Hyperbolic Heat Conduction Research Articles

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

• Analytical solution of the non-Fourier bio heat transfer equation. • Finite integral transform. • Time independent/dependent boundary conditions. • Coupling of the transient RTE and non-Fourier heat conduction models. • Thermal analysis of laser-irradiated biological tissue phantoms. A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients ( τ T ) and heat flux ( τ q ) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.

Phase change heat transfer during cryosurgery of lung cancer using hyperbolic heat conduction model

The paper reports a numerical study of phase change heat transfer process in lung cancer undergoing cryosurgery. A two dimensional hyperbolic bio-heat model with non-ideal property of tissue, blood perfusion and metabolism is used to analyze the problem. The governing equations are solved by finite difference method based on enthalpy formulation. Effects of relaxation time of heat flux in hyperbolic model on freezing process have been examined. A comparative investigation of two different models (hyperbolic and parabolic bio-heat models) is also presented.

An inspection to the hyperbolic heat conduction problem in processed meat

This paper analyzes a hyperbolic heat conduction problem in processed meat with the non-homogenous initial temperature. This problem is related to an experimental study for the exploration of thermal wave behavior in biological tissue. Because the fundamental solution of the hyperbolic heat conduction model is difficult to be obtained, a modified numerical scheme is extended to solve the problem. The present results deviate from that in the literature and depict that the reliability of the experimentally measured properties presented in the literature is doubtful.

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

Generalized variational principles for heat conduction models based on Laplace transforms

The classical variational principle does not exist for parabolic and hyperbolic heat conduction equations, which has led to the demand for special variational methods for heat conduction. O’Toole (1967) first used Laplace transforms for the variational principle only for Fourier’s law with the first type of boundary condition. In this paper, the Laplace transform strategy is extended to other parabolic and hyperbolic heat conduction models and other types of boundary conditions. Generalized variational principles are given for heat conduction models including Fourier’s law, the Cattaneo–Vernotte (CV) model, the Jeffrey model, the two-temperature model and the Guyer–Krumhansl (GK) model, based on Laplace transforms. The Laplace transform method transforms the heat conduction equations of these models into linear variational equations whose variational principles are already known. For the three standard types of boundary conditions, these generalized variational principles are strictly equivalent to the heat conduction equations for these models. The Laplace transform method has stronger convergence in infinite temporal domain problems. In physics, the Laplace transform method is understood as replacing the time dimension with the frequency of the temperature change and the rate of the entropy change.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Compressible Navier–Stokes Equations with hyperbolic heat conduction

We investigate the system of compressible Navier–Stokes equations with hyperbolic heat conduction, i.e. replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time [Formula: see text], global smooth solution exists for small initial data. Moreover, as [Formula: see text] goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier–Stokes equations.

Non-Fourier effect and inertia effect analysis of a strip with an induced crack under thermal shock loading

In this paper, the transient temperature fields and the dynamic stress intensity factors of a thermo-elastic strip containing an inner crack parallel to the heated surface under thermal shock are studied. The Biot number of the crack gap, hyperbolic heat conduction theory and equation of motion are considered to investigate the behavior of the temperature fields around the crack and the stress intensity factors. Fourier transform and Laplace transform are used to reduce this mixed boundary value problem. Numerical methods are used to solved the singular integrate equations. Finally, the numerical results are presented illustrating the influence of Biot number, non-Fourier effect and inertia effect on temperature field and stress intensity factors. It is found that the Biot number strongly affect the uniformity of the temperature field and the magnitude of the stress intensity factors. The stress intensity factors have higher amplitude and an oscillating feature comparing to those obtained under conventional Fourier thermal conduction condition and quasi-static hypothesis, which can help to better understand the crack behaviors of advanced materials under thermal impact loading.

A modified stochastic perturbation method for stochastic hyperbolic heat conduction problems

The stochastic hyperbolic heat conduction problem is addressed. The random field is approximated by using the Karhunen–Loeve expansion truncated after a finite numbers of terms, and a modified stochastic perturbation method (MSPM) is proposed for the stochastic hyperbolic heat conduction problems. The MSPM is based on the stochastic perturbation theory, and can provide the mean and covariance of the random temperature including some third- and forth-order terms. Compared with the original stochastic perturbation method, the MSPM is more accurate and is more convenient for the numerical implementation, and hence is more applicable for the stochastic hyperbolic heat conduction problem. Two numerical examples demonstrate that the proposed method can provide very accurate results and can be used for the stochastic hyperbolic heat conduction problems.

Analysis of hyperbolic heat conduction in 1-D planar, cylindrical, and spherical geometry using the lattice Boltzmann method

Analyses of hyperbolic heat conduction in an 1-D planar, cylindrical, and spherical geometry are analyzed using the lattice Boltzamnn method (LBM). Finite time lag between the imposition of temperature gradient and manifestation of heat flow causes the governing energy equation to be hyperbolic one. Temporal temperature distributions are analyzed for thermal perturbation of a boundary by suddenly raising its temperature and also by imposing a constant heat flux to it. Wave-like temperature distributions in the medium are obtained when constant temperature boundary condition is used. However, when constant heat flux boundary condition is used, temperature distribution fluctuates before it becomes stable. To check the accuracy of the LBM results, the problems are also solved using the finite difference method (FDM). LBM and FDM results compare exceedingly well. LBM has computational advantage over the FDM.

Analytical study for the estimation of thermal properties of processed meat based on hyperbolic heat conduction model

This study proposes an analytical method in conjunction with existing experimental temperature to estimate the unknown relaxation time and thermal diffusivity of processed meat based on the hyperbolic heat conduction model. This analytical method is a combination of the Laplace transform and least squares methods. The thermal contact resistance at the interface between adjacent samples at different temperatures is assumed to be negligible. The relaxation time is estimated from the temperature jump at a specific measurement location. The thermal diffusivity is determined from the definition of the dimensionless spatial coordinate and the resulting relaxation time. The results show that the relaxation time and thermal diffusivity obtained are in good agreement with the existing results. The obtained dimensionless temperature history at a specific measurement location is close to the experimental temperature data. This means that the Cattaneo–Vernottee (CV) model can be suitable for this study. The proposed analytical inverse method can be applied to determine a more accurate estimate of such problems. A comparison of the estimate obtained from CV and dual phase lag models is made.

An analytical solution for the hyperbolic unsteady thermal behaviour of micro-thermoelectric coolers with a suddenly time-dependent heat generation

The ultrafast thermal behaviour of a micro thermoelectric cooler under the hyperbolic heat conduction model is analysed. An important step forward is taken in that the instantaneous application of the electric current is modelled by means of the Heaviside function. Two terms of internal heat generation derive from this model: the former is time-independent, whereas the latter is time-dependent through Dirac’s delta function. A modified Separation of Variables method is used to solve analytically the governing equations. Also, the initial linear problem is split up into three sub-problems through the principle of superposition. In order to eliminate the numerical oscillations in the temperature profiles at early times in correspondence of the wave fronts, the semi-infinite hyperbolic body model is considered. The performance (COP) of the device is evaluated and then compared with that derived from the parabolic case. An increase of about 27% is observed at early times.

Effect of multi-pulses train on the thermomechanical response of a metal thin film

ABSTRACTUltrafast thermoelasticity based on hyperbolic two-step heat conduction model with electron-phonon interaction was applied to investigate the thermomechanical coupling responses of an immense homogeneous, isotropic gold film, which was irradiated by various ultrashort laser pulse trains. The solution to the problem was obtained by solving finite element governing equations. For the same energy injection, a new micromachining technique using laser pulse trains can offer certain advantages compared with a single laser pulse micromachining, such that the laser pulse train is provided with the advantages of long and short laser pulses. The effects of the pulse number and the pulse separation times for a laser pulse train on ultrafast thermomechanical coupling responses, i.e., the temperature, displacement as well as stress, are presented graphically and analyzed. It was found from the numerical results that the expected machining results (or thermomechanical responses) may be obtained by using pulse train technology with optimal pulse number and pulse separation time.

Exact Analytical Hyperbolic Temperature Profile in a Three-Dimensional Media Under Pulse Surface Heat Flux

AbstractIn this paper three-dimensional hyperbolic heat conduction equation in a cubic media with rectangular cross-section under a pulsed heat flux on the upper side has been solved analytically using the method of separation of variables and the Duhamel integral. The closed form solution of both Fourier and non-Fourier profiles are introduced with both modes of steady and pulsed fluxes. The results show the considerable difference between the Fourier and Non-Fourier temperature profiles. Then the answer procedure is used for modeling of interaction of a cubical tissue under a short laser pulse heating. The effects of pulse duration and laser intensity are studied analytically. Furthermore the results can be applied as a verification branch for other numerical solutions or laser treatments of biological tissues.

Numerical analysis of dual-phase-lag heat transfer for a moving finite medium subjected to laser heat source

To accommodate the micro-structural effect, this work applies the dual-phase-lag (DPL) heat transfer model to explore the transient heat transfer for a moving finite medium under the effect of a time-dependent laser heat source. Laser heating is modeled as an internal heat source. A numerical scheme has been developed to overcome the mathematical difficulties in dealing with the hyperbolic heat conduction equation. Comparison between present numerical results and the analytic solutions for the non-Fourier case is made to verify the accuracy of the present numerical method. Additionally, the effects of different medium parameters, for example, moving velocity, phase lags values of the heat flux and temperature gradient, on the behavior of heat transfer have been examined. It is found that there exists clear phase shifts in the temperature distributions due to the medium moving velocity. The heat-flux phase lag tends to induce thermal waves with sharp wave-fronts in the medium, the inclusion of temperature-gradient phase lag smoothens the sharp wave-fronts by promoting conduction into the medium, resulting in non-Fourier diffusion-like conduction.

Inertia effect analysis of a half‐plane with an induced crack under thermal loading using hyperbolic heat conduction

This paper considers the crack problem for a semi‐infinite plate subjected to a thermal shock by using hyperbolic heat conduction equations and equations of motion. The Laplace transform and Fourier transform are used to reduce the problem to a system of singular integral equations which are solved numerically. Numerical results are presented illustrating the influence of non‐Fourier effect and inertia effect on the stress intensity factor and normalized crack opening displacements. The results show that the stress intensity factors have higher amplitude and an oscillating feature comparing to those obtained under the conventional Fourier thermal conduction condition and quasi‐static hypothesis. These results can help understand the crack behaviors of advanced materials under thermal impact loading.

The dilemma of hyperbolic heat conduction and its settlement by incorporating spatially nonlocal effect at nanoscale

To model transiently thermal responses of numerous thermal shock issues at nano-scale, Fourier heat conduction law is commonly extended by introducing time rate of heat flux, and comes to hyperbolic heat conduction (HHC). However, solution to HHC under Dirichlet boundary condition depicts abnormal phenomena, e.g. heat conducts from the cold to the hot, and there are two temperatures at one location. In this paper, HHC model is further perfected with the aids of spatially nonlocal effect, and the exceeding temperature as well as the discontinuity at the wave front are avoided. The effect of nonlocal parameter on temperature response is discussed. From the analysis, the importance of size effect for nano-scale heat conduction is emphasized, indicating that spatial and temporal extensions should be simultaneously made to nano-scale heat conduction. Beyond that, it is found that heat flux boundary conditions should be directly given, instead of Neumann boundary condition, which does not make sense any longer for non-classical heat conductive models. And finally, it is observed that accurate solution to such problems may be obtained using Laplace transform method, especially for the time-dependent boundary conditions, e.g. heat flux boundary condition.

Heat transfer in infinite harmonic one-dimensional crystals

A closed system of differential-difference equations describing thermal processes in one-dimensional harmonic crystals is obtained in the paper. An equation connecting the heat flow and the kinetic temperature is obtained as a solution of the system. The obtained law of heat conduction is different from Fourier’s law and results in an equation that combines properties of the standard heat equation and the wave equation. The resulting equation is an analytic consequence from the dynamical equations for the particles in the crystal. Unlike equations of hyperbolic heat conduction, this equation is time-reversible and has only one independent parameter. A general analytical solution of this differential equations is obtained, and the analytical results are confirmed by computer simulations.

Numerical study of dual-phase-lag heat conduction in a thermal barrier coating with a hybrid method

In this study, the dual-phase-lag (DPL) heat conduction model is applied to study the non-Fourier heat transfer in a thermal barrier coating (TBC) structure subjected to heat flux on the exterior of the coating. An efficient numerical scheme involving the hybrid application of the Laplace transform and control volume methods in conjunction with hyperbolic shape functions is used to solve the hyperbolic heat conduction equations in the linearized form of DPL model. The transformed nodal temperatures are inverted to the physical quantities using numerical inversion of the Laplace transform. Parametric studies of properties of the substrate and the coating on the temperature distributions in the TBC are performed. A comparison between the present study and other work in the literature using the thermal wave model is also made. The results also show that the phase lag of heat flux tends to induce thermal waves with sharp wave fronts separating heated and unheated zones in the structure, while the phase lag of temperature gradient results in non-Fourier diffusion-like conduction and smooths the sharp wave fronts by promoting conduction into the medium.

On the non-Fourier thermal fracture of an edge-cracked cylindrical bar

The fracture behavior of a cylindrical bar containing a circumferential edge crack under transient thermal loading is analyzed in this paper using the dual phase lag heat conduction model. The lateral surface of the bar is subjected to an asymmetric, convective thermal boundary condition. The Laplace transform method is adopted to solve the one-dimensional, hyperbolic heat conduction equation in the un-cracked bar, and the corresponding thermal stress is obtained in the Laplace domain under plane strain condition. Then, based on the superposition method, the axial stress with minus sign is applied on the crack surface to form a mode I crack problem in the cylindrical coordinate. Love’s potential function is introduced to solve the crack problem and a singular integral equation is obtained by using the Fourier and Hankel transform methods. Furthermore, a numerical Laplace inversion technique is employed to transform all the results to the time domain. Finally, the effect of heat conduction model, phase-lag parameters, Biot number, and the crack depth on the transient temperature, thermal axial stress, stress intensity factor, and crack opening displacement is analyzed.