Fire Technology

, Volume 53, Issue 3, pp 1439–1469 | Cite as

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

  • Dallan Prince
  • Chen Shen
  • Thomas Fletcher


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.


Wildland fire modeling Shrub combustion Live fuels 

List of symbols

English letter symbols


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


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


Activation energy


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


Thermal conductivity


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


Moisture content (oven-dried basis)

\( n_{fuels} \)

Number of fuel elements (i.e. leaves)


Number of flames

\( N^{*} \)

Number of flames meeting meeting distance criterion for merging

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

Average Nusselt number over length L


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


Universal gas constant

\( R_{T} \)

Thermal flux ratio

\( Re_{L} \)

Reynolds number


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



\( T_{amb} \)

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


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


Fraction of initial mass released as volatiles

\( V_{\infty } \)

Ultimate yield of volatiles as fraction of initial mass

\( w \)

Leaf width


Fraction of initial mass released as water

\( X_{s} \)

Shrub conversion, or amount burned

Greek letter symbols

\( \alpha \)

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




Specific gravity


Flame opacity


Experimentally determined flame merging constant


Physics-based scaling factor for flame height




Density of convective gases


Boltzmann constant


Flame tilt angle (from vertical)


Convection coefficient correction factor for blowing


Kinematic viscosity



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


Refers to the cross-section


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


Refers to a flame


Refers to a fire scenario case


Indicates that flames are fully-merged


Indicates a maximum value


Refers to moisture (or water) component



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.


  1. 1.
    Rothermel RC (1972) A mathematical model for predicting fire spread in wildland fuels. Research Paper INT-115. USDA Forest Service, Intermountain Forest and Range Experiment Station Ogden, Utah, USAGoogle Scholar
  2. 2.
    Frandsen WH (1973) Effective heating of fuel ahead of spreading fire (trans: Station IFaRE). Research Paper INT-140. USDA Forest Service, Ogden, UTGoogle Scholar
  3. 3.
    Andrews PL (2007) BehavePLUS fire modeling system: past, present, and future. In: Proceedings of the 7th symposium on fire and forest meteorology, Bar Harbor, MaineGoogle Scholar
  4. 4.
    Andrews PL (2008) BehavePlus fire modeling system, version 4.0: Variable (trans: Station RMR). USDA Forest Service, Fort CollinsGoogle Scholar
  5. 5.
    Burgan RE, Rothermel RC (1984) BEHAVE: fire behavior prediction and fuel modeling system—FUEL subsystem. PMS 439-1 NFES 0275. US Department of Agriculture, Forest Service, Intermountain Forest and Range Experiment StationGoogle Scholar
  6. 6.
    Finney MA (1998) FARSITE: fire area simulator—model development and evaluation. Research Paper 0502-5001. USDA Forest Service Rocky Mountain Forest and Range Experiment Station, Ogden, UTGoogle Scholar
  7. 7.
    Finney MA (2002) Fire growth using minimum travel time methods. Can J Forest Res 32 (8):1420–1424. doi: 10.1139/X02-068 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Finney MA, Grenfell IC, McHugh CW, Seli RC, Trethewey D, Stratton RD, Brittain S (2011) A method for ensemble wildland fire simulation. Environ Model Assess 16(2):153–167. doi: 10.1007/s10666-010-9241-3 CrossRefGoogle Scholar
  9. 9.
    Linn R, Reisner J, Colman JJ, Winterkamp J (2002) Studying wildfire behavior using FIRETEC. Int J Wildland Fire 11 (3–4):233–246. doi: 10.1071/Wf02007 CrossRefGoogle Scholar
  10. 10.
    Clark MM (2008) Development and evaluation of a sub-grid combustion model for a landscape scale for 3-D wildland fire simulator. BYU, ProvoGoogle Scholar
  11. 11.
    Morvan D, Dupuy JL (2001) Modeling of fire spread through a forest fuel bed using a multiphase formulation. Combust Flame 127 (1–2):1981–1994CrossRefGoogle Scholar
  12. 12.
    Mell WE, Manzello SL, Maranghides A (2006) Numerical modeling of fire spread through trees and shrubs. In: Viegas DX (ed) V international conference on forest fire research, CoimbraGoogle Scholar
  13. 13.
    Zhou XY, Mahalingam S, Weise D (2007) Experimental study and large eddy simulation of effect of terrain slope on marginal burning in shrub fuel beds. Proc Combust Inst 31:2547–2555. doi: 10.1016/j.proci.2006.07.222 CrossRefGoogle Scholar
  14. 14.
    Finney MA, Cohen JD, Grenfell IC, Yedinak KM (2010) An examination of fire spread thresholds in discontinuous fuel beds. Int J Wildland Fire 19 (2):163–170. doi: 10.1071/WF07177 CrossRefGoogle Scholar
  15. 15.
    Tang W, Miller CH, Gollner MJ (2016) Local flame attachment and heat fluxes in wind-driven line fires. Proc Combust Inst. doi: 10.1016/j.proci.2016.06.064
  16. 16.
    Finney MA, Cohen JD, Forthofer JM, McAllister SS, Gollner MJ, Gorham DJ, Saito K, Akafuah NK, Adam BA, English JD (2015) Role of buoyant flame dynamics in wildfire spread. Proc Natl Acad Sci 112 (32):9833–9838. doi: 10.1073/pnas.1504498112 CrossRefGoogle Scholar
  17. 17.
    Fons WL (1946) Analysis of fire spread in light forest fuels. J Agric Res 72 (13):93–121Google Scholar
  18. 18.
    Pickett BM (2008) Effects of moisture on combustion of live wildland forest fuels. Ph.D., Brigham Young University, ProvoGoogle Scholar
  19. 19.
    Cole WJ, Dennis MH, Fletcher TH, Weise DR (2011) The effects of wind on the flame characteristics of individual leaves. Int J Wildland Fire 02:657–667. doi: 10.1071/WF10019 CrossRefGoogle Scholar
  20. 20.
    Engstrom JD, Butler JK, Smith SG, Baxter LL, Fletcher TH, Weise DR (2004) Ignition behavior of live California chaparral leaves. Combust Sci Technol 176:1577–1591CrossRefGoogle Scholar
  21. 21.
    Fletcher TH, Pickett BM, Smith SG, Spittle GS, Woodhouse MM, Haake E, Weise DR (2007) Effects of moisture on ignition behavior of moist California chaparral and Utah leaves. Combust Sci Technol 179:1183–1203CrossRefGoogle Scholar
  22. 22.
    Weise DR, Zhou X, Sun L, Mahalingam S (2005) Fire spread in chaparral—”go or no-go?”. Int J Wildland Fire 14 (1):99–106CrossRefGoogle Scholar
  23. 23.
    Pickett BM, Isackson C, Wunder R, Fletcher TH, Butler BW, Weise DR (2010) Experimental measurements during combustion of moist individual foliage samples. Int J Wildland Fire 19:1–10CrossRefGoogle Scholar
  24. 24.
    Lozano JS (2011) An investigation of surface and crown fire dynamics in shrub fuels. PhD Dissertation, University of California, RiversideGoogle Scholar
  25. 25.
    Smith SG (2005) Effects of moisture on combustion characteristics of live California chaparral and Utah foliage. M.S. Thesis, Brigham Young University, ProvoGoogle Scholar
  26. 26.
    Shen C, Fletcher TH (2015) Fuel element combustion properties for live Wildland Utah Shrubs. Combust Sci Technol 187:428–444CrossRefGoogle Scholar
  27. 27.
    Cole WJ, Pickett BM, Fletcher TH, Weise DR (2009) A semi-empirical multi-leaf model for fire spread through a manzanita shrub. In: 6th U.S. National Combustion Meeting, Ann Arbor, Michigan, May 18–20 2009Google Scholar
  28. 28.
    Shen C (2013) Application of fuel element combustion properties to a semi-empirical flame propagation model for live wildland Utah Shrubs. M.S. Thesis, Brigham Young University, ProvoGoogle Scholar
  29. 29.
    Gallacher JR (2016) The influence of season, heating mode and slope angle on wildland fire behavior. PhD Dissertation, Brigham Young University, ProvoGoogle Scholar
  30. 30.
    Prince DR, Fletcher ME, Shen C, Fletcher TH (2014) Application of L-systems to geometrical construction of chamise and juniper shrubs. Ecol Model 273:86–95. doi: 10.1016/j.ecolmodel.2013.11.001 CrossRefGoogle Scholar
  31. 31.
    Albini FA (1976) Estimating wildfire behavior and effects. General Technical Report INT-30. USDA Forest Service, OgdenGoogle Scholar
  32. 32.
    Prince DR (2014) Measurement and modeling of fire behavior in leaves and sparse shrubs. PhD Dissertation, Brigham Young University, ProvoGoogle Scholar
  33. 33.
    Fletcher TH, Pickett BM, Smith SG, Spittle GS, Woodhouse MM, Haake E, Weise DR (2007) Effects of moisture on ignition behavior of moist California chaparral and Utah leaves. Combust Sci Technol 179 (6):1183–1203. doi: 10.1080/00102200601015574 CrossRefGoogle Scholar
  34. 34.
    Shen C, Fletcher TH (2015) Fuel element combustion properties for live wildland Utah Shrubs. Combust Sci Technol 187(3):428–444. doi: 10.1080/00102202.2014.950372 CrossRefGoogle Scholar
  35. 35.
    Cole WJ, Dennis MH, Fletcher TH, Weise DR (2011) The effects of wind on the flame characteristics of individual leaves. Int J Wildland Fire 20:657–667. doi: 10.1071/WF10019 CrossRefGoogle Scholar
  36. 36.
    Albini FA (1981) A model for the wind-blown flame from a line fire. Combust Flame 43:155–174CrossRefGoogle Scholar
  37. 37.
    Fletcher TH, Pond HR, Webster J, Wooters J, Baxter LL (2012) Prediction of tar and light gas during pyrolysis of black liquor and biomass. Energ Fuel 26 (6):3381–3387. doi: 10.1021/Ef300574n CrossRefGoogle Scholar
  38. 38.
    Badzioch S, Hawksley PG (1970) Kinetics of thermal decomposition of pulverized coal particles. Ind Eng Chem Proc Dd 9 (4):521–530. doi: 10.1021/I260036a005 CrossRefGoogle Scholar
  39. 39.
    Stamm AJ (1964) Wood and cellulose science. Ronald Press Co., New YorkGoogle Scholar
  40. 40.
    Plumb OA, Spolek GA, Olmstead BA (1985) Heat and mass transfer in wood during drying. Int J Heat Mass Tran 28 (9):1669–1678CrossRefGoogle Scholar
  41. 41.
    Incropera FP, DeWitt DP, Bergman TL, Lavine AS (2007) Fundamentals of heat and mass transfer, 6th edn. J. Wiley, New YorkGoogle Scholar
  42. 42.
    Jenkins BM, Baxter LL, Miles TR, Miles TR (1998) Combustion properties of biomass. Fuel Process Technol 54 (1–3):17–46. doi: 10.1016/S0378-3820(97)00059-3 CrossRefGoogle Scholar
  43. 43.
    McCaffrey BJ (1979) Purely buoyant diffusion flames: some experimental results (trans: Center for Fire Research NEL). National Bureau of Standards, WashingtonCrossRefGoogle Scholar
  44. 44.
    Bird RB, Stewart WE, Lightfoot EN (2002) Transport phenomena, 2nd edn. Wiley International J. Wiley, New YorkGoogle Scholar
  45. 45.
    Lopez A, Molina-Aiz FD, Valera DL, Pena A (2012) Determining the emissivity of the leaves of nine horticultural crops by means of infrared thermography. Sci Hortic-Amsterdam 137:49–58. doi: 10.1016/j.scienta.2012.01.022 CrossRefGoogle Scholar
  46. 46.
    Sun L, Zhou X, Mahalingam S, Weise DR (2006) Comparison of burning characteristics of live and dead chaparral fuels. Combust Flame 144 (1–2):349–359CrossRefGoogle Scholar
  47. 47.
    Steward FR (1970) Prediction of height of turbulent diffusion buoyant flames. Combust Sci Technol 2 (4):203-212. doi: 10.1080/00102207008952248 CrossRefGoogle Scholar
  48. 48.
    Weng WG, Kamikawa D, Fukuda Y, Hasemi Y, Kagiya K (2004) Study on flame height of merged flame from multiple fire sources. Combust Sci Technol 176 (12):2105–2123. doi: 10.1080/00102200490514949 CrossRefGoogle Scholar
  49. 49.
    Liu N, Liu Q, Lozano JS, Shu L, Zhang L, Zhu J, Deng Z, Satoh K (2009) Global burning rate of square fire arrays: experimental correlation and interpretation. Proc Combust Inst 32 (2):2519–2526. doi: 10.1016/j.proci.2008.06.086 CrossRefGoogle Scholar
  50. 50.
    Lozano J, Tachajapong W, Weise DR, Mahalingam S, Princevac M (2010) Fluid dynamic structures in a fire environment observed in laboratory-scale experiments. Combust Sci Technol 182(7):858–878CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Chemical Engineering DepartmentBrigham Young UniversityProvoUSA

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