Wind-Aided Flame Spread Along a Horizontal Fuel Slab

Abstract

The American Society of Testing and Materials test E84 is widely employed in North America to characterize the rate of flame spread along samples of material proposed for use in construction of buildings; by this test, materials, in part, are qualified with respect to fire safety for classes of application. In this ten-minute-duration test, an 8 meter slab of the material comprises the ceiling of a sealed duct (termed a Steiner tunnel) through which hot vitiated air (up to about 1150 K) flows after time zero at about 1.2 m/s. Materials are rated according to the distance from the leading edge that the wind-aided flame propagates (or, alternatively for samples that become totally involved, by the elapsed time to flameover at the downwind end). An unsteady two-spatial-dimensional model of this test has been undertaken, as a first step toward the goal of anticipating behavior of a sample from knowledge of its chemical and physical properties. In the formulation, at any fixed position along the sample, as time increases, the sample is heated, by conduction and radiation, from its initial temperature to its pyrolysis temperature, but any further heat transfer from the gas phase to the sample (not conducted into the interior of the sample) contributes to gasification at that constant “pyrolysis temperature.” The gas-phase heat transferred to the sample is furnished by the initial vitiated-air enthalpy, supplemented by chemical exothermicity from a homogeneous diffusion flame at which fuel vapor pyrolyzed from the sample reacts with oxygen from the vitiated air stream. An isobaric Shvab-Zeldovich formulation of heat and species conservation [with treatment of the flame after Burke and Schumann; of convective transport, after Oseen, but with a nonlinear expression to account adequately for interphase mass transfer; and of diffusive transport, after Prandtl] proves amenable to an approach believed to be of general interest in Stefan problems with a split-type boundary condition. Split-type boundary condition here alludes to the fact that, at the two-phase interface, preheating boundary conditions (i.e., continuity of temperature and of heat flux) apply downwind of the pyrolysis front, but mass-transfer conditions (i.e., sublimation at a known, constant temperature) apply upwind of the pyrolysis front; the translation of the front with time must be found as part of the solution. The approach entails the use of the Fourier transform to integrate out the dependence on the coordinate transverse to the two-phase interface; with the parabolic nature of the full partial differential operator so removed, one may use method-of-characteristics solution in the interfacial plane for the Volterra integral equations derived from the hyperbolic suboperator. From knowledge of the dependence on time and stream-wise co-ordinate of the solution in the interfacial plane, the solution may be extended readily off the interfacial plane. The model presented here retains radiative transfer in simplistic form; in particular, the radiative cooling of the ambient hot air stream down the tunnel is included. Only by including such cooling may the model recover a result occasionally observed for a fire-retardant-treated foamed-plastic sample: visible flame propagates rapidly a finite distance along the sample, and then, abruptly, goes no further.