Periodic heat transfer through inhomogeneous media part 3: Hollow sphere

The paper presents the exact analytical solutions for periodic radial heat conduction through an inhomogeneous hollow sphere for a certain class of thermal-conductivity profile. The exact analytical solutions for some of these profiles, (including linear and quadratic) have been compared with those obtained by considering the spherical 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. It strengthens the confidence in applying the layered-structure method to the case of periodic heat conduction through an inhomogeneous hollow cylinder. Considering the inhomogeneous conducting medium to be made up of a number of spherical layers with a linear profile of thermal conductivity has been shown to be a more effective alternative method of accounting for any type of inhomogeniety; and it saves computation time, as the rate of convergence is much higher than for the homogeneous-layered structure method. 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 of the two surfaces of the hollow sphere.     

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.     

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.     

Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates

This study presents an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer. Here, it is supposed that the fibers are winded around the cylinder and their direction can be changed in each lamina. Laplace transformation is applied to change the domain of the solutions from time into the frequency. An appropriate Fourier transformation has been derived using the Sturm–Liouville theorem. Here, a set of equations for Fourier coefficients are obtained based on the boundary conditions both inside and outside the cylinder, and the continuity of temperature and heat flux at boundaries between adjacent layers. The exact solution of this set of equations is obtained using Thomas algorithm and Fourier coefficients are expressed by recessive relations. Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate. Some industrial examples are presented to investigate the ability of current solution for solving the wide range of applied steady and unsteady problems.     

最大热阻最小化的环形高导热通道三维体点导热最优构形的解析解法

基于构形理论,采用解析解法,以最大温差最小为优化目标,对基于环形高导热通道和圆柱形单元体的三维“体-点”导热模型进行构形优化,得到无量纲最大热阻最小的三维圆柱体最优构形.结果表明:增大高、低导热材料导热系数比、高导热材料占比和单元体数目均有助于提高圆柱形构造体的导热性能,但当单元体数目较大时,圆柱形构造体的导热性能改善...     

Entropy generation in a hollow cylinder with temperature dependent thermal conductivity and internal heat generation with convective–radiative surface cooling

This article investigates entropy generation in an asymmetrically cooled hollow cylinder with temperature dependent thermal conductivity and internal heat generation. The inside surface of the cylinder is cooled by convection on its inside surface while the outside surface experiences simultaneous convective–radiative cooling. The thermal conductivity of the cylinder as well as the internal heat generation within the cylinder are linear functions of temperature, introducing two nonlinearities in the one-dimensional steady state heat conduction equation. A third nonlinearity arises due to radiative heat loss from the outside surface of the cylinder. The nonlinear system is solved analytically using the differential transformation method (DTM) to obtain the temperature distribution which is then used to compute local and total entropy generation rates in the cylinder. The accuracy of DTM is verified by comparing its predictions with the analytical solution for the case of constant thermal conductivity and constant internal heat generation. The local and total entropy generations depend on six dimensionless parameters: heat generation parameter Q, thermal conductivity parameter β, conduction–convection parameters Nc₁ and Nc₂, conduction–radiation parameter Nr, convection sink temperature δ and radiation sink temperature η.     

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.     

Thermoviscoelastic Dynamic Behavior of a Double-Layered Hollow Cylinder Under Thermal Shocking

In this article, thermoviscoelastic dynamic behavior of a double-layered cylinder with a thermal barrier coating under radially symmetric mechanic and thermal loadings is investigated. The double-layered hollow cylinder is constructed of a viscoelastic layer and a homogenous layer, and the cylinder is subjected to thermal shocking. The material parameters of the cylinder are assumed to be temperature-dependent. The governing equation of the motion of the double-layered hollow cylinder under both dynamic mechanical and thermal loads is obtained based on the plane-stain theory, meanwhile, the transient heat transfer problems are solved by the finite difference method (FDM), Newmark method (NM), and iterative method. Numerical results show that mechanical load, boundary conditions, temperature field and whether considering the viscoelasticity of the inner layer each have a great influence on the dynamic behavior of the double-layered hollow cylinder.     

Inverse and efficiency of heat transfer convex fin with multiple nonlinearities

In this article, we first propose the novel semi‐analytical technique—modified Adomian decomposition method (MADM)—for a closed‐form solution of the nonlinear heat transfer equation of convex profile with singularity where all thermal parameters are functions of temperature. The longitudinal convex fin is subjected to different boiling regimes, which are defined by particular values of n (power index) of heat transfer coefficient. The energy balance equation of the convex fin with several temperature‐dependent properties are solved separately using the MADM and the spectral quasi‐linearization method. Using the values obtained from the direct heat transfer method, the unknown parameters of the profile, such as thermal conductivity, surface emissivity, heat generation number, conduction‐convection parameter, and radiation‐conduction parameter are inversely predicted by an inverse heat transfer analysis using the simplex search method. The effect of the measurement error and the number of measurement points has been presented. It is found that present measurement points and reconstruction of the exact temperature distribution of the convex fin are fairly in good agreement.     

A Unified Formulation for the Analysis of Temperature Field in a Thick Hollow Cylinder Made of Functionally Graded Materials With Various Grading Patterns

In this article, a unified formulation is presented to analyze the heat conduction and temperature field in a functionally graded thick hollow cylinder. To solve the heat transfer equation in a functionally graded thick hollow cylinder, an analytical method is presented using a nonlinear volume fraction law with variable exponent for mechanical properties. The new unified formulation is obtained based on various thermal boundary conditions such as heat convection at bounding surfaces and constant temperature of bounding surfaces. The functionally graded thick hollow cylinder is considered in axisymmetry condition and infinite length. The temperature field is discussed for various boundary conditions and different grading patterns in functionally graded materials. The presented results and data are verified with published literature.     

Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

Axisymmetric thermoelastic analysis in a finite hollow cylinder due to nonuniform thermal shock

An exact solution is obtained for two-dimensional elastodynamic analysis of finite hollow cylinder excited by nonuniform thermal shock. The cylinder is simply-supported at the two ends and is traction-free at the internal and external cylindrical surfaces. Based on the uncoupled linear thermoelastic theory, the solution is developed by employing the expansion of trigonometric series method and the separation of variables technique. The obtained solution contains two infinite series. One is trigonometric function and the other is Bessel function. The coefficients in the series solution are determined by the orthogonal properties of the functions. Numerical experiments are presented for describing the thermal dynamic behaviors of finite hollow cylinder.     

Heat transfer from hollow cylinder using optimal homotopy asymptotic method

Optimal homotopy asymptotic method (OHAM) is employed to investigate steady‐state heat conduction with temperature dependent thermal conductivity and uniform heat generation in a hollow cylinder. Analytical models are developed for dimensionless temperature distribution and heat transfer for two cases using mixed boundary conditions (Dirichlet, Neumann, and Robin). The inner cylinder is assumed to be insulated in both cases. In the first case, the outer cylinder is assumed to be isothermal whereas in the second case, the outer cylinder is convectively cooled by a fluid of temperature T₂ through a uniform heat transfer coefficient h. The effects of Biot number, dimensionless heat generation, and thermal conductivity parameters on the temperature distribution and heat transfer are determined analytically and validated numerically using MAPLE 14. In both cases, the results obtained by OHAM are found to be in good agreement with the numerical results. It is found that as the Biot number increases, the results approach that of the isothermal case. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20407     

A numerical study on the effect of a heated hollow cylinder on mixed convection in a ventilated cavity

The present work is aimed to study mixed convection heat transfer characteristics within a ventilated square cavity having a heated hollow cylinder. The heated hollow cylinder is placed at the center of the cavity. In addition, the wall of the cavity is assumed to be adiabatic. Flows are imposed through the inlet at the bottom of the left wall and exited at the top of the right wall of the cavity. The present study simulates a practical system such as air-cooled electronic equipment with a heat component or an oven with heater. Emphasis is sited on the influences of the cylinder diameter and the thermal conductivity of the cylinder in the cavity. The consequent mathematical model is governed by the coupled equations of mass, momentum and energy and solved by employing Galerkin weighted residual method of finite element formulation. A wide range of pertinent parameters such as Reynolds number, Richardson number, cylinder diameter and the solid-fluid thermal conductivity ratio are considered in the present study. Various results such as the streamlines, isotherms, heat transfer rates in terms of the average Nusselt number and average fluid temperature in the cavity are presented for different aforesaid parameters. It is observed that the cylinder diameter has significant effect on both the flow and thermal fields but the solid-fluid thermal conductivity ratio has significant effect only on the thermal field.     

Thermal stresses in layered thick cylindrical shells of infinite length

A layered infinite closed thick cylindrical shell subjected to steady thermo-loads on surfaces, which is treated as two-dimensional plane strain problem, i.e., a layered ring with rectangular cross section of unit width, was investigated based on exact thermoelasticity theory. An analytical method for the temperature, stress, and displacement fields in the shell was presented. First, the general solutions of temperature, displacements, and stresses in a single-layer ring were deduced, which exactly satisfy the two-dimensional heat conduction equation and the two-dimensional elasticity equations in polar coordinates by use of the Fourier series expansion. On the basis of continuities of physical quantities on the interface of two adjacent layers, the relationships of temperature, heat flux, displacement, and stress between the outermost surface and innermost surface of the layered ring were recursively obtained by use of the transfer matrix method. Excellent convergence of the solutions was observed. Comparing the present results with those obtained from the finite element method indicates the correctness of the present method. The influences of temperature loads, ring thickness, layer number, and material properties on the distributions of temperature, displacements, and stresses in the layered rings were studied in detail.     

分层岩土中地埋管换热器传热解析模型与分析

为准确预测地埋管换热器在分层岩土中的传热特征,采用分离变量法和格林函数法,基于单个圆环热源基本传热单元问题的解答,建立考虑岩土结构分层和横观各项同性特征的地埋管传热解析模型。该模型适用于工程中常见的垂直钻孔和桩基埋管换热器分层传热问题,具有较好的普适性。以2层岩土为例,利用模型解答对分层岩土中地埋管的传热特征以及分层参数对其影响规律进行研究。结果表明：均匀介质假设计算误差随作用时间的增加而逐渐增大,在靠近热源处误差更加明显,预测地埋管长时间温度响应时,应采用分层传热模型;在临界区域范围内,可用均质假设模型预测地埋管的传热特性,均质等效热物性参数取为对应岩土分层的热物性参数值;分层岩土导热系数对地埋管传热性能影响较大,岩土平衡温度随分层导热系数比的增大而明显降低;地埋管长度和直径的比值对地埋管传热性能有所影响,岩土平衡温度随长径比的增大而升高,且其影响程度随分层导热系数比的减小而增强。     

COMPUTER-AIDED CALCULATION OF APPROXIMATE CLOSED-FORM SOLUTIONS FOR LINEAR TRANSIENT HEAT CONDUCTION PROBLEMS IN MULTILAYERED BODIES

A solution procedure, based on the Kantorovich method, is presented for the derivation of approximate closed-form solutions for linear heat conduction problems in multilayered plane and cylindrical bodies using computers. Constant or space dependent initial conditions; linear time dependent boundary conditions of the first, second, or third kind; contact resistances between the layers; and a homogeneous distributed, time dependent volumetric heat source can be considered. The solution procedure is shown suitable for programming. In order to assess the approximate solution obtained, an error criterion is stated. The accuracy of the method is investigated through a numerical and an analytical example.     

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.     

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.