ANALYTICAL SOLUTION OF THE PROBLEM OF NON...

Heat Transfer Research 46(5), 447–464 (2015)

1064-2285/15/$35.00 © 2015 by Begell House, Inc. 447

1. INTRODUCTION

Early in 1822, the French mathematical physicist, Joseph Fourier studied many ex-perimental results on heat conduction, summarized them in his famous Fourier’s law, advancing a linear relationship between a heat fl ux and temperature gradient, at the thermal wave propagation velocity being infi nite. Subsequently, Fourier’s law

ANALYTICAL SOLUTION OF THE PROBLEM OF NON-FOURIER HEAT CONDUCTIONIN A SLAB USING THE SOLUTIONSTRUCTURE THEOREMS

Mohammad Akbari,1* Seyfolah Saedodin,1 Davood Toghraie Semiromi,2,* & Farshad Kowsari3

1Department of Mechanical Engineering, Semnan University, Semnan, Iran2Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran3Department of Mechanical Engineering, University of Tehran, Tehran, Iran*Address all correspondence to Mohammad Akbari E-mail: [email protected]

This paper studies an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction equation can be obtained by using the fundamental mathematics. In this paper, the non-Fourier heat conduction in a slab at whose a left boundary there is a constant heat fl ux and at the right boundary, a constant temperature Ts = 15, has been investigated. The complicated problem is split into multiple simpler problems that in turn can be combined to obtain a solution to the original problem. The original problem is divided into fi ve subproblems by sett ing the heat gen-eration term, the initial conditions, and the boundary conditions for diff erent values in each subproblem. All the solutions given in this paper can be easily proven by substituting them into the governing equation. The results show that the temperature will start retreating at approxi-mately t = 2 and for t = 2 the temperature at the left boundary decreases leading to a decrease in the temperature in the domain. Also, the shape of the profi les remains nearly the same aft er t = 4. The solution presented in this study can be used as benchmark problems for validation of future numerical methods.

KEY WORDS: structure theorem, non-Fourier, analytical solution, temperature components

Heat Transfer Research

448 Akbari et al.

has been proved valid in numerous engineering applications. Because the governing equation of Fourier’s law is parabolic, it leads to infi nite propagation velocities for thermal disturbances, which contradicts the basic physical principles. To resolve this problem, many researchers developed some modifi cations of Fourier’s law. The tech-nical circumstances in which the deviation from Fourier’s model becomes signifi cant may be encountered, for instanc, in microelectronic devices such as IC chips, laser pulse heating of extremely short duration or a very high heat fl ux for the annealing

NOMENCLATURE

dimensionless temperature,Tcoeffi cient in Fourier seriesankcT*/αfrcoeffi cient in Fourier seriesbndimensionless wall Tsthermal wave propagation ctemperature, kcTs

*/αfrspeed, m/stemperature, KT *specifi c heat, J/kg·Kcpwall temperature, KTs

*total internal heat generation fcoordinate, mx*in systemdimensionless space xreference laser power density, frcoordinate, cx*/2αW/m2

Greek symbolsdimensionless internal heatg

thermal diffusivity k/ρcp, m2/s∇generation

eigen value, (1 + 2n)π/2)βntransmitted energy strengthg0

eigen value, 2( ) 1π −nγninternal heat generation, W/m3′′′g

relative errorεlaser peak power density, W/m2I0dimensionless absorptionμthermal conductivity, W/m·Kkcoeffi cient, 2cτμ*heat fl ux, W/m2

q*

absorption coeffi cient, 1/mμ*dimensionless heat fl ux, q*/frqdensity, kg·m–3ρwall heat fl ux, W/m2qs

*

relaxation time α/c2, sτ0dimensionless wall heat fl ux, qsdimensionless initialφqs

*/frcondition functionsource termQdimensionless initial rate ofψsurface refl ectivity of a solidRtemperature change functiondimensionless time, c2t*/2αtdummy indexξtime, st*

449Non-Fourier Heat Conduction in a Slab

Volume 46, Number 5, 2015

of semiconductors, laser surgery in biomedical engineering, and impulse drying. In such systems, the predicted results cannot correspond satisfactorily to experimental data because the Fourier law includes the hypothesis of the heat disturbance infi nite propagation velocity (Qui et al., 1994). Almost 136 years after Fourier proposed the law of heat conduction, in 1958, a modifi ed non-Fournier heat fl ux equation has been developed by several researchers. Cattaneo (1958), Vernotte (1958), and Morse and Feshbach (1953) subsequently developed a thermal wave model by introducing a re-laxation time for a heat fl ux in the form

** *

0 *∂

+ τ = − ∇∂

qq k Tt

, (1)

where 0τ is the so-called relaxation time. The energy conservation law is given by

* *

* *∂ ∂′′′− + = ρ∂ ∂

pq Tg cx t

, (2)

where ′′′g denotes the internal energy generation rate per unit volume. Inserting Eq. (1) into Eq. (2), the hyperbolic heat transport equation takes the form

* 2 *2 *

0* *2 ( , )∂ ∂α∇ = + τ +

∂ ∂

T TT Q x tt t

, (3)

where Q(x, t) is the source term. Several solutions for fi nite media have been given in the literature. Tang and Araki solved the problem of non-Fourier heat conduction in a fi nite medium under harmonic periodic surface disturbance, laser-pulse (Tang and Araki, 1996) and pulse surface heating (Tang and Araki, 2000). Barletta and Zanchini (1997) examined analytically the hyperbolic conduction in an infi nite cylinder with internal heat generation produced by Joule effect and convection heat exchange with a surrounding fl uid. Using the CV heat fl ux equation they studied the propagation of thermal waves in a long solid cylinder whose boundary temperature undergoes a change (Barletta and Zanchini, 1996). The equations were solved analytically by La-place transform. Lewandowska and Malinowski (1998) have solved analytically the CV hyperbolic equation for a semi-infi nite body, with the heat source whose capacity is linearly dependent on temperature. Lewandowska and Malinowski (2006) presented an analytical solution for the case of a thin slab symmetrically heated on both sides, with the heating being treated as an internal source with the capacity dependent on coordinate and time. Jiang (2006) used the Laplace transform method for investigat-ing the hyperbolic heat conduction process in a hollow sphere with its two boundary surfaces subjected to sudden temperature changes. Moosaie considered the non-Fouri-er heat conduction in a fi nite medium and proceeded to the case of an arbitrary peri-odic (Moosaie, 2007) and nonperiodic (Moosaie, 2008) surface disturbances using the


450 Akbari et al.

Fourier integral representation of arbitrary nonperiodic functions. Babaei and Chen (2008)investigated the hyperbolic thermoelastic problem of an annular fi n whose base temperature is subjected to a sudden change by using an effi cient numerical scheme involving the hybrid application of Laplace transform. Their results showed that the application of hyperbolic shape functions can successfully suppress the numerical os-cillations in the vicinity of jump discontinuities. Zhou et al. (2008) present a two-di-mensional (2D), axisymmetric thermal wave model of bioheat transfer to investigate a laser-induced damage in biological tissues. They showed that the bioheat non-Fourier effect can be important when the thermal relaxation time of biomaterials is moderately long. The non-Fourier axsymmetric three-dimensional temperature fi eld within a hol-low sphere with general linear time-independent boundary conditions was analytically investigated by Moosaie (2009). The method of solution is the standard separation of variables. Ahmadikia and Rismanian (2011) obtained the analytical scheme in solving the problem of hyperbolic heat conduction in the fi n that is subjected to every peri-odic boundary condition using the Laplace transform method. Their results obtained from the hyperbolic heat conduction model successfully explained the non-Fourier thermal wave behavior in a small fi n for a fast phenomenon (high-frequency periodic boundary condition). Ahmadikia et al. (2012) analytically solved the Pennes bioheat transfer models by employing the Laplace transform method for small and large val-ues of refl ection power (albedo) during laser irradiation. They concluded that the non-Fourier effect should be considered during laser heating with low albedo, because errors in the predicted temperature values may occur. Ahmadikia and Moradi (2012) used the linear evolution of latent heat over the solidifi cation range for the nonisother-mal phase change. They obtained the infl uence of this discontinuity and the relaxation time on the temperature distribution through the subject tissues; the cooling rate and freezing position are studied and different results are obtained. Their results indicated that the enthalpy method is highly capable of solving heat conduction problems in the solidifi cation process. Bamdad et al. (2012) studied non-Fourier effects in extended surfaces. Their results showed that for all non-Fourier fi ns at initial times, the location of the discontinuity point depends only on the values of both the time and the relax-ation time. Also, the temperature of the discontinuity point depends only on the values of time, relaxation time, and cross-sectional area. Moreover, the cross-sectional area affects the amplitude of the refl ected thermal wave from the fi n tip in a way that there is no refl ected thermal wave in concave non-Fourier fi ns. Azimi et al. (2012) esti-mated the unknown base temperature in inverse non-Fourier fi n problems with differ-ent profi les, and also estimated two different time distributions of the unknown base temperature using an effi cient inverse method. Their results showed that the estima-tion of the unknown base temperature is not very sensitive to the measurement errors. Also, they showed that the ACGM is an accurate and stable inverse technique, which can successfully estimate the unknown base temperature in different non-Fourier fi ns without any a priori information of unknown function values or shapes. Kishor and



Kirtiwant (2012) considered a nonhomogeneous heat conduction problem in a thin hollow circular disk in an unsteady-state temperature fi eld due to internal heat gen-eration within it and discussed the temperature change and thermal defl ection. Their numerical results were compared with different metal disks. They concluded that, due to the internal heat generation in thin hollow circular disks, the thermal defl ection is inversely proportional to their thermal conductivity. Wang and Han (2012) studied an interface crack in a two-layered composite media under an applied thermal fl ux by using the hyperbolic heat conduction equation. They used the Laplace transform and its inversion and obtained a time-related solution. They concluded that the nonclassi-cal heat conduction model is important in studying the cracks and defects in modern devices and materials under strong thermal shock. Subhash and Sahai (2012) used the lattice Boltzmann method to analyze the non-Fourier heat conduction in 1D cylindri-cal and spherical geometries. They showed that when temporal and spatial resolutions tend to zero, the macroscopic form of the governing HHC equation is recovered from the LBM formulation.

Review of these articles show that various solutions are studied in various works. From each work, different aspects of non-Fourier heat conduction was studied and therefore different results were obtained each of which could be useful in its position. However, the lack of a general study in terms of nonzero boundary condition and nonzero initial conditions with internal heat generation in these studies is observed. Therefore, it seems that the research can make a good contribution to obviate the shortcomings inherent in other works.

Also, the experimental setup for modeling light sources and in particular lasers to melt and subsequently recrystallize thin semiconductor layers on insulators, such as oxidized wafers and bulk amorphous substrates (in this work, this layer is modeled as a slab composed of an isotropic heat conducting material) is very diffi cult. For over-coming this problem, we obtained an analytical solution for a non-Fourier problem in a slab with different boundary conditions at both sides. Our simulation has shown a good potential for applications to the commercial VLSI technology.

In this paper, an exact solution is presented to the problem of non-Fourier heat con-duction in a slab with nonzero boundary and initial conditions. As stated later, with application of the solution structure theorems along with the superposition technique, we can obtain analytical solutions to this problem. Finally, in this work some impor-tant observations are mentioned in the conclusion.

2. FORMULATION

2.1 Non-Fourier Heat Conduction

Let us consider a slab composed of an isotropic heat conducting material with differ-ent boundary conditions at both sides: a constant heat fl ux ′′sq on the left boundary


452 Akbari et al.

and a constant temperate Ts* on the right boundary (see Fig. 1). In order to get non-

dimensional equations, the following dimensionless variables are introduced into Eqs. (1), (2), and (3):

* 2 * * * 4, , , ,2 2

′′′α= = = = =

α α α r r r

cx c t kcT q gx t T q gf f cf

. (4)

The dimensionless governing equation can be written as follows:

2∂ ∂

+ = −∂ ∂q T qt x

, (5)

2∂ ∂

+ =∂ ∂T q gt y

, (6)

2 2

2 22 ( , )∂ ∂ ∂+ = +

∂∂ ∂

T T T f x ttt x

, (7)

1( , )2∂

= +∂gf x t gt

. (8)

The relevant boundary conditions are expressed in terms of the above dimensionless variables as

(1, ) = sT t T , (9)

(0, ) 2∂= −

∂ sT t q

x. (10)

FIG. 1: Contribution of temperature components at t = 0.1



The relevant initial conditions are assumed in terms of the above dimensionless vari-ables as

( , 0) ( )T x x= ϕ , (11)

( , 0) ( )2

∂= ψ =

∂T x gx

t. (12)

In this study, the internal energy generation term, g, for laser application is defi ned as (Ready, 1978)

0( , ) exp ( ) exp ( )= −μ −g x t g x t (13)

with

00

2 (1 )μ −=

r

I Rgf

, (14)

where I0 is the amplitude of laser peak power density, μ is the absorption coeffi cient, and R is the surface refl ectivity of the solid. This model assumes no spatial variations of g0 in the plane perpendicular to the laser beam.

2.2 Method of Superposition and Solution Structure Theorems

One of the oldest, simplest, and most widely used techniques for solving some types of heat conduction equations is the superposition technique. This method can be ap-plied to linear heat transfer problems with nonhomogenous terms. A partial differen-tial equation is called linear if the unknown function and its derivatives have no expo-nent greater than one, and if there are no cross terms. In this method, a complicated problem is split into multiple simpler problems which in turn can be combined to obtain a solution to the original problem. The method of superposition relies upon the assumption that the original problem, Eq. (7), can be divided into fi ve subproblems by setting the heat generation term, Eq. (8), the initial conditions, Eqs.(11) and (12), and the boundary conditions, Eqs. (9) and (10), at different values in each subproblem:

(1) ( , ) ( ) 0s sf x t x T q= ϕ = = = , (15a)

(2) ( , ) ( ) 0= ψ = = =s s f x t x T q , (15b)

(3) ( ) ( ) 0s s x x T qϕ = ψ = = = , (15c)

(4) ( ) ( ) ( , ) 0s x x f x t qϕ = ψ = = = , (15d)

(5) ( ) ( ) ( , ) 0s x x f x t Tϕ = ψ = = = . (15e)


454 Akbari et al.

Solutions to these suproblems are assigned, sequentially, as T1, T2, T3, T4, and T5. Therefore, the general solution to the original heat conduction equation, Eq. (17), is the sum of subproblems 1 through 5.

Note that T1, T2, T3, T4, and T5 represent the individual contributions of the initial rate of temperature change, initial condition, internal heat generation, and of the con-stant temperature *

sT to the right boundary and of the constant heat fl ux ′′sq to the left boundary in order to complete the temperature solution.

Subproblems 1 to 3 can be easily solved with application of the solution structure theorems (Lam and Fong, 2011) once the solution to subproblem 1 is known. Also, subproblems 4 and 5 can be easily solved by using the separation-of-variables ariables method. With application of the solution structure theorems, the solutions to subprob-lems 1 to 3 can be written as follows:

1( , ) ( , , ( ))= ψT x t F x t x , (16)

2 ( , ) 2 ( , , ( ))T x t F x t x

t∂⎛ ⎞= + ϕ⎜ ⎟∂⎝ ⎠

, (17)

3

0( , ) ( , , ( , ))= − ξ ξ ξ∫

tT x t F x t f x d , (18)

where T1(x, t) is obtained by using the Fourier method. The quantities T2(x, t) and T3(x, t) can readily be obtained by applying the solution structure theorem. It means that only the solution to subproblem 1 is required to fi nd the solutions to subproblems 2 and 3. Then T4(x, t) and T5(x, t) will be obtained by using the method of the separa-tion of variables. Finally, the general solution to the original heat conduction equation is the sum of subproblems 1 to 5.

2.3 Formulation of the Problem

There are a lot of examples of non-Fourier heat conduction problems with different boundary conditions, but these problems are limited to insulated boundaries or the boundaries with a zero temperature. In this paper, we obtain the general analytical solution to the problem of non-Fourier heat conduction in a slab where the left bound-ary is subjected to a nonzero constant heat fl ux and the right boundary to a nonzero constant temperature. Let us fi rst consider the solution to subproblem 1. This subprob-lem is solved with the condition ( , ) ( ) 0f x t x= ϕ = . Therefore, the equation of this subproblem and the initial and boundary conditions are written as

2 21 1 1

2 22∂ ∂ ∂+ =

∂∂ ∂

T T Ttt x

, (19a)



1(0, ) 0∂

=∂

T tx

, (19b)

1(1, ) 0=T t , (19c)

1( , 0) 0=T x , (19d)

1( , 0) ( )∂

= ψ∂

T x xt

. (19e)

By the theory of Fourier series expansion, the general solution to the equation is

1

10 0

20

2 20

2( , ) ( ) cos ( ) sin ( ) cos ( )

( sin ( ) sin ( ) cos ( ) ( , , ( )) .( )

∞ −

=

∞ − −μ

=

⎡ ⎤⎢ ⎥= ψ ζ β ζ ζ η β

η ⎢ ⎥⎣ ⎦

μ + β β= η β = ψ ζ

η β + μ

∑ ∫

∑

tn n n

nn

tn n

n nn n n

eT x t d t x

g e e t x F x t

(20)

Now, by using the solution structure theorems, we have

1

20 0

0

2( , ) ( ) cos ( ) [(sin ( ) cos( )] cos ( )

2 cos( ) 1 cos ( ) [(sin ( ) cos ( )] cos ( ) ,2( ) 2( )

∞ −

=

∞ −

=

⎡ ⎤⎢ ⎥= ψ ζ β ζ ζ η + γ η β

η ⎢ ⎥⎣ ⎦

⎡ ⎤β − π − β + π− η + γ η β⎢ ⎥η β − π β + π⎣ ⎦

∑ ∫

∑

tn n n n n

nn

tn n

n n n nn n nn

eT x t d t t x

e = t t x

1( )3

00 0

02 2 2

0

2( , ) { ( , ) cos ( ) cos ( )

sin ( ( ) cos ( )}

( sin ( )[1 cos ( )] cos ( ) ,( )

t tn n

nn

n n

tn n n n

n n n

eT x t f d

t x d

g e e t x=

∞ − −ζ

=

∞ − −μ

=

⎡ ⎤⎢ ⎥= ξ ζ β ξ β ξ ξ

η ⎢ ⎥⎣ ⎦

× η − ζ β ζ

μ + β β − η β

η β + μ

∑∫ ∫

∑

(22)

where

(21)


456 Akbari et al.

2 1η = β −n n , (23)

(2 1)2+ π

β =nn

. (24)

Finally, 4 ( , )T x t and 5( , )T x t can readily be obtained by using the separation-of-variables method. Subproblem 4 is solved by the condition

( ) ( ) ( , ) 0sx x f x t qϕ = ψ = = = , (25)

2 24 4 4

2 22∂ ∂ ∂+ =

∂∂ ∂

T T Ttt x

, (26a)

4 ( , 0) 0=T x , (26b)

4 ( , 0) 0∂

=∂

T xt

, (26c)

4 (0, ) 0∂

=∂

T tx

, (26d)

4 (1, ) = sT t T . (26e)

After some mathematical manipulations, one can obtain

41

0

2 sin 2 sin( , ) sin cos

cos( ) ,

∞−

=

⎛ ⎞− β β= + γ − γ⎜ ⎟β γ β⎝ ⎠

× β +

∑ t s n s ns n n

n n nn

n

T TT x t T e t t

x T (27)

where

12

00

1 (1 ) ( )2

−= − ψ ζ ζ∫tT e d , (28)

2(2 1) , 1 ,

2+ π

β = γ = β −n n nn (29)



Subproblem 5 is solved by the condition

( ) ( ) ( , ) 0ϕ = ψ = = =sx x f x t T , (30)

2 25 5 5

2 22∂ ∂ ∂+ =

∂∂ ∂

T T Ttt x

, (31a)

5( , 0) 0=T x , (31b)

5( , 0) 0∂

=∂

T xt

, (31c)

5(1, ) 0=T t , (31d)

5(0, ) 2∂

= −∂ s

T t qx

. (31e)

The solution to the above equation can be obtained straightforwardly:

5 21

4( , ) cos (cos 1)

[sin cos ] 2 (1 ) .

t sn n

n n n

n n n s

qT x t e x

t t q x

∞−

=

⎛ ⎞= β β −⎜ ⎟⎜ ⎟β γ⎝ ⎠

× γ + γ γ + −

∑ (32)

Finally, the temperature distribution within the slab can be expressed as

1 2 3 4 5( , ) ( , ) ( , ) ( , ) ( , ) ( , )= + + + +T x t T x t T x t T x t T x t T x t . (33)

3. RESULTS AND DISCUSSION

In this paper, the temperature distribution in a one-dimensional slab with nonzero initial and boundary conditions is examined. Even though this problem is limited to a one-dimensional slab, the method can easily be used as a building block and exten-sion to other multidimensional geometries in planar, cylindrical, and spherical coor-dinates. The solutions can also be extended for other types of non-Fourier problems subjected to different initial conditions, internal heat generation, and different bound-ary conditions.

In this study, the values g0 = 100 and 5μ = are selected according to the previous study by Lam and Fong (2011).


458 Akbari et al.

By applying the superposition technique, the partial differential equation governing the hyperbolic heat conduction problem is split into fi ve subproblems. Subproblems 1 through 5 represent the individual contributions of the initial rate of change in the temperature, initial condition, internal heat generation, constant temperature *

sT at the right boundary and constant heat fl ux ′′sq at the left boundary to the fi nal tem-perature, which are given by Eqs. (20)–(32), respectively. It can be noted from these equations that all the temperature profi les from 1( , )T x t to 5( , )T x t be in the form of an infi nite series. Figure 2 shows the contribution of various temperature components

FIG. 2

(a)

(b)



FIG. 2: Contribution of temperature components at t = 0.1 (a), t = 0.2 (b), t = 0.7 (c), andt = 1 (d)

(c)

(d)

at different times for a slab that has the left boundary with a constant heat fl ux and the right boundary with a constant temperature Ts = 15. From this fi gure we can see that at small times (t < 0.1), the contribution of 1T and 2T dominates over that of 3T that contributes little to the overall temperature. The quantity 5T plays a bigger role (especially near the left wall) and 4T plays the biggest role near the right wall. It can


460 Akbari et al.

be noted that the contribution of T4 is limited only to the region 0.75 < x < 1, and has the zero value in the range 0 0.75< <x at (t < 0.1). Hence, we conclude that for small time the contribution of 5T and 1T dominates over that of 2T , 3T , and 4T . This occurs because rapid temperature changes take place after 0>t . The heat input due to the gradient from the initial condition and heat input from the laser source atx = 0 contribute to the large 1T term. Also, the constant heat fl ux from the left bound-ary at 0=x contributes to the large 4T term. As time increases (Figs. 2b and 2c), the contribution of 3T to the total temperature rises. The rising of 3T is due to the incoming energy that comes from the left boundary. As time reaches t = 1 (Fig. 2d), the quantity 4T plays a more dominant role in the overall temperature. This means that the temperature of the right wall is penetrating into the domain and in this time, the whole domain will be affected by this temperature. Note that in these fi gures, all the temperature contributions are sloping to zero at x = 1 except for 4T . The tem-perature profi le within the slab at different exposure times from 0=t to 5=t is illustrated in Fig. 3. It can be observed from Fig. 3a that the temperature near the left wall will be affected rapidly compared to the downstream region of the slab. This oc-curs because the incoming energy and the constant heat fl ux are concentrated near this boundary and the effect of the right wall temperature (T5) on the temperature profi les is not felt. As time progresses, more energy will be transported to the downstream region of the slab. It is obvious that the temperature profi les have similar shapes at the mentioned time level. As shown in Fig. 3b, as time progresses from t = 0.1 to t = 1, the similarity in the profi les is broken down. The energy from the left boundary penetrates into the domain and an infl ection can be observed in the profi les. However, as the internal heat generation is a function of time, and due to the incoming energy from the right wall, the temperature will start retreating at approximately t = 2 (Fig. 3c). On the other hand, for t > 2 the temperature at the left boundary decreases and it will eventually decrease in the entire domain. It can be noted that the right hand side boundary was set at Ts = 15 and therefore the temperature would eventually reach Ts at all times. This result could be expected. Another result from the graphs is that the shape of the profi les remains nearly the same after t = 1.5.

Also, after t = 4, the shapes of the profi les do not change greatly and we can con-clude that the temperature profi les reach equilibrium conditions. This means that the temperature of the slab will keep increasing until the medium reaches equilibrium. In the case under consideration, the right-hand-side boundary was set at a constant temperature, and therefore it is expected that the overall temperature would eventually reach this constant value at large times. As mentioned previously, the internal heat generation is also a function of time, which decreases exponentially, so the tempera-ture will eventually decrease as time increases. It is clear that the solutions predict the transient behavior of temperature profi les quite well. Finally, we conclude that the obtained method provides a convenient, accurate, and effi cient solution to the non-Fourier equation, which is applicable to the analyses of various engineering situations.



FIG. 3: Temperature distribution from t = 0 to t = 0.1 (a), t = 0.1 to t = 1 (b), and t = 1 tot = 5 (c)

(a)

(b)

(c)


462 Akbari et al.

4. CONCLUSIONS

For the most practical purposes, the effects of non-Fourier conduction are negligible. As the size of the microelectronic devices decreases to tiny portions and the circuit speed increases, Fourier’s law cannot be used for heat transfer and temperature predic-tions. The wave character gives rise to the effects which do not occur under the classi-cal Fourier conduction. In the present study, the non-Fourier hyperbolic heat conduc-tion problem was solved for the slab that is subjected to nonhomogenous boundary conditions using analytical solutions and temperature distribution of the problem of nonhomogeneous heat conduction in a slab under unsteady-state temperature fi eld due to internal heat generation. With the application of the superposition technique along with the solution structure theorems, an analytical method has been presented in this paper for the analysis of the Cattaneo–Vernotte hyperbolic heat conduction equation. The temperature profi le inside a one-dimensional region can be obtained in the form of a series solution. The method is simple and requires only a basic background in applied mathematics. In the past, solution methods for non-Fourier heat conduction analysis primarily resorted to numerical techniques due to the complexity of the phys-ical governing hyperbolic equation. Only a few simple cases of hyperbolic heat con-duction problems were solved analytically. Therefore the present method provides a distinct advantage over other numerical methods. In this paper, the analytical solution to the problem of non-Fourier heat conduction in a one- dimensional slab is investi-gated. The purpose of this study is to present an analytical method to thermal wave phenomena in a fi nite slab under nonzero boundary conditions and nonzero initial conditions. The key fi ndings from the present solution are as follows:

• at small times the contribution of 1T and 2T dominates over that from 3T , with the latter contributing little to the overall temperature.

• for small time the contribution of 5T and 1T dominates over that from 2T , 3T , and 4T .

• at t = 0.7 the contribution of T4 and T5 dominates over that from T1, T2, and T3.• the temperature will start retreating at approximately 2=t .• for 2>t , the temperature at the left boundary decreases, leading to a decrease

in temperature in the domain.• the shape of profi les remains nearly the same after 4=t .The results presented here will be useful in engineering problems, particularly

in aerospace engineering for stations of a missile body not infl uenced by nose ta-pering.

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Ahmadikia, H., Moradi, A., Fazlaki, R., and Basiri Parsa, A., Analytical solution of non-Fourier and Fourier bioheat transfer analysis during laser irradiation of skin tissue, J. Mech. Sci. Technol., vol. 26, no. 6, pp. 1937–1947, 2012.



Ahmadikia, H. and Moradi, A., Non-Fourier phase change heat transfer in biological tissues during so-lidifi cation, Heat Mass Transfer, vol. 48, pp. 1559–1568, 2012.

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