Analytical expressions for the flame front speed in the downward combustion of thin solid fuels and comparison to experiments

Abstract

We derive analytical expressions for the propagation speed of downward combustion fronts of thin solid fuels with a background flow initially at rest. The classical combustion model for thin solid fuels that consists of five coupled reaction-convection-diffusion equations is here reduced into a single equation with the gas temperature as the single variable. For doing so we apply a two-zone combustion model that divides the system into a preheating region and a pyrolyzing region. The speed of the combustion front is obtained after matching the temperature and its derivative at the location that separates both regions. We also derive a simplified version of this analytical expression expected to be valid for a wide range of cases. Flame front velocities predicted by our analytical expressions agree well with experimental data found in the literature for a large variety of cases and substantially improve the results obtained from a previous well-known analytical expression.

Prediction of Three-Dimensional Downward Flame Spread Characteristics over Poly(methyl methacrylate) Slabs in Different Pressure Environments

“…In other words, gas velocity in the gaseous thermal region at preheated region is negligible. Thus, thermal conduction becomes the unique important heat transfer model in gas-phase [28], which may be roughly estimated as $q_s^{″}=k_g\frac{T_f-T_p}{l}$ where T f and T p are flame temperature and pyrolysis temperature, respectively; k g is the thermal conductivity; l is the normal distance from flame to preheated surface, which may be denoted by the thermal diffusion length $\overline{\mathrm{sans-serif\delta }}$ at the leading edge [8,15]. Based on the balance between convection and conduction terms in energy equation under naturally convective flow [29,30], the thermal diffusion length can be given by $\overline{\mathsf{\delta }}=\frac{\mathsf{\varphi }}{U_{ref}}$ where $\mathsf{\varphi }$ and $U_{ref}$ are the thermal diffusion coefficient and the reference velocity at the leading edge.…”

“…Based on the balance between convection and conduction terms in energy equation under naturally convective flow [29,30], the thermal diffusion length can be given by $\overline{\mathsf{\delta }}=\frac{\mathsf{\varphi }}{U_{ref}}$ where $\mathsf{\varphi }$ and $U_{ref}$ are the thermal diffusion coefficient and the reference velocity at the leading edge. $U_{ref}$ is induced flow velocity due to density variation near the flame which can be equated from conservation equations of buoyancy and inertia force [15,29,30] as $U_{ref}={false(\frac{gfalse({\mathsf{\rho }}_{\infty }-{\mathsf{\rho }}_{f}false)\mathrm{sans-serif\varphi }}{\mathsf{\rho }}false)}^{1/3}$ where ${\mathsf{\rho }}_{\infty }$ and ${\mathsf{\rho }}_f$ are the density of air at ambient temperature and flame temperature, respectively; $\mathsf{\varphi }$ and $\mathsf{\rho }$ in Equations (2) and (3) denote the quantities evaluated at an arithmetical average of ambient and adiabatic flame temperatures. Rearranging Equations (1)–(3), heat flux transferred to the preheated region is estimated as follows: $q_s^{″}=k_{g}false(T_f-T_{p}false){false[\frac{gfalse({\mathsf{\rho }}_{\infty }-{\mathsf{\rho }}_{f}false)}{\mathsf{\rho }}false]}^{1/3}{\mathsf{\varphi }}^{-2/3}$…”

“…Altenkirch et al [5] proposed a dimensionless flame spread rate for thermally thin paper sheets with Damkohler under different oxygen concentrations and pressures. Recently, Pujol and Comas et al [15,16] derived analytical expression of flame front spread by focusing on the gas-phase equations, which was much more simplified and accurate. Leventon et al [17] developed a comprehensive model to predict time to ignition and mass burning rate by coupling ThermaKin with empirical model of heat transfer.…”