Concurrent-flow upward flame spread over a vertical thin solid in pure buoyant flow is examined numerically. The computational model solves the two-dimensional, elliptic, steady and laminar conservation equations for mass, momentum, energy, and chemical species. Gas-phase combustion is modeled via a one-step, second order finite rate Arrhenius reaction. The simplified solid phase treatment assumes a zero order pyrolysis relation and includes radiative loss from the surface.
This work differs from previous treatments on upward flame spread (1) in that elliptic, rather than parabolic (boundary layer) equations are solved. The inclusion of the elliptic treatment and the finite rate chemical reaction enables us to study the entire flame profile and to examine the question of flame extinction. In addition, with the full momentum equations included, we are able to study and compare the differences between flame spreading in purely-buoyant and purely-forced flows.
The flame configuration is shown schematically in Fig 1. The coordinates are fixed with respect to the fuel burnout point (x=y=0) and steady solutions are sought. The mathematical formulation was given in (2) and will not be repeated here. The computation in (2) is restricted to purely forced flow. Mixed forced- and buoyant-flow cases were investigated in (3), but purely buoyant, concurrent-flow flame spread was not studied because of computational difficulty. In the present work, we complete the calculation with buoyant flow. Instead of a mixed elliptic-parabolic formulation used in (2,3) (elliptic equations in the flame stabilization zone near the solid burnout matched with a downstream parabolic solution), the entire region is solved with elliptic equations in this work. The entirely elliptic formulation consumes more computational time, but avoids the difficulty of solving boundary layer equations with a negative free stream velocity that occurs in a purely buoyant case where the spatial coordinates are attached to the flame.
Jiang, C.B., T'ien, J.S., Ferkul, P.V., Numerical Computation of Buoyant Upward Flame Spread and Extinction Over a Thin Solid in Reduced Gravity, Spring Technical Meeting of the Central States Session, The Combustion Institute.