RAPID TRANSIENT HEAT CONDUCTION IN MULTILAYER MATERIALS WITH PULSED HEATING BOUNDARY

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1 Numerical Heat Transfer, Part A, 47: , 25 Copyright # Taylor & Francis Inc. ISSN: print= online DOI: 1.18/ RAPID TRANSIENT HEAT CONDUCTION IN MULTILAYER MATERIALS WITH PULSED HEATING BOUNDARY J. Li Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, New York, USA P. Cheng School of Mechanical and Power Engineering, Shanghai Jiaotong University, Shanghai, People s Republic of China G. P. Peterson Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, New York, USA J. Z. Xu Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, People s Republic of China Rapid transient heat conduction in multilayer materials under pulsed heating is solved numerically based on a hyperbolic heat conduction equation and taking into consideration the non-fourier heat conduction effects. An implicit difference scheme is presented and a stability analysis conducted, which shows that the implicit scheme for the hyperbolic equation is stable. The code is validated by comparing the numerical results with an existing exact solution, and the physically unrealistic conditions placed on the time and space increments are identified. Using the validated model, the numerical solution of thermal wave propagation in multilayer materials is presented. By analyzing the results, the necessary conditions for observing non-fourier phenomena in the laboratory can be inferred. The results are also compared with the numerical results from the parabolic heat conduction equation. The difference between them is clearly apparent, and this comparison provides new insight for the management of thermal issues in high-energy equipment. The results also illustrate the time scale required for metal films to establish equilibrium in energy transport, which makes it possible to determine a priori the time response and the measurement accuracy of metal film, thermal-resistant thermometers. Received 2 May 24; accepted 1 October 24. This work was partially supported by Grant #HKUST614=2E from the Research Grant Council of Hong Kong and partially by the U.S. Office of Naval Research, Grant ONR N and the National Science Foundation CTS Address correspondence to G. P. Peterson, Department of Mechanical, Aerospace and Nuclear Engineering, Rennselaer Polytechnic Institute, 11 8th Street, Troy, New York 1218, USA. peterson@rpi.edu 633

2 634 J. LI ET AL. NOMENCLATURE a C C p q t t p thermal diffusivity sound speed specific heat at constant pressure heat flux time pulse duration T x k q s temperature coordinate thermal conductivity density thermal relaxation time INTRODUCTION Rapid transient heat conduction in multilayer materials with an inner-pulsed heating boundary is a fundamental phenomenon in fusion reactors, synchrotron sources that deliver high-brilliant X-ray beams, laser diagnostics and laser processing [1 4], superconducting electronic devices [5], and microelectromechanical systems (MEMS) technology, e.g., nanotip temperature sensors and nanometal film thermal sensors [6]. The operation of many laser techniques requires an accurate understanding and control of the energy deposition and its transport processes [1]. There also will be some thermal damage issues associated with these kinds of high-energy equipment, if we do not carefully consider the input energy density and pulse duration. In many MEMS applications, such as thermal ink-jet printers, an accurate prediction of the temperature variation in thin metal films is a critical parameter in determining the printing quality [7, 8]. Some previous experimental results on transient heat conduction in these types of materials indicate the presence of non-fourier heat conduction behavior [3, 9]. The formulation of the non-fourier heat conduction problem was first presented by Cattaneo and Vernottee [1, 11], who proposed a constitutive equation that was coupled with the local energy balance. To investigate such a problem, it is first necessary to solve the hyperbolic heat conduction equation. Although analytical solutions for the hyperbolic heat conduction equation have been obtained for a few relatively simple problems [3, 12], analytical methods capable of solving more complicated problems, such as the transient heat conduction in multilayer materials shown schematically in Figure 1, are not currently available. Considerable effort has been applied to numerical solutions of non-fourier heat conduction problems of this type. Yang [13] has applied the method of characteristics in conjunction with highorder, total variation diminishing schemes, to solve some specific hyperbolic heat conduction problems in an infinitely wide slab with constant properties. Chen and Lin [14] analyzed several one-dimensional problems involving either a semi-infinite medium or a plane, for the case of constant thermal properties. In this formulation, a Laplace transformation was used to remove the time-dependent terms from the governing equations. The discretized expressions of the transformed equations were then derived using a control-volume method. Glass et al. [15] employed MacCormark s predictor-corrector scheme to solve one-dimensional problems. In another article, Glass et al. [16] solved the hyperbolic heat conduction equation in a semi-infinite medium and considered three different kinds of thermal disturbance at the boundary surface: a sudden change of temperature, a sudden change of heat

3 HEAT CONDUCTION IN MULTILAYER MATERIALS 635 Figure 1. Schematic 1-D model of heat conduction in multilayer material. flux, and a pulsed heat source. These authors obtained a high degree of accuracy except for the case of sharp discontinuities, where numerical oscillations occurred. Pulvirenti et al. [17] employed MacCormark s predictor-corrector scheme to study hyperbolic heat conduction with temperature-dependent thermal conductivity, specific heat, and thermal diffusivity. As pointed out in [13, 15], the primary problems encountered in solving hyperbolic heat conduction equations numerically are due to the presence of fictitious numerical oscillations, particularly when sharp propagation fronts and reflective boundaries are involved, which are physically unrealistic. A majority of the previously discussed finite-difference schemes are based on the constitutive equation proposed by Cattaneo and Vernotte [1, 11] coupled with the local energy balance equation. Cattaneo and Vernotte s constitutive equation is a first-order hyperbolic equation, which can be solved by conventional numerical methods, as summarized by Yang [13]. Cattaneo and Vernotte s constitutive equation and the energy balance equation can be combined to give a second-order hyperbolic heat conduction equation in terms of temperature, to yield the thermal wave equation [3]. However, little has been done to obtain direct solutions of second-order hyperbolic heat conduction equations with finite-difference schemes [18 22]. Carey and Tsai [18] applied the central and backward difference schemes to derive the difference equation. Kar et al. [19] used a hybrid analytical-numerical method based on a Laplace transform method to remove the time-dependent terms. The direct numerical solution of the second-order hyperbolic heat conduction equation is helpful for understanding the fundamental physics behind the equations. In [18, 2 22], fictitious numerical oscillations were also observed; however, there was no discussion about the stability of the difference schemes for the second-order hyperbolic heat conduction equation solutions, nor the criteria used to select the time and space increments to avoid a physically unrealistic solution. In the current investigation, an implicit scheme is developed in order to solve second-order hyperbolic heat conduction equations numerically. After analyzing the stability of the implicit difference schemes, the effects of changes in the time

4 636 J. LI ET AL. and space increments on the physically realistic solution can be made by comparing the numerical results with an analytical solution developed by the authors. In this way, the fictitious numerical oscillations described above can be suppressed and ultimately eliminated, by regulating, i.e., decreasing, the time and space increments. This is true even for sharp propagation fronts (i.e., pulsed heating boundary conditions). The resulting method can then be applied to transient heat transfer in complex multilayer materials, similar to those shown in Figure 1 and compared with the numerical results obtained from the parabolic heat conduction equation. As anticipated, the calculations presented here resulted in a number of new observations and conclusions which provide new insights into the behavior of the thermal wave phenomena occurring in high-energy equipment or lower-power MEMS applications. By analyzing the results of the calculations, the conditions necessary to observe the predicted non-fourier phenomena in a laboratory setting can be inferred. MATHEMATICAL MODEL The Fourier law of heat conduction implies that transient conduction problems must satisfy the diffusion equation, and for a time-dependent boundary condition, the heat energy propagates instantaneously everywhere in the solid. However, when a pulsed electrical source or laser is used and the pulse duration is comparable to the mean free path of the energy carrier, nano- or microscale heat transfer effects must be taken into consideration. One of the approaches applicable in this situation is the wave theory of heat conduction, which uses the relaxation behavior to describe pulsed heat transport at short time intervals (see [3] for a review of the literature). For this situation, the governing heat transfer equation becomes qðx; t þ sþ ¼ krt ð1þ where s is the thermal relaxation time, which is related to the collision frequency of the molecules within the energy carrier. The thermal relaxation time, s, can be calculated using the thermal wave speed, C, and the thermal diffusivity, a, as s / a C 2 ð2þ If s is much smaller than the time interval for a particular transient process, Eq. (1) can be expanded using a first-order Taylor series expansion to yield q þ s qq qt ¼ krt ð3þ The above equation is called Cattaneo-Vernotte s constitutive equation. Combining Eq. (3) with the energy conservation equation, qc P qt qt þrq ¼ ð4þ

5 HEAT CONDUCTION IN MULTILAYER MATERIALS 637 yields the following unsteady hyperbolic heat conduction equation, qt qt þ s q2 T qt 2 ¼ ar2 T ð5þ which is referred to as the thermal wave equation [3]. Equation (5) is applicable to problems involving transient heat conduction heated by an internal pulsed heating source. It is also applicable to laser-induced bubble nucleation [23], thermal ink-jet printers [7, 8], laser-aided wafer surface cleanup, laser diagnostics, and laser processing [1 4]. There is little information in the literature regarding thermal wave propagation in a multilayer material, e.g., [24 27] et al., which considered only a sudden temperature rise in the external surface of one layer. Since the depth of thermal wave propagation is much smaller than the size of the objects, e.g., a metal film with 1 1-nm thickness on a 1 1-mm-sized thermal ink-jet printer head, the mathematical model for the heat conduction in multilayer material can be simplified as a 1-D model. For transient heat conduction in 1-D multilayer materials, Eq. (5) assumes the following form: qt qt þ s q 2 T i qt 2 ¼ a q 2 T i qx 2 ð6þ where i ¼ ; 1; 2 and denotes the different materials shown in Figure 1. This problem is solved subject to initial conditions, Tð; xþ ¼T ð7aþ qt qt ¼ ð7bþ t! and the boundary condition at x ¼ x (the pulsed incident energy is observed by the surface of layer and released into layer 1 and 2), X qt k i qq qx ¼ q þ s ð8aþ x¼x qt and q ¼ q s hðtþ hðt t p Þ ð8bþ where q s represents the pulse intensity of the heat flux and t P is the time duration of the pulse. Note that the quantity hðtþ in Eq. (8b) is the unit-step function defined by hðtþ ¼ for t ð8cþ 1 for t The boundary condition given in Eq. (8a) implies that at the inner location x ¼ x, the medium is subjected to a pulsed heat flux having a quadratic waveform, as shown

6 638 J. LI ET AL. Figure 2. Quadratic waveform heat flux at x ¼ x. in Figure 2. The boundary condition at the interface of two dissimilar materials x ¼ x 1 is Tj x¼x1 ¼ Tj x¼x1þ ð8dþ This boundary condition assumes a perfect thermal contact at the interface (e.g., using a sputtering method in microelectronic technology), as depicted by Frankel et al. [24]. The boundary condition at infinity is given by at x!1 T ¼ T ð8eþ If the following dimensionless variables are introduced, T T h ¼ k pffiffiffiffiffiffiffiffi q s a s e i ¼ s i s n ¼ t g ¼ p ffiffiffiffiffiffiffiffi x s a s r i ¼ a i a k i ¼ k i k / ¼ q q s Eqs. (6), (7), and (8) can be expressed in terms of these dimensionless quantities as qh qn þ e q 2 h i qn 2 ¼ r q 2 h i qg 2 ð9þ n ¼ hð; gþ ¼ qh qn ð1þ n! ¼ g ¼ x q/ p ffiffiffiffiffiffiffiffi / þ e i a s qn ¼ X qh k i qg / ¼ hðs nþ hðs n s n p Þ ð11aþ g 1 ¼ x 1 pffiffiffiffiffiffiffiffi hj a s g1 ¼ hj g1þ ð11bþ g!1 hðn; 1Þ ¼ ð11cþ

7 HEAT CONDUCTION IN MULTILAYER MATERIALS 639 Equations (9) (11) are the governing equations for the problem under consideration (refer to Figure 1). NUMERICAL SOLUTION In the present work, a numerical solution with a central difference scheme for the space variable and a fully implicit difference scheme is developed for the initialvalue problem. Thus, the finite-difference expression for Eq. (9) becomes h nþ1 j h n j Dn h nþ1 j þ e i 2h n j þ h n 1 j Dn 2 nþ1 hjþ1 2hnþ1 j þ h nþ1 j 1 ¼ r i Dg 2 p where j denotes the number of nodes and at g ¼ x = ffiffiffiffiffiffiffiffi a s, j ¼ 1 (here assuming x ¼ for convenience). Equation (12) can be rewritten as ð12þ a nþ1 j h nþ1 j ¼ a nþ1 jþ1 hnþ1 jþ1 þ anþ1 j 1 hnþ1 j 1 þ an j hn j þ a n 1 j h n 1 j ð13aþ with a nþ1 j ¼ 1 Dn þ e i Dn 2 þ 2r i Dg 2 ð13bþ a nþ1 jþ1 ¼ anþ1 j 1 ¼ r i Dg 2 ð13cþ a n j ¼ 1 Dn þ 2e i Dn 2 ð13dþ a n 1 j ¼ e i Dn 2 ð13eþ where a nþ1 j > ; a nþ1 jþ1 > ; an j > ; and a n 1 j < : The finite-difference equations given in Eqs. (13a) (13e) can be solved using a tri-diagonal matrix algorithm methodology (TDMA). The term h n 1 j will not appear in Eq. (13a) for the parabolic heat conduction equation in which all other coefficients are positive. Resolving the Boundary Conditions It is easy to solve the boundary condition given by Eq. (11) in finite difference form if q/=qn is finite. However, the pulse wave as shown in Figure 2 assumes a quadratic form where q/=qn is infinite at n ¼ and n ¼ t p =s. There are two methods to resolve this difficulty: (1) an approximate treatment, i.e., the quadratic wave is regarded as a very steep trapezoidal wave (the accuracy of the approximation depends on the choice of Dn); and (2) mathematical manipulation, i.e., transform the indefinite partial derivative into a solvable definite derivative [19, 28]. Because

8 64 J. LI ET AL. the second method is not convenient for complex situations and for numerical solutions, the first method is used here. Physically Realistic Solution of the Difference Scheme It is well known that the implicit difference scheme for parabolic heat conduction is unconditionally stable and convergent for the analytical results. Solving the first-order hyperbolic equation with the explicit difference scheme where the Courant number is less than unity, a stability condition exists [14, 16, 19, 29]. Based on the numerical results obtained in the current work, it is clear that the implicit difference scheme used in the numerical solution of the second-order hyperbolic equation is convergent, but may not be physically realistic depending on the time and space increments used. The principal reason for this undesirable behavior is due to the h n 1 j term in Eq. (13a). It is this term, h n 1 j, that causes the fictitious numerical oscillations under pulsed heating boundary conditions. In order to verify whether a solution of the difference scheme is a physically realistic solution of the governing equations, and to determine the most suitable values for the time and space increment for the numerical simulation, a simple case was examined numerically and compared with the analytical solution. For this purpose, consider a high-energy pulsed heat source, placed at the wall of a semiindefinite plate as shown in Figure 3. The mathematical model for transient heat conduction under pulsed heating is n ¼ qh qn þ q2 h qn 2 ¼ q2 h qg 2 hð; gþ ¼ ð14þ qh qn ð15þ n! ¼ g ¼ / þ q/ qn ¼ qh qg / ¼ hðs n Þ h s n s n p ð16aþ g!þ1 hn; ð þ1þ ¼ ð16bþ Figure 3. Schematical model for semi-indefinite plate.

9 HEAT CONDUCTION IN MULTILAYER MATERIALS 641 Equation (14), subject to the initial and boundary conditions (15) and (16a) (16b), has an explicit analytical solution given by Xu [2] (the main procedure for the analytical solution is shown in Appendix A): 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 Tðx; tþ T pffiffiffiffiffiffiffiffi t q s a s=k ¼ h x e t=2s t 2 x >< I 2 2s 4as >= C R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 t s e u=2s u 2 x >: I 2 2s 4as du >; 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 h t t p x e ðt tpþ=2s t t I p 2 x >< 2 2s 4as >= C R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t tp e u=2s u 2 x >: I 2 du >; þ 1 s 2s 4as ð17þ In the following paragraphs, the accuracy of the difference scheme proposed here for the numerical solution of the second-order hyperbolic differential equation given by Eq. (14) is presented and discussed. The detailed Von Neumann stability analysis of Eq. (13a) is given in Appendix B and illustrates that the implicit difference scheme given by Eq. (13a) is unconditionally stable for all Dn and Dg. However, it was also found that there existed fictitious oscillations if the step sizes were not chosen properly. In Figure 4 the solid line is the analytical solution for x! þ, which is calculated from the expression given by Eq. (17), while the dashed lines are the numerical solutions for Dg ¼ :476 and Dg ¼ :476 with the same value of Dn ¼ :1. After pulsed heating there is a temperature jump, p the ffiffiffiffiffiffiffiffi quantity of which can be determined from lim t! þ DTð; tþ!ðq s a sþ=k. At the end of the pulsed heating t ¼ t p, there is a temperature fall, the quantity of which is the same as the temperature jump when t! þ [2]. Because the quadratic wave is approximated as a very steep trapezoid, the choice of the time increment is important. From the calculated results, it is apparent that the smaller the time increment, the more accurate are the results. This calculation, however, requires considerably more computational time. The value of the space step also strongly influences the accuracy of the numerical results. By analyzing the partial differential equation for the hyperbolic heat conduction equation given in Eq. (14) with the boundary conditions in Eqs. (15) and (16a) (16b), and as pointed out as Chen and Lin [14], the primary difficulty encountered in the numerical solution of the hyperbolic heat conduction equation is numerical oscillations in the vicinity of sharp discontinuities, which is physically unrealistic. After many calculations, it was found that when a nþ1 jþ1 and anþ1 j 1 are larger than a n 1 j, the scheme shown in Eq. (13a) becomes stable and accurate. One possible reason may be due to a damping of the negative effect of a n 1 j on the stability of the scheme when ðr i =Dg 2 Þ > ðe i =Dn 2 Þ (here a nþ1 jþ1 ¼ anþ1 j 1 > a n 1 j ). However, it is difficult to obtain a rigorous proof on the stability of the problem with the complicated initial conditions and time-dependent boundary conditions, and is therefore beyond the scope of this article. From a comparison of different calculations shown in Figure 4, it is apparent that when ðr i =Dg 2 Þ > ðe i =Dn 2 Þ (here e and r are unity), the fictitious oscillation has been depressed and the numerical result agrees well with the analytical

10 642 J. LI ET AL. Figure 4. Comparison between the analytical result and the various numerical results. solution. As shown in Figure 4, the numerical results approach the analytical solution more accurately when the space increment, Dg, is changed from.476 to.476. No such problem exists in solving the parabolic heat conduction equation, because the term a n 1 j h n 1 j does not exist in the fully implicit difference scheme. In the following section, the problem proposed as shown in Figure 1 is examined further. First, the results of the numerical calculation with the hyperbolic heat conduction are presented, then the numerical results from the parabolic heat conduction with similar boundary and initial conditions are compared. RESULTS AND DISCUSSION In the present study, the multilayer material is composed of 3-nm platinum film (), cooling water (1), and quartz glass substrate (2). The numbers in parentheses represent the position in the material as labeled in Figure 1. The pulsed heating

11 HEAT CONDUCTION IN MULTILAYER MATERIALS 643 Table 1. Thermodynamic properties of the materials at atmospheric (25 C) sðsþ a kðw=m=kþ qðkg=m 3 Þ C p ðkj=kg KÞ Platinum 6: b , Water 1: 1 12a Glass 1: 1 1a ,19.74 a Estimated from Eq. (2) and [5, 9]. b Adopted from [28]. source is a quadratic pulse, q s ¼ W=cm 2 and t p ¼ 2: s. Some thermodynamic properties of the different materials at atmospheric conditions (25 C) are given in Table 1. Case1: Hyperbolic Heat Conduction in Multilayer Materials The heat wave propagation in the multilayer material with an inner heat source was calculated based on Eq. (13a) with a time increment of Dn ¼ :1 and a distance increment of Dg ¼ :476; the results of which are shown in Figure 4. The numerical results are also presented in Figure 5, where the two vertical dashed lines denote the thickness of the metal film (). Figure 5 illustrates the temperature distribution in the multilayer material for different times from n ¼ :1 ton ¼ 2: It is apparent that in the quartz glass with a larger thermal delay time, s, the thermal wave transmission will appear, and for water, which has a much smaller thermal delay time, s, the heat transmission will be dominated by Fourier s law. These results suggest that in materials with larger thermal delay times, it will be easier to observe the thermal wave propagation phenomena. Case 2: Parabolic Heat Conduction in Multilayer Materials Figure 6 illustrates the results of temperature variations calculated from the parabolic heat conduction equation under the same boundary conditions with different times ranging from n ¼ :1 ton ¼ 2. From Figures 5 and 6 we can see that the numerical results based on two different governing equations are different. At the beginning of the heating, the temperature variation from hyperbolic heat conduction is much more violent than that from the parabolic heat conduction equation as shown in Figures 5a and 6a, respectively. Just after the pulsed heating, when n ¼ 1, there is a temperature variation in the glass substrate with the parabolic heat conduction equation as shown in Figure 6b. With the hyperbolic heat conduction, the thermal wave transmits to only half of the depth of the metal film as shown in Figure 5b and until n ¼ 3 the thermal wave does not propagate into the glass substrate as shown in Figure 5c. From n ¼ 3ton¼2 as shown in Figures 5c 5f and Figures 6c 6f, the temperature distribution in water is almost the same for the two different kinds of governing equations. For small thermal delay time, when compared to the process time length, Fourier s law governs the heat conduction; but in a glass substrate the temperature distribution is totally different for the large thermal delay time in the quartz glass. With time, the developing thermal wave will propagate deeper into the glass with the hyperbolic heat conduction equation.

12 644 J. LI ET AL. Figure 5. Dimensionless temperature distribution in multilayer material with inner pulsed heating boundary calculated numerically from the hyperbolic heat conduction equation.

13 HEAT CONDUCTION IN MULTILAYER MATERIALS 645 Figure 6. Dimensionless temperature distribution in multilayer material with inner pulsed heating boundary calculated numerically from the parabolic heat conduction equation.

14 646 J. LI ET AL. Figure 7. Comparison of dimensionless temperature variation at the surface of metal film calculated from the hyperbolic heat conduction equation and the parabolic heat conduction equation. Figure 7 shows the comparison of the surface temperature variation of the p metal film at g ¼ x = ffiffiffiffiffiffiffiffi a s ðj ¼ 1Þ between the results calculated from the hyperbolic heat conduction equation and the parabolic heat conduction equation, respectively, under the same quadratic wave pulsed heating discussed above. It is clearly apparent that the heat transmission in multilayer materials as predicted by the thermal wave theory, is different from the Fourier law approach, and the temperature variation of the metal film will take on a complex and odd-shaped mode, based on thermal wave theory. Also, it is important to note that the transmission in the multilayer material is different from any single component material as shown in Figure 8. The temperature waves will propagate and superimpose with each other in multilayer material, and this interaction results in a more complex mode of temperature variation in the material. This result may be very important information for the management of the thermal issues occurring in high-energy devices and is also meaningful for the analysis of experimental data. From Figures 5 and 7, it can be found that the temperature in metal films will become uniform after n > 8: However, if the heat conduction is evaluated with the parabolic equation, such a uniform temperature distribution in metal film will be achieved just after n > 3 as shown in Figures 6 and 7. This implies that in MEMS technology, the fast transient

15 HEAT CONDUCTION IN MULTILAYER MATERIALS 647 Figure 8. Difference of dimensionless temperature variation at the surface of metal film between in a multilayer material and in a single-component material. temperature measurements obtained using thin metal film thermal sensors require longer duration times and should be larger than a value (here n > 8); otherwise, incorrect results may be obtained. The calculations with a quadratic pulse of small amplitude and long pulse duration, t p, were obtained by Li [3]. It was found that the numerical solutions based on the two governing equations are the same, hence under conventional conditions, the Fourier law can be used to describe the transient heat conduction accurately. By analyzing the hyperbolic heat conduction equation and the third kind of heating boundary, it appears that the non-fourier heat conduction behavior in materials is affected mainly by the properties of the materials and the time rate of change of the heat flux. Therefore, to observe the thermal wave propagation in transient heat conduction, the material with the largest thermal delay time, s, should be considered first and then the necessary boundary conditions formulated. From the boundary condition (8a), kðqt=qxþ q ¼ 1 þ sðqq=qtþ q ð18þ

16 648 J. LI ET AL. Table 2. Effect of boundary conditions on the appearance of the thermal wave (q s ¼ W=cm 2 at the surface of metal film and Dt ¼ 2: sþ sðsþ a ðdq=dtþ t!o þ ½sðDq=DtÞŠ=q Thermal wave phenomena Platinum 6: b Obvious Water 1: 1 12a Not obvious Glass 1: 1 1a ; Obvious a Estimated from ¼ Eq. (2) and [5, 9]. b Adopted from [28]. where the left-hand term is the dimensionless heat flux in the material. On the righthand side, the second term is the relative magnitude of the component of the thermal wave to the imposed heat flux. If sðqq=qtþ=q 1, the thermal wave can be observed in the experiment, as shown in Table 2, based on the caculations above, otherwise the transient heat conduction can be described by the diffusion equation. CONCLUSIONS A numerical method based on the implicit scheme was used to solve the hyperbolic heat conduction equation. The parameters which affect the magnitude of the thermal wave have been discussed. The numerical results from the hyperbolic conduction differential equations are compared with those results from the diffusion equation. The following conclusions are obtained. 1. A fully implicit solution scheme for a second-order hyperbolic equation may lead to physically unrealistic results, and the reliability of the results must be carefully analyzed. 2. The boundary conditions have a significant effect on the emergence of the thermal wave phenomena. Rapid increases in the rate at which the heat flux is added to the boundary surface is a necessary condition for the appearance of these thermal waves. 3. When the thermal delay time, s, is very small, the thermal wave phenomena can be observed, but are difficult to produce otherwise. 4. In a complex multilayer material with inner-pulsed heating, the thermal wave propagates differently, compared with that in a single material. This unusual behavior has not been discussed previously, and is a very important thermal issue in using high-energy equipment for heating purposes. REFERENCES 1. T. Q. Qiu and C. L. Tien, Femtosecond Laser Heating of Multi-layer Metal I. Analysis, Int. J. Heat Mass Transfer, vol. 39, no. 17, pp , J. Z. Xu, Non-Fourier Effect in Transient Heat Conduction Induced by a Pulsed Heating, Proc. China Thermophysics Congress, no , Beijing, pp. (I-77) (I-83), D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, Washington, DC, 1997.

17 HEAT CONDUCTION IN MULTILAYER MATERIALS D. W. Tang and N. Araki, On Non-Fourier Temperature Wave and Thermal Relaxation Time, Int. J. Thermophys., vol. 18, no. 2, pp , A. Bourdillon and N. X. Tan Bourdillon, High Temperature Superconductors: Processing and Science, Academic, San Diego, CA, L. Shi and A. Majumdar, Recent Developments in Microscale Temperature Measurement Techniques, in Heat Transfer and Transport Phenomena in Microscale System, pp , Begell House, New York, H. K. Park, X. Zhang, C. P. Grigoropoulos, C. C. Poon, and A. C. Tam, Transient Temperature during the Vaporization of Liquid on a Pulsed Laser-Heated Solid Surface, Trans. ASME, vol. 118, pp , A. Asai, Bubble Dynamics in Boiling under High Heat Flux Pulse Heating, ASME J. Heat Transfer, vol. 113, pp , A. Vadavarz, S. Kumar, and M. K. Moallemi, Significance on Non-Fourier Heat Wave in Microscale Conduction, ASME DSC, vol. 32, pp , P. Vernotte, Les paradoxes de la theorie continue de l equation de la chaleur, C. R. Acad. Sci., vol. 246, pp , C. Cattaneo, Sur une forme de l equation de la chaleur eliminant le paradoxe d ine propagation instantanee, C. R. Acad. Sci., vol. 247, pp , M. N. Ozisik and D. Y. Tzou, On the Wave Theory in Heat Conduction, ASME J. Heat Transfer, vol. 116, pp , H. Q. Yang, Characteristics-Based, High-Order Accurate and Nonoscillatory Numerical Method for Hyperbolic Heat Conduction, Numer. Heat Transfer B, vol. 18, pp , H. T. Chen and J. Y. Lin, Numerical Analysis for Hyperbolic Heat Conduction, Int. J. Heat Mass Transfer, vol. 36, pp , D. E. Glass, M. N. Ozisik, D. S. McRae, and B. Vick, On the Numerical Solution of Hyperbolic Heat Conduction, Numer. Heat Transfer, vol. 8, pp , D. E. Glass, M. N. Ozisik, D. S. McRae, and B. Vick, Hyperbolic Heat Conduction with Temperature-Dependent Thermal Conductivity, J. Appl. Phys., vol. 59, pp , B. Pulvirenti, A. Barletta, and E. Zanchini, Finite-Difference Solution of Hyperbolic Heat Conduction with Temperature-Dependent Properties, Numer. Heat Transfer A, vol. 34, pp , G. F. Carey and M. Tsai, Hyperbolic Heat Transfer with Reflection, Numer. Heat Transfer, vol. 5, pp , A. Kar, C. L. Chan, and J. Mazumder, Comparative Studies on Nonlinear Hyperbolic and Parabolic Heat Conduction for Various Boundary Conditions: Analytical and Numerical Solutions, ASME J. Heat Transfer, vol. 114, pp. 14 2, C. Weber, Analysis and Solution of the Ill-Posed Inverse Heat Conduction Problem, Int. J. Heat Mass Transfer, vol. 24, pp , S. H. Pulko, A. J. Wilkinson, and A. Saidane, TLM Representation of the Hyperbolic Heat Conduction Equations, Int. J. Numer. Modell., vol. 15, no. 3, pp , A. L. Koay, S. H. Pulko, and A. J. Wilkinson, Reverse Time TLM Modeling of Thermal Problems Described by the Hyperbolic Heat Conduction Equation, Numer. Heat Transfer B, vol. 44, pp , J. Li, Z. F. Zhang, and D. Y. Liu, Experimental and Theoretical Study on the Rapid Transient Nucleated Boiling Heat Transfer, Prog. in Natural Sci., vol. 11, no. 7, pp , J. I. Frankel, B. Vick, and M. N. Ozisik, General Formulation and Analysis of Hyperbolic Heat Conduction in Composite Media, Int. J. Heat and Mass Transfer, vol. 3, pp , 198.

18 65 J. LI ET AL. 25. J. I. Frankel, B. Vick, and M. N. Ozisik, Hyperbolic Heat Conduction in Composite Regions, in C. L. Tien, V. P. Carey, and J. K. Ferrell (eds.), Proc. 8th Int. Heat Transfer Conf., San Francisco, CA, vol. 2, pp , 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 Transfer, vol. 43, pp , H. T. Chen and K. C. Liu, Study of Hyperbolic Heat Conduction Problem in the Film and Substrate Composite with the Interface Resistance, Jpn. J. Appl. Phys., vol. 41, part 1, no. 1, pp , Z. Zhang and D. Y. Liu, Hyperbolic Heat Propagation in a Spherical Solid Medium under Extremely High Heating Rates, Proc. 7th AIAA=ASME Joint Thermophysics and Heat Transfer Conf., Albuquerque, NM, D. A. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, J. Li, Rapid Transient Nucleated Boiling Induced by Pulsed Laser, Ph.D thesis, Chinese Academy of Sciences, Beijing, Jan J. Tuma and R. A. Walsh, Engineering Mathematics Handbook, 4th Ed., McGraw- Hill, New York, APPENDIX A: ANALYTICAL SOLUTION GIVEN BY EQ. (17) OBTAINED FROM EQS. (14) (16) If the Laplace transform /ðx; ~ sþ ¼ R 1 /ðx; nþe sn dn is defined and applied, the Laplace transform of Eq. (14) with the initial condition given in Eq. (15) yields d 2 etðx; sþ dx 2 s a ð1 þ s sþ etðx; sþ ¼ ða-1þ where all variables are dimensional. The boundary condition, Eq. (16), becomes ~qð; sþ ¼ k d etð; sþ 1 e t ps s s eqð; sþ ¼q s dx s The solution of Eqs. (a-1) and (a-2) is ~Tðx; sþ pffiffiffiffiffiffiffiffi q s a s=k ¼ 1 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðx=cþ sðs þ 1=sÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s ssþ 1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðx=cþ sðs þ 1=sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðs þ 1=sÞ e t ps 1 þ 1 s s ða-2þ ða-3þ If we introduce the following inverse Laplace tranformation, similar to Carey and Tsai [18], 8 exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx=cþ qsðs þ 1=sÞ 9 < = r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! L 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : sðs þ 1=sÞ ; ¼ 1 e t=2s I t 2s 2 x 2 C H t x ¼ FðtÞ ða-4þ C

19 HEAT CONDUCTION IN MULTILAYER MATERIALS 651 and with [31] ( L 1 1 s exp½ ðx=cþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) Z sðs þ 1=sÞŠ t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F ðtþdt ða-5þ sðs þ 1=sÞ where I is a modified Bessel function of first kind, from Eq. (a-3), Eq. (17) is obtained. The more detailed derivation and calculations can be found in [2]. APPENDIX B: VON NEUMANN STABILITY ANALYSIS OF EQ. (13A) From Eq. (13a) and assuming the homogeneous properties in material and letting v nþ1 j ¼ h n j ; we have 8 1 Dn þ e þ 2r h nþ1 Dn 2 Dg 2 j >< r Dg hnþ1 jþ1 r Dg hnþ1 j 1 ¼ 1 Dn þ 2e h n Dn 2 j e v n ðb-1þ Dn 2 j >: v nþ1 j ¼ h n j Letting U¼ h and U v n ¼ V n e ikjh (here the problem is one-dimensional, so one of variables ik in the complex e ikjh is held constant) 1 Dn þ 1 1 e þ 2r r Dn 2 Dg B 2 Dg C AV nþ1 e ikjh B AV nþ1 e ikðjþ1þh r Dg B 2 1 AV nþ1 e ikðj 1Þh Dn ¼ þ! 2e e De Dn 2 V n e ikjh B 2 AV n e ikjh 1 Thus, after derivation and transforming, we have 1 Dn þ 1 e þð1 coskhþ 2r 1 Dn 2 2 AV nþ1 Dn þ 2e Dn The amplification factor is 1 Dn þ 1 e þð1 cos khþ 2r Dn Gðt; kþ ¼ 2 2 A Dn þ 2e Dn 2 e Dn 2 1 AV n e Dn 1 1 A ðb-2þ ðb-3þ ðb-4þ Following the expression (a-4), Gt; ð kþ ¼ cþ2b a b a A 1 ðb-5þ

20 652 J. LI ET AL. here, a ¼ 1 Dn þ e þ 1 cos kh 2 Dn ð Þ 2r Dg 2 b ¼ e Dn 2 c ¼ 1 Dn. The characteristic polynomial of the matrix Gðt; kþ is m 2 c þ 2b m þ b a a ¼ The characteristic roots of matrix Gðt; kþ are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1;2 ¼ c þ 2b ðc þ 2bÞ 2 4ab 2a Stability requires that m 1;2 1. From derivation, the following is obtained: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1;2 ¼ ð1=dnþþ2ðe=dn2 Þið1=DnÞ 1 ð8er=dg 2 Þð1 cos khþ 2½ð1=DnÞþðe=Dn 2 Þþð1 cos khþð2r=dg 2 ÞŠ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1=dnþþð2e=dn2 Þið1=DnÞ ð16er=dg 2 Þ sin 2 ðkh=2þ 1 ð2=dnþþð2e=dn 2 Þþð8r=Dg 2 Þ sin 2 ðkh=2þ p Its modulus is jmj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re al 2 þ Im aginary 2 ð1=dnþþð2e=dn 2 Þ R ¼ ð2=dnþþð2e=dn 2 Þþð8r=Dg 2 Þ sin 2 ðkh=2þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1=dnþ ð16er=dg 2 Þ sin 2 ðkh=2þ 1 I ¼ ð2=dnþþð2e=dn 2 Þþð8r=Dg 2 Þ sin 2 ðkh=2þ ðb-6þ ðb-7þ ðb-8þ ðb-9þ ðb-1þ From an order-of-magnitude analysis, it is obvious that jrj<1 and jij<1 for all Dn and Dg e and r are dimensionless properties p of the material; if it is homogeneous, these are unity. Thus, the modulus jmj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ b 2 < 1 is satisfied for all Dn and Dg. However, there is a practical limit on Dn and Dg because of truncation errors and the physically realistic solution as discussed in this article.

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