Hyperbolic Heat Conduction Equation Research Articles

Numerical simulation of non-Fourier effects in combined heat transfer

The non-Fourier effects on transient and steady temperature distribution in combined heat transfer are studied. The processes of coupled conduction and radiation heat transfer in grey, absorbing, emitting, scattering, one-dimensional medium with black boundary surfaces are analysed numerically. The hyperbolic heat conduction equation is solved by flux splitting method, and the radiative transfer equation is solved by P1 approximate method. The transient thermal responses obtained from non-Fourier heat conduction equation are compared with those obtained from the Fourier heat conduction equation. The results show that the non-Fourier effect can be important when the conduction to radiation parameter and the thermal relaxation time are larger. Further, the radiation effect is more pronounced at small values of single scattering albedo and conduction to radiation parameters. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature.

Effect of spatial variation of thermal conductivity on non-fourier heat conduction in a finite slab

The non-Fourier heat conduction problem in a finite slab is studied analytically. Dependence of thermal conductivity on space has been considered. The Laplace transform method is used to remove the time-dependent terms in the governing equation and the boundary conditions. The hyperbolic heat conduction (HHC) equation has been solved by employing trial solution method and collocation optimization criterion. Results show that the space-dependent thermal conductivity strongly affects the temperature distribution. A temperature peak on the insulated wall of the slab has been observed due to linear variation of thermal conductivity. It has been shown that the magnitude of the temperature peak increases with increasing the dimensionless relaxation time. To validate the approach, the results have been compared with the analytical solution obtained for a special case which shows a good agreement.

Frequency-modulated hyperbolic heat transport and effective thermal properties in layered systems

In this work heat transport in layered systems is analyzed using a hyperbolic heat conduction equation and considering a modulated heat source for both Dirichlet and Neumann boundary conditions. In the thermally thin case, with Dirichlet boundary condition, the well known effective thermal resistance formula is derived; while for Neumann problem only a heat capacity identity is found, due to the fact that in this case this boundary condition cannot become asymptotically steady when modulation frequency goes to zero. In contrast in the thermally thick regime, heat transport shows a strong enhancement when hyperbolic effects are considered. For this thermal regime, an analytical expression, for both Dirichlet and Neumann conditions, is obtained for the effective thermal diffusivity of the whole system in terms of the thermal properties of the individual layers. It is shown that the magnifying effects on the effective thermal diffusivity are especially remarkable when the thermalization time and the thermal relaxation time are comparable. The limits of applicability of our equation, in the thermally thick regime are shown to provide useful and simple results in the characterization of layered systems. Enhancement in thermal transport and in the effective thermal diffusivity is a direct consequence of having taken into account the fundamental role of the thermal relaxation time in addition to the thermal diffusivity and thermal effusivity of the composing layers. It is shown that our results can be reduced to the ones obtained using Fourier heat diffusion equation, when the thermal relaxation times tend to zero.

Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux

Presented is the analysis of non-Fourier effect in a hollow sphere exposed to a periodic boundary heat flux. The problem is studied by deriving an analytical solution of the hyperbolic heat conduction equation. Using the obtained analytical expression, the temperature profiles at outer and inner surfaces of the sphere are evaluated for various thermal relaxation times. By comparing the results of non-Fourier model with those obtained from Fourier heat conduction equation, the transition process from parabolic model to hyperbolic one is shown. The phase difference and amplitude ratio of boundary surfaces are calculated as functions of the thermal relaxation time and the results are depicted graphically.

A hybrid Green’s function method for the hyperbolic heat conduction problems

The present study is devoted to propose a hybrid Green’s function method to investigate the hyperbolic heat conduction problems. The difficulty of the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in the vicinity of sharp discontinuities. In the present study, we have developed a hybrid method combined the Laplace transform, Green’s function and ε-algorithm acceleration method for solving time dependent hyperbolic heat conduction equation. From one- to three-dimensional problems, six different examples have been analyzed by the present method. It is found from these examples that the present method is in agreement with the Tsai-tse Kao’s solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (1977) 343–345] and does not exhibit numerical oscillations at the wave front. The propagation of the two- and three-dimensional thermal wave becomes so complicated because it occur jump discontinuities, reflections and interactions in these numerical results of the problem and it is difficult to find the analytical solutions or the result of other study to compare with the solutions of the present method.

Simulation of thermal disturbances with finite wave speeds using a high order method

The classical heat diffusion theory based on the Fourier’s model breaks down when considering transient heat flow, for short times, extreme thermal gradients or at low temperatures. The hyperbolic heat conduction equation based on the Cattaneo model for the heat flux incorporates a relaxation mechanism in order to gradually adjust to a change in the temperature gradient. A spectral element method is applied for solving the hyperbolic system treating the heat flux as an independent variable in addition to temperature. The numerical solution is based on the time–space least squares spectral method. Numerical examples are included for discussing the effects of the thermal waves.

Integral transform solution for hyperbolic heat conduction in a finite slab

An analytical integral transformation of the thermal wave propagation problem in a finite slab is obtained through the generalized integral transform technique (GITT). The use of the GITT approach in the analysis of the hyperbolic heat conduction equation leads to a coupled system of second order ordinary differential equations in the time variable. The resulting transformed ODE system is then numerically solved by Gear's method for stiff initial value problems. Numerical results are presented for the local and average temperatures with different Biot numbers and dimensionless thermal relaxation times, permitting a critical evaluation of the technique performance. A comparison is also performed with previously reported results in the literature for special cases and with those produced through the application of the Laplace transform method (LTM), and the finite volume-Gear method (FVGM).

Hyperbolic heat-conduction equation. Mixed boundary-value problem

The integral representation of the exact solution of a mixed problem for a hyperbolic heat-conduction equation with the most general propositions within the framework of generalized thermomechanics has been obtained. The thermal state of a bounded medium with thermal effects at the boundary has been investigated numerically on a personal computer.

Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

A comparison of hyperbolic and parabolic models of phase change of a pure metal

The solidification history of individual thermal spray particles has been the subject of many experimental and theoretical studies. Yet it is customary to assume that solidification occurs at the equilibrium temperature, and that heat propagates according to Fourier’s Law. To account for a finite thermal diffusion speed, a hyperbolic heat conduction equation is usually adopted to analyze heat transfer. However, under certain circumstances, this equation can violate the second law of thermodynamics, and so others have modified the original hyperbolic equation via theories of extended irreversible thermodynamics. In this work, we study non-equilibrium effects of rapid solidification of a pure metal particle, and compare the so-called parabolic, hyperbolic and modified hyperbolic equations for heat transfer, to predict the interface undercooling due to thermal effects and velocity as a function of time, for different relaxation times. Results indicate that differences are limited to the early part of the solidification process, when undercooling is most significant, the interface velocity is highest, and non-equilibrium effects are most evident. As solidification progresses, the non-equilibrium effects wane and solidification can then be properly modeled as an equilibrium process.

A comparative study on various time integration schemes for heat wave simulation

The performance of several explicit and implicit time advancement schemes of first-order ODEs are examined for heat wave simulation with different boundary conditions. It is found that the boundary conditions have a considerable influence on the stiffness property of the hyperbolic heat conduction equation, due to the occurrence of thermal shock waves, and hence, according to the type of the enforced boundary conditions, a specific time integration scheme has to be performed in order to obtain an accurate and efficient solution. The results of the considered time integration schemes are also compared with analytical solution and based on the obtained results, some recommendations regarding the numerical simulation of hyperbolic heat conduction are presented.

Analytical Prediction of Quench Energies of Cooled Superconductors Based on the Hyperbolic Heat Conduction Model

The critical energy characteristics of cooled composite superconductors is analytically predicted based on the one-dimensional hyperbolic heat conduction model. The temperature dependence of the Ohmic heat generation, the finite speed of heat transfer, and the finite duration and finite length of the thermal disturbances are taken into account in the present model. The critical energies are calculated using a model based on the analytical solution of the hyperbolic heat conduction equation by the Laplace transformation method. The computational model results show that the critical energy depends on the relaxation time and disturbance duration. It is found that the hyperbolic conduction model predicts a lower-critical energy as compared to the predictions of the parabolic heat conduction model.

Bio-heat transfer analysis during short pulse laser irradiation of tissues

The objective of this paper is to analyze the temperature distributions and heat affected zone in skin tissue medium when irradiated with either a collimated or a focused laser beam from a short pulse laser source. Experiments are performed on multi-layer tissue phantoms simulating skin tissue with embedded inhomogeneities simulating subsurface tumors and as well as on freshly excised mouse skin tissue samples. Two types of lasers have been used in this study – namely a Q-switched pulsed 1064 nm Nd:YAG short pulse laser having a pulse width of 200 ns and a 1552 nm diode short pulsed laser having a pulse width of 1.3 ps. Experimental measurements of axial and radial temperature distribution in the tissue medium are compared with the numerical modeling results. For numerical modeling, the transient radiative transport equation is first solved using a discrete ordinates method for obtaining the intensity distribution and radiative heat flux inside the tissue medium. Then the temperature distribution is obtained by coupling the bio-heat transfer equation with either hyperbolic non-Fourier or parabolic Fourier heat conduction model. The hyperbolic heat conduction equation is solved using MacCormack’s scheme with error terms correction. It is observed that experimentally measured temperature distribution is in good agreement with that predicted by hyperbolic heat conduction model. The experimental measurements demonstrate that converging laser beam focused directly at the subsurface location can produce desired high temperature at that location compared to that produced by collimated laser beam for the same laser parameters. Finally the ablated tissue removal is characterized using histological studies as a function of laser parameters.

Axisymmetric non-Fourier temperature field in a hollow sphere

The non-Fourier axisymmetric (2+1)-dimensional temperature field within a hollow sphere is analytically investigated by the solution of the well-known Cattaneo–Vernotte hyperbolic heat conduction equation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The method of solution is the standard separation of variables method. General linear time-independent boundary conditions are considered. Ultimately, the presented solution is applied to a (1+1)—as well as a (2+1)—dimensional problem, and their respective non-Fourier thermal behavior is studied. The present solution can be reduced to special cases of interest by choosing appropriate boundary conditions parameters.

Two-dimensional non-Fourier heat conduction with arbitrary initial and periodic boundary conditions

In this paper, the (2+1)-dimensional hyperbolic heat conduction equation is analytically solved under the influence of arbitrary initial conditions for a rectangular plate with homogeneous boundary conditions of first-kind. The temperature field is obtained as a double Fourier series. The presented solution is valid even for discontinuous but integrable initial conditions. Afterwards, the solution is generalized by means of a transformation to cover problems with inhomogeneous first-kind boundary conditions. Another interesting issue is that the obtained solution can be considered as a solution to the Klein–Gordon equation under the influence of arbitrary initial conditions by means of a simple transformation.

Nonlinear mathematical model for intensive steel quenching and its analytical solution in closed form

AbstractIn the present paper the nonlinear mathematical model for intensive steel quenching processes is considered. This model involves the hyperbolic heat conduction equation, the initial conditions and the Newton type nonlinear boundary conditions characterizing the boiling process on quenchant surface. The reduction of the original problem to the Volterra type nonlinear integral equation is the essence of offered approach. Besides, in the present work some analytical procedure for solving of obtained nonlinear equation is proposed. It is proved that this analytical procedure permits finding the unique solution of the original nonlinear model. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique

In this paper, a non-Fourier heat conduction problem is analyzed by employing newly developed theory. Application of conventional numerical schemes leads to strong oscillations of the results around discontinuities in solution domain. To overcome this difficulty the variational formulation of the Laplace-transformed hyperbolic heat conduction equation is developed. The results were used for evaluation of parameters used in approximate transformed temperature profiles. To validate the approach the results were compared with the exact analytical solution solved at special case and with an approach previously reported in the literature. Both showed a close agreement with the proposed approach.

On hyperbolic heat conduction in solids: minimum mean free path of energy carriers and a method of estimating thermal diffusivity and heat propagation characteristics

The paper gives the analytical solution to the one dimensional hyperbolic heat conduction equation in an insulated slab-shaped sample that is heated uniformly on the front face with δ or laser impulse. The solution results in a formula that enables to estimate the minimum mean free path of energy carriers in the sample to detect the second sound (i.e. the thermal wave) at the sample rear face. A method of experimental data evaluation at the second sound effect is proposed, which gives the thermal diffusivity of the sample and the parameters of heat propagation.

Numerical solution of hyperbolic heat conduction in thin surface layers

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 32 (1989) 2065–2073], 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) 343–345] 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.

Heat-Transfer Mechanism of Sample-Probe in Variable-Temperature Scanning Probe Microscope and Influence of Temperature on Tunneling Current

The macroscopical breakage and nature change of material usually originate in nanometer scale, and the nature of some materials depends on temperature strongly. So it is extremely important for the research of heat transfer in micro-scale and nanometer technology to understand and control the influence of temperature on the nature of material, and developing advanced measuring technique can also improve our knowledge on theories involved in such problem. Due to the usage of variable-temperature sample stage, utilizable range of temperature is widely extended, and thus topography images of materials changing with temperature can be observed so that the thermal properties of materials can be studied further. Accordingly, the theoretical basis of heat-transfer and the influence of temperature on tunneling current owing to temperature variation should be presented. In the view of microscale heat-transfer, this paper describes the problem that heat current transfers from surface of sample to tip of probe through layer of air by means of Boltzmann theory, which can be expressed by the hyperbolic equation of heat conduction when the time is nearly equivalent to slack time and the scale is much larger than the characteristic scale of local thermodynamic equilibrium. The mode of heat-transfer on probe is also analyzed and the analytical solution is obtained in the paper. In addition, the influence of temperature increment of sample on tunneling current is discussed in detail and the change trend of tunneling current with temperature is also obtained in a limited temperature range. Due to such factors, which are possible to disturb the topography images of sample, there is no doubt that many difficulties will be brought to developing new technology such as detecting displacement and strain of materials at microscale by using the images of sample heated and unheated. To understand and control such factors has important advantage for making progress in the advanced scientific fields.