## 2 Previous Models
### Non-Local Radiation: Some 3D Effects  
In fact, the model in Section 2.4 was not implemented as-is, since another important improvement, the non-local radiation, was simultaneously incorporated. This is an essential improvement to reach the current version of the model, and it allows incorporating certain three-dimensional effects while maintaining the simplicity of a two-dimensional model. In a first approximation, the idea of non-local radiation was introduced by means of a convolution operator, simultaneously maintaining the local radiation represented with the nonlinear term of the diffusive term of the energy equation; see [30]. However, based on the results, it was decided to simplify the energy equation and choose only non-local radiation [32]. This was a challenge from a computational point of view as it involved solving the radiation intensity equation in a 3D domain, the layer of air on the surface in which the fire develops. The equations of the above models are defined over the surface \(S\) where the fire takes place, defined by the mapping:  
\[ \begin{array}{rcl}S\,:\,d&\longmapsto&\mathbb{R}^{3}\\ (x,y)&\longmapsto&(x,y,h(x,y)),\end{array}  \tag{15}\]  
where \(h(x,y)\) is a known function representing the topography of the surface \(S\). In order to take into account some 3D effects, specifically non-local radiation, we shall consider the following 3D domain, representing the air layer of thickness \(\delta\) on the surface \(S\):  
\[D{=}\{(x,y,z)\colon x,y\in d,h(x,y){<}z{<}h(x,y)+\delta\}.\]  
The non-dimensional simplified equations defined on the surface \(S\) are  
\[\frac{\partial}{\partial t}e+\mathbf{w}\cdot\
abla e+\alpha u=r, \tag{16}\]  
\[e\in G(u), \tag{17}\]  
\[\frac{\partial}{\partial t}c=-g(u)c, \tag{18}\]  
completed again with homogeneous Dirichlet boundary conditions, assuming that the surface \(S\) is large enough so that the fire does not reach the boundary during the simulation interval \(t_{max}\), the initial conditions representing the initial fuel load and fire source. The unknowns \(e=\frac{E}{MK_{\infty}}\), the dimensionless enthalpy, \(u=\frac{T-T_{\infty}}{T_{\infty}}\), the dimensionless temperature of the solid fuel, and \(c=\frac{M}{M_{0}}\), the mass fraction of solid fuel, are bidimensional variables defined in \(S\times(0,t_{max})\). Note that we abandoned the Frank-Kamenetskii change of variable since an Arrhenius-type expression no longer appears in the energy equation. The physical quantities \(E\), \(T\), and \(M\) are the enthalpy, the temperature of solid fuel, and the fuel load, respectively; \(C\) is the heat capacity of solid fuel; \(T_{\infty}\) is a reference temperature, namely the ambient temperature; \(M_{0}\) is the initial fuel load. The multivalued maximal monotone operator \(G\) is slightly different:  
\[G(u)=\left\{ \begin{array}{ll}u&\text{if}\qquad u<u_{v},\\ \left[u_{v}\,\ u_{v}+\lambda_{v}\right]&\text{if}\qquad u=u_{v},\\ u+\lambda_{v}&\text{if}\quad u_{v}<u<u_{p},\\ \left[u_{p}+\lambda_{v}\,\ \infty\right]&\text{if}\qquad u=u_{p},\end{array} \right. \tag{19}\]  
where, again, \(u_{v}\), \(u_{p}\), and \(\lambda_{v}\) are the same as above. We simplified the model assuming that the maximum value that the dimensionless temperature of the solid fuel \(u\) reaches is the dimensionless solid fuel pyrolysis temperature \(u_{p}\). We must specify that, in this model, only the solid phase of the combustion process is considered: the mass fraction of solid fuel \(c\) is a dimensionless variable between \(0\) and \(1\), and as just mentioned, the maximum value of the dimensionless solid fuel temperature \(u\) is the dimensionless pyrolysis temperature \(u_{p}\). Note that this is not the fire temperature, since the gaseous phase in this model is parameterized through the flame temperature \(T_{f}\) and the flame height \(F\) in the radiation term, which at the moment are considered input data dependent on the type of fuel. In Section 2.6, we explain how to improve this parameterization.