# The Simulation of Different Combustion Stages of Micron-Sized Aluminum Particles
## 2 Model Formation
### Surface Formalism  
The mathematical formalism to describe surface kinetics for events such as adsorption, desorption, surface reactions and deposition is developed in this section. There are three types of species in surface formalism: gas-phase, surface and bulk. The first is a species in the gas phase above the particle surface. A surface species denoted by (S), is defined to be the chemical species on the top-most layer of the particle which means the liquid-gas interface. For example, the evaporation and absorption process (Al + Al(L) = Al(B) + Al(L)) of aluminum are shown in Figure 3. The symbol Al represents the gas atom of Al; Al(L) represents the open site at the surface, which will be occupied by gas species to form surface species; Al(B) represents the liquid aluminum particle. During the combustion of the aluminum, the two processes are going on at the same time, and when equilibrium is reached, the evaporation rate will be much greater than the absorption rate.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Species**} & \multirow{3}{*}{\begin{tabular}{c} **Molecular** \\ **Geometry** \\ \end{tabular} } & \begin{tabular}{c} **Lennard-Jones** \\ **Potential Well** \\ **Depth,** \\ \(\varepsilon_{k}/k_{p}\) **(K)** \\ \end{tabular} & \begin{tabular}{c} **Lennard-Jones** \\ **Collision** \\ **Diameter,** \\ \(\epsilon\left(\hat{\mathbf{A}}\right)\) \\ \end{tabular} & \begin{tabular}{c} **Dipole** \\ **Moment,** \(\mu\) \\ **(Debye)** \\ \end{tabular} & \
\begin{tabular}{c} **Polarizability,** \\ **\(\alpha\left(\hat{\mathbf{A}}^{3}\right)\) \\ \end{tabular} &
\begin{tabular}{c} **Rotational** \\ **Relaxation** \\ **Collision Number** \\ **Z\({}_{rot}\) at 298 K** \\ \end{tabular} \\ \hline Al & 0 & 2836 & 2.655 & 0 & 8.34 & 1 \\ AlO & 1 & 1381.2 & 3.203 & 4.32 & 6.7 & 1 \\ AlO\({}_{2}\) & 1 & 984 & 3.564 & 0 & 9.33 & 1 \\ Al\({}_{2}\  
The detailed surface reaction mechanism employed to model the aluminum oxidation is taken from [22] and is shown in Table 3. The surface reactions can be represented in the general form:  
\[\sum_{k=1}^{K_{g}+K_{s}}\ \ v^{\prime}{}_{ki}A_{k}\Leftrightarrow\sum_{k=1}^{K_{g}+K _{s}}\ v^{\prime}{}_{ki}A_{k}\ \ \ \ \ (i=1,\ldots,N_{s}) \tag{13}\]  
In any surface reaction, gas-phase species can participate in surface chemistry as well. The rate-of-progress variable for the \(i^{th}\) surface reaction can also be written in the similar form as gas-phase reaction:  
\[q_{i}=k_{f,j}\prod_{k=1}^{K_{g}+K_{s}}\ \big{[}X_{k}\big{]}^{\
u^{\prime}{}_{ki} }-k_{r,i}\prod_{k=1}^{K_{g}+K_{s}}\ \big{[}X_{k}\big{]}^{\
u^{\prime}{}_{ki}} \tag{14}\]  
The only difference is that \([X_{k}]\) can represent either a volume (mol/m\({}^{3}\)) or a surface molar concentration (mol/m\({}^{2}\)). Surface molar concentration is calculated as:  
\[[X_{k}]=Z_{k}\Gamma/\sigma_{k} \tag{15}\]  
\(\Gamma\) and \(\sigma_{k}\) is obtained from quantum chemistry [30; 31]. For bulk species, \([X_{k}]\) is taken as 1. For absorption reactions, their forward rate \(k_{f,j}\) is calculated by stick coefficient r in following equation:  
\[k_{f,i}=\frac{\gamma_{i}}{1-\gamma_{i}/2}\frac{\prod_{k=1}^{K_{s}}\sigma_{k} ^{\
u_{ki}}}{\Gamma^{m}}\sqrt{\frac{RT}{2\pi W_{k}}} \tag{16}\]