Numerical solution of hyperbolic heat conduction in thin surface layers

Transcription

1 International Journal of Heat and Mass Transfer 50 (007) 9 Numerical solution of hyperbolic heat conduction in thin surface layers Tzer-Ming Chen * Department of Vehicle Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao East Road, Taipei 1063, Taiwan, ROC Received 6 February 006; received in revised form 8 October 006 Available online 11 May 007 Abstract The purpose of the present paper is to propose a new hybrid method investigating the effect of the surface curvature of a solid body on hyperbolic heat conduction. The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have developed a new hybrid method combined the Laplace transform, the weighting function scheme [Shong-leih Lee, Weighting function scheme and its application on multidimensional conservation equations, Int. J. Heat Mass Transfer 3 (1989) ], and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. Four different examples have been analyzed by the present method. It is found from these examples that the present method is in good agreement in the analytical solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) 33 35] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex. Ó 006 Elsevier Ltd. All rights reserved. Keywords: Hyperbolic heat conduction; Hybrid method; Thin surface layers 1. Introduction In the last decades, study of the hyperbolic heat conduction equation has received considerable interest, because of it wide applicability in engineering applications, such as laser-aided material processing, cryogenic engineering, the high-intensity electromagnetic irradiation of a solid and the high-rate heat transfer in rarefied media. The solutions of the hyperbolic heat conduction can be found in a number of publications, such as one-dimensional given by [ 13] and two-dimensional solutions given by [1 16]. Baumeiser and Hamill [], Taitel [3], Ozisik and Vick [], and Wu [5] obtained an analytical solution of one-dimensional HHC, for a semi-infinite medium or in a finite medium with convection, or radiation at the wall surface. Carey and Tai [6] applied the central and backward difference schemes to examine the oscillation of numerical solution * Tel.: address: tmchen@ntut.edu.tw at the reflected boundary. To remedy the numerical difficulty encountered, many numerical schemes such as the predictor-corrector scheme [7], the transfinite element formulation [8], the technique based on the Galerkin finite element and mixed implicit-explicit scheme [9], the characteristic method [10], and the hybrid scheme [11] have been proposed. Glass et al. [1] and Yeung and Tung [13] studied the effect of the surface radiation on thermal wave propagation in a one-dimensional slab. The objective of the present study is to propose a new hybrid method investigating the influence of the surface curvature of a solid body on hyperbolic heat conduction. The present method combine the Laplace transform, weighting function scheme, and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. The Laplace transfer method is used to remove the time-dependent terms from the governing equation, and then the discretized expression of the transformed equation is given by the weighting function scheme and the hyperbolic shape function. It is found from four examples that the present method is in good /$ - see front matter Ó 006 Elsevier Ltd. All rights reserved. doi: /j.ijheatmasstransfer

2 T.-M. Chen / International Journal of Heat and Mass Transfer 50 (007) 9 5 Nomenclature C propagation velocity of thermal wave c p specific heat E r dimensionless radiation parameter a sra fr 3 k c f r reference heat flux k thermal conductivity q heat flux R the average radius of curvature at x =0, y =0 R 1,R the principal radii of curvature at x =0,y =0 s Laplace transform parameter T temperature T 0 surrounding temperature z W f (z) weighting function, 1 e z z parameter of the weighting function x, y, z coordinators Greek symbols k a thermal diffusivity, qc p a s surface absorptivity c c ¼ o 1 þ o 1 ox oy ¼ x¼0;y¼0 1 R 1 þ 1 R g dimensionless length, c1 a h dimensionless temperature, ðt T 0Þkc _ af r h the previously calculated surface temperature q density r Stefan Boltzmann constant B length, z f(x,y) n dimensionless time, c t a Superscript the Laplace transform agreement in the analytical solutions [1] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-fourier effect is significant.. Analysis Consider a coordinate system, as shown as in Fig. 1. The x y plane forms a tangential plane at the surface point of interest. The surface of the body can be described by an equation of the form z = f(x, y). The hyperbolic heat conduction equation is given by 1 o T C ot þ 1 ot a ot ¼ o T ox þ o T oy þ o T ð1þ oz Let us introduce a new independent variable B = z f(x,y) and neglect terms of order ð d R Þ, where d is the heat penetration length and R is the average radius of curvature at x =0,y = 0. The equation at x = 0 and y = 0 is given by 1 o T C ot þ 1 ot a ot ¼ o T o1 þ c ot 1 where c ¼ o 1 ox þ o 1 ¼ 1 þ 1 oy x¼0;y¼0 R 1 R Fig. 1. Coordinate system. ðþ ð3þ R 1 and R are the principal radii of curvature at x =0, y =0. For convenience of numerical analysis, let us define by the following dimensionless variables: n ¼ C t ðþ a g ¼ C1 ð5þ a hðg; nþ ¼ kcðt T 0Þ ð6þ af r Qðg; nþ ¼ q ð7þ f r The resulting equation becomes o h on þ oh on ¼ o h o g þ e oh og where e ¼ ca C 3. Numerical scheme ð8þ ð9þ To remove the n-dependent terms, taking the Laplace transform of Eq. (8) with respect to n gives d h dg þ e d h dg ðs þ sþ h ¼ 0 ð10þ Consider a homogeneous second-order ordinary differential of the form h 00 þ ah 0 k h ¼ 0 ð11þ Combining the weighting function scheme and hyperbolic shape function, the resulting discretized form is given ðh 00 þ ah 0 k hþ i ¼ a W h i 1 þ a P h i þ a E h iþ1 ð1þ

3 6 T.-M. Chen / International Journal of Heat and Mass Transfer 50 (007) 9 where kw ð z i 1 Þ a W ¼ Dx i 1 sinhðkdx i 1 Þ ð13þ kw ðz i Þ a E ¼ Dx i sinhðkdx i Þ ð1þ a P ¼ a W a E þ kðcoshðkdx iþ 1Þ Dx i sinhðkdx i Þ ð15þ and z W ðzþ ¼ 1 e z ð16þ z i ¼ a iþ 1 Dx i ð17þ Using these Eqs. (1) (17) for Eq. (10) leads to the discretized expression of Eq. (10) as a W h i 1 þ a P h i þ a E h iþ1 ¼ 0 ð18þ ðs þ sþ 1 W ð z i 1 Þ a W ¼ h i ð19þ Dx i 1 sinh ðs þ sþ 1 Dx i 1 ðs þ sþ 1 W ðz i Þ a E ¼ h i ð0þ Dx i sinh ðs þ sþ 1 Dx i a P ¼ a W a E þ ðs þ sþ 1 fcosh½ðs þ sþ 1 Dx Š 1g h i Dx i sinh ðs þ sþ 1 Dx i ð1þ z i ¼ edx i ðþ The rearrangement of Eqs. (18) () in conjunction with the boundary conditions can yield the following matrix system ½KŠf hg¼ff g ð3þ The nodal dimensionless temperature h i can be determined by using the application of the Gaussian elimination algorithm and the numerical inversion of the Laplace transform technique [17].. Results and discussion Example 1. Prescribed wall temperature. The initial and oh ðg; 0Þ ¼0 on ðþ hð0; nþ ¼1; hðg!1; nþ ¼0 ð5þ The analytical solution [1] of this example is expressed as 8 1 >< g Z n I 1 hðg;nþ¼e g e e g þ 1 e e s g >: h i1 ð1 e Þðs g Þ ðs g Þ 1 9 >= ds Uðn gþ >; ð6þ Table 1 lists the comparison of the present method solutions by Dg = 0.05 and analytical solutions for the problem at t = 0.5 and t = 1.0. From Table 1, it is seen that Table 1 Comparison of the present method and analytical solution resulting from a prescribed wall temperature x Present method Analytic solution Eq. (6) e = 0.1 e = 0.0 e = 0.1 e = 0.1 e = 0.0 e = 0.1 t = t =

4 T.-M. Chen / International Journal of Heat and Mass Transfer 50 (007) 9 7 Example. Prescribed wall heat flux. The initial and ohðg; 0Þ ¼ 0 on ð7þ Qð0; nþ ¼1; Qðg!1; nþ ¼0; Qðg; 0Þ ¼0 ð8þ The analytical solution [1] of this example is expressed as ( ( hðg; nþ ¼ a ) 1 e C e g e n I 0 1 e ðn s Þ Fig.. The influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed wall temperature by Dg = 0.05 at t = 0.5, t = 1.0, t = 1.5 and t =.0. the present method solutions are in agreement with the analytical solution using the Eq. (6). Fig. shows the influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed wall temperature by Dg = 0.05 at t = 0.5, t = 1.0, t = 1.5, and t =.0. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex. þ Z n g ( ) ) 1 e s I 0 1 e ðs g Þ ds Uðn gþ ( a Z ( 1 ) 1 C e e g e f e I 0 1 e ðn f Þ Uðn fþdf þ Z 1 g g Z n e f e 0 ( ) ) 1 e f I 0 1 e ðs f Þ Uðs fþ dsdf ð9þ The boundary condition for the Laplace transform of the dimensionless temperature at surface g = 0 can be obtained d h þ ð0; sþ ¼ s dg s ð30þ Table Comparison of the present method solutions and analytical solutions resulting from a prescribed wall heat flux x Present method Analytic solution Eq. (9) e = 0.1 e = 0.0 e = 0.1 e = 0.1 e = 0.0 e = 0.1 t = t =

5 8 T.-M. Chen / International Journal of Heat and Mass Transfer 50 (007) 9 Fig. 3. The influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed wall heat flux by Dg = 0.05 at t = 0.5, t = 1.0, t = 1.5 and t =.0. Fig.. The effect of the surface curvature of a finite slab body on hyperbolic heat conduction problem by Dg = 0.01 at t = 0., t = 0.5, t = 1. and t = 1.5. And the discretized form is represented as ðs þ sþ 1 W ðz 1 Þ n h i cosh½ðs þ sþ 1 Dx sinh ðs þ sþ 1 1 Š h 1 þ o h Dx 1 ¼ s þ ð31þ s Table shows the comparison of the present method solutions by Dg = 0.05 and analytical solutions [1] resulting from a prescribed wall heat flux problem at t = 0.5 and t = 1.0. It is observed that the present method solutions well agree with the analytical solutions. Fig. 3 represents the influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed wall heat flux by Dg = 0.05 at t = 0.5, t = 1.0, t = 1.5, and t =.0. It can be seen that the present method solutions do not exhibit numerical oscillations at the wave front. ohðg; 0Þ ¼ 0 on ð3þ Qð0; nþ ¼ E r h þ 1; Qðg!1; nþ ¼0; Qðg; 0Þ ¼0; where E r ¼ a sra fr 3 k c ð35þ The boundary condition is linearized by the Taylor s series approximation and the Laplace transform of the dimensionless temperature at surface g = 0 can be obtained d h dg ð0; sþ ¼ ðs þ Þ E r " _! # h _ 3 3h h þ 1 s s ð36þ where h _ is the previously calculated surface temperature. Example 3. Prescribed in a finite slab. The initial and hð0; nþ ¼1; oh ðg; 0Þ ¼0 on ð3þ oh ð1; nþ ¼0 og ð33þ Fig. illustrates the effect of the surface curvature of a finite slab body on hyperbolic heat conduction problem by Dg = 0.01 at t = 0., t = 0.5, t = 1. and t = 1.5. Example. Prescribed surface radiation. The initial and Fig. 5. The influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed surface radiation by Dg = 0.015, E r = 0.1, E r = 1.0 and E r =10att = 0.5 and t = 1.0.

6 T.-M. Chen / International Journal of Heat and Mass Transfer 50 (007) 9 9 And the discretized form is represented as ðs þ sþ 1 W ðz 1 Þ n h i h i cosh ðs þ sþ 1 Dx h1 sinh ðs þ sþ 1 1 þ o h Dx 1 " ¼ðsþÞ E r h _ 3 3h _! # h þ 1 s s ð37þ Fig. 5 shows the influence of the surface curvature of a solid body on hyperbolic heat conduction problem with a prescribed surface radiation by Dg = 0.015, E r = 0.1, E r = 1.0, and E r =10att = 0.5 and t = Conclusions The new hybrid has shown success in solving the hyperbolic heat conduction problem with a conservation term. To illustrate the accuracy and efficiency of the new method, four different examples have been analyzed. It is found from these examples that the present method is in good agreement in the analytical solutions [1] and does not exhibit numerical oscillations at the wave front. And the influence of surface curvature of a solid surface with hyperbolic heat conduction is also shown in Figs. 5. These results represent how the surface temperature is modified by the surface curvature during the short period when the non- Fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex. References [1] Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) [] K.J. Baumeister, T.D. Hamill, Hyperbolic heat conduction equation a solution for the semi-infinite body problem, ASME J. Heat Transfer 91 (1969) [3] Y. Taitel, On the parabolic hyperbolic and discrete formulation of the heat conduction equation, Int. J. Heat Mass Transfer 15 (198) [] M.N. Ozisik, B. Vick, Propagation and reflection of thermal waves in a finite medium, Int. J. Heat Mass Transfer 7 (198) [5] C.Y. Wu, Integral equation solution for hyperbolic heat conduction with surface radiation, Int. Commun. J. Heat Mass Transfer 15 (1988) [6] G.F. Carey, M. Tai, Hyperbolic heat transfer with reflection, Numer. Heat Transfer 5 (198) [7] D.E. Glass, M.N. Ozisik, D.S. McRae, B. Vick, On the numerical solution of hyperbolic heat conduction, Numer. Heat Transfer 8 (1985) [8] K.K. Tamma, S.B. Railkar, Specially tailored transfinite-element formulations for hyperbolic heat conduction involving non-fourier effects, Numer. Heat Transfer 15 (Part B) (1989) [9] K.K. Tamma, J.F. D Costa, Transient modeling analysis of hyperbolic heat conduction problems employing mixed implicit explicit method, Numer. Heat Transfer 19 (Part B) (1991) [10] H.Q. Yang, Characteristics-based, high-order accurate and nonoscillatory numerical method for hyperbolic heat conduction, Numer. Heat Transfer 18 (Part B) (1990) 1 1. [11] H. T Chen, J.Y. Lin, Numerical analysis for hyperbolic heat conduction, Int. J. Heat Mass Transfer 36 (11) (1993) [1] D.E. Glass, M.N. Ozisik, B. Vick, Hyperbolic heat conduction with surface radiation, Int. J. Heat Mass Transfer 8 (1985) [13] W.K. Yeung, T.L. Tung, A numerical scheme for non-fourier heat conduction, Part I: one-dimensional problem formulation and applications, Numer. Heat Transfer 33 (Part B) (1998) [1] H.Q. Yang, Solution of two-dimensional hyperbolic heat conduction by high-resolution numerical methods, in: AIAA, 7th, Thermophysics Conference, July 6 8, Nashville, TN, 199, (AIAA, 9-937). [15] H.T. Chen, J.Y. Lin, Numerical solution of two-dimensional nonlinear hyperbolic heat conduction problem, Numer. Heat Transfer 5 (Part B) (199) [16] S.H.W. Shen, A numerical solution of two-dimensional hyperbolic heat conduction with non-linear boundary conditions, Int. J. Heat Mass Transfer 39 (003) [17] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transform, J. Comput. Appl. Math. 10 (198)