Hyperbolic Heat Conduction Equation Research Articles

A unified solution of several heat conduction models

Energy transport within a finite thin film subjected to either symmetric or asymmetric heating at the boundaries is investigated. Non-Gaussian heat sources are modeled as time-varying and spatially-decaying laser incidences. Comparison of heat transfer mechanisms based on the classical diffusion, Cattaneo–Vernotte, simplified thermomass, and dual-phase-lag models is undertaken. This study presents a single generalized analytical temperature profile in a finite thin film in terms of an infinite series, utilizing the superposition technique in conjunction with the solution structure theorems, which is found to be applicable to solutions of all the aforementioned models. The temperature solution for a particular model can be obtained directly by applying appropriate coefficients as they appear in the proposed generalized governing heat conduction equation. Through analyses and numerical examples, the time history of heat transfer behaviors due to the collision of energy from both sides of the thin film is compared among these models. It reveals that the method provides a convenient and efficient solution to the classical heat diffusion equation as well as other forms of hyperbolic heat conduction equations.

Analytical Solution of the Hyperbolic Heat Conduction Equation for a Moving Finite Medium Under the Effect of Time Dependent Laser Heat Source

This paper presents an analytical solution of the hyperbolic heat conduction equation for a moving finite medium under the effect of a time-dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by g(x,t) = I(t) (1 – R)μe−μx while the finite body has an insulated boundary. The solution is obtained by the Laplace transforms method, and the discussion of solutions for two time characteristics of heat source capacities (instantaneous and exponential) is presented. The effect of the dimensionless medium velocity on the temperature profiles is examined in detail. It is found that there exists clear phase shifts in connection with the dimensionless velocity U in the spatial temperature distributions: the temperature curves with negative U values lag behind the reference curves with zero U, while the ones with positive U values precedes the reference curves. It is also found that the phase differences are the sole products of U, with increasing U predicting larger phase differences.

Thermal Modeling and Analysis of a Thermal Barrier Coating Structure Using Non-Fourier Heat Conduction

This paper presents a numerical solution of the hyperbolic heat conduction equation in a thermal barrier coating (TBC) structure under an imposed heat flux on the exterior of the TBC. The non-Fourier heat conduction equation is used to model the heat conduction in the TBC system that predicts the heat flux and the temperature distribution. This study presents a more realistic approach to evaluate in-service performance of thin layers of TBCs typically found in hot sections of land based and aircraft gas turbine engines. In such ultrafast heat conduction systems, the orders of magnitude of the time and space dimensions are extremely short which renders the traditional Fourier conduction law, with its implicit assumption of infinite speed of thermal propagation, inaccurate. There is, therefore, the need for an advanced modeling approach for the thermal transport phenomenon taking place in microscale systems. A hyperbolic heat conduction model can be used to predict accurately the transient temperature distribution of thermal barrier structures of turbine blades. The hyperbolic heat conduction equations are solved numerically using a new numerical scheme codenamed the mean value finite volume method (MVFVM). The numerical method yields minimal numerical dissipation and dispersion errors and captures the discontinuities such as the thermal wave front in the solution with reliable accuracy. Compared with some traditional numerical methods, the MVFVM method provides the ability to model the behavior of the single phase lag thermal wave following its reflection from domain boundary surfaces. In addition, parametric studies of properties of the substrate on the temperature and the heat flux distributions in the TBC revealed that relaxation time of the substrate material, unlike the thermal diffusivity and thermal conductivity has very little effect on the transient thermal response in the TBC. The study further showed that for thin film structures subject to short time durations of heat flux, the hyperbolic model yields more realistic results than the parabolic model.

Non-Fourier Thermoelastic Analysis of an Annular Fin with Variable Convection Heat Transfer Coefficient

This paper numerically investigates the hyperbolic thermoelastic problem of an annular fin. The ambient convection heat transfer coefficient of the fin is assumed to be spatially varying. The major difficulty in dealing with such problems is the suppression of numerical oscillations in the vicinity of a jump discontinuity. An efficient numerical scheme involving hybrid application of Laplace transform and control volume method in conjunction with hyperbolic shape functions is used to solve the linear hyperbolic heat conduction equation. The transformed nodal temperatures are inverted to the physical quantities by using numerical inversion of the Laplace transform. Then the stress distributions in the annular fin are calculated subsequently. The results in the illustrated examples show that the application of hyperbolic shape functions can successfully suppress the numerical oscillations in the vicinity of jump discontinuities.

Response of functionally graded cylindrical shells under moving thermo-mechanical loads

The transient thermoelastic analysis of functionally graded (FG) cylindrical shells under moving boundary pressure and heat flux is presented. The material properties are assumed to be temperature-dependent and graded in the radial direction. The hyperbolic heat conduction equations are used to include the influence of finite heat wave speed (i.e., the non-Fourier effect). To benefit from the high accuracy and low computational efforts of the differential quadrature method (DQM) in conjunction with the effectiveness of the finite element method (FEM) in general geometry, loading and systematic boundary treatment, a combination of these methods is employed to discretize the governing equations in the spatial domain. The resulting system of differential equations is solved using Newmark's time integration scheme in the temporal domain. The presented formulation and method of solution are validated by showing their fast rate of convergence and by comparing the results, in the limit cases, with those obtained using the commercial finite element package ANSYS and some other available solutions in the literature. Then, the effects of different geometrical, material and load parameters on the transient thermoelastic behavior of the FG cylinders under moving mechanical and thermal loads are studied.

Non-Fourier heat conduction in layered composite materials with an interface crack

This paper studies an interface crack in a two-layered composite media under an applied thermal flux by using the hyperbolic heat conduction equation. The problem is solved by Laplace transform and integral equation technique. It is found that the crack tip thermal flux is infinity and the singularity is described by the transient thermal flux intensity factor. The crack tip field is expressed in terms of the thermal flux intensity factor in closed-form. The solution of the problem is presented in non-dimensional form. A characteristic length parameter and a characteristic time parameter are identified to control the time-varying behavior of the thermal flux intensity factor. The results demonstrate that the crack considerably enhance the local thermal flux and temperature distributions. It is important to consider the non-Fourier effect in studying the thermally induced cracking problem in layered composite materials.

AN EXACT SOLUTION FOR THERMAL ANALYSIS OF A CYLINDRICAL OBJECT USING A HYPERBOLIC HEAT CONDUCTION MODEL

In this paper, the hyperbolic heat conduction equation in a cylinder subjected to a special heat flux boundary condition is solved. Equations are solved by deriving the analytical solution using separation of variables method. The temperature layers and profiles of sample calculations are presented. The wavy nature of this kind of heat conduction can easily be seen in the temperature profiles. It is found that with the increasing Vernotte number, a point can get to higher temperature during the process. Also, it can be perceived from temperature profiles that it is possible that the temperature of different points of object becomes even lower than the initial temperature.

Analysis of dispersion relations of a coupled thermoelasticity problem with regard to heat flux relaxation

Dispersion relations for a coupled thermoelasticity problem including a hyperbolic heat conduction equation are derived, and their asymptotic analysis is performed. Dependences of the wave number and characteristics of the vibration damping rate on frequency are obtained and compared with similar diagrams in the classical model.

Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is generated at the base of the fin, which moves toward the tip of the fin and is reflected from the tip. The current study of various parameters on the thermal shock location shows that relaxation time has a great influence on the temperature distribution on the fin. An unsteady boundary condition in the base fin caused the shock, which is generated continuously from the base and has interacted with the other reflected thermal shocks. Results of the current study show that the hyperbolic heat conduction equation can violate the second thermodynamic law under some unsteady boundary conditions.

Tracking a sharp crested wave front in hyperbolic heat transfer

In this article we investigate the numerical oscillations encountered when approximating the solution to the hyperbolic heat conduction equation. We consider a benchmark problem and show that it is not well-posed, unless a jump condition is specified. The alternative is to “smooth” the jump which leads to a sharp crested wave front, but with no discontinuity. To track the wave front we split the problem into auxiliary problems and solve these using different methods. The resulting solution is oscillation-free.

A finite element/finite difference scheme for the non-classical heat conduction and associated thermal stresses

This paper develops a finite element code based on the hyperbolic heat conduction equation including the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for time. The time-related responses are obtained by solving the system of differential equations via finite difference technique. A relationship for the time step length and the element size was obtained to ensure that numerical oscillation in temperature be suppressed. Temperature-dependent material properties are taken into account in the proposed analysis model. In addition to the temperature field, the thermal stresses are also obtained from the developed method. The thermal stresses associated with the non-classical heat conduction are found to be considerably difference from those associated with classical heat conduction.

Application of solution structure theorem to non-Fourier heat conduction problems: Analytical approach

This study presents an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction (HHC) equation can be obtained using fundamental mathematics. The method is applied to investigate two one-dimensional HHC problems with different boundary and initial conditions to study the effect of spatially-decaying and time-varying laser incidence at the surface. The methodology provides a convenient, accurate, and efficient solution to the HHC equation, which is applicable to a variety of HHC analyses for various engineering applications. The examples presented in this study can be used as benchmark problems for future numerical method validations.

METHOD OF LINES AND FINITE DIFFERENCE SCHEMES WITH THE EXACT SPECTRUM FOR SOLUTION THE HYPERBOLIC HEAT CONDUCTION EQUATION

This paper is concerning with the 1-D initial–boundary value problem for the hyperbolic heat conduction equation. Numerical solutions are obtained using two discretizations methods – the finite difference scheme (FDS) and the difference scheme with the exact spectrum (FDSES). Hyperbolic heat conduction problem with boundary conditions of the third kind is solved by the spectral method. Method of lines and the Fourier method are considered for the time discretization. Finite difference schemes with central difference and exact spectrum are analyzed. A novel method for solving the discrete spectral problem is used. Special matrix with orthonormal eigenvectors is formed. Numerical results are obtained for steel quenching problem in the plate and in the sphere with holes. The hyperbolic heat conduction problem in the sphere with holes is reduced to the problem in the plate. Some examples and numerical results for two typical problems related to hyperbolic heat conduction equation are presented.

Heat diffusion vs. wave propagation in solids subjected to exponentially-decaying heat source: Analytical solution

The classical parabolic heat conduction equation based on Fourier’s law of heat conduction and the hyperbolic heat conduction equation based on the Cattaneo–Vernotte (C–V) constitutive relation are compared in this study. The present investigation considers the effect of thermal diffusion and wave propagation in solids subjected to a time-varying and spatially-decaying laser irradiation. The incoming energy is simulated as internal heat generation inside the medium. Temperature profiles are presented as closed-form series solutions for both models by using the method of superposition and the solution structure theorem. Results demonstrate that temperatures, based on the classical diffusion theory, are under-predicted at small and moderate times as compared to the wave model under high heat flux and rapid change in temperature situations. The C–V model predicts a higher peak temperature inside the medium and forms a small wave beneath the incident surface due to the lagging behavior in energy transport when compared to the diffusion model at extremely short time periods. The diffusion model may show an overall smaller temperature in the beginning; however, it over-predicts the peak temperature as compared to the results with the wave model at large times.

Entropy analyses for hyperbolic heat conduction based on the thermomass model

This paper is divided into three major sections with the first one introducing the concept of generalized entropy in extended irreversible thermodynamics briefly, that is, the entropy of a non-equilibrium system depend not only on the classical variables but also on the dissipative fluxes, which makes the hyperbolic equation of heat conduction based on the Cattaneo–Vernotte model compatible with the second law of thermodynamics. The second section deals with the hyperbolic heat conduction based on the thermomass model. According to the Einstein’s mass-energy relation, the phonon gas in dielectrics can be viewed as a kind of weighty compressible fluid, and the momentum equation of the phonon (thermomass) gas in the dielectrics, which consists of the driving force, inertia and resistance of phonon (thermomass) gas, is just the damped thermal wave equation. In the third section our analyses show that the contribution of the kinetic energy of the phonon gas in the expression of extended entropy based on the thermomass model is identical with that of the heat flux in the expression of generalized entropy in extended irreversible thermodynamics. It implies that the hyperbolic heat conduction based on the thermomass model is compatible with the second law of thermodynamics.

Analytical solution of hyperbolic heat conduction equation in relation to laser short-pulse heating

In the present study, the hyperbolic heat conduction equation is derived from the Boltzmann transport equation and the analytical solution of the resulting equation appropriate to the laser short-pulse heating of a solid surface is presented. The time exponentially decaying pulse is incorporated as a volumetric heat source in the hyperbolic equation to account for the absorption of the incident laser energy. The Fourier transformation is used to simplify the hyperbolic equation and the analytical solution of the simplified equation is obtained using the Laplace transformation method. Temperature distribution in space and time are computed in steel for two laser pulse parameters. It is found that internal energy gain from the irradiated field, due to the presence of the volumetric heat source in the hyperbolic equation, results in rapid rise of temperature in the surface region during the early heating period. In addition, temperature decay is gradual in the surface region and as the depth below the surface increases beyond the absorption depth, temperature decay becomes sharp.

Analysis of the Thermal Response in Micromachine under the Thermal Shock

The thermal response for given micromachine with the boundary surface exposed to sudden temperature change is studied by deriving an analytical solution of the hyperbolic heat conduction equation. Using the obtained analytical expression, the temperature profiles at the outer surface and interior of the micro beam are evaluated for various thermal relaxation times. The behaviors of hyperbolic heat propagation in micro beam are analyzed and possible anomalies are discussed by comparing the thermal behaviors of Fourier heat conduction.

Modelling of heat processes during the surface treatment of inhomogeneous metal bodies by a high-temperature moving pulse source

A mathematical model capable of describing the thermal action of concentrated energy fluxes on inhomogeneous metal bodies subjected to electromechanical treatment (EMT) is developed. The peculiarities of formulation and solution of a three-dimensional heat conduction problem under high-speed heating and cooling conditions, including the description of the spatial and time configurations of a moving pulse heat source and the account of the latent heat of phase transformations and the temperature dependence of the thermophysical characteristics of a body in specific structural areas, are considered. The obtained numerical results allowed us to study the temperature distribution mechanisms and the rate of temperature change caused by the action of the moving pulse source on the steel sample with a surface layer or covering subjected to EMT. It is shown that a consideration of the temperature dependence of thermophysical coefficients leads to quantitative and qualitative changes in the distribution of temperature and its gradients. Application of a hyperbolic heat conduction equation to the analysis of the EMT processes provides only an inessential refinement of the problem solution.

Non-Fourier 3-Omega Method for Thermal Conductivity Measurement

The 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation is presented. Mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated. Results obtained from Fourier and hyperbolic heat conduction equations based on parameterized dielectric thermophysical parameters are compared. The analysis is applied to a thin Chemical vapor deposited (CVD) diamond film and bologna meat and it is shown in these cases that by neglecting the non-Fourier effects one can underestimate the thermal conductivity by 50%.

A finite element method for non-Fourier heat conduction in strong thermal shock environments

Non-Fourier effect is important in heat conduction in strong thermal environments. Currently, generally-purposed commercial finite element code for non-Fourier heat conduction is not available. In this paper, we develop a finite element code based on a hyperbolic heat conduction equation, which includes the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for the time. The transient responses are obtained by solving the system of differential equations, based on the finite difference, mode superposition, or exact time integral. The code is validated by comparing the numerical results with exact solutions for some special cases. The stability analysis is conducted and it shows that the finite difference scheme is an ideal method for the transient solution of the temperature field. It is found that with mesh refining (decreasing mesh size) and/or high-order elements, the oscillation in the vicinity of sharp change vanishes, and can be essentially suppressed by the finite difference scheme. A relationship between the time step and the space length of the element was identified to ensure that numerical oscillation vanishes.