## 4 Discussion
\begin{table}
\begin{tabular}{l c c c c c c} \hline
**Time \(t_{j}\) (s)** & **0.256** & **0.284** & **0.320** & **0.357** & **0.424** & **0.569** \\ \hline Global carbon conversion \(X_{\rm C}\big{(}t_{j}\big{)}\) & 0.11 & 0.21 & 0.35 & 0.47 & 0.63 & 0.84 \\ Particle temperature \(T_{p}\big{(}t_{j}\big{)}\) (\({}^{\circ}\)C) & 1292 & 1364 & 1382 & 1339 & 1291 & 1228 \\ Particle radius \(R_{p}\big{(}t_{j}\big{)}\) (\(\mu\)m) & 65.0 & 65.0 & 65.0 & 65.0 & 59.4 & 50.8 \\ Outer volume number \(N_{\rm S}\big{(}t_{j}\big{)}\) & 500 & 500 & 500 & 500 & 457 & 391 \\ \hline \end{tabular}
\end{table} 
Table 3: Calculated quantities at six times of the combustion of a coal char particle in the drop tube furnace. Regulation temperature: 1200 \({}^{\circ}\)C.  
The oxygen concentration curves, Figure 4b, are in complete agreement with the local carbon conversion ones. The four curves corresponding to the four first times end at the particle surface with close but different oxygen concentrations, due to the evolution of the particle temperature with respect to time. These curves are in agreement with the values of the efficiency factor \(\eta_{p}\) above-discussed, confirming that the combustion is controlled by both diffusion and kinetics of combustion. Figure 4c shows the variations of the local value of the product \((A_{C}\rho_{C})(i,. )\), which represents the area of the carbon surface available for combustion per volume unit, versus the number of the volume element. In a given volume element, the local product \((A_{C}\rho_{C})(i,. )\) increases from its initial value, according to the increase of the specific surface area \(A_{C}\) up to its maximum 300,000 m\({}^{2}\).kg\({}^{-1}\) and the decrease of the carbon density \(\rho_{C}\) due to the local consumption of carbon, while the specific surface area \(A_{C}\) is forced to stay at its maximum. Every volume element undergoes a very similar variation. The maxima of this product are identical for all volume elements because the specific surface area \(A_{C}\) reaches the same maximum corresponding to the same limit (300,000 m\({}^{2}\).kg\({}^{-1}\)) for the same local carbon conversion (\(X_{C}=0.11\)). After this maximum, the decrease of the carbon density \(\rho_{C}\) controls the decrease of the product \(A_{C}\rho_{C}\) until the end of this local combustion. During the combustion propagation, this process is successively followed by every volume element, but at different times and with different durations due to the temperature variations. This explains the succession of widening peaks along the radius, whose amplitude is limited by the same maxima. The two first curves (\(t=0.256\) and 0.284 s) present a particular shape because at those times, the outer element number 500 already started to evolve. Figure 4: Local carbon conversions \(X_{C}\) (**a**), oxygen concentrations \([O_{2}]\) (**b**), local reactive surface per unit volume \(A_{C}\rho_{C}\), (**c**), and local carbon consumption rates \(r_{C}\) (**d**), at the six chosen times indicated in Table 3, and for the regulation temperature of the reactor equal to 1200 \({}^{\circ}\)C.  
Figure 4d shows the variations of the local combustion rate \(r_{\rm C}\) per volume unit, defined in (11). The local combustion rate contains the kinetic constant \(k_{s}\) which only depends on the temperature, the local product \((A_{\rm C}\rho_{\rm C})(i,. )\) and the local oxygen concentration \([O_{2}](i,.)\). The variations of the local combustion rate versus the volume element combine the effects of the variations of these quantities. They also exhibit successive peaks but with a curved decreasing maximum when the element number \(i\) decreases. This is explained by the decrease of the oxygen concentration when the combustion front moves toward the particle center and by the decrease of the particle temperature when the combustion zone reaches the elements with numbers less than 300.