Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction

Cai, Ruixian; Gou, Chao; Li, Hongqiang

doi:10.1016/j.ijthermalsci.2005.12.007

Cited by 18 publications

(16 citation statements)

References 14 publications

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“…The initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards as the procedures presented in [24][25][26][27][28][29][30][31][32]. Some special assumptions are adopted during the derivation to accomplish variables separation.…”

Section: Discussionmentioning

confidence: 99%

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Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

et al. 2008

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This paper presents two exact explicit solutions for the three dimensional dual-phase lag (DLP) heat conduction equation, during the derivation of which the method of trial and error and the authors' previous experiences are utilized. To the authors' knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.

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Section: Discussionmentioning

confidence: 99%

“…According to the method of trial and error and the authors' experiences in deducing exact explicit solutions [26][27][28][29][30][31][32], it is first assumed that…”

Section: The First Exact Solutionmentioning

confidence: 99%

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

et al. 2008

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“…In addition, they are of theoretical significance since they correspond to physically important situations. However, in spite of developing analytical approaches dedicated to the DPL model, most solutions still available in the literature are obtained by numerical methods, and there are few exact solutions [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…A non-linear three-dimensional hyperbolic heat conduction equation is solved analytically using an algebraically explicit method in [11]. The authors expressed that the main aim of their work is to obtain some possibly explicit analytical solutions as the benchmark for computational heat transfer.…”

Section: Introductionmentioning

confidence: 99%

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Ghazanfarian

Abbassi

2012

Int J Thermophys

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Analytical and numerical solutions of the 2D transient dual-phase-lag (DPL) heat conduction equation are presented in this article. The geometry is that of a simplified metal oxide semiconductor field effect transistor with a heater placed on it. A temperature jump boundary condition is used on all boundaries in order to consider boundary phonon scattering at the micro-and nanoscale. A combination of a Laplace transformation technique and separation of variables is used to solve governing equations analytically, and a three-level finite difference scheme is employed to generate numerical results. The results are illustrated for three Knudsen numbers of 0.1, 1, and 10 at different instants of time. It is seen that the wave characteristic of the DPL model is strengthened by increasing the Knudsen number. It is found that the combination of the DPL model with the proposed mixed-type temperature boundary condition has the potential to accurately predict a 2D temperature distribution not only within the transistor itself but also in the near-boundary region.

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“…Schnaid [36] attempted to derive the governing equation for heat conduction with a finite speed of heat propagation directly from classical thermodynamics. Cai et al [37] presented algebraically explicit analytical solutions of hyperbolic type heat conduction equations in three dimensions. Lin and Chen [38] sought numerical solutions of hyperbolic heat conduction in cylindrical and spherical systems.…”

Section: Introductionmentioning

confidence: 99%

Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

Sharma

2009

Int J Thermophys

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The expression for the transient temperature during damped wave conduction and relaxation developed by Baumeister and Hamill by the method of Laplace transforms was further integrated. A Chebyshev polynomial approximation was used for the integrand with a modified Bessel composite function in space and time. A telescoping power series leads to a more useful expression for the transient temperature. By the method of relativistic transformation, the transient temperature during damped wave conduction and relaxation was developed. There are four regimes to the solution. These include: (i) a regime comprising a Bessel composite function in space and time, (ii) another regime comprising a modified Bessel composite function in space and time, (iii) the temperature solution at the wave front was also developed separately, and (iv) the fourth regime at a given location X in the medium is at times less than the inertial thermal lag time. In this regime, the temperature was found to be unchanged at the initial condition. The solution for the transient temperature from the method of relativistic transformation is compared side by side with the solution for the transient temperature from the method of Chebyshev economization. Both solutions are within 12 % of each other. For conditions close to the wave front, the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2 % of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface, the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solution for transient surface heat flux from the parabolic Fourier heart conduction model and the hyperbolic damped wave conduction and relaxation models are compared with each other. For τ > 1/2 the parabolic and hyperbolic solutions are within 10 % of each other. The parabolic model has a "blow-up" as τ → 0, and the hyperbolic model is devoid of singularities. The transient temperature from the Chebyshev economization is within an average of 25 % of the error function solution for the parabolic Fourier heat conduction model. A penetration distance beyond which there is no effect of the step change in the boundary is predicted using the relativistic transformation model. KeywordsDamped wave conduction and relaxation · Laplace transforms · Microscale heat transfer · Parabolic and hyperbolic models · Relativistic transformation Nomenclature erf (r ) Error function erf(r ) = 2 √ π r 0 e −r 2 dr H (τ ) Space integrated temperature I 0 (x) Modified Bessel function of the first kind and zeroth order I 1 (x) Modified Bessel function of the first kind and first order J 0 (x) Bessel function of the first kind and zeroth order k Thermal conductivity of semi-infinite medium (W · m −1 · K −1 ) K 0 (x) Modified Bessel function of the second kind and zeroth order mInitial temperature (K) T s Surface temperature (K) T n (r ) Chebyshev polynomial (Tables 1, 2) uBesse...

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Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction

Cited by 18 publications

References 14 publications

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

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