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.