Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

Identification of boundary heat fluxes in a falling film experiment using high resolution temperature measurements

In this paper, we consider a three-dimensional inverse heat conduction problem (IHCP) in a falling film experiment. The wavy film is heated electrically by a thin constantan foil and the temperature on the back side of this foil is measured by high resolution infrared (IR) thermography. The transient heat flux at the inaccessible film side of the foil is determined from the IR data and the electrical heating power. The IHCP is formulated as a mathematical optimization problem, which is solved with the conjugate gradient method. In each step of the iterative process two direct transient heat conduction problems must be solved. We apply a one step θ-method and piecewise linear finite elements on a tetrahedral grid for the time and space discretization, respectively. The resulting large sparse system of equations is solved with a preconditioned Krylov subspace method. We give results of simulated experiments, which illustrate the performance and tuning of the solution method, and finally present the estimation results from temperature measurements obtained during falling film experiments.     

A SERIAL SOLUTION FOR THE 2-D INVERSE HEAT CONDUCTION PROBLEM FOR ESTIMATING MULTIPLE HEAT FLUX COMPONENTS

A serial algorithm for the inverse heat conduction problem (IHCP) has been developed to estimate the individual flux components, one by one, at the unknown boundary, based on the function specification method. The sensitivity coefficient defined in this algorithm brings out the influence of the heat flux components independent of each other. The objective function minimizes the difference in the measured temperature and the contribution of the individual flux component to the thermal field at the sensor location. The serial algorithm developed here could be used with data from both overspecified and underspecified sensors with respect to the number of flux components. The method was tested for delineating independent heat fluxes at the boundary of a two-dimensional solid for both space- and time-varying heat fluxes. Simulated thermal histories obtained from direct solution were used as inputs for the inverse problem for characterizing the new algorithm.

Three types of analyses were done on the results of the IHCP, focused on (1) the convergence of error in estimated temperatures at the different sensor locations, (2) overall error in estimated temperatures for the whole domain, and (3) the total heat energy transferred across the boundary. It is shown that the optimum configuration of independent unknown fluxes is given by the one with minimum energy estimates across the boundary, for both cases.     

Two dimensional steady-state inverse heat conduction problems

In this work we estimate the surface temperature in two dimensional steady-state in a rectangular region by two different methods, the singular value decomposition (SVD) with boundary element method (BEM) and the least-squares approach with integral transform method (ITM). The BEM method is efficient for solving inverse heat conduction problems (IHCP) because only the boundary of the region needs to be discretized. Furthermore, both temperature and heat flux at the unknown boundary are estimated at the same time. The least-squares technique involves solving the equations constructed from the measured temperature and the exact solution. The measured data are simulated by adding random errors to the exact solution of the direct problem. The effects of random errors on the accuracy of the predictions are examined. The sensitivity coefficients are also presented to illustrate the effect of sensor location on the estimated surface conditions. Numerical experiments are given to demonstrate the accuracy of the present approaches.     

Thermocouple Data in the Inverse Heat Conduction Problem

The presence of thermocouples inside a heat-conducting body will distort the temperature field in the body and may lead to significant bias in the temperature measurement. If temperature histories obtained from thermocouples are used in the inverse heat conduction problem (IHCP), errors are propagated into the IHCP results. The bias in the thermocouple measurements can be removed through use of appropriate detailed thermocouple models to account for the dynamics of the sensor measurement. The results of these models can be used to generate correction kernels to eliminate bias in the thermocouple reading, or can be applied as sensitivity coefficients in the IHCP directly. Three-dimensional and axisymmetric models are compared and contrasted and a simple sensitivity study is conducted to evaluate the significance of thermal property selection on the temperature correction and subsequent heat flux estimation. In this paper, a high-fidelity thermocouple model is used to account for thermocouple bias in an experiment to measure heat fluxes from solidifying aluminum to a sand mold. Correction kernels are obtained that are used to demonstrate the magnitude of the temperature measurement bias created by the thermocouples. The corrected temperatures are used in the IHCP to compute the surface heat flux. A comparison to IHCP results using uncorrected temperatures shows the impact of the bias correction on the computed heat fluxes.     

Reconstruction of local heat fluxes in pool boiling experiments along the entire boiling curve from high resolution transient temperature measurements

In this paper, we consider a transient inverse heat conduction problem (IHCP) defined on an irregular three-dimensional (3D) domain in pool boiling experiments. Heat input to a circular copper heater of 35 mm diameter and 7 mm thickness is provided by a resistance heating foil pressed to the bottom of the heater. The heat flux at the inaccessible boiling side is estimated from a number of temperature readings in the heater volume. These temperatures are measured by some high-resolution microthermocouples, which are mounted 3.6 μm below the surface in the test heater. The IHCP is formulated as a mathematical optimization problem and solved by the conjugate gradient (CG) method. The arising partial differential equations (PDEs) are solved using the software package DROPS. A simulation case study is used to validate the performance of the solution approach. Finally, we apply the solution approach to the IHCP in pool boiling experiments. The procedure enables the reconstruction of local instantaneous heat flux distribution on the heater surface at different locations along the boiling curve.     

A data method using the inner temperature difference to improve the stability of the inverse heat conduction problems

Abstract

This article proposes a method to construct two series systems for improving the stability of the inverse heat conduction problems (IHCP) in a finite slab. The transfer function between the surface heat flux or temperature and the inner temperature difference is respectively obtained by Laplace transform technique firstly. Then the series systems which can solve IHCP based on the inner temperature difference are constructed by replacing the unsuitable zero and pole points of the transfer function approximated by è approximation. Finally the effects of the series systems are evaluated by a typical example. The results of the evaluation show that this method can obtain the surface heat flux and temperature by the inner temperature difference, and enhance the response speed of the measurement system at the same time. In addition this method can also improve the signal to noise ratio (SNR) of the inverse solutions by selectively amplifying the high SNR parts of the inner temperature difference. The present work provides an effective method to improve the stability of IHCP.     

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.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

A two-dimensional inverse heat conduction problem in estimating the fluid temperature in a pipeline

An inverse heat conduction problem (IHCP) was investigated in the two-dimensional section of a pipe elbow with thermal stratification to estimate the unknown transient fluid temperatures near the inner wall of the pipeline. An inverse algorithm based on the conjugate gradient method (CGM) was proposed to solve the IHCP using temperature measurements on the outer wall. In order to examine the accuracy of estimations, some comparisons have been made in this case. The temperatures obtained from the solution of the direct heat conduction problem (DHCP) using the finite element method (FEM) were pseudo-experimental input data on the outer wall for the IHCP. Comparisons of the estimated fluid temperatures with experimental fluid temperatures near the inner wall showed that the IHCP could accurately capture the actual temperature in form of the frequency of the temperature fluctuations. The analysis also showed that the IHCP needed at least 13 measurement points for the average absolute error to be dramatically reduced for the present IHCP with 37 nodes on each half of the pipe wall.     

Practical application of inverse heat conduction for wall condition estimation on a rotating cylinder

The solution of the linear, inverse, transient heat conduction problem (IHCP) in a cylindrical geometry is analysed. The rotating cylinder under investigation is experiencing boiling convection induced by the impingement of a water jet. The initial temperature is known, additional temperature measurements in time are taken with sensors positioned at a constant radius within the solid material, and the estimation of the wall heat flux at the external radius is sought. First, simulated temperature measurements inside the cylinder are processed in order to be used to estimate the wall heat flux. When noise is present in the data, some of the simulated results obtained using the least squares method exhibit oscillatory behavior, but these large oscillations are substantially reduced by the implementation of a regularization technique. Real experimental data are also used for the wall condition estimation and for the subsequent building of local boiling curves are plotted and discussed. The question of the possible effect of a temperature dependent conductivity on the reconstructed wall condition is also considered.     

Inverse problem of coupled thermoelasticity for prediction of heat flux and thermal stresses in an annular cylinder

This study presents a means of solving the inverse problem of coupled thermoelasticity in an annular cylinder. While knowing the temperature history at any point of the body, the boundary time-varying heat flux can be computed, and subsequently the distributions of temperature and thermal stresses in the cylinder can be determined as well. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained on the surface and at an interior location of the cylinder. Numerical results demonstrate that excellent estimation for the heat flux and thermal stresses distributions can be obtained for all the test cases considered in this study.     

Heat flux estimation of a plasma rocket helicon source by solution of the inverse heat conduction problem

A solution of the inverse heat conduction problem (IHCP) by the steepest descent method is carried out in order to determine the waste heat flux from a helicon plasma discharge using transient surface temperature measurements obtained from infrared thermography. The infrared camera data is calibrated against thermocouple data and mapped to real locations on the observed surface. The magnitude and distribution of the heat flux to the gas containment tube in the helicon is investigated as the applied power, gas flow rate, magnetic field distribution and neutral gas are varied.     

Two-dimensional non-linear inverse heat conduction problem based on the singular value decomposition

In this paper an efficient sequential method is developed in order to estimate the unknown boundary condition on the surface of a body from transient temperature measurements inside the solid. This numerical approach for solving an inverse heat conduction problem (IHCP) takes into account two-dimensional problems, planar or axisymmetric cylindrical, composite materials with irregular boundaries and temperature-dependent thermal properties. The unknown surface condition is assumed to have abrupt changes at unknown times. The regularization procedure used for the solution of the IHCP is based on the singular value decomposition technique. An overall estimate of error is defined in order to find the optimal estimation in the 2D IHCP (linear and non-linear). The stability and accuracy of the scheme presented is evaluated by comparison with the Function Specification Method. This comparative study has been carried out using numerically simulated data, and the parameters considered include shape of input, noise level of measurement, size of time step and temperature-dependent thermal properties. A good agreement was found between both methods. Beside this, the slight differences on estimations and number of future temperatures are discussed in this paper.     

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.     

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.     

Determination of thermal contact resistance from transient temperature measurements

The temperature distribution in combustion engine components is highly influenced by thermal contact resistance. For the prediction and optimisation of the thermal behaviour of modern combustion engines knowledge about the contact heat transfer is crucial.Available correlations to predict the contact resistance are simplifications of the real geometric conditions and only tested for moderate pressures up to 7 MPa. Typical combustion engine applications include contact pressures up to 250 MPa.The experimental approach presented here to derive the thermal contact resistance in terms of contact heat transfer coefficients for high temperature and high pressure conditions is based on transient infrared temperature measurements. Two bodies initially at two different temperatures are brought in contact and the surface temperature histories are recorded with a high-speed infrared camera. The contact heat flux is calculated by solving the related inverse problem. From the contact heat flux and from the measured temperature jump at the interface the contact heat transfer coefficient is calculated.The inverse method used for the calculation of the heat flux is based on the analytical solution for a semi-infinite body and a step response to a Neumann boundary condition. This method provides an algorithm that is used in a sequential manner. The use of “future” temperature data greatly improve the stability of the governing equations and reduce the sensitivity to measurement errors.     

用共轭梯度法与蒙特卡洛法求解三维系统的辐射热负荷反问题

采用蒙特卡洛法与共轭梯度法确定三维矩形炉膛中燃烧器或加热表面的温度分与热负荷分布。讨论了测量误差对计算结果的影响，结果表明：当测量值不含误差时，用共轭梯并法计算得到的加热表面的温度值和热流量值与其真实值吻合很好。实验测量误差对计算精度的影响与加热表面及受热表面之间的辐射热交换系数有关，辐射热交换系数越大，测量误差对计算结果的影响越大。     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric inverse heat conduction problem (IHCP). In the radially symmetric IHCP data on an inner fixed boundary is determined from Cauchy data given on an outer boundary. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach of Johansson et al. (2008) for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.