## 4 Discussion
Instead of [PERSON]'s expression (4) of the ratio \(\tau\) between the instantaneous rates of CO and CO\({}_{2}\) formation, that of [PERSON] could be used, [3], which is defined as:  
\[\tau(t)=10^{2}5\mathrm{exp}\left(-\frac{25,000}{R_{g}T_{p}(t)}\right). \tag{14}\]  
The impact of the expression of the ratio \(\tau\) on the radiation, convection and enthalpy terms in the right-hand side of Equation (3) is analyzed in Figure 5.  
During particle combustion which occurs between 0.2 and 0.8 s, most of the energy is exchanged by the particle through the enthalpy of the reaction, the hyphened line, according to the area lying between the enthalpy curve and the time axis. This exchange contributes to approximately 50% of the total exchanged energy (absolute values considered). The maximum of the enthalpy power (which contains the ratio \(\tau\)) increases from 1.97 \(\times\) 10\({}^{-2}\) W with [PERSON]'s equation (at 0.298 s) to 2.41 \(\times\) 10\({}^{-2}\) W with [PERSON]'s equation (at 0.307 s), corresponding to a 22% increase. This result is in agreement with the drastic decrease of the ratio \(\tau\) when changing [PERSON]'s equation to [PERSON]'s one. For the radiative power, the effect is more pronounced with an increase from 5.41 \(\times\) 10\({}^{-3}\) to 7.32 \(\times\) 10\({}^{-3}\) W, representing a 35% increase, when passing from [PERSON]'s to [PERSON]'s equations. The convective power increases from 1.39 \(\times\) 10\({}^{-2}\) to 1.64 \(\times\) 10\({}^{-2}\) W, representing a 18% increase. The radiative power is more affected than the two others by the change of equation for \(\tau\) because of the temperature dependence in \(T^{4}\). To conclude this Discussion section, a comparison is established between the results of the present study which considers a shrinking coal particle and that of the previous study [20] which considered a coal char particle with a fixed radius along the combustion process. [PERSON]'s equation (14) of the ratio \(\tau\) between the instantaneous CO and CO\({}_{2}\) formation rates was used in the previous study [20]. Consequently, the simulations are also performed with the present model involving [PERSON]'s equation. The particle  
Figure 5: Values of the radiation (solid line), convection (dotted line) and enthalpy (hyphened line) terms in the right-hand side of (3) versus time, according to [PERSON]’s (**a**) and [PERSON]’s (**b**) equations, and for the regulation temperature of the reactor equal to 1200 \({}^{\circ}\)C.  
temperature and the global carbon conversion obtained with the two models are compared in Figure 6, for the regulation temperature of the reactor equal to 1200 \({}^{\circ}\)C.  
The curves of Figure 6a are presented in the time range 0.2-1.0 s, to highlight the active combustion time range, as previously discussed. The maximal particle temperatures returned by the two models show an acceptable agreement (difference approximately equal to 31 \({}^{\circ}\)C) between both models, but the difference between the times at which these maxima are reached is significantly high (0.09 s), Figure 6a. The maximal differences between the simulated and experimental particle temperatures are quite the same (approximately 70 \({}^{\circ}\)C). They are higher than that obtained when considering [PERSON]'s Equation (4) for the ratio \(\tau\). In the time range 0.25-0.60 s, the shape of the curve presenting the particle temperature simulated through Model 2 L with [PERSON]'s equation is closer to that of the experimental temperature than that returned by Model 1. The comparison of the two global carbon conversions shows a significant difference, Figure 6b. The simulated carbon conversion curve obtained with Model 2 L is always higher than that obtained with the previous model. The first experimental point indicated in [13] (time equal to 0.25 s) is largely above the two simulated curves. The second experimental point (time equal to 0.5 s) lies between the two curves.