Algebraically Explicit Analytical Solution of Three- Dimensional Hyperbolic Heat Conduction. Equation

Transcription

1 Adv. Theor. Appl. Mech., Vol. 3, 010, no. 8, Algebraically Explicit Analytical Solution of Three- Dimensional Hyperbolic Heat Conduction Equation Seyfolah Saedodin Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran Mohsen Torabi Department of Mechanical Engineering, Faculty of Engineering Semnan University, Semnan, Iran Abstract In this paper, the three-dimensional hyperbolic effect subjected to a cosine heat flux boundary condition is carried out. Equations in rectangular coordinates are solved. Analytical solution with method of separation of variables is derived. The main aim of this paper is to obtain some possibly explicit analytical solution of the (1+3)-dimensional hyperbolic heat conduction equation for given initial and boundary condition with method of separation of variables. The temperature layers and profiles for various relaxation times in three different examples are calculated. Also, the reflection of the heat wave in the solid in some examples is shown. Keywords: Hyperbolic heat conduction; Relaxation time; Separation of variables method; Heat wave reflection 1. Introduction In the last decades, several experiments have shown that the classical theory of heat conduction in solids based on Fourier's law may fail when unsteady processes are involved. Indeed, the parabolic heat implies an infinite propagation speed of

2 370 S. Saedodin and M. Torabi the heat wave. Wave-lie thermal response with finite propagation speed, was first observed experimentally in solid 4 He by Acerman et al. [1]. In situations which include extremely high temperature gradients, extremely large heat fluxes and extremely short transient duration, the heat propagation speeds are finite. In order to obtain a theory of heat conduction, compatible with a finite propagation speed of heat wave, Cattaneo [5], and Vernotte [] proposed a modification of Fourier's law. Which, is now well nown as Cattaneo-Vernotte's constitutive equation q q + τ = T (1) t Where q is the heat flux vector, τ is the thermal relaxation time, is the constant thermal conductivity of the material and T is the temperature gradient. If equation (1), combined with the conservation of energy gives the hyperbolic heat conduction equation T T + τ = αδt t t Where α =, ρ, c ρc and Δ are thermal diffusivity, mass density, specific heat capacity and Laplace s differential operator, respectively. Many analytical and numerical solutions of equation () have been solved in the literature. Most researches applied hyperbolic heat conduction equation (HHCE), in one-dimensional. Roetzel and Das [15] and Sahoo and Roetzel [17] calculated the heat exchanger problems. Tang and Arai [0] computed the hyperbolic fin problems under the periodic thermal conditions. As well, the thin film problems are also investigated by Tan and Yang [19]. Mullis [13] discussed the rapid solidification problems. Barletta and Pulvirenti [] have obtained the temperature field in a solid cylinder with constant as well as exponentially decaying boundary heat flux. Lor and Chu [9] analyzed the problem with the interface thermal resistance. Zhang et al. [4] processed the hyperbolic conduction model with heat source term analytically by the method of Laplace Transform. Torii and Yang [1] numerically solved the case of a thin film subjected to a symmetrical heating on both sides and Lewandowsa and Malinowsi [8] solved the same problem analytically by the method of the Laplace transform. Moosaie [1] investigated the hyperbolic conduction in a finite medium with the arbitrary initial conditions. Chen [7] combined the Laplace transform, weighting function scheme and the hyperbolic shape function to solve the time dependent HHCE with a conservation term. Saleh and Al-Nimr [18] employed Laplace transform, software pacage MATLAP and Taylor series, to solve the one-dimensional hyperbolic equation. Few papers have been analyzed HHCE in two or three dimensions. Yang [3] developed the two-dimensional HHCE in an arbitrary body-fitted coordinate. Chen and Lin [6] formulated a numerical scheme, involving the Laplace transform technique and the control volume method for the problem. Zhou et al. [5] presented a thermal wave model of bioheat transfer, together with a seven-flux model. Saedodin and Torabi [16] investigated two-dimensional hyperbolic heat ()

3 Algebraically explicit analytical solution 371 conduction in a cylinder. To the authors' nowledge, there is only one paper that has been solved three-dimensional hyperbolic equation analytically. Barletta and Zanchini [3] analytically investigated the HHCE in three-dimensions. They applied the Laplace transform technique. In this paper, an analytical expression of temperature field is obtained for a threedimensional problem of the hyperbolic heat conduction. In fact, the solution of the problem is obtained by employing the method of separation of variables. Using our analytical solution, we performed sample calculation of temperature surfaces and profiles. Also, on those samples, we mention the reflection of the thermal wave. Nomenclature A, B, C, am, c1, c, C1, D1, D, C nfg Constant coefficients c heat capacity Gree symbols Fo Fourier number α Thermal diffusivity Thermal ρ Mass density conductivity Laplace's differential Δ operator L Height of the solid Gradient operator l 1 Length of the solid τ Thermal relaxation time l Width of the solid θ Dimensionless temperature Ve Vernotte number ξ, ε, ω Dimensionless spatial x y, z, Spatial coordinate ψ ( ξ, coordinates Function employed in Eq. (1) X ( ξ ), Y ( ε ), Functions employed Function employed in Eq. Z( ω), T ( φ ( ξ, ω) (1) in Eq.(19) q Heat flux vector t Temporal coordinate β m, γ, λn, η f, μg Eigenvalues Parameter defined by Eq. T Temperature κ (35a) Ambient Parameter defined by Eq. T κ i temperature (35b) Parameter defined by Eq. T i Initial temperature ϑ l (33). Problem statement There are many inds of heat flux boundary conditions in natural and technology. For heat flux, Mirahmadi et al. [10] utilized Gaussian distribution heat flux. Moosaie [11] utilized the time-dependence cosine heat flux. In this paper, we

4 37 S. Saedodin and M. Torabi utilized space-dependence cosine function as a heat flux boundary condition in one of the six boundary surfaces..1. Governing differential equation Consider a solid, as shown as Fig. 1. For this case, a three-dimensional unsteady HHCE without any heat generation can then be expressed as: T T T T T + τ = α( + + ) t (3) t x y z.. Boundary conditions For this case the boundary conditions are: T ( 0, y, z, t) x (4a) T ( 1, y, z, t) = T l (4b) T ( x,0, z, t) y (4c) T ( x,, z, t) = T l (4d) T x, y,0, t) = T ( (4e) x y T ( x, y, L, t) = q cos( ) cos( ) z l 1 l (4f).3. Initial conditions Consider the solid initially has been at the ambient temperature. Then: T i = T (5) Hence the initial conditions are: T x, y, z,0) = T ( (6a) T ( x, y, z,0) t (6b) 3. Analytical solution We introduce the following dimensionless quantities:

5 Algebraically explicit analytical solution 373 T T x y θ =, ξ =, ε =, z αt ω =, Fo =, Lq l 1 l L L Ve ατ L = (7) Where θ is dimensionless temperature and ξ, ε, ω are dimensionless coordinates. Fo is the Fourier number, Ve is the Vernotte number. In our treatment, we assume: l 1 = l = L (8) By introducing the dimensionless quantities and Eq. (8), we expressed HHCE as: θ θ θ θ θ + = + + Fo Fo ξ ε ω Ve (9) Also, the boundary conditions are: θ( 0, ξ (10a) θ ( 1, ε, (10b) θ ( ξ,0, ε (10c) θ ( ξ,1, (10d) θ ( ξ, ε,0, (10e) θ ( ξ, ε,1, ) = cos( ξ )cos( ε ) ω Fo (10f) and the initial conditions are: θ ( ξ, 0) (11a) θ ( ξ, ε, 0) Fo (11b) If we want to apply the separation of variables method, first we should split up eq. (9) into a set of simpler problems. Carslaw and Jaeger [4] and Özisi [14] determined the solution of Eq. (9) from: θ ( ξ, = ψ ( ξ, + φ( ξ, ω) (1) Where the temperature φ ( ξ, ε, ω) is taen as the solution of the following Eqs.:

6 374 S. Saedodin and M. Torabi φ φ φ + + ξ ε ω φ( 0, ε, ω) ξ (13) (14a) φ ( 1, ε, ω) (14b) φ( ξ,0, ω) ε (14c) φ ( ξ,1, ω) (14d) φ ( ξ, ε,0) (14e) φ( ξ, ε,1) = cos( ξ ) cos( ε ) (14f) ω Where the temperature ψ ( ξ, is taen as the solution of the following Eqs.: ψ ψ ψ ψ ψ Ve + = + + Fo Fo ξ ε ω (15) ψ ( 0, ξ (16a) ψ ( 1, ε, ψ ( ξ,0, ε ψ ( ξ,1, ψ ( ξ, ε,0, ψ ( ξ, ε,1, ω ψ ( ξ, 0) = φ ( ξ, ω) ψ ( ξ, ε, 0) Fo The solution of the partial differential equation (13) becomes ( ξ, ε, ω) = [ cos( )cos( )sinh( am βmξ γ ε ω βm + γ )] m= 0= 0 and γ are eigenvalues of Eqs. cos m (16b) (16c) (16d) (16e) (16f) (16g) (16h) φ (17) Where βm β and cos γ, respectively. Using boundary condition (14f) and Utilizing orthogonality condition, the constant a m is given as following Eq.:

7 Algebraically explicit analytical solution 375 a m = β m + γ 1 cosh( β m + γ sin(1 + β m ) sin(1 β m ) + ) 1+ β 1 β m m sin(1 + γ ) sin(1 γ ) + 1+ γ 1 γ To solve the partial differential equation (15), we should use the following separation ansantz: ψ ( ξ, ε, X ( ξ ) Y ( ε ) Z ( ω) T ( (19) By substituting the Eq. (19) into the Eq. (15) and subtracting to (19): 1 d T 1 dt 1 d X 1 d Y 1 d Z ( Ve + ) = ± λ (0) T dfo T dfo X dξ Y dε Z dω Here + λ is suitable to our problem. Finally, the problem separately expressed in ξ - and ε - and ω - and Fo-directions as follows: (18) d X + λ X dξ (1) d X ( 0) dξ (a) X ( 1) (b) d Y + η Y dε d Y ( 0) dε ( 1) (3) (4a) Y (4b) d Z + μ Z dω ( 0) (5) Z (6a) d Z ( 1) dω d T dt Ve + + ( + η + μ ) T dfo dfo d T ( 0) dfo (6b) λ (7) Solving the Eqs. (1), (3) and (5) we obtain: X ( ξ ) = Acos( λnξ ) (9) Y ( ε ) = Bcos( η f ε ) (30) Z ( ω) = Csin( μgω) (31) Where λ n, η f and μ g are eigenvalues of Eqs. cos λ n, cos η f and cos μ g, respectively. For the Eq. (7), if 1 4Ve ϑ > 0, we obtain: Fo T ( = e Ve Where κfo κfo ( c1 sinh( ) + c cosh( )) Ve Ve l (8) (3)

8 376 S. Saedodin and M. Torabi ϑ l λn + η f + μg = (33) l Fo ( ) Fo Fo T Fo e Ve κ ( c1 sin( i κ = ) + c cos( i )) Ve Ve And if 1 4Ve ϑ < 0 Where κ = 1 4Ve ϑ l (35a) κ = iκ i (35b) By substituting the Eqs. (3) and (34) into initial condition (8) to eliminating c 1 or c : Fo Ve Fo Fo e 1 κ κ sinh( ) + cosh( ) κ = real κ Ve Ve T ( = C1 Fo (36) i Ve κ Fo κ + ifo e 1 sin( ) cos( ) κ = iκi κi Ve Ve Substituting the Eqs. (9-31) and (36) into (19), the following equation for ψ ( ξ, obtain: N F G Fo 1 κfo κfo ψ( ξ, = Cnfg exp( ) sinh( ) cosh( ) n 0 f 0 g 0 Ve + κ Ve Ve = = = cos( λnξ )cos( η fε)sin( μgω) (37) Fo 1 κifo κifo + Cnfg exp( ) sin( ) cos( ) cos( λ ξ)cos( η ε)sin( μ ω) + n f g N+ 1 F+ 1 G+ 1 Ve κ Ve Ve i Where N, F, G are maximum value of n, f, g when the κ is real for each loop, respectively. Finally, using the Eq. (16g) and orthogonality condition sin(1 + λ ) sin(1 ) sin(1 ) sin(1 ) ( 1) g n λn + η f η f C nfg = λn 1 λn 1 η f 1 η f λ (38) + + n + η f + μ g As a good comparison, we should solve the three-dimensional parabolic equation. If Fourier s law holds, i.e. in the limitve 0, the values of κi are always real and therefore, ψ ( X, T, Z, becomes ψ ( ξ, Fo C nfg Fo (34) ) = exp( ϑl )cos( λnξ )cos( η f ε )sin( μ gω) (39) n= 0 f g= 0 4. Result and discussion Using our analytical solution, we performed sample calculations of temperature surfaces and profiles, based on Eqs. (1), (17), (37) and (39). The results of calculations are presented in Figs. -5.

9 Algebraically explicit analytical solution Comparison of surface temperature evolvement obtained from Fourier and non-fourier model with same Fourier number Fig. shows the surface temperature profiles for the two cases. The Fourier number and ε that we simulated for are 0.4 and zero, respectively. It can be perceived from Fig. a that, in Fourier model the speed of propagation is infinite.hence, all of the solid can touch the heat flux. But it can be perceived from Figs. b and c that, because of the non-fourier effect, the heat wave cannot touch the other side of the solid at the moment. In addition, it can be seen that, if the Vernotte number increases, the traverse distance of the heat wave decreases. Furthermore, Fig. shows that, the more Vernotte number, the less thermal penetration depth. 4.. Comparison of temperature distribution for non-fourier model with same Fourier number but at different Vernotte number along the ω direction Fig. 3 shows temperature profiles along the ω direction at Fo. 4, ξ = ε. 5. It can be seen from Fig. 3 that, while Vernotte number is bigger than Fourier number, there are many points that don t feel the effect of heat flux. It can be seen that, until the heat flux dosen t touch the other side of the solid, while the Vernotte number increases, the dimensionless temperature decreases. But after the heat flux touches the other side of the solid, if the Vernotte number increases, the dimensionless temperature is decreases Comparison of temperature distribution for the non-fourier model with different Vernotte number for the unique point Fig. 4 shows temperature profiles at the point ξ = ε = ω. 5. It can be seen from Fig. 4 that, the higher Vernotte number causes each point to be at initial temperature, more. Moreover, it can be perceived again from Fig. 4 that, as much as the Vernotte number is higher, the point can get to higher temperature during the process. Also the thermal wave reflection, causes the existence of a fracture in the thermal profile of the point. In the Fig. 5, the fracture of the thermal wave can be observed, for the two Vernotte numbers, 0.3 and 0.5. As can be observed in the Fig. 5, the reflective wave with the Vernotte number 0.5, could touch and pass the point. Therefore, the temperature profile fracture can be seen in the Fig. 4, for this Vernotte number. But for the Vernotte number 0.7, the reflective wave cannot touch the point till Fo = 1. Hence in Fig. 4, any fracture cannot be observed in the temperature profile. 5. Conclusion In this paper, the three-dimensional HHCE was solved analytically with the spacedependence cosine boundary condition by the separation of variables method. We perceived that, the more the Vernotte number, the more the Fourier number needed for the solid to reach an equilibrium temperature. In addition we

10 378 S. Saedodin and M. Torabi investigated the wave reflection and we observed that due to the difference of a Vernotte number in a particular Fourier number, the wave reflection can reach the considered point, pass that or before reaching this point, the wave reflection occurs in the solid and propagates in the solid. Acnowledgment The authors would lie to acnowledge the support of the department of mechanical engineering and the office of gifted of Semnan University for funding the current research grant. References [1] C.C. Acerman, B. Berman, H.A. Fairban and R.A. Guyer, Second sound in solid helium, Phys. Rev. Lett. 16 (1966) [] A. Barletta and B. Pulvirenti, Hyperbolic thermal waves in a solid cylinder with a non-stationary boundary heat flux, Int. J. Heat Mass Transf.41 (1998) [3] A. Barletta and E. Zanchini, Three-dimensional propagation of hyperbolic thermal waves in a solid bar with rectangular cross-section, Int. J. Heat Mass Transf.4 (1999) [4] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, nd ed., Oxford University Press, New Yor, 000. [5] C. Cattaneo, Sur une forme de l équation de la chaleur éliminant le paradoxe d une propagation instantanée, C.R. Acad. Sci. 47 (1958) [6] H.T. Chen and J. Y. Lin, Analysis of two-dimensional hyperbolic heat conduction problems, Int. J. Heat Mass Transf.37 (1993) [7] T.M. Chen, Numerical solution of hyperbolic heat conduction in thin surface layers, Int. J. Heat Mass Transf.50 (007) [8] M. Lewandowsa and L. Malinowsi, 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 Transf. 33 (006) [9] W.B. Lor and H.S. Chu, Effect of interface thermal resistance on heat transfer in a composite medium using the thermal wave model, Int. J. Heat Mass Transf.43 (000)

11 Algebraically explicit analytical solution 379 [10] A. Mirahmadi, S. Saedodin and Y. Shanjani, Numerical Heat Transfer Modeling in Coated Powder as Raw Material of Powder-Based Rapid Prototyping Subjected to Plasma Arc, Numer. Heat Transf. Part A 51 (007) [11] A. Moosaie, Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance, Int. Commun. Heat Mass Transf. 34 (007) [1] A. Moosaie, Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions, Int. Commun. Heat Mass Transf.35 (008) [13] A. M. Mullis, Rapid solidification within the framewor of a hyperbolic conduction model, Int. J. Heat Mass Transf.40 (1997) [14] M. N. Ozisi, Heat conduction, nd ed., John Wiley &Sons, New Yor, [15] W. Roetzel and S. K. Das, Hyperbolic axial dispersion model: concept and its application to a plate heat exchanger, Int. J. Heat Mass Transf.38 (1995) [16] S. Saedodin and M. Torabi, Analytical Solution of Non-Fourier Heat Conduction in Cylindrical Coordinates, Int. Rev. of Mech. Eng. 3 (009), [17] R. K. Sahoo and W. Roetzel, Hyperbolic axial dispersion model for heat exchangers, Int. J. Heat Mass Transf.45 (00) [18] A. Saleh and M. Al-Nimr, Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique, Int. Commun. Heat Mass Transf.35 (008) [19] Z.M. Tan and W.J. Yang, Heat transfer during asymmetrical collision of thermal waves in a thin film, Int. J. Heat Mass Transf.40 (1997) [0] D.W. Tang and N. Arai, Non-Fourier heat conduction in a finite medium under periodic surface thermal disturbance, Int. J. Heat Mass Transf. 39 (1996) [1] S. Torii and W. J. Yang, Heat transfer mechanisms in thin film with laser heat source, Int. J. Heat Mass Transf. 48 (005) [] P. Vernotte, Les paradoxes de la théorie continue de l équation de la chaleur, C.R. Acad. Sci. 46 (1958) [3] H.Q. Yang, Solution of two-dimensional hyperbolic heat conduction by highresolution numerical method, Numer. Heat Transf. Part A 1 (199)

12 380 S. Saedodin and M. Torabi [4] D. Zhang, L. Li, Z. Li, L. Guan and X. Tan, Non-Fourier conduction model with thermal source term of ultra short high power pulsed laser ablation and temperature evolvement before melting, Physica B 364 (005) [5] J. Zhou, Y. Zhang and J. K. Chen, Non-Fourier Heat Conduction Effect on Laser-Induced Thermal Damage in Biological Tissues, Numer. Heat Transf. Part A 54 (008) Fig. 1. The solid configuration ω ξ (a)

13 Algebraically explicit analytical solution 381 ω ξ (b) ω ξ (c) Fig.. The surface temperature evolution with Fo. 4 for different Vernotte number (a) Fourier model ( Ve ) (b) Ve. 7 (c) Ve. 9

14 38 S. Saedodin and M. Torabi ω Fig.3. The distribution for the non-fourier model with the same Fourier number, but at different Vernotte number along the ω direction Fig. 4. The temperature distribution for non-fourier model with different Vernotte number for the unique point

15 Algebraically explicit analytical solution 383 ω (a) ω (b) Fig. 5. The surface temperature evolution and reflection of the heat wave at ξ = ε.5 for different Vernotte number (a) Ve. 5 (b) Ve. 7 Received: May, 010