An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

The present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\textquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.     

Transient analytical solution to heat conduction in composite circular cylinder

An analytical method leading to the solution of transient temperature filed in multi-dimensional composite circular cylinder is presented. The boundary condition is described as time-dependent temperature change. For such heat conduction problem, nearly all the published works need numerical schemes in computing eigenvalues or residues. In this paper, the proposed method involves no such numerical work. Application of ‘separation of variables’ is novel. The developed method represents an extension of the analytical approach derived for solving heat conduction in composite slab in Cartesian coordinates. Close-formed solution is provided and its agreement with numerical result is good which demonstrates a good accuracy of the developed solution form.     

Transient analytical solution to heat conduction in multi-dimensional composite cylinder slab

The analytical solution for the problem of transient heat conduction in multi-dimensional composite cylinder slab is developed for a time-dependent boundary condition. For such problems, numerical programs are needed to obtain eigenvalues and residues in most of the published papers. The numerical schemes may become unstable due to the existence of imaginary eigenvalues in multi-dimensional cases. In this paper, the proposed analytical method involves no numerical complications. By a novel application of the methods of the Laplace transform and separation of variables together with variable transformations, the residue calculation is avoided. The developed analytical method is powerful which represents extension of the analytical approach derived for the heat conduction problem in Cartesian coordinates. A closed form solution is provided. Calculation examples show that the analytical solutions predict good agreement with the numerical results.     

Analysis and solution of the ill-posed inverse heat conduction problem

The inverse conduction problem arises when experimental measurements are taken in the interior of a body, and it is desired to calculate temperature and heat flux values on the surface. The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. A special solution procedure is developed for the one-dimensional case which replaces the heat conduction equation with an approximating hyperbolic equation. If viewed from a new perspective, where the roles of the spatial and time variables are interchanged, then an initial value problem for the damped wave equation is obtained. Since the formulation is well-posed, both analytic and numerical solution procedures are readily available. Sample calculations confirm that this approach produces consistent, reliable results for both linear and nonlinear problems.     

Numerical analysis of peridynamic and classical models in transient heat transfer,employing Galerkin approach

The main scope of this research is to study transient heat transfer based on combined peridynamic theory and classical model. The principle of the energy conservation is used to develop the governing equation in case of existing both classical and peridynamic heat transfer mechanisms. The integro‐differential equation is developed containing the classical coefficient of heat conduction and peridynamic kernel function. To solve the equation, a spectral method based on a series solution in accordance with Gelerkin approach is implemented. Trapezoidal method is employed to compute the integral terms. Implicit Euler method is used to numerically solve the problem in time domain. In part of numerical studies, different kernel functions are attributed to the peridynamic coefficient of heat transfer. Relevant constraints are applied for making equivalency between classical and each of the peridynamic kernel functions. To ensure the accuracy of the numerical results, adequate convergence analyses are conducted. The results related to a few cases are compared to relevant data reported in the open literature showing good agreements. Moreover, comparisons are made to observe probable variations versus time. In case studies, effects of the nonlocal range on the heat transfer were studied.     

Transient Temperature Distribution Within a Flat Sheet During the Welding Process — Analytical Solution

An analytical study of transient heat conduction process with moving heat source/sink was considered. In this study a closed-form solution for the heat conduction equation is obtained. Using the obtained solution, the spatial temperature distribution and the temperature time history could be obtained. The obtained results are compared with numerical and experimental data in the literature. The comparison shows that the present solution can be used do determine the effect of different heat flow parameters on the temperature pattern and history in any similar heat conduction problem.     

General formulation and analysis of hyperbolic heat conduction in composite media

Some recent experimental results show the existence of reflections of thermal waves at the interface of dissimilar materials in superfluid helium. In light of these results, a theoretical investigation of thermal waves in composite is provided to give a theoretical foundation to the observed phenomenon. A general one-dimensional temperature and heat flux formulation for hyperbolic heat conduction in a composite medium is presented. Also, the general solution, based on the flux formulation, is developed for the standard three orthogonal coordinate systems. Unlike classical parabolic heat conduction, heat conduction based on the modified Fourier's law produces non-separable field equations for both the temperature and flux and therefore standard analytical techniques cannot be applied in these situations. In order to alleviate this difficulty, a generalized finite integral transform technique is proposed in the flux domain and a general solution is developed for the standard three orthogonal coordinate systems. The general solution is applied to the case of a two-region slab with a pulsed volumetric source and insulated exterior surfaces which displays the unusual and controversial nature associated with heat conduction based on the modified Fourier's law in composite regions.     

DPL Model Analysis of Non—Fourier Heat Conduction Restricted by Continuous Boundary Interface

IntroductionAs widely known, the hahonal Fourier law isbased on a large quantity of regular heat transfer (i.e. thethermal bine scale is comparatively lOng and the heatflux density is comparatively small) experiments and it'sjust a phenomenological descriphon of regular thermalProcesses. The Fourier law itself mpes an infinitespeed of Propagation of thermal distUrbance, indicatingthat a local change in tempera~ causes aninstantaneous per'tUrbation in the temperatore at eachPOint in the medi…     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting method

Abstract

In this article, an explicit Lie-group shooting method (LGSM) is developed to solve the time-dependent heat source and the initial data for backward heat conduction problems. To recover both unknown data simultaneously, it is very difficult to obtain a stable solution by explicit or implicit schemes. To solve these problems by using conventional numerical schemes, numerical iterative regularization techniques and numerical integration techniques are necessary. To avoid these numerical techniques and to increase the computational efficiency, an explicit LGSM is developed. According to the solution of the quadratic equation of the LGSM, the initial condition can be directly obtained by using the final condition and boundary conditions at the initial time and final time. Using the reciprocal relationship of the solutions for the initial condition and the final condition, the proposed algorithm can avoid numerical integration and numerical iteration. Additionally, a closed-form formula from a two-point Lie-group equation can be directly used to calculate the heat source term. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and stable solution, even with noisy measurement data, by comparing the estimation results with the existing literature.     

Application of Noise Reduction Technique in the Inverse Analysis Using Kalman Filter

Abstract

The inverse techniques usually employ the sensor measurement to estimate the unknown quantities. Regardless of sensor accuracy, the measurements contain some degrees of uncertainty and error, inadvertently. Inasmuch as, the inverse problems are ill-conditioned in general term, the measurement errors cause instabilities, perturbations, and excursions in the solution procedure. To handle the noise difficulties, a novel approach is proposed in the current study. In this method, the measurement errors are filtered to alleviate the noise priori to utilization of inverse method. The Kalman filter is implemented to remove the noise from the original sensor readings. Thereafter, the Levenberg–Marquardt method is implemented to predict the unknown. To evaluate the accuracy and robustness of the developed approach, a high nonlinear test case containing moving boundary heat conduction problem is investigated. Comparing the obtained results illustrates the improvement of inverse solution procedure by employing the noise filtering technique.     

A BOUNDARY CONDITION DISSECTION METHOD FOR THE SOLUTION OF BOUNDARY-VALUE HEAT CONDUCTION PROBLEMS WITH POSITION-DEPENDENT CONVECTIVE COEFFICIENTS

A boundary condition dissection method is developed on the postulation that a condition imposed on the boundary of a heat conduction problem may be realized in practice by using conditions not of the imposed kind. Thus, a Robin condition imposed on the boundary may be dissected as a linear combination of the boundary heal flux and temperature, arid by doing so, heat conduction problems with position-dependent convective coefficients can be solved by the separation-of-variables technique. The method leads to the solution of a Fredholm integral equation of the second kind with a degenerate kernel, and this equation may be solved by using simultaneous algebraic equations. The method is superior to the finite-difference method and the methods of weighted residuals, which have been conventionally used in solving suck problems. Extension of the method to the solution of other heat conduction problems is also possible and mentioned in the paper.     

Bayesian Estimation of Temperature-Dependent Thermophysical Properties and Transient Boundary Heat Flux

In this article, we apply a Bayesian approach for the simultaneous identification of volumetric heat capacity, thermal conductivity, and boundary heat flux, in a one-dimensional nonlinear heat conduction problem. The Markov chain Monte Carlo sampling approach, implemented in the form of the Metropolis–Hastings algorithm, was used for the solution of the inverse problem. Simulated temperature measurements were used in the inverse analysis in order to examine the accuracy and stability of the overall approach. Independent measurement data were used to construct the prior model for the coefficients to be estimated. The approach is also applied to experiments involving the heating of a reference material with an oxyacetylene torch.     

Rewetting of Hot Vertical Surfaces by a Falling Liquid Film

In this work, a transient heat conduction model is developed for rewetting a hot wall surface by a falling liquid film. In the model, the heat conduction in the rewetted wall is assumed to be two‐dimensional. Convection heat transfer from the hot surface to rewetting fluid is considered negligible in the dry surface region ahead of the wet front. The numerical solution indicates that the rewetting process is mainly controlled by two‐dimensional heat conduction in the rewetted wall, even for the walls of low Biot number, especially at low initial temperatures. The effects of Biot number and initial wall temperature on the rewetting velocity are investigated. Comparison of the results with previous studies is presented.     

Meshless method based on the local weak-forms for steady-state heat conduction problems

In this article, the meshless local Petrov–Galerkin (MLPG) method is applied to compute two steady-state heat conduction problems of irregular complex domain in 2D space. The essential boundary condition is enforced by the transformation method, and the MLS method is used for interpolation schemes. A numerical example that has analytical solution shows the present method can obtain desired accuracy and efficiency. Two cases in engineering with irregular boundary are computed to validate the approach by comparing the present method with the finite volume method (FVM) solutions obtained from a commercial CFD package FLUENT 6.3. The results show that the present method is in good agreement with FVM. It is expected that MLPG method (which is a truly meshless) is very promising in solving engineering heat conduction problems within irregular domains.     

Fractional heat conduction in a thin hollow circular disk and associated thermal deflection

The time nonlocal generalization of the classical Fourier law with the “Long-tail” power kernel can be interpreted in terms of fractional calculus and leads to the time fractional heat conduction equation. The solution to the fractional heat conduction equation under a Dirichlet boundary condition with zero temperature and the physical Neumann boundary condition with zero heat flux are obtained by integral transform. Thermal deflection has been investigated in the context of fractional-order heat conduction by quasi-static approach for a thin hollow circular disk. The numerical results for temperature distribution and thermal deflection using thermal moment are computed and represented graphically for copper material.     

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.     

Analytical solution of unsteady heat conduction in a two-layered material in imperfect contact subjected to a moving heat source

An analytical approach of transient heat conduction in two-layered material, of finite depth, with an imperfect thermal contact, subjected to a moving gaussian laser beam was developed.The method consists of deriving the solution of the homogeneous part of the heat equation by using the well known separation of variables method and expressing the source term in series form. The porous aspect of granular coating layer on substrate was also taken into account earlier in this modelling work. This model has been successfully applied on a practical system; laser cladding of electronic copper tracks on alumina substrates. This analytical model can be used also for estimation of the thermal contact resistance between layers.     

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.     

TRANSIENT LINEAR AND NONLINEAR HEAT CONDUCTION IN WEAKLY MODULATED DOMAINS

Heat conduction in two-dimensional domains with spatially periodic boundary is addressed in this study. The periodic modulation is assumed to be weak, but is of arbitrary shape. A regular perturbation approach is implemented to determine the temperature and heat flux throughout the domain. It is observed that the validity of the perturbation approach extends to include geometries of practical importance. Transient linear as well as steady nonlinear heat conduction problems are examined. The periodic domain is mapped onto the rectangular domain. For both steady and transient linear heat conduction, a fully analytical spectral solution becomes possible. The nonlinear problem is shown to reduce to a set of ordinary differential equations of the two-point-boundary-value type, which is solved using a variable-step-size finite-difference scheme. The perturbation approach is validated upon comparison with conventional methods; excellent agreement is obtained against the boundary- and finite-element methods.