Analytical method of two‐dimensional inverse heat conduction problem using Laplace transformation: Effect of number of measurement points

An analytical method has been developed for the inverse problem of two‐dimensional heat conduction using the Laplace transform technique. The inverse problem is solved for only two unknown surface conditions and the other surfaces are insulated in a finite rectangular body. In actual temperature measurement, the number of points in a solid is usually limited so that the number of temperature measurements required to approximate the temperature change in the solid becomes too small to obtain an approximate function using a half polynomial power series of time and the Fourier series of the eigenfunction. In order to compensate for this lack of measurement points, the third‐order Spline method is recommended for interpolating unknown temperatures at locations between measurement points. Eight points are recommended as the minimum number of temperature measurement points. The calculated results for a number of representative cases indicate that the surface temperature and the surface heat flux can be predicted well, as revealed by comparison with the given surface condition. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 618–629, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10116     

A solution to the two‐dimensional inverse heat conduction problem

A simple method is developed in this paper to solve two‐dimensional nonlinear steady inverse heat conduction problems. The unknown boundary conditions can be numerically obtained by using the iteration and modification method. The effect of measurement errors of the wall temperature on the algorithm is numerically tested. The results prove that this method has the advantages of fast convergence, high precision, and good stability. The method is successfully applied to estimate the convective heat transfer coefficient in the case of a fluid flowing in an electrically heated helically coiled tube. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 113–119, 2000     

Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

Any solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10117     

Analytical approach with Laplace transform to the inverse problem of one‐dimensional heat conduction transfer: Application to second and third boundary conditions

An analytical method using Laplace transformation has been developed for one‐dimensional heat conduction. This method succeeded in explicitly deriving the analytical solution by which the surface temperature for the first kind of boundary condition can be well predicted. The analytical solutions for the surface temperature and heat flux are applied to the second and third of the boundary conditions. These solutions are also found to estimate the corresponding surface conditions with a high degree of accuracy when the surface conditions smoothly change. On the other hand, when these conditions erratically change such as the first derivative of temperature with time, the accuracy of the estimation becomes slightly less than that for a smooth condition. This trend in the estimation is similar irrespective of any kind of boundary condition. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 29–41, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10069     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

Analytical solution of the inverse unsteady wall heat conduction problem and experimental application

Based on the analytical solution of the unsteady heat conduction differential equation, a solution procedure is presented for the inverse unsteady wall heat conduction problem, i.e. for the calculation of the thermal properties of structural elements of existing buildings under real transient conditions, using on-site temperature measurements. Previous procedures, which were based on the finite-difference method, required a considerable number of temperature measurements in space and time within the wall. The advantage of the present analytical procedure is that it requires only two temperature measurements, apart from some information on the outdoor and indoor temperature variations. The two temperature measurements may be taken on the outdoor and indoor wall surfaces at the same time level, or on one of these surfaces at two different time levels. The proposed analytical procedure provides the values of the thermal conductivity and heat capacity of structural elements, and therefore it may be used in practice for ex post checking of the materials used by the constructor, or for load calculation when heating or cooling systems are to be installed in old buildings of unknown wall properties. Experimental examples are presented which show that the proposed analytical procedure may be applied in practice with very good accuracy.     

A Novel Variational Formulation of Inverse Problem of Heat Conduction with Free Boundary on an Image

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Flow field and heat transfer of a two‐dimensional impinging jet disturbed by an elastically suspended circular cylinder

The flow field around a circular cylinder elastically suspended with a cantilever‐type plate spring in the jet impingement region was visualized to investigate the mechanism of the impingement heat transfer. The impingement distance H was kept constant at 3 or 5 times as large as the jet slot width, h = 15 mm.The Reynolds number was fixed at 10,000, or 5000 in the case of flow visualization. The self‐induced periodic swing motion of the cylinder across the jet axis was caused by the interaction between the jet and the elastically suspended cylinder. It was found that this swing motion has direct effects on the flow and heat transfer characteristics of the stagnation region. The ensemble‐averaged values of the flow velocity and its fluctuations depended on the cylinder diameter and the impingement distance. The local Nusselt number in the case of H/h = 3 with the oscillating cylinder of the smallest diameter D = 4 mm was increased to 1.15 times as large as that without the cylinder. The interesting patterns of the intermittency function defined with a certain threshold level of turbulence intensity were obtained under the above experimental conditions. © 2001 Scripta Technica, Heat Trans Asian Res, 30(4): 313–330, 2001     

Solution of inverse heat conduction problem using the lattice Boltzmann method

In this paper the D2Q9 lattice Boltzmann method (LBM) was utilized for the solution of a two-dimensional inverse heat conduction (IHCP) problem. The accuracy of the LBM results was validated against those obtained from prevalent numerical methods using a common benchmark problem. The conjugate gradient method was used in order to estimate the heat flux test case. A complete error analysis was performed. As the LBM is attuned to parallel computations, its use is recommended in conjugation with IHCP solution methods.     

New Type of Basic Functions of FEM in Application to Solution of Inverse Heat Conduction Problem

The work presents the application of heat polynomials for solving an inverse problem. The heat polynomials form the Treffetz Method for non-stationary heat conduction problem. They have been used as base functions in Finite Element Method. Application of heat polynomials permits to reduce the order of numerical integration as compared to the classical Finite Element Method with formulation of the matrix of system of equations.     

One‐dimensional transient heat conduction in a bilayer finite slab: A case study in the printing process

In this work, a typical case of heat distribution is examined during a paper printing process, based on one‐dimensional transient heat conduction in two‐layer finite slabs with an insulated free surface, and a constant temperature free surface. Analytical solutions were obtained in non‐dimensional form. Various examples of applying these solutions are presented. The accuracy of the solutions, with respect to time, is analyzed considering the eigenvalues of their infinite solutions. It is observed that the larger the number of eigenvalues in consideration, the better the accuracy of the solutions. The model related to a two‐layer slab describes the simplified case in which all heat transfer occurs only by conduction. The solutions obtained are finally compared with the solutions for heat conduction in two semi‐infinite solids. The comparison between the two solutions shows that results are in good agreement only during short time scales. The heat distribution study is expected to be helpful in knowing the effectiveness of various mediums to be used as the reciever during the printing process; however, there is scope for development of more robust models.     

Integral transform methods for inverse problem of heat conduction with known boundary of a thin rectangular object and its stresses

The three-dimensional inverse transient thermoelastic problem for a thin rectangular object is considered within the context of the theory of generalized thermoelasticity. The upper surface of the rectangular object occupying the space D: a≤x≤a; b≤y≤b; 0≤z≤h; with the known boundary conditions. Laplace and Finite Marchi-Fasulo transform techniques are used to determine the unknown temperature, temperature distribution, displacement and thermal stresses on upper plane surface of a thin rectangular object. The distributions of the considered physical variables are obtained and represented graphically.     

Analysis of two‐dimensional porous fins with variable thermal conductivity

Analysis of porous fins for their higher heat transfer in comparison with solid fins with identical volumes has attracted significant attention. In this paper, a two‐dimensional thermal analysis of a porous fin having variable thermal conductivity coefficient is performed using finite difference method. Heat transfer through porous media is simulated using passage velocity from Darcy's model. The thermal conductivity of the solid phase is considered as a linear function of temperature. It is found that the temperature profile of the fin is completely two‐dimensional even for high Rayleigh and Darcy numbers (Ra = 10³～10⁴, Da = 0.01), because the temperature changes significantly along the transverse axis especially for lower Rayleigh and Darcy numbers. Also, the effects of important nondimensional parameters such as Rayleigh and Darcy numbers, porosity, Nusselt, thermal conductivity, and aspect ratio on the temperature profile are investigated. The results demonstrate that the temperature distribution is strongly dependent on the Rayleigh and Darcy numbers.     

Numerical simulation of separated flow transition and heat transfer around a two‐dimensional rib

Numerical simulations of separated flow transition and heat transfer around a two‐dimensional rib mounted in a laminar boundary layer were performed. The separated shear layer becomes unstable due to the Kelvin–Helmholtz instability and generates a two‐dimensional vortex. This vortex becomes three‐dimensional and collapses in the downstream part of the separation bubble. As a result, transition from laminar to turbulent flow occurs in the separated shear layer. Streamwise vortices exist downstream of the reattachment flow region. The low‐frequency flapping motion and transition of the separated shear layer are influenced by three‐dimensional dynamics upstream of the separation bubble. Large‐scale vortices around the reattachment flow region have substantial effects on heat transfer. Downstream of the reattachment point, the surface friction coefficient and Nusselt number are different from their profiles in the laminar boundary layer and approach the distributions seen in the turbulent boundary layer. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(8): 513–528, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20177     

Numerical Methods for Inverse Problem of Heat Conduction with Unknown Boundary Based on Variational Principles with Variable Domain

In this article a variable-domain variational approach to the entitled problem is presented.A pair of comple-mentary variational principles with a variable domain in terms of temperature and heat-streamfunction are firstestablished.Based on them,two methods of solution—generalized Ritz method and variable-domain FEM—both capable of handling problems with unknown boundaries,are suggested.Then,three sample numericalexamples have been tested.The computational process is quite stable,and the results are encouraging.Thisvariational approach can be extended straightforwardly to 3-D inverse problems as well as to other problems inmathematical physics.     

Application of the variational iteration method to nonlinear non‐Fourier conduction heat transfer equation with variable coefficient

In this paper, a variational iteration method (VIM) has been applied to nonlinear non‐Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results of VIM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems, and it is predicted that VIM can find a wide application in new engineering problems. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20362     

Solution to shape identification problem of unsteady heat‐conduction fields

This paper presents a numerical analysis method for shape determination problems of unsteady heat‐conduction fields in which time histories of temperature distributions on prescribed subboundaries or time histories of gradient distributions of temperature in prescribed subdomains have prescribed distributions. The square error integrals between the actual distributions and the prescribed distributions on the prescribed subboundaries or in the prescribed subdomains during the specified period of time are used as objective functionals. Reshaping is accomplished by the traction method that was proposed as a solution to shape optimization problems of domains in which boundary value problems are defined. The shape gradient functions of these shape determination problems are derived theoretically using the Lagrange multiplier method and the formulation of material derivative. The time histories of temperature distributions are evaluated using the finite‐element method for a space integral and the Crank–Nicolson method for a time integral. Numerical analyses of nozzle and coolant flow passage in a wing are demonstrated to confirm the validity of this method. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(3): 212–226, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10086     

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...     

Stability and accuracy of power‐series method for one‐dimensional heat conduction with non‐uniform grid systems

The power‐series method, a finite analytic approach to heat transfer and fluid flow problems that is based on power‐series expansion, was applied to a one‐dimensional heat‐conduction problem to evaluate its stability and accuracy. Application to a specific heat‐conduction problem with non‐uniform grid systems showed that it had stability within the ranges 10⁻⁵<Δt,Δx_E, and Δx_W,a<10⁵, and 10⁻⁵<β<10⁵. Comparison of its solutions with those by the fully implicit and Stefanovic–Stephan methods showed that this method yielded more accurate and robust solutions. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(7): 470–480, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20085     

Boundary shape determination of steady‐state heat‐conduction fields

In this paper, a practical method of numerical analysis for the boundary shape design of steady‐state heat‐conduction fields to control the temperature distribution to a prescribed distribution is presented. Reshaping was accomplished by the traction method proposed by one of the authors as a solution to domain optimization problems in which elliptic boundary value problems were defined. In this study, we formulated a temperature square error norm minimization problem between the prescribed distribution and the actual distribution on prescribed boundaries and theoretically derived a shape gradient function for this problem. We developed a convenient numerical method using a general‐purpose FEM program for the temperature prescribed problem. The successful results for the two‐dimensional problems of an erosion surface on a blast furnace hearth and a coolant flow passage in a wing demonstrate the validity of the presented method. © 2001 Scripta Technica, Heat Trans Asian Res, 30(3): 245–258, 2001