A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

Meshless method for solving radiative transfer problems in complex two-dimensional and three-dimensional geometries

A meshless method is presented to solve the radiative transfer equation in complex 2D and 3D geometries. In order to avoid numerical oscillations, the even parity formulation of the discrete ordinates method is used. A moving least squares approximation meshless method is used to solve the second order partial differential equations. Prediction results of radiative heat transfer problems obtained by the proposed method are compared with reference in order to assess the correctness of the present method.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

An engineering foundation for controlling heat transfer in one-dimensional transient heat conduction problems

This paper develops an engineering foundation for controlling heat transfer in one dimensional transient heat conduction problems based upon concepts borrowed from vibration control problems. The foundation distinguishes between modal control, distributed control, discrete control and direct feedback control and then singles out direct feedback control because its simplicity. An example demonstrates modal control and direct feedback control of the variation of the transient temperature field in a one dimensional slab geometry.     

Effect of radiation and convection heat transfer on cooling performance of radiative panel

The radiative panel is an equipment combining the solar heating and nocturnal radiant cooling technology. This study conducted the thermal performance of radiative panels for both radiation and convection cooling. Using the cover test by the mirror polished aluminum plate, the net cooling capacity of radiative panel was tested. The net cooling capacity of the radiative panel and contribution degree of the radiation heat transfer and convection heat transfer to the net cooling capacity was computed using the simulation model, and the influences of the cloud, ambient temperature and inclination angle on the radiation cooling were discussed. From the experimental results, the net cooling capacity was 45–70 W/m² when the radiative panel wasn’t covered, and the net cooling capacity was 10–30 W/m² when the mirror polished aluminum plate existed on a clear night in February in Tianjin. From the simulation results, the net cooling capacity of the radiative panel was about 50–70 W/m², and the radiation cooling was about 45 W/m², being responsible for 64%–90% of the net cooling capacity. The temperature differences between radiative panel and environment were the main influencing factors for the radiation cooling capacity. With an increase of the temperature difference, the radiation cooling capacity increased, and when the variation 5 °C of the temperature difference, the radiation cooling capacity will increase about 10–20 W/m². When it was partly cloudy, the radiation cooling capacity was about 50 W/m² and the fall rate of the radiation cooling capacity was less than 24%. With an increase of the cloud, the radiation cooling will decrease significantly. When it was overcast, the radiative panel even absorbed heat around 45 W/m² from the environment. When the tilt angle of radiative panel was less than 30°, the fall rate of the radiation cooling capacity was less than 11.3%. When the tilt angle was greater than 30°, the radiation cooling decreased significantly. In the case of being placed vertically, the radiation cooling capacity reduced by 84.8%.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

Effect of gas radiation on radiative heat transfer in urban street canyon model

In this paper, we describe the results of numerical simulation of radiative heat transfer between the human body and an urban street canyon (building walls, pavement, and the sky) in the presence of participating non‐gray gas mixtures consisting of H₂O and CO₂. The ambient temperature in typical summer conditions and the concentration of gas mixtures during summer in Tokyo were assumed. Further, the parallel infinite plane model and simple urban street canyon model were used. The results show that the participating gas significantly affects the infrared radiation field in an urban street canyon. The radiation flux emitted by the participating gas is approximately 35% of the total radiation flux incident on the human body surface. This causes a homogenization of the infrared radiation field surrounding the human body. Gas radiation plays an important role in the heat transfer between the human body and the environment under hot and humid summer conditions. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20258     

Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivity

This paper deals with the effect of the temperature dependent thermal conductivity on transient conduction and radiation heat transfer in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. The thermal conductivity of the medium is assumed to vary linearly with temperature. The radiative part of the energy equation was solved using the collapsed dimension method. To facilitate solution of the energy equation, which is a highly nonlinear one, time linearization was done first and then the equation was solved using the alternating direction implicit scheme. Results for the effects of the variable thermal conductivity were found for temperature and heat flux distributions.     

Estimation of slab surface radiative emissivities by solving an inverse coupled conduction,convection, and radiation problem

Slab surface radiative emissivities severely affect the radiative heat transfer in a reheating furnace, as well as the slabs’ coupled conduction, convection, and radiation. Accurate evaluation of these parameters is of significance to ensure the high accuracy of the mathematical model for a reheating furnace, which is beneficial to the energy saving. However, it is difficult to directly and accurately measure these parameters. In this article, slab surface radiative emissivities in a reheating furnace are estimated by solving a nonlinear inverse problem, which is an inverse coupled conduction, convection, and radiation problem. An efficient and accurate gradient method, i.e., Levenberg–Marquardt algorithm, is applied to obtain the solution of the inverse problem. First, a finite difference method and the complex-variable-differentiation method are used for sensitivity analysis, and the inversion accuracy coupled with the efficiency is demonstrated. Then, effects of initial guesses, measurement errors, and measurement locations on estimated slab surface radiative emissivities are investigated in detail. Finally, conclusions are drawn based on the results and analysis.     

Combined mode conduction and radiation heat transfer in a spherical geometry with non-Fourier effect

This article deals with the analysis of combined mode non-Fourier conduction and radiation heat transfer in a concentric spherical enclosure containing a conducting–radiating medium. The finite volume method (FVM) has been employed to calculate the volumetric radiative information and also to solve the governing energy equation, which is of hyperbolic nature. The non-Fourier effect which manifests in the form of a sharp discontinuity in the temporal temperature distribution and propagates with a finite speed has been investigated. As time progress, the discontinuity in the temperature distribution decays and in the steady-state, results with and without non-Fourier effect are the same. Detailed study of the effect of various parameters such as the extinction coefficient, the scattering albedo, the conduction radiation parameter, the emissivity and the anisotropy factor has been carried out. Results of the present work have been compared with the steady-state response of the combined mode Fourier conduction–radiation problems available in literature. Results have been found to agree well.     

Local least–squares element differential method for solving heat conduction problems in composite structures

Abstract

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.     

Composite heat transfer with thermal radiation in non-grey medium part I: Interaction of radiation with conduction

The energy equation including thermal radiation is a non-linear high order integro-differential equation and the spectroscopic constants involved are usually complex functions of frequency. Accordingly, it is formidable to solve the equation rigorously. On this basis many investigators have introduced the assumption of the grey gas that spectroscopic constants are independent of wavelength. This assumption, however, might smear the essential feature of radiative heat transfer. Alternatively dividing a spectral band into parts of center and wings and estimating an appropriate effective absorption coefficient in each part, the opaque (Rosseland) approximation is applicable to the central part in a band and the transparent approximation to the part of wings. Such an analytical procedure reduces to simple treatment despite of taking into account of non-grey behaviour. The current study considers a simple interaction problem between conduction and radiation excluding the convection in the mediums. Numerical calculations are performed on carbon monoxide and carbon dioxide.     

Combined conduction and radiation heat transfer with variable thermal conductivity and variable refractive index

This article deals with the solution of conduction–radiation heat transfer problem involving variable thermal conductivity and variable refractive index. The discrete transfer method has been used for the determination of radiative information for the energy equation that has been solved using the lattice Boltzmann method. Radiatively, medium is absorbing, emitting and scattering. To validate the formulation, transient conduction and radiation heat transfer in a planar participating medium has been considered. For constant thermal conductivity and constant and variable refractive indices, results have been compared with those available in the literature. Effects of conduction–radiation parameter and scattering albedo on temperature have been studied for variable thermal conductivity and constant and/or variable refractive index. Lattice Boltzmann method and the discrete transfer method have been found to successfully deal with the complexities introduced due to variable thermal conductivity and variable refractive index.     

A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity

A new radial integration boundary element method (RIBEM) for solving transient heat conduction problems with heat sources and variable thermal conductivity is presented in this article. The Green’s function for the Laplace equation is served as the fundamental solution to derive the boundary-domain integral equation. The transient terms are first discretized before applying the weighted residual technique that is different from the previous RIBEM for solving a transient heat conduction problem. Due to the strategy for dealing with the transient terms, temperature, rather than transient terms, is approximated by the radial basis function; this leads to similar mathematical formulations as those in RIBEM for steady heat conduction problems. Therefore, the present method is very easy to code and be implemented, and the strategy enables the assembling process of system equations to be very simple. Another advantage of the new RIBEM is that only 1D boundary line integrals are involved in both 2D and 3D problems. To the best of the authors’ knowledge, it is the first time to completely transform domain integrals to boundary line integrals for a 3D problem. Several 2D and 3D numerical examples are provided to show the effectiveness, accuracy, and potential of the present RIBEM.     

Simultaneous radiation and conduction heat transfer in a graded index semitransparent slab with gray boundaries

The simultaneous radiation and conduction heat transfer in a semitransparent slab of absorbing-emitting gray medium is solved in this paper. The refractive index of the medium spatially varies in a linear relationship, and the two boundary walls are diffuse and gray. A curved ray tracing technique in combination with a pseudo-source adding method is employed to deduce the radiative intensities on gray walls. Resorting to some of the results presented by Ben Abdallah and Le Dez, an exact expression of the radiative flux in medium is deduced. The influences on the temperature and radiative flux fields are examined, which are caused by the refractive index distribution, absorbing coefficient, thermal conductivity and the boundary wall emissivities. The results display the significant influences of the refractive index distribution and boundary wall emissivities on the radiative flux and temperature in medium.     

Element differential method with the simplest quadrilateral and hexahedron quadratic elements for solving heat conduction problems

In this article, two types of new quadrilateral and hexahedron quadratic isoparametric elements are proposed for the element differential method (EDM) for solving heat conduction problems. These elements, called as the Ultra elements, have the minimum numbers of nodes comparing with the existing elements and have the feature that a central node is included inside them, which is necessary for the EDM analysis. The EDM is a strong-form method, which does not require control volumes and any integration. In the previous EDM for solving heat conduction problems, the Lagrange elements were used, which had many elemental nodes. The proposed new types of elements can circumvent this deficiency, in which only a few nodes are required. The shape functions for these elements are constructed for the first time, and the first and the second order derivatives of the shape functions with respect to intrinsic and global coordinates are analytically derived. Several 2D and 3D numerical examples are given to demonstrate the effectiveness, the accuracy and the efficiency of the newly proposed elements for the EDM for solving heat conduction problems.     

Efficient finite-difference scheme for solving some heat transfer problems with convection in multilayer media

An efficient finite-difference method for solving the heat transfer equation with piecewise discontinuous coefficients in a multilayer domain is developed. The method may be considered as a generalization of the finite-volumes method for the layered systems. We apply this method with the aim to reduce the 3D or 2D problem to the corresponding series of 2D or 1D problems. In the case of constant piecewise coefficients, we obtain the exact discrete approximation of the steady-state 1D boundary-value problem.     

Experimental research of radiative heat transfer in fluidized beds

A new method to measure the radiative heat transfer in fluidized beds was presented. Experiments were carried out on a 0.8 th⁻¹ fluidized bed combustion boiler. The residual slag of fired coal was operated in a fluidized bed at room temperature. As the radiative heat transfer at room temperature is insignificant, its contribution at high temperatures might be obtained by the comparison of experimental results at high and low temperatures. On experimental study, a radiative contribution was given as a function of bed temperature and particle size. The results were compared with those in other references.     

粒子辐射换热计算中散射相函数处理方法的探讨

对粒子散射相函数的各种处理方法进行了总结归纳,并以黑体平行大平壁均匀粒子介质层的辐射换热问题为研究对象,在辐射平衡条件下,对比研究了采用Mie散射理论和线性散射相函数近似处理粒子散射相函数时介质内的辐射热流及温度分布情况。辐射传递方程采用离散坐标法求解,并在求解过程中对散射相函数进行了重新归一化处理。研究表明,Mie散射相函数计算过程复杂费时,均匀粒子的Mie散射相函数随散射角强烈波动,这使辐射传递方程的求解更加困难;线性散射相函数近似简单易行,当所选线性系数基本符合Mie散射相函数前向或后向散射特征时,采用线性相函数近似可以大大简化计算,并可正确估算粒子介质内的辐射热流与温度分布情况,是一种较好的处理散射相函数的方法。     

Combined conduction and radiation heat transfer in a gray anisotropically scattering planar medium with diffuse-specular boundaries

Coupled conduction and radiation heat transfer in a gray planar nonlinearly anisotropic scattering medium bounded between two plane parallel surfaces reflecting both diffusely and specularly is analyzed. The governing integrodifferential equations are solved by a numerical iterative method consisting of Numerov's method to solve the energy equation and Chandarsekhar's discrete ordinates method in conjunction with the Crank-Nicolson method to solve the radiative transfer equation. Convergence of the solution is enhanced by Ng-acceleration. The numerical algorithm described is found to be fast and reliable. Numerical results based on S₃₂ method indicate that anisotropy plays an important role, and difference between the diffuse and specular reflections is found to be insignificant.