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.     

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.     

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.     

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.     

A robust and fast algorithm for three-dimensional transient inverse heat conduction problems

Despite numerous studies of inverse heat conduction problems (IHCP) over the last several decades, their solutions still suffer from the mathematical difficulties and the bottleneck of currently available numerical methods for large-scale problems. In this paper, we present a robust and efficient algorithm for the solution of a specific type of three-dimensional (3D) IHCP commonly involved in various engineering applications. The solution method incorporates the Tikhonov regularization for tackling the severe ill-posedness and the conjugate gradient (CG) method for solving the resulting minimization problems. A model function approach is used to significantly reduce the effort needed to find the optimal Tikhonov regularization parameter. The proposed solution method requires no a priori knowledge of the measurement noise and is much more computationally efficient than the traditional Tikhonov regularization-based inversion approaches. Thus, it can be used for the efficient solution of large-scale practical problems. Two simulation case studies of practical significance are presented to validate and assess the performance of the proposed method. Finally, the solution method is successfully applied to the reconstruction of instantaneous heat fluxes from experimentally measured temperature data.     

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.     

Effect of pressure on heat transfer coefficient at the metal/mold interface of A356 aluminum alloy

The aim of this paper is to correlate interfacial heat transfer coefficient (IHTC) to applied external pressure, in which IHTC at the interface between A356 aluminum alloy and metallic mold during the solidification of casting under different pressures were obtained using the inverse heat conduction problem (IHCP) method. The method covers the expedient of comparing theoretical and experimental thermal histories. Temperature profiles obtained from thermocouples were used in a finite difference heat flow program to estimate the transient heat transfer coefficients. The new simple formula was presented for correlation between external pressure and heat transfer coefficient. Acceptable agreement with data in literature shows the accuracy of the proposed formula.     

An inversion approach for the inverse heat conduction problems

The inverse heat conduction problems (IHCP) analysis method provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. Successful applications of the IHCP method depend mainly on the efficiency of the inversion algorithms. In this paper, a generalized objective functional, which has been developed using a generalized stabilizing functional and a combinational estimation that integrates the advantages of the least trimmed squares (LTS) estimation and the M-estimation, is proposed. The objective functional unifies the regularized M-estimation, the regularized least squares (LS) estimation, the regularized LTS estimation, the regularized combinational estimation of the LTS estimation and the M-estimation, and the regularized combinational estimation of the LS estimation and the M-estimation into a concise formula. The filled function method, which is coupled with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, is developed for searching a possible global optimal solution. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. Favorable numerical performances and satisfactory results are observed, which indicates that the proposed algorithm is successful in solving the IHCP.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

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.     

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.     

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.     

Training based, moving digital filter method for real time heat flux function estimation

In this paper the neural networks is utilized to estimate the “filter coefficients” needed to estimate heat flux in a particular system. In developing the training phase of the network inspiration is drawn from the Burgraff's exact solution of the IHCP as well as the filter method. Thus, the estimation phase neither requires any temperature field nor the sensitivity coefficients calculations. The neural network used in this work is a 2-layer perceptron. It is shown via classical triangular heat flux test cases that the method can yield very accurate, very efficient as well as stable estimations.     

Inverse heat conduction estimation of inner wall temperature fluctuations under turbulent penetration

Turbulent penetration can occur when hot and cold fluids mix in a horizontal T-junction pipe at nuclear plants. Caused by the unstable turbulent penetration, temperature fluctuations with large amplitude and high frequency can lead to time-varying wall thermal stress and even thermal fatigue on the inner wall. Numerous cases, however, exist where inner wall temperatures cannot be measured and only outer wall temperature measurements are feasible. Therefore, it is one of the popular research areas in nuclear science and engineering to estimate temperature fluctuations on the inner wall from measurements of outer wall temperatures without damaging the structure of the pipe. In this study, both the one-dimensional(1D) and the two-dimensional(2D) inverse heat conduction problem(IHCP) were solved to estimate the temperature fluctuations on the inner wall. First, numerical models of both the 1D and the 2D direct heat conduction problem(DHCP) were structured in MATLAB, based on the finite difference method with an implicit scheme. Second, both the 1D IHCP and the 2D IHCP were solved by the steepest descent method(SDM), and the DHCP results of temperatures on the outer wall were used to estimate the temperature fluctuations on the inner wall. Third, we compared the temperature fluctuations on the inner wall estimated by the 1D IHCP with those estimated by the 2D IHCP in four cases:(1) when the maximum disturbance of temperature of fluid inside the pipe was 3℃,(2) when the maximum disturbance of temperature of fluid inside the pipe was 30℃,(3) when the maximum disturbance of temperature of fluid inside the pipe was 160℃, and(4) when the fluid temperatures inside the pipe were random from 50℃ to 210℃.     

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.     

A technique for uncertainty analysis for inverse heat conduction problems

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the uncertainty analysis. An example calculation is presented. Parameter importance factors are defined and computed for the example problem; the volumetric heat capacity is the dominant parameter and an explanation is offered. Thoughts are presented on extending the analysis to include the non-linear problem of temperature dependent properties.     

液态金属浇注过程中流动与传热耦合数值模拟

浇注系统的设计是铸造工艺设计的主要内容。利用数值模拟技术,可对浇注系统进行优化设计。建立了液态金属流动和传热的数学模型,讨论了边界条件、初始条件的选择,对液态金属浇注过程中的流动与传热耦合问题进行了数值模拟,并对模拟结果进行了讨论。所得结论对于浇注系统和过程的优化设计具有重要的实用价值。     

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.     

General solution of one-dimensional,time dependent coupled heat flow systems

A new, powerful method of analysis, involving the combined use of finite integral transform and finite element techniques, is presented for the solution of time dependent heat flow systems composed of many one-dimensional elements connected through the nodes. This method leads to an eigenvalue problem which is not of the conventional Sturm-Liouville type. A procedure for the determination of the eigenvalues is described. The solution obtained is in the form of an infinite series and contains quasi-steady and transient terms. The general solution obtained can be applied in the mathematical modelling of many engineering applications such as the determination of the penetration of the daily temperature cycle into buildings, the analysis of heat transfer in array of extended surfaces in compact heat exchangers, and many others.