## 2 Mathematical Model  
### Physical Problem Statement  
The ignition of a sphere-shaped double gas hydrate particle was investigated using three heat supply schemes: convective, radiant, and conductive. When stating the problem for convective heating, an air flow with a temperature that varied in the range of 973-1273 K was assumed to be supplied from below to the hydrate sample (methane-propane, particle radius \(R_{\mathrm{g}}=0.005\) m), located in the geometric solution domain with the following coordinates (X-position = \(10R_{\mathrm{g}}\); Y-position = \(10R_{\mathrm{g}}\)). An increase in the hydrate temperature occurred as a result of the heat transfer from the outside air, whose temperature was much higher than the initial sample temperature. When the particle temperature exceeded the equilibrium temperature (at a pressure of 1 bar), the gas hydrate dissociated into gas and ice at a constant dissociation rate with the subsequent release and ignition of methane. Heating was conducted at a constant air flow rate \(U_{\mathrm{v}}\) of 1 m/s. The time interval from when the heating started until the ignition conditions were met was the ignition delay time \(\tau\). When stating the problem for radiant heating, it was assumed that a spherical gas hydrate particle is introduced into a high-temperature air medium (973-1273 K). A gas hydrate particle is heated (primarily due to radiant heat supply), and its surface temperature reaches the equilibrium temperature of dissociation. This is followed by the formation of an ice melting front with a boundary moving to the center (deep layers) of the hydrate particle. When the vaporization conditions are met, the water film evaporates. When the temperatures in the vicinity of the gas hydrate and the concentrations of the mixture (fuel, water vapors and inert gas) components rise to levels sufficient for ignition, the gas hydrate is ignited. Preliminary research has shown that depending on the ambient temperature, these conditions can be achieved before the ice melting or water film formation. When stating the problem for conductive heating, it was assumed that a spherical gas hydrate particle (\(R_{\mathrm{g}}=0.005\) m) with an initial temperature of 203 K was placed on a metal cylinder surface heated to high temperatures. The volume percent of fuel (methane) in the gas hydrate is known. The near-surface layer of the fuel is heated. At certain temperatures, dissociation begins, with the hydrate breaking up into ice and methane. With a temperature rise, the ice that is the gas hydrate component starts melting. On reaching the vaporization conditions, water evaporates from the gas hydrate surface. When dissociation intensifies, methane and propane enter the area filled with inert gas. A gas-vapor mixture with a fuel and inert gas emerges in the close vicinity of the gas hydrate particle. When reaching the temperatures and component concentrations sufficient for ignition, the mixture ignites. In accordance with the contemporary theory of condensed substance combustion, two simultaneously satisfied criteria used to estimate the ignition delay times were adopted: (1) the energy released as a result of a chemical reaction of the fuel oxidation (methane-propane) is greater than the heat transferred from the heating source and emerging gas-vapor mixture; and (2) the temperature of the mixture of gaseous components of the fuel in the oxidation reaction zone exceeds the initial temperature of the heating source [31]. Key assumptions: (1) the particle remained spherical with a constant size \(R_{\mathrm{g}}\) despite melting, evaporation, and combustion; (2) the dependence of the thermophysical characteristics on temperature was not taken into account; and (3) the mass exchange processes in the gas hydrate pores were not taken into account. These assumptions do not place any significant restrictions on the problem statement generality and are true for fast processes.