Two-dimensional Inverse Heat Conduction Problem Research Articles

Regularization strategies for a two-dimensional inverse heat conduction problem

We consider a two-dimensional inverse heat conduction problem for a slab. This is a severely ill-posed problem. Two regularization strategies, one based on the modification of the equation, the other based on the truncation of high frequency components, are proposed to solve the problem in the presence of noisy data. Error estimates show that the regularized solution is dependent continuously on the data and is an approximation of the exact solution of the two-dimensional inverse heat conduction problem. The relation of these two and other regularization strategies is also discussed.

Parallel genetic algorithms for the solution of inverse heat conduction problems

A parallel genetic algorithm (PGA) is proposed for the solution of two-dimensional inverse heat conduction problems involving unknown thermophysical material properties. Experimental results show that the proposed PGA is a feasible and effective optimization tool for inverse heat conduction problems.

Lattice-free finite difference method for numerical solution of inverse heat conduction problem

Inverse heat conduction problem consists of finding an initial temperature distribution from the knowledge of a distribution of the temperature at the present time. Here, we assume that the associated boundary conditions are known. The heat conduction problem backward in time is a typical example of ill-posed problems in the sense that the solution exists only for regular functions of some kind describing the present temperature distribution and also the solution is unstable for the present temperature distribution function. Conventional numerical methods often suffer from instability of the problem itself when high accuracy is intended in the approximation. Our aim is to create a meshless method which is applicable to the ill-posed inverse heat conduction problem. We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our new method we show a tool in using exponential functions in Taylor's expansion. From numerical experiments we confirmed that our method is effective for solving two-dimensional inverse heat conduction problem numerically subject to mixed boundary conditions.

Estimation of Heat Transfer Coefficient in Two-Dimensional Inverse Heat Conduction Problems

The finite-difference method in conjunction with the least-squares scheme, cubic spline, and temperature measurements is applied to predict the distribution of the heat transfer coefficient on a surface exposed to a moving fluid. In the present study, the functional form of the heat transfer coefficient is unknown a priori. The whole space domain of the unknown heat transfer coefficient can be divided into several analysis subintervals and then the cubic spline is introduced to estimate the unknown values. In order to show the accuracy of the present inverse method, a comparison among the present estimates, previous results, and exact solution is made. The results show that the present inverse scheme not only can reduce the number of the thermocouples but also can increase the accuracy of the estimated results. Also, the present estimates are not very sensitive to the measurement locations. Good estimation of the heat transfer coefficient can be obtained from the knowledge of the temperature recordings even for the case with measurement errors.

Structured Multiblock Grid Solution of Two-Dimensional Transient Inverse Heat Conduction Problems in Cartesian Coordinate System

ABSTRACT In this study a structured multiblock grid is used to solve two-dimensional transient inverse heat conduction problems. The multiblock method is implemented for geometric decomposition of the physical domain into regions with blocked interfaces. The finite-element method is employed for direct solution of the transient heat conduction equation in a Cartesian coordinate system. Inverse algorithms used in this research are iterative Levenberg-Marquardt and adjoint conjugate gradient techniques for parameter and function estimations. The measured transient temperature data needed in the inverse solution are given by exact or noisy data. Simultaneous estimation of unknown linear/nonlinear time-varying strengths of two heat sources in two joined surfaces with equal and different heights is obtained for the solution of the inverse problems, and the results of the present study for unknown heat source functions are compared to those of exact functions. This study is an attempt to challenge the goal of combining a multiblock technique with inverse analysis methods. In fact, the structured multiblock grid is capable of providing accurate solutions of inverse heat conduction problems (IHCPs) in industrial configurations, including composite structures. In addition, the multiblock IHCP solver is suitable to estimate unknown parameters and functions in these structures.

Identification of unifying heat transfer mechanisms along the entire boiling curve

The interfacial geometry and the associated heat transfer mechanisms close to and at the boiling surface are identified rigorously along the entire boiling curve from wall temperature and two-phase flow sensor probe data taken during boiling of isopropanol. It is conjectured, that the liquid–vapor interface of the wetting structure provides the key mechanism for heat removal due to evaporation along the entire boiling curve. Local fluctuations of surface temperature superheat and heat flux are inferred by the solution of a two-dimensional inverse heat conduction problem employing wall temperature data. Peak heat fluxes of several MW⋅m −2 are estimated. In a next step, the interfacial geometry close to the surface is identified. It is shown that the size of the dry spots and the nucleation site density are well connected to the vapor fraction and the interfacial area density as well as to the contact angle of the wetting fluid. Further, we show that a suitably designed single four-sensor optical probe could provide the necessary data required to exactly identify the wetting structure using the wetting structure geometry model proposed here. We further find that interfacial area density accounts for the contact angle of the fluid and correlates well with the boiling heat flux. In conclusion, the results show that essential mechanisms of heat removal along the entire boiling curve can be attributed to interfacial presence and evolution. A single measure, namely the interfacial area density, is suspected to provide the essential mechanism of the heat removal in the boiling process. Finally, the results are discussed with respect to the candidate models for the interfacial geometry and heat transfer mechanisms proposed in the literature. Only, specifically designed and refined boiling experiments as well as identification methods along the line presented in this paper are needed to further verify this postulate.

Implementation of an analytical two-dimensional inverse heat conduction technique to practical problems

Two improvements to practical implementation of a solution to the two-dimensional inverse heat conduction problem are presented. The first concept is useful for experimental data with strong or irregular fluctuations in time. The second procedure improves the spatial resolution for problems where the source of the surface heat flux distribution is moving along the surface. The method is tested against analytical solutions and data from quench cooling experiments. Both procedures are found to enhance the quality of the inverse solution results.

A stability estimate for a Cauchy problem for an elliptic partial differential equation

A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y = 0, and boundary data are for x = 0 and x = 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y = 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle.

A two‐dimensional inverse heat conduction problem for estimating heat source

This note considers the problem of estimating unknown time‐varying strength of the temporal‐dependent heat source, from measurements of the temperature inside the square domain, when the prior knowledge of the source functions is not available. This problem is an inverse heat conduction problem. In this process, the direct problem will be solved by using the heat fundamental solution. Then a sequential algorithm is developed to solve a Volterra integral equation, which has been produced by using unknown source term and overposed data conditions. This algorithm is based on the piecewise linear continuous functions. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient.

Identification of source terms in 2-D IHCP

We introduce a stable numerical space marching scheme based on discrete mollification—implemented as an automatic adaptive filter—for the approximate identification of temperature, temperature gradient, and source terms in the two-dimensional inverse heat conduction problem (IHCP). The stability and error analysis of the algorithm, together with some numerical examples, are provided.

Experimental evaluation of flow boiling incipience of subcooled fluid in a narrow channel

Experimental investigations dealt with heat transfer of R123 flowing through a narrow vertical channel of 1 mm height with one wall heated uniformly and others approximately adiabatic. Particular attention was paid to boiling incipience conditions and nucleate hysteresis phenomena. An inverse problem was dealt with as, on the basis of temperature measurement on external side of the heating foil (with thermography technique) and the measurement of the electric power supplied to this foil, it was possible to determine local heat transfer coefficients on the contact surface of the heating wall and the fluid flowing along the narrow channel. Two models of heat transfer through the heating foil and the isolating glass into the boiling liquid were proposed and discussed: one- and two-dimensional. In order to solve the two-dimensional inverse heat conduction problem the method of thermal polynomials was used with application of the least square and Trefftz computational techniques. Some exemplary results of numerical calculations on the basis of experimental data were presented for both models. The correlations concerning Nusselt number determination for R123 boiling incipience in vertical narrow channel were developed.

A Maximum Entropy Solution for a Two-Dimensional Inverse Heat Conduction Problem

Abstract A solution scheme based on the maximum entropy method (MEM) for the solution of two-dimensional inverse heat conduction problems is established. MEM finds the solution which maximizes the entropy functional under the given temperature measurements. The proposed method converts the inverse problem to a nonlinear constrained optimization problem. The constraint of the optimization problem is the statistical consistency between the measured temperature and the estimated temperature. Successive quadratic programming (SQP) facilitates the numerical estimation of the maximum entropy solution. The characteristic feature of the proposed method is investigated with the sample numerical results. The presented results show considerable enhancement in resolution for stringent cases in comparison with a conventional method.

A dual reciprocity BE-based sequential function specification solution method for inverse heat conduction problems

In this work, a method based on the Dual Reciprocity Boundary Element along with Sequential Function Specification scheme is suggested for solving two-dimensional inverse heat conduction problems involving unknown time and space varying boundary heat flux estimation. The measured transient temperature data utilized in the solution may be from locations inside the body or from locations on its inactive boundaries. The spatial variation of the unknown heat flux is approximated using polynomials, which is shown to conveniently reduce the number of unknowns in the inverse algorithm. The effectiveness and the reliability of the method are verified via a set of test data obtained by simulated experiments, which take into consideration various boundary heat flux characteristics.

A differential quadrature solution of the two-dimensional inverse heat conduction problem

A numerical procedure using a combined differential quadrature and Taylor series approach is presented for the solution of the transient two-dimensional inverse heat conduction problem. The method is relatively easy to program, compared to other more complex solutions currently available, and gives good results for both exact and inexact (noisy) data. The method is suitable to provide estimates of unknown boundary conditions, in advanced inverse procedures.

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

AbstractAn 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

Estimation of Boundary Conditions in Nonlinear Inverse Heat Conduction Problems

A sequential method is proposed to estimate boundary conditions for nonlinear heat conduction problems. An inverse solution is deduced from a e nite element method, the concept of the future time, and a modie ed Newton‐ Raphson method. The undetermined boundary condition at each timestep is denoted as theunknown variablein a set of nonlinear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of nonlinear equations. No selected function isneeded to represent the undetermined function in advance. Two examples are used to demonstrate the characteristics of the proposed method. Thecloseagreementbetween theexactvaluesand theestimated resultscone rmsthevalidity and accuracy of the proposed method. The results show that the proposed method is an accurate, stable, and efe cient method to determine the boundary conditions in two-dimensional nonlinear inverse heat conduction problems.

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.

Chaos-regularization hybrid algorithm for nonlinear two-dimensional inverse heat conduction problem

A numerical model of nonlinear two-dimensional steady inverse heat conduction problem was established considering the thermal conductivity changing with temperature. Combining the chaos optimization algorithm with the gradient regularization method, a chaos-regularization hybrid algorithm was proposed to solve the established numerical model. The hybrid algorithm can give attention to both the advantages of chaotic optimization algorithm and those of gradient regularization method. The chaos optimization algorithm was used to help the gradient regularization method to escape from local optima in the hybrid algorithm. Under the assumption of temperature-dependent thermal conductivity changing with temperature in linear rule, the thermal conductivity and the linear rule were estimated by using the present method with the aid of boundary temperature measurements. Numerical simulation results show that good estimation on the thermal conductivity and the linear function can be obtained with arbitrary initial guess values, and that the present hybrid algorithm is much more efficient than conventional genetic algorithm and chaos optimization algorithm.

Controlled cooling of a hot plate with a water jet

Experiments are performed, under steady-state conditions, to study the boiling heat transfer from a hot plate to a planar jet of water. Temperature control of the heating surface enables the determination of entire boiling curves and the identification of each boiling regime from forced convection to film boiling. By solving the two-dimensional inverse heat conduction problem the local heat flux and corresponding temperature can be computed up to a distance of 55 mm from the stagnation line. Measurements are carried out under atmospheric pressure in a heater temperature range from 100 to 650 °C. The jet temperature is varied to study the influence of subcooling in any regime of the boiling curve. Furthermore experiments are carried out with variable jet velocity and distance between nozzle and heater, for an immersed jet and for a free surface jet, respectively. The transition from nucleate to film boiling at the stagnation point can clearly be determined. Film boiling occurs, e.g., at about 450 °C surface temperature at a jet velocity of 0.8 m/s and a subcooling of 16 K.

Simultaneous space diffusivity and source term reconstruction in 2D IHCP

A numerical marching scheme is presented for the simultaneous recovery of the diffusivity coefficient, spatial source term, temperature, and heat flux distributions in the two-dimensional inverse heat conduction problem when noisy data at the active boundary and the initial measured temperature distribution are given. No information about the quality and/or quantity of the noise is assumed. A proof of stability and convergence of the algorithm are provided together with several numerical examples of interest.