## 2 Materials and Methods
### Description of the Model Accounting for the Coal Char Combustion Experiments  
Throughout the presentation of the model, no distinction will be introduced between the letters representing local and global quantities (see also the Nomenclature at the end of the manuscript). The variables attached to the quantities in the successive equations of the model certainly allow for understanding the local or global character of these quantities. To start with the description of the model under consideration in the present study, the position of the particle is calculated as a function of the time parameter through the integral: \(x_{p}(t)=\int_{0}^{t}u_{g}(s)ds\) (m), where \(u_{g}\) is the mean velocity of the injected gas, see Section 2.3, Hypothesis 1. The evolution of the uniform particle temperature \(T_{p}\) with respect to time is governed by the differential equation:  
\[ \begin{split}(m_{C}(0)(1-X_{C}&(t))C_{p,C}+m_{ash}( t)C_{p,ash})\frac{dT_{p}}{dt}(t)\\ &=4\pi R_{p}^{2}(t)\epsilon_{p}\sigma\Big{(}T_{wall}^{4}(t)-T_{p} ^{4}(t)\Big{)}+4\pi R_{p}^{2}(t)h\big{(}T_{g}(t)-T_{p}(t)\big{)}\\ &+(\tau(t)\big{|}\Delta_{R}H_{CO}^{0}\big{|}(t)+\Big{|}\Delta_{R} H_{CO_{2}}^{0}\Big{|}(t))\frac{\Phi_{\text{O}_{2}}(t)}{1+\tau(t)/2},\end{split}  \tag{3}\]  
where \(m_{C}(0)\) is the initial mass of carbon in the coal char particle, \(X_{C}\) the global carbon conversion, \(C_{p,C}\) the calorific value of carbon, \(m_{ash}\) the ash mass within the particle, \(C_{p,ash}\) the calorific value of ash, \(\epsilon_{p}\) the char particle emissivity, \(\sigma\) Stefan-Boltzmann constant, \(T_{wall}\) the temperature of the reactor walls, \(h\) the convective heat transfer coefficient, \(\Delta_{R}H_{CO}^{0}\) and \(\Delta_{R}H_{CO_{2}}^{0}\) the enthalpies respectively expressed in terms of the particle temperature as: \(\Delta_{R}H_{CO}^{0}=-110.5\times 10^{3}-6.38\times\big{(}T_{p}-298\big{)}\), \(\Delta_{R}H_{CO_{2}}^{0}=-393.5\times 10^{3}-13.1\times\big{(}T_{p}-298\big{)}\) (J.mol\({}^{-1}\)), \(\Phi_{\text{O}_{2}}\) the oxygen molar flux at the particle surface and \(\tau\) the ratio between the instantaneous rates of CO and CO\({}_{2}\) formation. In the present study, this ratio \(\tau\) is computed through the expression proposed by [PERSON] (2013):  
\[\tau(t)=10^{-\frac{2\left|\Omega\right|\Omega\right>+10^{3}}{T_{p}(t)}+2.3996}. \tag{4}\]  
The global carbon conversion \(X_{C}\) is computed as the solution to the differential equation:  
\[\frac{dX_{C}}{dt}(t)=\frac{R_{C}(t)}{m_{C}(0)}, \tag{5}\]  
where \(R_{C}(t)\) is the global carbon consumption rate at time \(t\), which is expressed as:  
\[R_{C}(t)=M_{C}k_{s}(t)\int_{0}^{R_{p}(t)}A_{C}(r,t)\rho_{C}(r,t)[O_{2}](r,t)r^{ 2}dr, \tag{6}\]  
\(M_{C}\) being the molar weight of carbon, \(k_{s}\) the kinetic constant for carbon combustion associated to the reaction (2) and which is given an Arrhenius expression, \(A_{C}\) the local specific surface area of the carbonaceous part, \(\rho_{C}\) the local apparent density of the carbonaceous part and \([O_{2}]\) the local oxygen concentration, the solution to the partial differential equation written in spherical coordinates:  
\[\frac{1}{r^{2}}\frac{\partial}{\partial r}\bigg{(}r^{2}D_{O_{2}}^{\epsilon}( r,t)\frac{\partial[O_{2}](r,t)}{\partial r}\bigg{)}=\frac{k_{s}(t)A_{C}(r,t) \rho_{C}(r,t)[O_{2}](r,t)}{\gamma(t)}, \tag{7}\]  
following Hypothesis 4 concerning the preservation of the spherical symmetry of the particle. Here \(\gamma\) is the stoichiometric factor of the reaction of carbon combustion, \(D_{O_{2}}^{\epsilon}\) the effective diffusivity of oxygen in the particle, \(k_{d}\) the oxygen mass transfer coefficient and \([O_{2}]_{\infty}\) the oxygen concentration at infinity, which is computed as: \([O_{2}]_{\infty}(t)=\big{(}P_{O_{2}}\big{)}_{\infty}/\big{(}R_{g}T_{g}(t)\big{)}\), with \(\big{(}P_{O_{2}}\big{)}_{\infty}=0.12\times 1.013\times 10^{5}\) (Pa).