Transient Inverse Heat Conduction Problem Research Articles

Structured Multiblock Body-Fitted Grids Solution of Transient Inverse Heat Conduction Problems in an Arbitrary Geometry

The aim of this study is to develop iterative regularization algorithms based on parameter and function estimation techniques to solve two-dimensional/axisymmetric transient inverse heat conduction problems in curvilinear coordinate system. The multiblock method is used for geometric decomposition of the physical domain into regions with patched-overlapped interface grids. The central finite-difference version of the alternating-direction implicit technique together with structured body-fitted grids is implemented for numerical solution of the direct problem and other partial differential equations derived by inverse analysis. The approach of estimating unknown parameters and functions is iterative inverse analysis algorithms based on the Levenberg-Marquardt method and conjugate gradient methods. The temperature histories are delivered by noisy/non-noisy sensors located on the lower boundary of the domain. Three cases are considered, (1) a time-space-varying upper boundary condition, (2) two time-varying heat sources, and (3) a time-varying heat source and a time-space-varying boundary condition, within a two-dimensional/axisymmetric solid body of arbitrary geometry. The simultaneous estimation of two unknown functions is performed to assess the performance of inverse analysis methods to solve complicated inverse heat conduction problems. The results of the present study are compared to those of exact heat sources and boundary condition, and good agreement is achieved. However, the results show that the accuracy of the upper boundary condition identification is dependent on the uncertainty of the measured temperature data, the number of grid points, and the curvature of the geometric configuration as well.

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.

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.

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.

Restoring boundary conditions in heat conduction

The restoration of boundary conditions in one-dimensional transient inverse heat-conduction problems (IHCP) is described. In the formulation, the boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain, together with the space or time-dependent ambient temperature of the environment surrounding the heat conductor, are found from additional boundary-temperature or average boundary-temperature measurements. Numerical results obtained using the boundary-element method are presented and discussed.

Investigation of heat transfer coefficient in two‐dimensional transient inverse heat conduction problems using the hybrid inverse scheme

AbstractA hybrid scheme of the Laplace transform, finite difference and least‐squares methods in conjunction with a sequential‐in‐time concept, cubic spline and temperature measurements is applied to predict the heat transfer coefficient distribution on a boundary surface in two‐dimensional transient inverse heat conduction problems. In this study, the functional form of the heat transfer coefficient is unknown a priori. The whole spatial domain of the unknown heat transfer coefficient is divided into several analysis sub‐intervals. Later, a series of connected cubic polynomial function in space and a linear function in time can be applied to estimate the unknown surface conditions. Due to the application of the Laplace transform, the unknown heat transfer coefficient can be estimated from a specific time. In order to evidence the accuracy of the present inverse scheme, comparisons among the present estimates, previous results and exact solution are made. The results show that the present inverse scheme not only can reduce the number of the measurement locations but also can increase the accuracy of the estimated results. Good estimation on the heat transfer coefficient can be obtained from the knowledge of the transient temperature recordings even in the case with measurement errors. Copyright © 2007 John Wiley & Sons, Ltd.

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.

Precise integral algorithm based solution for transient inverse heat conduction problems with multi-variables

By modeling direct transient heat conduction problems via finite element method (FEM) and precise integral algorithm, a new approach is presented to solve transient inverse heat conduction problems with multi-variables. Firstly, the spatial space and temporal domain are discretized by FEM and precise integral algorithm respectively. Then, the high accuracy semi-analytical solution of direct problem can be got. Finally, based on the solution, the computing model of inverse problem and expression of sensitivity analysis are established. Single variable and variables combined identifications including thermal parameters, boundary conditions and source-related termsetc. are given to validate the approach proposed in 1-D and 2-D cases. The effects of noise data and initial guess on the results are investigated. The numerical examples show the effectiveness of this approach.

Heat flux characterisation in hot jet and flame/wall interaction by IHCP resolution coupled with infrared measurements

The aim of this work is to develop a method dedicated to the identification of heat fluxes between turbulent flames and 3D solids in order to improve the conception of aeronautic structures designed to endure and sustain fire events. Transient temperature modelling inside the impinged structure can only be realistic if the convective and radiative heat fluxes used as boundary conditions of the numerical model are accurately known. To do so, these two different heat transfer contributions are to be separated from the total heat flux information, obtained by solving an IHCP (Inverse Heat Conduction Problem). The inverse technique that was implemented for this study is based on adding back-face infrared measurements data as observation equation, thus closing this ill-posed problem. Infrared camera provides an accurate temperature map, allowing high resolution heat flux mapping without perturbing the impingement zone. The number of unknowns initially equal to the camera resolution is reduced by solving the problem in the cosine space thanks to Discrete Cosine Transform. As for the resolution of this multidimensional transient IHCP, Beck's inverse algorithm has been coupled to future time-steps stabilisation method. Flame to wall heat transfer knowledge is improved thanks to the study of a dedicated test bench composed of a propane / air burner on the one hand, and a 14kW air heater on the other hand. To take into account the spatial non-uniformity of the adiabatic wall temperature used in Nusselt calculation, a new adiabatic wall temperature computation is presented and compared to Nusselt calculations based on constant reference temperature. In the scope of this study, we present the comparison between propane/air flame and hot round isothermal air jet impinging perpendicularly on a titanium plate.

A differential quadrature solution of the two-dimensional inverse heat conduction problem

A numerical procedure using a combined differential quadrature and Taylor series approach is presented for the solution of the transient two-dimensional inverse heat conduction problem. The method is relatively easy to program, compared to other more complex solutions currently available, and gives good results for both exact and inexact (noisy) data. The method is suitable to provide estimates of unknown boundary conditions, in advanced inverse procedures.

Some Analytical and Numerical Solutions to Inverse Problems Applied to Optimizing Phase-Transformation Tracking in Gas Quenching

A transient inverse heat conduction problem focused on gas quenching of steel plates and rings is posed and solved, both analytically and numerically. The quenching objective is to calculate the transient convective heat transfer coefficient which would produce an optimized phase transformation cooling curve. The governing nonlinear heat equation is nondimensionalised, and a small parameter, the reciprocal of the Fourier number, is identified. This allows the construction of an analytic solution in the form of an asymptotic series. For higher values of the reciprocal Fourier number, a numerical scheme incorporating the function specification and Keller Box methods is used to generate solutions. Comparison of the results proves favorable, and suggests that for this inverse problem asymptotic methods provide an attractive alternative to solely numerical ones.

A THREE-DIMENSIONAL INVERSE PROBLEM OF ESTIMATING THE VOLUMETRIC HEAT GENERATION FOR A COMPOSITE MATERIAL

A 3-D transient inverse heat conduction problem is solved in this study by using the conjugate gradient method (CGM) and the general-purpose commercial code CFX4.2-based inverse algorithm to estimate the strength of the unknown heat generation in a 3-D irregular composite medium. The advantage of calling the CFX4.2 code as a subroutine in the present inverse calculation is that many difficult but practical 3-D inverse problem can be solved under this construction because the general-purpose commercial code has the ability to solve the direct problem easily. Results obtained by using the CGM to solve this 3-D inverse problems are justified based on the numerical experiments with the simulated exact and inexact measurements. It is concluded that accurate heat generation can be estimated by the CGM except for the final time. The reason and improvement of this singularity are addressed.

An overdeterrnined two-dimensional transient inverse heat conduction problem

The method of solving the two-dimensional nonlinear inverse heat conduction problems is presented. The time- and space temperature distribution inside a solid and heat transfer coefficient distribution on the boundary of the solid is determined based on the temperature histories measured in several selected inside locations. The problem is overdetermined because the number of measuring points is higher than the number of calculated parameters (components of heat transfer coefficient).

A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method

In the present study a three-dimensional (3-D) transient inverse heat conduction problem is solved using the conjugate gradient method (CGM) and the general purpose commercial code CFX4.2-based inverse algorithm to estimate the unknown boundary heat flux in any 3-D irregular domain. The advantage of calling CFX4.2 as a subroutine in the present inverse calculation lies in that many difficult but practical 3-D inverse problems that can be solved under this construction. Results obtained by using the conjugate gradient method to solve these 3-D inverse problems are justified based on the numerical experiments. It is concluded that accurate boundary fluxes can be estimated by the CGM except for the final time. The reason and improvement of this singularity are addressed. Finally, the effects of the measurement errors on the inverse solutions are discussed.

A method to invert MUPUS. Temperature recordings for the subsurface temperature field of P/Wirtanen

The thermal sensors on the penetrator of the MUPUS experiment package selected for the ESA Rosetta mission will enable us to determine the near-surface energy balance of the nucleus of comet P/Wirtanen by measuring the subsurface temperature profile and the thermal conductivity of the near-surface layers. Model calculations suggest that the penetrator itself will perturb the ambient temperature field such that the temperature profile will be smoothed if the thermal diffusivity of the nucleus is significantly smaller than that of the penetrator tube. It is possible, however, to calculate the undisturbed temperature profile from the data using a method based on a solution of the transient inverse heat conduction problem. Our model calculations show that a satisfactory estimate of the undisturbed temperature field can be obtained in comparatively little computing time by calculating the temperature distribution in the model volume from temperature histories at discrete points representing the penetrator temperature sensors.

Transient inverse heat conduction problem solutions via Newton's method

Analytical first and second-order sensitivities are derived for a general, transient nonlinear problem and are then used to solve an inverse heat conduction problem (IHCP). The inverse analyses use Newton's method to minimize an error function which quantifies the discrepancy between the experimental and predicted responses. These Newton results are compared to results obtained from the first-order variable metric Broyton-Fletcher-Goldfarb-Shanno (BFGS) method. Inverse analyses are performed for both linear and nonlinear thermal systems. For linear systems, Newton's method converges in one iteration. For nonlinear systems, Newton's method sometimes diverges apparently due to a small radius of convergence. In these cases a combined BFGS-Newton's method is used to solve the IHCP. The unknown data fields are parameterized via the eigen basis of the Hessian to illustrate the need for regularization. Regularization is then incorporated and the IHCP is solved with Newton's method. All heat transfer analyses and sensitivity analyses are performed via the finite element method.

A Control Theory Method for Solutions of Inverse Transient Heat Conduction Problems

A Control Theory Method for Solutions of Inverse Transient Heat Conduction Problems

Two-dimensional linear transient inverse heat conduction problem - Boundary condition identification

This article deals with the identification of unknown time- and space-dependent boundary conditions for systems driven by the heat equation. We first consider a one-dimensiona l and single-input problem, dealing with the identification of the time-dependent heat flux on one side of a one-dimensional linear thermal wall, from temperature and heat flux measurements on the other side. We then focus on a quenching process; our interest is to identify the time- and space-dependent heat flux on the boundary of a metal piece from temperature measurements performed inside the material (two-dimensional geometry). Those inverse problems are solved by use of a regularization method, and the solution is obtained by minimization of a quadratic criterion. Because of the linearity of the input-output relationship, the solution of this minimization is derived from the linear quadratic optimal control theory (resolution of a nonstationary Riccati equation). The robustness of the method for very small signal-to-noise ratio is shown. In the two-dimensional multi-input problem, the identification sensitivity to the localization of the measurement points is analyzed. In both cases, we consider input that are discontinuous in time, in order to show the method accuracy in the high-frequency domain.

Estimation of aerodynamic heat transfer in free flight at Mach number up to 4.7

A one-dimensional inverse transient heat conduction problem with a radiation boundary condition is analysed using implicit finite difference approximation. An iterative procedure is described which predicts unknown surface conditions from the measured thermocouple temperature. A unified thermal analysis (temperature/heat flux prediction) of the sounding rocket aerothermodynamic environment is presented using flight data which are transmitted to ground test station.

A method and a computer program for determining the thermal diffusivity in a solid slab

The authors describe a method for computing the thermal diffusivity of a solid, based on a computer assisted evaluation of the solution of the transient inverse heat conduction problem. The program computes either the unknown diffusivity or simulates the one-dimensional unsteady heat transfer problem. The user may model the boundary conditions by a choice of different functions. The program provides instruction and information at all stages of input and provides tabular output of results. It may be used by anybody wishing to solve or simulate heat transfer processes.