# A Physical Model and Improved Experimental Data Correlation for Wind Induced Flame Drag in Pool Fires

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## Abstract

The flames over a burning liquid fuel are observed to spill over the downwind edge of the liquid pool in a wind. Empirical correlations in the literature relate the total base dimension of the fire (diameter + the spill over) with the wind Froude number. This leads to erroneous and physically incorrect (negative) value for the flame spillover at low wind speeds and/or in large diameter fires. The data from laboratory scale (0.1–0.6 m) to field scale (up to 35 m) fires of different hydrocarbon fuels on the wind induced flame “drag” or “spillover” were re-examined. The ratio of the flame spillover distance with the pool diameter is seen to vary in direct proportion to the square root of the Froude number but with different proportionality constants for different fuels. A physical model was developed to analyze the phenomena that occur at the base of a pool fire subject to a wind. This model indicates that the non-dimensional downwind flame spillover distance is directly proportional to the square root of the Froude number, inversely proportional to the square root of the dimensionless heat of combustion of the fuel and directly proportional to the 1/4th power of the ratio of vapor density to air density. Available experimental data are synthesized into a single correlation when plotted on the basis of the non-dimensional parameters from the model. This correlation includes the Froude number, the Damkohler number (dimensionless heat of combustion of the fuel), the wind flow Reynolds number and the fuel vapor-to-air density ratio.

## Keywords

flame spillover flame drag extent froude number Reynolds number entrainment pool fire## Nomenclature

- C
Specific heat at constant pressure of gases (J/kg K)

- D
Pool (or dike) equivalent diameter (m)

- ΔD
Ground-level extension of the fire beyond the downwind edge of the dike (m)

*Fr*_{0}Evaporation Froude number = \( {\frac{{u_{0}^{2}}}{{g\,R_{0}}}} \)

*Fr*_{W}Wind Froude number = \( {\frac{{U_{W}^{2}}}{g\,D}} \)

- \( Fr_{10} = {\frac{{U_{W,10}^{2}}}{g\;D}} \)
Froude number based on wind speed measured 10 m above ground

- g
Acceleration due to gravity (m/s

^{2})- ΔH
_{c} Heat of combustion (lower value) of the fuel in air (J/kg)

- \( \dot{m}(z) \)
Vertical mass flow rate at a plume cross section at height z (kg/s)

- \( \dot{m}_{a}^{\prime} (z) \)
Mass rate of entrainment of air per unit axial length at height z (kg/m s)

- \( \dot{m}_{a} (z) \)
Mass rate of entrainment of air into the fire plume up to height z (kg/s)

- \( \dot{M}(z) \)
Momentum flux at any cross section at height z (kg m/s)

- \( \tilde{M} \)
Non-dimensional momentum flux = \( {\frac{{\dot{M}}}{{\dot{M}_{0}}}} \)

- P
Atmospheric pressure (absolute value) (Pa)

- r
Air to fuel mass ratio for stoichiometric combustion of the fuel (kg/kg)

- R
Fire plume radius at any height z (m)

- R
_{0} Radius of the fire base = D/2 (m)

- \( \text{Re}_{D} \)
Reynolds number = \( {\frac{{U_{w} D}}{{\nu_{a}}}} \)

- \( \Re_{u} \)
Universal gas constant (J/mole K)

- t
_{ch} Characteristic time scale = R

_{0}/u_{0}(s)- T
Temperature of the gas (K)

- u
Cross section averaged upward velocity of the gases at any height z (m/s)

- \( \tilde{u} \)
Non-dimensional upward velocity = u/u

_{0}- U
_{w} Wind speed (m/s)

- U
_{w,10} Wind speed at 10 m above ground (m/s)

- \( \dot{V} \)
Volumetric flux of gas flow at any section (at height z) (m

^{3}/s)- \( \tilde{V} \)
Non-dimensional volume flux = \( {\frac{{\dot{V}}}{{\dot{V}_{0}}}} \)

- Z
Height above the fire base (m)

## Greek Letters

- α
Turbulent entrainment coefficient for air entrainment

- β
Fraction of the air mass entrained that burns with its stoichiometric mass of fuel

- η
Non-dimensional radius of the fire = R/R

_{0}- μ
Molecular weight of the gas (kg/kmole)

- \( \nu_{a} \)
Kinematic viscosity of air (m

^{2}/s)- ρ
Density of the gases in the fire plume or outside in air (kg/m

^{3})- σ
Non-dimensional gas density (with respect to base vapor) = \( {\frac{\rho}{{\rho_{0}}}} \)

- σ
_{a} Non-dimensional air density (with respect to base vapor) = \( {\frac{{\rho_{a}}}{{\rho_{0}}}} \)

- τ
Non-dimensional time = t/t

_{ch}- ζ
Non-dimensional height through the fire plume = Z/R

_{0}- \( \omega \)
Inverse volumetric expansion ratio of gases due to combustion = \( {\frac{1}{{\left({1 + {\frac{{\beta \,\Updelta H_{c}}}{{rC_{a} T_{a}}}}} \right)}}} \)

## Subscripts

- a
Air

- ch
Pertaining to characteristic parameters

- cr
Critical condition

- v
Vapor property close to the (boiling) liquid surface

- 0
Initial condition (at the fire base)

- 10
Pertaining to conditions at a height of 10 m above ground

## Acronyms

- LHS
Left hand side (of an equation)

- LNG
Liquefied natural gas

- NFDE
Non-dimensional flame drag extent

- RHS
Right hand side (of an equation)

## References

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