than 100 μm burning in air, which was confirmed by the Fourier transform infrared (FTIR) online measurement experiment and detailed modeling work using Sobolev’s kinetics of CO oxidation directly measured at the flame front rather than in postflame regions.
at the particle surface producing CO and/or CO2. The second is the double-film model, proposed by Burke and Schumann,8 assuming that carbon is consumed because of the C-CO2 reaction, while the CO thus formed is oxidized at the flame sheet to form CO2. These models are both used widely in practical applications, but their assumptions are too simple to describe all of the complex combustion phenomena. The single-film model completely neglects that the primary product of surface oxidation, CO, burns further in the particle boundary layer. On the other hand, the double-film model, assuming that all of the reactions are infinitely fast, is also not realistic. Thus, these models would have some restrictions for their applications because of the neglecting of some of the physical realities of the complex combustion. Therefore, it looks attractive for the researchers to develop new models or methods, both rationally reflecting the physical realities and being suitable for engineering application. Makino put forward an approximate explicit expression for the combustion rate of a carbon particle in three limiting combustion conditions: the frozen mode, the flame-attached mode, and the flame-detached mode,9 as seen in Figure 1. The frozen mode corresponds to a zero burning rate of CO, and the flame-attached and flame-detached modes both correspond to infinitely fast burning of CO. The real combustion rate of the carbon particle would range between the limiting conditions because of the finite-rate homogeneous reaction of CO. Fu et al., on the other hand, presented a group of “universal” curves for calculating the burning rate of carbon particles in air,10 described jointly by the burning rate-controlling parameter (Fb) and the homogeneous-reaction Damk€ ohler number (Dag), as shown in Figure 2. The method is more instructive than applicable for
1. Introduction The combustion of char or carbon particles at the temperature of a pulverized coal flame is a complex physicochemical process, which involves the surface heterogeneous reactions (C-O2 and C-CO2 reactions) and the CO-O2 homogeneous reaction with very strong coupling of heat and mass transfer in the particle boundary layer. ðIÞ CðsÞ þ 1=2O2 f CO CðsÞ þ CO2 f 2CO
ðIIÞ
CO þ 1=2O2 f CO2
ðIIIÞ
The continuous-film model could accurately describe the combustion conditions and progress of a burning carbon particle by solving the transport and energy conservation equations in the boundary layer of the particle.1-4 However, it is not popularly used for industrial pulverized coal flames, because of its complexity and difficulty to couple with more comprehensive computing codes. The classical theory for quasi-steady combustion of a carbon particle consists of two simplified models. One is the single-film model, proposed by Nusselt5 and perfected by Essenhigh6 and Field,7 assuming that only oxidation occurs *To whom correspondence should be addressed. E-mail: mczhang@ sjtu.edu.cn. (1) Makino, A.; Law, C. K. Proc. Combust. Inst. 1986, 21, 183–191. (2) Amundson, N. R.; Mon, S. Diffusion and reaction in carbon burning. In Dynamics and Modeling of Reactive Systems; Academic Press: New York, 1980; pp 353-374. (3) Daniel Hsuen, H. K.; Sotirchos, S. V. Chem. Eng. Sci. 1989, 44, 2653–2665. (4) Yu, J.; Zhang, M. C.; Zhang, J. Proc. Combust. Inst. 2009, 32, 2037–2042. (5) Nusselt, W.; Ver, Z. Deut. Ing. 1924, 68, 124–128. (6) Essenhigh, R. H. Inst. J. Fuel 1961, 34, 239–244. (7) Field, M. A.; Gill, D. W.; Morgan, B. B.; Hawksley, P. G. W. Combustion of Pulverized Coal: British Coal Utilisation Research Association (BCURA): Cheltenham, U.K., 1967; p 186. r 2009 American Chemical Society
(8) Burke, S. P.; Schuman, T. E. W. Proc. Int. Conf. Bitum. Coal 1932, 2, 485–509. (9) Makino, A. Combust. Flame 1992, 90, 143–154. (10) Fu, W. B.; Zhang, B. L. Combust. Sci. Technol. 1993, 89, 405–412.
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predictions and the experimental measurements of Young and Niksa,15 which could not be obtained using the single-film model. Moreover, the MFF model was further verified by the Fourier transform infrared (FTIR) online measurement experiment.14 The MFF model adopts the assumption of an infinitely fast CO-O2 homogeneous reaction. However, CO burns at a finite rate in nature. Some experimental investigations were devoted to study the kinetics of CO combustion. Howard et al. summarized and put forward a global equation to describe CO combustion kinetics in the postflame region over the temperature range of 840-2360 K.16 On the other hand, Sobolev measured the rate of CO burning in Bunsen flames.17 Despite the kinetics of CO oxidation suggested by refs 16 and 17 having a big difference, which will be shown later, the burning rates of CO are finite. The main objective of this work is to extend the MFF model applicable to the cases with a finite-rate homogeneous reaction. Using the different homogeneous-reaction kinetics suggested,16,17 a prediction comparison between the amended MFF model and other combustion models will be conducted. In addition, questioning the CO flame appearance for small carbon particles will be discussed too.
Figure 1. Makino’s explicit solutions of combustion rates for three limiting modes.9
2. Brief Introduction and Further Analysis to the MFF Model Because quasi-steady formulation of the MFF model has already been performed by Zhang et al.,11-14 only the final solution is presented here. 1 1 a þ 1Ks, O2 Kd, O2 b Pg, CO2 Pg, O2 þ 1 1 a þ 1Ks, CO2 Kd, CO2 b q ¼ 1 1 a þ 1Ks, O2 Kd, O2 b 1 1 a a þ 1þ 1 1 a bKd, CO2 Ks, O2 Kd, O2 b þ 1Ks, CO2 Kd, CO2 b ð1Þ aq bK d, CO2 q00 ¼ 1 1 a þ 1Ks, CO2 Kd, CO2 b
ð2Þ
q0 ¼ q -q00
ð3Þ
Pg, CO2 þ
Figure 2. Effects of the homogeneous reaction kinetics to the burning rate of the carbon particle in air.10
engineering calculation, because it can only be used for the cases in air. Recently, a new method, called the moving flame front (MFF) model, was put forward and developed by Zhang et al.11-14 The MFF model takes into account the effect of the CO flame in the particle boundary layer and the finite-rate C-O2 and C-CO2 heterogeneous reactions on the carbon surface. In the model, the combustion rate and the particle surface temperature are expressed in algebraic forms; thus, the computational efforts are greatly decreased. Despite its simplicity, predictions by this model faithfully reproduce many of the characteristics obtained by the rigid continuous-film model. Good agreement was also obtained between the model
a a a2 0 0 00 00 Ts ¼ Tg þ ðq H þ q H -Hr Þ 1 þ ðqH 000 -Hr Þ b λm bλm ð4Þ a2 Tf ¼ Tg þ ðqH 000 -Hr Þ bλm
ð5Þ
where Ks,O2 =k0 exp(-E0 /(RTs)), Kd,O2 =2MCDm,O2/(aRTm), Ks,CO2=k00 exp(-E00 /(RTs)), Kd,CO2=MCDm,CO2/(aRTm), and Hr =εσ(Ts4 - Tw4). According to the assumptions to the CO flame in the MFF model, there are only two limiting cases occurring:11 (1) Prior
(11) Zhang, M. C.; Yu, J.; Xu, X. C. Combust. Flame 2005, 143 (3), 150–158. (12) Zhang, M. C.; Yu, J.; Zhang, J. Proceedings of the 6th Asia-Pacific Conference on Combustion, Nagoya, Japan, 2007; pp 158-161. (13) Zhang, M. C.; Yu, J.; Qi, Y. F. Proceedings of the 6th AsiaPacific Conference on Combustion, Nagoya, Japan, 2007; pp 162-165. (14) Yu, J.; Zhang, M. C. Energy Fuels 2009, 23, 2874–2885.
(15) Young, B. C.; Niksa, S. Fuel 1988, 67, 155–164. (16) Howard, J. B.; Williams, G. C.; Fine, D. H. Proc. Combust. Inst. 1973, 14, 975–985. (17) Sobolev, G. K. Proc. Combust. Inst. 1959, 7, 386–391.
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In Makino’s formulation for the coupling functions,9 β was ~ - 1, then defined as β=exp(m) ~ ¼ lnð1 þ βÞ m ð12Þ
to particle igniting, CO does not burn in the boundary layer, which can be calculated by the single-film model assuming only CO formed. (2) Following ignition, an infinitesimally thin flame front would form around the particle. The effect of the CO flame on the combustion rate of a carbon particle is calculated by substituting the CO flame front location b (or a/b) to eqs 1-5 shown above. Detailed derivation of the flame front location for a general situation is shown in Appendix A. Now, let us look at some special situations, corresponding to Makino’s definitions for three explicit solutions.9 (1) The flame-attached mode. The flame front is located at the surface of the particle, and then b=a. Substituting a/b=1 to eq 1 gives Ks, O2 Pg, O2 þ Ks, CO2 Pg, CO2 ð6Þ q ¼ 2Ks, O2 Ks, CO2 1þ Kd, O2 Kd, CO2
It was found that the explicit solutions were very close to those ~ predicted by the rigid continuous-film model.9 In the case of m ~ ≈ β, which may lead to the same expressions , 1, we obtain m ~ and β. The approximate assumption m ~ ≈ β was also used for m by Libby and Blake,18 a long time ago. From the identities of eqs 6-8 and eqs 9-11, it might be inferred that the little ~ and β was due to consideration or not of difference between m Stefan flow, with the latter actually performed in conformations of the MFF and single-film models. ~ MFF and β for the Although the identical expressions of m flame-detached mode have been obtained, it should be pointed out that this can be reached only with the assumption of no oxygen passing through the flame front. However, it is not really necessary for both the concentrations of oxygen and CO to be zero at the flame front with an infinitely fast homogeneous reaction, at least in the sense of mathematics. In other words, the assumption of the flame-detached mode with pure surface reduction might bring some additional error to its predictions. When the transitions from the fameattached mode are compared to the flame-detached mode, shown in Figures 1 and 2, the difference of sudden changeover and smooth transform might be attributed to the reason mentioned above. To calculate the carbon combustion rate, Makino’s explicit expressions need to know the particle temperature at first; it will not be convenient for practical engineering applications. In contrast, the MFF model is more advantageous from the fact that all of these parameters are directly calculated from the ambient conditions.
(2) The flame-detached mode (with surface reduction only). The CO flame front is located somewhere apart from the particle surface (b > a). Supposing oxygen does not diffuse to the particle surface crossing the flame front, as Makino did,9 then only the C-CO2 reaction occurs at the particle surface and q0 =0, which leads to Kd, O2 Pg, O2 Kd, O2 Pg, O2 þ Kd, CO2 Pg, CO2 ¼ ð7Þ q ¼ Kd, CO2 a 1þ 2 Ks, CO2 b re (3) The frozen mode. This mode means neglecting the CO-O2 homogeneous reaction completely, which corresponds to the situation described by the MFF model before particle ignition. However, it is also one of the bounding conditions for an ignited carbon particle with a finite rate of the CO burning environment. This situation can be easily reached, if we locate the CO flame front far from the particle surface infinitely, viz. putting a/b=0 in eq 1. It leads to the extended single-film model producing CO. q ¼
Pg, O2 1 1 þ K s, O 2 K d , O 2
þ
Pg, CO2 1 1 þ Ks, CO2 Kd, CO2
3. Extended Application of the MFF Model with a Finite-Rate Homogeneous Reaction Taking the finite CO oxidation rate into account, the combustion rate of a coal particle should range between these limiting conditions, just as shown in Figure 2. However, it is beyond the prediction capability of the original MFF model. To investigate the influence of the finite-rate homogeneous reaction, we introduce a concept of “characteristic combustion rate of CO relative to its generation” or simply “CO relative combustion rate” at first. The CO relative combustion rate (j) is defined as a characteristic ratio of the CO combustion rate integrated in a certain length scale (l) near the carbon particle to the CO generation rate at the particle surface. For simple and clear use of this concept, the CO generation rate is supposed to be calculated by the single-film model producing CO, as eq 8. In addition, l = a is used, and then the CO relative combustion rate is wl MC wa MC j ¼ ¼ ð13Þ q MCO qSF MCO
ð8Þ
To compare these results with Makino’s explicit solutions easily, the combustion rate q above can be written as the ~ MFF = qa/(FgDg). Interesting enough, dimensionless form, m ~ MFF are it was found that the formula descriptions of m exactly the same as the transfer number β in Makino’s explicit expressions as follows (the deduction is shown in Appendix B): Y~ O¥ AsO β≈ δ 1 þ 2AsO -AsP Y~ P¥ AsP þ ðfor flame-attached modeÞ ð9Þ δ 1 þ 2AsO -AsP
The CO relative combustion rate is supposed to be calculated on the basis of the gas concentration and the temperature at the particle surface, which are also given by the single-film model producing CO Eg nO nA nF w ¼ MCO kg exp CCO CO CH2 O 2 RTs nF nO nA Eg Ps, CO Ps, O2 Ps, H2 O ¼ MCO kg exp ð14Þ RTs ðRTs ÞnF þnO þnA
Y~ O¥ þ Y~ P¥ AsP β≈ ðfor flame-detached modeÞ ð10Þ δ 1 þ AsP β≈
Y~ O¥ AsO Y~ P¥ AsP þ ðfor frozen modeÞ δ 1 þ AsO δ 1 þ AsP
ð11Þ
where AsO =Da0s(T~g/T~s) exp(-θ0s/T~s), AsP =Da00s (T~g/T~s) exp0 0 ~ (-θ00s /T~s), T=(R FcpT)/HCO, θs=(RFcpE)/(RHCO), Das=(aBs)/ Dg, Da00s =(aB00s )/Dg, δ=MCO2/MC, Y~ O,¥ =ROYg,O2, Y~ P,¥ = Yg,CO2, RO=(vCO2MCO2)/(vO2MO2), RF=(vCO2MCO2)/(vCOMCO).
(18) Libby, P. A.; Blake, T. R. Combust. Flame 1979, 36, 139–169.
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where the gas partial pressures at the particle surface are qSF qSF , Ps, CO ¼ Pg, CO þ , Ps, H2 O ¼ Pg, H2 O Ps, O2 ¼ Ks, O2 Kd, CO It is obvious that the characteristic combustion rate of CO relative to its generation j takes the value from zero to infinite, which corresponds to the frozen mode and the mode of infinitely fast CO oxidation, respectively. On the other hand, the real impact of the CO flame with a finite oxidation rate on the combustion of a carbon particle can be defined by a “normalized effectiveness coefficient of CO oxidation”, γ. ~ SF -m ~ m γ ¼ ð15Þ ~ SF -m ~ MFF m
Figure 3. Relations between γ and Dag directly from Figure 2,10 and calculated by eq 16.
The range of γ will be (0, 1). When CO oxidation is in frozen mode, γ=0. When the reaction is infinitely fast, then γ=1. Our task is to find an approximate relationship between j and γ, which can then be used to estimate the real impact of the CO flame from j. It can be seen from Figure 2, given by Fu et al.,10 that the combustion rate curves with different homogeneous reaction Damk€ ohler numbers (Dag) are dense near the two limiting cases while sparse in the middle. In other words, near the two limiting cases, the impact of Dag on the combustion rate is relatively weak; but away from the limiting cases, the impact of Dag is much greater. Thus, we choose an approximate function of CO relative combustion rate (j) to calculate the effectiveness coefficient (γ) γ ¼ 0:5fth½lnððRjÞδ Þ þ 1g
Table 1. Kinetic Parameters for CO Oxidation Sobolev Howard
Eg (J/kmol)
nF
nO
nA
1.25 10s 1.25 10s
1 1
0.25 0.5
0 0.5
4. Predictions by the Amended MFF Model Quasi-steady calculations of carbon particle combustion were carried out below. The partial pressures of oxygen and water vapor are 0.21 and 0.1 atm at the ambient. Relatively high water vapor partial pressure was used in the calculations to properly reflect its influence on the CO combustion rate for a pulverized coal flame. However, the water vapor gasification, C þ H2O=CO þ H2, was still ignored here. An additional calculation showed that this reaction is very slow compared to the C-O2 or C-CO2 reactions by 1-2 orders of magnitude under general combustion conditions. Particle diameters are 50, 100, and 200 μm, respectively. Two descriptions of CO oxidation kinetics (Sobolev17 and Howard16) were used, shown in Table 1. The kinetic parameters of heterogeneous reactions were E0 = 1.80 108 J/kmol, E00 =2.70 108 J/kmol, B0 =6.2 106 m/s, and B00 =6.2 107 m/s. The calculation results by the amended MFF model were compared to those predicted by the continuous-film model, the single-film model producing CO, and the original MFF model. To show the essential difference of predictions by these models more clearly rather than the influence brought from consideration of ~ calculated by the conStefan flow or not, the results (m) tinuous-film model have been transferred to the corresponding value of β applying eq 12. The results are shown in Figures 4 and 5. It can be seen from Figures 4 and 5 that all of the predictions by the amended MFF model with R=3.0 and δ=0.25 agree well with those by the rigid continuous-film model, no matter what kinetics of homogeneous reaction are taken and no matter what the diameters of carbon particles are. In addition, the predictions of particle ignition temperature by the amended MFF model are more accurate than the original one too. With regard to the other two models, the prediction errors for different kinetics of the homogeneous reaction are quite different. With a relatively higher reaction rate from Sobolev’s kinetics, predictions by the continuous-film model are better approximated by those from the original MFF model than by the single-film model. With a lower reaction rate from Howard’s kinetics, the predictions by the continuous-film model are better approximated by the single-film model, viz. the frozen mode, for
ð16Þ
where R and δ are adjustable parameters. Supposing that the lowest curve with Dag = 109, the highest value in Figure 2, corresponds to infinitely fast CO oxidation ~ MFF, then γ can be calculated directly from giving the values of m eq 15 with different Dag and Fb, as shown in Figure 3. Inspiring enough, the relations between γ and Dag with different Fb are very close, which means that a universal relationship between γ and j might be found. Taking the same calculation conditions of Fu and Zhang,10 j was calculated first, and then by comparing the γ calculated from eq 16 with different values of R and δ, an approximate best fit with R=3.0 and δ=0.25 was reached, as shown in the same figure. Because the formula of the MFF model could also be used for the frozen mode setting (a/b)SF=0, a simple and effective method to reflect the effects of the finite-rate homogeneous reaction is adjusting the flame front location in MFF as follows: a a a a ¼γ þ ð1 -γÞ ¼γ ð17Þ b b MFF b SF b MFF With the finite-rate homogeneous reaction, the particle ignition temperature is supposed to be predicted using the same method too. An effective heterogeneous heat release and a modified diffusion reaction rate coefficient are supposed to be used in the single-film model with iterative calculations. Heq ¼ H 0 þ γðH 000 -H 0 Þ ð18Þ Kd, O2 ¼ ð2 -γÞMC Dm, O2 =ðaRTm Þ
kg 1.7 1010 (m3/kmol)0.25 s-1 1.3 1011 m3 kmol-1 s-1
ð19Þ
According to the hypothesis of “surface oxidation induced CO ignition”,13 which means the heterogeneous ignition must accompanied with an obvious temperature jump for the particle, the iterative calculation should start from the initial value γ=1 for the case Tig,CO2 > Tex,eq. Otherwise, iteration is not needed for Tig,CO2 < Tex,eq but taking γ 0. The MFF model works for quasi-steady-state combustion. Thus, principally above-mentioned checking is needed for every calculation step. If size reduction undergoes with an ignited burning particle but at a certain moment Tig,CO2 > Tex,eq is not satisfactory anymore, then the CO flame will become extinct. 874
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Figure 4. Comparison of predictions by different models with Sobolev’s kinetics.
the particles less than 100 μm. The influences of CO flame are shown more obvious for the particles over 100 μm. These results, to some extent, are in accordance with Makino’s point of view, deducted from the alike or same kinetics.9 To check if the empirical constants R=3.0 and δ=0.25 are also suitable for the cases that oxygen partial pressure are not 0.21 atm, calculations for 100 μm particle at oxygen partial pressures of 0.10 and 0.32 atm were also performed. The results are shown in Figures 6 and 7. It can be seen from these figures that the values of R and δ shown above can be considered as universal constants now. For all of these cases mentioned above, the ignition temperatures of a carbon particle calculated by the amended MFF model and the rigid continuous-film model were compared in Table 2 and Figure 8. It can be seen that the new method for ignition temperature calculation, viz. eqs 18 and 19 used for the single-film model, is also satisfactory enough for engineering application.
5. Further Discussion on CO Flame Appearance for Small Carbon Particles As mentioned before, Howard et al. put forward a global expression16 to describe CO combustion kinetics based on a series of experimental investigations, including both their own and those from the literature. However, in all of these experiments, the burning rates of CO were measured in the postflame region. Differently, Sobolev et al. measured the burning rate at the flame front of a Bunsen lamp.17 The CO burning rates measured at the flame front are much higher than those given by Howard’s expression. Sobolev ascribed the differences to the fact that the conditions in the flame front might possibly deviate from thermal and chemical equilibrium substantially, unlike the case in postflame. It can be seen from the text above that the differences of predictions using different kinetics for CO oxidation are also quite enormous. For small particles or under very low oxygen concentrations, the primary product from the surface reactions, CO, might not be ignited in the particle boundary layer. To 875
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: DOI:10.1021/ef900994k
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Figure 5. Comparison of predictions by different models with Howard’s kinetics.
calculations have been carried out using different models, as shown in Figure 9. It was found that the higher reaction rate of CO from Sobolev’s kinetics was needed to accurately reproduce the experimental results. In addition, perhaps, most of the researchers who used the thermal explosion theory to predict the ignition temperature of a carbon particle assuming CO2 as the only product of the surface oxidation22-24 need Sobolev’s kinetics to support too. In summary, we uphold to take the higher reaction rate from Sobolev’s kinetics for the calculation of CO burning in the particle boundary layer, which should be different from those measured in the postflame region. Also, because we are taking the higher reaction rate from Sobolev’s kinetics, predictions of the original MFF model would also be satisfying enough.
describe this phenomenon, Makino and Law put forward the ignition and extinction criterion of a burning carbon rod.19 The ignition and extinction states were determined as functions of the surface temperature, the oxygen concentration, and the velocity gradient of the flow. Applying this criterion and the kinetics of CO oxidation similar to Howard’s, they concluded that carbon particles less 100 μm would burn in the frozen mode. In addition, it seemed to be supported by Smith’s and BCURA’s experimental data.20,21 However, our FTIR online measurement experiment captured successfully the elevated particle temperature caused by the ignition of CO near the particle surface for 77 μm carbon particles.14 To explain this experimental data, theoretical (19) Makino, A.; Law, C. K. Combust. Sci. Technol. 1990, 73, 589– 615. (20) Smith, I. W. Combust. Flame 1971, 17, 303–314. (21) British Coal Utilisation Research Association (BCURA). Laboratory studies of combustion. BCURA Annual Report, Cheltenham, U.K., 1966.
(22) Bandyopadyay, S.; Bhaduri, D. B. Combust. Flame 1972, 18, 411. (23) Karcz, H.; Kordlyleuski, W. Fuel 1980, 59, 799. (24) Xieu, D. V.; Masuda, T.; Cogili, J. G.; Essenhigh, R. H. Proc. Combust. Inst. 1984, 18, 1461–1468.
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Figure 6. Comparison of predictions by different models for oxygen partial pressures of 0.10 atm.
Figure 7. Comparison of predictions by different models for oxygen partial pressures of 0.32 atm.
extended to be used for the finite-rate cases, applying the concept of “characteristic combustion rate of CO relative to its generation” and a universal “effectiveness-transforming formula”. The predictions by the amended MFF model agree
6. Conclusion In this paper, the original MFF model with the assumption of an infinitely fast homogeneous reaction was successfully 877
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explicit solutions for three limiting conditions, viz. the flame-attached mode, the flame-detached mode, and the frozen mode.
Table 2. Comparison of the Ignition Temperature Predicted by the Amended MFF Model and the Continuous-Film Model ignition temperatures (K) calculation conditions
continuousfilm model
amended MFF model
Sobolev’s kinetics, Xo2=0.21, 50 μm Sobolev’s kinetics, Xo2=0.21, 100 μm Sobolev’s kinetics, Xo2=0.21, 200 μm Howard’s kinetics, Xo2=0.21, 50 μm Howard’s kinetics, Xo2=0.21, 100 μm Howard’s kinetics, Xo2=0.21, 200 μm Sobolev’s kinetics, Xo2=0.10, 100 μm Sobolev’s kinetics, Xo2=0.32, 100 μm Howard’s kinetics, Xo2=0.10, 100 μm Howard’s kinetics, Xo2=0.32, 100 μm
1212 1160 1114 1249 1185 1143 1216 1126 1308 1140
1215 1155 1105 1245 1185 1135 1220 1125 1315a 1150
a
Acknowledgment. The authors gratefully acknowledge financial support from the Chinese National Natural Science Fund (Grant 50476018) and The Special Fund of The Key Fundamental Research of China (Grants 2004CB217703 and 2006CB200303).
Appendix A The partial pressures of O2 and CO2 at the flame front can be written as14 " # 2aq 1 1 a q0 ð20Þ Pf , O2 ¼ Pg, O2 ¼ þ 1bKd, O2 Ks, O2 Kd, O2 b
Ignition without temperature jumping.
Pf , CO2 ¼ Pg, CO2 þ
aq bKd, CO2
ð21Þ
Under a hypothetical condition of no consumption of CO and O2 for the CO-O2 homogeneous reaction, the partial pressures of CO at the flame front are obtained applying Fick’s law.25 aðq0 þ 2q00 Þ 2aðq0 þ q00 Þ Pf , CO ¼ Pg, CO þ ≈ Pg, CO þ ð22Þ bKd, CO bKd, O2 where Kd,CO = (MCDm,CO)/(aRTm) and Dm,CO ≈ Dm,O2 approximately. If only the C-O2 oxidation reaction occurs at the particle surface (q00 =0), it is assumed that the flame front lies on the spherical surface, where CO and O2 concentrations are in accordance with the stoichiometric ratio of the reaction. This assumption was also used in Annamalai’s model for homogeneous ignition of a carbon particle.26 Pf , CO ¼ 2Pf , O2 ð23Þ
Figure 8. Comparison of ignition temperature predictions.
Thus, b is obtained by a ¼ b ox
Kd, O2 þ1 K s, O 2
!
2þ
1Pg, CO 12 Pg, O2
1Pg, CO 2 Pg, O2
! ð24Þ
(1) When Pg,CO=0, under complete diffusion control, Ks,O2 . Kd,O2 and then box =2a, and under partial diffusion control, 6 0 with Kd,O2/Ks,O2 > 0 and then box < 2a. (2) When Pg,CO ¼ complete diffusion control, box > 2a. If only the C-CO2 reduction reaction occurs at the particle surface (q0 =0), CO2 diffusion should be continuous at both sides of the flame front. Then, b is obtained by Kd, CO2 1þ Ks, CO2 a ! ¼ ð25Þ b re Pg, CO2 Kd, CO2 2 1þ Pg, O2 Kd, O2
Figure 9. Theoretical calculations for the experimental data in ref 14.
well with those by the rigid continuous-film model, no matter what kinetics of the homogeneous reaction are taken and no matter what the particle diameters are. In addition, the predictions of the particle ignition temperature are accurate too. The difference of predictions to the carbon combustion rate is obvious using different kinetics for CO oxidation. Under certain conditions, the CO flame may yet appear in the boundary layer of a small carbon particle less than 100 μm burning in air, which is in accordance with the FTIR online measurement of the particle temperature and the detailed modeling work of the authors. In addition, this could attribute to the use of a higher reaction rate of CO from Sobolev’s kinetics, which were directly measured at the flame front of a Bunsen lamp, and should be different from those measured in the postflame region. In addition, strict theoretical deduction was carried out confirming that the MFF model can also yield Makino’s
(1) When Pg,CO2=0, under complete diffusion control, Ks,CO2 . Kd,CO2 and then bre = 2a, and under partial diffusion (25) Zhang, J.; Zhang, M. C.; Yu, J. Proceedings of the 6th International Symposium on Coal Combustion, Wuhan, China, 2007; pp 137-141. (26) Annamalai, K.; Durbetaki, P. Combust. Flame 1977, 29 (2), 193– 208.
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: DOI:10.1021/ef900994k
Zhang et al.
b=radius of the CO flame front (m) B=pre-exponential factor of the surface reaction (m/s) Bg =pre-exponential factor of the homogeneous reaction C=species concentration (kmol/m3) cp =specific heat (J kg-1 K-1) D=diffusion coefficient (m2/s) Da=Damk€ ohler number E=activation energy of the surface reaction (J/kmol) Eg = activation energy of the homogeneous reaction (J/kmol) H=heat of the heterogeneous reaction (J/kg of C) HCO=heat of the CO þ 1/2O2 f CO2 reaction (J/kg of CO) Heq =effective heterogeneous reaction heat (J/kg of C) Hr =radiative heat flux (W/m2) k=pre-exponential factor of the reaction (kg m-2 s-1 Pa-1) Kd=diffusion-limited reaction rate coefficient (kg m-2 s-1 Pa-1) Ks =surface reaction rate coefficient (kg m-2 s-1 Pa-1) M=molecular weight (kg/kmol) ~ m=dimensionless reaction rate P=partial pressure (Pa or atm) q = reaction rate of carbon per unit surface area (kg m-2 s-1) R=universal gas constant (J kmol-1 K-1) T=temperature (K) T~ =dimensionless temperature v=stoichiometric coefficient of the homogeneous reaction w=gas consumption rate (kg m-2 s-1) X=volume fraction Y=mass fraction Y~ =dimensionless mass fraction β=transfer number in Makino’s explicit solution δ=stoichiometric product (CO2)/carbon mass radio ε=emissivity of the particle surface θ=dimensionless activation temperature λ=thermal conductivity (W m-1 K-1) F=gas density (kg/m3) σ=Stefan-Boltzmann constant (W m-2 K-4) j=CO relative combustion rate γ=normalized effectiveness coefficient of CO oxidation
control, Kd,CO2/Ks,CO2 > 0 and then bre < 2a. (2) When Pg,CO2 6¼ 0 with complete diffusion control, bre > 2a. Considering C-O2 and C-CO2 reactions simultaneously, b can be calculated in mass-weighted average. 0 00 a a q a q ¼ þ ð26Þ b b ox q b re q Thus, the flame-front location is determined by five parameters, viz. Pg,CO/Pg,O2, Pg,CO2/Pg,O2, Kd,O2/Ks,O2, Kd,CO2/Ks,CO2, and Kd,CO2/Kd,O2. During the combustion process of a carbon particle, the location of the flame front is affected by the ambient gas (O2, CO, and CO2) concentrations and the heterogeneous reaction rates at the particle surface. Appendix B Supposing that the approximation can be used in the particle boundary layer. ð27Þ Dm =Tm ¼ Dg =Tg ¼ constant In the deduction of Makino’s continuous-film model, the approximation FmDm=FgDg=constant has been used. These two approximations are coincident for the ideal gas. Because of Dalton’s law of partial pressure and the ideal gas equation, the relation of the mass fraction and partial pressure at the particle surface is Mi ð28Þ Ys, i ¼ Ps, i RFs Ts In Makino’s continuous-film model, the O2 consumption rate of the C-O2 reaction is described as wO2 ¼ B0 Fs Ys, O2 expð-E 0 =ðRTs ÞÞ ð29Þ In the MFF model, another kinetic description is used. ð30Þ q0 ¼ Ks, O2 Ps, O2 ¼ Ps, O2 k0 expð-E 0 =ðRTs ÞÞ wO2 ¼
q0 MO2 2MC
Thus, the conversion between B0 and k0 is k0 RTs B0 ¼ 2MC
ð31Þ
ð32Þ
Subscripts d=diffusion eq=equivalent ex=extinction f=flame front g=ambient i=gas species ig=ignition m=average in the particle boundary layer MFF=moving flame front model ox = only the C-O2 oxidation reaction at the particle surface re = only the C-CO2 reduction reaction at the particle surface s=particle surface SF=single-film model
Similarly, the conversion between B00 and k00 is obtained for the C-CO2 reaction. k00 RTs ð33Þ B00 ¼ MC Because of those expressions above, then we have K s, O 2 AsO ¼ Kd , O 2 AsP ¼
Ks, CO2 Kd, CO2
ð34Þ ð35Þ
Therefore, the sameness of eqs 6-8 and eqs 9-11 can be obtained using eqs 27, 28, and 32-35.
Superscripts
Nomenclature
0
=reaction C þ 1/2O2 f CO =reaction C þ CO2 f 2CO 000 =reaction C þ O2 f CO2
a=radius of the particle (m) ASO =parameter in Makino’s explicit solution ASP =parameter in Makino’s explicit solution
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