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Electric Power Systems Research

j o ur na l ho me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r

Mathematical modeling of flashover mechanism due to deposition of fire-produced soot particles on suspension insulators of a HVTL

Emad H. El-Zohri

^a,∗

, M. Abdel-Salam

^b

, Hamdy M. Shafey

^c

, A. Ahmed

^b

aElectricalEngineeringDepartment,FacultyofIndustrialEducation,SohagUniversity,Sohag,Egypt

bElectricalEngineeringDepartment,FacultyofEngineering,AssiutUniversity,Assiut,Egypt

cMechanicalEngineeringDepartment,FacultyofEngineering,AssiutUniversity,Assiut,Egypt

a rt i c l e i n f o

Articlehistory:

Received4January2012

Receivedinrevisedform11August2012 Accepted17September2012

Keywords:

Mathematicalmodelingofagriculturalfires Sootdeposition

Fire-induced-flashover

a b s t r a c t

Thispaperpresentsamathematicalintegratedmodelthatsimulatesthecoupledeventscausingflashover duetothedepositionofsootparticlesonsuspensioninsulatorsofhighvoltagetransmissionlines(HVTL).

Themodelconsidersnon-steadythree-dimensionalmulti-phaseflowofagriculturalfireproducingthe sootparticles.Inaddition,themodeldescribesindetailthemechanismofthesootdepositioncombined withthedevelopingoftheelectricfield.Themodelequationsaresimultaneouslysolvedusinganiterative finite-volumenumericaltechniquetogetherwiththeindirectboundaryelementandchargesimulation methods.Themodelvalidityandaccuracyareverifiedthroughthediscussionoftheresultsforarepre- sentativecasestudyofa15kVcap-and-pininsulatorstring.Thediscussionincludesacomparisonofthe presentnumericalpredictionsforcharacteristicsofthedepositedsootlayer,electricfielddistribution, andcharacteristicsofflashoveroccurrence,withtheavailableresultsintheliterature.

Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction

High voltagetransmissionlines(HVTL)usuallycrossagricul- tural fields in which accidental fires may occur. Also, farmers intentionallyburncropresiduesinsuchagriculturalfieldsasahar- vestingaid.Theoccurringagriculturalfiresresultinenvironmental impactrepresentedbythermalandgaseouspollutionaccompanied withtheproductionofsootparticles.Thesootparticlesdeposited ontheinsulatorsurfaceofthehighvoltagetransmissionlinesare oftenreportedtocauseflashoverandconsequently,theoutages oftheselines.Insometropicalcountries,thenumberoflineout- agesduetofires,whetherintentionaloraccidental,canbeupto hundredsayearperline[1].Thisleadstoagreateconomiclossfor boththeutilityandusers.

Accuratemodelingoftheflashovermechanismrequiresdeep understanding and appropriate formulation of the equations governingthephysical and chemical processes associated with combustion,spreadoffire,and depositionoffire-producedsoot particles.Thepresenceofelectricfieldoftheenergizedtransmis- sionlineaffectstheflowanddepositionofsootparticlesasthesolid phaseofthemulti-phaseflowofthefireproducts.Thedeposited soot layer results in an increasing leakage current over the

∗Correspondingauthor.Tel.:+20882299037;fax:+2088332553.

E-mailaddress:[email protected](E.H.El-Zohri).

insulatorsurfacewhichbyturnleadstotheflashoverofhighvolt- ageinsulators.

Mostofthepublishedworks[2–6]onmathematicalmodeling ofcombustionandfirespreadweregenerallylimitedduetosim- plifyingassumptionswhichcannotbeextendedtomany ofthe realcases.Recently,thepresentauthorsdevelopedamorereal- isticmathematicalthree-dimensionalnon-steadyfiremodel[7]as apreparatorystageofthepresentwork.

The researchworks on topics related to fire spread investi- gate models and measurements of turbulent gas–particle flow and the soot particles deposition [8–10]. Current approaches commonlyusedtosimulateturbulentgas–particleflowincom- putational fluid dynamics (CFD) are the Eulerian–Eulerian and theEulerian–Lagrangianmodels[11,12].IntheEulerian–Eulerian approach,boththegasandparticleflowsaretreatedascontinu- ousfluidflowandregardedasinteractingwitheachother.Inthe Eulerian–Lagrangianapproach,theEulerianequationsofthegas phasearesolvedandtheLagrangianequationsofparticlemotion areintegratedbytrackingindividualparticlethroughtheflowfield.

Tian[13] investigated the performance of thetwo gas–particle models,anddevelopedparticle-wall collisionmodelsdescribing theassociatedboundaryconditions.ZhangandChen[14]useda modified Lagrangianmethodtopredictparticledepositiononto indoorsurfaces.ValentineandSmith[15]describedamodelcou- pledwithaparticlecloudtrackingtechniqueforpredictingparticle deposition in turbulent flow fields. Cohan [16] experimentally

0378-7796/$–seefrontmatter.Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.epsr.2012.09.009

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E.H.El-Zohrietal./ElectricPowerSystemsResearch95 (2013) 232–246 233

Nomenclature

a absorptioncoefficientthegas/sootmixture,m⁻¹ A,B,C cosinedirectionsofavector

D theeffectivediffusioncoefficientofanytransported variable

E magnitudeofelectricfieldstrength,Vm⁻¹ fvs sootvolumefraction

g_i gravityvector,ms⁻²

h enthalpyofthegasmixture,Jkg⁻¹ h heatofreaction,Jkg⁻¹

I current,A J irradiance,Wm⁻²

k turbulentkineticenergy,m²s⁻² m^acc_p massofaccumulatedsootparticles,kg p pressureofthegasmixture,Pa Pr Prandtlnumber

Q_i discretecharges,C R resistance,

R universalgasconstant Re Reynoldsnumber Sc Schmidtnumber

t time,s

T temperatureofthegasmixture,K TSL temperatureofsootlayer,K u_i velocityvectorofgasphase,ms⁻¹ up_i velocityvectorofsootparticles,ms⁻¹ Vp electricpotentialatanypointp,V

x_j x,y,zCartesiancoordinatesintensornotation Y massfraction

Greeksymbols

ıSL sootlayerthickness,m

ε turbulentkineticenergydissipationrate surfacechargedensity,Cm⁻²

viscosity

˙

ω reactionorprocessrate,kgm⁻³s⁻¹ anytransportedvariable

densityofthegasphase _p densityofthesootparticles

Stephan–Boltzmannconstant

turbulent Prandtl/Schmidtnumber for anytrans- portedvariable

SL electricconductivityofsootlayer,Sm⁻¹ timeconstant

incidentangleofsootparticles Subscriptsandsuperscripts

˛ gasspecies(˛=CO,CO₂,O₂,H₂OandN₂)

∞ freestreamvalue L leakagepathlength,m n normalcomponent p sootparticles

anytransportedvariable t tangentialcomponent SL sootlayer

validated a more specific model for thesoot deposition in fire dynamicssimulator (FDS). Important factors in soot deposition modelingare theparticledepositionvelocity and sootmaterial properties.Thesefactorscanbedeterminedusingmeasurements, models,andpropertiesindifferentpublishedworks[8,17,18].

Numerous works have been devoted to understand the phenomenaleadingtoflashoverofpollutedinsulatorsinorderto

elaborateamodelallowingonetopredictaccuratelythecritical flashovervoltage.Acommonlimitationformostproposedmodels [19–23]isthesimplifiedstaticrepresentationofpropagatingarcin serieswiththeresistanceofthepollutedlayer.Proposedmodelsin thelastthreedecadesindividuallyconsideredvariousparameters suchasarcdynamics[24–28],thechemicalnatureofthepollut- ants[29],andmultiplearcs[30].Thediscussedpublishedworks arevaluableasafirstsimpleapproachfortheseparatetopics(fire model,sootdeposition,andflashovermechanism)involvedinthe realcomplexcaseoffire-inducedflashover.

Theaimofthepresentworkistoformulateandsolveanon- steadythree-dimensionalmathematicalintegratedmodelforthe flashovermechanismduetothedepositionoffire-producedsoot particlesonhighvoltageinsulators.Thismodelaccuratelysimu- lates thecoupledreal eventsof themulti-phaseflow produced by agriculturalfires occurringbeneath a highvoltagetransmis- sionline.Also,themodelintroducesadetailedtreatmentofsoot particlesdepositionontheinsulatorstringunitswithaprecisely describedgeometry.Thefeaturesofthisgeometryarekeyfactors inflowanddepositionofsootparticles,electricfielddistribution, and consequently, flashovercriterion. The model equations are simultaneouslysolvedusinganiterativefinite-volumenumerical technique togetherwithcharge simulationand indirectbound- aryelementmethodsforcalculatingtheelectricfielddistribution.

Model numerical predictionsarepresented and discussed for a representativecasestudytocheckthemodelvalidityandaccu- racy.Thesepredictionsincludecharacteristicsofthedepositedsoot layer, electricfield distribution, and characteristicsof flashover occurrence.

2. Mathematicalformulation

The present mathematical model analyzes the transport phenomenaincludingfireproductsleadingtotheflashover,speci- fiestheconditionsfortheagriculturalfiremodel,anddescribesthe transmissionlineinsulatorboundaryandtheassociatedprocesses occurringatthisboundary.Themodelincludesthebasicsystem of differential and integral equations that governthetransport phenomena,withcorrespondinginitialandboundaryconditions.

The model equations aresimultaneously solvedusing anitera- tivefinite-volumenumericaltechnique.Thefollowingsub-sections describethedetailsofthepresentmathematicalmodel.

2.1. Theoreticalmodelandbasicassumptions

Fig. 1shows theimportant features describingthetheoreti- calmodelincludingtheagriculturalfiremodelandtheinsulator stringmodel.Thefigureshowsthecoordinatesystem,dimensions, andtheassociatedouterboundariesofthewholecomputational domain.Awind-drivenfirepropagatesintheagriculturalfuelbed ofadepthıM,widthWM,andlengthLMformingafiremodelwith aflameofaheightHM.Thewindflowisinthex-directionwitha verticalvelocitydistributionasshowninFig.1.Thedimensionsof thewholecomputationaldomainaremainlyexpressedintermsof thosefortheconceptualstructurerepresentingtheagriculturalfire model.Theseselecteddimensionsarelargeenoughtoagreewith thepracticalconsiderationsofcomputationalfluiddynamics(CFD) forwindenvironmentaroundstructures[31].Theinsulatorstring consistingofNidenticalrotationallysymmetricunitsisstressedby asystemvoltageVsystemataheightHstwithrespecttotheground plane.Theverticalaxisoftheinsulatorstringislocatedinthesym- metryplaneofthewholecomputationaldomain,atahorizontal lengthLstfromthefiremodel.Usually,theheightH_Moftheoccur- ringfirebeneaththetransmissionlineislessthantheheightHst

withenoughdistanceresultinginsmallvaluesoftheelectricfield

Fig.1. Generalfeaturesanddimensionsofthecomputationaldomainincludingtheagriculturalfiremodelandtheinsulatorstringofthetransmissionline.

atverticallevelsdowntoH_M.Thefollowingbasicassumptionsare consideredinthepresentmathematicalformulation.

(1)Addedtothestatedwinddirectionandlocationoftheinsulator stringaxis,theagriculturalfuelbedusuallyhasuniformstruc- tureandproperties.Accordingly,thespatialvariationsofthe flowpropertiesareconsideredidenticalaroundthesymmetry planeofthewholecomputationaldomain.

(2)Theeffectoftheelectricfieldcanbeneglectedin thespace downtotheflame heightHM.Accordingly,inthis spacethe sootparticlesaresimplyincludedinthegasphaseofthemulti- phasemediumofthefiremodel.Ontheotherhand,thesoot particlesaretreatedasaseparatesolidphaseintheotherspace ofthewholecomputationaldomainwheretheelectricfieldis effective.

(3)Thecontinuityandmomentumequationsareonlyconsidered forthesolidphaseofthesootparticleswhichareinthermal equilibriumwiththeassociatedgasphase.Inaddition,thedif- fusionandtheturbulencetermsinthemomentumequations areneglectedduetothesmalldensityofthesolidphasecom- paredwiththegasphasedensity.

(4)Nochemicalreactionsareconsideredinthespaceofthewhole computationaldomainabovetheflameheighteitherforthe carriergasphaseorforthedispersedsolidphaseofthesoot particles.

(5)Theeffectsofanyobjects(e.g.buildings,towers,trees...),exist- ingwithinthewholecomputationaldomain,areneglectedand attentionisfocusedonlyontheinsulatorstring.

Thestatementofassumption(1)suggeststhatonlyhalfofthe wholecomputationaldomainissufficienttodescribepreciselythe spatialvariationsofthewholeflow,withlesscomputations.More- over,foreasy butaccuratetreatmentofthewholeflow,halfof thecomputationaldomaincanbedivided,accordingtoassump- tion(2),intotwodistinctSub-domainsasshowninFig.2.Thetwo Sub-domainshavethecommonouterboundariesofinflow,outflow 1,outflow2,andthesymmetryplane.Besidestheseboundaries, theSub-domainIisboundedinverticaldirectionbytheground boundaryandtheimaginaryinterfaceplane,whiletheSub-domain IIis boundedbytheimaginaryinterfaceplaneand theoutflow 3boundary.TheSub-domainIconcernswiththeagriculturalfire modelrepresentingthemulti-phaseflowwhosesolidphaseisthe

stagnantagriculturalsolidfuelparticles.Ontheotherhand,the Sub-domainIIconcernswiththemulti-phaseflowofthefireprod- uctswhosesolidphaseisthesootparticles.

2.2. TreatmentoftheagriculturalfiremodelinSub-domainI Thefiremodelinthepresentstudyisrepresentedbyamulti- phasemedium,includingmulti-class agriculturalsolid fuel.The model assumptions, basic equations, and associated initialand boundaryconditionsareformulatedandpresentedindetailbythe present authors[7]. Inthis model, thedegradation ofthesolid fuel iscontrolled bydehydration, pyrolysis, and charoxidation processeswhosereactionratesareexpressedbyArrhenius-type equations.Thedegradationofthesolidfuelismainlycharacter- izedby thetimevariation of watercontent,dry fuel,char,and ashmassfractions.Theevolutionofthesemassfractionsbesides thefuel volume fraction,density,and temperature are allgov- ernedbyasetofsevenfirst-orderordinarydifferentialequations [7].

Theflowbehaviorofthegasphaseinthepresentfiremodelis describedbytheturbulencemodeledbalanceequationsformass, momentum,energy,andchemicalspeciesmassfraction.Thegas phaseisamixtureresultingfromthethermaldegradationofthe solidphasematrixandthecombustionreactions.Thesefirepro- cesses,normallycontributesoot andmajorfivegaseousspecies namely,CO,CO2,O2,H2O,andN2.Therefore,massfractionsofthese fivegasesbesidethesootvolumefractionhavebeenconserved.An ellipticpartialdifferentialequationinthegenericform[4–6]isused toexpressthebalancesofvariousflowpropertiesfornon-steady three-dimensionalgasflowofthefiremodel.

2.3. Basicequationsforthemulti-phaseflowofthefireproducts inSub-domainII

2.3.1. Gasphase

TheflowbehaviorofthegasphaseinSub-domainIIistreatedin amannersimilartothatofthefiremodelinSub-domainI.Thegas phaseinSub-domainIIisconsideredasamixtureofthesamefive combustiongasesCO,CO₂,O₂,H₂O,andN₂.Thesootparticlesare consideredasthesolidphaseofSub-domainIIwhoseflowchar- acteristicsarediscussedinthenextsub-section.Thegeneralform

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E.H.El-Zohrietal./ElectricPowerSystemsResearch95 (2013) 232–246 235

Fig.2.Maindifferencesbetweenthecomputationalsub-domains.

ofthebalanceequationsforthegasphasecanbeexpressedbythe followingellipticpartialdifferentialequation:

∂()

∂t +∂(u_j)

∂x_j = ∂

∂x_j

D∂

∂x_j

+S, (1)

whereisthegenericformofthetransportfluidpropertyhav- ingacorrespondingdiffusiveexchangecoefficientDandasource termS.Therepresentationofthegenericfluidpropertyandthe associatedexpressionsforDandSaresummarizedinTable1.

Thefollowingequationofstateisusedtocalculatethepressure pofthegasphaseassumingitamixtureofidealgases.

p=RT

˛

Y˛

W˛

, (2)

whereisthegasphasedensity,Ristheuniversalgasconstant,T isthegastemperature,Y˛isthemassfractionofgasspecies,and W˛isthemolecularweightofgasspecies.

Thelastterminthesourceexpressionformomentumequations inTable1representsthemomentumexchangebetweenthegas phaseandthesolidphasewithvelocitycomponentsup_i.Thetime constantinthistermiscalculatedas

= 18

pd²_soot, (3)

whereisthegasviscosity,pisthesolidphasedensity,anddsootis thediameterofthemono-sizedsootparticle,assumedtobe1␮m.

Thespecificenthalpyofthegasphasehisafunctionofthemass fractionsY˛andthegastemperatureT,givenby

h=

˛

Y˛

h⁰_˛+

T 0

C˛(T)dT

. (4)

The irradiance Jappearing in theenergy (specificenthalpy) equationiscalculatedbyintegratingthethermalradiationintensity inoveralldirections.Thisintensityisobtainedfromthesolutionof thethree-dimensionalradiativetransferequation[7].

Expressionsforturbulentviscositytandtheturbulencefunc- tionsP_k,W_k,andRarefoundelsewhere[6].

TheconstantsinTable1havethefollowingvalues.

Cε1 =1.42,Cε2=1.68,Cε3=1.5, _h= fvs= Y˛=0.7,

k=0.7179, _ε=1.3,Sc=Pr=0.71.

2.3.2. Solidphase

Thedispersedmono-sizedsootparticlesrepresentingthesolid phasecanbetreated asa quasi-fluidcontinuum[13] of aden- sityp,flowingwithvelocitycomponentsup_i.Theflowbehavior ofthis solidphaseiscontrolledbythefollowing continuityand momentumequations,intensornotation.

∂p

∂t +∂(pup_j)

∂x_j =0 (5)

∂(pup_i)

∂t +∂(pupjupi)

∂x_j =pg_i+p(u_i−up_i)+ε0

∂E²

∂x_i (6) Therighthandsideof Eq.(6) consistsofthree sourceterms representingthegravitationalforce,thedragforce(momentum exchange),andtheelectrophoreticforcewithEisthemagnitude ofelectricfieldstrengthandε₀isthedielectricconstantofair.

2.4. Initial,outerboundary,andcontinuityconditionsfor Sub-domainII

2.4.1. Initialconditions

TheinitialconditionsinSub-domainIIrepresentthevaluesof theflowpropertiesforbothgasandsolidphases,justatthebegin- ninginstantofthefire(att=0)whentheflowpropertiesarethose forthefreestream.Thus,theinitialconditionscanbesummarized asinTable2.

2.4.2. Outerboundaryandcontinuityconditions

In thepresentwork,theouterboundariesareplanes(Fig.1) enclosingthewholecomputationaldomain.Theinnerboundary isrepresentedbythesurfacesoftheinsulatorstringunits(Fig.2).

Table1

Summaryoftheconservationequations,Eq.(1),forgasphaseexpressedinthegenericform.

Fluidflowproperty ␾ D S

Mass 1 0 0

Momentum ui +t

−∂p

∂xi

+gi−2 3

∂

∂xj

eff∂uk

∂xk

−k

ıij

+ ∂

∂xj

eff

∂u_j

∂xi

−p(ui−up_i)

Specificenthalpy h _Pr+^t

h ag(J−4 T⁴)

Kineticenergyofturbulence k +^t

k Pk+Wk−ε

Rateofdissipationofturbulentenergy ε +_ε^t (Cε1−R)^ε_kP_k−Cε2^ε_k²+Cε3ε kW_k

Speciesmassfraction Y˛

Sc+ ^t

Y˛ 0

Table2

InitialconditionsofflowpropertiesinSub-domainII.

Gasphase Solidphase

u=v=w=0, T=T_∞, YCO=Y_CO,∞, YCO₂=YCO₂,∞, YO₂=YO₂,∞, YH₂O=YH₂O,∞, p=p_∞,

YN2=1−(Y_CO,∞+YCO2,∞+YO2,∞+YH2O,∞), ε=10⁻⁶m²s⁻³, k=10⁻⁶m²s⁻².

Theinitialvalueofthegasdensity()is deducedusingtheequationofstateofthe idealgas.

p=0, up=vp= wp=0.

Inthissub-sectiontheouterboundaryconditionsandthecom- monconditionsattheinterfaceplane(continuityconditions)are discussedandformulated.Thedescriptionandformulationofthe innerboundaryconditionsarediscussedindetailinSection2.5.

TheconditionsatthefiveouterplanesboundingtheSub-domain IIcanbeformulatedinthefollowingmannerasshowninTable3.

Thecommoncontinuityconditionsattheinterfaceplanearefor- mulatedandstatedinTable4.

2.5. Innerboundaryconditionsattheinsulatorstring

This sub-section is devoted for satisfactory modeling of the insulatorboundarygeometry,aswellasthoroughdescriptionof variousprocessesoccurringatthisboundary.Thedetaileddiscus- sionontheinsulatorboundaryenablesaccuratesimulationofthe correspondingboundaryconditionsaffectingthemulti-phaseflow Table3

OuterboundaryconditionsofflowpropertiesinSub-domainII.

Boundarynameand geometry^a

Boundaryconditions

Symmetry:

0≤x≤LC,y=0, HM<z≤HC

v=0, vp=0,∂

∂y =0 for =/v,

∂up

∂y =∂wp

∂y =0,∂p

∂y =0

Outflow1:

0≤x≤LC, y=

W_C

2, HM<z≤HC

p=p∞, ∂

∂y =0 for all,

∂up

∂y =∂vp

∂y =∂wp

∂y =0, ∂p

∂y =0

Outflow2:

x=LC, 0≤y≤

WC

2, HM<z≤HC

p=p∞, ∂

∂x =0 for all,

∂up

∂x =∂vp

∂x =∂wp

∂x =0, ∂p

∂x =0

Outflow3:

0≤x≤LC, 0≤y≤

WC 2, z=HC

p=p_∞, ∂

∂z =0 for all,

∂up

∂z =∂vp

∂z =∂wp

∂z =0, ∂p

∂z =0

Inflow:

x=0, 0≤y≤

WC

2, HM<z≤HC

v=w=0, Y˛=Y˛,∞, T=T∞, p=p∞, up=vp=wp=0, p=0,

Equationsofreference[6]areusedtocalculatethe airinflowproperties;u,k,andεintermsofsurface roughnessz0andwindvelocityUwatagiven heightHw.

aLC=5HM+LM+Lst+15HM,WC=5HM+WM+5HM,HC=HM+5HM.

insideSub-domainII.Thediscussionalsofocusesontheanalysis towardsthemaintargetofthepresentwork,namely,deposition ofsootparticles(solidphase)andtheassociatedflashovermecha- nism.

2.5.1. Descriptionoftheinsulatorgeometry

TheinsulatorstringconsistsofNidenticalunitsconnectedto each otherasshown in Figs.1 and2 whose surfacesrepresent theinsulatorboundary.Consequently,arepresentativegeometric descriptionofthesurfaceforoneunitisdemonstrated.Fig.3shows themainfeaturesofthegeometryfortheunitmodel.Theunitcan beapproximatedbyanoblatespheroid(Fig.3a).Thespheroidis locatedatacenterO’(xu,yu,zu)asshowninFig.3b.Thespheroid surfaceisobtainedbyrotatingthehalfofanellipseaboutitsminor axis(z’).Theangleofrotation(0–2)andtheangleofellipse generationˇ(−/2to/2)arethesurfacedescriptionparameters (Fig.3b).Thehalf-lengthsaand bofthemajorand minoraxes, respectively,areobtainedfromtherealdimensionsoftheinsula- torunit.Theformulationoftheboundaryconditionsattheunit surfacerequiresthemathematicaldescriptionofthetangentplane andtheoutwardnormalvectorwithordinate(AppendixA)ata generalpointP(x’,y’,z’)onthesurfaceasshowninFig.3a.

2.5.2. Boundaryconditionsforthegasphaseattheinsulatorunit surface

Considering the gasflow, theinsulator represents a bound- aryofcurvedsolidnon-porouswall.Atthisboundary,theno-slip conditionforthevelocitycomponentsandturbulenceapplies.An appropriatethermalconditionattheinsulatorsurfaceisthatthe gastemperatureTequalsthesootlayertemperatureT_SL.Thevalue ofTSLisdeterminedlaterinSection2.7.Theimpermeabilitycondi- tionisappliedforthemassfractionsofthegasphasespecies.Thus, thegasphaseboundaryconditionscanbewrittenas

u=v=w=0,k=0,ε=0,p=p_∞,T=T_SL,∂Y˛

∂ =0 (7) 2.5.3. Boundaryconditionsforthesolidphaseattheinsulator unitsurface

The conditions of the solid phaseparticles at the insulator surfacecanbeidentifiedthroughinter-relationsfortheparticles velocityvectoranddensity.Theapplicationofthisconceptdeals

Table4

Commoncontinuityconditionsforthetwosub-domainsattheinterfaceplane

0≤x≤LC, 0≤y≤^W₂^C, z=HM

.

Solidphase Gasphase

(up)_II=(u)I,(vp)_II=(v)I, (wp)_II=(w)_I,(p)_II=soot(fvs)_I

(u)II=(u)I,(v)II=(v)I,(w)II=(w)I, ()_II=()_I,(Y˛)_II=(Y˛)_I,(T)_II=(T)_I Note.ThesubscriptIreferstoSub-domainI,thesubscriptIIreferstoSub-domainII, thesootvolumefractionfvsisaflowpropertyforsub-domainI,andsootisthesoot density.

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E.H.El-Zohrietal./ElectricPowerSystemsResearch95 (2013) 232–246 237

Fig.3. Mainfeaturesofthegeometricmodelfortheinsulatorunit:(a)thespheroidsurfaceofthemodeland(b)surfaceparametersanddimensions.

withthechoiceofacontrolspaceadjacenttothespecifiedpointP ontheinsulatorsurface.Fig.4showssuchacontrolspacewhich isboundedbythetangentplaneatthepointPandanimaginary parallelplaneataheighth(muchlargerthanthethicknessofthe depositedsootlayer).Thesootparticlesarrivingattheinsulator surfacewithincidentvelocityvectorVwand densityp,wsuffer reboundingwithrestitutioncoefficientsassociatedwiththeinward normalcomponentu^p_n,w andthetangentialcomponentu^p_t,w.Both incidentandreflectedvelocitiesconstitutetheflowvelocityfield inthecontrolspaceadjacenttothepointP.Also,massofthesoot particlesisconservedforthiscontrolspace.

Fig.4. Thecontrolspacewiththesootparticlevelocityvectorcomponents.

Neglectingtheeffectofsootdepositiononmassconservation, theboundaryconditionscanbeidentifiedbythefollowinggeneric equation[13]:

aw+b∂

∂

w

=c =

u^p_n,u^p_t,p

(8)

Theassociatedcoefficientsa,b,andcareexpressedintermsof restitutioncoefficientsenandet.Thesearefunctionsoftheincident angle=tan⁻¹

u^p_t,w/u^p_n,w

[13],andaregivenby

en=0.993−1.76+1.56²−0.49³ (9) and

et=0.988−1.66+2.11²−0.67³ (10) Theinwardnormalvelocityincidentattheinsulatorsurfaceu^p_n,w mustexceedavalueknownascapturevelocity,ucforthesootpar- ticlestoreboundfromtheinsulatorsurface.So,foru^p_n,w≤ucboth oftherestitutioncoefficientsenandetareequaltozero.Thecapture velocityucdependsonmanyfactors(e.g.particlesizeandmaterial, insulatorsurfaceconditions...),andistobedeterminedexperi- mentally.Basedontheavailabledataintheliterature,areasonable valueofuc=0.001ms⁻¹forsootparticlesisadopted[8–10].The differentialformofthegenericequation(Eq.(8))istransformed toalinearalgebraicequationintheunknownw.Thisisachieved bysubstitutingforthederivative ^∂_∂

w

withitsforwardnumerical formulaintermsofthevalueofat=h.Thealgebraicformof thegenericequationhandlesthevelocitycomponentsu^p_n andu^p_t. However,thecouplingoftheinnerboundaryconditionsisthrough theCartesiancomponentsup_iusedintheflowmomentumequation (Eq.(6)).Thus,atwo-waytransformationbetweenthetwosystems ofvelocitycomponentsmustbeperformed(AppendixB).

2.6. Evaluationofthedepositedsootlayerontheinsulatorstring Thefire-inducedpollutionissensitivetothefiremodelparam- eters. These parameters include the horizontal lengthbetween insulatorstringandfiremodel(Lst),lengthoffiremodel(L_M),width offiremodel(WM),heightoffiremodel(HM),depthoffuelbed

Fig.5.Thedifferentialcontrolvolumeinthesootlayerandthethermalenergyrates.

(ıM),windvelocity(Uw),andinitialfuelmoisturecontent(FMC).

Asthedimensionsoftheburntfieldconfigurationincrease,thefire intensityincreasesandconsequently,theproduced-sootsothat thechanceforhighrateofdepositionincreases.Asthewindveloc- ityincreases,themomentumofsoot particlesincreasessothat thechanceforhighrateofdepositiondecreases.Also,astheini- tialfuelmoisturecontentincreases,thefireintensitydecreasesso thattheproduced-sootdecreasesandconsequently,lowdeposition rate.

Thesootparticleswithnormalvelocitiesu^p_n,wlessthanorequal tothecapturevelocity uc are liabletodeposit ontheinsulator surface.Therefore,thelocalinstantaneousdepositionfluxofsoot particles(DF)pcanbeestimatedby

(DF)_p=_p,wu^p_n,w, u^p_n,w≤u_c (11) The localinstantaneousaccumulatedmassofdeposited soot particlesperunitaream^acc_p iscalculatedas

m^acc_p =

t

0

(DF)_p dt (12)

Thelocalinstantaneoussoot-layerthicknessı_SLontheinsulator surfaceisestimatedas

ı_SL= m^acc_p SL

, (13)

wherethesoot-layerdensitySL=(1−ε_v)soot.Thevoidageratio εvforloosepackingofsootparticlescanbetakenas0.45andthe bulkdensityofthesootmaterialsootistakentobe1800kgm⁻³ [5].

Thethermalinnerboundaryconditionofthegasphaseatthe insulatorsurface,T=T_SLinEq.(7),requiresaccurateknowledgeof thelocalinstantaneoussoot-layertemperatureT_SL.Thevalueof TSLisafunctionofthesoot-layerthicknessıSLandvarieswithtime andlocationontheinsulatorsurface.Thetemperaturedistribution functionwithinthedepositedsoot layercanbeobtainedinthe followingmanner.

2.7. Temperaturedistributionwithinthedepositedsootlayer Fig.5showsadifferentialcontrolvolumeinthesootlayerata generalpointP(x’,y’,z’)onthespheroid(insulatorunit)surface.

ThecontrolvolumehasaheightıSL,adifferentiallengthdL,anda differentialwidthdS.ThelengthListhetotallengthontheperiph- eryofthehalf-ellipsesdescribingtheprofileoftheinsulatorstring unitsmeasuredfromthelowestpoint.Thislengthisreferredas

theleakagepathlength.ThelengthSisthelengthonacircleof rotation.ExpressionsfortheleakagepathlengthLandthelength SonthesurfaceoftheinsulatorstringaregiveninAppendixAin termsofthesurfacedescriptionparametersˇand.Thediffer- entialcontrolvolumeisexposedtodifferentthermalenergyrates withitslowersurfaceisassumedthermallyinsulatedasshownin Fig.5.Thethermalenergyratesareclassifiedasgenerationrate qgen,heattransferratesbyconductionqL,qS,q_L+dL,andq_S+dS,and heattransferratebyconvectionqconv.Thethermalenergygenera- tionisaresultoftheleakagecurrentflowingthroughthesoot-layer alongtheabove-mentionedleakagepath.Expressionsfortheheat transferratesarederivedusingFourier’slawforconductionand Newton’slawofcoolingforconvection[32].Thesethermalenergy ratesarebalancedwiththerateofchangeofthestoredenergyin thedifferentialcontrolvolume.

Applyingthefirstlawofthermodynamicsonthecontrolvol- umeofFig.5andsubstitutingforthedifferentenergytermsby theirexpressions,thefollowingtwo-dimensionalnon-steadydif- ferentialheattransferequationforthesootlayertemperatureTSL

isobtained.

∂T_SL

∂t =D_SL

1 ıSL

∂

∂L

ı_SL∂T_SL

∂L

+ 1 ıSL

∂

∂S

ı_SL∂T_SL

∂S

+ ^SLE²_t,L K_SL + Kgas

ı_SLtK_SL

∂T

∂

w

,

wherethethermaldiffusityofthesootlayerDSL=KSL/SL·cSL,KSL

(=1Wm⁻¹K⁻¹)[32]isthethermalconductivityofthesootlayer, c_SL(=800Jkg⁻¹K⁻¹)[32]isthespecificheatofthesootlayer, _SLis thelocalinstantaneouselectricconductivityofthesootlayer,Et,L

isthetangentialcomponentoftheelectricfieldstrengthalongthe leakagepath,K_gasisthethermalconductivityofthegasmixture, and ^∂T_∂

w

isthegastemperaturegradientintheoutwardnormal directionattheinsulatorsurface.

TheinitialconditionassociatedwithEq.(14)canbewrittenas theinsulatorstringsurfaceisinitiallyatambienttemperatureT_∞. Thecorrespondingboundaryconditionsarebasedontwo facts:

symmetryconditions(at=0and=),andthermalinsulation conditionsatconnectingpointsbetweeninsulatorstringunits(at ˇ=−/2andˇ=/2).

Thevariationoftheelectricconductivityofthesootlayer SL

withitstemperatureT_SLisobtainedby

SL= ^SL∞

[1+˛_SL(T_SL−T_∞)] (15)

where SL∞(=1500Sm⁻¹)[32]istheelectricconductivityofthe sootlayeratT_∞,˛_SL(=−0.0005K⁻¹)[32]isthetemperaturecoeffi- cientofthesootlayer.

2.8. Electricfielddistribution

Thepresenceofthedepositedsootlayerpollutingtheinsulator surfacedistortsthecapacitivepotentialdistributionandtheelec- tricfieldisnolongercapacitivebutmaybecapacitive-resistiveor resistive,dependingontheseverityofthesurfacepollution.The algorithmdevelopedforcalculatingtheelectriccapacitive-resistive fielddistributionisbasedontheindirectBoundaryElementMethod andtheChargeSimulationMethod.Theinsulatorunderinvestiga- tion(Fig.3)issurroundedbyairandisstressedbetweenapair ofelectrodes(capandpin).Thetopelectrodeistakenasgrounded whilethebottomelectrodeisstressedbytheappliedvoltage.Since theinsulator-electrodesassemblyisarotationallysymmetriccon- figuration, ring charges[33] are employed for simulation. Each electrodeis simulatedbya setofNe ringchargesplacedwithin theelectrode.Theinsulator–airinterfaceissimulatedbytwosets

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E.H.El-Zohrietal./ElectricPowerSystemsResearch95 (2013) 232–246 239

Fig.6.Spheroidalelement.

of2N_dringcharges,onesetplacedininsulatorandothersetinair.

Imagesof thefictitiousringchargeswithrespecttotheground planeare alsoconsidered in thepresent algorithm. For power frequencycapacitive-resistivefieldcalculation,complexfictitious chargesvaryingsinusoidallywithtimeareemployedtogivethe instantaneousfielddistribution[34,35].Theelectrodesandinsu- latorboundariesarediscretizedintoseveralboundaryelements andasuitablepolynomialisintroducedfortheequivalentsurface chargesalongthediscreteboundaryelements.Thentheelectric fieldintheregionofinterestisconsideredtobecausedbytheequiv- alentsurfacechargesalongtheboundaryelements.Thepotential V_p,dataspacepointpwithordinaterduetothesurfacecharges withinafiniteelementofareaA(Fig.6)isexpressedbytheFred- holmintegralequationofthefirstkindas

V_p,d= 1 4ε

A

(ˇ,)

r− r

^{dA, (16)}

wheredAisthedifferentialareacenteredatasurfacepointP(ˇ,) withspaceordinater, (ˇ,)isthecorrespondingsurfacecharge density,andεisthedielectricconstant.Thesurfacechargedensity (ˇ,)canbeexpressedasasimple4terms-polynomial[36].Sub- stitutingfor (ˇ,),r,anddAintermsofˇand,Eq.(16)canbe re-writtenas

V_p,d= 1 4ε

ˇ₂ ˇ₁

_{2 1}

(k₀+k₁ˇ+k₂+k₃ˇ)(a cosˇ

a²sin²ˇ+b²cos² ˇdˇd)

(xu+a cosˇcos−x)²+(yu+a cosˇsin−y)²+(zu+b sinˇ−z)²

(17)

Thefourcoefficientsofthispolynomial(k0,k1,k2,andk3)canbe givenintermsofthecoordinates(ˇ₁,ˇ₂,₁,and₂)andthecharge densities( ₁, ₂, ₃,and ₄)ofthefourverticesoftheelement A(Fig.6).Integratingandsurveyingallelementsoftheinsulator string,onecangetthepotentialV_p,datthepointpduetoallsurface chargedensitiesasasummationover2N_dunknownsurfacecharge densities _i,i=1,2,...,2N_d eachmultipliedbyacorresponding potentialcoefficientfunction.Asimilarsummationcanbeconsid- eredforthecontributionofthepotentialVp,eatthepointpdueto 2NeunknowndiscretechargesQ_i,i=1,2,...,2Nesimulatingthe electrodes.ThusthetotalpotentialVpcanbeexpressedas

Vp=Vp,e+V_p,d=

2Ne

i=1

P_e,i·Q_i+

2Nd

i=1

P_d,i· i, (18)

whereP_e,iandP_d,iarethepotentialcoefficientfunctionsduetodis- cretechargessimulatingtheelectrodesandduetosurfacecharge densitiesontheinsulatorsurface,respectively.

The electricfield strength atthepoint p canbeobtainedas Ep=− ∇Vp,andthetangentialcomponentalongtheleakagepath isgivenbyE_t,L=∂V_p/∂L.

TheunknowndiscretechargesQ_iandsurfacechargedensities _i inEq.(18)arechosensuchthattheysatisfythefollowingboundary conditions.

Dirichlet’sconditionontheelectrodesurfaceis

Vp=Ve, (19)

where Ve is the known potentialon theHV electrodeand the groundedelectrodewithrespectivevaluesofsystemvoltageVsystem

andzero.

Potential continuity condition on theinsulator–air interface (insulatorsurface)is

Vp|ε=ε_insulator=Vp|ε=ε₀ (20)

Continuityconditionofthenormalcomponentoftheelectric fluxdensityDnontheinsulatorsurfaceis

Dn|ε=εinsulator−Dn|ε=ε0 = true, (21)

where trueisthetruesurfacechargedensity.Thenumericalvalue ofthetruesurfacechargedensity _true,ijatasurfacepointp,node i,j,iscalculatedintermsofthepotentialsatthespecifiednodeand thesurroundingnodesas

true,ij = 1 iω

1 A_ij

(Vp)_i−1,j−(Vp)_i,j

R_i −(Vp)_i,j−(Vp)_i+1,j R_i+1

+(V_p)_i,j−1−(V_p)_i,j

R_j −(V_p)_i,j−(V_p)_i,j+1 R_j+1

, (22)

wherei=√

−1,ωistheangularfrequency,andA_ij=(L·S)_ijis theelementsurfaceareaatthenodei,j.

TheresistancesR_i,R_i+1,R_j,andR_j+1inEq.(22)arecalculatedin termsofthelocalsurfaceresistanceR_SLatthedifferentnodes,and theassociateddifferencesofLandS.Thelocalsurfaceresistance R_SListhereciprocaloftheproductofthelocalsootlayerthickness ıSLandconductivity SL.

2.9. Flashovercriterion

Theleakagecurrentincreasesduetotheaccumulateddeposi- tionofthesootlayer.Thisincreaseintheleakagecurrentcontinues

totheextentcausing startofpartialarcs(discharges)alongthe leakagepath.Withfurtherincreaseintheleakagecurrent,these partialarcselongateandjointocauseflashoveroccurrence.This requiresaconditionthatthearclengthbridgesasmuchas67%of thetotalleakagepathlength[37].

Thestartofdischargeataspecifiedpointontheinsulatorsurface iscontrolledbythearcpropagationcriterion(Et,L>Earc,L),where thearcelectricfieldstrengthalongtheleakagepathE_arc,Lcanbe expressedintermsoftheleakagecurrentI_Lby

E_arc,L=AI_L⁻ⁿ (23)

Forairenvironment,thearcconstantstakethevaluesA=63and n=0.76[28].TheleakagecurrentILcanbeestimatedastheaverage

Fig.7. Timeevolutionofthedepositedsootlayerthicknessalongtheleakagepathlengthfordifferentlocationsontheinsulatorstring.

Fig.8. Timeevolutionofthedepositedsootlayertemperaturealongtheleakagepathlengthfordifferentlocationsontheinsulatorstring.

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E.H.El-Zohrietal./ElectricPowerSystemsResearch95 (2013) 232–246 241

Fig.9.Timeevolutionforthesurfaceresistanceofthedepositedsootlayeralongtheleakagepathlengthfordifferentlocationsontheinsulatorstring.

ofitslocalnumericalvaluesI(Li)(passingthroughfinitecircular ringsalongtheleakagepath),calculatedby

I(L_i)=2

m j=1

SLE_t,Lı_SLS

ij (24)

wheremisthenumberofthejthnodesovertheithfiniteringand (S)_ijisthedifferencelengthonthecorrespondinghalfring.

2.10. Numericalsolution

Thedifferentialequationsofthepresentmathematicalmodel are simultaneously solved using an iterative finite-volume

Fig.10.Effectofvaryingthesystemvoltageonthedepositionofthesootlayer.

numericaltechnique.Thesetofthefirst-orderordinarydifferen- tialequationsforthesolidphaseinthefiremodelaresolvedusing thefourth-orderRunge–Kuttamethod.Thegasphaseequationsfor thefiremodelinSub-domainIandthegasphaseequationsaswell asthesolidphaseequationsinSub-domainIIarealldiscretized ona staggered,nonuniformcartesianthree-dimensional gridof finitecells.Asecond-orderbackwardEulerschemeisusedfortime integration.Asecond-ordercentraldifferenceschemeisusedto approximatethediffusionterms.Theresultingdiscretizedequa- tionsareasystemoflinearalgebraicequationswhicharesolved iterativelyusingtheline-by-linetridiagonalmatrixalgorithm[38].

Inordertoaccelerateconvergence,allthegasvariablesareunder- relaxedusinginertialrelaxation.Adiscreteordinatemethodisused asapartofthenumericalsolutionfortheradiativetransferequa- tion[39].Thechargesimulationmethodandtheindirectboundary elementmethodareusedforthenumericalcalculationoftheelec- tricfielddistributionasformulatedindetailinSection2.8.Usually, itisneededtopredictnumericalresultsforaspecificcasewithin assignedtime.Therunningofcalculationsiscompletedunlessthe conditionofflashoveroccurrenceissatisfiedbeforethistime.

3. Resultsanddiscussion

Thepresentmathematicalmodelwassimulatedbyacomputer programwhichhasbeendevelopedandprocessedtotestthemodel validityandaccuracy.Arepresentativecasestudyofsingle-phase high voltage transmission linewith a single insulator string is selectedtoperformthistask.Thetransmissionlinecrossesanagri- culturalfuelbedofnaturaltallgrasslandinwhichawind-driven fire propagates. The main input datarelevant tothe transmis- sionlineandtheagriculturalfiremodelarelistedinTable5.The selectedvaluesoftheparametersinTable5forsimulationsare

Fig.11.Timeevolutionforpotentialalongtheleakagepathlengthfordifferentlocationsontheinsulatorstring.

Fig.12. Timeevolutionforelectricfieldstrengthalongtheleakagepathlengthfordifferentlocationsontheinsulatorstring.