## 2 Mathematical Model
### Mathematical Model and Numerical Methods  
The solution domain for the heat transfer problem when using convective heating is presented in Figure 1a. A system of transient partial differential equations such as in [31] was used to describe the interrelated processes of heat transfer in the \"hydrate particle-external gas medium\" system under the conditions of chemical reactions, exothermic, and endothermic phase transformations at \(0<\tau<\tau_{\mathrm{d}}\). Figure 1: A scheme of the solution domain for different heat exchange schemes: (**a**) convective; (**b**) radiant; (**c**) conductive. Nomenclature: 1—gas medium; 2—hydrate particle. When solving the problem of modeling the process of gas hydrate ignition, the authors used the conjugation of the Cartesian and spherical systems. This approach is widely used in modeling. For a more detailed description of the basic pairing procedures, see [37]. For the mixture of water vapors, combustible gas, and air (\(0<x<H\), \(0<y<L\)):  
Continuity equation  
\[\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}= -\omega,\ \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \tag{1}\]  
Equation of motion  
\[ \begin{array}{l}\frac{\partial\omega}{\partial t}+u\frac{\partial\omega}{ \partial x}+v\frac{\partial\omega}{\partial y}=\gamma(\frac{\partial^{2} \omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}}),\ u=\frac{ \partial\psi}{\partial y},\ v=-\frac{\partial\psi}{\partial x},\\ \omega{=}\ \mathrm{rot}_{z}\overset{\rightarrow}{v}=\frac{\partial v}{ \partial x}-\frac{\partial u}{\partial y}\end{array}  \tag{2}\]  
Energy equation  
\[(\frac{\partial T_{1}}{\partial t}+u\frac{\partial T_{1}}{\partial x}+v\frac{ \partial T_{1}}{\partial y})=a_{1}(\frac{\partial^{2}T_{1}}{\partial x^{2}}+ \frac{\partial^{2}T_{1}}{\partial y^{2}})+\sum_{i=1}^{n}Q_{i}W_{i},\ W_{i}=A_ {ri}T_{1}{}^{\beta_{ri}}e^{-E_{ni}/R_{i}T_{1}}; \tag{3}\]  
Hydrate gas (\(\mathrm{C_{g}}\)) and water vapor (\(\mathrm{C_{w}}\)) diffusion equation  
\[\rho_{1}(\frac{\partial\mathrm{C_{g}}}{\partial t}+u\frac{\partial\mathrm{C_{ g}}}{\partial x}+v\frac{\partial\mathrm{C_{g}}}{\partial y})=\rho_{1}D_{1}( \frac{\partial^{2}\mathrm{C_{g}}}{\partial x^{2}}+\frac{\partial^{2}\mathrm{C _{g}}}{\partial y^{2}})-W_{i} \tag{4}\]  
\[\rho_{1}(\frac{\partial\mathrm{C_{w}}}{\partial t}+u\frac{\partial\mathrm{C_{ w}}}{\partial x}+v\frac{\partial\mathrm{C_{w}}}{\partial y})=\rho_{1}D_{1}( \frac{\partial^{2}\mathrm{C_{w}}}{\partial x^{2}}+\frac{\partial^{2}\mathrm{C _{w}}}{\partial y^{2}}) \tag{5}\]  
\[C_{\mathrm{w}}+C_{\mathrm{g}}+C_{\mathrm{o}}=1 \tag{6}\]  
Energy equation for hydrate particles (\(0<r<\mathrm{\textit{R}}_{g}\), \(0<\varphi<2\pi\)):  
\[\frac{\partial T_{2}}{\partial t}=a_{2}\left[\frac{\partial^{2}T_{2}}{ \partial x^{2}}+\frac{2}{r}\frac{\partial T_{2}}{\partial r}\right] \tag{7}\]  
The volume percentage of the gas-vapor mixture components was calculated from their mass concentrations using the equations:  
\[ \begin{array}{l}\eta_{11}=\frac{\mathrm{C_{g}}/\rho_{11}}{\mathrm{C_{g}}/ \rho_{11}+\mathrm{C_{w}}/\rho_{12}+\mathrm{C_{o}}/\rho_{13}},\ \eta_{12}=\frac{\mathrm{C_{w}}/\rho_{12}}{\mathrm{C_{g}}/\rho_{11}+\mathrm{C_{ w}}/\rho_{12}+\mathrm{C_{o}}/\rho_{13}},\\ \eta_{11}+\eta_{12}+\eta_{13}=1.\end{array}  \tag{8}\]  
The thermophysical characteristics of the gas-vapor mixture as a heterogeneous system were given by  
\[ \begin{array}{l}\lambda_{1}=\lambda_{11}\cdot\eta_{11}+\lambda_{12}\cdot \eta_{12}+\lambda_{13}\cdot\eta_{13},\ C_{1}=C_{11}\cdot\eta_{11}+C_{12}\cdot \eta_{12}+C_{13}\cdot\eta_{13},\\ \rho_{1}=\rho_{11}\cdot\eta_{11}+\rho_{12}\cdot\eta_{12}+\rho_{13}\cdot\eta_{1 3}\end{array}  \tag{9}\]  
Initial conditions (\(t=0\)): \(T=T_{\mathrm{g0}}\), \(C_{\mathrm{o}}=1\), \(C_{\mathrm{g}}=0\), \(C_{\mathrm{w}}=0\), \(\psi=0\), \(\omega=0\) at \(0<x<H\), \(0<y<L\); \(T=T_{\mathrm{h0}}\) at \(0<r<\mathrm{\textit{R}}_{g}\), \(0<\varphi<2\pi\).