DevelopmentandcomparisonoftheDTM,theDOMandtheFVMformulationsfortheshort-pulselasertransportthroughaparticipatingmediumSubhashC.Mishraa,*,PranshuChugha,PranavKumara,KunalMitrabaDepartmentofMechanicalEngineering,IndianInstituteofTechnologyGuwahati,Guwahati781039,IndiabDepartment of Mechanical and Aerospace Engineering, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL 32901-6975, USAReceived5May2005;receivedinrevisedform9September2005Availableonline19January2006AbstractThepresentarticledealswiththeanalysisoftransientradiativetransfercausedbyashort-pulselaserirradiationonaparticipatingmedium. A general formulation of the governing transient radiative transfer equation applicable to a 3-D Cartesian enclosure has beenpresented. To solve the transient radiative transfer equation, formulations have been presented for the three commonly used methods inthestudyofradiativeheattransfer,viz.,thediscretetransfermethod,thediscreteordinatemethodandthenitevolumemethod.Toshow the uniformity in the formulations in the three methods, the intensity directions and the angular quadrature schemes for computingthe incident radiation and heat ux have been taken the same. To validate the formulations and to compare the performance of the threemethods, eect of a square short-pulse laser having pulse-width of the order of a femtosecond on transmittance and reectance signals incase of an absorbing and scattering planar layer has been studied. Eects of the medium properties such as the extinction coecient, thescattering albedo and the anisotropy factor and the laser properties such as the pulse-width and the angle of incidence on the transmit-tance and the reectance signals have been compared. In all the cases, results of the three methods were found to compare very well witheachother.Computationally,thediscreteordinatemethodwasfoundtobethemostecient.2005ElsevierLtd.Allrightsreserved.1.IntroductionIntherecentpast, researchontheapplicationoftran-sient radiative heat transfer in participating media hasgained a momentum. This owes to the availability ofshort-pulse lasers combinedwiththe rapiddevelopmentin electronics to analyze the signals having temporal varia-tions of the order of a femtosecond to a picosecond. Appli-cations of short-pulse laser transport in participating mediainclude,butarenotlimitedto,opticaltomographyoftis-sues [14], remote sensing of oceans and atmospheres[57], laser material processingof microstructures [810]andparticle detectionandsizing[11]. Adetailedreviewdealing with various aspects of the transient radiativetransfer caused by the irradiation of short-pulse lasershasbeengivenbyKumarandMitra[10].To analyze radiative transport problems involvingsteady-state diuse and/or collimated radiation, more thana dozen methods are available [12,13]. Depending upon themediumtypes, geometriesinvolvedanditsapplicationinthe presenceof othermodesof heat transfer,eachmethodhas some strong and weak points [12,13]. However, amongthe available methods, the discrete transfer method (DTM)[14], the discrete ordinate method (DOM) [15] andthenitevolumemethod(FVM)[16] areextensivelyusedforthe transient or the steady-state conjugate mode (radiation,conductionand/orconvection) heat transfer problems inwhichtransportofradiationisconsideredtobeasteady-stateprocess[12,13].Inthepresentdecade,withthestartofresearchinareaof transient radiative transfer, applications of various0017-9310/$-seefrontmatter 2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.ijheatmasstransfer.2005.10.043*Correspondingauthor.Tel.:+913612582660;fax:+913612690762.E-mailaddress:scm_iitg@yahoo.com(S.C.Mishra).www.elsevier.com/locate/ijhmtInternationalJournalofHeatandMassTransfer49(2006)18201832radiative transfer methods were extended to transient stud-ies [1731]. Kumar et al. [17], Mitra et al. [18] used the P-1approximation to model the transient radiative transferin1-Dand2-Drectangularenclosures. Integral equationformulationwas usedbyTanandHsu[19,20] andWuand Wu [21]. Wu and Ou [22] applied the dierentialapproximationtoanalyzetheproblems. TheMonteCarlomethodwas usedtomodel thetransient radiativetrans-fer bySchweiger et al. [23] andGuoet al. [24,25]. GuoandKumar[26] andSakami etal. [27] extendedapplica-tions of the DOMto the 2-Drectangular enclosure.ApplicationoftheDOMtothe3-Drectangularenclosurewas extendedbyGuoandKumar [28]. Rathet al. [29]usedthe DTMtostudy the transient radiative transferprocessinaplanarmedium. Theirresultscomparedverywell withthose of the DOMandthe integral equationformulation. The application of the radiation elementmethodtostudythetransientradiativetransferinvariousrectangulargeometrieswasextendedbyGuoandKumar[30]. TheFVMwasappliedtostudytheshort-pulselasertransport inparticipating media by Chai [31] andChaietal.[32].A comparative study of the four methods, viz., the P-Napproximation, the two-ux, the DOMand the directnumerical integration methods in predicting transient radi-ativetransferinaplanarmediumwasdonebyMitraandKumar[33].Theyshowedthatallmethodsdonotpredictthewave propagationspeeds correctly. WuandOu[22]compared the modied dierential approximations, theintegral equation formulation and the modied P-1approximation in the study of a planar medium. Theyfound good agreements between dierential approxima-tionsandtheintegralequationformulation.It has been pointed above that the DTM, the DOM andFVMarethewidelyusedmethodsinanalysisofthermalproblems inwhichradiationis consideredtobe inthesteady-state, and recently these methods have alsobeenappliedtotransient radiativetransfer. However, asfar as comparison of these methods is concerned, no studyhas been performed so far to check the accuracy andNomenclatureA areaa anisotropyfactorc speedoflightG incidentradiationH HeavisidefunctionI intensityIbblackbodyintensity,rT4p^i;^j;^k unitvectorsinx-,y-,z-directions,respectivelyM numberofdiscretedirectionsp scatteringphasefunction^n outwardnormalq heatuxr positionS sourceterms geometricdistanceinthedirectionoftheinten-sityT temperaturet timetppulse-widthV volumeofthecellX, Y, Z dimensionsofthe3-Drectangularenclosurex, y, z CartesiancoordinatedirectionsGreeksymbolsa nite-dierenceweighingfactorb extinctioncoecient(=ja + rs)d Dirac-deltafunctionjaabsorptioncoecientl directioncosinewithrespecttothex-axise emissivityh polaranglen directioncosinewithrespecttothey-axisg directioncosinewithrespecttothez-axisr StefanBoltzmann constant = 5.67 108W/m2K4rsscatteringcoecients opticaldepthX direction,(h, /)DX solidangle,sin hdhd/x scatteringalbedo rsb/ azimuthalangleSubscriptsav average0 forcollimatedradiationc collimatedd diuseE, W, N, S, F, Beast,west,north,south,front,backw wall/boundarye exiti inletP cellcentreref reectances startx, y, z forx-,y-,z-facesofthecontrolvolumet totaltr transmittanceSuperscriptsD downstreampointm indexforthediscretedirectionU upstreampoint*dimensionlessquantityS.C.Mishraetal./InternationalJournalofHeatandMassTransfer49(2006)18201832 1821computational eciency of one over the other. The presentworkis,therefore,aimedatcomparingtheresultsandthecomputational ecienciesofthesethreepopularmethodsforvariousparameters.Intheliterature,theformulationsof these methods, for the steady [12,13] as well as for tran-sient[2632]statesarepresentedinamannerwhichseemtobeentirelydierent. Oneotherobjectiveofthepresentarticle is thus also to present a general formulation inwhichinthethreemethodsthesameintensitydirections,expressionsforthesourcetermsandthequadraturesfortheangularintegrationsof theincidentradiationandheatux can be used. Further, the ray tracing algorithms in theDOMandtheFVMarealsoexactlythesame. Theonlydierence between the two methods being that in theFVMunliketheDOM, intensityoveranelemental solidangleis not consideredisotropic. Theraytracingproce-dures inthe DTMis dierent fromthose of the DOMandtheFVM. However, forthetransientstudy, thetimemarchingprocedureinthethreemethodsisthesame.To validate the formulations in the DTM, the DOM andtheFVM, eect of asquareshort-pulselaser irradiationon transmittance and reectance signals froma planarmediumhasbeenstudiedandcompared. Thesecompari-sonshavebeenmadefortheeectsoftheextinctioncoef-cient, the scattering albedo, the anisotropy factor, theangle of incidence and the pulse-width of the laser irradia-tion. CPUtimes in the three methods have also beencompared.2.FormulationLet us consider anabsorbing, emittingandscatteringthree-dimensional mediumas showninFig. 1. Its northboundaryis subjectedtoacollimatedradiationIcat anangleX0.TheradiationpulseIcatthisboundaryisavail-able only for a durationtpwhichis of the order of anano-second. The incident radiation pulse travels withthespeedoflightc,andatanylocationinthemedium,itremains available for the durationof the pulse-widthtp(Fig.2).Sincethe mediumisparticipating andthebound-ariesareat nitetemperatures, thediuseradiationalsobecomes time dependent. Inthis situation, the radiativetransfer equation(RTE) inanydirection^s identiedbytheangleXabout theelemental solidangleDXis givenby[12]1c_ _oIot dIds bI jaIb rs4p_4pIpX; X0 dX01where s is the geometric distance in the direction ^s, ka is theabsorptioncoecient, bistheextinctioncoecient, rsisthe scattering coecient and p is the scattering phasefunction.InEq. (1), theintensityI withinthemediumis com-posedoftwocomponents,viz., thecollimatedintensityIcandthediuseintensityId.I Ic Id2Thevariationof thecollimatedcomponent Icwithinthemediumisgivenby[12]dIcds bIc3Fig. 1. A 3-D rectangular geometry and the coordinates underconsideration.Fig. 2. (a) A planar mediumsubjected to collimated square pulseirradiationat its topboundary. (b) Time of arrival of the collimatedsquare pulse at dierent locations in a planar medium for a normal angleofincidence, h0 = 0.1822 S.C.Mishraetal./InternationalJournalofHeatandMassTransfer49(2006)18201832SubstitutingEqs.(2)and(3)inEq.(1)yields1c_ _oIcot 1c_ _oIdot dIdds bId jaIb rs4p_4pIdpX; X0 dX0 rs4p_4pIcpX; X0 dX04Eq.(4)canbewrittenas1c_ _oIcot 1c_ _oIdot dIdds bId Sc Sd bId St5where Sc and Sd are the source terms resulting from the col-limatedandthediusecomponents of radiation, respec-tively.InEq.(5),St = Sc + Sdisthetotalsourceterm.At any point in the medium, the collimated radiationIcremains available only for the pulse durationtp. For asquarepulse, thevalueof Icremainsconstant duringthetimedurationtpandzeroat other times. Thus thetimederivativeofIcbecomeszero(thisfactcanbeunderstoodwith the help of the illustration given in Fig. 2 for a planarmedium).Eq.(5)canthenbewrittenas1c_ _oIdot dIdds bId St6The source termSc resulting from the collimated radiationIcisgivenbySct rs4p_4pX00IcX; tpX; X0 dX07For a linear anisotropic phase function pX; X0 1 a cos h cos h0,thesourcetermScisgivenbySct rs4p_ __2p0_p0Ich; /; t1 acos hcos h0 sinhdhd/8where hand /arethepolarandazimuthalangles,respec-tively. In terms of the incident radiationG and heat ux q,Eq.(8)canbewrittenasSct rs4pGct a cos hqct 9whereGcandqcaregivenbyGc Ich0; /0 10aqc Ich; / cos h0 Ich0; /0 cos h0dh h0d/ /010bInEq.(10b), distheDiracdeltafunctiondenedasdh h0 1; forh h00; forh 6 h0_11InEq.(10)above,IcisgivenbyIch; /; t Ich0; /0; t expbds0Hbct ds0 f g H bct ds0 bctp_ _ _ dh h0d/ /0 12where ds0 dz= cos h0is the geometric distance in thedirectionX0of the collimatedradiationand t*= bct isthenon-dimensionaltime.Eq.(12)canbewrittenasIch; /; t Ich0; /0; t expbds0 Ht bds0 f g Hft bds0 tpg dh h0d/ /0 13In the above equations, H is the Heaviside function denedasHy 1; y> 00; y< 0_14Radiation travels with the speed of lightc and hence takessomenitetimetoreachaparticularlocation. Therefore,thetimeof arrival of thepulseat that point will dependuponitslocation. Moreover, atanylocationinthemed-ium, Icremains available only for the durationtp(seeFig. 2). These eects are takencare, mathematicallybythe introductionof the Heaviside functioninthe aboveequations.In Eq.(5), thesourcetermSdresultingfromthediuseradiationIdisgivenbySdt jaIbt rs4p_4pX00IdtpX; X0 dX015IntermsofGandq,forlinearanisotropicphasefunctionpX; X0 1 a cos h cos h0,Eq.(15)isgivenbySdt jaIbt rs4pGdt a cos hqdt 16InEq.(16),Gdandqdaregivenby[34]Gdt _4pX0Idt dX_2p/0_ph0Idh; /; t sin hdhd/%

M/k1

Mhl1Idhl; /k; t2 sin hlsinDhl2_ _D/k17whereMhandM/arethenumberofdiscretepointscon-sidered over the complete span of the polar angle h(0 6 h 6 p) and azimuthal angle / (0 6 / 6 2p), respectively.qdt _4pX0Idt cos hdX_2p/0_ph0Idh; /; t cos hsinhdhd/%

M/k1

Mhl1Idhl;/k; t2sinhlcos hlsinDhlD/k18S.C.Mishraetal./InternationalJournalofHeatandMassTransfer49(2006)18201832 1823ForaboundaryhavingtemperatureTwandemissivityew,theboundaryintensityId(rw, t*)isgivenbyandcomputedfromIdrw; t erT4wp1 ep_ __^nw^s