Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation

Transcription

1 Excerpt frm the Prceedngs f the COMSOL Cnference 9 Bstn Nanscale Heat Transfer usng Phnn Bltzmann Transprt Equatn Sangwk Shn *, and Ajt K. Ry Ar Frce Research Labratry, Unversty f Daytn Research nsttute *3 Cllege Park, Daytn, OH , sangwk@stanfrdalumn.rg Abstract: COMSOL Multphyscs was used t slve a phnn Bltzmann transprt equatn (BTE) fr nanscale heat transprt prblems. One-dmensnal steady-state and transent BTE prblems were successfully slved based n fnte element and dscrete rdnate methds fr spatal and angular dscretzatns, respectvely, by utlzng the bult-n feature f the COMSOL, Ceffcent Frm f PDE. A senstvty study was cnducted wth varus dscretzatn refnements fr dfferent values f the Knudsen number, whch s a measure f the nanscale regme. t was fund that suffcent refnement fr angular dscretzatn s crtcal n btanng accurate slutns f the BTE. Keywrds: Nanscale heat transprt, phnn Bltzmann transprt equatn, fnte element methd, dscrete rdnates methd.. ntrductn Fr the last tw centures, many heat cnductn prblems have been mdeled successfully by a Furer dffusve equatn (FDE). The equatn can be derved usng a cnservatn law f energy and Furer s lnear apprxmatn f heat flux usng a temperature gradent. The FDE s a parablc equatn reflectng a dffusve nature f heat transprt. An underlyng assumptn s that the heat s effectvely transferred between lcalzed regns thrugh suffcent scatterng events f phnns wthn a medum. Therefre, the FDE des nt hld when the number f scatterngs s neglgble, whch culd happen when a mean free path f phnn s smlar r a larger rder f magntude than a dman sze f nterest, sgnfcant bundary scatterng ccurs at materal nterfaces causng thermal resstance, etc. Anther prblem f the FDE s that t admts an nfnte speed f the heat transprt, whch s cntradctry t the thery f relatvty. Therefre, the FDE s napprprate fr the heat transfer prblems at small tme and spatal scales. T reslve the ssue f the nfnte speed f heat carrer, hyperblc equatnns, such as a Cattane equatn (Telegraph equatn) [] r a relatvstc heat equatn [], are ften used t reflect a wave nature f the heat transprt fr cases f small speed f heat prpagatn (e.g., lw temperature, lw cnductve medum, thermal nsulatr, phase transtn, etc.) and/r hgh speed f heat flux (e.g., pc- r femtsecnd pulsed-laser heatng.) Despte sme ssues [], these hyperblc equatns can be used t cnsder the fnte speed f phnns fr a shrt tme scale. Hwever, they stll cannt be used fr small spatal scale. Fr the nanscale heat transfer analyss, ne needs equatns and methds fr small-scale smulatn n terms f bth tme and space. A mlecular dynamcs (MD) smulatn can be a useful and accurate methd n ths regard. Hwever, the MD smulatn s usually cmputatnally expensve, s t s sutable fr systems havng a few atmc layers r several thusand atms, but nt sutable fr devce-level thermal analyss. Alternatvely, a phnn Bltzmann transprt equatn (BTE), a frstrder partal dfferental equatn fr a phnn dstrbutn functn, has been used fr ths purpse [3-5]. The dstrbutn functn s a scalar quantty n the sx-dmensnal phase space (three space crdnates and three wavevectr crdnates). The phnn BTE s als called an equatn f phnn radatve transfer (EPRT) when the phnn dstrbutn functn s replaced wth a phnn ntensty functn [3]. The BTE s knwn t be dffcult t slve, and thus ften smplfed wth a relaxatn tme apprxmatn. The BTE can predct a ballstc nature f heat transfer under an assumptn that partcle-lke behavrs f phnn are much mre sgnfcant than ts wave-lke behavrs, whch make the BTE vald fr structures larger than the wavelength f phnns. The BTE can be slved analytcally fr smple gemetres [4, 6, 7], r numercally fr cmplex gemetres by usng ether determnstc methds (e.g., dscrete rdnates methd, sphercal harmncs methd,

2 Reprt Dcumentatn Page Frm Apprved OMB N Publc reprtng burden fr the cllectn f nfrmatn s estmated t average hur per respnse, ncludng the tme fr revewng nstructns, searchng exstng data surces, gatherng and mantanng the data needed, and cmpletng and revewng the cllectn f nfrmatn. Send cmments regardng ths burden estmate r any ther aspect f ths cllectn f nfrmatn, ncludng suggestns fr reducng ths burden, t Washngtn Headquarters Servces, Drectrate fr nfrmatn Operatns and Reprts, 5 Jeffersn Davs Hghway, Sute 4, Arlngtn VA -43. Respndents shuld be aware that ntwthstandng any ther prvsn f law, n persn shall be subject t a penalty fr falng t cmply wth a cllectn f nfrmatn f t des nt dsplay a currently vald OMB cntrl number.. REPORT DATE OCT 9. REPORT TYPE 3. DATES COVERED --9 t TTLE AND SUBTTLE Nanscale Heat Transfer usng Phnn Bltzmann Transprt Equatn 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNT NUMBER 7. PERFORMNG ORGANZATON NAME(S) AND ADDRESS(ES) Ar Frce Research Labratry,3 Cllege Park,Daytn,OH, PERFORMNG ORGANZATON REPORT NUMBER 9. SPONSORNG/MONTORNG AGENCY NAME(S) AND ADDRESS(ES). SPONSOR/MONTOR S ACRONYM(S). DSTRBUTON/AVALABLTY STATEMENT Apprved fr publc release; dstrbutn unlmted 3. SUPPLEMENTARY NOTES Presented at the 9 COMSOL Cnference Bstn, 8- Oct, Bstn, MA. SPONSOR/MONTOR S REPORT NUMBER(S) 4. ABSTRACT COMSOL Multphyscs was used t slve a phnn Bltzmann transprt equatn (BTE) fr nanscale heat transprt prblems. One-dmensnal steady-state and transent BTE prblems were successfully slved based n fnte element and dscrete rdnate methds fr spatal and angular dscretzatns, respectvely by utlzng the bult-n feature f the COMSOL Ceffcent Frm f PDE. A senstvty study was cnducted wth varus dscretzatn refnements fr dfferent values f the Knudsen number, whch s a measure f the nanscale regme. t was fund that suffcent refnement fr angular dscretzatn s crtcal n btanng accurate slutns f the BTE. 5. SUBJECT TERMS 6. SECURTY CLASSFCATON OF: 7. LMTATON OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THS PAGE unclassfed Same as Reprt (SAR) 8. NUMBER OF PAGES 3 9a. NAME OF RESPONSBLE PERSON Standard Frm 98 (Rev. 8-98) Prescrbed by ANS Std Z39-8

3 fnte vlume methd [8], etc.) r statstcal methds (e.g., Mnte Carl smulatn [9, ]). n general, slvng the BTE s much mre effcent than the MD, and the predctns agree well wth expermental data [, ]. n the present study, we wll slve the BTE wth the relaxatn tme apprxmatn fr nedmensnal heat transprt prblems usng the COMSOL Multphyscs. We wll cmbne the fnte element (FE) capablty f the COMSOL wth the DOM t slve bth steady-state and transent heat transfer prblems. The numercal results wll be cmpared wth analytcal nes fr the steady-state prblem. A senstvty study wll be cnducted fr varus scale measures t shw the dfference between the BTE slutns wth the cnventnal FDE nes. Furthermre, we wll determne a sutable dscretzatn scheme fr spatal and angular spaces n rder t btan relable transent slutns n usng the COMSOL.. Use f COMSOL Multphyscs The BTE equatn under a relaxatn tme apprxmatn s wrtten as f f f f v f () t t scat τ, where f s the carrer dstrbutn functn, f the equlbrum dstrbutn, v the carrer grup velcty vectr, and τ the relaxatn tme. The EPRT n terms f the phnn ntensty s wrtten as v () t τ, where the phnn ntensty s expressed as ( t, v, r) v hf ( t, v, r) D( ) / 4π, where h s the reduced Planck cnstant, the phnn angular frequency, and D () the densty f state per unt vlume. An equlbrum phnn ntensty n Eq. () s wrtten as dω 4π Ω 4π. (3) Fr ne-dmensnal prblems, Eq. () becmes vcsθ (4) t x τ, where θ s a plar angle between a phnn prpagatn drectn and the glbal heat transfer drectn ( x ). A slutn prcess f the -D BTE n Eq. (4) alng wth Eq. (3) can be summarzed as fllws: Wth an ntal value f, Eq. (4) s slved fr the phnn ntensty feld, ( t, x), fr a partcular phnn prpagatn angle, θ. After btanng the slutns fr all pssble sld angles, 4 π, can be calculated usng Eq. (3). Wth ths updated value f, the slutn prcess repeats untl the cnvergence f and. Therefre, the BTE and thus the EPRT are hghly nnlnear equatns t slve. Once the phnn ntensty s slved, heat flux and an equvalent temperature can be btaned by ntegratng the phnn ntensty ver the sld angles, 4 π, such that q csθ dω and (5) T u c Ω 4π 4π dω c v c v Ω 4π, (6) where c s the vlumetrc specfc heat. One f the mst wdely used methds fr slvng the BTE numercally s the dscrete rdnates methd (DOM). The DOM s a tl t transfrm the equatn f radatve transfer nt a set f smultaneus partal dfferental equatns. Ths s based n a dscrete representatn f drectnal varatn f ntensty. A slutn t the transprt prblem can be fund by slvng the equatn f radatve transfer fr a set f dscrete drectns spannng the entre sld angle. The ntegrals ver the sld angle can be apprxmated by numercal ntegratn usng Gauss-Legender quadratures. The BTEs n Eqs. (3)-(4) were slved by the DOM usng a bult-n feature f Ceffcent Frm PDEs n COMSOL Multphyscs. The spatal dman ( x ) was dscretzed nt n elements as a nrmal FE mesh refnement, whle the angular dman ( Ω ), referred t as rdnates, was dscretzed n the drectnal crdnates by dvdng the angular space nt a fnte number, m. The angular dman was dscretzed at Gaussan quadrature pnts wthn µ csθ. Fr each plar angle ( θ m ), we slved the system f nnlnear equatns fr the phnn ntensty, ( t, x, µ ), and then updated m the equlbrum phnn ntensty, ( t, x), usng

4 fr all angles. Eqs. (3) and (5) can be wrtten n dscrete frms as ( t, x) ( t, x, µ m ) w and m (7) m q ( t, x) π ( t, x, µ ) µ w m m m m m, (8) respectvely. The weght satsfes w. The slver used fr the steady-state and transent analyses was a bult-n drect slver, UMFPACK. Default values were used fr the slver parameters except that strct tme steps were taken by the slver and a maxmum BDF rder was set t. When the relatve errr f the calculated value f the equvalent equlbrum ntensty between tw teratn steps was less than a present tlerance, we cnsdered the prblem was cnverged and then calculated the equvalent temperature and heat flux usng Eqs. (6) and (8), respectvely. 5. Results and Dscussn We have slved the BTE equatns wth the COMSOL under bth steady-state and transent states. Spatal and angular dmans were dscretzed nt n elements and m quadrature pnts, respectvely. A quadratc Lagrange element was used fr the spatal FE mesh. The ttal degree f freedm fr the -D prblem becmes m ( n ). Fr the steady-state prblems, t was bserved that the wellcnverged slutns were btaned wth a relatvely carse mesh (5 elements) whle usng 6 quadrature pnts. The senstvty f mesh and quadrature pnts fr the transent analyss wll be stated later n ths sectn. Fgure shws steady-state temperature dstrbutns va nndmensnal emssve pwer ( e * ) alng the -D dman fr varus Knudsen numbers ( Kn ). Fr cmparsn purpse, an analytcal slutn was als btaned by usng an analgy t the radatn heat transfer [5] by slvng an ntegral equatn, ξ * * e η η η η η η ( ) E( ) e ( ) E( ) d (9), where η x L s the nndmensnal crdnate, where L s the length f the dman, n E ( x) µ exp( x µ ) dµ the expnental n ntegral, ξ the ptcal thckness, whch s an m nverse f the Knudsen number, Kn Λ L, where Λ s the average mean free path f * phnns, and e ( η ) [ e ( η) J q] [ J q J q] s the emssve pwer nndmensnalzed by a dfference f bundary heat fluxes ( J and J ) at bth ends f the -D dman. n Fgure, the analytc and numercal slutns fr varus Kn (.,,, ) were drawn wth markers and lnes, respectvely, whch shws an excellent agreement f the slutns wth each ther. Whle the cnventnal Furer slutn wuld yeld the temperature gradent varyng frm at η t at η, the slutn frm the BTE yelds a reduced temperature gradent. The larger the Knudsen number, the smaller the temperature gradent, whch s attrbuted frm mre ballstc phnn transprt as cmpared t the dffusve ne. Nndm. emssve pwer, e* q Kn. Kn Kn Kn Fgure. Steady-state nndmensnalzed emssve pwer dstrbutn fr varus Knudsen numbers. We studed the effects f refnements n terms f spatal FE meshes and angular quadrature pnts n the tme-dependent transent BTE slutns. The transent tme ( t ) was nndmensnalzed by the relaxatn tme f phnns ( τ ), s that t * t τ. Fgure shws nndmensnalzed temperature dstrbutns at t *. alng the -D dman fr Kn. We used three dfferent FE meshes (5, 6 and elements) and a fxed angular dvsn wth 6 Gaussan pnts ( n gp 6 ). The three FE meshes yeld nearly dentcal temperature slutns wth a decreasng trend frm the ht sde ( η ) t the cld sde ( η ). q

5 Smth slutns were btaned except at a regn near the ht bundary. An nset rectangular area n Fgure wthn η. 3 was zmed n and repltted n Fgure 3, whch clearly shws sgnfcant wggles near the ht bundary. Therefre, the FE mesh refnement des nt help smthng ut the wggles at all. The wgglng f the slutn, called a ray effect, s knwn t be attrbuted t nsuffcent refnement n the angular dman, and thus cannt be smthed ut wth the FE mesh refnement n the spatal dman. Nndmensnal temperature, θ elements 6 elements elements Fgure. Transent nndmensnal temperature dstrbutn predcted wth 5, 6 and fnte elements and 6 Gaussan pnts. Nndmensnal temperature, θ elements 6 elements elements...3 t *. alng the -D dman fr Kn predcted wth sx dfferent Gaussan quadrature pnts ( n gp 4, 8, 6, 3, 64, 8 ). We utlzed a Matlab-scrptng feature f COMSOL t mplement the large n gp n buldng the system f equatns. A fxed spatal dvsn wth 4 FE elements was used n these calculatns. An nset rectangular area wthn η. 3 was magnfed and repltted n Fgure 5. Althugh a hghly refned FE mesh (4 elements) was used, the carse angular dvsns allevate the wgglng f the slutn (e.g., n gp 4 ). The wggles near the ht end gradually decrease wth the ncrease f n gp, and thus the ray effect. Therefre, we fund that the spatal and angular refnements are ndependent f each ther, and thus d nt recmmend t use the hghly refned spatal mesh wth the carse angular mesh. Nndmensnal temperature, θ ngp4 ngp8 ngp6 ngp3 ngp64 ngp Fgure 4. Transent nndmensnal temperature dstrbutn predcted wth 4 fnte elements and sx Gaussan quadrature pnts ( n ) fr angular gp dscretzatn. Fgure 3. A magnfed vew f Fgure wthn η.3. T elmnate the ray effect, we made further refnement n the angular dman by usng mre quadrature pnts. Fgure 4 shws nndmensnalzed temperature dstrbutns at

6 Nndmensnal temperature, θ ngp4 ngp8 ngp6 ngp3 ngp64 ngp8...3 Fgure 5. A magnfed vew f Fgure 5 wthn η.3. Fgure 6 and Fgure 7 shw the temperature and heat flux dstrbutns, respectvely, at varus tme scales wth three dfferent Knudsen numbers. The result successfully reprduced the ne reprted earler by Chen [4], but ths tme usng the FE methd usng the COMSOL. The fgures shw the heat transfer frm the ht sde t the cld sde wth the fnte speed f phnn as the shrt term (small t * ) slutns ndcate, whch cannt be predcted by the FDE. Whle cnstant temperatures were appled as bundary cndtns at bth sdes, we can bserve the temperature jump at these bundares. The hgher the Kn, the larger the jump. The reasn fr the temperature jump at these emssve bundares was well explaned n earler wrk [7]. Fr a small Knudsen number at Kn., the temperature slutn appraches that f the steady-state FDE as the tme ncreases, whle fr a large Knudsen number at Kn, the temperature slutn appraches a nearly hrzntal lne, whch ndcates the ballstc heat transfer rather than the dffusve ne, and results n a small temperature gradent and thus a large thermal cnductvty. 6. Cnclusns We have slved the phnn Bltzmann transprt equatn fr the nanscale heat transfer prblems wth the COMSOL Multphyscs. One-dmensnal steady-state and transent prblems were successfully slved usng FEM and DOM fr spatal and angular dscretzatns, respectvely, by utlzng the bult-n feature f the COMSOL, Ceffcent Frm f PDE. A senstvty study was cnducted wth varus dscretzatn refnements fr dfferent values f the Knudsen number, whch s a measure f the nanscale regme. Sgnfcant ray effects were fund wth carse angular dscretzatns. Hwever, false scatterng due t carse spatal dscretzatn was nt severe wth the COMSOL slutn. t was fund that the spatal and angular refnements were ndependent f each ther, and t was nt recmmended t use the hghly refned spatal mesh wth the carse angular mesh n slvng the BTE wth the COMSOL. The present success wth the -D smulatn encurages us t apply t fr multdmensnal cases. (a) Nndmensnal temperature, θ t*. t*. t* t* (b) Nndmensnal temperature, θ t*. t*. t* t* (c) Nndmensnal temperature, θ t*. t* t* t* Fgure 6. Temperature dstrbutns at dfferent tme scales. (a) Kn, (b) Kn and (c) Kn..

7 (a) Nndmensnal heat flux, q* t*. t*. t* t* (b) Nndmensnal heat flux, q* t*. t*. t* t* (c) Nndmensnal heat flux, q* t*. t* t* t* Fgure 7. Heat flux dstrbutns at dfferent tme scales. (a) Kn, (b) Kn and (c) Kn.. 8. References. C. Ba, et al., On Hyperblc Heat Cnductn and the Secnd Law f Thermdynamcs, Jurnal f Heat Transfer, 7(), (995).. Y. M. Al, et al., Relatvstc Heat Cnductn, nternatnal Jurnal f Heat and Mass Transfer, 48(), (5). 3. A. Majumdar, Mcrscale Heat Cnductn n Delectrc Thn Flms, Jurnal f Heat Transfer, 5(), 7-6 (993). 4. G. Chen, Ballstc-Dffusve Equatns fr Transent Heat Cnductn frm Nan t Macrscales, Jurnal f Heat Transfer, 4(), 3-38 (). 5. G. Chen, Nanscale Energy Transprt and Cnversn: A Parallel Treatment f Electrns, Mlecules, Phnns and Phtns, Oxfrd Unversty Press (5). 6. G. Chen, Nnlcal and Nnequlbrum Heat Cnductn n the Vcnty f Nanpartcles, Jurnal f Heat Transfer, 8(3), (996). 7. R. Yang, et al., Smulatn f Nanscale Multdmensnal Transent Heat Cnductn Prblems Usng Ballstc-Dffusve Equatns and Phnn Bltzmann Equatn, Jurnal f Heat Transfer, 7(3), (5). 8. J. Y. Murthy, et al., Cmputatn f Sub-Mcrn Thermal Transprt Usng an Unstructured Fnte Vlume Methd, Jurnal f Heat Transfer, 4(6), 76-8 (). 9. M.-S. Jeng, et al., Mdelng the Thermal Cnductvty and Phnn Transprt n Nanpartcle Cmpstes Usng Mnte Carl Smulatn, Jurnal f Heat Transfer, 3(4), (8).. S. Mazumder, et al., Mnte Carl Study f Phnn Transprt n Sld Thn Flms ncludng Dspersn and Plarzatn, Jurnal f Heat Transfer, 3(4), ().. G. Chen, Sze and nterface Effects n Thermal Cnductvty f Superlattces and Perdc Thn- Flm Structures, Jurnal f Heat Transfer, 9(), -9 (997). 9. Acknwledgements Ths wrk was perfrmed under U.S. Ar Frce Cntract N. FA865-5-D-55.

8 Presented at the COMSOL Cnference 9 Bstn Nanscale Heat Transfer usng Phnn Bltzmann Transprt Equatn Sangwk Shn Ar Frce Research Labratry, Materals and Manufacturng Drectrate, AFRL/RX, Wrght-Pattersn AFB, OH, Unversty f Daytn Research nsttute, Daytn, OH emal : sangwk.shn@udr.udaytn.edu Ajt K. Ry Ar Frce Research Labratry, Wrght-Pattersn AFB, OH 9 COMSOL Cnference, Bstn, MA Octber 8-, 9

9 Outlne Backgrund nfrmatn. Descrptn f phnn Bltzmann transprt equatn (BTE). Mdelng and slutn prcedure f BTE usng COMSOL. Results Steady-state and transent prblems. ssues f refnement n spatal and angular dmans. Summary and cnclusns.

10 Furer Equatn (FE) Fr last tw centures, heat cnductn has been mdeled by Furer Eq (FE). T Cnservatn f energy: ρc q t Dffusve Furer s lnear apprxmatn f heat flux: q k T L >> Λ T T T α t x Parablc equatn > Dffusve nature f heat transprt. T Heat s effectvely transferred between lcalzed regns thrugh suffcent scatterng events f phnns wthn medum. Des nt hld when number f scatterng s neglgble. e.g., mean free path ~ devce sze (chp-package level). Bundary scatterng at nterfaces causng thermal resstance. Admts nfnte speed f heat transprt > Cntradct wth thery f relatvty. L Furer Equatn cannt be used fr small tme and spatal scales. 3

11 Hyperblc Heat Cnductn Equatn (HHCE) Reslve the ssue f the Furer equatn wth the nfnte speed f heat carrer. C t T T α t Defntn f heat flux: T q τ q k T, ( τ : relaxatn tme) t Hyperblc equatn > Wave nature f heat transprt. Called as Cattane equatn r Telegraph equatn. Fnte speed f heat carrers. Ad hc apprxmatn f heat flux defntn. Vlates nd law f thermdynamcs. f heat surce vares faster than speed f sund, heat wuld appear t be mvng frm cld t ht. ( C α τ ) HHCE: culd be used fr shrt tme scale, but nt fr shrt spatal scale. 4

12 Small Scale Heat Transprt (Tme & Space) Furer Equatn cannt be used fr small tme and spatal scales. HHCE: culd be used fr shrt tme scale, but nt fr shrt spatal scale. Needs equatns and methds fr small scale smulatn n terms f bth tme and space. Mlecular dynamcs smulatn. Accurate methd. Cmputatnally expensve. Sutable fr systems havng a few atmc layers r several thusands f atms. Nt sutable fr devce-level thermal analyss. Bltzmann Transprt Equatn (BTE). Ballstc-Dffusve Equatn (BDE). Smlar t Cattane Eq. (HHCE) wth a surce term. Derved frm BTE. Gd apprxmatn f BTE wthut nternal heat surce, dsturbance, etc. 5

13 Bltzmann Transprt Equatn (BTE) BTE: als called as equatn f phnn radatve transfer (EPRT). Equatn fr phnn dstrbutn functn: f t v f Can predct ballstc nature f heat transfer. Neglects wave-lke behavrs f phnn. Vald fr structures larger than wavelength f phnns (~ RT). Slutn methds: f t scat Determnstc: dscrete rdnates methd, sphercal harmncs methd. Statstcal: Mnte Carl. Much mre effcent than MD. f f τ Agrees well wth expermental data. L << Λ x T Ballstc T L 6

14 Detals f Bltzmann Transprt Equatn (BTE) Phnn ntensty: ( t, v, r) v f ( t, v, r) D( ) / 4π BTE becmes EPRT: t v τ, 4π Ω 4π dω *Equlbrum phnn ntensty determned by Bse-Ensten statstcs Fr -D, t v csθ x τ x θ Ι L Fr each angle (θ ), slve nn-lnear equatn wth teratns fr Slvng fr Ι. Updatng Ι ο. Heat flux: q Ω D csθ d dω 4π nternal energy: u D ( ct ) f D( ) d dω 4 π Ω 4π v d dω 7

15 Mdelng & Slutn Prcedure usng COMSOL t v csθ x τ Use a bult-n feature f COMSOL, Ceffcent Frm PDEs. The spatal dman s dscretzed usng FE mesh. The angular (mmentum) dman s dscretzed usng Gaussan quadrature pnts. Fr each angle (θ ), set up the BTE wth crrespndng ceffcents (µ csθ ) and BCs (Neumann vs. Drchlet). Calculate equlbrum phnn ntensty ( ) by numercal ntegratn f ( ) usng Gaussan quadratures. Slve. Drect slver (UMFPACK). Max. BDF rder. Pstprcess and vsualze the results. 8

16 Orgnal -D BTE: Nndmensnalze wth Splt nt () and (-) drectns: Dscretze angular space at Gaussan quadrature pnts: After FE run, pstprcess: 9 Detals f Slutn Prcedure < > 4 4 * *, Drchlet BCs : ) (, ) (, η η η η π σ π σ µ η µ µ η µ T T Kn t Kn t ) cs (, θ µ τ µ x v t L Kn L x t t Λ,, * η τ ), ( ), ( gp gp gp gp n n n n w w t q w w t µ µ π η η Kn t η µ *

17 Ceffcent Frm fr BTE (6 Fnte elements, 6 Gaussan Pnts) Fle Edt Optns Draw Physcs Mesh Slve Pstprcessng Multphyscs Help fvldel Tree F t: f:: < > 3 Geml DE, Ceffcent Frm (c) ;.. POE, Ceffcent Frm (c) :.. POE, Ceffcent Frm (c3)! POE, Ceffcent Frm (c4) ; POE, Ceffcent Frm (cs) : POE, Ceffcent Frm (c6)! POE, Ceffcent Frm (c7)! POE, Ceffcent Frm (c8) : POE, Ceffcent Frm (c9)! POE, Ceffcent Frm (clo)! POE, Ceffcent Frm (el l ) ~ POE, Ceffcent Frm (c)! POE, Ceffcent Frm (c3) ' POE, Ceffcent Frm (c4) :.. POE, Ceffcent Frm (cs) :.. PDE, Ceffcent Frm (c6) Equatn V'{-cV'ul -au! y) au! ~ V'ul f Subdman selectn Ceffcents POE ceffcents Ceffcent Value/ Expressn Descrptn l Dffusn ceffcent a t Absrptn ceffcent Surce term e. l Mass ceffcent v d. t Dampng/Mass ceffcent Grup: Select by grup ~ Actve n ths dman l Cnservatve flux cnvectn ceff. ~!Kn*mu(l) Cnvectn ceffcent y l Cnservatve flux surce term OK l [ Cancel [ Apply [ Help POE, Ceffcent Frm (c) Dependent varables: ul Default element type: Lagrange- Qua Wave extensn: Off Weak cnstrants: Off.. Mesh cnssts f 5 elements. Mesh cnssts f 3 el ements. ~~~--~:_~>~~M~e~s~h_:c~nssts f 6 el ements. lllr---r Nrmal.9 v!memry: (6/)

18 Steady-State Prblem: Analytc vs. Numercal Slutns Emssve pwer ~ Temperature Gradent e * ( η) e ( η) J J q J q q * * * e e ( η ) e ( η ) Nndm. emssve pwer, e* Kn Λ L Small temperature gradent at smaller scale Kn. Kn Kn Kn Gradent f nndm. emssve pwer, e* Analytc Numercal (ngp6) Numercal (ngp8) Knudsen number, KnΛ/L

19 Steady-State Prblem: Analytc vs. Numercal Slutns Heat flux Thermal cnductvty q * q J q J q k * q * L * e Nndmensnal heat flux, q* Analytc Numercal (ngp6) Numercal (ngp8). Knudsen number, KnΛ/L k* q*l/ e* Analytc Numercal (ngp6) Numercal (ngp8). Knudsen number, KnΛ/L

20 Transent Prblem: Effect f Spatal Refnement (Mre Fnte Elements) Refne spatal (x-) drectn wth n fnte elements (n5, 6, ). Dvde angular drectn wth 6 Gaussan pnts (ngp6). Nndmensnal temperature, θ t*. 5 elements 6 elements elements Nndmensnal temperature, θ elements 6 elements elements Ray effect....3 Spatal refnement leads t a smther slutn. Hwever, t des nt slve ray effect. 3

21 Transent Prblem: Effect f Angular Refnement (Mre Gaussan Pnts) Refne spatal (x-) drectn wth 4 FE elements. Dvde angular drectn wth ngp Gaussan pnts (ngp4,8,6,3,64,8). Nndmensnal temperature, θ t*. ngp4 ngp8 ngp6 ngp3 ngp64 ngp8 Nndmensnal temperature, θ ngp4 ngp8 ngp6 ngp3 ngp64 ngp Angular refnement reslve ray effect. Spatal and angular refnements are ndependent. Hghly refned spatal mesh wth carse angular mesh allevates slutn. 4

22 Results f BTE (Temperature & Heat Flux Dstrbutns wth Tme ncrease) Nndmensnal temperature, θ Nndmensnal heat flux, q* t*. t*. t* t* t*. t*. t* t* Nndmensnal temperature, θ Nndmensnal heat flux, q* t*. t*. t* t* Kn Kn Kn t*. t*. t* t* Nndmensnal temperature, θ Nndmensnal heat flux, q* t*. t* t* t* t*. t* t* t*

23 Cmparsns f FE, HHTC, BDE vs. BTE (Temperature & Heat Flux Dstrbutns fr 3 Kn) Nndmensnal temperature, θ BTE BDE Furer Cattane Kn t* Nndmensnal temperature, θ BTE BDE Furer Cattane Kn t* Nndmensnal temperature, θ BTE BDE Furer Cattane Kn. t* Nndmensnal heat flux, q* BTE BDE Furer Cattane Nndmensnal heat flux, q* BTE BDE Furer Cattane Nndmensnal heat flux, q* BTE BDE Furer Cattane

24 Summary and Cnclusns Nanscale smulatn s cnducted fr phnn heat transfer usng Bltzmann transprt equatn. Phnns fr delectrc, thermelectrc, semcnductr materals. Electrns fr metals. Gas mlecules fr rarefed gas states. Numercal slutn f BTE has been btaned fr -D prblem (bth steady-state and transent prblems). Temperature and heat flux dstrbutns frm the nanscale smulatn yeld cmpletely dfferent results frm the slutns frm Furer and Cattane equatns. Thermal cnductvty wll be dfferent, t. 7