## 4 Discussion  
The model which is proposed in the present study also gives access to local quantities and their variations within the coal char particle during its combustion. To better understand how the combustion evolves within the particle, the behavior of different local quantities is analyzed here during the combustion process. The local oxygen concentration \([O_{2}](i,. )\)(mol.m\({}^{-3}\)) and the product \((A_{C}\rho_{C})(i,. )\) (m\({}^{2}\).m\({}^{-3}\)) between the local carbonaceous material density \(\rho_{C}(i,. )\) (kg.m\({}^{-3}\)) and the local specific area \(A_{C}(i,. )\), which is the area of the reactive carbon surface per volume unit, are the key parameters for the combustion rates. These local quantities allow the local carbon consumption rate per volume unit \(r_{c}(i,. )\) (kg.s\({}^{-1}\).m\({}^{-3}\)) to be analyzed, given through:  
\[r_{c}(r,t)=M_{\rm C}k_{s}(t)A_{\rm C}(r,t)\rho_{\rm C}(r,t)[O_{2}](r,t) \tag{12}\]  
and to the local carbon conversion \(X_{\rm C}(i,. )\) computed through:  
\[X_{\rm C}(r,t+dt)=X_{\rm C}(r,t)+M_{\rm C}k_{s}(t)A_{\rm C}(r,t)[O_{2}](r,t)(1- X_{\rm C}(r,t))dt. \tag{13}\]  
Six times \(\big{(}t_{j}\big{)}_{j=1,...,6}\) are chosen along the combustion process to analyze the behavior of the above-indicated local quantities within the particle. Their values, together with that of the corresponding global carbon conversion, particle temperature, particle radius and label of the outer volume element, returned for a regulation temperature of the reactor equal to 1200 \({}^{\circ}\)C, are gathered in Table 3. These six chosen times are sufficient to analyze the trends of the different local quantities previously indicated, whence the combustion behavior of the particle. Figure 4 presents the values of the local carbon conversion, oxygen concentration, relative surface per unit volume and carbon consumption rates at the six chosen times indicated in Table 3, in terms of the volume element number and computed according to Equations (12) and (13). The curves of the different local quantities present quite the same shape for the six chosen times, being mainly translated. Looking at the local carbon conversion curves and especially at the right extremities of the four curves corresponding to the four lower chosen times of 0.256, 0.284, 0.320 and 0.357 s, Figure 4a, the outer element volume exhibits a local carbon conversion ranging from 0.4 to approximately 1. This is because this outer volume element is progressively consumed and close to disappearing. The local carbon conversions drastically decrease from the right extremity (which corresponds to the number of the surface volume element) toward the particle center, up to the left extremity of the curve where the local carbon conversions are equal to 0. The distance between these two extremities on the horizontal axis defines the zone of the particle where the combustion is active at time \(t\). The combustion is more active close to the particle surface and decreases down to 0 at the left extremity of the curve (center of the particle). At every chosen time \(t_{j}\), only a spherical shell (containing roughly 150 volume elements) is burning under a significant oxygen concentration. The thickness of the combustion zone in which combustion occurs at a given time does not highly vary, while this zone progresses toward the particle center. It mainly occurs at the particle surface due to the particle shrinking. Just a small broadening is observed. This is in relation with the decrease of the particle radius and the increase of the porosity which makes the oxygen diffusion more efficient in a neighborhood of the particle surface.