## 2 Materials and Methods  
The methods followed along this research were based on a hybrid assessment, analytical and numerical, together with a validation exercise with real testing data. The modelling set of equations were obtained based on the mass conservation principles. The nitrogen gas was supplied by a manifold with several nozzles located along the fuel tank (see [14] for a complete description). The gases were dynamic together with the nitrogen nozzle location and the particular tank geometries led to an heterogeneous gas distribution along the tank. This heterogeneous condition has been modelled considering the stratified two-phase flows in [15], making use of the volume of fluid (VOF) model (see [16; 17] for a detailed description) by solving the phase continuity equation. In the modelling exercise developed in this paper, it was considered that the heterogeneous nitrogen discharge in complex tank geometries provoked gas dispersion. As a consequence, a diffusion process emerges in response to the differences in the gas concentration along the tank zones. The selection of an appropriate diffusion principle is of relevance and leads to a whole significant discussion (see [18] and the references therein). For our purposes, the considered set of equations is non-linear in general and aims to characterize the propagation of the interacting gases. Such interactions are of significance in tank zones not influenced by the advection, so that the nitrogen propagates by diffusion. This progressive propagation in a diffusive front has been used for modelling purposes in other fields to simulate porosity effects [19; 20]. In our case, the propagation dynamic was conceived as an inherent feature introduced by the travelling waves solutions. In addition to the diffusion, the rate of nitrogen discharge along the domain induces a forced convection within the tank airspace. To obtain the modelling equations, we consider the following dynamics between the nitrogen concentration (\(N\)) and the oxygen concentration (\(\Theta\)): initially, the fuel tank usage is not saturated with nitrogen, so it is considered that the nitrogen time rate (\(N_{t}\)) is high. As long as the nitrogen concentration increases, the oxygen is expelled and the nitrogen time rate decreases. Then:  
\[N_{t}=\Theta^{n}, \tag{2}\]  
where \(n\in(0,1)\) shall be obtained for each particular case. In the same manner, the oxygen concentration decreases for increasing nitrogen quantities:  
\[\Theta_{t}=-N^{m}. \tag{3}\]  
The expressions (2) and (3) will be referred to as the reaction (\(\Theta^{n}\)) and absorption (\(-N^{m}\)) in the modelling set of equations  
Note that both the nitrogen and oxygen are functions of space and time \(N(x,t)\) and \(\Theta(x,t)\) for \(0<t<\infty\), \(x\in\mathds{R}^{3}\) (\(d=3\), in this case). Consider a virtual domain within the fuel tank \(\Sigma\in\mathds{R}^{3}\). The mass conservation principle establishes that the time rate of nitrogen and oxygen concentrations (\(N_{t},\Theta_{t}\)) shall be equal to the concentration flux through the boundaries of \(\Sigma\) (\(\partial\Sigma\)) plus the absorption or reaction that act as independent terms. We admit that the vectors \(\Psi_{N}\in\mathds{R}^{3}\) and \(\Psi_{\Theta}\in\mathds{R}^{3}\) are the concentration fluxes along the boundary (\(\partial\Sigma\)) per unit of area and per unit of time through the borders of the subdomain \(\Sigma\). Consider the vector \(\pi\in\mathds{R}^{3}\) as the normal local at any point of \(\partial\Sigma\). We admit that the absorption and reaction terms per unit of time and volume are given by the generic functions \(G_{N}(N,\Theta,x,t)\) for the nitrogen and \(G_{\Theta}(N,\Theta,x,t)\) for the oxygen. Each \(\Psi_{N}\), \(\Psi_{\Theta}\), \(G_{N}\), \(G_{\Theta}\) is continuous with continuous derivatives.