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Semi-empirical Model for Fire Spread in Shrubs with Spatially-Defined Fuel Elements and Flames

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Abstract

A semi-empirical model was developed which forms shrub geometries from distinct fuel elements (e.g. leaves) and describes flame spread from element to element. Ignition, flame growth and flame decay patterns were based on combustion data of single leaves. Extension of the model to various heating conditions was achieved by scaling the flame growth parameters using physics-based heat transfer models. The resulting model offers a novel approach to examine fire spread and to explicitly describe both distinct fuel elements and fire behavior. This approach balances computational speed and modeling detail while providing a unique perspective into fire spread phenomena. Comparisons of the tuned model to fire spread behavior measured in an open-roofed wind tunnel benchmarked the model’s ability to simulate fire spread in manzanita shrubs.

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Abbreviations

A :

Pre-exponential factor

\( A_{{{\text{c}},{\text{local}}}} \) :

Cross-sectional area normal to the flame axis affected by heat release

\( A_{c} \) :

Cross-sectional area

\( A_{face} \) :

Two-sided face area of a leaf

B :

Experimentally determined flame merging constant

\( c_{{w_{1} }} \) :

Concentration of water at location 1

\( c_{{w_{2} }} \) :

Concentration of water at location 2

\( c_{p,dry} \) :

Specific heat of dry leaf mass

\( c_{p,gas} \) :

Specific heat of passing gases

\( c_{p,water} \) :

Specific heat of water

\( c_{p} \) :

Specific heat

\( d_{f} \) :

Downward flame extension (below leaf)

\( {\mathcal{D}}_{S} \) :

Drying diffusivity of water in wood

E :

Activation energy

g :

Gravitational constant

\( h_{1} \) :

Flame height of a solitary flame

\( h_{f} \) :

Flame height of a leaf

\( h_{f,group} \) :

Collective flame height of a group of flames

\( h_{f,max} \) :

Maximum flame height

\( h_{f,max,0} \) :

Maximum flame height at base conditions (same value as \( h_{f,max} \))

\( h_{f,max,F} \) :

Maximum flame height at fire spread scenario

\( \bar{h}_{L} \) :

Average heat convection coefficient for length L

k :

Thermal conductivity

l :

Leaf length

\( L_{1} \) :

Flame length of a solitary flame

\( L_{f} \) :

Merged flame length for some flame configuration

\( L_{m} \) :

Fully-merged flame length (no separation between flames)

\( m_{0,w} \) :

Initial mass of water

\( m_{dry} \) :

Oven-dry mass

\( m_{water} \) :

Mass of water remaining in a leaf

MC :

Moisture content (oven-dried basis)

\( n_{fuels} \) :

Number of fuel elements (i.e. leaves)

N :

Number of flames

\( N^{*} \) :

Number of flames meeting meeting distance criterion for merging

\( \overline{Nu}_{L} \) :

Average Nusselt number over length L

Pr :

Prandtl number

\( q_{conv} \) :

Convective heat transfer (units of power)

\( q_{evap} \) :

Heat of vaporization of water (units of power)

\( q_{rad} \) :

Radiation heat transfer (units of power)

\( r_{f} \) :

Flame radius

R :

Universal gas constant

\( R_{T} \) :

Thermal flux ratio

\( Re_{L} \) :

Reynolds number

S :

Separation distance (gap width) between flames

\( \hat{S} \) :

Normalized separation between flames in three dimensions

\( t_{bo} \) :

Burnout time (from start of heating)

\( t_{burn} \) :

Shrub burn time (from first ignition to last extinction)

\( t_{end,0} \) :

End time determined by physics-based model for base condition

\( t_{end,F} \) :

End time determined by physics-based model for fire spread scenario

\( t_{h} \) :

Time of maximum flame height (from start of heating)

\( t_{ig} \) :

Time of ignition (from start of heating)

\( t_{x,0} \) :

Flame times (e.g. t ig or t h ) at base condition, where x indicates ig, h, or bo

\( t_{x,F} \) :

Flame times at fire spread scenario

T :

Temperature

\( T_{amb} \) :

Ambient temperature (i.e. far away from fire activity)

\( T_{conv} \) :

Bulk convective gas temperature

\( T_{leaf} \) :

Leaf temperature

\( T_{local} \) :

Convective gas temperature near the leaf (considers volatile combustion)

\( T_{soot} \) :

Radiation temperature of soot

\( T_{surr} \) :

Radiation temperature of surroundings

U :

Wind speed local to a leaf

\( U_{bulk} \) :

Bulk wind speed

\( v \) :

Convective gas velocity at the leaf surface

\( \bar{v}_{U,\theta } \) :

Mean component of wind speed in the flame axis direction

\( \bar{v}_{z,\theta } \) :

Mean component of buoyant velocity in the flame axis direction

V :

Fraction of initial mass released as volatiles

\( V_{\infty } \) :

Ultimate yield of volatiles as fraction of initial mass

\( w \) :

Leaf width

W :

Fraction of initial mass released as water

\( X_{s} \) :

Shrub conversion, or amount burned

\( \alpha \) :

Thermal diffusivity, or parameter in beta distribution

\( \beta \) :

Parameter in beta distribution

\( \Delta F_{0} \) :

Fractional change in dry mass of base case

\( \Delta F_{F} \) :

Fractional change in dry mass of new fire condition

\( \Delta H_{comb} \) :

Heat of combustion

\( \Delta m_{dry,i} \) :

Change in dry mass over time step \( \Delta t \) of component i

\( \Delta m_{dry} \) :

Change in dry mass over time step \( \Delta t \)

\( \Delta m_{water} \) :

Mass released of water during \( \Delta t \)

\( \Delta x \) :

Leaf thickness

\( \Delta t \) :

Time step

\( \Delta z_{{f,{ \hbox{max} }}} \) :

Maximum flame height above shrub

ε :

Emissivity

γ :

Specific gravity

κ :

Flame opacity

Λ :

Experimentally determined flame merging constant

φ h :

Physics-based scaling factor for flame height

ρ :

Density

ρ conv :

Density of convective gases

σ :

Boltzmann constant

θ :

Flame tilt angle (from vertical)

θ T :

Convection coefficient correction factor for blowing

υ :

Kinematic viscosity

0:

Refers to an initial state, base case, or reference value

c :

Refers to the cross-section

dry :

Specifies an oven-dried quantity or a dry-matter component

f :

Refers to a flame

F :

Refers to a fire scenario case

m :

Indicates that flames are fully-merged

max :

Indicates a maximum value

w :

Refers to moisture (or water) component

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Acknowledgements

This work was supported in part by JFSP contracts 11-JV-11272167-044 and 11-JV-11272167-054 and NSF Grant CBET-093842. Any opinions, findings, and conclusions or recommendations expressed in this dissertation are those of the graduate student and advisor and do not necessarily reflect the views of the National Science Foundation or any other government funding agency. Neither the NSF nor any other agency has approved or endorsed the content in this article.

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  1. Chemical Engineering Department, Brigham Young University, Provo, UT, 84602, USA

    Dallan Prince, Chen Shen & Thomas Fletcher

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  1. Dallan Prince
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Correspondence to Thomas Fletcher.

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Prince, D., Shen, C. & Fletcher, T. Semi-empirical Model for Fire Spread in Shrubs with Spatially-Defined Fuel Elements and Flames. Fire Technol 53, 1439–1469 (2017). https://doi.org/10.1007/s10694-016-0644-9

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