## 1 Introduction  
Modeling the combustion process of coal particles and of coal char particles in furnaces or boilers is still a complicated task, although such a combustion process was already analyzed and simulated by many authors for a long time and through different models or tools. The combustion of coal particles may be decomposed in two main stages, which can superimpose: a devolatilization one, leading to coal char particles, and a combustion one. A complete review of the different models for coal pyrolysis, combustion and gasification was presented in [1], together with the corresponding mechanisms and structure modifications. A Thiele analysis and a random pore model were combined to describe the combustion of coke particles in an atmospheric fluidized bed combustor [2]. A model was proposed by [PERSON] to simulate the physical changes occurring during the combustion of char particles [3]. Local values of the parameters were considered here, which allow the local use of the random pore model for the description of the evolution of the specific surface area. The devolatilization of a single coal particle and the combustion of the residual char were modeled in [4]. For the devolatilization stage, the authors assumed that the activation energy obeys a Gaussian distribution and they wrote an energy balance which describes the evolution of the temperature inside the spherical particle (with fixed radius), with respect to time. For the further char combustion stage, the authors used the pore evolution model introduced in [5]. They wrote conservation laws for the gas species, a total molar balance of gas mixture, an energy balance and a total mass balance of carbon in the spherical char particle. In the gas boundary layer, they wrote conservation equations. For the resolution of this large system of evolution equations, the authors built an integrated Fortran code. The combustion of a gasified semi-char in a drop tube furnace was analyzed in [6], through a pore development process and the use of the Thiele modulus method, based on the following expression of the effectiveness factor:  
\[\eta=\left(\coth(3\Phi)-\frac{1}{3\Phi}\right)\frac{1}{\Phi}, \tag{1}\]  
where \(\Phi\) is the Thiele modulus, and on a simple expression of the oxygen concentration at the particle surface. Here, the authors assumed a constant particle temperature. A random pore model with intraparticle diffusion was introduced in [7], to describe the combustion of char particles. A coal combustion model was presented in [8], which was based on a kinetic model, the chemical reaction rate being taken as a function of the coal intrinsic reactivity and fuel mass for a global reaction of order \(n\). Different models of coal combustion were compared in [9]. A reduced and one-dimensional model was proposed in [10], to simulate the combustion of a spherical char particle. This model was based on a constant particle radius and uniform values of the different structural parameters within the particle and at any time of the combustion process, allowing the use of the Thiele modulus method, through Equation (1). The authors evaluated the impact of ash on the combustion behavior. More recently, the characteristics of char burnout of a biomass/coal blend were analyzed through a simplified single particle reaction model [11]. The authors here defined an intrinsic reactivity coefficient as a function of the char conversion rate and of the effectiveness factor which involves the Thiele modulus, according to Equation (1). In [12], the authors analyzed and simulated the combustion of a coal char particle in a drop tube furnace under different mixtures of O\({}_{2}\), CO\({}_{2}\), Ar and N\({}_{2}\). Their model was based on a heat transfer equation and two heterogeneous reactions. They computed the carbon consumption rate. For the effectiveness factor, the authors used Equation (1). The spherical particle was allowed to shrink keeping its spherical symmetry.