## 1 Introduction
For the validation of their model, the authors compared their simulations with experimental carbon conversion values. In [13], the authors considered the combustion of a single coal char particle in a drop tube furnace under O\({}_{2}\)/H\({}_{2}\)O conditions. They considered two reactions for the char-O\({}_{2}\) combustion reaction, two for the char-H\({}_{2}\)O combustion reaction, three for the char-H\({}_{2}\)O gasification reaction and six further gasification or combustion reactions. They associated kinetic relationships to these reactions. They then wrote a total molar balance of gas mixture, a heat transfer equation and a mass transfer equation. They used the Maxwell-Stefan diffusion equation for the mass transfer in a multicomponent system (O\({}_{2}\), CO, CO\({}_{2}\), H\({}_{2}\)O and H\({}_{2}\)). They added initial and boundary conditions. For the resolution of their model, the authors used the finite volume method. For the validation of their model, they compared their simulations with experimental data concerning the global carbon conversion. More complex models are proposed in the literature to simulate the combustion of char particles falling in a drop tube furnace, which use Computational Fluid Dynamics (CFD) tools. The purpose here is to also simulate the trajectory of the particle during its fall in the drop tube furnace. Such models require an appropriate meshing of the reactor and dedicated software to solve the complex model, which usually involves different chemical reactions [1]. Numerical simulations of coal combustion performed in three furnaces were described in [14]. Mainly focusing on the devolatilization stage, the authors used the tabulated-devolatilization-process model developed in [9] and the FLASHCHAIN model developed for example in [15], to simulate this devolatilization stage. In [16], the authors simulated the combustion of spherical and non-deformable coal particles in a large-scale laboratory furnace. They distinguished between the devolatilization and char combustion stages. In [17], the Large Eddy Simulation tool was used to simulate the combustion of pulverized coal in a large-scale laboratory furnace. The authors also distinguished between the devolatilization and char combustion stages. They used an in-house code to solve the systems of equations. In [18], a flamelet model was proposed for coal combustion involving a quite similar tool. The oxy-fuel combustion of Victorian brown coal in a drop tube furnace and in a 3 MW pilot scale burner was simulated in [19] with CFD tools. The authors here presented a complete set of equations and they considered a particle random trajectory model. In the present study, a simple model is proposed to simulate the shrinking behavior of a spherical coal char particle burning during its fall in a drop tube furnace. The model thus focuses on the \"second\" stage of the combustion of a coal particle under high temperature ramps. It is based on two main hypotheses: the non-uniformity of structural quantities (particle density, porosity, mean pore radius and specific surface area) and the preservation of the spherical symmetry of the coal char particle during its combustion. The model mainly consists of two differential equations which respectively describe the evolution with respect to time of the particle temperature and that with respect to the particle radius of the oxygen concentration inside the particle. This model is solved using an in-house Fortran code, thus requiring no commercial or CFD software, or CFD skills. It remains much simpler than many available models especially that based on CFD tools. Instead of Fortran, every scientific software able to solve partial differential equations can be used. To validate the model, comparisons between the experimental and simulated temperatures of a coal char particle introduced in the drop tube furnace are first proposed at three constant regulation temperatures of the reactor (1100, 1200 and 1300 \({}^{\circ}\)C).