## 2 Materials and Methods
### Hypotheses of the Model Accounting for the Coal Char Combustion Experiments  
The hypotheses of the model proposed in the present study are the following:  
**Hypothesis 1:**_A mean velocity \(u_{g}\) of the injected gas is considered in each horizontal section of the reactor, which is calculated in terms of the gas flow \(F_{g}^{0}\) and gas temperature \(T_{g}\), according to the formula: \(u_{g}=\left(T_{g}/298\right)F_{g}^{0}/\left(\pi(R_{r})^{2}\right)\) (m.s\({}^{-1}\)). The gas temperature \(T_{g}\) is measured along the reactor axis and set in the model as a function of the particle position._  
**Hypothesis 2:**_The coal char particles may obey a transient velocity regime during a very short time length after their injection, but it is assumed that they are carried through the reacting zone of the drop tube furnace at the gas velocity._  
**Hypothesis 3:**_The coal char particles enter the reactor at 323 K._  
**Hypothesis 4:**_The coal char particles are supposed to keep their spherical symmetry during their fall in the drop tube furnace._  
**Hypothesis 5:**_A mean value of the initial coal char particle radius is considered: \(R_{p}(0)=65\times 10^{-6}\) m, as determined through measurements._  
**Hypothesis 6:**_Due to the small radius of the particle, the particle temperature \(T_{p}(t)\) is supposed to be uniform within the particle at any time \(t\) of the combustion process, in agreement with the discussion in [PERSON] et al. (2013)._  
**Hypothesis 7:**_The heat transfer modeling includes convection between the particle and gas, radiations between the particle and reactor walls and heat release by particle combustion._  
**Hypothesis 8:**_The ash presence inside the coal char particle is accounted for, which is supposed to be homogeneously distributed within the particle at any time of the combustion process._  
**Hypothesis 9:**_The oxygen diffusion coupled with the reaction within the porous structure is modeled to determine the local oxygen concentration along the particle radius at any time of the combustion process. External diffusional limitations through a boundary layer are also accounted for. Due to oxygen internal diffusion limitations, the combustion is not homogeneous within the particle along the combustion process. For the numerical resolution of the system of equations, the initial particle is divided into \(N=500\) concentric volume elements, with equal thickness: \(dr=R_{p}(0)/N=1.3\times 10^{-7}\ m\). The volume element which lies between the spheres of radii \((i-1)dr\) and \(idr\) is labeled with its subscript \(i\), \(1\leq i\leq N\). For the oxygen diffusion model, a unique pore model is locally used within every volume element of the spherical particle, which allows estimating the different local pore radii and their evolution versus time. The different physical quantities (particle density, carbonaceous material density, porosity, pore radius and specific surface area) are supposed to be initially uniformly distributed within the particle._  
**Hypothesis 10:**_The quasi-stationarity of the oxygen diffusion process is assumed, due to the very small values of the time derivatives (corresponding to accumulations terms) within the diffusion differential equation._  
**Hypothesis 11:**_The diminution of the particle radius along the combustion process is accounted for, assuming that the outer volume element of the particle labelled \(N_{\mathrm{s}}(t)\), with \(1\leq N_{\mathrm{s}}(t)\leq N\), is removed from the particle when the local carbon conversion in this volume element becomes higher than a chosen threshold: \((X_{\mathrm{C}})_{N_{\mathrm{s}}}^{lim}=0.99\). At the beginning of the combustion process, \(N_{\mathrm{s}}(0)=N=500\). The radius \(R_{p}(t)\) of the particle thus evolves with respect to the time parameter \(t\) according to the expression: \(R_{p}(t)=R_{p}(0)N_{\mathrm{s}}(t)/N\).