Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation

Transcription

1 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

2 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.

3 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

4 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

5 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

6 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

7 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

8 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

9 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 : ) (, ) (, π σ π σ µ µ µ µ T T Kn t Kn t ) cs (, θ µ τ µ x v t L Kn L x t t Λ,, τ ), ( ), ( gp gp gp gp n n n n w w t q w w t µ µ π Kn t µ

10 Ceffcent Frm fr BTE (6 Fnte elements, 6 Gaussan Pnts)

11 Steady-State Prblem: Analytc vs. Numercal Slutns Emssve pwer ~ Temperature Gradent e ( ) e ( ) J J q J q q e e ( ) e ( ) 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) Nndmensnal crdnate,. Knudsen number, KnΛ/L

12 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 ql/ e Analytc Numercal (ngp6) Numercal (ngp8). Knudsen number, KnΛ/L

13 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 crdnate, Nndmensnal temperature, θ elements 6 elements elements Ray effect....3 Nndmensnal crdnate, Spatal refnement leads t a smther slutn. Hwever, t des nt slve ray effect. 3

14 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 ngp8...3 Nndmensnal crdnate, Nndmensnal crdnate, Angular refnement reslve ray effect. Spatal and angular refnements are ndependent. Hghly refned spatal mesh wth carse angular mesh allevates slutn. 4

15 Results f BTE (Temperature & Heat Flux Dstrbutns wth Tme ncrease) Nndmensnal temperature, θ Nndmensnal heat flux, q t. t. t t Nndmensnal crdnate, t. t. t t Nndmensnal temperature, θ Nndmensnal heat flux, q t. t. t t Kn Kn Kn Nndmensnal crdnate, t. t. t t Nndmensnal temperature, θ Nndmensnal heat flux, q t. t t t Nndmensnal crdnate, t. t t t Nndmensnal crdnate, Nndmensnal crdnate, Nndmensnal crdnate, 5

16 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 crdnate, Nndmensnal crdnate, Nndmensnal crdnate, Nndmensnal heat flux, q BTE BDE Furer Cattane Nndmensnal crdnate, Nndmensnal heat flux, q BTE BDE Furer Cattane Nndmensnal crdnate, Nndmensnal heat flux, q BTE BDE Furer Cattane Nndmensnal crdnate, 6

17 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