## 2 Materials and Methods
### Numerical Procedure
#### 2.2.2 Governing Equation
\[\dot{m_{\text{vl}}}=coeff\cdot\alpha_{\text{v}}\rho_{\text{v}}\frac{(T_{\text{ sat}}-T_{\text{v}})}{T_{\text{sat}}}\] (6)  
where \(\alpha_{\text{v}}\) is the volume fraction of the phase; \(\rho\) is the density of the phase, kg/m\({}^{3}\); \(T_{\text{l}}\) is the temperature of the liquid phase, K; \(T_{\text{v}}\) is the temperature of the gas phase, K; \(T_{\text{sat}}\) is the saturation temperature, K; and \(coeff\) represents a coefficient, which equals to reciprocal relaxation time. The numerical process of the energy source term \(S_{\text{h}}\) is shown in Equation (7). \[S_{\text{h}}=-\dot{m}\cdot L_{\text{v}} \tag{7}\]  
where \(m\) is the quality, kg; \(L_{\text{v}}\) is the latent heat of evaporation, kJ/g. Equations (8)-(10) are used to calculate the evaporative latent heat of leaking marine oil in the fishing vessel's engine room. \[L_{\text{v}}=h_{\text{q}}^{\text{s}}-\ h_{\text{p}}^{\text{s}} \tag{8}\]  
\[h_{\text{p}}^{\text{s}} =h_{\text{p}}^{\text{f}}+\int_{T_{\text{ref}}}^{T_{\text{sat}}} cp_{\text{p}}\text{d}T\] (9) \[h_{\text{q}}^{\text{s}} =h_{\text{q}}^{\text{f}}  
where \(Y_{\rm ig}\) is the mass fraction of a passive species representing radicals which form when marine oil in the domain breaks down; \(S_{\rm ct}\) is the turbulent Schmidt number; and \(S_{\rm ig}\) is the source term for the ignition species.  
The abovementioned source term for the ignition species can be developed as below:  
\[S_{\rm ig}=\int_{t=t_{0}}^{t}\frac{1}{T_{\rm ig}}\rm dt \tag{12}\]  
where \(t_{0}\) is the time at which marine oil is introduced into computational domain, s; \(T_{\rm ig}\) is the ignition delay with the units of time, s.  
When modeling the ignition delay of the leaked marine oil on the heated surface, the chemical reactions were allowed to take place when the ignition species reached a value of 1 in the domain. For the ignition delay model, the relevant equations were built into FLUENT, which reproduces Arrhenius correlations. If the ignition species is less than 1 when using the ignition delay model, the chemical source term is suppressed by not activating the combustion model at that particular time step. It means that the energy release will be delayed. This approach does not apply to solving for typically low-temperature chemistry, but it is reasonable for a high-temperature chemical model in current research. The equations related to the ignition delay model in this work can be shown by Equation (13). \[T_{\rm delay}=\left(\frac{C_{1}+0.22\overline{S_{\rm p}}}{6N}\right)\exp\left[ E_{\rm a}\left(\frac{1}{RT}-\frac{1}{17,190}\right)+\left(\frac{21.2}{p-12.4} \right)^{\rm ep}\right] \tag{13}\]  
where \(T_{\rm delay}\) is the ignition delay time, s; \(C_{1}\) is a pre-exponential constant number with a value of 0.36; \(N\) is the operation speed of a device that forms a heated surface, m/s; \(E_{\rm a}\) is the effective activation energy, kJ/mol; and \(e_{\rm p}\) is the pressure exponent, Pa. The equation for the effective activation energy \(E_{\rm a}\) is expressed by Equation (14). \[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{i}}(u_{i}(\rho E +p))=\frac{\partial}{\partial x_{i}}\left(k_{\text{eff}}\frac{\partial T}{ \partial x_{i}}-\sum_{j}h_{j}J_{j}+\ u_{j}\big{(}\tau_{ij}\big{)}_{\text{eff} }\right)+\ S_{\text{h}} \tag{3}\]0  
where \(E_{\rm h}\) is the activation energy, kJ/mol; \(CN\) is the cetane number.