An Inverse Heat Conduction Problem

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

Introduction.- Mathematical Preliminaries.- Regularization of moment problems by trancated expansion and by the Tikhonov method.- Backus-Gilbert regularization of a moment problem.- The Hausdorff moment problem: regularization and error estimates.- Analytic functions: reconstruction and Sinc approximations.- Regularization of some inverse problems in potential theory.- Regularization of some inverse problems in heat conduction.- Epilogue.- References.- Index.

Solution of an inverse transient heat conduction problem in a part of a complex domain

Purpose – The purpose of this paper is to reduce the destructive effects of existing unavoidable noises contaminating temperature data in inverse heat conduction problems (IHCP) utilizing the wavelets.

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

In [1] the application of a method of concentrated sources to heat conduction problems was presented. In the present paper, the method is extended to include thermal displacements and stresses. The method is applicable to direct and inverse problems as well. The extension of the method to thermoelasticity consists in introducing, besides “fictitious” concentrated heat sources, “fictitious” concentrated mechanical forces outside a considered region. In a direct problem of heat conduction theory (or thermoelasticity), we look for a solution to partial differential equations subject to suitable initial-boundary conditions. In an inverse problem of heat conduction, we look for a boundary temperature (among other things), assuming that a temperature is prescribed on a subregion of the body and for each time. In the case of an inverse thermoelastic problem, we may look for a boundary thermomechanical load when some of the thermomechanical fields, such as temperature, displacement, or stress, are prescribed over a subregion of the body and for each time.

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

This study introduces a Bayesian spatiotemporal modeling approach to solve inverse heat conduction problems (IHCPs), employing penalized splines within a spatiotemporal forward model. The complexity and ill-posed nature of IHCPs, characterized by potential nonexistence, nonuniqueness, or instability of solutions, pose significant challenges for traditional methods. Addressing this, our study presents a spatiotemporal forward model that incorporates spatial, temporal, and interaction terms, accurately capturing the intricate dynamics inherent in IHCPs and using this information as a leverage to solve the inverse problem. We adopted a Bayesian inference framework for the subsequent parameter estimation problem and developed a Gibbs sampling algorithm to sample from the posterior distribution of the model's parameters, enhancing the estimation process. Through case studies on a one-dimensional (1D) heat simulation and a pool boiling experiment using multisensor thermocouple data for heat flux reconstruction, we demonstrate the model's superiority over traditional methods. The inclusion of the spatiotemporal interaction term significantly enhances model performance, indicating its potential for broader application in solving IHCPs. The application of this method in both simulated and real-world scenarios highlights its effectiveness in capturing the spatiotemporal complexities of IHCPs. This work contributes to the field by offering a robust methodology for addressing the spatial and temporal complexities inherent in IHCPs, supported by a comprehensive Bayesian inference framework and the use of a Gibbs sampling algorithm for parameter estimation.

Two-dimensional Inverse Heat Conduction Problem Using a Meshless Manifold Method

Inverting methods for thermal reservoir evaluation of enhanced geothermal system

Estimation of unknown surface conditions using measured temperature time history inside or back wall of a solid body is known as inverse heat conduction problem. The purpose of present paper is to solve inverse heat conduction problem using iterative method in conjunction with measured thermocouple data. Exact solution of one-dimensional heat conduction equation is obtained to calculate temperature distribution inside a finite slab. An analytical expression of the sensitive coefficient is derived for constant wall heat flux. It is observed that the sensitive coefficient varies linearly with wall heat flux. Experiments were conducted in Kinetic Heating Facility to obtain temperature-time data. The facility works employing with close loop feed-back system with thyristor controlled thermac. Experimental results are compared with the estimated values and are found in good agreement. The Kinetic Heating Facility can easily be adopted to estimate the thermal diffusivity of material.

Resolving inverse heat conduction problems (IHCPs) based on interior measured temperatures is a reliable approach for aerodynamic heat prediction. In most IHCP studies, it is generally assumed that the data obtained by temperature sensors are the positional temperature, whereas in actual measurements, the thermocouple undergoes a transient heat transfer process, leading to a slight bias in its readings compared to the desired temperature at the target location. In this paper, a thermocouple model is established to simulate the sensor dynamic response effect. Then, a traditional space marching method (SMM) and the calibration integral equation method (CIEM) are applied to investigate the sensor dynamic response on surface heat flux predictions. The Gauss filtering method and future time regularization method are used as regularization schemes for SMM and CIEM, respectively. For achieving optimal predictions, phase-plane and cross-correlation analyses are applied to extract the optimal regularization parameter. The results indicate that delay and attenuation will occur in surface heat flux prediction using SMM, and this negative phenomenon can be eliminated by using CIEM. In general, CIEM has great merits for resolving IHCPs compared to traditional approaches, which eliminate the impact of dynamic response and improve the accuracy of aerodynamic heat predictions. The results of this study can provide a reference for transient IHCPs that should take the influence of sensor dynamic response on inverse prediction into consideration.

A linear inverse model for the temperature-dependent thermal conductivity determination in one-dimensional problems

The inverse heat conduction problem with many internal responses is considered. After discretization with respect to time, the problem is described by a system of the Helmholtz equations in a recurrent form. An approximate solution of a heat conduction problem in an integral form is shown. Then, an approximate solution of an inverse heat conduction problem in a flat slab is presented for many internal temperature responses. Stability of the solution with respect to the internal response errors is investigated for two cases: when the integrals are calculated with the use of the average value theorem and when they are calculated numerically. Analysis of the norm of a matrix that is essential for the solution stability shows that in the case of three sensors the norm slightly changes with change of the middle sensor location. If more than three sensors are taken into consideration, the results practically will not change comparing to the case of three sensors. The internal temperature response errors are suppressed if the time step is greater than a certain critical value.

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.