Analytical solution of non-fourier heat conduction problem on a fin under periodic boundary conditions

Transcription

1 Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 DOI 0.007/s Analytical solution of non-fourir hat conduction problm on a fin undr priodic boundary conditions H. Ahmadikia,* and M. Rismanian Dpartmnt of Mchanical Enginring, Univrsity of Isfahan, Isfahan, , Iran Dpartmnt of Mchanical Enginring, Bu-Ali Sina Univrsity, Hamadan, Iran (Manuscript Rcivd March 6, 00; Rvisd Jun 8, 0; Accptd July 0, 0) Abstract Fourir and hyprbolic modls of hat transfr on a fin that is subjctd to a priodic boundary condition ar solvd analytically. Th diffrntial quation in Fourir and non-fourir modls is solvd by th Laplac transform mthod. Th tmpratur distribution on th fin is obtaind using th rsidual thorm in a complx plan for th invrs Laplac transform mthod. Th thrmal shock is gnratd at th bas of th fin, which movs toward th tip of th fin and is rflctd from th tip. Th currnt study of various paramtrs on th thrmal shock location shows that rlaxation tim has a grat influnc on th tmpratur distribution on th fin. An unstady boundary condition in th bas fin causd th shock, which is gnratd continuously from th bas and has intractd with th othr rflctd thrmal shocks. Rsults of th currnt study show that th hyprbolic hat conduction quation can violat th scond thrmodynamic law undr som unstady boundary conditions. Kywords: Analytical solution; Conduction; Fin; Hyprbolic; Priodic Introduction Classical Fourir hat conduction modl is applid to prdict th tmpratur distribution in gnral nginring problms undr rgular conditions []. Th parabolic charactristic of Fourir s laws implis that hat flux is causd simultanously with th cration of th tmpratur gradint. This assumption cannot b tru bcaus w know that two phnomna that ar rlatd to ach othr cannot xist simultanously []. Such an immdiat rspons lads to a local chang in tmpratur that causs instantanous tmpratur prturbations in all rgions; thus, hat propagation spd will b infinit []. Fourir s law has a nonphysical conclusion for situations that ar involvd with vry high tmpratur gradints, xtrmly short tims, vry low tmpraturs, and vry small structural dimnsions, among othrs. Th mathmatical dscription of non-fourir hat conduction law, which rprsnts th tim lag of hat wavs, is a hyprbolic typ of diffrntial quation. Non-Fourir hyprbolic hat transfr in th fins at short tims undr priodic boundary conditions has a wid application in micro-dvics, such as hating and cooling This papr was rcommndd for publication in rvisd form by Associat Editor Dongsik Kim * Corrsponding author. Tl.: , Fax.: addrss: ahmadikia@ng.ui.ac.ir KSME & Springr 0 of microlctronic lmnts, micro-fabrication tchnology, and micro-hat xchangrs, among othrs. Numrous studis ar dvlopd to solv th analytical and numrical hat conduction quations in th fins. Howvr, applying analytical mthods to solv hat transfr in th fins with complicatd boundary conditions, variabl physical proprtis, and thrmal discontinuity that ar producd in th hyprbolic quations is difficult. Thus, numrical schms ar usd in most studis. Th major problm of numrical solutions is th prsnc of oscillations nar th thrmal discontinuitis, whras analytical mthods do not hav unrasonabl oscillations nar th thrmal discontinuitis. Analytical solutions ar usd to chck th accuracy and convrgnc of th numrical mthods. Various analytical and numrical mthods of th hyprbolic hat conduction quation subjctd to priodic boundary conditions wr prsntd in Rfs. [3-3] and many othrs. Most studis solvd th fin problms in th Fourir domain by applying th numrical mthods. Yn and Wu [4] solvd th hyprbolic hat conduction in a finit slab with surfac radiation and priodic hat flux using th Laplac transform mthod. Chang and Juhng [5] analytically solvd th hyprbolic hat conduction in a slab undr th sinusoidal priodic surfac hating procss. Aziz and Na [6] adoptd a prturbation mthod to solv th fins with various thrmal proprtis. Convctiv hat transfr in th fin undr a priodic boundary

2 90 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 condition is analytically solvd by Yang [7]. Th problm of priodic boundary conditions in th hyprbolic hat conduction was invstigatd by Tang and Araki [8] and Abdl- Hamid [9]. Priodic boundary conduction in matrials with non-fourir hat conduction modl for a on-dimnsional slab was xamind by Cossali [0] through th transfr function mthod. Abdallah [] invstigat th analytic mthod of a boundary valu problm for a smi-infinit mdium with traction-fr surfac hatd by a high-spd lasr puls. H usd a Dirac lasr puls boundary that was not priodic. With rgard to th priodic boundary condition, most studis hav dalt with conduction hat transfr using th parabolic (Fourir s law) hat quation or numrical schms, whras som rlid on th hyprbolic hat quation. Howvr, th hyprbolic modl of th hat transfr cannot accuratly prdict th tmpratur in a mdium. Th prsnt work focuss on th analytical schm in solving th hyprbolic hat conduction in th fin that is subjctd to vry priodic boundary condition using th Laplac transform mthod. Unlik othr numrical mthods, this analytical mthod is fr of oscillations around th thrmal discontinuitis. Th objctiv of th prsnt work is to invstigat th ffcts of rlaxation tim by having various boundary conditions on th tmpratur distribution in th fin, and by assssing th scond thrmodynamic law in th hyprbolic hat quation modl.. Physical modl and hat transfr in th fin Phonons propagat at th sound spd dpnding on th typ of solid mdium. Thus, a rspons tim with vry small ordr implis a submicron dpth pntration, thrby ncssitating a simultanous considration of th microscopic ffct in spac. To attain th rliabl prformanc of th microdvics, th ffctiv mans for hat rmoval at short tims must b nsurd. Th rspons tim of th thrmal and rlaxation tim of th nrgy carrirs rsulting in high tmpratur at short tims and causing arly-tim thrmal damag bfor stady stat oprations can occur. Microscopic modls such as th phonon-lctron intraction modl [], phonon scattring modl [3], and phonon radiativ transfr modl [4] rsultd from th solutions of th smi-classical Boltzmann transport quation. Th classical Fourir diffusion modl dscribs th corrlation btwn th hat flux and tmpratur gradint in a macroscal hat transfr. Th thrmal wav modl (CV wav) dpicts a tmpratur disturbanc propagating as a wav, with thrmal diffusivity acting as a damping ffct in hat propagation. Th fractal modl [5] is mployd for dscribing th conducting path in amorphous matrial and th scattring of phonons ovr th corrlation lngth on a small scal. Th DPL modl [6] includs th dlay tim ffcts du to microscal ffcts on th transint rspons. In this study, w us a modifid hat flux proposd by Vrnott [4] and Cattano [7] to solv hat transfr in th fin with tim-dpndnt boundary Fig.. Th fin configuration. conditions. Th thrmal wav modl givn by Cattano and Vrnott is applid for micro-solid matrials at vry short tims. Wang t al. [8] showd that th CV wav modl can b usd for th thrmomass gas. Thy built a thrmomass gas modl basd on hyprbolic hat conduction thory to dscrib th fluid-flow-lik hat conduction procss in a mdium. Wang and Guo [9] also prsntd nw govrning quations for non-fourir hat conduction in nanomatrials basd on th concpt of thrmomass. Considr a straight fin with uniform thicknss b, width w, and lngth L, which has an initial tmpratur T 0 (s Fig. ). Th ratio b/l is a small valu, and th fin tip (x=l) is adiabatic. At a spcific tim, a priodic tmpratur boundary condition is applid to th fin bas (x=0). b( b, m ω b, m T 0, t) = T + ACos( t)( T T ) () whr T b, T, and T b,m ar priodic bas tmpratur, ambint tmpratur, and man bas tmpratur, rspctivly. A is th input tmpratur amplitud and ω is th tmpratur oscillation frquncy. Th latral surfacs of th fin dissipat hat to th nvironmnt by convction hat transfr cofficint. Th hyprbolic hat conduction quation for th fin is givn by Eq. (): (, ) h ( ) = x b (, ) (, ) T x t k T T T x t T x t h τρc + ρc + τ T T t t b t whr T(x, t) rprsnts tmpratur; k, ρ, and C ar th thrmal conductivity, dnsity, and spcific hat capacity in a mdium, rspctivly. τ is th rlaxation tim, which mans that th fr path λ is ovr phonon vlocity and ν (spd of sound in th mdium). Rlaxation tim illustrats that thr is a finit lag tim for th onst of a thrmal currnt aftr a tmpratur gradint is imposd on a mdium. In th absnc of rlaxation tim (τ = 0, Eq. () is rducd to th classical Fourir s law. Eq. () is a hat wav quation that propagats a tmpratur disturbanc in th form of a hat wav; this quation is dampd using th diffusivity cofficint α. Th following dimnsionlss quantitis, i.., dimnsionlss tmpraturθ, dimnsionlss convctiv hat transfr H, dimnsionlss tim ξ, dimnsionlss spac η, dimnsionlss frquncy of th tmpratur oscillation Ω, and dimnsionlss ()

3 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 9 tim rlaxation, ar introducd as: x αt ατ ωl hl η =, ξ =, =, Ω=, H = (3) L L L α bk T T0 T T0 θ =, θ =. (4) T T T T bm, 0 bm, 0 Eq. () and th rlvant boundary conditions ar xprssd in trms of th abov dimnsionlss variabls as: H θ θ θ + ( + ) = + H( θ θ) ξ ξ η ) θ (0, ξ = + Acos( Ω ξ ) (6) ) θη (, ξ = 0 (7) θη (,0) = θξ ( η,0) = 0. (8) 3. Th analytical procdur of tmpratur priodic boundary condition Th Laplac transform mthod is usd for solving hyprbolic hat transfr in th fin that is subjctd to thrmal priodic boundary conditions. Th main problm of this mthod is th invrs Laplac transform. In this study, w us th invrs thorm by applying th rsidu thorm in th complx plan. Aftr taking th Laplac transform of Eq. (5), th following ordinary diffrntial quation is obtaind. Θ d Hθ s ( s H 0 dη Θ+ = s whr Θ(η, s) is Laplac transform of θ (η, ξ). By solving Eq. (9) and applying th boundary conditions (6) and (7), w would hav Hθ Bs Hθ Cosh m( η ) Θ ( η, s) = +. + (0) sm s ( s +Ω) sm Cosh( m) Eq. (0) is solvd using th invrs imag functions by calculating rsidus. Function of θ (x, t) is th invrs Laplac transform of Θ(x, s) that is obtaind from th complx intgral: γ + (, ) (, ) lim il ts θ x t = Θ x s ds () πi L γ + il which is known as th invrs thorm of Laplac transform mthod [30]. This intgral is takn along th infinit lin L (lin x=γ) and half circl C R, whr all singular points S j (j =,,, N) in circl C R of radius R nclos th whol intgral pols. If Θ(x, s) is analytic, xcpt for a numbr of N pols (5) (9) that ar all to th lft of som lin x=γ, w complt th contour of Eq. (0) by a big contour L+C R and by nclosing th whol intgral pols. If Θ(x, s) is analytic (xcpt for a numbr of pols that ar all to th lft of som lin x=γ) and if it has a branch point at z=s j, thn w complt th contour of th invrsion intgral, including a loop along th cut and around th branch point by introducing a cut along th lft sid of lin x=γ (for mor dtails, s Rf. [30]). Aftr applying this thorm to Eq. (), an accurat tmpratur distribution in th fin is calculatd by th ral part of Eq. (). Hξ ξ / ξ / ( ) θ θηξ (, ) = ral m + ( ) H Cosh ( η ) ( iω+ )( iω+ H ) iωξ ( m) + B Cosh ( iω+ )( iω+ H ) whr snξ ξ / Hξ ξ / Hθ an n 0 s n s n + H s n = + H Bs n s an n sn ( sn +Ω ) ξ n= n sn = + H ± + H H + λ () 4 (3) π λ n = (n + ) (4) m= Cosh( η ) H / Cosh H Cos ( η ) λnλn an =. n ( ) ( sn + + H ) (5) (6) 4. Th arbitrary priodic tmpratur boundary condition If th boundary condition at th fin bas is priodic with an arbitrary function, w can writ it in th Fourir s sris form. For xampl, if th boundary condition is in th stp function form shown in Fig., thn th Fourir s sris is: θ nπξ nπξ = A + A cos + B sin p ( ξ) 0 b n n p n= n= (7) whr p is th priod of boundary condition and Ω=nπ/p. W can us th suprposition thorm bcaus th govrning quation and boundary conditions ar linar. Thrfor, this problm is dividd into thr sub-problms with th following boundary conditions:

4 9 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 cosh i ( η ) ( iω+ )( iω+ θ( η, ξ) = ral cosh ( iω+ )( iω+ cos ( η ) λnλn sn sn ξ + n n 0 ( s ) s = n n + + H +Ω Ωξ (0) If th boundary condition at th fin bas is sinusoidal θ b3 with θ =0, th solution namd θ3 ( ηξ, ) will b a ral part of Eq. (). Fig.. Stp function boundary condition. b3 ( 0, ξ) = A ( 0, ξ) = cos = ( n p) θb 0 θb θ 0, ξ sin πξ /. (8) On of th boundary conditions abov (.g., constant boundary condition) is solvd with θ, whras th othr boundary conditions ar solvd without θ (or θ =0) bcaus Eq. (5) is nonhomognous du to trm θ. Whn th boundary condition at th fin bas has a constant valu θ b, (including θ ), th solution can b obtaind by applying th Laplac transform and th invrs thorm in complx variabls. Th solution of this cas is xprssd as follows: θ( η, ξ) =A0cosh ( η) H / cosh H ξ ξ Hξ cosh H η + θ ( ) H cosh ( H ) cos ( η ) λnλn A0 snξ + ral n n 0 ( s ) s = n n + + H cos ( η) λnλn Hθ (9) n 0 ( ) (sn + + ) sn( sn n= H + ξ ξ snξ Hξ ( ) ( ) sn + H whr s n, λ n, and m ar dfind in Eqs. (3)-(5), rspctivly. Solving th govrning quation with cosin boundary condition θ b without θ is similar to solving th quations mntiond in sction 3, which is th ral part of Eq. (0). i cosh ( η ) ( iω+ )( iω+ h) xp( Ω i ξ) θ3( η, ξ) = ral cosh ( iω+ )( iω+ h) cosh ( η ) ( iω+ )( iω+ + xp( iωξ ) cosh ( iω+ )( iω+ cos ( η ) λnλn Ω s n ξ + n n 0 ( s ) s = n n + + K +Ω () Thrfor, th solution of th hyprbolic hat transfr in th fin with arbitrary priodic boundary condition is: θ ( ηξ, ) = θ( ηξ, ) + Aθ( ηξ, ) + B θ( ηξ, ). n n 3 n= n= () 5. Th hating flux priodic boundary condition In this sction, w assum that th hat flux boundary condition at th fin bas is: θ ( ηξ, ) = θ( ηξ, ) + Aθ( ηξ, ) + B θ( ηξ, ). n n 3 n= n= (3) This boundary condition can b a part of th arbitrary priodic boundary condition. θ =0 is also assumd. Onc th Laplac transform mthod in Eq. () undr th boundary condition (3) is applid, thn th tmpratur will b: ( ) ( η ) A Ω i+ cosh m s=ω i iωξ θηξ (, ) = ral m s=ω i sinh m s=ω i A Ωi cosh m ( η ) s=iω Ω i ξ + m s=iω sinh m s=iω (4) 3 ξ A + Ω A H + Ω HΩ Ω H H +Ω H ( n Ω ) λn ( η ) ( ) As cos n snξ + s n n +Ω sn + H + =

5 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 93 Fig. 3. Th tmpratur distribution on th fin subjctd to th boundary condition Eq. (6). unstady boundary condition Eq. (6) is shown at various dimnsionlss tims, dimnsionlss rlaxation tim =5, and undr conditions A=, H=.9, θ =, and Ω=0.8. At dimnsionlss tim ξ=0., th thrmal wav is clos to th bas of th fin du to its finit vlocity. At tim ξ=, th thrmal wav movs to th tip of th fin, and th fin tmpratur is incrasd aftr th thrmal wav bcaus it has a convctiv hat transfr with th nvironmnt. With an incras in tim (at tim ξ=3), th thrmal shock rachs th tip of th fin whr it is rflctd. Hr, th fin tmpratur is incrasd aftr th thrmal shock du to th convctiv hat transfr. Th thrmal shock is rflctd back aftr it rachs th tip of th fin. Som othr thrmal wavs ar gnratd bcaus th bas tmpratur changs rapidly. On on hand, th thrmal wav kps on hating bcaus th fin tip is coolr than that of th nvironmnt. Consquntly, th fin tip gts warmr than its bas and brings about a hat flux toward th fin bas. Anothr hat wav is gnratd that movs to th fin bas bcaus th thrmal wav spd is finit (du to rlaxation tim). Th thrmal wavs thn mov back and forth until thy ar dampd. At tim ξ=5 (a long tim), th thrmal wav is compltly uniform, and that w will not s any thrmal wavs in th fin. Th dimnsionlss location of th thrmal shock η s can b obtaind by: η = ξ/. (5) s Fig. 4. Tmpratur distribution on th fin at =5. whr m= H sinh ( η ) H / cosh H. 6. Discussion of rsults 6. Th priodic boundary condition tmpratur Th fin tmpratur subjctd to th boundary condition θ(0, ξ)=+acos(ωξ) with dimnsionlss paramtrs A=, H=.9, and θ = is calculatd by Eq. (3). Th tmpratur distribution at dimnsionlss tim ξ=0.5 and dimnsionlss rlaxation tim =0.5 for various frquncis of tmpratur oscillations is shown in Fig. 3. Th thrmal shock is causd in th tmpratur bcaus th govrning quation is hyprbolic. Fig. 3 shows that th bas tmpratur frquncy dos not hav any influnc on th location and span of thrmal shock. Th tmpratur bfor thrmal shock influncs th bas tmpratur frquncy. Th tmpratur aftr thrmal shock is not influncd du to th bas tmpratur aftr th shock and th finit hat propagation spd. No paramtr xrts an influnc on th tmpratur aftr th shock, xcpt a tmpratur incras du to convctiv hat transfr with th nvironmnt. In Fig. 4, th tmpratur distribution on th fin undr th If η s is gratr than th unit (fin lngth), w should considr th rflctd wav from th fin tip; thus, thrmal wav location can b obtaind by: ηas, = ηs ηs (6) whr [η s ] is th brackt of η s, and η a, s is th actual location of th thrmal shock. If [η s ] is an vn numbr, thn it is valuatd from th fin bas; if [η s ] is an odd numbr, thn it is valuatd from th fin tip. Th tmpratur distribution corrsponding to th analytical solution Eq. () at various non-dimnsional rlaxation tims and dimnsionlss tim ξ=0.5 is prsntd in Fig. 5. With a dcras in rlaxation tim, th shock wav location in th fin movs to th right sid du to an incras in shock wav vlocity and a dcras in rlaxation tim. At th rlaxation tim =0.00, th dimnsionlss tmpratur tnds to gt closr to th tmpratur obtaind from th Fourir s law modl; hnc, thr is no thrmal shock in this cas. Th tmpratur distribution undr boundary condition Eq. (6), dimnsionlss tim ξ=00 (a long tim), and various rlaxation tims is shown in Fig. 6. Th rlaxation tim has a grat influnc on tmpratur distribution. This fact shows that vn for long priods, th variation of rlaxation tim brings about grat ffcts on th tmpratur distribution in th fin. Thus, applying Fourir hat quation for slightly high rlaxation tim can lad to significant rror.

6 94 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 5 ξ=0.3 ξ=0.5 ξ=.0 ξ =.75 Dimnsionlss Tmpratur Dimnsionlss spatial coordinat Fig. 5. Tmpratur distribution on th fin at various rlaxation tims and ξ=0.5, Ω=.0, A=, H=.9, and θ =.0. Fig. 6. Tmpratur distribution on th fin at various rlaxation tims, and ξ=00, Ω=.0, A=, H=.9, and θ = Studying th accuracy of th hyprbolic hat quation A problm rlatd to th hyprbolic hat quations is th cration of thrmal shocks. Basd on th assumption that th spd of thrmal wav in th Fourir modl is infinit, th thrmal shock will not b gnratd. According to Fig. 5, hat flux is not infinit in this location with rgard to th infinit tmpratur gradint. By chcking Eq. (3), this infinit tmpratur gradint is du to th tim drivativ of th hat flux in th fin. Thus, crating infinit tmpratur gradint cannot b a good rason for proving th invalidity of th hyprbolic hat quation modl. To study th accuracy of th thrmodynamic laws, considr th following xampl: th thrmal distribution for hat flux priodic boundary condition (3), H=0 (no dissipat hat transfr to th ambint), Ω=0, and ξ=.75 ar shown in Fig. 7. For th priodic boundary condition q=cos(ωξ) at ξ=0, which is th first quartr of th unit circl, hat is continuously injctd into th fin bas with a positiv valu whil th tmpraturs dcras at th initial tims as shown in Fig. 7. Thus, for th tim intrval whr cos(ωξ) has a positiv valu, both Fig. 7. Dimnsionlss tmpratur distribution on th fin in th hat flux boundary condition at ξ=.75, =6, Ω=.0, A=, and H=0. th hat flux and tmpratur gradint hav positiv valus. Thrfor, hat flows from a highr tmpratur to a lowr tmpratur, which is a violation of th scond law of thrmodynamics. By incrasing th rlaxation tim and frquncy of th priodic boundary condition, th violation of th scond law of thrmodynamics is incrasd. W can now xprss a priodic function for th boundary condition, which continus picwis in trms of both sins and cosins. Thrfor, w can find an intrval that hyprbolic quations violat th scond law of thrmodynamics. In Fig. 7, w obsrv that th tmpratur of fin has a ngativ valu (a blow ambint tmpratur). This shows that for a dimnsionlss tim from 0.5 up to.75, both th hat flux and tmpratur gradint at th fin bas ar positiv, which can violat th scond law of thrmodynamics. According to this viwpoint, tmpratur dcrass whil hat is xposd to th fin. Thrfor, w conclud that hyprbolic hat quation violats th scond thrmodynamic law. Howvr, this phnomnon occurs during a vry short intrval. Morovr, thrmodynamic laws attmpt to dscrib quilibrium, whras non-fourir conduction sks to prsnt a corrct dscription of th transint bhavior. 7. Conclusion For th most practical purpos, th ffcts of non-fourir conduction ar ngligibl. As th siz of th microlctronic dvics dcrass to tiny portions and th circuit spd incrass, Fourir s law cannot b usd in hat transfr and tmpratur prdiction. Th wav charactr givs ris to th ffcts, which do not occur undr classical Fourir conduction. In th prsnt study, th non-fourir hyprbolic hat conduction was solvd in th straight small fin that is subjctd to thrmal and hat flux priodic boundary conditions using analytical solutions. Th non-fourir thrmal wav bhavior in th small fin for fast phnomnon (high frquncy priodic boundary condition) is succssfully xplaind by th rsults obtaind from th hyprbolic hat conduction modl. Th

7 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 95 ffcts of various paramtrs on th shock wav show that only th rlaxation tim has an influnc on th location and movmnt of th shock wavs. Th frquncy and amplitud of th priodic boundary condition and diffusivity cofficint of th fin hav a high influnc on th strngth of th thrmal shock wavs. Th parabolic (classical diffusion) and hyprbolic quations fail to captur th microscal rsponss during an unstady boundary condition. From a physical viwpoint, both modls violat th scond thrmodynamic law in th short-tim transint boundary condition. Th hyprbolic modl is rndring an undrstimatd tmpratur in th unstady hat flux boundary condition. Rsults show an inductiv bhavior, discontinuitis in th thrmal stp rspons, and ngativ (sub ambint) tmpraturs during th hating procss. Nomnclatur A b C H h K L q T T 0 T T b T b,m t w x Grk symbols : Amplitud of th input tmpratur : Thicknss of th fin : Spcific hat capacity : Dimnsionlss convctiv hat transfr : Convctiv hat transfr cofficint : Thrmal conductivity : Lngth of th fin : Hat flux : Tmpratur : Initial tmpratur of th fin : Ambint tmpratur : Priodic boundary condition : Man bas tmpratur : Tim : Width of th fin : Spatial coordinat α : Diffusivity cofficint : Dimnsionlss rlaxation tim η : Dimnsionlss spatial coordinat η s, η a,s : Dimnsionlss location of th thrmal shock λ : Mans fr path btwn phonons ν : Sound spd of in th mdium ρ : Dnsity τ : Rlaxation tim ω : Frquncy of th tmpratur oscillation Ω : Dimnsionlss frquncy of th oscillation ξ : Dimnsionlss tim θ : Dimnsionlss tmpratur : Dimnsionlss ambint tmpratur θ Rfrncs [] K. Fushinobu, K. Hijikata and Y. Kurosaki, Hat transfr rgim map for lctronic dvics cooling, Intrnational Journal of Hat and Mass Transfr, 39 (996) [] V. A. Cimmlli, Hyprbolic hat conduction at cryognic tmpraturs, Rniconti Dl Circolo Matmatico Di Palrmo, 45 (996) [3] G. Krzysztof, M. J. Cialkowski and H. Kaminski, An invrs tmpratur fild problm of th thory of thrmal strsss, Nuclar Enginring, 64 (98) [4] Y. W. Yang, Priodic hat transfr in straight fins, Journal of Hat Transfr, 94 (97) [5] R. G. Eslingr and B. T. F. Chung, Priodic hat transfr in radiating and convcting fins or fin arrays, AIAA Journal, 7 (979) [6] A. Aziz and T. Y. Na, Priodic hat transfr in fins with variabl thrmal paramtrs, Intrnational Journal of Hat and Mass Transfr, 4 (98) [7] S. A. Al-Sana and A. A. Mujahid, A numrical study of th thrmal prformanc of fins with tim-indpndnt boundary conditions including initial transint ffcts, Warm Stoffubrtrag, 8 (993) [8] J. Y. Lin, Th non-fourir ffct on th fin prformanc undr priodic thrmal conditions, Applid Mathmatical Modlling, (8) (998) [9] C. Y. Wu, Hyprbolic hat conduction with surfac radiation and rflction, Intrnational Journal of Hat and Mass Transfr, 3 (989) [0] A. Kar, C. L. Chan and J. Mazumdr, Comparativ studis on nonlinar hyprbolic and parabolic hat conduction for various boundary conditions: analytic and numrical solutions, Journal of Hat Transfr, 4 (99) 4-0. [] D. W. Tang and N. Araki, Th wav charactristics of thrmal conduction in mtallic films irradiatd by ultra-short lasr pulss, J. Phys. D: Applid Physics, 9 (996) [] H. S. Chu, S. Lin and C. H. Lin, A nw numrical mthod to simulat th non-fourir hat conduction in a singl-phas mdium, J. of Quantitativ Spctroscopy & Radiativ Transfr, 73 (00) [3] C. H. Huang and H. H. Wu, An itrativ rgularization mthod in stimating th bas tmpratur for non-fourir fins, Intrnational Journal of Hat and Mass Transfr, 49 (006) [4] C. C. Yn and C. Y. Wu, Modlling hyprbolic hat conduction in a finit mdium with priodic thrmal disturbanc and surfac radiation, Applid Mathmatical Modlling, 7 (003) [5] J. C. Chang and W. N. Juhng, Non-Fourir hat conduction in a slab subjctd to priodic surfac hating, Journal of th Koran Physical Socity, 36 (000) [6] A. Aziz and T. Y. Na, Priodic hat transfr in fins with variabl thrmal paramtrs, Intrnational Journal of Hat and Mass Transfr, 4 (98) [7] C. Y. Yang, Estimation of th priodic thrmal conditions on th non-fourir fin problm, Intrnational Journal of Hat and Mass Transfr, 48 (005) [8] D. W. Tang and N. Araki, Wavy, wavlik, diffusiv thr-

8 96 H. Ahmadikia and M. Rismanian / Journal of Mchanical Scinc and Tchnology 5 () (0) 99~96 mal rsponss of finit rigid slabs to high-spd hating of lasr-pulss, Intrnational Journal of Hat and Mass Transfr, 4 (999) [9] B. Abdl-Hamid, Modlling non-fourir hat conduction with priodic thrmal oscillation using th finit intgral transform, Applid Mathmatics Modl, 3 (999) [0] G. E. Cossali, Priodic conduction in matrials with non- Fourir bhavior, Intrnational Journal of Thrmal Scincs, 43 (004) [] I. A. Abdallah, Maxwll-Cattano hat convction and thrmal strsss rsponss of a smi-infinit mdium du to high spd lasr hating, Progrss in Physics, 3 (009) - 7. [] R. A. Guyr and J. A. Krumhansl, Solution of th linarizd Boltzmann quation, Physical Rviw, 48 (966) [3] A. Majumdar, Rol of fractal gomtry in th study of thrmal phnomna, Annual Rviw of Hat Transfr, 4 (99) 5-0. [4] P. Vrnott, Ls panadoxs d la thori continu d l'quation d la chalur, C.r.acad.Sci.Paris, 46 (958) [5] D. D. Josph and L. Prziosi, Hat wavs, Rviws of Modrn Physics, 6 (989) [6] D. Y. Tzou, Macro-to-microscal hat transfr: Th lagging bhavior, Taylor and Francis, Washington DC., USA (996). [7] M. C. Cattano, Sur un form d l'quation d la chalur liminant l paradox d'un propagation instantan, Compts Rndus Hbd. Sancs Acadmic. Scinc, 47 (958) [8] M. Wang, B. Cao and Z. Guo. Gnral hat conduction quation basd on thrmomass thory, Frontirs in Hat and Mass Transfr, (00) [9] M. Wang and Z. Guo. Undrstanding of siz and tmpratur dpndncs of ffctiv thrmal conductivity of nanotubs, Physics Lttr A, 374 (00) [30] V. S. Arpaci, Conduction hat transfr, Addison Wsly Publication, Nw York, USA (966). Hossin Ahmadikia is an assistant profssor of Mchanical Enginring at th Univrsity of Isfahan, Isfahan, Iran. H rcivd his B.Sc. dgr in Frdosi Univrsity, Mashad, Iran in 990. H rcivd his M.Sc. and Ph.D dgrs from Isfahan Univrsity of Tchnology, Isfahan, Iran in 993 and 000, rspctivly. His rsarch focuss on biological hat transfr and turbulnc modling. Milad Rismanian rcivd his B.Sc. dgr in Mchanical Enginring from Bu-Ali Sina Univrsity, Hamadan, Iran in 009. H is currntly an M.Sc. studnt in Sharif Univrsity, Iran. His rsarch intrsts includ nano-micro mchanics and biological hat transfr.