ANALYTICAL SOLUTION OF THE PROBLEM OF NON-FOURIER HEAT CONDUCTION IN A SLAB USING THE SOLUTION STRUCTURE THEOREMS

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1 Heat Transfer Research 46(5), (2015) ANALYTICAL SOLUTION OF THE PROBLEM OF NON-FOURIER HEAT CONDUCTION IN A SLAB USING THE SOLUTION STRUCTURE THEOREMS Mohammad Akbari, 1* Seyfolah Saedodin, 1 Davood Toghraie Semiromi, 2, * & Farshad Kowsari 3 1 Department of Mechanical Engineering, Semnan University, Semnan, Iran 2 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran 3 Department of Mechanical Engineering, University of Tehran, Tehran, Iran * Address all correspondence to Mohammad Akbari m.akbari.g80@gmail.com 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 flux and at the right boundary, a constant temperature T s = 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 five subproblems by setting the heat generation term, the initial conditions, and the boundary conditions for different 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 approximately 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 profiles remains nearly the same after 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 1. INTRODUCTION Early in 1822, the French mathematical physicist, Joseph Fourier studied many experimental results on heat conduction, summarized them in his famous Fourier s law, advancing a linear relationship between a heat flux and temperature gradient, at the thermal wave propagation velocity being infinite. Subsequently, Fourier s law /15/$ by Begell House, Inc. 447

2 448 Akbari et al. a n b n c c p coefficient in Fourier series coefficient in Fourier series thermal wave propagation speed, m/s specific heat, J/kg K NOMENCLATURE T T s T * dimensionless temperature, kct * /αf r dimensionless wall temperature, kct * s /αf r temperature, K f r f total internal heat generation wall temperature, K in system x * coordinate, m reference laser power density, x dimensionless space W/m 2 coordinate, cx * /2α g g 0 g I 0 dimensionless internal heat generation transmitted energy strength internal heat generation, W/m 3 laser peak power density, W/m 2 T s * Greek symbols thermal diffusivity k/ρc p, m 2 /s eigen value, (1 + 2n)π/2) β n γ n ε eigen value, relative error 2 ( nπ) 1 k thermal conductivity, W/m K μ dimensionless absorption q * heat flux, W/m 2 coefficient, 2cτμ * q dimensionless heat flux, q * /f μ * r absorption coefficient, 1/m wall heat flux, W/m 2 ρ density, kg m 3 q s * q s Q R t t * dimensionless wall heat flux, q * s /f r source term surface reflectivity of a solid dimensionless time, c 2 t * /2α time, s τ 0 φ ψ ξ relaxation time α/c 2, s dimensionless initial condition function dimensionless initial rate of temperature change function dummy index has been proved valid in numerous engineering applications. Because the governing equation of Fourier s law is parabolic, it leads to infinite propagation velocities for thermal disturbances, which contradicts the basic physical principles. To resolve this problem, many researchers developed some modifications of Fourier s law. The technical circumstances in which the deviation from Fourier s model becomes significant may be encountered, for instanc, in microelectronic devices such as IC chips, laser pulse heating of extremely short duration or a very high heat flux for the annealing Heat Transfer Research

3 Non-Fourier Heat Conduction in a Slab 449 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 infinite propagation velocity (Qui et al., 1994). Almost 136 years after Fourier proposed the law of heat conduction, in 1958, a modified non-fournier heat flux 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 relaxation time for a heat flux in the form * * q * q +τ 0 = k T, (1) * where τ 0 is the so-called relaxation time. The energy conservation law is given by * * q T + g = ρc * p, (2) * x 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 * T T α T = + τ * 0 + Q( x, t), (3) *2 where Q(x, t) is the source term. Several solutions for finite media have been given in the literature. Tang and Araki solved the problem of non-fourier heat conduction in a finite 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 infinite cylinder with internal heat generation produced by Joule effect and convection heat exchange with a surrounding fluid. Using the CV heat flux 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 Laplace transform. Lewandowska and Malinowski (1998) have solved analytically the CV hyperbolic equation for a semi-infinite 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 investigating the hyperbolic heat conduction process in a hollow sphere with its two boundary surfaces subjected to sudden temperature changes. Moosaie considered the non-fourier heat conduction in a finite medium and proceeded to the case of an arbitrary periodic (Moosaie, 2007) and nonperiodic (Moosaie, 2008) surface disturbances using the Volume 46, Number 5, 2015

4 450 Akbari et al. Fourier integral representation of arbitrary nonperiodic functions. Babaei and Chen (2008)investigated the hyperbolic thermoelastic problem of an annular fin whose base temperature is subjected to a sudden change by using an efficient numerical scheme involving the hybrid application of Laplace transform. Their results showed that the application of hyperbolic shape functions can successfully suppress the numerical oscillations in the vicinity of jump discontinuities. Zhou et al. (2008) present a two-dimensional (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 field within a hollow 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 fin that is subjected to every periodic 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 fin 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 values of reflection 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 solidification range for the nonisothermal phase change. They obtained the influence 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 solidification process. Bamdad et al. (2012) studied non-fourier effects in extended surfaces. Their results showed that for all non-fourier fins at initial times, the location of the discontinuity point depends only on the values of both the time and the relaxation 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 reflected thermal wave from the fin tip in a way that there is no reflected thermal wave in concave non-fourier fins. Azimi et al. (2012) estimated the unknown base temperature in inverse non-fourier fin problems with different profiles, and also estimated two different time distributions of the unknown base temperature using an efficient inverse method. Their results showed that the estimation 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 fins without any a priori information of unknown function values or shapes. Kishor and Heat Transfer Research

5 Non-Fourier Heat Conduction in a Slab 451 Kirtiwant (2012) considered a nonhomogeneous heat conduction problem in a thin hollow circular disk in an unsteady-state temperature field due to internal heat generation within it and discussed the temperature change and thermal deflection. 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 deflection 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 flux 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 nonclassical 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 cylindrical 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 difficult. For overcoming 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 conduction 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 important 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 different boundary conditions at both sides: a constant heat flux q s on the left boundary Volume 46, Number 5, 2015

6 452 Akbari et al. FIG. 1: Contribution of temperature components at t = 0.1 and a constant temperate T s * on the right boundary (see Fig. 1). In order to get nondimensional equations, the following dimensionless variables are introduced into Eqs. (1), (2), and (3): * 2 * * * cx c t kct q 4αg x =, t =, T =, q =, g =. (4) 2α 2α αf r f r cf r The dimensionless governing equation can be written as follows: q T + = 2q, (5) x T + q = g, (6) y T T T + 2 = + f( x, t), (7) 2 2 t x f( x, t) 1 g = + 2 g. (8) The relevant boundary conditions are expressed in terms of the above dimensionless variables as T(1, t) = T s, (9) T(0, t) x = 2qs. (10) Heat Transfer Research

7 Non-Fourier Heat Conduction in a Slab 453 The relevant initial conditions are assumed in terms of the above dimensionless variables as T( x, 0) =ϕ ( x), (11) T( x, 0) =ψ ( x) = g. (12) 2 In this study, the internal energy generation term, g, for laser application is defined as (Ready, 1978) gxt (, ) = g0 exp ( μx) exp ( t ) (13) with g0 = 2 I0μ(1 R), (14) fr where I 0 is the amplitude of laser peak power density, μ is the absorption coefficient, and R is the surface reflectivity of the solid. This model assumes no spatial variations of g 0 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 applied to linear heat transfer problems with nonhomogenous terms. A partial differential equation is called linear if the unknown function and its derivatives have no exponent 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 five 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) f( x, t) =ϕ ( x) = Ts = qs = 0, (15a) (2) f ( x, t) =ψ ( x) = Ts = q s = 0, (15b) (3) ϕ ( x) = ψ ( x) = Ts = qs = 0, (15c) (4) ϕ ( x) = ψ ( x) = f( x, t) = q s = 0, (15d) (5) ϕ ( x) = ψ ( x) = f( x, t) = T s = 0. (15e) Volume 46, Number 5, 2015

8 454 Akbari et al. Solutions to these suproblems are assigned, sequentially, as T 1, T 2, T 3, T 4, and T 5. Therefore, the general solution to the original heat conduction equation, Eq. (17), is the sum of subproblems 1 through 5. Note that T 1, T 2, T 3, T 4, and T 5 represent the individual contributions of the initial rate of temperature change, initial condition, internal heat generation, and of the constant temperature T s to the right boundary and of the constant heat flux q s 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 subproblems 1 to 3 can be written as follows: T1( x, t) = F( x, t, ψ( x )), (16) T2( x, t) = 2 + F( x, t, ϕ( x)), (17) t T3 ( x, t ) = F ( x, t ξ, f ( x, ξ )) d ξ, (18) 0 where T 1 (x, t) is obtained by using the Fourier method. The quantities T 2 (x, t) and T 3 (x, t) can readily be obtained by applying the solution structure theorem. It means that only the solution to subproblem 1 is required to find the solutions to subproblems 2 and 3. Then T 4 (x, t) and T 5 (x, t) will be obtained by using the method of the separation of variables. Finally, the general solution to the original heat conduction equation is the sum of subproblems 1 to 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 boundary is subjected to a nonzero constant heat flux and the right boundary to a nonzero constant temperature. Let us first consider the solution to subproblem 1. This subproblem is solved with the condition f( x, t) =ϕ ( x) = 0. Therefore, the equation of this subproblem and the initial and boundary conditions are written as 2 2 T1 T1 T1 + 2 =, (19a) 2 2 t x Heat Transfer Research

9 Non-Fourier Heat Conduction in a Slab 455 T1(0, t) x = 0, (19b) T1(1, t ) = 0, (19c) T1( x, 0) = 0, (19d) T1( x, 0) =ψ( x). (19e) By the theory of Fourier series expansion, the general solution to the equation is t 1 2e T1 ( x, t) = ψ( ζ) cos ( βnζ) dζ sin ( ηnt) cos ( βnx) n= 0 ηn 0 2t μ ge 0 ( μ+βne sin ( βn) = sin ( η ) cos ( β ) = (,, ψ ( ζ )). 2 2 nt nx F x t n= 0 ηn( β n + μ ) (20) Now, by using the solution structure theorems, we have t 1 2e T2 ( x, t) = ψ( ζ) cos ( βnζ) dζ [(sin ( η nt) + γn cos( ηnt)] cos ( βnx) n= 0 ηn 0 t 2e cos( βn π) 1 cos ( β n + π) = [(sin ( η ) + γ cos ( η )] cos ( β ), = 0 η 2( β π) 2( β + π) nt n nt nx n n n n (21) t ( t ζ) 1 2e T3 ( x, t) = { f( ξ, ζ) cos ( βnξ) cos ( βnξ) dξ 0 n= 0 ηn 0 sin ( ηn( t ζ) cos ( βnx)} dζ (22) = t μ ge 0 ( μ+βne sin ( βn)[1 cos ( ηnt)] cos ( βnx), n= 0 η n( β n + μ ) where Volume 46, Number 5, 2015

10 456 Akbari et al. 2 η = β 1 n n, (23) β = n (2n + 1) π. (24) 2 Finally, T4( x, t ) and T 5 ( x, t ) can readily be obtained by using the separation-ofvariables method. Subproblem 4 is solved by the condition ϕ ( x) = ψ ( x) = f( x, t) = q s = 0, (25) 2 2 T4 T4 T4 + 2 =, (26a) 2 2 t x T4( x, 0) = 0, (26b) T4( x, 0) T4(0, t) x = = 0, (26c) 0, (26d) T4(1, t) = T s. (26e) After some mathematical manipulations, one can obtain 2 sin β 2 sin β 4(, ) = + t T s n T T x t Ts e sin γnt s n cos γnt n= 1 βnγn βn cos( β nx) + T0, (27) where (1 t T = e ) ψ ( ζ ) d ζ, (28) 2 0 (2n + 1) π 2 β n =, γ n = β n 1, 2 (29) Heat Transfer Research

11 Non-Fourier Heat Conduction in a Slab 457 Subproblem 5 is solved by the condition ϕ ( x) = ψ ( x) = f( x, t) = T s = 0, (30) 2 2 T5 T5 T5 + 2 = 2 2 t x, (31a) T5( x, 0) = 0, (31b) T5( x, 0) = 0, (31c) T5(1, t ) = 0, (31d) T5(0, t) x = 2qs. (31e) The solution to the above equation can be obtained straightforwardly: t 4q s T5 ( x, t) = e cos βnx (cos β 1) 2 n n= 1 β nγn (32) [sin γ nt + γn cos γ nt] + 2 qs(1 x). Finally, the temperature distribution within the slab can be expressed as T( x, t) = T1( x, t) + T2( x, t) + T3( x, t) + T4( x, t) + T5( 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 extension to other multidimensional geometries in planar, cylindrical, and spherical coordinates. The solutions can also be extended for other types of non-fourier problems subjected to different initial conditions, internal heat generation, and different boundary conditions. In this study, the values g 0 = 100 and μ= 5 are selected according to the previous study by Lam and Fong (2011). Volume 46, Number 5, 2015

12 458 Akbari et al. By applying the superposition technique, the partial differential equation governing the hyperbolic heat conduction problem is split into five 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 T s at the right boundary and constant heat flux q s at the left boundary to the final temperature, which are given by Eqs. (20) (32), respectively. It can be noted from these equations that all the temperature profiles from T1( x, t ) to T 5 ( x, t ) be in the form of an infinite series. Figure 2 shows the contribution of various temperature components (a) (b) FIG. 2 Heat Transfer Research

13 Non-Fourier Heat Conduction in a Slab 459 (c) (d) FIG. 2: Contribution of temperature components at t = 0.1 (a), t = 0.2 (b), t = 0.7 (c), and t = 1 (d) at different times for a slab that has the left boundary with a constant heat flux and the right boundary with a constant temperature T s = 15. From this figure we can see that at small times (t < 0.1), the contribution of T 1 and T 2 dominates over that of T 3 that contributes little to the overall temperature. The quantity T 5 plays a bigger role (especially near the left wall) and T 4 plays the biggest role near the right wall. It can Volume 46, Number 5, 2015

14 460 Akbari et al. be noted that the contribution of T 4 is limited only to the region 0.75 < x < 1, and has the zero value in the range 0 < x < 0.75 at (t < 0.1). Hence, we conclude that for small time the contribution of T 5 and T 1 dominates over that of T 2, T 3, and T 4. This occurs because rapid temperature changes take place after t > 0. The heat input due to the gradient from the initial condition and heat input from the laser source at x = 0 contribute to the large T 1 term. Also, the constant heat flux from the left boundary at x = 0 contributes to the large T 4 term. As time increases (Figs. 2b and 2c), the contribution of T 3 to the total temperature rises. The rising of T 3 is due to the incoming energy that comes from the left boundary. As time reaches t = 1 (Fig. 2d), the quantity T 4 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 figures, all the temperature contributions are sloping to zero at x = 1 except for T 4. The temperature profile within the slab at different exposure times from t = 0 to t = 5 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 occurs because the incoming energy and the constant heat flux are concentrated near this boundary and the effect of the right wall temperature (T 5 ) on the temperature profiles is not felt. As time progresses, more energy will be transported to the downstream region of the slab. It is obvious that the temperature profiles 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 profiles is broken down. The energy from the left boundary penetrates into the domain and an inflection can be observed in the profiles. 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 T s = 15 and therefore the temperature would eventually reach T s at all times. This result could be expected. Another result from the graphs is that the shape of the profiles remains nearly the same after t = 1.5. Also, after t = 4, the shapes of the profiles do not change greatly and we can conclude that the temperature profiles 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 temperature will eventually decrease as time increases. It is clear that the solutions predict the transient behavior of temperature profiles quite well. Finally, we conclude that the obtained method provides a convenient, accurate, and efficient solution to the non- Fourier equation, which is applicable to the analyses of various engineering situations. Heat Transfer Research

15 Non-Fourier Heat Conduction in a Slab 461 (a) (b) (c) FIG. 3: Temperature distribution from t = 0 to t = 0.1 (a), t = 0.1 to t = 1 (b), and t = 1 to t = 5 (c) Volume 46, Number 5, 2015

16 462 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 predictions. The wave character gives rise to the effects which do not occur under the classical Fourier conduction. In the present study, the non-fourier hyperbolic heat conduction 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 field 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 profile 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 physical governing hyperbolic equation. Only a few simple cases of hyperbolic heat conduction 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 investigated. The purpose of this study is to present an analytical method to thermal wave phenomena in a finite slab under nonzero boundary conditions and nonzero initial conditions. The key findings from the present solution are as follows: at small times the contribution of T 1 and T 2 dominates over that from T 3, with the latter contributing little to the overall temperature. for small time the contribution of T 5 and T 1 dominates over that from T 2, T 3, and T 4. at t = 0.7 the contribution of T 4 and T 5 dominates over that from T 1, T 2, and T 3. the temperature will start retreating at approximately t = 2. for t > 2, the temperature at the left boundary decreases, leading to a decrease in temperature in the domain. the shape of profiles remains nearly the same after t = 4. The results presented here will be useful in engineering problems, particularly in aerospace engineering for stations of a missile body not influenced by nose tapering. REFERENCES 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 , Heat Transfer Research

17 Non-Fourier Heat Conduction in a Slab 463 Ahmadikia, H. and Moradi, A., Non-Fourier phase change heat transfer in biological tissues during solidification, Heat Mass Transfer, vol. 48, pp , Ahmadikia, H. and Rismanian, M., Analytical solution of non-fourier heat conduction problem on a fin under periodic boundary conditions, J. Mech. Sci. Technol., vol. 25, pp , Azimi, A., Bamdad, K., and Ahmadikia, H., Inverse hyperbolic heat conduction in fins with arbitrary profiles, Numer. Heat Transfer, Part A: Applications: An Int. J. Comput. Methodol., vol. 61, pp , Babaei, M. H. and Chen, Z. T., Hyperbolic heat conduction problem in a functionally graded hollow sphere, Int. J. Thermophys., vol. 29, pp , Bamdad, K., [Azimi, A., and Ahmadikia, H., Thermal performance analysis of arbitrary-profile fins with non-fourier heat conduction behavior, J. Eng. Math., DOI /s , Barletta, A. and Zanchini, E., Hyperbolic heat conduction and thermal resonance in a cylindrical solid carrying a steady-periodic electric field, Int. J. Heat Mass Transfer, vol. 39, pp , Barletta, A. and Zanchini, E., Thermal-wave heat conduction in a solid cylinder which undergoes a change of boundary temperature, Heat Mass Transfer, vol. 32, pp , Cattaneo, C., Sur une former de l equation de la chaleur elinant le paradoxe d une propagation instance, C.R. Acad. Sci., vol. 247, pp , Jiang, F., Solution and analysis of hyperbolic heat propagation in hollow spherical objects, Heat Mass Transfer, vol. 42, pp , Kishor, R., and Kirtiwant, P. G., Non-homogeneous heat conduction problem and its thermal deflection due to internal heat generation in a thin hollow circular disk, J. Thermal Stresses, vol. 35, pp , Lam, T. T. and Fong, E., Application of solution structure theorem to non-fourier heat conduction problems: Analytical approach, Int. J. Heat Mass Transfer, vol. 54, pp , Lewandowska, M. and Malinowski, L., An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides, Int. Commun. Heat Mass Transfer, vol. 33, pp , Lewandowska, M. and Malinowski, L., Hyperbolic heat conduction in the semi-infinite body with the heat source which capacity linearly depends on temperature, Heat Mass Transfer, vol. 33, pp , Moosaie, A., Axisymmetric non-fourier temperature field in a hollow sphere, Arch. Appl. Mech., vol. 79, pp , Moosaie, A., Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance, Int. Commun. Heat Mass Transfer, vol. 34, pp , Moosaie, A., Non-Fourier heat conduction in a finite medium subjected to arbitrary non-periodic surface disturbance, Int. Commun. Heat Mass Transfer, vol. 35, pp , Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, New York: McGraw-Hill, pp , Qui, T., Juhasz, T., Suarez, C., Bron, W. E., and Tien, C. L., Femto second laser heating of multi-layer II experiments, Int. J. Heat Mass Transfer, vol. 37, pp , Ready, J., Industrial Application of Lasers, New York: Academic Press, Subhash, C. M. and Sahai, H., Analyses of non-fourier heat conduction in 1D cylindrical and spherical geometry An application of the lattice Boltzmann method, Int. J. Heat Mass Transfer, vol. 55, pp , Volume 46, Number 5, 2015

18 464 Akbari et al. Tang, D. W. and Araki, N., Analytical solution of non-fourier temperature response in a finite medium under laser-pulse heating, Heat Mass Transfer, vol. 31, pp , Tang, D. W. and Araki, N., Non-Fourier heat conduction behavior in finite mediums under pulse surface heating, Mater. Sci. Eng., vol. 292, pp , Vernotte, P., Les paradoxes de la théorie continue de l équation de la chaleur, C.R. Acad. Sci., vol. 246, pp , Wang, B. L. and Han, J. C., Non-Fourier heat conduction in layered composite materials with an interface crack, Int. J. Eng. Sci., vol. 55, pp , Zhou, J., Zhang, Y., and Chen, J. K., Non-Fourier heat conduction effect on laser-induced thermal damage in biological tissues, Numer. Heat Transfer, vol. 54(A), pp. 1 19, Heat Transfer Research