Transient analytical solution to heat conduction in composite circular cylinder

An analytical method leading to the solution of transient temperature filed in multi-dimensional composite circular cylinder is presented. The boundary condition is described as time-dependent temperature change. For such heat conduction problem, nearly all the published works need numerical schemes in computing eigenvalues or residues. In this paper, the proposed method involves no such numerical work. Application of ‘separation of variables’ is novel. The developed method represents an extension of the analytical approach derived for solving heat conduction in composite slab in Cartesian coordinates. Close-formed solution is provided and its agreement with numerical result is good which demonstrates a good accuracy of the developed solution form.     

Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique

In this paper, a non-Fourier heat conduction problem is analyzed by employing newly developed theory. Application of conventional numerical schemes leads to strong oscillations of the results around discontinuities in solution domain. To overcome this difficulty the variational formulation of the Laplace-transformed hyperbolic heat conduction equation is developed. The results were used for evaluation of parameters used in approximate transformed temperature profiles. To validate the approach the results were compared with the exact analytical solution solved at special case and with an approach previously reported in the literature. Both showed a close agreement with the proposed approach.     

The effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows

Axial heat conduction effects within the fluid can be important for duct flows if either the Prandtl number is relatively low (liquid metals) or if the dimensions of the duct are small (micro heat exchanger). In addition, axial heat conduction effects in the wall of the duct might be of importance. The present paper shows an entirely analytical solution to the extended Graetz problem including wall conduction (conjugate extended Graetz problem). The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is relatively simple to compute and valid for all Péclet numbers. The analytical results are compared to own numerical calculations with FLUENT and good agreement is found.     

Eigenvalue analysis of temperature distribution in composite walls

The transient heat conduction problem in two‐layer composite wall is solved analytically using spectral analysis. Eigenvalues and corresponding eigenfunctions of the spectral problem for the temperature distribution in composite walls are analysed using the Rouche Theorem. The number of eigenvalues is obtained and the temperature distribution of this complicated problem is given by a formula with calculated eigenvalues. The analytical solution obtained is in explicit form and provides easy determination of temperature rise in heating and thawing applications of composite materials. Copyright © 2001 John Wiley & Sons, Ltd.     

Transient Temperature Distribution Within a Flat Sheet During the Welding Process — Analytical Solution

An analytical study of transient heat conduction process with moving heat source/sink was considered. In this study a closed-form solution for the heat conduction equation is obtained. Using the obtained solution, the spatial temperature distribution and the temperature time history could be obtained. The obtained results are compared with numerical and experimental data in the literature. The comparison shows that the present solution can be used do determine the effect of different heat flow parameters on the temperature pattern and history in any similar heat conduction problem.     

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.     

The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows

Axial heat conduction effects within the fluid can be important for duct flows if the Prandtl number is relatively low (liquid metals). In addition, axial heat conduction effects within the flow might also be important, if the heating zone is relatively short in length. The present paper shows an entirely analytical solution to the extended Graetz problem with piecewise constant wall temperature boundary conditions. The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is as simple to compute as the one without axial heat conduction. The analytical results are compared to available numerical calculations and good agreement is found.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

The analytical solution of the one alloy solidification problem

In this paper we obtain the analytical solution for a semi-infinite solidifying alloy. Thus, a three-phase problem including solid, solid–liquid, and liquid phases is analytically solved. Linearization of the heat conduction equation for an alloy is based on the method proposed in our recent papers.Note that the method does not allow one to solve the problem of solidification of an alloy with the given function λ(T) (liquid fraction). The dependence λ(T) is determined from the condition of linearization of the heat conduction equation within the mush zone.The analytical solution presented is an important test example for analysis of the numerical schemes used for systems with moving boundaries, e.g., for programs simulating vacuum arc remelting.     

GREEN'S FUNCTION APPROACH TO THREE-DIMENSIONAL HEAT CONDUCTION EQUATION OF FUNCTIONALLY GRADED MATERIALS

A Green's function approach based on the laminate theory is adopted to solve the three-dimensional heat conduction equation of functionally graded materials (FGMs) with one-directionally dependent properties. An approximate solution for each layer is substituted into the governing equation to yield an eigenvalue problem. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the three-dimensional transient temperature. The eigenvalues and the corresponding eigenfunctions are determined from the homogeneous boundary conditions at outer sides and from the continuous conditions of temperature and heat flux at the interfaces. A three-dimensional transient temperature solution with a source is formulated by the Green's function. Numerical calculations are carried out for an FGM plate, and the numerical results are shown in tables and figures.     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

Temperature distribution in a rectangular plate heated by a moving heat source

In this paper, a solution to the problem of heat conduction in a rectangular plate subjected to the activity of a moving heat source is presented. The temperature of the plate changes because a limited area on the plate surface is heated by a heat source. The heat source moves along an elliptical trajectory which always remains within the plate area. An exact solution to the problem in an analytical form is obtained by applying the Green’s function method. Exemplary results of numerical calculations to determine the temperature distribution in the plate are presented.     

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.     

FINITE ANALYTIC NUMERICAL METHOD FOR TRANSIENT HEAT DIFFUSION IN LAYERED COMPOSITE MATERIALS

The finite analytic (FA) method, which has been recently developed and used in fluid flow and heat transfer problems, is presented and extended to the solution of the problem of transient heat conduction in a one-dimensional muMlayered composite slab. The basic idea of the FA method is to incorporate a local analytic solution of the governing equation in the numerical solution of the boundary-value problem. In thii: study, the local analytic solutions are obtained by the standard integral-transformation technique and the associated eigenvalue problem is solved by the Galerkin method. Some numerical examples are computed to demonstrate the applicability of the FA method in dealing with heat diffusion in a composite material. It is shown that the FA method is accurate and provides reductions in computational costs.     

ANALYSIS/FINITE-ELEMENT COMBINED METHODOLOGY ON TRANSIENT HEAT CONDUCTION OF A FINITE DOMAIN WITH PLANAR-DISCRETE SOURCES

Abstract

A useful method, involving the combined use of the analysis and the finite-element methods, is successfully extended to the transient heat conduction problem with isolated heat sources. The results are compared in tables with exact solutions and other numerical data, and the agreement is found to be good. Previously reported analysis /finite-element combined method has been confined to the slow convergence in series solution of analytical method. By using the third Aitken's delta-squared process for accelerating the convergence of infinite series, this restriction is removed, and the new method provides a more powerful solution to transient problems with heat sources     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Lattice Boltzmann simulation of non-Fourier heat conduction with phase change

The lattice Boltzmann method (LBM) combined with the enthalpy method is a very effective method to solve the solid–liquid phase transition problem. However, when the heat flux is very high or the time of the process is in the same order of magnitude as the relaxation time, it is necessary to consider the non-Fourier effect in heat conduction. At this time, whether the LBM-BGK format based on Bhatnagar-Gross-Krook (BGK) approximation is still valid remains to be discussed. In this paper, the hyperbolic lattice Boltzmann method (HLBM) is combined with the enthalpy method to solve the non-Fourier solid–liquid phase change problem. By solving the non-Fourier heat conduction problem and the Fourier solid–liquid phase change problem, the numerical solution is compared with the analytical solution to verify the accuracy of the algorithm. The effect of different relaxation times on the solid–liquid phase transition is analyzed. In addition, the effect of changes in thermal diffusivity due to state changes on the non-Fourier solid–liquid phase transition is discussed.     

Hybrid formulation and solution for transient conjugated conduction–external convection

This work presents a hybrid numerical–analytical solution for transient laminar forced convection over flat plates of non-negligible thickness, subjected to arbitrary time variations of applied wall heat flux at the fluid–solid interface. This conjugated conduction–convection problem is first reformulated through the employment of the coupled integral equations approach (CIEA) to simplify the heat conduction problem on the plate by averaging the related energy equation in the transversal direction. As a result, an improved lumped partial differential formulation for the transversally averaged wall temperature is obtained, while a third kind boundary condition is achieved for the fluid from the heat balance at the solid–fluid interface. From the available steady velocity distributions, a hybrid numerical–analytical solution based on the generalized integral transform technique (GITT), under its partial transformation mode, is then proposed, combined with the method of lines implemented in the Mathematica 5.2 routine NDSolve. The interface heat flux partitions and heat transfer coefficients are readily determined from the wall temperature distributions, as well as the temperature values at any desired point within the fluid. A few test cases for different materials and wall thicknesses are defined to allow for a physical interpretation of the wall participation effect in contrast with the simplified model without conjugation.     

用渐近分析法研究蜂窝蓄热体温度分布

1前言在冶金、机械和石化工业锻造炉、均热炉、连续加热炉、热处理炉、钢包烘烤炉、辐射管和熔铝炉上应用的高温空气燃烧(High Temperature Air Combustion,Hi-TAC)[1],具有热效率高、低NOx排放和燃烧放热均匀等特点。大多数的HiTAC应用了蜂窝蓄热系统[2]。温度变化和温度效率(     

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.