Mathematical Modeling of Forest Fuel Ignition by the Focused Sunlight

Physical Model

The following scheme of the process under study was adopted. On the underlying surface there is a forest fuel layer, on a small area of ​​which the flux of solar radiation is focused. The processes occurring in the focusing element are not modeled due to the lack of both experimental data and the results of theoretical studies on this problem. A layer of forest fuel is heated and thermally decomposed with the formation of gaseous pyrolysis products. The composition of the gas mixture is taken in three components (fuel - carbon monoxide, oxidizing agent - oxygen, inert components). The pyrolysis products diffuse into the region of the gas mixture. At certain temperatures and concentrations of the reacting gases, the mixture ignites. The following ignition criteria are adopted: 1) the heat input from the chemical reaction exceeds the heat flux from the heated surface to the region of the gas mixture; 2) the temperature in the gas mixture reaches a critical value. The convective transfer of heat and matter is not taken into account, since during the ignition period, the thermal and diffusion relaxation lengths are several orders of magnitude longer than the convective (Vilyunov, 1984).

Designations: where T_i, ρ_i, c_i, λ_i are temperature, density, heat capacity, thermal conductivity (1 is the layer of forest fuel, 2 is the air); С_i, M_i are concentration and molar mass (4 is the oxygen, 5 is the combustible gas, 6 is the inert components of air); q_p is the thermal effect of the pyrolysis reaction of forest fuel; k₁ is the pre-exponent of the pyrolysis reaction of forest fuel; E₁ is the activation energy of the pyrolysis reaction of forest fuel; R is the universal gas constant; φ is the volume fraction of dry organic matter of forest fuel; q₅ is the thermal effect of the oxidation reaction of carbon monoxide; ν₅ is the fraction of heat absorbed by a layer of forest fuel; R₅ is the mass rate of the oxidation reaction of carbon monoxide; α₁ is the heat transfer coefficient; α₂ is the heat transfer coefficient; k₅ is the pre-exponent of the oxidation reaction of carbon monoxide; E₅ is the activation energy of the oxidation reaction of carbon monoxide; D is the diffusion coefficient, Y₅ is the mass flow of combustible pyrolysis products, x_i are auxiliary variable; q_s is the flux of focused solar radiation. x,y,z are spatial coordinate. t is the time coordinate. The indices es, ea, n correspond to environmental parameters in the soil, air, and at the initial time.

Initial data: ρ₁=500 kg/m³; ρ₂=0.1 kg/m³; c₁=1400 J/(kg×K); c₂=1200 J/(kg×K); λ₁=0.102 W/(m×K); λ₂=0.1 W/(m×K); q_p=1000 J/kg; k₁=3.63×10⁴; E₁/R=9400 K; φ_1н=1; q₅=10⁷ J/kg; k₅=3×10¹³ s^-1; E₅/R=11500 K; ν₅=0.3; α₁=20 W/(m²×К); α₂=80 W/(m²×К); D=10^-6; M₄=0.032; M₅=0.028; M₆=0.044.