International Journal of Heat and Mass Transfer 56 (2013) 653–666

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

A unified solution of several heat conduction models

Tung T. Lam ⇑The Aerospace Corporation, El Segundo, CA 90245-4691, USA

a r t i c l e i n f o

Article history:Received 9 July 2012Received in revised form 11 August 2012Accepted 30 August 2012Available online 27 October 2012

Keywords:Heat diffusion modelCattaneo–Vernotte modelDual-phase-lag modelSimplified thermomass modelSuperposition methodSolution structure theorems

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08

⇑ Tel.: +1 310 336 5408; fax: +1 310 336 2270.E-mail address: tung.t.lam@aero.org

a b s t r a c t

Energy transport within a finite thin film subjected to either symmetric or asymmetric heating at theboundaries is investigated. Non-Gaussian heat sources are modeled as time-varying and spatially-decay-ing laser incidences. Comparison of heat transfer mechanisms based on the classical diffusion, Cattaneo–Vernotte, simplified thermomass, and dual-phase-lag models is undertaken. This study presents a singlegeneralized analytical temperature profile in a finite thin film in terms of an infinite series, utilizing thesuperposition technique in conjunction with the solution structure theorems, which is found to be appli-cable to solutions of all the aforementioned models. The temperature solution for a particular model canbe obtained directly by applying appropriate coefficients as they appear in the proposed generalized gov-erning heat conduction equation. Through analyses and numerical examples, the time history of heattransfer behaviors due to the collision of energy from both sides of the thin film is compared among thesemodels. It reveals that the method provides a convenient and efficient solution to the classical heat dif-fusion equation as well as other forms of hyperbolic heat conduction equations.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Conduction is a mode of heat transport that takes place in solidsin which thermal energy transfers from a region of higher temper-ature to that of lower temperature by the kinetic motion or directimpact of molecules without material mass movement. The funda-mental rate equation, governed by the first law of thermodynam-ics, describes the aforementioned mechanism based on themacroscopic diffusion theory which is generally credited to theFrench mathematical physicist, Joseph Fourier, who publishedthe theory in his book titled, Théorie Analytique de la Chaleur, in1822 [1]. The classical Fourier parabolic heat equation was usedfor all analyses until the 1950s even though it assumes that ther-mal energy travels within a solid medium at a non-physical infinitespeed. This is a valid assumption for most applications, but itbreaks down in situations that include low temperature conditionsor engineering applications with high-power for short durations, asthe Fourier law of heat conduction is applicable only at the spatialand temporal macroscales.

In the middle of the twentieth century, Morse and Feshbach [2],Cattaneo [3] and Vernotte [4] independently proposed a new formof the heat conduction equation in separate investigations by add-ing a relaxation term to account for the increase in the heat fluxvector due to phonon collisions in a duration of the mean free time,denoted by sCV. This became the classical thermal wave model

ll rights reserved..055

which is often referred to as the C–V hyperbolic heat conductionmodel, in which the mean free time is referred to as the relaxationtime (sCV) and is defined as the ratio of the effective mean free pathto the phonon speed (the speed of sound). As a result of this mod-ification, the original energy conservation equation assuming par-abolic heat conduction is transformed into a hyperbolic waveequation. The C–V hyperbolic heat conduction model addressesshort time scale effects over a spatial macroscale. Detailed reviewsof the thermal relaxation in the wave theory of heat propagationwere performed by Joseph and Preziosi [5] and Ozisik and Tzou [6].

With the emergence of micro- and nano-engineering technol-ogy in recent years, there is a strong manufacturing industry de-mand for better and more sophisticated analytical capabilities todescribe transient process of heat transport for extremely shorttime scales. Consequently, other heat conduction models havebeen developed based on different types of relations between theheat flux density vector and the temperature gradient as well asother types of constitutive relations.

Many new simulation models have recently been developed inorder to study the mechanisms of heat conduction in the micro/nanoscale that cannot be described by Fourier’s law. These studiesinclude phonon–electron interaction in metal films [7,8], phononscattering in dielectric crystals [9], insulators and semiconductors[10–12], etc. Most of the new developments have different physicalbases such as microscale rather than macroscale phenomenologi-cal approaches. The dual-phase-lag model [13] describes thermallag to accommodate relaxation behaviors of both the heat flux vec-tor and the temperature gradient at different times in the heattransfer process. Thermomass model [14] was developed by

Nomenclature

A constant, Eq. (16)an Fourier series coefficientB constant, Eq. (16)bn Fourier series coefficientb dimensionless parameter, Eq. (9)C constant, Eq. (16)C1 . . . C12 constants, Eqs. (39) and (42)c thermal wave propagation speed, m/scp specific heat, J/kg KD constant, Eq. (16)f total energy in systemfr reference heat fluxF functional form for Sub-problem 1 (hyperbolic) or 2

(parabolic)Gl energy magnitude factor at the left boundaryGr energy magnitude factor at the right boundaryg dimensionless incident energy at the boundarygl dimensionless energy strength at the left boundary,

Glð1� RÞJolgr dimensionless energy strength at the right boundary,

Grð1� RÞJolJ0 dimensionless laser fluencek thermal conductivity, W/m Kl unit length parameterM constant, Eq. (17)N constant, Eq. (17)R surface reflectanceq dimensionless heat fluxt dimensionless timetp dimensionless energy pulse timeT dimensionless temperatureT0 dimensionless temperature solution for n = 0T1 dimensionless temperature solution due to w function

contribution

T2 dimensionless temperature solution due to u functioncontribution

T3 dimensionless temperature solution due to f functioncontribution

Tn dimensionless time-dependent portion of the tempera-ture solution

x dimensionless spatial coordinate

Greek symbolsa thermal diffusivity, k/qcp, m2/sbn eigenvalue, Eq. (25c)cn eigenvalue, Eq. (33a)e relative errorf dummy variable for time#n eigenvalue, Eq. (25b)kn eigenvalue, Eq. (26b)l absorption coefficient, m�1

n dummy variable for spaceq density, kg/m3

qd density of dielectrics, kg/m3

s relaxation time constant, su initial condition functionw initial rate of temperature change function-n constant, Eq. (33b)

Superscript⁄ dimensional parameter

SubscriptCV Cattaneo–Vernotten series solution indexq heat fluxT temperatureTM thermomass

654 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

deriving the constitutive equation for heat transport based on thephonon gas equation of motion. This implies that both the thermaland mechanical fields consist of phonons, which are energy carri-ers with a finite mass. The constitutive equation derived fromthe thermomass model provides some knowledge on thermal wavepropagation through the phonon gas, thus resulting in slower ther-mal waves as compared to those determined by using the C–Vmodel. In a recent study [15], the thermomass model can also bederived from a microscopic basis with the Boltzmann transportequation. The resulting generalized heat conduction law was basedon both the macroscopic thermomass theory and the microscopicphonon Boltzmann method. Furthermore, a latest study [16] basedon the thermomass model also confirmed that the Fourier law ofheat conduction broke down in applications when subjected tovery high heat flux due to the spatial inertia of heat.

This paper presents closed-form temperature solutions basedon the parabolic and hyperbolic heat conduction equations for aone-dimensional medium in which both external sides are insu-lated. The medium under consideration is a thin film subjectedto a time-varying and spatially-decaying laser energy source inci-dent on both surfaces that is modeled as an internal heat genera-tion term inside the solid. The initial condition takes the form ofa sine function. The wave models selected in the study includethe C–V model, thermomass model, and dual-phase-lag model.The governing equations for all these models are solved by themethod of superposition with the aid of the solution structure the-orems. The temperature profiles take the form of a series solution.

Temperature distributions within the one-dimensional thin film,subjected to the aforementioned heating conditions are presented.Comparisons of the heat propagation behavior and the transientcharacteristics between the diffusion and the other three wave-based models are made.

The outline of the paper is as follows. In Section 2, the governingheat conduction equations based on the diffusion, C–V, simplifiedthermomass, and dual-phase-lag models are presented. Section 3presents a generalized heat conduction model, the boundary con-ditions and initial conditions. Section 4 describes the concept ofthe solution method for the generalized heat conduction problem,which is comprised of the superposition technique and the solu-tion structure theorems. Section 5 derives the general temperaturesolution which is applicable to various models for the physicalproblem given in Section 3. The application of the generalized heatconduction model is presented in Section 6. Sets of coefficients tothe governing partial differential equation for different heat con-duction models are summarized therein such that by selecting anappropriate set of coefficients, the solution to a particular heat con-duction model can be readily obtained. Since the Fourier heat con-duction diffusion model is different from other types of thermalwave models, Section 7 is devoted to the diffusion model. Section 8presents numeric case studies, in which temperature distributionswithin a thin film are also compared for the diffusion model andthe three wave-based models, i.e., the C–V model, thermomassmodel, and dual-phase-lag model. Concluding remarks on thisstudy are summarized in Section 9.

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 655

2. Heat conduction models

2.1. Diffusion model

The parabolic Fourier law heat conduction equation specifiesthat the heat flux conducted across a solid is proportional to thetemperature gradient taken in a direction normal to the materialsurface in question. The equation is based on the diffusion theory,which means that the entire temperature field reacts instanta-neously to a temperature variation within the solid. This isacceptable for most engineering applications even though theapproximation implies a nonphysical infinite speed of energytransfer. The Fourier law is given as

q� ¼ �k@T�

@x�ð1Þ

The energy conservation law is given by

qcp@T�

@t�¼ � @q�

@x�þ g� ð2Þ

where g⁄ denotes the internal energy generation rate per unit vol-ume inside a medium. By substituting Eq. (1) into Eq. (2), it yields

qCp@T�

@t�¼ k

@2T�

@x�2þ g� ð3Þ

Eq. (3) is the classical diffusion model which governs thermal en-ergy transport in solids. For convenience, the governing equation,Eq. (3), is non-dimensionalized by using the following parameters

x ¼ cx�

2a; t ¼ c2t�

2aT ¼ kcT�

afr; q ¼ q�

fr; g ¼ 4ag�

cfr; sCV

¼ ac2 ; l ¼ 2al�

c; J0 ¼

c2J�0afr

ð4Þ

After some mathematical manipulations, the governing partial dif-ferential equations for one-dimensional heat conduction repre-sented by Eq. (3) can be expressed in a dimensionless form as

2@T@t¼ @

2T@x2 þ g ð5Þ

2.2. Cattaneo–Vernotte (C–V) model

The classical diffusion heat conduction model, Eq. (5), simulatesenergy transport as a result of temperature variation within the so-lid. This approximation implies a simultaneous and nonphysicalinfinite speed of energy transfer. This assumption is acceptablefor most engineering applications; however, it breaks down inlow temperature or high power, short duration applications oncethe finite speed of thermal propagation becomes significant. Asa result of this deficiency, Cattaneo [3] and Vernotte [4] proposeda modified version of the heat conduction equation by adding arelaxation term to represent the lagging behavior of energy trans-port within the solid, which takes the form

sCV@q�

@t�þ q� ¼ �k

@T�

@x�ð6Þ

In Eq. (6), sCV (=a/c2) is the relaxation parameter that relates thethermal diffusivity to wave propagation speed. One can see that ifenergy travels at an infinite propagation speed (i.e., c ?1), Eq.(6) will be reduced to the classical Fourier heat conduction equa-tion. The equation governing the propagation of thermal energycan be derived by using the conservation law, Eq. (2), and the mod-ified Fourier’s law, Eq. (6). With the elimination of the heat flux, thehyperbolic heat conduction equation is obtained

@T�

@t�þ sCV

@2T�

@t�2¼ a

@2T�

@x�2þ a

kg� þ sCV

@g�

@t�

� �ð7Þ

Eq. (7) is the C–V model which governs heat conduction in solidsand takes the form of wave propagation. Again, it can be non-dimensionalized by using parameters given in Eq. (4), whichbecomes

2@T@tþ @

2T@t2 ¼

@2T@x2 þ g þ 1

2@g@t

ð8Þ

2.3. Thermomass model

Heat conduction in dielectrics is due to the motion of the ‘‘pho-non gas’’. By relating the similarity between the mass, pressure,and inertial force of the phonon gas with the fluid properties, heatflux, temperature and wave speed of the flow field, Guo and Hou[14] developed the thermomass model for phonon transport. Forbrevity, the one-dimensional thermomass heat conduction equa-tion is presented as

@T�

@t�þ sTM

@2T�

@t�2¼ að1� bÞ @

2T�

@x�2� 2l @2T�

@x�@t�

þ 1qdcp

g� þ sTM@g�

@t�þ l @g�

@x�

� �ð9Þ

where sTM, and l are the characteristic time and characteristiclength parameter, respectively. Additionally, b represents thesquare of the thermal Mach number, which is the ratio of the driftvelocity to the sound speed, for the phonon gas. The terms with band l represent the spatial inertia of heat transport caused by theheat flux. Details of these parameters are available in Guo andHou [14]. In Eq. (9), one may obtain the simplified thermomassmodel by neglecting the terms with coefficients l and b to yield

@T�

@t�þ sTM

@2T�

@t�2¼ a

@2T�

@x�2þ 1

qdcpg� þ sTM

@g�

@t�

� �ð10Þ

Eq. (10) can be non-dimensionalized by using the parameters givenin Eq. (4) to obtain

2@T@tþ sTM

sCV

@2T@t2 ¼

@2T@x2 þ g þ 1

2sTM

sCV

@g@t

ð11Þ

Eq. (11) represents the simplified thermomass wave model which issimilar to the C–V model with an extra coefficient sTM

sCVappearing in

the equation. This term depicts the ratio of the relaxation time inthe thermomass and the relaxation time in the C–V model. Themodel reduces to the C–V model if the ratio becomes unity.

2.4. Dual-phase-lag model

The C–V thermal wave model is an improvement over the Fou-rier diffusion heat conduction model as it avoids the unrealisticassumption of energy transfer at a nonphysical infinite speed asa result of temperature variation in a solid, allowing for a delayedresponse between the heat flux vector and the temperature gradi-ent. However, the C–V wave model still assumes an immediate re-sponse between the temperature and the energy transport. Inother words, the response would take place right after a tempera-ture gradient has been established across a material control vol-ume. Therefore, the C–V wave model still assumes aninstantaneous heat flow within a solid and one may conclude thatthe temperature gradient is always the cause while the heat flux isalways the effect during the transport process. As a result of thisdeficiency, Tzou [17–19] proposed the dual-phase-lag model to in-clude the cause-and-effect of the temperature gradient and heatflux relationship to remove the aforementioned assumption made

Table 1Coefficients for the generalized heat conduction equation.

Thermal model Coefficients

A B C D M N

Cattaneo–Vernotte 2 1 1 0 1 12

Simplified thermomass 2 sCVsTM

1 sCVsTM

0 sCVsTM

12

Dual-phase-lag 2 1 1 sT2sq

1 12

656 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

in the C–V thermal wave model. The end result allows either thetemperature gradient (cause) to precede the heat flux vector (ef-fect) or vice versa during the energy transport process. Theassumption can be written in a mathematical form as

q�ðx; t þ sqÞ ¼ �k@T�ðx; t þ stÞ

@x�ð12Þ

where sT and sq are the phase lags of the temperature and heat flux,respectively. For sT > sq (the temperature relaxation time is largerthan the heat flux relaxation time) the temperature gradient estab-lished across the solid is a result of the heat flow such that the heatflux is the cause while the temperature gradient is the effect. On theother hand, the opposite is true when sT > sq (the temperature gra-dient is the cause while the heat flux is the effect), which impliesthat the temperature gradient within the solid is established first,faster than the heat flux in this case. The first-order approximationof Eq. (12) results in the following expression

q�ðx; tÞ þ sq@q�ðx; tÞ@t�

¼ �k@T�ðx; tÞ@x�

þ sT@2T�ðx; tÞ@t�@x�

" #ð13Þ

Combining Eq. (13) with the energy equation designated by Eq. (3)yields

@T�

@t�þ sq

@2T�

@t�2¼ a

@2T�

@x�2þ asT

@3T�

@x�2@tþ a

kg� þ sq

@g�

@t�

� �ð14Þ

Once more, Eq. (14) can be re-written in a dimensionless formwith the parameters designated by Eq. (4), which yields

2@T@tþ @

2T@t2 ¼

@2T@x2 þ

1sT

2sq

@3T@x2@t

þ g þ 12@g@t

ð15Þ

3. Generalized heat conduction model

3.1. Governing equations

All heat conduction models presented in Section 2, exhibit somesimilarities. In this section, a generalized model is developed suchthat any of these models can be deduced from the proposed modelwith some slight modifications of the coefficients as they appear inthe original governing partial differential equations. The tempera-ture solution to a particular thermal model may then be obtainedfrom a generalized heat conduction equation.

The dimensionless governing heat conduction equation can bewritten in a generalized form as

A @T@tþ B @

2T@t2 ¼ C

@2T@x2 þD

@3T@x2@t

þ f ðx; tÞ ð16Þ

The above generalized heat conduction partial differential equationhas been non-dimensionalized by using the dimensionless parame-ters given by Eq. (4). By using Eq. (16), the C–V, simplified thermo-mass, and dual-phase-lag models can readily be obtained with aproper selection of coefficients A;B; C and D and a forcing functionf(x, t).

In the present study, an isotropic thin film, 0 6 x 6 1, with uni-form thickness and constant thermophysical properties is as-sumed. Initially, the thin film is at temperature T(x,0) = u(x). Fortime t > 0, the boundary at x = 0 and x = 1 are both kept insulated.At the same time, either of the boundaries can be subjected toexternal energy incidences symmetrically or asymmetrically.

3.2. Internal thermal energy generation

Sources of heating within the system are modeled as internalheat generations which are induced by time-varying and

spatially-decaying laser incidences at one or both boundaries.The forcing function denoted by

f ðx; tÞ ¼ Mg þN @g@t

ð17Þ

3.3. Boundary conditions

During the short period of heating, heat losses from the bound-aries are assume negligible [7,8], which implies

@Tð0; tÞ@x

¼ 0 ð18aÞ

and

@Tð1; tÞ@x

¼ 0 ð18bÞ

3.4. Initial conditions

It is assumed that the thin film is initially at temperature u(x)

Tðx;0Þ ¼ uðxÞ ð19aÞ

As the hyperbolic heat conduction equation is second order in time,therefore another initial condition is required to obtain a solution.The second initial condition can be deduced from Eq. (2). Sincethe heat flux within the solid is zero initially, the temporal temper-ature variation within the solid at t = 0 can be written as

@Tðx;0Þ@t

¼ gðx;0Þ2¼ wðxÞ ð19bÞ

Note that the boundary conditions and initial conditions pre-sented above are selected arbitrarily. They may not represent prac-tical situations in most engineering applications. Nevertheless, therationale for the selection is to verify the solution structure methodas stated in the upcoming section to the fullest extent possible.They are chosen for completeness and whether they are strictlypractical is not the intent of this work.

3.5. Generalized heat conduction equation coefficients

Equations governing the energy transport process, based on dif-ferent forms of thermal models, as presented in Section 2 are uti-lized in the analysis. For brevity, these equations will not berepeated herein. The governing equation for these models canreadily be obtained by selecting the appropriate set of coefficientsfrom Table 1.

4. Solution method

4.1. Method of superposition

One of the most common and rather simple analytical tools inthe study of heat conduction is the superposition technique.Superposition can be applied to linear heat transfer problems withnon-homogeneous terms [20]. The technique is developed by

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 657

splitting a complicated problem into multiple simpler problemswhose solutions can be combined to give a solution to the originalproblem. The benefit of this method is that one can split up a tran-sient problem with several nonhomogeneous terms and nonzeroinitial conditions into several problems, each having just one non-homogeneous term and a zero initial condition, or a homogeneousproblem with a nonzero initial condition. The physical problembeing investigated herein meets the aforementioned criteria andcan therefore be solved by superposition.

With the application of the principle of superposition, the originalproblem designated by Eq. (16) can be divided into three sub-problems by setting the heat generation term, Eq. (17), and the initialconditions, Eq. (19), to different values in each sub-problem: (1)f(x, t) = u(x) = 0 , (2) f(x, t) = w(x) = 0, and (3) u(x) = w(x) = 0. Solu-tions to these sub-problems are designated as T1, T2 and T3 respec-tively. Therefore the general solution of the original hyperbolicheat conduction, Eq. (16), is the sum of Sub-problems 1–3, which is

Tðx; tÞ ¼ T1ðx; tÞ þ T2ðx; tÞ þ T3ðx; tÞ ð20Þ

Note that T1, T2 and T3 represent the individual contributions ofthe initial rate of temperature change, initial condition, and inter-nal heat generation to the complete temperature solution, respec-tively. These sub-problems can be obtained by some conventionalmethods, i.e., Fourier method, method of separation of variables,method of Laplace transformation, etc.

4.2. Solution structure theorems

It is well understood that the aforementioned methods can alsobe applied to Sub-problems 2 and 3; however, it is time consumingand laborious. These two sub-problems can be easily solved withthe application of the solution structure theorems [21,22] oncethe solution to Sub-problem 1 is known. In this study, we will dem-onstrate that incorporating the solution structure theorems alongwith the superposition technique will save a great deal of effort.Only one sub-problem is required to be solved in detail while theremaining sub-problem solutions can be readily obtained. Withthe application of the solution structure theorems, the solutionsto all of these sub-problems can be written as follows [22]

T1ðx; tÞ ¼ F x; t;wðxÞð Þ ð21aÞ

T2ðx; tÞ ¼ A þ @

@t

� �F x; t;uðxÞð Þ þ DF x; t; k2

nuðxÞ� �

ð21bÞ

T3ðx; tÞ ¼Z t

0F x; t;�f; f ðx; fÞð Þdf ð21cÞ

Note that the constants A and D are different for each model, whichwill be defined later to differentiate the various models. In addition,the eigenvalue kn can be determined by solving the governing equa-tion or obtained directly from Table 2.1 in [22, p. 57] for specificboundary conditions.

In this study, the solution to Sub-problem 1, T1, given by Eq.(21a), will be solved by using the Fourier method. It follows thatSub-problems 2 and 3, given by Eqs. (21b) and (21c), can then bereadily solved by applying the solution structure theorems to thesolution of Sub-problem 1. The applicability and proof of thesolution structure theorems for heat transfer problems will notbe repeated in this paper but interested readers should refer toWang et al. [22].

5. Solution of the generalized heat conduction model

This section is devoted to the solution of the generalized heatconduction model as given in Section 3. Let us first determine

the contribution of the initial condition (Eq. (21a)) to the temper-ature profile (T1(x, t)). For the same initial and boundary conditionsas presented in Section 3, the solution to Sub-problem 1 can besolved as follows. Using Fourier series, one can write the solutionto the governing equation as

T1ðx; tÞ ¼X1

n

T1nðtÞ cosðknxÞ þ sinðknxÞ½ � ð22Þ

With the application of the boundary conditions, Eq. (18), the sineterm is deleted from Eq. (22) and it can be simplified as

T1ðx; tÞ ¼X1

n

T1nðtÞ cosðknxÞ ð23Þ

By substituting Eq. (23) into Eq. (16) and after some mathematicalmanipulations, one can obtain the following differential equation

B @2T1n

@t2 þ AþDk2n

� � @T1n

@tþ Ck2

nT1n ¼ 0 ð24Þ

The solution to Eq. (24) takes the form

T1nðtÞ ¼ e#nt ½an sinðbntÞ þ bn cosðbntÞ� ð25aÞ

with #n and bn are defined as

#n ¼ � AþDk2n

� �=2B ð25bÞ

and

bn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCk2

n=B � AþDk2n

� �2=4B2

qð25cÞ

and the coefficients an and bn in the corresponding series need to bedetermined. By substituting Eq. (25a) into Eq. (23), the temperaturewithin the thin film for Sub-problem 1 can be expressed as

T1ðx; tÞ ¼X1

n

e#nt an sinðbntÞ þ bn cos bnt½ � cosðknxÞ ð26aÞ

where

kn ¼ np ð26bÞ

Since a cosine term appears at the end of Eq. (26a), n = 0 is also asolution which will be determined separately. The constants an

and bn can be determined with the application of the initial condi-tion as given in Eq. (19a). Recall that Sub-problem 1 is solved withthe following conditions, f(x, t) = u(x) = 0, thus one can concludethat

bn ¼ 0 ð27Þ

With the application of the initial condition Eq. (19b),

wðxÞ ¼ anbn cosðknxÞ ð28Þ

By using the orthogonality property and performing the integrationover the interval 0 6 x 6 1, one may obtain

an ¼2bn

Z 1

0wðxÞ cosðknxÞdx ð29Þ

Once the values of the constants an and bn are known, the solutionto Eq. (26a) for the case with n > 0 is complete. However, the n = 0case must be treated separately because the integrands are differentfor this case. For n = 0, the derivation restarts from Eq. (24) follow-ing the same procedure above,

B @2T0

@t2 þA@T0

@t¼ 0 ð30Þ

With the application of the initial conditions and the orthogonalitycondition for Sub-problem 1, one can obtain the following solutionfor the n = 0 case:

658 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

T0ðx; tÞ ¼BA 1� e�

ABt

Z 1

0wðxÞdn ð31Þ

By combining Eqs. (26a) and (31), the final temperature forSub-problem 1 takes the form

T1ðx; tÞ ¼BA 1� e�

ABt

Z 1

0wðnÞdnþ 2

X1n¼1

e#nt

bn

Z 1

0wðnÞ

� cosðknnÞdf sinðbntÞ cosðknxÞ ð32aÞ

With the application of solution structure theorems provided by Eq.(21), Sub-problems 2 and 3, can readily be written as follows

T2ðx; tÞ ¼ B þ ð1� BÞe�ABth i Z 1

0uðxÞdn

þ 2X1n¼1

e#ntZ 1

0uðnÞ cosðknnÞdn

� �ðA þ #n þDk2

nÞsinðbntÞ

bn

�

þ cosðbntÞ� cosðknxÞ ð32bÞ

T3ðx; tÞ ¼BA

Z t

01� e�

ABðt�fÞ

h i Z 1

0f ðn; fÞdndf

þ 2X1n¼1

1bn

Z t

0e#nðt�fÞ sin½bnðt � fÞ�

Z 1

0f ðn; fÞ cosðknnÞdn

� �df

�cosðknxÞ ð32cÞ

Note that the temperature solution presented above is validwhen the eigenvalue bn is real. However, when the discriminantin Eq. (25c) has a purely imaginary value such that bn = cni, thesin(bnt) and cos(bnt) terms appearing in Eqs. (32b) and (32c) needto be modified with the following transformations in thecalculation

cn ¼ �#n-n ð33aÞ

where

-n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4BCk2

n

ðA þ Dk2nÞ

2

sð33bÞ

Thus, the trigonometric functions in the temperature solutions maybe replaced with the following expressions

e#nt sinðbntÞbn

¼ e#nð1þ-nÞt � e#nð1�-nÞt

2#n-nð34aÞ

e#nt cosðbntÞ ¼ e#nð1�-nÞt � e#nð1þ-nÞt

2ð34bÞ

0.0

1.0

2.0

3.0

4.0

0 1 2 3 4 5

Tem

por

al N

on-G

auss

ian

Hea

ting

Pro

file

t/tp2 e

xp(-

t/ tp)

tp = 0.1, 0.2, 0.3, 0.4, 0.5, 1.0

Time, t

Fig. 1. Temporal portion of the non-Gaussian heat source intensity profile.

6. Application of the generalized heat conduction model to laserincidence on a thin film

Recall from Eqs. (16) to (17) the generalized dimensionless heatconduction equation takes the form

A @T@tþ B @

2

@t2 ¼ C@2T@x2 þD

@3T@x2@t

þ f ðx; tÞ ð16Þ

where

f ðx; tÞ ¼ Mg þN @g@t

ð17Þ

and M and N are specific thermal model related constants whichcan be found in Table 1 and g represents the dimensionless internalenergy generation. To demonstrate the applicability of the general-ized heat conduction for the solutions of the C–V, thermomass anddual-phase-lag models, a special set of energy source, boundary and

initial conditions is selected for a case study. Again, the boundaryconditions and initial conditions are selected arbitrarily such thatit is possible to verify the solution structure method to the fullestextent. In other words, they are chosen for completeness andwhether they are strictly practical is not the intent of this work.

6.1. Pulsed laser heat source

As stated previously, the thin film is subjected to simultaneous(either symmetric or asymmetric) heating at both boundaries.Sources of heating are modeled as internal heat generations whichare induced by time-varying and spatially-decaying laser energyincidences. Similar to the studies by Tang and Araki [23,24], a mod-ified energy source term, g⁄, can be written

g�ðx; tÞ ¼ ð1� RÞJ�0l� Gle�l�x� þ Gre�l

�ðL�x�Þ� �ðt�=t�

2

p Þ�e

t�t�p ð35Þ

to account for energy incident on both sides of the thin film. In Eq.(35), tp represents the pulsed time while the symbols Gl and Gr rep-resent the magnitude factor of the heat source at the left (x = 0) andright boundaries (x = 1), respectively. In a dimensionless form Eq.(35) can be re-written in a non-Gaussian form as

gðx; tÞ ¼ gle�lx þ gre

�lð1�xÞ� �ðt=t2

pÞe� t

tp ð36Þ

The temporal portion of the heat source appearing in Eq. (36) is de-picted for various impulse times (tp) in Fig. 1. For small impulsetimes, one may envision that the incoming energy level will concen-trate at the beginning due to the nature of the peak intensity profile.As the impulse time increases, the peak of the intensity profile de-creases and therefore energy is spread out over a longer timeinterval.

By utilizing the energy function given by Eq. (36), the forcingfunction, Eq. (17), can be written as

f ðx; tÞ ¼ 1t2

p

glee�lx þ gre

�lð1�xÞ� �Mt þNð1� t=tpÞ� �

e�t

tp ð37Þ

The terms inside the first set of square brackets in Eq. (37) representthe magnitude of the spatially decaying incidence of the transmit-ted energy source on the left-hand-side and right-hand-side bound-aries, respectively. This portion of the equation captures theattenuation of the energy source intensity in the through-thicknessdirection. In terms of temporal variation, an exponential functiondesignated by the remaining terms provides a pulse profile that ac-counts for the pulsed time, tp, making it more realistic than a stepfunction or delta function.

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 659

6.2. Boundary and initial conditions

As both boundaries of the thin film are insulated, the tempera-ture gradients at both boundaries are zero, according to Eqs. (18a)and (18b). It is arbitrarily assumed that the thin film is initially attemperature

uðxÞ ¼ 0:01 sinðpx=2Þ ð38aÞ

Since the hyperbolic heat conduction equation is second order intime, another initial condition is required to obtain a solution. Thesecond initial condition can be deduced from Eq. (2). As the heatflux within the solid is zero initially, the temporal temperature var-iation within the solid at t = 0 can be written as

wðxÞ ¼ 0 ð38bÞ

6.3. Temperature solutions

By using the internal heat generation given by Eq. (36) with thefunctional form f(x, t) defined in Eq. (37), the temperature solutionsto the three sub-problems are as follows

T1ðx; tÞ ¼ 0 ð39aÞ

T2ðx; tÞ ¼0:02 B þ ð1� BÞe�ABt

h ip

þ 0:04X1n¼1

e#n tð1� 4n2Þp

ðA þ #n þDk2nÞ

sinðbntÞbn

þ cosðbntÞ� �

cosðknxÞ ð39bÞ

T3ðx; tÞ ¼BC1

At2p

ðC2e�t=tp þ C3Þ þ2t2

p

X1n¼1

tpðC4 � C5Þe�t

tp

n

� e#nt tpC6 cosðbntÞ � ðC7 þ C8ÞsinðbntÞ

bn

� ��C9

C10cosðknxÞ

ð39cÞ

where

C1 ¼ðgr þ g1Þð1� e�lÞ

lð39dÞ

C2 ¼Mtpð�AB ttp þ tp þ tÞ � N A

B tpðtp � tÞ þ t� �� �

ðAB tp � 1Þ2þ tðN

�MtpÞ �Mt2p ð39eÞ

C3 ¼Mt2p �

t2p M�N A

B� �

e�ABt

AB tp � 1� �2 ð39fÞ

C4 ¼Mtp t2pð#

2nt þ 2#n þ b2

ntÞ þ 2tpð#nt þ 1Þ þ th i

ð39gÞ

C5 ¼ N t2pðt � tpÞð#2

n þ b2nÞ þ 2#ntpt þ tp þ t

h ið39hÞ

C6 ¼ 2Mt2pð#ntp þ 1Þ � N tp½1� t2

pð#2n þ b2

nÞ� ð39iÞ

C7 ¼Mt2p 2#ntp þ 1þ t2

pð#2n � b2

nÞh i

ð39jÞ

C8 ¼ N tp ð#ntpÞ3 þ 2ð#ntpÞ2 þ #ntp½ðbntpÞ2 þ 1� þ 2ðbntpÞ2n o

ð39kÞ

C9 ¼ le�l ðgrel � glÞ cosðknÞ � gr þ gle

l½ �= l2 þ ðknÞ2h i

ð39lÞ

C10 ¼ ð#ntpÞ2 þ 2#ntp þ ðbntpÞ2 þ 1h i2

ð39mÞ

By using Eq. (39), the temperature solution to the C–V, thermomass,and dual-phase-lag models can readily be obtained with a properset of coefficients A;B; C;D;M;N and a forcing f(x, t) as tabulatedin Table 1.

As a reminder, if the discriminant of the eigenvalue bn in Eq.(25c) has a negative value, it needs to be changed to cn insteadas shown in Eq. (33a) and the sine and cosine functions appearingin Eq. (39) must also be modified by using the exponential func-tions as provided in Eq. (34).

7. Diffusion model

The classical Fourier heat conduction model takes the form of aparabolic partial differential equation which is slightly differentfrom the other three models (C–V, thermomass and dual-phase-lag) as they are hyperbolic in nature. Therefore the generalizedheat conduction model solution given in Section 5 cannot be ap-plied directly to the diffusion model.

In this study, the classic diffusion model will also be applied tothe thin film case study described in Section 6 utilizing the sameinitial and boundary conditions. The dimensionless governingequation for the classical diffusion model is given by Eq. (5). Asthe parabolic system is a first degree in time and second degreein space, the solution to the parabolic heat conduction equation re-quires only one initial condition which is given by Eq. (38a); theinitial temperature gradient, Eq. (38b), is not needed. The completetemperature solution consists of two sub-problems: one accountsfor the effects of the initial condition, u(x) and the second onefor the heat generation, f(x, t).

7.1. Solution of the diffusion model

The solution to the governing heat conduction equation for thediffusion model can be obtained by following the same approach asgiven in Section 6 for the hyperbolic system. By following the samesolution procedure, the temperature distribution for a one-dimen-sional thin film with both insulated boundaries based on the diffu-sion model, Eq. (5), can easily be derived and is available in the textby Wang et al. [22, p. 117]. The two sub-problems mentioned inthe previously paragraph can readily be written as

T2ðx; tÞ ¼Z 1

0uðnÞdnþ 2

X1n¼1

Z 1

0uðnÞ cosðknnÞdn

� �e�k2

nt=2

� cosðknxÞ ð40aÞ

T3ðx; tÞ ¼Z t

0

Z 1

0f ðn; fÞdndf

þ 2X1n¼1

Z t

0

Z 1

0e�k2

nðt�fÞ=2 cosðknnÞf ðn; fÞdndf

� �cosðknxÞ

ð40bÞ

It is clear that once the initial condition ðuÞ and the source function(f) that appear in Eqs. (40a) and (40b) become available, the T2 andT3 sub-problems can readily be evaluated. The final solution of thediffusion model, assuming a temperature profile in which the thinfilm’s boundaries are insulated, can be obtained by summing theindividual T2 and T3 solutions given by these two equations. Sub-problem 1 is not required since the initial condition which specifiedthe rate of change in temperature, w(x), does not exist.

7.2. Temperature solutions

Based on Eqs. (5) and (36), the heat generation takes the form as

660 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

f ðx; tÞ ¼ gle�lx þ gre

�lð1�xÞ� � t2t2

p

!e�

ttp ð41Þ

The solutions to the two sub-problems, Eqs. (40a) and (40b), can bewritten as follows

T2ðx; tÞ ¼0:02pþ 0:04

X1n¼1

e�k2nt=2

pð1� 4n2Þ

" #cosðknxÞ ð42aÞ

T3ðx; tÞ ¼0:5C1C11

t2p

þ 1t2

p

X1n¼1

C9C12 cosðknxÞ ð42bÞ

and

C11 ¼ t2p � tpðtp þ tÞe�

ttp ð42cÞ

C12 ¼ð2tpÞ2e�k2

nt=2 � 2tp 2ðtp þ tÞ � k2ntpt

� �e�t=tp

ð2� k2ntpÞ2

ð42dÞ

and C1 and C9 are given by Eqs. (39d) and (39l), respectively.

8. Results and discussion

This section presents numerical results for two case studies inorder to demonstrate the differences in the heat transfer mecha-nism using different heat conduction models. The distinct charac-teristics of each will also be examined and compared. The heatsource magnitudes at the left (gl) and right (gr) boundaries, energypulse time (tp), relaxation times (sCV,sT,sq,sTM) and energy attenu-ation (l) are the dominant parameters affecting heat transfer.

In this study, except noted otherwise, the dimensionless tem-perature profile for a range of dimensionless times t and positionsx were selected for the distinct cases: (1) gl = 10, gr = 10, tp = 0.001,and l = 10; (2) gl = 10, gr = 0, tp = 0.01, and l = 10. Results aregenerated using the single generalized closed-form temperaturesolution. Magnitudes of the heat sources and energy attenuationeffects will not be explored in this investigation, as they have beenstudied by Lam [25] for a similar physical configuration with theC–V model by using the Godunov scheme.

The solution method utilizing the superposition techniques andthe solution structure theorems on the study of heat conduction insolids has successfully been used by Lam and Fong [26,27]. Thetemperature profile is presented in a series form, therefore the

Table 2Temperature comparisons between diffusion, C–V and DPL models.

t Diffusion and DPL comparison

x Diffusion DPL (sT/2sq = 0.5) Absolute diff

0.1 0.0 0.1294E+01 0.1291E+01 3.16E�030.2 0.1041E+01 0.1039E+01 1.05E�030.4 0.5556E+00 0.5558E+00 1.77E�040.6 0.2087E+00 0.2094E+00 7.46E�040.8 0.6345E�01 0.6442E�01 9.81E�041.0 0.3040E�01 0.3145E�01 1.05E�03

0.5 0.0 0.5915E+00 0.5886E+00 2.83E�030.2 0.5752E+00 0.5736E+00 1.58E�030.4 0.5326E+00 0.5323E+00 2.89E�040.6 0.4800E+00 0.4808E+00 7.99E�040.8 0.4376E+00 0.4391E+00 1.52E�031.0 0.4213E+00 0.4231E+00 1.77E�03

1.0 0.0 0.5136E+00 0.5123E+00 1.21E�030.2 0.5122E+00 0.5115E+00 7.21E�040.4 0.5086E+00 0.5084E+00 1.59E�040.6 0.5041E+00 0.5045E+00 3.47E�040.8 0.5005E+00 0.5012E+00 6.96E�041.0 0.4991E+00 0.5000E+00 8.21E�04

accuracy of the current method depends only on the number ofterms used in the series expansion process. One needs only toincrease the number of terms to achieve the desired accuracy.Since the solution takes the form of an infinite series with a simplealgebraic expression, the computation time is fast and efficient.The calculation process of the series solution termination is basedon a preselected criteria with relative error e. The stopping crite-rion for this series is based on a relative error test commonly usedin engineering and scientific computations. If Tn+1 and Tn are twosuccessive partial sums for the temperature, then the relative errortest can be stated as

relative error ¼ jTnþ1 � tnjjTnþ1j

6 e ð43Þ

The iteration process for the temperature reaches closure when theerror criterion is e 6 10�6.

8.1. Model validation

To validate the models presented in this study, a comparison ofthe temperature profiles in a thin film based on the diffusion, C–V,and DPL models is investigated for the Case 2 condition. By lettingthe relaxation constants sT = sq or sT/2sq = 0.5 within the DPL mod-el, one may obtain the thin film temperatures which are equivalentto the solution for a diffusion model. Table 2 presents the temper-ature profiles for three time intervals based on these two modelswith the aforementioned assumption in the DPL model. It is obvi-ous that temperature results from these two models comparedwell. Furthermore, the DPL model can be reduced to pure waveC–V model results by setting sT/2sq = 0 in the generalized model.Temperature results presented in Table 2 confirms that the gener-alized model is indeed valid for the C–V and DPL models. Based onthe results as shown, one can conclude that the diffusion and gen-eralized heat conduction models as formulated in this study isvalid.

8.2. Diffusion model

Fig. 2 shows the temporal progression of the dimensionlesstemperature profile for Case 1 using the diffusion model, Eq.(42). This case simulates a situation in which both boundariesare subjected to symmetrical energy exposures. Due to externalheating, the exposed surface temperature is considerably higher

C–V and DPL comparison

erence x C–V and DPL (sT/2sq=0) Absolute difference

0.0 0.2362E+01 00.2 0.9035E+00 00.4 0.1277E+00 00.6 0.2446E�01 00.8 0.1160E�01 01.0 0.1035E�01 0

0.0 0.4305E+00 00.2 0.4896E+00 00.4 0.1013E+01 00.6 0.5353E+00 00.8 0.8031E�01 01.0 0.2718E�01 0

0.0 0.3453E+00 00.2 0.3431E+00 00.4 0.3390E+00 00.6 0.3556E+00 00.8 0.5586E+00 01.0 0.1793E+00 0

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 661

than that at the mid-section of the medium. Once the pulse hasended, the absorbed-transmitted energy moves toward the centerof the thin film; as a result, this causes the peak temperatures atthe boundaries to move slower as time increases. The wholeapproaches an equilibrium state at large values of t.

Fig. 3 presents the temperature history at several locations forCase 2. In this case, only the left-hand-side boundary is subjectedto external heating. Unsurprisingly, the peak temperature occursat locations near the incident boundary. As t increases andincoming energy from the left diffuses inward toward the right-hand-side of the boundary, the temperature at the left surfacedecreases while the opposite side rises. The temperature withinthe medium eventually reaches an equilibrium state at large valuesof t as the absorbed energy reaches a steady-state condition.

8.3. C–V model

Solutions of the hyperbolic heat conduction equation (C–V mod-el) were solved by Tan and Yang [28,29], Torii and Yang [30] andLewandowska and Malinowski [31] for the case of a thin film sym-metrically or asymmetrically heated on both sides. The former twostudied the collision of thermal waves in a thin film due to asym-metrical temperature change on both sides while the latter twoinvestigations analyzed the symmetrical heating effects of aninternal heat source. The present work studies a similar problem

Tem

per

atur

e, T

Time, t

x = 0.01

0.5

0.3

1

0.1

0.8

Fig. 3. Diffusion model – temporal temperature profile at various positionssubjected to asymmetric heating for Case 2.

Tem

per

atur

e, T

Position, x

t = 0.5

0.01

0.1

0.05

Fig. 2. Diffusion model – spatial temperature profile for various values of timesubjected to symmetric heating for Case 1.

in which the thin film is heated either symmetrically or asymmet-rically by a pulse heat flux on both surfaces with a temporal profilein a non-Gaussian form and with spatially-decaying internal heatgeneration.

Fig. 4 shows the temporal progression of the dimensionlesstemperature profile for Case 1, solved according to the Cattaneo–Vernotte model, Eqs. (32a) to (32c), with coefficients from the firstrow of Table 1. For small values of tp, two sharp waves appear nearthe boundaries as both boundaries are exposed to symmetric en-ergy loads. Due to lagging behavior, two thermal waves travelslowly toward the center of the thin film after the heat sourcescease operations. As energy moves inward, both waves meet atthe center of the thin film at t = 0.5 which creates a new peak tem-perature at this location. Note that the peak at the center is greaterthan the initial temperature peaks at either boundary. The ‘‘in-crease’’ in peak temperature is due to constructive interference ofsomewhat elastic waves, is only momentary, and is unstable. Thetemperatures at the boundaries are dropping while the peak tem-peratures are traveling with the wave fronts. At the end of this firstencounter, the exposed surface temperature is considerably lowerthan that at the mid-section of the medium. The accumulatedenergy at the center will reflect back toward the boundaries tocomplete another cycle of propagation. The cycle will repeat itselfuntil the temperature within the medium approaches an equilib-rium state at large values of t.

Fig. 5 presents the temperature history at several locations forCase 2. In this case, only the left-hand-side boundary is subjectedto external heating. The peak temperature occurs at locations nearthe incident boundary, x = 0, as expected. As shown previously, theabsorbed energy will propagate in a wave form toward the right-hand-side boundary and then reflect backward to the left side. InFig. 5, the initial peak at x = 0.01 appears at t near 0, while the sec-ond and third peaks pass through this same location at t = 2 and 4.At x = 1.0, the wave arrives at this location and thus creates a peaktemperature at t = 1 and 3. Waves passing other locations are alsoshown in Fig. 5 for reference. The thermal waves will travel backand forth until the medium reaches an equilibrium condition at alarge value of t once the absorbed energy has achieved steady-stateas shown in Fig. 5.

Note that the wave propagation and reflection phenomenon isclearly shown in Fig. 5. For example, at x = 0.3, the thermal wavemakes its first pass at �t = 0.3 while the reflected energy fromthe right-hand-side boundary passes this same location at�t = 1.7. This constitutes a full cycle of the incoming energytraveling through x = 0.3. Once the return energy reaches the

Tem

per

atur

e, T

Position, x

8

t = 0.5

0.1

21

4

0.3

Fig. 4. C–V model – spatial temperature profile for various values of time subjectedto symmetric heating for Case 1.

Tem

per

atur

e, T

Position, x

6

t = 8

7

2

4

Fig. 6. Simplified thermomass model – spatial temperature profile for variousvalues of time subjected to symmetric heating for Case 1.

Tem

per

atur

e, T

Time, t

x = 0.01

0.5

0.3 1

0.1

0.8

Fig. 5. C–V model – temporal temperature profile at various positions subjected toasymmetric heating for Case 2.

ratu

re, T

1

x = 0.01

662 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

left-hand-side boundary and is reflected toward the right-hand-side surface again, the energy makes the third pass throughx = 0.3 at �t = 2.3. The reflected wave makes the fourth and fifthpasses through x = 0.3 at �t = 3.7 and 4.3. Hence, the periodictemperature peaks can be seen at the given locations shown in thisfigure. Similar phenomena for energy passing through x = 0.01, 0.1,0.5, 0.8, and 1 are also shown in Fig. 5 for comparison.

Tem

pe

Time, t

Fig. 7. Simplified thermomass model – temporal temperature profile at variouspositions subjected to asymmetric heating for Case 2.

8.4. Simplified thermomass model

Fig. 6 shows the temperature profile for a range of dimension-less time t for Case 1. The two characteristic time parametersselected for this simulation are sCV = 4.65 � 10�12 s andsTM = 1.38 � 10�10 s [14]. As for the C–V model, even though bothsides of the boundaries are subjected to symmetrical energyexposures, the boundaries may not be the hottest locations dueto thermal wave propagation. Peak temperatures are located inthe interior of the film instead of at the exposed surfaces.Fig. 6 depicts two waves traveling away from the boundaries to-ward the center. After collision, the waves continue to travel to-ward the boundaries to complete the first pass. Thereafter,reflections take place and the reflected waves propagate towardtheir initial origin. As the pulsed energy stops, thermal propaga-tion and reflection take place throughout the thin film, the entiremedium eventually will come to an equilibrium state at a largevalue of t, similar to the C–V model prediction.

Fig. 7 presents the temperature history for Case 2 at two loca-tions, x = 0.01 and 1, to demonstrate the propagation and reflectionphenomenon similar to Fig. 5 for the C–V model. The peak waveamplitude is reduced as time, t, increases. In this case the overalltemperature in the thin film approaches to a steady-state temper-ature value of T = 0.5 at t > 80.

8.5. Dual-phase-lag model

Al-Khairy [32,33] investigated thermal wave propagation in afinite medium heated on both sides symmetrically by a heat sourcewith Gaussian distribution in both the temporal and spatialdomain using the dual-phase-lag model. The analytical solutionis obtained by using the finite integral transforms and the methodof variation of parameters. Recall that a non-Gaussian heat sourceis employed in the present study. The temperature profile withinthe thin film based on the dual-phase-lag model is presented inFig. 8 for Case 1 with sT/2sq = 0.002 for various t, while the temper-ature history at several locations is shown in Fig. 9 for Case 2, with

sT/2sq = 0.2. Since the C–V, thermomass, and dual-phase-lag mod-els are all developed based on wave theory for the study of energytransport in solids, their predicted temperatures exhibit similarwave-form behavior. A close examination of Figs. 8 and 9 revealsthat the temperature results using the dual-phase-lag model dis-play similar physical characteristics as discussed in the previousparagraphs with the C–V and thermomass models. The only differ-ence is that the magnitude of the peak temperature and its fre-quency of occurrence may be different. The effect of the sT/2sq

parameter on the temperature response will be discussed in moredetail in Fig. 10. The energy transport mechanism is similar to theprocess previously discussed in the thermomass model section.Fig. 8 also presents the temperature history at several locationsto demonstrate the propagation and reflection phenomenon as dis-cussed above in Fig. 5 for the C–V model.

The major distinction between the dual-phase-lag model andother heat conduction wave models lies in the parameter sT/2sq,which is defined as the ratio between the temperature relaxationtime and the heat flux relaxation time; this parameter determinesthe dominant mode of heat transport. The temperature responseexhibits wavelike behavior when 0 < sT/2sq < 0.5. When sT/2sq = 0, the temperature response is reduced to the response pre-dicted with the C–V model. The ‘‘wavy’’ feature in the temperatureprofile disappears when sT/2sq equals to 0.5 as the energy trans-port mechanism transitions from wave propagation to diffusion.

Tem

per

atur

e, T

Time, t

x = 0.01

0.5

0.3

1

0.1

0.8

Fig. 9. Dual-phase-lag model – temporal temperature profile at various positionssubjected to asymmetric heating for Case 2 with sT/2sq = 0.2.

Tem

per

atur

e, T

Position, x

τ τ 0.0, 0.02, 0.1, 0.3, 0.5

Fig. 10. Dual-phase-lag model – effect of relaxation time constant (sT/2sq) onspatial temperature profile subjected to symmetric heating with gl = 10, gr = 10,tp = 0.1, t = 0.5 and l = 10.

Tem

per

atur

e, T

Position, x

τT/2τq = 0

0.002

0.02

0.1, 0.15, 0.2, 0.3, 0.5, 1.0

Fig. 11. Dual-phase-lag model – effect of relaxation time constant (sT/2sq) onspatial temperature profile subjected to asymmetric heating with gl = 10, gr = 0,tp = 0.001, t = 0.4 and l = 10.

Tem

per

atur

e, T

Position, x

2

t = 0.5

8

0.1

1

4

0.3

Fig. 8. Dual-phase-lag model – spatial temperature profile for various values oftime subjected to symmetric heating for Case 1 with sT/2sq = 0.002.

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 663

When sT/2sq is greater than 0.5, the temperature profile exhibits athermal diffusion-like effect which results in rapid temperature re-sponse in a relatively shorter time, similar to the classical diffusion

model; thus it is termed ‘‘over-diffusion.’’ The behavior of this phe-nomenon is clearly shown in Fig. 10. Similar behavior for the effectof sT/2sq on the temperature profile can be seen in Fig. 11 whentemperatures are plotted as a function of x for various values ofsT/2sq. Results indicate that as sT/2sq increases, thermal diffusiondominates which implies that a rapid temperature response willtake place in shorter time. Consequently, a temperature equilib-rium condition within the film similar to heat diffusion behavioris achieved at a longer time.

Another detailed examination of the effect of the relaxation timeparameter, sT/2sq, on the mode of heat transport is further pre-sented in Figs. 12a–12d. The temperature distribution for variousvalues of relaxation time at various times is presented withgl = 10, gr = 0, tp = 0.1, and l = 10. Fig. 12a shows the temperatureprofiles for a pure wave propagation phenomenon, where sT/2sq = 0. It is evident that the temperature profiles exhibit waveforms within the medium at early times. The arrows designatethe energy propagation directions. Thermal propagation and reflec-tion in the thin film due to energy pulsation at the surfaces can beclearly seen. At early times, energy travels from the incident surfaceon the left-hand-side boundary and then propagates toward theright-hand-side boundary. The arrival energy is then reflected backto the left-hand-side boundary. The propagation and reflection cy-cles will continue until the system has reached its equilibrium state.

Fig. 12b examines the case with sT/2sq = 0.1. It is expected thattemperatures within the medium will exhibit wavelike profiles.The only difference between the current case and the case pre-sented in Fig. 12a is that the peak temperatures are lower sincethe relaxation time constant is slightly higher (0.1) as comparedto the former case (0).

Figs. 12c and 12d represent the temperature profiles for diffu-sion and over-diffusion conditions which have relaxation time con-stants (sT/2sq) of 0.5 and 1.5, respectively. Since both casessimulate the diffusion phenomenon, the temperature profiles arepurely diffused without wave forms. The main difference betweenthese two cases is that the magnitude of the peak temperature de-creases as the relaxation time constant sT/2sq deviates further fromthe pure wave case (sT/2sq = 0). A close examination of Fig. 12c and12d reveals that the total energy content of the thin film increaseswith time. Refer to the temporal portion of the non-Gaussian heatsource intensity profile shown in Fig. 1. For small impulse times,the incoming energy level is concentrated at the beginning. Asthe impulse time increases, the peak of the intensity profiledecreases and energy is more spread out over time, resulting in adecrease in energy concentration at the wave front. This behavior

Tem

per

atur

e, T

Position, x

t = 0.3

0.5

0.7

1.01.5

2.0

2.5

1.8

Fig. 12a. Dual-phase-lag model – spatial temperature profile (pure wave propaga-tion, sT/2sq = 0) for various values of time subjected to asymmetric heating withgl = 10, gr = 0, tp = 0.1, and l = 10.

Tem

per

atur

e, T

Position, x

t = 0.3 0.5, 1.0, 1.5, 2.0, 2.5

Fig. 12d. Dual-phase-lag model – spatial temperature profile (over-diffusion, sT/2sq = 1.5) for various values of time subjected to asymmetric heating with gl = 10,gr = 0, tp = 0.1, and l = 10.

τΤ/2τ = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5

Tem

per

atur

e, T

Time, t

Fig. 13a. Dual-phase-lag model – effect of the relaxation time constant (sT/2sq) onthe temporal temperature profile subjected to asymmetric heating with gl = 10,gr = 0, tp = 0.1, x = 0.0 and l = 10.

Tem

per

atur

e, T

Position, x

0.5

0.7

1.0

1.5

t = 0.3

Fig. 12c. Dual-phase-lag model – spatial temperature profile (diffusion, sT/2sq = 0.5) for various values of time subjected to asymmetric heating with gl = 10,gr = 0, tp = 0.1, and l = 10.

Tem

per

atur

e, T

Position, x

t = 0.3

0.5

0.7

1.0

1.5

2.0

2.5

Fig. 12b. Dual-phase-lag model – spatial temperature profile (wavelike, sT/2sq = 0.1) for various values of time subjected to asymmetric heating with gl = 10,gr = 0, tp = 0.1, and l = 10.

664 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

is clearly shown in Figs. 12c and 12d, for longer impulse times, thewavy behavior flattens out. In other words, the wave characteris-tics of the temperature disappeared.

The effect of the relaxation time constant on the temperaturetime history at the boundaries, x = 0.0 and 1.0, is shown inFigs. 13a and 13b, respectively. The simulations are investigatedwith gl = 10, gr = 0, tp = 0.1, x = 0.0 and l = 10. Again, the tempera-ture profile exhibits a strong wave form when sT/2sq = 0 and tran-sitions to a pure diffusion form as the relaxation time constant (sT/2sq) is departed further away from zero. The major difference isthat the magnitudes of the peak temperatures are higher for thecase at x = 0, Fig. 13a, as compared to the location at x = 1,Fig. 13b, due to the pulsed energy incident at the location x = 0.Therefore, the peak temperature is expected to be higher at theincident boundary. As the heat propagates toward the oppositeside, energy is lost along the way as it warms the medium. Themagnitude of the peak temperature is therefore lower at theright-hand-side boundary as compared to that at the left-hand-side boundary.

8.6. Comparison of models

A comparison of the classical diffusion heat conduction modeland three other thermal wave models (e.g., C–V, simplified thermo-mass, and dual-phase-lag) are made herein. In the followingdiscussion, the values for the time characteristic parameters,

Tem

per

atur

e, T

Position, x

Diffusion

Dual-Phase-Lag

Thermomass C-V

Fig. 14b. Comparison of models – effect of laser pulse time on spatial temperatureprofile subjected to symmetrical heating with gl = 10, gr = 10, tp = 0.1, t = 0.1 andl = 10 (sT/2sq = 0.02).

τΤ/2τ = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5

Tem

per

atur

e, T

Time, t

Fig. 13b. Dual-phase-lag model – effect of the relaxation time constant (sT/2sq) onthe temporal temperature profile subjected to asymmetric heating with gl = 10,gr = 0, tp = 0.1, x = 1.0 and l = 10.

Tem

pera

ture

, T

Time, t

Dual-Phase-Lag

Thermomass

Diffusion

C-V

Fig. 15. Comparison of models – temporal temperature profile subjected toasymmetrical heating with gl = 10, gr = 0, tp = 0.01, and l = 10 at x = 0.01 (sT/2sq = 0.02).

T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666 665

sCV and sTM, used in the simplified thermomass model simulationare sCV = 4.65 � 10�12 s and sTM = 1.38 � 10�10 s, respectively,while the ratio of sT/2sq is set to 0.02 in the dual-phase-lag model.

Fig. 14a presents a dimensionless temperature profile as a func-tion of x with gl = 10, gr = 10, tp = 0.001, t = 0.1 and l = 10 for allthree wave models and the diffusion model. It reveals that peaktemperatures predicted with the simplified thermomass modelare higher than those with the C–V and dual-phase-lag models,and significantly higher than those with the diffusion model. Thepredicted temperatures from the simplified thermomass and C–Vmodels remain the same throughout the region with a slight differ-ence near the peaks.

In the thermomass model simulation, the value of the relaxa-tion time constant ratio sCV/sTM = �0.034 is selected. This meansthat the relaxation times, sTM, is much greater than the value ofsCV (i.e., the thermal transport velocity in the thermomass modelis much lower than that in the C–V). The much faster heat transportpredicted by the CV model leads much lower local temperaturepeaks than that those predicted by the thermomass model. Sincethe heating time, tp = 0.001, is much smaller than the characteris-tics relaxation times sTM and sCV, the diffusion model wouldexperience the highest thermal transport velocity which leads toa much lower localized peak temperature. The results are clearlyrevealed in Fig. 14a.

Tem

per

atur

e, T

Position, x

Diffusion

Dual-Phase-Lag

Thermomass

C-V

Fig. 14a. Comparison of models – effect of laser pulse time on spatial temperatureprofile subjected to symmetrical heating with gl = 10, gr = 10, tp = 0.001, t = 0.1 andl = 10 (sT/2sq = 0.02).

By increasing the pulse time tp from 0.001 to 0.1, a snap shot ofthe temperature profile is shown in Fig. 14b. It reveals that increas-ing the pulse duration flattens the peak temperature wave as a re-sult of a decrease in energy concentration throughout the thin film.In this simulation, temperatures within the medium predictedwith the diffusion model have almost reached equilibrium as ithas the fastest transport velocity. The C–V temperature predictionstill shows significant peaks when compared to the diffusion mod-el prediction as it has a lower heat transport velocity. Resultsreveal that the thermal energy waves predicted with thedual-phase-lag model and C–V model travel faster than with thethermomass model with sCV/sTM = �0.034.

Fig. 15 compares the temporal temperature responses of thefour models. Dimensionless temperature profiles are plotted as afunction of t with gl = 10, gr = 0, tp = 0.01, and l = 10 at x = 0.01(sT/2sq = 0.02). There are significant differences between modelsas the magnitude and location of the peak temperatures are quitedifferent. The disparity between the results of these models lies inthe relaxation time constants sCV, sT, sq and sTM; they play animportant role in heat conduction.

The inter-relationship of these relaxation time constants de-serves further explanation. The constants sT/2sq and sTM/sCV appear

666 T.T. Lam / International Journal of Heat and Mass Transfer 56 (2013) 653–666

in the dual-phase-lag and thermomass models, respectively. Aspreviously discussed in Section 8.1, the dual-phase-lag model canbe reduced to the C–V and diffusion models by setting the constantto 0 and 0.5, respectively. Results for the C–V model can beobtained from the thermomass model if the constant sTM/sCV isset to unity. Therefore, the magnitude and speed of the tempera-tures within the solids are heavily dependent on the selection ofthese constants.

9. Summary

An analytical equation, in the form of an infinite series, has beendeveloped for the temperature response in a finite thin film sub-jected to heating by a time-varying and spatially-decaying internalheat source. The solution was obtained utilizing the superpositiontechnique, solution structure theorems, and Fourier series expan-sion. By using the generalized equation along with an appropriateset of dimensionless parameters, the temperature profile withinthe medium can be generated easily for the Cattaneo–Vernotte,dual-phase-lag, and thermomass models. Note that the proposedgeneralized equation can also be utilized for the study of heat con-duction in thin films based on the two-step phonons-electronsinteraction model [13, p. 121]. For brevity, this model was not ex-plored in this study. Temperature data for the aforementionedmodels was presented to demonstrate the applicability and versa-tility of the generalized temperature equation for the solution ofthese models. Comparisons were also made to examine the varia-tions of these models, as well as the diffusion model, since each hasits own subtlety in nature.

In this study, the effects of the energy pulse time and relaxationtime on energy transport within a solid were examined. The max-imum temperature occurs at internal locations, rather than at theexternal surfaces for all models, with the exception of the diffusionmodel. The reason for this occurrence is that the diffusion modelassumes energy transport at an infinite speed such that the incom-ing energy would dissipate instantly throughout. However, sincehyperbolic models assume that energy propagates as thermalwaves, the temperature profiles require a relaxation time, creatinglocalized higher temperature spots along the heat conduction path.This is the main distinction between thermal wave models and thediffusion model. Consequently, transient temperatures in solidsmay be under-predicted with the diffusion model. This is particu-larly true in applications where extremely rapid, high pulse rate,and high intensity heat transfer phenomena are involved in the de-sign process. The rationale is that temperature peaks resultingfrom the wave behavior in various wave models may vary fromeach other as a result of different wave velocities used by differentmodels. The spatial difference and time variation of the peak tem-peratures are critical to the performance and longevity of mostengineering devices. Therefore, careful consideration must be ta-ken to select a proper thermal model to predict accurately the ther-mal behavior in micro- and nano-engineering applications – animportant role in the design of electronic components.

Even though the sample problems investigated in this paper arelimited to a one-dimensional planar medium with insulatedboundaries, the method could be used as a building block andextended to other multi-dimensional geometries and boundaryconditions. The present study reveals that the generalized temper-ature solution presented in this paper provides a convenient andefficient solution to the C–V, simplified thermomass, anddual-phase-lag heat conduction models.

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