Modeling fire plumes

Abstract

Theoretical treatments for turbulent diffusion flames and for the strongly heated regions of fire plumes in a still environment may be based on those developed for weakly buoyant plumes, but appropriate modifications must be made to allow for the high temperatures and the large variations in density involved. A discussion is given of some of the modifications that are needed, and the effects of large variations in density on the plume dynamics and aspects of heat transfer by radiation are presented separately. The entrainment into strongly buoyant plumes depends on the local ratio ρ/ρ0 of mean plume to ambient densities as well as on the mean plume velocity u , and dimensional arguments can do no more than define a local entrainment function (ρ/ρ0) n E 0 , where n is undetermined and E 0 is the well-established entrainment constant for weakly buoyant plumes. The value n =1/2 is suggested by the dependence of lateral diffusion on the Reynolds stresses, and the form E =(ρ/ρ0) 1/2 E 0 is adopted here; comparable assumptions have been made previously. Equations for the conservation of mean mass flux, mean momentum flux, and mean heat flux along the turbulent diffusion column of the plume are obtained making full allowance for the dynamic effects of large variations in density. Provided that the level of intensity of the plume turbulence is not too high, these equations can be reduced approximately to a form directly related to the set of equations used previously in the study of weakly buoyant forced plumes. These sets of equations are related by the transformation ρ 1/2 a =ρ0 1/2 b , where a and b are local length scales (essentially plume radii) for the strongly buoyant and weakly buoyant plumes, respectively. Hence, the behavior of strongly buoyant plumes can be described in terms of existing solutions for weakly buoyant plumes, and in general it can be seen that strong plumes spread less rapidly than weak ones at first, although they are soon reduced to weakly buoyant behavior unless the large temperature differences are maintained, as by combustion. Fire plumes are often rich in smoke and soot from imperfect combustion, and in such cases when the mean free path for radiation is small in relation to the plume diameter the opaque radiation approximation may be adopted. In this case, the heat transfer by radiation can be divided into a vertical flux along the column of a diffusive character, and the outwards radiation from the edges of the plume through transparent air to the distant environment. It is shown that in many practical cases the vertical flux by radiation is small in relation to vertical convection and to edge radiation, and may be neglected for small-values of the parameter 8πE 0 2 /3kb a , where k is the absorption coefficient within the plume and b 8 the source radius. Indeed, when flow velocities are high, even the cooling effect of radiation from the plume edges may be small relative to cooling by turbulent entrainment of cold ambient air. A solution of this type is presented. The purpose of this paper is to discuss questions involved in the formulation of fire plume theories; therefore, few solutions are given in detail and no attempt is made to link the dynamical effects with those due to radiation. A wider range of solutions will be published elsewhere. Theoretical treatments for plumes of very hot gas in a uniform still environment and for turbulent diffusion flames may be based on an extension of the treatments for weakly buoyant plumes, provided that due allowance is made for the large variations in density and for the effects of radiation. This paper gives a discussion of the modifications needed for a theory of very hot plumes, and the dynamic and radiative effects are treated here separately. In strongly buoyant plumes, a modified entrainment function must be used, and variations of mass and inertia per unit volume with varying temperature must be taken into account. The full equations for strongly buoyant plumes may be transformed approximately into a set of equations already used to study weakly buoyant forced plumes, and this reduction shows that very hot plumes have a reduced rate of spread at first. In an opaque plume, where the flow is turbulent and the length scale of the energy-containing eddies is much larger than the mean free path for radiation, heat transfer by radiation can be separeted into a vertical radiative diffusion and radiation from the plume boundaries to the distant environment. The vertical transfer is often much less than that from the boundaries and may be neglected except in the immediate neighborhood of the source. In cases of extreme cooling by radiation, longitudinal diffusion may no longer be neglected.