non-Fourier Heat Conduction Equation Research Articles

Influence of Circular through Hole in Pt-Rh Bushing on Temperature Propagation at High Temperature.

In the fiberglass industry, Pt-Rh bushings made of platinum and rhodium have very good characteristics, such as high temperature resistance, corrosion resistance, oxidation resistance, and creep resistance. In this paper, a semi-infinite lath structure model is constructed, and the expression of the surface temperature distribution of a Pt-Rh alloy plate with a circular through hole is obtained based on the non-Fourier heat conduction equation, complex function method and conformal mapping method. At the same time, the influence of the position of the circular through hole in the Pt-Rh bushing and the parameters of the incident light source (Non-diffusion incident wave number and relative thermal diffusion length) on the surface temperature distribution of the Pt-Rh bushing is studied by using this formula. It is found that: 1. heat concentration and fracture are occur easily at the through hole; 2. when the through hole is in the asymmetric center, the greater the asymmetry, the smaller the maximum temperature amplitude; 3. when the buried depth of the through hole increases, the maximum temperature amplitude decreases; 4. when the incident wave number and the relative thermal diffusion length of the incident light source are larger, the maximum temperature amplitude is smaller. The numerical results are almost consistent with those of ANSYS thermal simulation. The expression of the surface temperature distribution of the semi-infinite lath structure proposed in this paper can effectively reduce the loss of precious metal materials and the time of thermal simulation in the experimental process, as well as provide important significance for structural design, quality inspection, process optimization, and service life improvement of Pt-Rh bushings.

Thermal Wave Scattering by an Elliptic Subsurface Hole Buried in a Block, Based on the Non-Fourier Equation.

With the application to engineering practice, the study of the scattering of thermal waves using innovative and comprehensive methods is becoming increasingly important. The thermal wave scattering by an elliptic subsurface hole in a block with two boundaries is discussed based on the non-Fourier heat conduction equation, employing the complex function method and the conformal mapping method, and a general solution for the thermal wave scattering is given. The numerical results of temperature distributions around a subsurface hole are presented and the effects of geometrical and physical parameters on the temperature distributions were analyzed. The wave number, the shape and position of the hole, the scale of the block, and the frequency of the heat load were found to have great effects on distributions and variations of temperature. The findings of this study could be helpful in providing better understandings of infrared thermal wave imaging, the physical inverse problem, and the evaluation of internal holes in materials.

Effect of hyperbolic heat conduction on the linear and nonlinear vibration of CNT reinforced size-dependent functionally graded microbeams

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton–Raphson method is used to solve the hyperbolic (non-Fourier) heat conduction equations to obtain temperature, displacements and nonlinear frequency in the functionally graded (FG) nanocomposite Timoshenko microbeam. To do so, we need to follow two steps: (1): solving the hyperbolic heat conduction to obtain the temperature in the spatial and temporal domains by using DQM and Newton–Raphson method; (2): implementation of the obtained temperature in thermoelastic equations of microbeam to obtain displacements and frequency at each time step by direct iterative method. The material length scale parameter is introduced in the non-classical Timoshenko beam model, to interpret the size effect in microstructures. The material properties of the FG nanocomposite beam are estimated using the Eshelby-Mori-Tanaka approach and carbon nanotubes (CNTs) are randomly distributed within the composite. The nonlinear governing equations and boundary conditions are derived using the Hamilton principle and von Kármán geometric nonlinearity. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the beams. All material properties such as Young modulus (E), heat capacity (Cp), relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of temperature and CNT volume fraction. The effects of temperature change, thermal conductivity, CNTs volume fraction, length to span ratio, heat wave speed, heat flux, and end support conditions on the nonlinear vibration of the beam are discussed in detail. Unlike all previous publications, the present results show that increasing thickness-to-length scale ratio (h/l) will increase the frequency.

Hyperbolic heat conduction and thermoelastic solution of functionally graded CNT reinforced cylindrical panel subjected to heat pulse

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton Raphson method is used to solve the non-Fourier heat conduction equations to obtain temperature, displacements, and stresses in a cylindrical panel made of functionally graded, carbon nanotubes (CNTs) reinforced composite. The heat conduction of a domain made of nanocomposites subjected to heat generation is simulated with a finite heat wave speed. Furthermore, based on the three-dimensional elasticity theory, the thermoelastic analysis of the nanocomposite cylindrical panel subjected to the transient heat conduction is presented. The dynamic Young's modulus of Single-Walled Carbon Nanotubes (SWCNT) can be expressed as a function of loading rate and environmental temperature. All material properties such as heat capacity (Cp), thermal relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of both temperature and CNT volume fraction. The hyperbolic heat conduction is solved to obtain temperature in the spatial and temporal domains. Then by implementing the obtained temperature in thermoelastic equations of the cylindrical panel, the displacements and stresses will be obtained at each time step. The proposed method marches in the time direction block by block. In each block, there are several time levels, and the numerical results at these time levels are obtained simultaneously. Through this way, the numerical solution at the (n + 1)th time level depends on the solutions at its previous levels from the 1st to the nth levels. The final results in the temporal domain are obtained using the Newton–Raphson method. Accuracy of the present solution is confirmed by comparing with some available results in the literature. A detailed numerical study is conducted to examine the effects of heat wave speed and heat flux, CNT volume fraction and the geometrical parameters on the deflection of the cylindrical panel.

Analytical solution of dual-phase-lag heat conduction in a finite medium subjected to a moving heat source

In the present study, the dual-phase-lag heat conduction model was employed to describe the thermal behavior of a square plate which is heated by a moving temporally non-Gaussian heat source. The Green's function approach was utilized to develop a general solution for the non-Fourier heat conduction equation in a two-dimensional finite medium. The temperature responses of the medium obtained from Fourier's law, hyperbolic heat conduction model and dual-phase-lag heat conduction model were compared to show the influences of the phase lag parameters. In addition, the effect of scanning speed of heat source on the temperature distribution was discussed. The vector diagrams of heat flux were illustrated to show its propagation.

Investigation of the Effects of Non-Linear and Non-Homogeneous Non-Fourier Heat Conduction Equations on Temperature Distribution in a Semi-Infinite Body

In this paper, the non-Fourier heat conduction in a semi-infinite body was examined. The heat wave non-Fourier heat conduction model was used for thermal analysis. Thermal conductivity was assumed temperature-dependent which resulted in a non-linear equation. The heat source was also considered temperature-dependent which resulted in a non-homogeneous equation. The Mac-Cormack predictor-corrector numerical method was employed to solve the equations. It was concluded that, the non-linear analysis of the non-Fourier heat transfer problems is of great importance. Also, the case which assumed a temperature-dependent heat source had a considerable difference with the case where a constant heat source was assumed.

A novel SPH method for the solution of Dual-Phase-Lag model with temperature-jump boundary condition in nanoscale

This paper proposes a newly developed smoothed-particle hydrodynamics (SPH) method for the solution of one-dimensional heat conduction problem within a nanoscale thin slab for Knudsen numbers of 0.1 and 1 under the effect of Dual-Phase-Lag (DPL) model. A novel temperature-jump boundary condition is applied to the Lagrangian particle-based mesh-free SPH method in order to take into account the boundary phonon scattering phenomenon in the micro- and nano-scales. The formulation and discretization of the non-Fourier DPL heat conduction equation containing a third-order combined spatial-time derivative together with a temperature-jump boundary condition are presented and then a proper nanoscale time-stepping of the SPH method has been introduced. The dimensionless temperature and heat flux distributions have shown a good agreement with the existing numerical and analytical data for different dimensionless times, temperature to heat flux phase-lag ratios, and the Knudsen numbers. It is found that the developed SPH method have accurately simulated the complex behavior of the DPL model with relatively low computational cost.

Non-Fourier two-temperature heat conduction model used to analyze ultrashort-pulse laser processing of nanoscale metal film.

This paper utilizes non-Fourier two-temperature heat conduction model to investigate the temperature field in nanometer-sized thin films irradiated by an ultrashort-pulse laser. Ultrashort-pulse laser processing for nanometer-sized devices is usually applied in MEMS and nanotechnology. For ultrashort-pulse laser interaction with metals, the two-temperature model was proposed to describe the heat transport in metals due to a substantial nonequilibrium between the electron and lattice temperature. For heat conduction in nanoscale devices, Fourier law is inadequate for describing the heat conduction in nanoscale due to the boundary scattering and the finite relaxation time of heat carriers. Therefore, in this work, the Non-Fourier two-temperature heat conduction model used to analyze ultrashort-pulse laser processing of nanoscale metal film. The result obtained from non-Fourier heat conduction equations is compared with the available experimental data. The parametric effects are also discussed.

Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

Temperature distributions at the surface of functionally graded materials containing a cylindrical defect

The recognition of subsurface defects is an important aspect of inverse heat transfer problems. The solution of direct problems is the basis of the inverse problems. In this paper, based on the non-Fourier heat conduction equation and using the image method and the wave function expansion method, an analytical method to study the multiple scattering of thermal waves and temperatures from the subsurface defect in semi-infinite functionally graded materials (FGMs) is presented. A general solution of the scattered fields of thermal waves is obtained. Without loss of generality, the incidence of thermal waves excited by the periodically modulated laser and the subsurface defect are treated as the line heat source and a cylindrical cavity with thermal insulation in the modelling, respectively. The amplitude of the changes in temperature at the frontal surface in the cylindrical defect due to different physical and geometrical parameters is analysed, discussed, and presented graphically using the analytical method. The significant changes in the temperature were found to occur at the frontal surface because of the existence of conducting defects. In addition, the influence of the defect state on the temperature was found to also be related to the nonhomogeneous parameters of FGMs, the wavenumber, the frequency of the thermal waves, and the depth of the buried defects. The numerical results and method proposed in this paper are expected to be applied for the analysis of infrared thermal waves in non-destructive evaluations of functionally graded materials and inverse problems of mathematical physics.

Current Research Progress in Non-Classical Fourier Heat Conduction

Classical Fourier law can accurately describe most heat conduction problems. But for ultrafast heat conduction process and micro/nanoscale heat conduction problems, non-classical Fourier (non-Fourier) effect may become dominated. The paper gives a review on the current progress on non-Fourier heat conduction in engineering. It includes basic concept, physical models, thermal relaxation effect, and related experiments. Also introduced are the solution methods of non-Fourier heat conduction equations, including closed-form solution, finite difference method, finite element method, molecular dynamics simulation, variational method, and other hybrid methods. Some challenging issues are discussed at the conclusion of the paper.

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 heat conductions in nanomaterials

We study the non-Fourier heat conductions in nanomaterials based on the thermomass theory. For the transient heat conduction in a one-dimensional nanomaterial with a low-temperature step at both ends, the temperature response predicted by the present model is consistent with those by the existing theoretical models for small temperature steps. However, if the step is large, the unphysical temperature distribution under zero predicted by the other models, when two low-temperature cooling waves meet, does not appear in the predictions by the present model. The steady-state non-Fourier heat conduction equation derived by the present model has been applied to predict the effective thermal conductivities of nanomaterials. The temperature and size dependences of effective thermal conductivities of nanofilms, nanotubes, and nanowires from the present predictions agree well with the available data from experiments in the literature and our molecular dynamics simulation results, which again proves the validity of the proposed heat conduction equations. The present analysis suggests that the inertial effect of high-rate heat and the interactions between heat and surface in confined nanostructures dominate the non-Fourier heat conduction in nanomaterials.

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.

An approximate analytic solution of the heat conduction equation at nanoscale

The Adomian decomposition method (ADM) and the Adomian double decomposition method (ADDM) for solving the 3D non-Fourier heat conduction equation at nanoscale based on the dual-phase-lag framework are proposed. We show that the noise terms that appear in ADM solution can be removed, if the ADDM is employed.

Fuel rod model based on Non-Fourier heat conduction equation

In this paper we explore the applicability of a fuel rod mathematical model based on Non-Fourier transient heat conduction as constitutive law for the Light Water Reactors transient analysis (LWRs). In the classical theory of diffusion, Fourier law of heat conduction is used to describe the relation between the heat flux vector and the temperature gradient assuming that the heat propagation speeds are infinite. The motivation for this research was to eliminate the paradox of an infinite thermal wave speed. The time-dependent heat sources were considered in the fuel rod heat transfer model. The close of the Main Steam Isolated Valves (MSIV) transient in a Boiling Water Reactor (BWR) was analyzed by different relaxation times. The results show that for long-times the heat fluxes on the clad surface under Non-Fourier approach can be important, while for short-times and from the engineering point of view the changes are very small. Some results from transient calculations are examined.

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

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.

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.

Non-Fourier heat conduction in a finite medium subjected to arbitrary non-periodic surface disturbance

The non-Fourier heat conduction equation is solved for a finite medium under arbitrarily chosen surface disturbances by utilizing the principle of superposition along with the Fourier integral representation of arbitrary non-periodic functions. The obtained solution is such that for a given non-periodic disturbance, we just have to introduce the Fourier integral coefficients of that function to the presented general solution and perform some definite integrations, analytically if possible, but in general numerically which is a straightforward computational task.

Dynamic effective thermal properties of functionally graded fibrous composites using non-Fourier heat conduction

A micromechanics-based thermo-dynamical model is proposed to predict the distribution of dynamic effective thermal properties of functionally graded fibrous composites resulting from thermal waves. Generally speaking, in functionally graded materials there exist two microstructurally distinct zones: fiber-matrix zone and a transition zone. In the fiber-matrix zone, based on the non-Fourier heat conduction equation in materials, the dynamic effective thermal properties for any macroscopic material points are determined by employing the effective medium method in the corresponding microstructural representative volume element (RVE). In transition zone, a transition function is introduced to make the wave fields continuous and differentiable. Numerical examples of the dynamic effective thermal properties in the gradation direction under different parameters are graphically presented. Analysis shows that the material properties of each phase, the incident frequency, and the gradation parameter of materials have great effect on the distribution of dynamic effective properties in the gradation direction. In different material zones, the effect displays great difference. In the region of higher frequencies, the relaxation time of the two phases also has great effect the dynamic thermal properties. At last, the results are discussed in detail.