Inverse analysis with integral transformed temperature fields: Identification of thermophysical properties in heterogeneous media

The objective of this work is to introduce the use of integral transformed temperature measured data for the solution of inverse heat transfer problems, instead of the common local transient temperature measurements. The proposed approach is capable of significantly compressing the measured data through the integral transformation, without losing the information contained in the measurements and required for the solution of the inverse problem. The data compression is of special interest for modern measurement techniques, such as the infrared thermography, that allows for fine spatial resolutions and large frequencies, possibly resulting on a very large amount of measured data. In order to critically address the use of integral transformed measurements, we examine in this paper the simultaneous estimation of spatially variable thermal conductivity and thermal diffusivity in one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the Generalized Integral Transform Technique (GITT). The inverse problem is handled with Bayesian inference by employing a Markov Chain Monte Carlo (MCMC) method. The unknown functions appearing in the formulation are expanded in terms of eigenfunctions as well, so that the unknown parameters become the corresponding series coefficients. Such projection of the functions in an infinite dimensional space onto a parametric space of finite dimension also permits that several quantities appearing in the solution of the direct problem be analytically computed. Simulated measurements are used in the inverse analysis; they are assumed to be additive, uncorrelated, normally distributed, with zero means and known covariances. Both Gaussian and non-informative uniform distributions are used as priors for demonstrating the robustness of the estimation procedure.     

Enhanced convergence of eigenfunction expansions in convection-diffusion with multiscale space variable coefficients

ABSTRACT

A convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.     

Eigenfunction expansions for transient diffusion in heterogeneous media

The Generalized Integral Transform Technique (GITT) is employed in the analytical solution of transient linear heat or mass diffusion problems in heterogeneous media. The GITT is utilized to handle the associated eigenvalue problem with arbitrarily space variable coefficients, defining an eigenfunction expansion in terms of a simpler Sturm-Liouville problem of known solution. In addition, the representation of the variable coefficients as eigenfunction expansions themselves has been proposed, considerably simplifying and accelerating the integral transformation process, while permitting the analytical evaluation of the coefficients matrices that form the transformed algebraic system. The proposed methodology is challenged in solving three different classes of diffusion problems in heterogeneous media, as illustrated for the cases of thermophysical properties with large scale variations found in heat transfer analysis of functionally graded materials (FGM), of abrupt variations in multiple layer transitions and of randomly variable physical properties in dispersed systems. The convergence behavior of the proposed expansions is then critically inspected and numerical results are presented to demonstrate the applicability of the general approach and to offer a set of reference results for potentials, eigenvalues, and related quantities.     

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples.     

Stresses in radiative annular fin under thermal loading and its inverse modeling using sine cosine algorithm (SCA)

This work presents a novel mathematical model for the analysis of thermal stresses in a radiative annular fin with temperature-dependent thermal conductivity and radiative parameter. An approximate analytical solution for thermal stresses is derived using a homotopy perturbation method (HPM)-based closed-form solution of steady-state nonlinear heat transfer equation, coupled with classical elasticity theory. The effect of thermal parameters on the temperature field and the thermal stress fields are discussed. The various thermal parameters, such as a parameter describing the temperature-dependent thermal conductivity, coefficient of thermal expansion, coefficient of radiative parameter, and the variable radiative parameter, are inversely estimated for a given stress field. For inverse modeling, a population-based sine cosine algorithm (SCA) was employed to estimate the thermal parameters. The inverse modeling is verified by using the estimated thermal parameters in the closed-form solution of stress field. The reconstructed stress fields obtained from the inversely estimated parameters are then compared with the reference stress field. Results show a very good agreement between the reference stress field and the inversely estimated stress fields.     

NUMERICAL ALGORITHM FOR ESTIMATING TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract

The hybrid scheme of the Laplace transform technique and the central difference approximation is applied to estimate the temperature-dependent thermal conductivity by utilizing temperature measurements inside the material at an arbitrary specified time. In the present study the functional form of the thermal conductivity is not known a priori. Thus, this problem can be regarded as the functional estimation in inverse calculation. The accuracy of the predicted results is examined from various illustrated cases using simulated exact and inexact temperature measurements obtained within the medium. Results show that a good estimation on the thermal conductivity can be obtained with any arbitrary initial guesses of the thermal conductivity. The advantage of the present method in the inverse analysis is that, for most types of boundary conditions, the relation between the thermal conductivity and temperature at any specified time can be determined without measuring the early temperature data.     

Bayesian estimation of thermophysical parameters of thin metal films heated by fast laser pulses

In this paper, we apply the Markov Chain Monte Carlo method, within the Bayesian framework, for the estimation of parameters appearing in the heat conduction model in metals under the condition of thermal non-equilibrium between electrons and lattice. Such non-equilibrium can be experimentally observed in a time scale of up to few picoseconds, during the heating of thin metal films with laser pulses of the order of femtoseconds. Simulated measurements containing random errors are used for the solution of the inverse problem. Results are presented for the simultaneous estimation of the electron–phonon coupling factor, the thermal conductivity and the heat capacity of the electron gas.     

Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

The present work addresses the direct and inverse problems for convective heat transfer with incompressible laminar gas flow in micro-channels, within the range of validity of the slip-flow regime. The direct problem analysis combines the classical integral transform method and the generalized integral transform technique (GITT), by analytically solving the two-dimensional steady-state convection problem and finding a hybrid numerical-analytical solution for the required eigenvalue problem. The inverse problem analysis makes use of the accuracy and robustness of the direct problem solution and focus on the simultaneous identification of the momentum and thermal accommodation coefficients, related to gas flow and heat transfer within micro-channels, besides the usually unknown boundary condition parameters, here represented by the external Biot number. The inverse analysis is based on the availability solely of temperature measurements at the channel external wall, along its length, as obtained for instance via infrared camera thermography. A Bayesian inference approach is adopted in the solution of the identification problem based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings sampling algorithm. A typical example of slip flow in parallel-plates micro-channel is selected to illustrate both the direct and inverse problems solution approaches.     

Explicit boundary heat flux reconstruction employing temperature measurements regularized via truncated eigenfunction expansions

The present work is aimed at developing and validating an explicit inverse problem solution for boundary heat flux reconstruction in thermally thin plates, supposing a large amount of temperature measurements is available, both in space and time, such as typically obtained by modern infrared thermography systems with fine spatial resolution and high frequency. In order to handle the magnification of the measurement errors when applying the derived explicit inversion formula, a regularized representation is proposed for the measured temperatures, consisting of truncated eigenfunction expansions in which only the most important modes, based on the discrepancy principle, are employed. Two applications are considered, the first one consists of a spatially uniform but time varying heat flux with fast transitions, and the second one consists of a spatially varying heat flux concentrated in two small spots. Good results are obtained with very little computational effort, indicating the feasibility of the proposed approach. This work adds to the available standard inverse problems tools, either as an alternative approach or as a companion in obtaining a starting approximate solution for iterative methods.     

A MODE-I CRACK PROBLEM FOR AN INFINITE SPACE IN GENERALIZED THERMOELASTICITY

ABSTRACT

In this work, we solve a dynamical problem of an infinite space with a finite linear crack inside the medium. The Fourier and Laplace transform techniques are used. The problem is reduced to the solution of a system of four dual integral equations. The solution of these equations is shown to be equivalent to the solution of a Fredholm integral equation of the first kind. This integral equation is solved numerically using the method of regularization. The inverse Laplace transforms are obtained numerically using a method based on Fourier expansion techniques. Numerical values for the temperature, stress, displacement, and the stress intensity factor are obtained and represented graphically.     

Heat transfer in a reaction–diffusion system with a moving heat source

A general formulation is presented for a moving boundary problem in which heat is generated at the boundary due to an exothermic reaction involving a species which diffuses into a dispersed phase from an external medium of finite volume. The speed of the moving boundary is prescribed based on the solution of the mass diffusion problem and an analysis is presented of the thermal dynamics of the system. The set of equations describing heat transport leads to a Green’s function type problem with time dependent boundary conditions and the Galerkin finite element method is employed to develop a numerical solution. Transformations are introduced to freeze the moving boundary and partition the domain for ease of computation, and an iterative scheme is defined to satisfy the heat flux jump boundary condition and match the temperature field across the moving boundary. The numerical results are used to set the limits of applicability of an analytical perturbation solution. Essential aspects of thermal dynamics in the system are described and parametric regions resulting in a local temperature hot spot are delineated. Computed contour plots describing thermal evolution are presented for different combinations of parameter values. These may be of utility in the prediction of thermal development, for control and avoidance of hot spot formation, and in physical parameter estimation.     

An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity

An inverse analysis utilizing the conjugate gradient method of minimization and the adjoint equation is used for simultaneously estimating the temperature-dependent thermal conductivity and heat capacity per unit volume of a material. No prior information is used for the functional forms of the unknown thermal conductivity and heat capacity in the present study, thus, it is classified as the function estimation by inverse calculation. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained within the medium. Results show that the CPU time used on a VAX-9420 computer is within 1.4–4.46 s for all the test cases considered here. Moreover, excellent estimations on the thermal properties can be obtained when a good initial guess of either thermal conductivity or heat capacity is given before the inverse calculations.     

A new ANN driven MCMC method for multi-parameter estimation in two-dimensional conduction with heat generation

This paper proposes a hybrid approach, wherein Markov Chain Monte Carlo (MCMC) simulations are used in a Bayesian framework, in conjunction with artificial neural networks (ANN) for solving an inverse heat conduction problem. The proposed algorithm was tested for the problem of steady state two-dimensional heat conduction from a square slab with uniform volumetric internal heat generation. Two variants of this problem namely: (a) estimation of the convection heat transfer coefficient, h and the thermal conductivity of the material, k given the rate of heat generation q_v and (b) estimation of k and q_v given the heat transfer coefficient h, are considered. For both the problems, temperature data at certain fixed locations in the slab serves as the input. For the purposes of establishing the soundness and efficacy of the algorithm, temperatures obtained by a numerical solution to the governing equation for known values of the parameters to be retrieved are treated as “measured” data. However, white noise was added to these data not only to make the analysis realistic but also to ascertain the robustness of the retrieval methodology. In order to significantly reduce the computational time associated with the MCMC simulations, first, a neural network was trained with limited number of solutions to the forward model. This network was used to replace the forward model (conduction equation) during the process of retrievals with Markov Chain Monte Carlo simulations in a Bayesian framework, thereby making the retrievals hybrid. The performance of the proposed hybrid technique was evaluated for different priors and various levels of noise. Comparisons with retrievals done directly by ANN revealed that the performance of the hybrid method is demonstrably superior, particularly with noisy data.     

Estimation of parameters in multi-mode heat transfer problems using Bayesian inference – Effect of noise and a priori

Parameter estimation problems and heat source/flux reconstruction problems are some of the most frequently encountered inverse heat transfer problems. These problems find their application in many areas of science and engineering. The primary focus of this paper is on the heat transfer parameter estimation for a two-dimensional unsteady heat conduction problem with (a) convection boundary condition and (b) convection and radiation boundary condition. The paper demonstrates the effect of a priori model on the performance of the algorithm at different noise levels in the measured data. The inverse problem is solved using three different a priori models namely normal, log normal and uniform. The posterior PDF is sampled using the Metropolis–Hastings sampling algorithm. Both single-parameter estimation and multi-parameter estimation problems are addressed and the effects of corresponding a priori models are studied. It was found that the mean and maximum a posteriori estimates for thermal conductivity and the convection heat transfer coefficient were insensitive to the a priori model at all the considered noise levels for the single-parameter estimation problem. At high noise levels in the two-parameter estimation problem, the estimates for thermal conductivity and convection coefficient were sensitive to the a priori model. It was also found that the standard deviation of the samples was correlated to the error in estimation in the single-parameter estimation case. In three parameter estimation case, alternate solutions to the same problem were retrieved due to a strong correlation between the convection coefficient and the emissivity. However, a more informative a priori model could address this issue.     

Prediction of the thermal conductivity of metal hydrides – The inverse problem

With sustainability as an important and driving theme, not merely of research, but that of our existence itself, the effort in developing sustainable systems takes many directions. One of these directions is in the transport sector, particularly personal transport using hydrogen as fuel, which logically leads on to the problem of hydrogen storage. This paper deals with the prediction of the effective conductivity of beds of metal hydride for hydrogen storage. To enable modeling of the effective thermal conductivity of these systems, it is necessary to arrive at the functional dependence of the thermal conductivity of the solid hydride on its hydrogen concentration or content. This is the inverse problem in thermal conductivity of multiphase materials. Inverse methods in general are those where we start from known consequences in order to find unknown causes. Using published and known data of the effective thermal conductivity of the hydride–hydrogen assemblage, we arrive at the unknown hydride conductivity by analysis. Among the models available in the literature for determination of the effective conductivity of the bed from the properties of the constituent phases, the model of Raghavan and Martin is chosen for the analysis as it combines simplicity and physical rigor. The result is expected to be useful for predicting the thermal conductivity of hydride particles and determining the optimum heat transfer rates governing the absorption and desorption rates of hydrogen in the storage system.     

Inverse determination of heat generation sources in two-dimensional homogeneous solids: Application to orthotropic medium

An implementation of the Network Simulation Method for determining the time-dependent heat generation of a source in a 2-D heat conduction problem under first and second type boundary conditions is presented. The thermal properties of the orthotropic solid are temperature-dependent. No prior information of the functional dependence of the time-dependent source was required. Input data for this inverse problem comprise the temperature history (measurements) at a particular location of the boundary. The common iterative least-squares approach, typical for this kind of ill-posed problem, is used in the estimation, while a piecewise continuous function is proposed as an approximate solution for the estimations. An application to estimate the triangular time dependence of the heat source generation, with error-affected measurements, is shown.     

A simplex search method for a conductive–convective fin with variable conductivity

An inverse problem is solved for simultaneously estimating the convection–conduction parameter and the variable thermal conductivity parameter in a conductive–convective fin with temperature dependent thermal conductivity. Initially, the temperature field is obtained from a direct method using an analytical approach based on decomposition scheme and then using a simplex search minimization algorithm an inverse problem is solved for estimating the unknowns. The objective function to be minimized is represented by the sum of square of the error between the measured temperature field and an initially guessed value which is updated in an iterative manner. The estimation accuracy is studied for the effect of measurement errors, initial guess and number of measurement points. It is observed that although very good estimation accuracy is possible with more number of measurement points, reasonably well estimation is obtained even with fewer number of measurement points without measurement error. Subject to selection of a proper initial guess, it is seen that the number of iterations could be significantly reduced. The relative sensitiveness of the estimated parameters is studied and is observed from the present work that the estimated convection–conduction parameter contributes more to the temperature distribution than the variable conductivity parameter.     

Neural Computation to Estimate Heat Flux in an Atmospheric Plasma Spray Process

Temperature is a key parameter in the thermal spray process and is a consequence of the heat flux experienced by the workpiece. This paper deals with the estimation of the heat flux transmitted to a workpiece from an atmospheric plasma spray torch during the preheating process often implemented in thermal spraying. An inverse heat conduction problem solution using a conjugate gradient method was considered to determine the heat flux starting from a known temperature distribution. Results from the later method were used to train an artificial neural network to discover correlations between selected processing parameters and heat flux.     

One-step GPS for the estimation of temperature-dependent thermal conductivity

In this paper we are concerned with the estimation of temperature-dependent thermal conductivity of a one-dimensional inverse heat conduction problem. First, we construct a one-step group-preserving scheme (GPS) for the semi-discretization of quasilinear heat conduction equation, and then derive a quasilinear algebraic equation to determine the unknown thermal conductivity under a given initial temperature and a measured temperature perturbed by noise at time T. The new method does not require any prior information on the functional form of thermal conductivity. Several examples are examined to show that the new approach has high accuracy and efficiency, and the number of iterations spent in solving the quasilinear algebraic equation is smaller than five even in a large temperature range.     

On the inverse transient heat-transfer problem in structural elements exposed to solar radiation

The inverse transient heat-transfer problem in walls, whether or not exposed to solar radiation, i.e. the estimation of the thermal properties of a wall if the transient temperature and/or heat flow fields are known, is interesting both from the theoretical and practical point of view. An attempt is made to analyse and solve this problem. Methods are developed for estimating the thermal properties of structural elements which are already parts of existing buildings, i.e. under real transient, non-periodic conditions. The finite-difference and experimental examples presented show that the thermal diffusivity, the thermal conductivity and the overall heat transfer coefficient may be estimated with very good accuracy by taking on-site temperature and heat flow measurements.