Simple 1-D semi-infinite heat conduction problems enable to demonstrate the  potential of the fractional calculus in determination of transient thermal  impedances under various boundary conditions imposed at the interface (x=0).  The approach is purely analytic and very effective because it uses only  simple semi-derivatives (half-time) and semi-integrals and avoids  development of entire domain solutions. 0x

El Ganaoui M.

Hristov Jordan

International audienceSimple 1-D semi-infinite heat conduction problems enable to demonstrate the potential of the fractional calculus in determination of transient thermal impedances under various boundary conditions imposed at the interface (x = 0). The approach is purely analytic and very effective because it uses only simple semi-derivatives (half-time) and semi-integrals and avoids development of entire domain solutions

Hristov, Jordan

El Ganaoui, Mohammed

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Thermal impedance estimations by semi-derivatives and semi-integrals: 1-D semi-infinite cases

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THERMAL IMPEDANCE ESTIMATIONS BYSEMI-DERIVATIVES AND SEMI-INTEGRALS1-D semi-infinite casesbyJordan HRISTOV a* and Mohammed El GANAOUI baDepartment of Chemical Engineering ,University of Chemical Technology and Metallurgy,Sofia, BulgariabLongwy University Technology Institute, Université Henri Poincaré-Nancy,Cosnes et Romain, FranceOriginal scientific paperDOI: 10.2298/TSCI120522211HSimple 1-D semi-infinite heat conduction problems enable to demonstrate the po-tential of the fractional calculus in determination of transient thermal impedancesunder various boundary conditions imposed at the interface (x = 0). The approachis purely analytic and very effective because it uses only simple semi-derivatives(half-time)andsemi-integralsandavoidsdevelopmentofentiredomainsolutions.Key words: thermal impedance, fractional calculus, half-time fractionalderivativesIntroductionA thermal source heating a half-space is a general academic formulation of high tem-perature source interaction with materials in the material processing and precision manufactur-ing. Although this is a very old subject [1, 2 ], it remains scientifically important and at the mi-croscopic scale, difficult tomasterinsomeconfigurations [3], such astransitional arrangementsand the presence of heat convection, cooling of electronic devices both at the package and sys-temlevel,andcooling ofpowersemi-conductorsusingheatsinks[4].Themainproblememerg-ing under such circumstances is about the transient materialbehaviour and the heat spreading indepth of the processed materials [2, 5, 6]. In many cases, the time-depended behaviour relevanttothe thermalimpedanceisofprimaryimportance[7-10]. Someworksontransient temperaturefieldduestoheatspotswithuniform[3]orcontinuous ortime-dependent heatsources[11]havebeendeveloped. Thethermalimpedancehasbeenestimatedbyeitherexactanalytical [1,3,5,6,9, 12-14] or numerical solutions [4, 9].Case 1Concerning the 1-D energy equation in a heated semi-infinite materials heated by aline (plane) isothermal source at x = 0, we have:¶¶¶¶TtTxTT x t  a22 ,( , ) (1)with boundary and initial conditions:l ¶¶Txqx txtTtTT xxs00000 00,,,,, ,(,) , (,)    Tt ,0 (2)Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589 581* Corresponding author; e-mail: jordan.hristov@mail.bgT(x, 0) indicates an undisturbed initial temperature field of the heated materials.The solution eq. (1) is well known [1]:Txt Tqt xtxtxt(,) e x p 	   2 14 2122s erflaa aa p     (3)Setting x = 0 we get the surface temperature Ts = T(0, t) that allows to define the ther-mal impedance of the semi-infinite medium Z =( Ts – T)/qs. In this simple case we have Z1 ==( 2 at)/(l p).Case 2Moreover, iftheenergyeq.(1)isgoverned byaprescribed temperatureboundary con-dition, namely:T(0,t ) =T s,t > 0, T=T ,t > 0 (4)with exact solution [1]:Txt TTTxtq TxTs xs (,),    ss erf12 0 all ¶¶Ttp a(5)Then, the thermal impendence denoted as Z2 is Z2 = p( at/l). In both examples, dueto the fixed position of the heat source, the transient resistance does not reach the static case [9,15].These simplecases only create the background of the problem developed in this work.Theprinciplequestioninsolvingsuchproblemsis:whytheentiredomainsolutionT(x,t)shouldto be developed and then setting x = 0, through T(0, t) and q(0, t), the thermal impedance has tobe determined?This work refers to determination of thermal impedances in 1-D semi-infinite prob-lemsdue to constant and time-dependent heat sources acting on ahalf-plane areawithout devel-opment of entire domain solutions. The latter type of heat sources are results of runaway chemi-cal reaction, uncontrolled nuclear processes and accidental flame effect on materials under fire.The main approach developed in this work is based on relationships between the local values oftheheatgradient(flux)andthetemperaturebasedonfractionalhalftimederivativeandintegrals[10,13,16,17]withoutdevelopmenttheentiredomainsolutions. Thefractional derivatives andintegrals are not local [16] widely encountered in applications to transient rheology [18, 19],heat[10, 13],andmasstransfer[20, 21],non-linear diffusion inporousandgranularmedia[22],Stefan problem [23-25] manifest this technique as a power tool for efficient engineering solu-tions of complex problems.Theoretical background:Fractional-time derivative and integralsFollowingtheresultsofKulishandLage[17](seealsoref.13)thetemperatureandtheflux governed by the model (1) are related by:qxtTxttTt(,)(,) // la¶¶ p1212 (6a)TxtqxttT (,)[(,) ] // 	al¶¶1212  (6b)Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...582 THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589The operators 1/2/t1/2 and –1/2/t–1/2are half-time fractional derivative and integral inthe Riemann-Liouville sense [16, 26], namely:D ddd tt TxtttTxutuu 1212120112///(,)(/)(, ) 	 ¶¶G(7a)Dd tt TxttTxutuu 	 1212120112///(,)(/)(, ) ¶¶G(7b)were u is a dummy variable and G(1/2) = p.It is important to stress on the fact that the relationships (2.1a,b) come from the equa-tion: ¶¶¶¶TtTx1212//  a (8)that can be obtained by representing eq. (1) as a product of two operators [16, 27] , namely:¶¶¶¶¶¶¶¶12121212 0//// tx txT   	   aa (9)Only the second term has physical meaning [16].The relationships (8) and (9) arenon-local and valid over the entire semi-infinite domain.Results: thermal impedance due stationary line sourcesGeneral relationshipsFrom the general definitions of the thermal impedance we get for every x and t:Zx tTxt Tqxtqxttqxt11212(,)(,)(,)[(,) ](,// al¶¶)() with l ¶¶Tzqtxs0(10a)Zx tTxt TqxtTxt TTxtt2 1212(,)(,)(,)(,)(,) // la¶¶TtTtT tpwith s (,) () 0 (10b)Now, we will demonstrate how this approach works with some classical boundaryconditions defining thermal loads at x =0 .Time-independent boundary conditions atx=0Letusconsider again thetwosimplecasesthat wecommentedatthebeginning, wherewith x = 0 we get:(1) with  l ¶¶Tzqxs0const. (11a)TtqttTqt T s (,)[( ,) ] // 0021212 	  	alal¶¶ p (11b)(2) with T(0,t ) =T s = const. (12a)Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589 583qq tTttTttTT ss    (,)(,)(// 00 1 1212lala¶¶ pp) (12b)Therefore,Z t Z t122() () , frac frac pp alal(13a, b)Theresultsarethesameasthoseobtainedbytheentire-domainsolutions(3)and(5).Time-depended line sources atx=0More interesting and practically cases emerge with time-dependent loads at x =0 .W econsider some classic case widely used in transient heat transfer problems with many analyticaland numerical solutions. This is the background of the results developed further in this workdemonstrating the efficiency of the method.Ramp (linear rise) thermal loadsConsider, two time-dependent analogues of the problems exemplified above.Case 3The thermal load at x =0i sT(0, t)=Ts= T + ktt.Taking into account thatDt Cm() =[C/G(1–m)]t–mandDt Ctm() =Cm[dm(Ct)]/d(Ct)m [16]for m = 1/2 we get:qtTk ttTtk stt ()// (,)()frac 021212 	  lala¶¶ p tp(14)Zktk tk tTttt222(– ) ramp laappl(15)Case 4Prescribed flux problem with a ramp rise q(0, t)=qs = kqt.Further with Dt C 12 / () =2 C t/p and Dt Ct 12 / () = (1/ C)[d–1/2(Ct)/d(Ct)–1/2] [16]with q(0, t)=qs = kqt the surface temperature:TtqttTk t T (,)[( ,) ] // 00 431 12123 	 	alal¶¶ p q (16)Then,ZTtTkT ktq 10 431()(,)  rampq qpal(17)Exponential temporal thermal sources atx=0Taking into account that [16]:De r f t ee e e kttkk tk t 12 1 / exp( ) exp( ) 	p(18a)De r f t eeee ktkkt kt   12 1 / exp( ) exp( ) (18b)DD DD 1/2 1/2 1/2 1/2tt tt Cf C f Cf C f () , ()  Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...584 THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589Case 5With T(0, t)=Texp(kett) we have:qtTtT k kt ktTtet et et sf r a c erf () (,) e x p ( ) ) 0 	 lapp (19a)ZTk tT k kt kt kqetet et et e21(e x p )exp( )[e x p ( ) ]  laerf te t tk tt 1erfal(19b)Case 6Similarly with q(0, t)=q0exp(kett), we have:TtqxttTqkkteqeq (,)[(,) ]exp( )// 0 1 1212 0 	 alal¶¶ erf kt T eq	  (20a)Zqkkt ktqkTeqeq eqe2001(e x p )exp( )exp( alerfqeqeq tktktt)erf al(20b)Periodic time-dependent boundary conditionsx=0Consider now more complicated thermal loads [17] expressed by time-depended andvarying functions, namely:Case 7With Ts(t)=T + TAsin(wt)a tx = 0 the boundary flux is:qt D T T tTts t () [ s i n ( ) ] / 	  law la120Ap(21a)qt T t ts() s i n 	 laww wApp 42 2 L (21b)andZ t ttt tTt 242 2 	 sin( )sin( )sinalwww w ppL(22)Case 8With qs(t)=qAsin(wt)a tx = 0 we have:Tt q t q t s t () [ s i n ( ) ] s i n /   	  alw alww w D AA1242 2 p W tp (23a)Zt ttTqt 142 2  	 sin( )sinsin( )alww wwppWqt Asin( ) w(23b)Long-time behavioursThe auxiliary Fresnel function L[16, 28] in (21a) controls the initial transient regime.ForlongtimesenoughtoholdL(t)0at,t~10p/wi.e.whenmax. / L 21 0 3 wt p   ,wetheme-Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589 585diumattainthesteady-periodicregime[17].Hence(Case7)thesteady-periodicheatfluxisqs(t)==( l/ a)TA w/2[sin(wt)+c o s ( wt)]. Then, the steady-periodic thermal impedance Z Tt 2 sin( ) is:Z t tt ttTt 22 	sin( )sin( )[sin( ) cos( )]alww ww(24a)Similarly,fortheotherauxiliary Fresnelfunction W[16, 28]ineq.(23a)becomesneg-ligible at t ~4 0p/w, i. e. when max. / W 24 1 0 2 w p   [17]. The corresponding steady-peri-odic thermal impedance is:Z t tttqt 14   sin( )sinsin( )alwwwpTqt4A sin( ) w(24b)Other time-dependent thermal sources atx=0The results of two frequently encountered time-dependent boundary conditions aresummarized in tab. 1 as good examples of the outcomes of the developed method.Table 1. Thermal impedances and surface temperature/flux expressions for sometime-dependent thermal loadsFunction Prescribedtemperature Flux Prescribed flux Temperatureft  TTb t st 	 Eq. (6a)qb t s  qEq. (6b)qbst la 2p TbtT sq 	al 2p Ztal2pZtalp2f = t2 Ts = T + ctt2 qctst la833pqs = cqt2 Tc t T s 	al16155pq Ztal38p Ztal1615 pCommentariesThe method developed here shows direct links between two interrelated issues re-quired to calculate the transient thermal impedance of heated bodies. The common approach istodeveloptheentiredomainsolutions,thencalculating thefieldgradientatx=0ofthetempera-ture at the same point is prescribed or vice versa. In most of cases, the entire domain solutionsare cumbersome, needs time-wasting techniques, special function, etc. finally, and those solu-tions reduce to pre-factors depending on the imposed boundary conditions, when the spaceco-ordinate issettozero.Hence,thefractional calculus approach avoids thedevelopment ofen-tire-domain solutions because both the temperature and flux are related through fractional halftime derivative and integral in the Riemann-Liouville sense.The 1-D half-plane problems were especially chosen to demonstrate the approach be-causesuchproblemareclassicintheliteratureandinmanycaseseitherexactorapproximateso-Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...586 THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589lutions exist, depending on the imposed boundary conditions, of course. Therefore, the resultsare directly comparable to those developed by the classical methods using entire-domain solu-tions. All results were especially present as products of at/l which is a dimension of a “ther-mal impendence” [KW–1m–2] and terms depending on the type of the imposed boundary condi-tion. This is a natural outcome of the basic Fourier law, because we have to mention that insemi-infinite problems with missing natural length scale, the role is played by at having a di-mensions of length. Obviously, no limit can be attained since the area is infinite, too.Cases of finite 1-D bodies can be also solved by fractional calculus methods but thisneeds different techniques and approaches beyond the scope of the present work.ConclusionsThe article presents a method for estimation and calculation of transient thermal im-pedancesof1-Dsemi-infiniteareas(bodies)undervarioustime-dependingboundaryconditionsconcerning both prescribed temperature and prescribed flux problem. The method is based onrelationships basedonRiemann-Liouville half-timefractionalderivatives andintegrals.Theap-proach is simple and avoids development of entire-domain solutions.AppendixFractional derivative-definition and propertiesFractional calculus is the branch of calculus that generalizes the derivative of a func-tion to non-integer order, allowing calculations such as deriving a function to 1/2 order. Despite“generalized” would be a better option, the name “fractional” is used for denoting this kind ofderivative. The Riemann-Liouville derivative is the most used generalization of the derivative.Itisbased onCauchy'sformulaforcalculating iterated integrals (seerefs.16and26formorede-tails):Dffd mm m()()()()xtxttx 	 11 10 G(A-1)Fractional derivatives satisfy quite well all the properties that one could expect fromthem, despite some of them are only characteristic of integer order differentiation and someother have restrictions. Assuming further m = 1/2 for seek of clarity of the explanation in themain text we have:– linearity Df g D f D g mm m [ () () ] () () ax bx a x b x 	 	 (A-2)– composition rule DDf ddf mgmgmg () () xxx 		– with m = 1/2 (a semi-derivative) used in the workdd121212 ///xC Cx p(A-3)Dfdfd(12 121212//// ()())Cx CCxCx (A-4)where C is any constant.AcknowledgmentsThe work was partially supported by the grant N10995/2012 of University of Chemi-cal Technology and Metallurgy (UCTM), Sofia, Bulgaria.Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589 587References[1] Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, London, 1959[2] Breaux, H. J., Schlegel, P. T., Transient Heating of Thin Plates, Int. J Heat Mass Transfer, 13 (1970), 1,pp. 218-211[3] Muzychka, Y. 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Heat Transfer, 122 (2000), 2, pp. 372-376Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...588 THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589Nomenclaturebt,b q – rate constant, [Ks–1/2]ct – rate constant, [Ks–2]cq – rate constant, [Wm–2s–2]kt,k q,k e,k et,k eq– rate constants, [s–1]q – heat flux density, [Wm–2]qA – heat flux amplitude, [Wm–2]qs – surface heat flux (x = 0), [Wm–2]T – temperature, [K]TA – temperature amplitude, [K]T – temperature of the undisturbed– medium, [K]t – time, [s]x – space co-ordinate, [m]Z – thermal impedance, [Km2W–1]Z1 – thermal impedance of a prescribed flux– problem, [Km2W–1]Z2 – thermal impedance of a prescribed– temperature problem, [Km2W–1]Z1(frac),Z 2(frac) – thermal impedances developed by– fractional derivative/ integral,– [Km2W–1]Greek symbolsa – thermal diffusivity, [m2s–1]l – thermal conductivity, [Wm–1K–1]G – gamma functionL and W – auxiliary Fresnel integral (functions f– and g in Abramowitz and Stegun [28]w – frequency, [s–1]Subscriptss – surface (at x =0 )T – prescribed temperature problemq – prescribed flux problem[18] Siddique, I., Vieru, D., Stokes Flows of a Newtonian Fluid with Fractional Derivatives and Slip at theWall, Int. 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Heat Mass Transfer, 53 (2010), 23-24, pp. 5622-5625[25] Liu, J., Xu,M., SomeExact Solutionsto Stefan Problemswith Fractional Differential Equations, J. Math.Anal. Appl., 351 (2010), 2, pp. 536-542[26] Debnath, L., Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhouser, Boston,Mass., USA, 1997[27] Babenko,Yu.I.,Heat-MassTransfer,MethodsforCalculationofThermalandDiffusionalFluxes(inRus-sian), Khimia Publ., Moscow, 1984[28] Abramovitz, M., Stegun, I. A., Handbook of Mathematical Functions, Dover, New York, USA, 1964Paper submitted: May 22, 2012Paper revised: November 13, 2012Paper accepted: November 20, 2012Hristov, J., El Ganaoui, M.: Thermal Impedance Estimations by Semi-Derivatives ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 581-589 589

Thermal impedance estimations by semi-derivatives and semi-integrals: 1-D semi-infinite cases

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