Framework for submodel improvement in wildfire modeling

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Framework for submodel improvement in wildfire

modeling

Mohamad El Houssami, Aymeric Lamorlette, Dominique Morvan, Rory

Hadden, Albert Simeoni

To cite this version:

Mohamad El Houssami, Aymeric Lamorlette, Dominique Morvan, Rory Hadden, Albert Simeoni.

Framework for submodel improvement in wildfire modeling. Combustion and Flame, Elsevier, 2018,

190, pp.12-24. �10.1016/j.combustflame.2017.09.038�. �hal-02114000�

Framework

for

submodel

improvement

in

wildfire

modeling

Mohamad

El

Houssami

a

_,

_Aymeric

_Lamorlette

b

_,

_Dominique

_Morvan

c

_,

_{Rory M.}

_Hadden

a

_,

Albert

Simeoni

c,∗

a The University of Edinburgh, Edinburgh EH9 3JL, UK

b Aix-Marseille Université, CNRS, CentMarseille, M2P2, 13451 Marseille, France c Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA

a

r

t

i

c

l

e

i

n

f

o

Keywords: Multiphase Forced flow Porous LES

a

b

s

t

r

a

c

t

Anexperimentaland numericalstudy wascarriedouttoassessthe performanceofthedifferent sub-modelsand parametersused todescribetheburning dynamicsofwildfires.Amultiphaseformulation wasusedand comparedtostaticfiresofdriedpitchpineneedlesofdifferentbulkdensities.The sam-pleswereexposedtoanexternalheatfluxof50kW/m2 _{intheFMGlobalFirePropagationApparatus} andsubjectedtodifferentairflows,providingacontrolledenvironmentandrepeatableconditions. Sub-modelsforconvectiveheattransfer,dragforces,andcharcombustionwereinvestigatedtoprovidemass lossrate, flamingduration,and gasemissions.Good agreementofpredicted masslossrates and heat releaserateswasachieved,whereallthesesubmodelswereselectedtosuitthetestedconditions. Sim-ulatedflamingtimesfordifferentflowconditionsanddifferentfuelbulkdensitiescomparedfavorably againstexperimentalmeasurements.Thecalculationofthedragforcesandtheheattransfercoefficient was demonstratedto influencegreatly theheating/cooling rate,the degradationrate,and theflaming time.ThesimulatedCOandCO2valuescomparedwellwithexperimentaldata,especiallyforreproducing thetransitionbetweenflamingandsmoldering.Thisstudycomplementsapreviousstudymadewithno flowtoproposeasystematicapproachthatcanbeusedtoassesstheperformanceofthesubmodelsand tobetterunderstandhowspecificphysicalphenomenacontributetothewildfiredynamics.Furthermore, thisstudyunderlined theimportanceofselectingrelevant submodelsandthenecessityofintroducing relevantsubgrid-scalemodellingforlargerscalesimulations.

1. Introduction

Understandingtheburningbehaviorofwildlandfuelsis essen-tial to develop a complete understanding of wildfire spread. For thelast decades,numerousstudieshavebeenattemptedtobetter describethefiredynamicsofsolidfuels [1–3],includingpolymers [4–8],cellulosicmaterials[9,10],anddifferentspeciesofwood[11– 14]. The transfer of this knowledge to describe the burning be-haviorofwildlandfuels,whicharenaturalpolymers,isnottrivial because they differ from classical solid fuels, asthey often con-sistin highly porous media where the solid phase is very small compared to the gas phase. The main challenge arises fromthe porousnatureofthefuelsanditscouplingwiththetransient sur-roundingenvironment,whichstronglyinfluencestheheattransfer, degradationrates,and theburningdynamics [15]. Indeed,during combustionofahighlyporousbed,theheatcapturedbythesolid

∗ _{Corresponding author.}

E-mail address: asimeoni@wpi.edu (A. Simeoni).

phase is a result ofthe energy balance fromthe flame feedback byradiationandfromthefreshairentrainedintothefuellayerby convection,butalsoreactingwiththepyrolysisproductsandwith thechar residue.As a consequence,thesehomogeneousand het-erogeneouscombustionscontributetotransferenergytothesolid phase.Thisbehaviorisdifferentfromthatobservedforsolidfuels forwhichthe radiativeandconvective contributionsof theflame can be considered to be transferred intothe solid by conduction [16]. This highly porous configuration is typically found in pine needlebeds,whichareoftenfoundonthe forestfloors.Theycan typicallybefoundaccumulatednearstructuresintheWildland Ur-ban Interface (WUI), increasing the fire risk [17]. As the need to understandtheburningdynamicsofwildlandfuels isamatterof greaturgencyinordertoimprovewildfireassessmenttools,many studieshavebeenperformedoverthelast decades.Among those, Dimitrakopoulos and Papaioannou [18] used the cone colorime-terto determinetheignitiontime andtheflammabilityoftypical Mediterraneanforestfuelsandproposedfourflammabilityclasses. Schemel et al. [19] reported a calorimetric study of pine needle bedsusingtheFMGlobalFirePropagationApparatus(FPA),during

Nomenclature

C Convectiveheattransferconstants CD Dragforcecoefficient

D Equivalentdiameter

ECHAR Energyactivationforcharoxidation

HRR AverageHeatReleaseRate k Airthermalconductivity

KCHAR pre-exponentialcoefficientforcharoxidation

m0 Initialmass

m,n Convectiveheattransfercoefficientconstants Nu Nusseltnumber

P Pressure

Pr Prandtlnumber

Q_char(s) Charoxidationsourceterm

Q_CONV(S) Convectiveheattransfersourceterm R Idealgasconstant

Re Reynoldsnumber

T,Ts Gasandsolidphasetemperatures

U Velocity

YO₂ Oxygenmassfraction

Greeksymbols

α

g,

α

s Volumefractionofgasandsolidphases



hCO,



hCO₂ HeatofcombustionofCOandCO2

ρ

Gasdensity

σ

s Surfacetovolumeratio

ϕ

Splitfunction

ϕ

(S) char ,

ϕ

( S) DRY ,

ϕ

( S)

H₂O Massfractionofchar,drypine and

wa-terinsolidphase

χ

Convectiveheattransfercoefficient ˙

ω



char,

ω

˙



pyr,

ω

˙vap Volumetric rate of charring, pyrolysis,

andvaporization

whichitwasshownthatcalculatingtheheatreleaserate(HRR)by meansofcalorimetryforpineneedlescanbereinforcedbytheuse ofmasslossrateandbyknowingtheheat ofcombustioninwell ventilatedtestconditions.Bartolietal.[20]demonstratedthat per-meabilitydrivestheburningdynamicsinporousbedsandthatthe energy released increases with permeability. Additionally it was shownthat fora givenpermeability,the fuelspecieshave an in-fluenceon timesto ignitionandduration offlames.Mindykowski etal.[21]conductedexperimentsinthesameexperimentalsetup toinvestigatethepilotedignitiontimeoflitterscomposedofoven Mediterraneanpinespeciesandkermesoakleaves.

The porous bed can be described through CFD modeling us-ing a multiphase formulationthat allows the fire rateof spread (ROS)[22–24],andtemperaturefieldtobeestimated[25].For in-stance, Padhi et al. [26] described the flow field generated due to the steadyburning of a shrub using the multiphase formula-tion.Velocityandtemperatureprofileswere analyzedinthe con-tinuousflame region andthermalplume regionandit was high-lighted that the porous nature of the shrub affects air entrain-ment intothe shrub, thus affecting thenature ofthe fire plume. Hence, itisimportanttoevaluate themodel’scapabilitiesto sim-ulate air entrainment and air mixture in the fuel bed. The use of themultiphase formulationis not new,it hasbeen appliedin numerous studies to describe the interaction between the atmo-spheric boundary layerwitha canopy problemsusing isothermal flows[27–33].However,themultiphaseformulationisusually ap-plied to simulate large-scale wildfires under complex conditions withoutthoroughlypre-validatingorverifyingiftheselected sub-models(i.e.theclosuremodels)arewelladaptedtothefuel condi-tionsandtothesurroundingenvironmentinwhichtheyareused,

andifthey allow the relevantphysics to be captured[15,34–37]. Forinstance,degradationratesare providedfromTGAanalysisat amuchsmallerscale(microscopic)thanreality[15],anddrag co-efficientsarederivedfromwindtunneltestsforsimplegeometries at ambient temperatures [38]. In a recent study, Hoffman et al. [39]evaluatedcrownfirespreadrateusingFIRETEC[40]andWFDS [41](both includea multiphase approach) andcompared the re-sultstoa compilationofwildfire observationsinNorth American forests[42].Itwasnotedthattheabilitytotestthemodelwas lim-itedbyalackofenvironmentalandfueldatasuchaserrors associ-atedwithpoint-to-pointcomparison,whichisalimitationoflarge scalefires.Thisaspectisessentialbecausetherateofspread pre-dictionsfromdetailedphysics-basedmodelsaresensitivetosmall variationsinboththespatialpatternofthefuelsandthe environ-ment. Therefore, this study recommended further assessment of detailedphysics-basedmodels,particularlybyprovidingadditional data regarding fuel and environmental characteristics (i.e. wind). Thisiswhyitisnecessarytoestablishaframeworktailoredtothe developmentoffiremodelingwiththemultiphaseapproach.It al-lowstostudyelementaryaspectsoftheproblemandtogradually movetowards complexity(bychangingthe fuelorenvironmental properties).Thismethodologyfollowsabuildingblockapproachto modeldevelopmentandfacilitatesabetterunderstandingofforest fuelflammabilityandofitscorrespondingfiredynamics.

Thisstudyincludeswelldocumented fireexperimentsthatare conductedinacontrolledenvironment,providingprecise measure-ments for different fuel and ambient conditions, to quantify the influenceoftheparameters onthemodels numericalpredictions. As these conditions change, some parameters and submodels in thenumericalmodelarenolongervalidandneedtobeadjusted. This parametric study is often overlooked at larger scale due to the excessiveuncertainties causedby the large variationsof fuel properties(i.e.specialdistribution,fuelmoisturecontent),the un-stablewind(i.e.gusts),andthestrongcouplingbetween submod-els,whichmakesitimpossibletopinpointexactlywhichsubmodel isnotbehavingphysically.Inthisstudy,experimentswere under-takenintheFMGlobalFirePropagationApparatus(FPA)[43], simi-larlyasin[19,20]whereitwasdemonstratedtoprovingrepeatable conditions.Pineneedlebedsofvaryingporosityweresubjectedto aradiativeheatandvariousvaluesofairflowstoobservethe burn-ingbehaviorunderdifferentconditionsrelevanttowildfirespread. Pineneedlebedswereusedasareferencefuelbecausetheyallow repeatablefuelbedpropertiesunderlaboratorysettings.Massloss rates,heatreleaserates,flamingtimes,gasemissions,and temper-aturefields were used to comparethe experiments to the simu-lations.Particular attentionwasgivento forcedflowconditions -extendingprevious work on naturalconvection[44] -because in wildfires,theflowisusuallyhighlyunstableandnotwelldefined. Consequently,localvariationsinthe flowcanhaveimportant im-pactsonthelocalfireregime.However,theflowiscontrolledand can be well characterized in the FPA. This framework, with the appropriatemodificationssupportsthedevelopmentoflargescale CFDmodeling by providing inputsandindications forthe neces-sarysubscalemodeling.

2. Methods

2.1. Experimentaldetails

Experimentswere conductedby burningbedsof pineneedles inthe FPA [43] (Fig. 1). The apparatus provides a controlled en-vironmentwithrespectto airflowandrepeatable,uniform heat-ing conditions.Samples were exposed to a radiative heat flux of 50 kW/m2 _{applied to the top surface of the sample, throughout}

theentireexperiment mimickinga strongflame feedbackfroma largerfiresurroundingthesample.Thisvalueofreceivedheatflux

Fig.1. Overview of the Fire Propagation Apparatus (FPA). Table 1

Summary of bulk densities and porosities. Mass [g] Bulk density [kg/m 3 ] Porosity _α

g [-]

8.7 23 0.96

11.4 30 0.95

15.0 40 0.93

isrepresentativeoftypicalfire front propagation[45].Deadpitch pine(Pinusrigida)needles werepackedincylindricalporous bas-kets(67%%openingfraction)of12.6cmdiameterand3.0cmdepth followingthe protocolused in [46]. Fuelmoisture content (FMC) wasdeterminedbyconditioningneedlesat60°Cfor24h.The av-erageFMCwas7%%ofthedryweight.Thesurfacetovolumeratio (

σ

s)is7295m−1,andthedensity(

ρ

s)is607kg/m3.

Airflow was introduced at the inlet under the sample and passedthrough andaroundthe poroussample. Threeinlet fluxes of air were used No Flow (NF), Low Flow (LF) (50L/min, cor-responding to a free stream velocity of 6.67cm/s), and High Flow (HF) (200L/min, corresponding to 26.8cm/s). A hot wire anemometer(Kimo®_{AMI301witha0.01m/sresolution)wasused}

toestimatetheaveragedvelocityoftheflowpenetrating,andthe flowcirclingaroundtheporoussample.Eveniftheflowconditions usedinthisstudyarerelativelylowcomparedtothemean veloci-tiesthatcanbefoundinwildfires[47,48],itisimportanttoassess the model’s performance under these specific conditions, which couldalsobefoundlocallyduringafire.Followingthisframework, otherflowregimesandmoremoderateflowscouldbestudied.

Mass lossrate(MLR) wasderived frommeasurements ofmass usinga loadcell (0.001g resolution) andthe exhaustgaseswere analyzed forcomposition (CO, CO2 and O2) using non-dispersive

infraredandparamagnetic techniques ata samplingrateof 2Hz. Flamingtimewasmeasuredwitha0.5serrormarginandtheend offlameouttimewasreachedwhentheCOconcentrationdropped under4ppm.

Three bulk densities were tested: 23, 30, and 40kg/m3_{, as}

showninFig.2.Thecorrespondingmassesandporositiesarelisted inTable 1. 40kg/m3 _{is the maximumdensity that can fit in the}

samplebasketwithoutovercompressionandbreakingofthe nee-dles.23kg/m3 _{corresponds to theminimal load that needs tobe}

usedtoavoidsignificantreflectionofthebasketwhenheated[44]. Thesevaluesareinthesamerangeastypicalfuelloadsmeasured

intheforest [49].Moredetailsoftheexperimentalsetupare pro-videdin[44].Allexperimentalresultswereaveragedover3 repeti-tionsormore,andtheerrorbarsrepresentthestandarddeviation forallexperiments.

2.2. Computationalmodeling

In this study, simulations were carried out using ForestFire-FOAM (FFF) [44]. It has been previously demonstrated that FFF allows modeling the fire-induced behavior of a porous, reactive andradiativemedium[44].Conservationequations(mass, momen-tum,andenergy) aresolved foran averaged control volume ata scale sufficient to contain both coexisting gas and solid phases, and considering strong coupling between the phases [34]. Pro-cessessuchasdrying,pyrolysis,andcharcombustionaredescribed throughtemperature-dependentinteractionbetweenthesolidand gasphases.Thegasphasecombustionisrepresentedusingan ex-tensionoftheEddyDissipationConcept(EDC),inwhichthe char-acteristictimescaleoffuel–airmixingisdifferentunderturbulent andlaminarflowconditions[50].Thiscombustionmodelwasused in a study that focused on fully-developed turbulent region of a wallflamewherethediffusiontimeusedforthelaminarflowwas a placeholderforthe nearwall region andwasnot verified.It is recognizedthatthevalidityofthemodifiedEDCmodelforlaminar flamestudies stillneeds tobe established,butthisonewasused intheabsenceofabetterandsimplemodelforlaminarflamesin LES.A completedescription ofFFFis available in[44,51]andthe governingequationsarepresentedintheAppendix.The computa-tionaldomain representingthe FPA ina two dimensional config-urationis sketchedinFig.3.The overalldomainis arectangleof 5mwideand2mhigh.Themeshwascomposedprimarilyof hex-ahedralcells.Basedonameshsensitivitystudy,ameshresolution of 0.001_{× 0.001} m2 _{was determined asproviding converging}

re-sultsinthevicinityofthesampleandnearthelamps.Themeshis stretched beyondthezone ofinterestuntiltheboundaries reduc-ingcomputationaltimewithoutaffectingtheresults.Adaptivetime stepiscalculatedbasedonCourant–Fredrichs–Lewy(CFL)number [52],witha maximumvalue of 0.7toachieve temporal accuracy andnumericalstability.

Sub-modelsforheattransfer,degradation,combustion,andgas emissionhavebeenimplementedtoallowbetterrepresentationof these processes. However, in order to appropriately simulate the conditionspresentedabove,specificsubmodelshadtobeadjusted. Thesesubmodelsare presentedhereafterandall othersubmodels thatarenotpresentedarediscussedinthepreviousstudy[44]. 2.2.1. Dragforces

In the multiphase model, the drag force acting on the solid-phase is often [22,23,36,53] approximated using the correlation proposed by Clift et al.[54] forestimating thedrag force coeffi-cient(CD)ofspheres. CD= 24



1+0.15Re0.687



Re × 3 8; 1<Re<1000 (1)

Tomorecloselyrepresentthecylindricalgeometryofthepine needles, the drag coefficient was approximated using a pseudo fluidmodelforarraysofemergingcircularcylinders[55]: CD =11Re−0.75+0.9



1− exp



−1000 Re



+1.2



1− exp



− Re 4500



; 0.02 <Re <2× 105 ₍₂₎

The use of a correlation forarrays of cylinders is justified by the“sheltering” effectthatdiminishesthedragdownstreaman el-ement [56,57].As elementspacingdecreases, thebulk drag coef-ficientdecreases [58].Hence, correlationsestablished for isolated

Fig. 2. Pitch pine needle samples with (a) 23 kg/m 3 ; (b) 30 kg/m 3 ; (c) 40 kg/m 3.

Fig. 3. Computational domain of the FPA (not to scale).

elementsare nolongereffectivebecause ofthisstrongsheltering effect.

2.2.2. Convectiveheattransfer

The term representing the contribution of convective heat transfer Q_CONV(S) betweengas andsolid phases isdescribed as fol-lowsinthegasphaseenergybalanceequation(inAppendix): Q_CONV(S) =

α

s

σ

s

χ

(

T− Ts

)

(3)

With T and Ts the gas and solid phase temperatures,

respec-tively.Theheattransfercoefficient

χ

isestimatedfromtheNusselt number[38]:

Nu=

χ

D

k =CRe

m_Prn ₍₄₎

where,kistheairthermalconductivity(W/m.K)andDthe equiv-alentdiameterofafuelparticle(cylindricalneedle)approximated as4/

σ

s.D isalsoused asthe characteristiclengthforcalculating

the Reynoldsnumber. As demonstratedin [59],despite the com-plexvegetativestructure,asimplecorrelation(i.e.Eq.4)is appro-priatetorepresenttheconvectiveterm,aslongasthecoefficients C,m andnare adapted torepresentthe observedpacking effect, or“sheltering” effect[56].Sincesamplesarepreparedbystacking

pine needles over each other, one can assume that the formula-tion of the convective heat transfer coefficient is similar to that of array of staggered cylinders in cross flow. Many values for C, m, and n are proposed to representthe Nusselt (Nu) number in thesespecific orsimilarconditions[60,61].ValuesforC,m,andn wereproposedbyŽukauskas[60](C₌₌1.04;m₌₌0.4;n₌₌0.36), Colburn [38] (C==0.33; m==0.6; n==1/3), Incropera and De-Witt[38] (C==0.683; m==0.466;n==1/3), andothers [61]for flowacrossbanksofstaggeredtubes,for10ormorerowsoftubes andfor differentReynolds regimes. All the cited correlationsare widelyapplied instudiesthat are usingthemultiphase approach [36,37,51,53],buttherearenoindicationsonwhichonesaremore suitable.

2.2.3. Charoxidation

Previous work [44] has shown that using a 1st order Arrhe-nius submodel underestimates char oxidation rate(

ω

˙_char) due to thelowinduced massflux ofoxygen.Consequently,thechar oxi-dationreactioncould notbe sustainedafterflameout,despite the constantlampradiation.Inthepresentstudy,thesamemodelwas testedagain,butthistimeinforcedflowconditionstoverifyifitis sufficienttoinitiatecharcombustionwithoutusingmoredetailed modeling or additional forcing term depending on the Reynolds numbertoaccountforblowingeffects[23].Thecharoxidationrate isestimated fromthe following Arrhenius equation, where ECHAR

is the energy activation of char oxidation and KCHAR is the

pre-exponentialcoefficient,allderivedfromTGAanalysis[15].

˙

ω

 char=

α

s

σ

s

υ

O2

α

g

ρ

YO2KCHARe _−E CHAR RTs  (5)

Charoxidationrepresentsanimportantsourceofheatreleased duringsmoldering. Hence,it is consideredthat any materialthat formschar duringits thermaldecompositioncan potentially sus-tainasmolderingprocess[62].Thisheterogeneousoxidationis in-completeandemitsahigheryieldofCOthanthegas-phase com-bustion[63].ByimplementingthemodelofEvansetal.[64]into the multiphase formulation,the contribution of char combustion totheenergyequation ofthesolidphase(in Appendix)becomes:

α

sgQ_char(s) =[

(

2

ϕ

− 1

)



hCO2+2

(

1−

ϕ

)



hCO]

ω

˙



char (6)

with

α

sg==0.5, assuming that 50%% of the heterogeneous

com-bustiontakesplaceatthesurfaceofthesolidphaseandtheother 50%%takesplaceinthegasphase[22,23,34,51].

ϕ

= 2+CO/CO2 2CO/CO2+2

Fig. 4. Drag force coefficient estimation using different submodels. and CO CO2 =2500exp



−6240 Ts



(8)

The splitfunction (

ϕ

) allowspredictionoftheratioofCOand CO2concentrationsproducedduringsmoldering[65].However,the

gasphase combustion submodel (EDC) is designedto oxidizeall COmixedwithoxygen,includingthe COproduced bysmoldering combustion.Butinrealitynot allCOisoxidized,particularlythat originatedfromsmoldering,duetothelow temperature.To sepa-ratebetweenthetwotypesofCO,an“inert” COphasewas intro-ducedasasmolderingproduct.

3. Resultsanddiscussion

Theestimationofthedragcoefficient(Eq.(2))affectsthemass andthevelocityoftheflowthat enteredthesample. A misrepre-sentationofthisvaluecanleadtoamisrepresentationofthetotal dragforce resultingfromtheinteraction betweenthe solidphase andthegas phase. Thisresults inincorrect heat transfer estima-tion,andpoorlyrepresentedburningdynamics.Regardingthedrag force coefficient estimation, the outcome of the aforementioned submodels(Eq.(1)and(2)) are presentedinFig.4.The Reynolds (Re) number was calculated using the kinematic air viscosity of dryairat300K(1.568× 10−5_m2 _s−1_{),resultinginRe}LF_==2.3and

ReHF_{==9.3. For Re>4, both submodels provide similar}

estima-tions, butfor lower Reynolds numbers (Re<4), corresponding to flowslowerthan13.14cm/s(orHF/2),thevalueofCDishigher

us-ingCliftetal.modelforasphere[54].However,flowvelocityfields were successfullysimulatedusing the pseudofluidmodel atboth lowflow(LF)andathighflow(HF),matchingthemeasured veloc-itiesontop,andaroundthesamplefortestswithoutcombustion (Fig.5aandb).Anoverestimationofthedragforceleadstoan un-derestimationoftheflowinthefuelbed,whichaffecttheburning dynamicsthrough cooling, airmixing, combustion rate, char oxi-dationrate,amongothers.Whenmodelinglargescale forestfires, thesecorrelationsare usually appliedto estimate themean drag force generated from both wind/litter and wind/tree interactions ina control volume larger than 1m _{× 1}m [40,66]. By doing so, theflowandconsequently,thefire behaviorofthelitterare mis-represented.Hence, vegetationelements producingdifferent drag coefficientshavetobeseparated.

Once ignition occurred, the observed flame wasmostly lami-naratthe base,asshowninFig. 6a.Transientbehavior wasalso

observed(Fig.6b),butonlyintheintermittentzone oftheflame, whichwasmostlyduetotheentrainedair.

Thedifferencebetweenthethreeconvectiveheattransfer coef-ficientsoutcomespresentedinFig.7isnonnegligible.Foratypical HF inlet,the measured flowinside the fuel bedisapproximately 0.2m/s, corresponding to ReHF_{==9.3. The correlations proposed}

by Zukauskas etal.and DeWittetal.give convective coefficients

χ

>40W/m2_{K.However,thisvalueishighandcompeteswiththe}

radiative heat transfer source term;thereby preventingthe solid temperaturefromrising,degradationtooccur,andultimately igni-tiontohappen.Moreover,thesecorrelationswereinitiallyreported formoderateflowtemperatures[60]andwerenotverifiedathigh temperatures.

On the other hand, coefficient proposed by Colburn et al. [38] corresponded to

χ

≤ 20 W/m2 _{K, providing a more}

moder-ateheating/coolingrate.We canjustifythatthesecoefficientsare moreadapted toaccountforthesheltering effectproducedwhen elementsare close together andact asa bulk. Eventhough, this value is relatively low, it was demonstrated that the packing of needles generallycauses the heat transfer coefficient to decrease [59].

The reason why the other correlations worked in previously citedstudiesisprobablybecausetheenergybalancewas compen-sated by an overestimationof theradiative heat transfer. In con-trast,theradiativeheattransferoftheFPAconfigurationwas care-fullyexaminedandquantified[44].

Figure8aillustratesthepredictedtemperaturefieldofthe cor-responding experiment during flaming (HF, 40kg/m3_).

Tempera-turerangeswereinthesameorderofmagnitudeasother flames fromwildland fuels [67,68]. The shape of the simulatedflame is consistent withthe experimental one (Fig. 6a). The evolution of the temperatureinthe solid phase ispresented inFig.8b atthe top ofthe sample, themiddleand thebottom usingthe correla-tion proposed by Colburn etal.[38].The temperature on top,in the middle, and in the bottom increased at a slower rate than observed in NF conditions [44] before ignition due to the con-vective cooling.After theinitial ignition, the middletemperature promptlyincreased.At12s,themiddletemperaturereached800K, followedbyan increaseinthetemperaturerate,corresponding to alocalizedignition.As theflamepropagateddownwards,the bot-tomtemperatureincreaseduntilitwasfullyinvolvedintheflame. Temperaturemeasurementsinsidethefuelbedwerenotobtained inthisstudyduetoexperimentallimitations.Twomainissues oc-curredwithtemperaturemeasurementswithforcedflow.First,the vibration ofthe thermocouples inside the fuel bedhadan influ-enceon the pine needlearrangement in the sample, which it is importantgreatcareinhavingsimilaronesforeveryexperimental procedure, inorder to havebetter repeatable conditions. Second, thewiring ofthe thermocouplesinthe FPAcombustion chamber obstructs the inflow. However, the simulated temperature evolu-tion is consistent withtemperature measurements conductedfor Pinusstrobus(

σ

s==10,788m−1) intheFPAundersimilartesting

conditions(heatfluxof50kW/m2_)[69].

MeasuredandsimulatedflamingtimesareplottedinFig.9for differentbulkdensitiesandinletflows.Figure9showsthat flam-ingtimeincreasedwithincreasingbulkdensity.Flamingtimewas longerastherewasmorefuel toburn.It alsoincreasedwith de-creasingflowvelocity,andtheflamesweremaintainedthelongest (45s) under natural convection at maximum bulk density. The forcedflowenhanced the mixingpyrolysis gaseswithairand in-creasedthe burningrate. Forlow bulk densities,the influenceof theforcedflowwasnotasdominantasforhigherbulk densities: forallflow conditions,flamingtimesvariedbetween13and17s. Since the samplewasvery porous (96%%), the induced air easily penetratedandprovidedenough oxygento obtainwell-ventilated combustionconditions.Flaming time linearlyincreasedwithbulk

Fig. 5. Measured and simulated air velocity on the peripheral free space (gap) and on top of the fuel (sample) in the FPA at HF for (a) 23 kg/m 3 and (b) 40 kg/m 3 .

Fig. 6. Pine needles burning in the FPA (a) low view angle (b) high view angle.

Fig. 7. Convective heat transfer coefficient estimation.

densityforvariousflowratessincetheaircontributionisdirectly relatedtoanincreaseinthecombustionrate,asthelatterbecomes limitedbythefuelavailable.Incontrast,undernaturalconvection the relation between flaming time and bulk density is non lin-ear,because thecombustion rateislimited byboth the available fuelandtheoxygen,especiallyforhighbulkdensities(92%%). For clarity, simulationsare onlyplottedforNF andHF, whicharethe

lower andupperboundingconditions. Also, goodagreement was foundforLF.Simulations slightlyoverpredictedtheflamingtime by afew seconds. Itis a directconsequence fromusinga highly efficientgasphasecombustionmodel,whereallavailablefuelwas consumedwhenmixedwithOxygen. Inrealitytheefficiencyrate isnotmaximum(duetoairdilutionandcooling).

One of the main advantages of numerical simulations is the abilityofseparatingtheprocessesinvolvedintheevolutionofthe solidphase,such asinFig.10,inwhich,themeasuredand simu-latedtimeevolutionofthemasssample(normalizedbytheinitial mass)wereplottedforabulkdensityof40kg/m3_andHF.The

sim-ulatedmasslosswassplitintoa dryfraction,amoisture content fraction,andacharfraction.Pilotedignitionwasobserved experi-mentally.However,thepilot(inFig.6a)wasnotsimulatedbecause thecombustionmodelimpliesthat flamingcombustionoccursas soon as fuel and oxidizer are mixed. As ignition occurred faster (5s)inthe simulation,allthe curveswereshifted tosynchronize bothignitiontimes.Thisisnotsurprisingbecausepilotedignition isamarginal eventthat anysmallvariationintheexperimentor inthenumericalconditioncaninfluenceit.Moreimportantly,we are interestedin comparing the combustionphases following ig-nition.One can observe that the simulated (total) massloss fol-lowedthemeanexperimentalmasslossclosely.Thedifferent dot-tedlinescorrespondtothethreesimulatedfractions,givingmore insightonthe masslossmechanisms.In thebeginning,the mass fractionwasonlycomposedofmoisture contentanddryfraction. Thenthe moisturecontent evaporatedduringthe first13s ofthe simulation.Moredrymaterial waslost(via pyrolysis)when igni-tionoccurred,duringwhichthedryfractiondecreasedsteeply,and

Fig. 8. (a) Temperature field during flaming (b) Simulated solid temperature at the top, middle and bottom of the sample corresponding to the three blue sampling points in (a), for HF and 40 kg/m 3 . Vertical line marks ignition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

article).

Fig. 9. Measured and simulated flaming time for different bulk densities and inlet flow. Filled markers: experiments, empty markers: simulations, error bars: standard deviations for experiments.

thecharfractionformed(correspondingtoflaming).Thetransition tosmolderingcombustionoccurredafter30s(nearflameouttime), duringwhich thecharoxidation reactionwasdominantuntilthe end.

Mass loss rates (MLR) were calculatedfrom experimental and numericalmass loss data andare shown inFig. 11 fortwo bulk densities:23kg/m3 _{(Fig. 11a)and40kg/m}3 _{(Fig.11b) atHF.A 5s}

moving average wasused to smooth experimental curves. Inthe first10sofbothcases,thesmallerslope correspondsto pyrolysis andevaporationthat occurredbefore ignition.When ignition oc-curred,theslopeincreasedsteeply.Thesubsequentdecrease corre-spondsto flameout(verticallinearound30s),whereaftermainly charoxidationoccurred. Thenumericalresultsover predictedthe

Fig. 10. Experimental and simulated mass loss, including simulated mass loss of dry, water, and char phases. Vertical dotted line corresponds to experimental and simulated ignition times. Solid vertical lines correspond to simulated and experi- mental flameout times respectively. (HF - 40 kg/m 3 ).

maximumvalue byaround 20%%,butfollowedthesametrendas intheexperimentsunderbothconditions.

Theradiation attenuationcoefficientcan drasticallychange in-fluencethe radiationdistributioninthe fueldepthandaffectthe distributionofthemasslossrate.Inthesesimulations,the extinc-tioncoefficientwaskeptas4/

α

s

σ

s,similarlyasinapreviousstudy

[44].However,itwasdemonstratedinAcemetal.[70]thatthe es-timation ofthistheoretical coefficient could be could be affected bythegeometricalshapeby10%%.

Forbettercomparisonbetweenthedifferentbulkdensitiesand flowconditions,MLRwereaveragedforeachconfiguration,andare presentedseparatelyduringflamingandafterflameoutinFig.12a andb,respectively.Duringflaming(Fig.12a),onecanobservethat

Fig. 11. Mass loss rates (MLR) at HF for different bulk densities (a) 23 kg/m 3 and

(b) 40 kg/m 3 . Error bars: standard deviation. Vertical line: flameout time.

the average MLR increased with an increasing inlet flow for all tested conditions because, generally the flow enhances the mix-ingofpyrolysisgaseswithair,increasingthecombustionrate.The followingobservationscanbemade:

- UnderHFconditions,theaveragevalueincreasedalmost lin-earlywiththebulkdensity.Thisisaresultofthedirect in-fluenceoftheairflowonthecombustionrate.

- For NF conditions, the average MLR increasesat 30kg/m3_,

because there is more fuel to burn. However, it decreased atthe highestbulkdensity, mostlyduetothe low ventila-tionconditionsinducedbynaturalconvection,whichslowed downthereaction.

- As forLF condition,themeanMLRremains non-monotonic as a result of the competition between radiation heating more of the solid phase with increasing bulk density (in-creasing

α

s),andthedecreasingoxygen availabletosustain

thereaction.

- Atthelowestbulkdensity(23kg/m3_{),whichcorrespondsto}

thehighestporosity (Table 1),the meanMLRissimilar re-gardlessoftheflowconditions,becauseaircaneasily pene-trateandprovidewell-ventilatedcombustionconditions.

- For higher bulk densities, the sample was more compact, causingmorecontactbetweenpineneedles,andmore shad-owingeffect.Therefore,therewaslesscontactwiththegas phase.Astheextinctionlength(_{∼ 4/}

α

s

σ

s)[71]changesfrom

13.7mm to7.8mm,between23and40kg/m3_{, respectively,}

radiationcannot penetrate deeperin thesamplebody.The opticalthicknessdefinedastheratiobetweenthe depthof thesample(3cm) andtheextinctionlengthshowsthat the sampleisopticallythick(2.19and3.84for23and40kg/m3_,

respectively).

- Whenthetotalamountofradiationisdistributedona shal-low layer the limited factor is the amount of fuel avail-ableforpyrolysis,However, when thesame amountof en-ergy is distributed deeper in the fuel (i.e. low bulk den-sity),moremassundergoespyrolysis.Ultimately,pyrolysisof adenserfuelwillbelimitedbytheheattransferred down-wards,whichiscorrelatedtothebulkdensity.

- Between experiments and simulations with and without flow, we can highlight the existence of two regimes. The oxygenlimited oxygen regime (no wind)andthe fuel lim-itedregime(withwind)[72].BecausethemeanMLRisnot amonotonicfunctionofthedensityinNFandLFconditions, wecandeterminethatthecombustionislimitedbyoxygen supply(viasaturation).

Overall,simulations areconsistentwithmeasurements, regard-lessofthepeakvaluesthatareslightlyoverestimatednumerically, asshowninFig.11aandb,andin[44].

Concerning the average MLR after flameout presented in Fig 12b,wecansafelyassumethatmasslossisdrivenbysmoldering. Sincethelatteroccursatveryslowreactionrate,theaverageMLR isoneorderofmagnitudesmallerthantheoneofflaming,which makes itdifficult to distinguishtendencies. Globally,one can ob-servethat theaverage MLRincreaseswithbulkdensityandwith theflow,becausethereismorefuelandmoreair,respectively.This isconsistentwithEq.(5),wherethecharoxidationrateis propor-tional to the oxygen concentration and to the bulk density. The numerical predictions in HF compared well with measurements, asa resultoftheproperflow andheattransferestimation. How-ever,resultsunderNFconditionsdidnotmatchtheexperimental results because the charoxidation was not sustained due to the lowairsupplyin naturalconvection,asexplainedin [44].Hence, thereactiononlyoccurredinthefewsecondsafterflameoutthen dropped,resultinginahighaverageMLRcomparedtothe exper-iments,whichlasted longer.The smoldering time inNF couldbe

Fig. 13. Measured and simulated Heat Release Rates at HF for different bulk densities (a) 23 kg/m 3 and (b) 40 kg/m 3 . Error bars: standard deviation. Vertical line: flameout

time.

Fig. 14. Averaged values for measured and simulated heat release rates (HRR) per unit of initial fuel mass (a) during flaming (b) after flameout.

enhancedby eithertestingamorecomprehensivemodelfor esti-matingthecharcombustionrate,orbyaccountingfortheincrease ofthe surface to volume ratioin the solid phase (which is con-stant)asaresultofthecharporesthatareformedinthefuel.

The measuredandsimulatedHeatReleaseRate (HRR)are pre-sentedin Fig. 13 for two bulk densities atHF. Computed curves weretime-shifted sothat theycoincidewithexperimental curves whenignitionoccurred.ThepeakHRRwasslightlyoverestimated forboth cases,a resultof thehigh MLRpeaks.Nevertheless, the overalltrendismatchedandthetotalheatreleasedissimilar.The peakHRRat40kg/m3 _{(Fig.13b)isatapproximately11kW,which}

iscomparabletoresultsin[20,69]forAleppopine(Pinus halepen-sis) testedundersimilarconditions.The transitionbetween flam-ingandsmoldering(atflameout)wasbetterpredictednumerically forthehigherbulkdensity(Fig.13b).However,forthelowerbulk density(Fig.13a),an abrupttransitioncan benoticed onlyinthe simulation. This behavior can be justified by the larger amount offuelburning(i.e.highersolid fraction)availabletoincreasethe heattransferfromsolidtosolid,andtobettersustainingchar oxi-dationafterflameout.

For better illustration of the influence of the porosity in the HRR,averaged valuesformeasured andsimulatedHRRwere nor-malizedover their corresponding initial fuel mass (HRR/m0) and

are presentedduring flamingand after flameout, inFig. 14a and b,respectively.Thenormalizationallowsbettercomparisonofthe energyratereleased betweendifferentbulk densities. In Fig14a,

HRR_/m0 globally increases with the flow. This is consisted with

the observed MLR (Fig. 12a) and withthe shorter flaming times forhigher flowconditions. Essentially,Higher HRR isreachedfor shorter flamingtime due to thebetter mixing, and consequently the enhanced the combustion rate. The observed HRR/m0 trends

todecreasewithan increasingbulk density,whichisalso consis-tentwiththeaforementioned observationsaboutthecompetition betweenradiation penetration andoxygen availability. As forthe numericalpredictions,thesetendencies are well matchedathigh flow butcannotbe obtainedunderNF conditionsduetothe low combustionrateestimated[44].

Regarding HRR/m0 after flameout (Fig. 14b), overall the

mea-sured values are also an order of magnitude lower than during flaming.Theydecreasewithanincreasingbulkdensity,sincemore air can reach the reacting solid phase for lower bulk densities. HRR/m0 is consistently slightly higher at HF, especially for the

higher bulk density. Simulations of HF conditions matched the measurements.However,forNFcondition,ahighaverageMLRwas calculated after flameout (Fig. 12b). Consequently, the calculated HRR_/m0 wasoverestimated compared to measurements only

be-cause a higher HRR was released over a shorter period of time. However,thetotalenergyreleasedissmaller,sincenotallthechar wasoxidizedinNF.

Figure15a,b,andcdescribetheevolutionofthefueland oxy-genmassfractionsinside thefuelbedunderHFconditionsat ig-nition,5sand15safterignition,respectively.Itcanbenotedthat

Fig. 15. Fuel and oxygen mass fractions along the vertical and at the center of the sample (hatched area) during flaming. Figures a, b, and c correspond to distributions at ignition, 5 s and 15 s, respectively for HF conditions. Figures d, e, and f correspond to distributions at ignition, 20 s and 40 s, respectively for NF conditions.

thepropagationoccursfromtoptobottom,whichisinagreement withtheexperimentalobservations.Moreover,theevolutionofthe massfractions(step)arerepresentativeofatypicaldiffusionflame, meaningthatnopartialmixingininvolved.Itisclearthatthe oxy-geniscompletelyconsumedinthefirstcellofthereactionzone.

Ontheother hand,Fig.15d,e,andfrepresenttheevolutionof themassfractioninNFconditions.InFig.15d,theconditionsat ig-nitionaresimilartothoseofencounteredwithHF(Fig.15a).Asthe flamepropagatesslowerinNF,Fig.15eandfshowthemass frac-tions20and40safterignition.Itappearsthattheentirefuel sam-ple isembeddedin thefuel richzone andthat there isno suffi-cientoxygeninsideittoensureflamingcombustion.Anecdotal ob-servationsofexperimentsusingpineneedlesinthecone calorime-terundernaturalconvectionallowedtoseetheflameburning be-low thesample, whichis characteristic of thebehavior of a fuel richdiffusionflameandconfirmsthepredictionfromFig.15eand f.ThisbehaviorwasalsoobservedintheFPAinnaturalconvection butitwashardertoseebecauseofthelightoftheheaters.

Figures16and17presenttheexperimentalandsimulatedtime evolution of CO and CO2 generation for a fuel bulk density of

40kg/m3_{inHFandNF,respectively.Theexperimentalvalueswere}

measuredintheexhaustduct,similarlythesimulatedvalueswere obtained at the outlet boundary. Useful information can be ex-tractedfromthesefiguresandthreephasescanbedefined.During thefirstphase,whichoccursjustafterignition,onlyflamingis in-volvedasCO2productionincreases.Duringthesecondphase,

flam-ingandsmolderingoccur simultaneously.Finally,duringthethird phasewhichisafterflameout,onlyheterogeneouscombustion oc-curs. BycomparingFigs 16and17,it canbe notedthat thetime ofeachphasecanbeverydifferent.Innaturalconvection(Fig17),

Fig. 16. Measured and simulated CO and CO 2 production (HF - 40 kg/m 3 ). The ver-

tical lines represent the experimental (red dashed) and the numerical (black solid) flameout times. Plots are shifted to have ignition at 0 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

thetransitionfromflamingtosmoldering(secondphase)occursin ashortertimethaninforcedflow(Fig16).Thisbehaviorismainly duetothe additionaloxygen suppliedinside thefuel bedby the forcedflow, whichallowsflamingandsmoldering tooccur simul-taneously.Incontrast,smoldering occursmainlyafterflameout in naturalconvection.Thedifferencesin COcurvesbetweenHFand

Fig. 17. Measured and simulated CO and CO 2 production (NF - 40 kg/m 3 ). The ver-

tical lines represent the experimental (red dashed) and the numerical (black solid) flameout times. Plots are shifted to have ignition at 0 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

NF(Figs 16 and17) demonstratethe changingin behaviorin the combustion process. COconcentration is an indicator of the two stagesofcombustionwhen correlated tovisual observations.The firststepincreaseinFig.17isduetotheignitionofthesample.A steadyproductionofCOfollows,whichcorrespondstotheflaming stage(first phase) withpossiblyan oxidation ofCOinthe flame. Theconsequent increase is duetothe decrease oftheflame and theinitiationof smoldering combustion.The two processes over-lapduringthisstage(secondphase).Afterflameout,COproduction peaksduetothesoleoccurrenceofsmolderingcombustion(third phase).Incontrast,thesteadystateoftheCOdisappearsinHF.A fastercombustiontimewasobservedwithembersstartingtoglow beforeflameout.Thesetendenciesconfirmthemeasurements and theobservationsdescribedbySchemeletal.[19].

Concerning thenumericalpredictions, all themain tendencies werecapturedqualitatively by themodel,eventhough the quan-titativepredictions were not accurate.In both configurations,the simulationsoverestimatedtheCO2 productionduringflaming

be-cause the entire pyrolysis products were consumed in the gas phaseduetothenatureofthecombustionmodel.InFig.16, smol-deringwasnotactivatedduringflaming,therefore,noCOwas de-tectedinthesimulation.Inreality,notallCOfrompyrolysisis en-tirely oxidized,due to dilution withair and cooling. Indeed,the concentrationofmeasuredCOincreasedastheflamewas quench-ingandmoresmolderingwasoccurring.Anextinctionmodelcould beaddedtoaccount fortheunburnedgasesandbetterrepresent theincompletecombustionintheflame,especiallyatalargerscale [73].However, its absencedidnot severely influencethe burning dynamics of the pine needle bed (which was the focus of this work), asit was mainly driven by theincident radiation coming fromthelamps. In Fig.17,numerical COis detected even before flameout time, whichis dueto an early activation ofthe smol-dering process, as it was observed experimentally. Hence, both burningdynamics inNF andHF are captured by the model.The experimental peak value for CO was 25ppm for HF, whereas it was35ppm for the simulation. This is also due to the combus-tionmodelthatoverestimatedthecombustionefficiencyinthegas phaseandhadlittleinfluenceontheburningdynamics.An adjust-mentwill certainly be neededforlarge fires,whereextinction is likelyplayarole,especiallyifthemodelistobeusedtoestimate fireemissions.

4. Conclusion

The goal of the presented framework was to assess the rele-vance andthe performance of the submodels used to close CFD models particularly with the multiphase approach for wildfires. Somesubmodelsweresuccessfullyadaptedinaspecific and real-isticrangeofconditions,whichallowedimprovingburningrate es-timationsandbetterdescribing theunderlyingphysics. Moreover, theimportanceandtheimplicationsofusingappropriate submod-els were also demonstrated. By comparing simulations to exper-iments with various fuel bulk densities and various inlet flows, at a constant applied heat flux with a well defined fuel, it was shownwhichsubmodelsneedtobeappropriatelydefinedinorder to provideacceptable predictions. Because ofthe strongcoupling betweenthedifferentsubmodels,theproposedframeworkofusing controlled experimentsandmatchingsimulations atasmallscale isnecessaryforchoosingtheadequatesubmodels.Thiswillensure that thephysics involved inthe burningofthe fuel arecorrectly capturedbefore predictingfirespread atalarger scale.The main resultsaresummarizedasfollows:

• Theestimationofthedragforcecoefficient affectstheflow field inside the porous fuel,consequently it affects the as-sociated burning dynamicsduringflaming and smoldering. Theproposedcorrelationscanbeusedforsubgridmodeling andincludedinlargescalesimulationstorepresentthedrag forcethroughlitterbedsseparatelyfromtheonefortrees. • For the convective heat transfer coefficient, we have

veri-fied that Colburn coefficientsare appropriate for these ex-perimentalconditions, whereas other cited coefficients can preventthefuelfromigniting.

• Under forced flow conditions, the char oxidation model based on a single step Arrhenius equation is sufficient for characterizing smoldering combustion, and especially mass lossafterflameout.However,thismodelisnotadapted un-der natural convection, and further investigation is neces-sary.

• The additional split function in the char oxidation allows predictingan acceptablegasemissionandadequately mod-eling thetransitionbetweenflaming andsmoldering emis-sion.

• Overall,theselectedgasphasecombustionmodelalloweda successful predictionof the burning dynamics ofthe solid phaseinthe chosen rangeofconditions.However,it needs furtherimprovementsforsupportingemissionestimations.

Acknowledgments

The authors gratefully acknowledge the Fire Dynamics Group atFMGlobal forthe helpprovidedinthedevelopmentof Forest-FireFOAM.InsightsofferedbyG.Accaryaremostappreciated.This work was granted access to the HPC resources of Aix-Marseille Université financedbytheprojectEquip@Meso ( ANR-10-EQPX-29-01) oftheprogram« Investissementsd’Avenir» supervisedbythe FrenchAgenceNationalepourlaRecherche.

Appendix. GoverningequationsinFFF

Thegasphaseisgovernedbyasetoftransportequations rep-resentingthemassbalanceequations:

∂

α

g

ρ

¯

∂

t +

∂

α

g

ρ

¯u˜j

∂

xj =

(

1 −

ν

ash

)

ω

˙  char+

ω

˙  vap+

(

1 −

ν

char

)

ω

˙  pyr (9)

where ∼ is the Favre filter operator,

ν

char and

ν

ash are the char

∼ 0.05[15,74].Themomentumequationisdefinedas:

∂

ρ

¯u˜i

∂

t +

∂

(

α

g

ρ

¯u˜iu˜j

)

∂

xj =−

∂

¯p

∂

xi +

∂

∂

xj ×



α

g

ρ

¯

(

ν

+

ν

t

)



∂

u˜i

∂

xj +

∂

u˜j

∂

xi −2 3

∂

u˜k

∂

xk

δ

i j

+

α

g

ρ

¯gi− FD (10)

ν

isthemolecularviscosity,

ν

tisthesubgridscaleviscosity,andp

isthepressure.FD representsthedragforcesourcetermresulting

fromtheinteractionbetweenthegasflowandthesolidphase,and isdefinedas:

FD=

α

g

ρ

CD

α

s

σ

s

2

|

u

|

ui (11)

Since thermal equilibrium is not assumed between the solid fuelparticleandgaseousphase,thetemperatureinthesolidphase issolvedseparately,inthefollowingequation:

Cp(S)

α

s

ρ

s dTs dt =Q (S) ± −



hvap

ω

˙  vap−



hpyr

ω

˙  pyr−

α

sgQ_char(S) (12)

withQ_±(S) theenergy balance onthe solid phase exchanged with thegaseousphasebyconvectionandradiation.Cp(S)isthespecific

heatcapacityofthesolidphase,



hchar,



hpyr,



hvaparetheheat

of reaction for charring, pyrolysis, and evaporation, respectively. Thetime evolutionofthefuelischaracterizedbythevariationof its dry,water, andcharmassfractions. Theycanbe describedby thefollowingthreeordinarydifferentialequations:

d

α

s

ρ

s

ϕ

_H(S) 2O dt =− ˙

ω

 vap (13) d

α

s

ρ

s

ϕ

_DRY(S) dt =− ˙

ω

 pyr (14) d

α

s

ρ

s

ϕ

_char(S) dt =

(

ν

char−

ν

soot

)

ω

˙  pyr−

 ν

ash

ν

char +1



ω

˙_char (15)

Theglobalmassbalanceequationforthesolidphaseis: d

(

α

s

ρ

s

)

dt =

(

ν

char−

ν

soot− 1

)

ω

˙  pyr− ˙

ω

 char− ˙

ω

 vap (16)

Assumingthatthesolid consumptionisonlyduetochar com-bustion.Thebalanceequationforthesolidfractionisresolved fol-lowing: d

α

s dt =− ˙

ω

 char

ρ

s (17)

Finally,theenergybalanceofthegasphaseiswrittenas:

∂



α

g

ρ

¯h˜



∂

t +

∂



α

g

ρ

¯u˜j˜h



∂

xj =D¯p Dt +

∂

∂

xj



α

g

ρ

¯



α

D+

ν

t Prt

 ∂

_h˜

∂

xj

−Q(S)

CONV+QRAD+QCOMB−

(

1−

α

sg

)

Q_char(s) (18)

where h is the enthalpy,

α

D is the thermal diffusivity

(consider-ing aunity Lewisnumberapproximation).QCOMBandQRAD arethe

sourcetermforcombustionandradiationinthegasphase, respec-tively,asdescribedin[44].

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