3.2.15. Fuel Boundary Condition Submodel

In most cases, fires are the result of burning fuel vapor in air. Exceptions include oxygenated and energetic materials that embody both fuel and oxidizer. The source of fuel vapor may be a gas release, the vapor which forms over a liquid surface due to its vapor pressure, liquid fuel which is heated above its vaporization temperature, or solid materials which are heated to the point where combustible gases are released due to pyrolysis reactions. The purpose of this submodel is to provide the mass flux and temperature of fuel vapor which enters the computational domain at the boundaries. This submodel is only required if the source of fuel is a solid or liquid since gas releases can be specified as a flow boundary condition. Since the generation of fuel vapor from these materials involves, as a minimum, representing thermal transport within the material including phase change, a simplified approach is taken here to serve the basic need of present generation fire models. The development of improved, validated models is presently underway. Present generation models are limited to liquid fuels in the form of pools (i.e., a defined amount of fuel constrained in a pool with fixed, known geometry) and spills onto non-absorbing substrates. (See Martinez and Hopkins [71] for a model of fuel spill in a porous medium.) Although the form of the submodel will allow first order estimates of fire growth rates, data acquired to date (Saito et al. [72]) tend to show that relevant flame spread mechanisms include features which occur at lengths scales several orders of magnitude below the resolution of present grids. Additional submodels will be therefore be required to predict flame spread with confidence. The following quantities are required:

$T_{fuel,vap}$ , the vaporization temperature of the fuel (K).

$h_{fg}$ , the heat of vaporization of the fuel (KJ/kg),

$C_{p_l}$ , the specific heat of the liquid fuel (KJ/kg-K),

$T_{fuel,init}$ , the initial temperature of the liquid fuel (K),

$\alpha_{fuel,liq}$ , the absorptivity of the liquid fuel,

$q_{rad}''$ , the radiative heat flux incident on the fuel surface,

$q_{conv}''$ , the convective heat flux incident on the fuel surface.

The fuel boundary condition submodel generates the following output:

$\dot{m}''$ , the mass flux of fuel (kg/ ${\rm m}^2$ -s).

The fuel pool will be modeled as a mass of liquid that is gradually converted to vapor which in turn enters the flow field as a distinct species. The fuel vapor generation rate is based on the incident heat flux to the pool surface. Data for heavy hydrocarbon fuels (Gritzo, et al. [73, 74]) show the following:

After the initial transient (which includes flame spread) the fuel burning (and hence vaporization) rate is steady.

Heating of the fuel is limited to the top 1.5 cm (which greatly exceeds the penetration depth for combined thermal transport in semitransparent media).

Fuel transport occurs within the pool due to the preservation of a fuel free surface and the presence of a non-uniform heat flux to the fuel surface.

The temperature at the free surface of the fuel is spatially uniform and approximately equal to the mean of the distillation curve for multi-component fuels.

Given these observations, the present submodel includes two options for calculating the fuel vaporization. These options are used for both pool and spill fires.

3.2.15.1. Option 1: Constant, Specified Mass Flux

In this option, the output of the submodel will be specified directly by the user. Fuel will be released at the boundaries defined by a fuel free surface. Since the burning rate is constant, the mass flux can be considered constant. Fuel burn rate data (for example, Blinov and Khudiakov [75]) are available as a function of pool size for a variety of fuels. This option neglects the physical process of fuel heating and is therefore only appropriate for steady burning fires. The spatial variation of fuel vaporization is also neglected.

3.2.15.2. Option 2: Mass Flux as a Function of Incident Heat Flux

Before to the surface of the fuel reaches its vaporization temperature, the pool model splits the incoming flux ( $q''$ ) between the heating of the fuel and vaporization of the fuel. The fraction of heat transferred into the material ( $f_s$ ) is determined using a linearized approximation for the temperature distribution in the media by (using a user-specified upper bound of $p_s$ )

(3.498) $f_s = \min \left( p_s, \frac{T_b - T_f}{\Delta T_{bf}} \right)h$

This leads to an evaporation rate of

(3.499) $\dot{m}'' = \frac{q''(1 - f_s)}{h_{fg}}.$

and a rate of change in the pool temperature ( $T_f$ ) and height ( $H$ ) of

(3.500) $\frac{d T_f}{d t} = \frac{q'' f_s}{H \rho_f C_{p,f}},$

(3.501) $\frac{d H}{d t} = -\frac{\dot{m}''}{\rho_f}.$

Due to low diffusivity and high opacity of hydrocarbon fuels, the temperature gradient in the liquid fuel develops quickly, is considerably larger than the linear approximation, and does not extend to the lower surface of the fuel, so the fuel temperature variation with respect to height is neglected. The transient fuel heating occurs at the same short time and length scales as flame spread. The inclusion of this feature is not suggested until a more rigorous technique for modeling flame spread can be developed.