Periodic heat transfer through inhomogeneous media part 2: Hollow cylinder

The paper presents exact analytical solutions for periodic radial heat conduction through an inhomogeneous hollow circular cylinder for a certain class of thermal conductivity profile. The exact analytical solutions for some of these profiles (including linear and quadratic) are compared with those obtained by considering the cylindrical medium to be made up of a number of homogeneous layers with different thermal conductivities, varying from layer to layer, and using the layered-structure (or matrix-multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. This gives confidence in the application of the layered-structure method to periodic heat conduction through an inhomogeneous hollow cylinder. Assuming the inhomogeneous hollow cylinder to be made up of a number of cylindrical layers with a linear profile of thermal conductivity has also been shown to be a more effective alternative method of considering any type of inhomogeniety; it saves computation time, as the rate of converegence is much higher than for the homogeneous-layer structure method. Numerical results are presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and the heat flux on the two sides of the cylinder.     

Periodic heat transfer through inhomogeneous media. Part 1. Slab

The paper presents exact analytical solutions of one-dimensional periodic heat conduction through an inhomogeneous slab for a certain class of thermal conductivity profiles (including linear and exponential). The exact analytical solutions for some of these profiles have been compared with those obtained by considering the slab to be made up of a number of homogeneous layers with different thermal conductivities varying from layer to layer and using the layered structure (or matrix multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. This gives confidence in the application of the layered-structure method to periodic heat conduction through inhomogeneous slabs. The numerical results have been presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and heat flux on the two sides of the slab.     

Use of linear profile layers for the evaluation of the admittance matrix of an inhomogeneous slab

It is seen that the convergence of values of the elements of the admittance matrix for periodic heat conduction through an inhomogeneous slab with a number of layers (layered structure method) is much faster when the conductivity profile in the layer is taken as linear (with least-square fit), than when the layer is considered as homogeneous.     

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.     

Temperature transients in spherical medium irradiated by laser pulse

Temperature wave solution predicted by hyperbolic heat conduction model is developed for a hollow sphere exposed to laser pulse heating. Using the obtained analytical solution, the temperature distribution, the propagation and reflection of the temperature wave due to such heat pulse is investigated for different thermal relaxation time and laser pulse duration. The effect of geometry on the temperature profile is also studied and the results of the hyperbolic and Fourier model are compared.     

A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium

Many non-equilibrium heat conduction processes can be described by the macroscopic dual-phase lag model (DPL model). In this paper, a numerical method, which combines the dual reciprocity boundary element method (DRBEM) with Laplace transforms, is constructed to solve such mathematical equation. It is used to simulate the non-Fourier phenomenon of heat conduction in a single-phase medium, then numerically predict the differences between the thermal diffusion, the thermal wave and the non-Fourier heat conduction under different boundary conditions including pulse for one- and two-dimensional problems. In order to check this numerical method's reliability, the numerical solutions are still compared with two known analytical solutions.     

The symbolic operator representation applied to the derivation of solutions of unsteady heat diffusion problems

The method of the Symbolic Operator Representation, first introduced by Lord Rayleigh in the solution of the wave equation, is applied to the unsteady heat conduction problem. The general equations for the temperature field around particle of arbitrary shape are expressed in terms of a symbolic operator Ξ Hence, the analytical solution for the temperature field around a sphere and for the rate of heat transfer from a spherical particle are obtained, when the temperature of the particle is a time-dependent function. As an example for the use of this method, analytic solution is derived in an explicit form for the case of a linear field around the spherical particle. The solutions obtained by this method are exact.     

Numerical solutions for mixed controlled solidification of phase change materials

Kinetics of solidification of phase change materials (PCM) was analyzed for combined heat conduction in the PCM and container wall, and convection in the cold fluid. Stable and convergent numerical methods were derived after transformation to normalized size scale, and corresponding immobilization of the moving boundary. The accuracy was confirmed by comparing numerical solutions and corresponding analytical solutions for control by heat conduction in solidifying layers. The proposed method was used to assess when solidification is controlled solely by conduction in solidified layer, and to analyze relative roles of conduction through the container and convection in the cold fluid.     

Application of CESE method to simulate non-Fourier heat conduction in finite medium with pulse surface heating

This study employs the space–time conservation element and solution element (CESE) method to simulate the temperature and heat flux distributions in a finite medium subject to various non-Fourier heat conduction models. The simulations consider three specific cases, namely a single phase lag (SPL) thermal wave model with a pulsed temperature condition, a SPL model with a surface heat flux input, and a dual phase lag (DPL) thermal wave model with an initial deposition of thermal energy. In every case, the thermal waves are simulated with respect to time as the thermal wave propagates through the medium with a constant velocity. In general, the simulation results are found to be in good agreement with the exact analytical solutions. Furthermore, it is shown that the CESE method yields low numerical dissipation and dispersion errors and accurately models the propagation of the wave form even in its discontinuous portions. Significantly, compared to traditional numerical schemes, the CESE method provides the ability to model the behavior of the SPL thermal wave following its reflection from the boundary surface. Further, a numerical analysis is performed to establish the CESE time step and mesh size parameters required to ensure stable solutions of the SPL and DPL thermal wave models, respectively.     

Transient Piezothermoelasticity of a Two-Layered Composite Hollow Cylinder Constructed of Isotropic Elastic and Piezoelectric Layers Due to Asymmetrical Heating

This article is concerned with the theoretical treatment of transient piezothermoelastic problem involving a two-layered hollow cylinder constructed of isotropic elastic and piezoelectric layers due to asymmetrical heat supply. The transient two-dimensional temperature is analyzed by the method of Laplace transformation. By using the exact solutions for piezoelectric hollow cylinder and isotropic hollow cylinder, the theoretical analysis of transient piezothermoelasticity is developed for a two-layered composite hollow cylinder under the state of plane strain. As an example, numerical calculations are carried out for an isotropic elastic hollow cylinder made of steel, bonded to a piezoelectric layer of cadmium selenide. Some numerical results for the temperature change, the stress and the electric potential distributions in a transient state are shown in figures. Furthermore, the influence of thickness of the piezoelectric layer or the isotropic elastic layer upon the temperature change, stresses and electric potential is investigated.     

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.     

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.     

Thermal and mechanical stresses in a functionally graded thick sphere

In this paper, a general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material is presented. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. The material properties, except Poisson's ratio, are assumed to vary along the radius r according to a power law function. The analytical solution of the heat conduction equation and the Navier equation lead to the temperature profile, radial displacement, radial stress, and hoop stress as a function of radial direction.     

Analytical investigations for heat conduction problems in anisotropic thin-layer media with embedded heat sources

This article develops the analytical rigorous solution of a fundamental problem of heat conduction in anisotropic media. The steady-state temperature and heat flux fields in a thin-layer medium with anisotropic properties subjected to concentrated embedded heat sources or prescribed temperature on the surface are analyzed. A linear coordinate transformation is used to transform anisotropic thin-layer problems into equivalent isotropic problems without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, exact closed-form solutions of the specific problems are presented in series forms. The complete solutions of heat conduction problems for the thin-layer medium consist only of the simplest solutions for an infinite homogeneous medium with concentrated heat sources. The numerical results of the temperature and heat flux distributions are provided in full-field configurations.     

蜂窝蓄热体温度特性数学解析

基于蜂窝蓄热体气固传热精确解，研究蓄热体温度变化和切换周期设计方法，忽略沿气流流动方向的固体导热影响，建立了周期传热数学模型，并求出了气固温度分布精确解。和数值计算相比，半解析解可信，按炉内低氧稳定燃烧和蓄热体低温端不结业的要求，可进行切换周期优化设计，从而为低氧弥散燃烧设计和操控优化提供一种高效、经济、准确的解析研究方法。     

Explicit full field analytic solutions for two-dimensional heat conduction problems with finite dimensions

This study presents explicit analytical solutions of heat conduction problems for isotropic media with finite dimensions. The geometry configurations considered in this study include composite layer, wedge and circular media. The boundary conditions are assumed to be either thermal isolation or isothermal. The full field analytical solutions of temperature and heat fluxes for the composite layered media subjected to an embedded heat source are derived first by Fourier transform technique in conjunction with the image method. The corresponding problems of composite wedge and circular media are constructed by conformal mapping and the solutions of composite layer media. The explicit full field solutions are expressed in simple closed-forms which can be easily used in engineering applications. The numerical calculations of the temperature and heat fluxes distributions are provided in full field configuration base on the available analytic solutions.     

Numerical analysis of unsteady heat conduction in regular solid bodies comprising natural convection to nearby fluids

The present paper addresses unsteady, unidirectional heat conduction in regular solid bodies (vertical plate, horizontal cylinder, and sphere) that exchange heat by natural convection with a neighboring fluid. From thermal physics, natural convection constitutes a worst-case scenario for forced convection cooling. Under the premises of natural convection heat transfer, the unsteady, 1-dimensional heat conduction equation consists in a linear parabolic partial differential equation with a dominant natural convection boundary condition represented by the mean convective coefficient that depends upon temperature. As expected, the nonlinear unsteady, unidirectional heat conduction problem is complex and does not admit an exact, analytical solution. Instead, the nonlinear unsteady, unidirectional heat conduction problem forcibly necessitates approximate numerical treatment with the finite difference method. The computed dimensionless center, surface, and mean temperatures varying with dimensionless time are obtained numerically and are graphed for 3 solids: iron, aluminum, copper exposed to 3 fluids: air, water, oil; the 6 media are used in numerous engineering applications.     

Effect of FGM core on dynamic response of a buried sandwich cylindrical shell in poroelastic soil to harmonic body waves

The two-dimensional dynamic interaction of progressive plane seismic waves with an arbitrarily thick, isotropic, and functionally graded cylindrical shell of infinite extent embedded in a boundless fluid-saturated porous elastic medium is investigated. The inhomogeneous shell is approximated by a laminate model, for which the solution is expected to gradually approach the exact one as the number of layers increases. Continuity of the displacement and stress components at the interfaces of neighboring layers is applied to form a system global transfer matrix, ultimately leading to determination of the modal scattering and transmission coefficients. The analytical results are illustrated with numerical examples in which an air-filled steel–zirconia FGM shell, buried in a water-saturated Ridgefield Sandstone formation, is insonified by fast compressional or shear waves at normal incidence. The effects of material compositional gradient and FGM layer thickness on the basic dynamic field quantities are evaluated and discussed. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

Evaluation of the interfacial conduction heat transfer coefficient in two-temperature macroscopic models of homogenous porous media using a fully developed unsteady microscopic model in periodic unit cells

A method is proposed for the evaluation of the interfacial conduction heat transfer coefficient in two-temperature macroscopic models of homogeneous fluid-saturated porous media. It is based on the numerical solutions of a microscopic model of unsteady conduction heat transfer in periodic unit cells, with different uniform initial temperatures of the fluid and solid. A novel formulation of the microscopic model in the fully developed regime is also proposed. Results for the variation of interfacial conduction Nusselt number with porosity, fluid–solid thermal conductivity ratio, and fluid–solid thermal diffusivity ratio are presented and discussed for four two-dimensional and two three-dimensional cases.     

Numerical study on melt fraction during melting of phase change material inside a sphere

This paper presents a numerical study on the constrained melting of phase change material (PCM) inside a sphere to investigate the effect of various factors on the melt fraction. A mathematical model of melting processes of the PCM inside a sphere is developed. And experiments are conducted to verify the numerical method. On the basis of the model, the effects of the sphere radius, the bath temperature, the PCM thermal conduction coefficient and the spherical shell material on the melt fraction of PCM inside a sphere are discussed. The results show that the PCM inside a sphere melts fast as the sphere radius is small, the bath temperature increases, and the PCM thermal conductivity is high. And the metal shell with high thermal conductivity should be adopted preferentially. The present study provides theoretical guidance for the design and operation of the phase change heat storage unit with sphere containers.