One-dimensional Inverse Heat Conduction Problem Research Articles

Solutions of one-dimensional inverse heat conduction problems: a review

Estimations using the inverse conduction approach to predict temperature and heat flux at the exposed surface lead to indirect measurements away from the exposed surface within the solid. The approach is extremely useful when access to direct measurements is not possible due to various working conditions, and thereby provides estimates without disturbing the flow under the real flow condition over the surface. The approach is useful not only for heat transfer applications but also for numerous engineering applications, including fluid mechanics and furnace applications. The approach requires the time history of effective parameters at the strategic locations away from the exposed surface to be known. In the present paper, a review of sequential development of various inverse heat conduction methods is presented to get solutions in different geometries.

Shannon-Taylor technique for solving one-dimensional inverse heat conduction problem

The Shannon-Taylor interpolation technique was introduced by Butzer and Engels in 1983. In this work, the sinc-function is replaced by a Taylor approximation polynomial. In this work, we implement the Shannon-Taylor approximations to solve a one-dimensional heat conduction problem. One of the major advantages of this approach is that the resulting linear system of equations of the approximation procedure has an explicit coefficient matrix. This is not the case of the classical sinc methods due to finite integrals involving e^{-x^2}. We establish rigorous error estimates involving an additional Taylor’s series tail. Numerical illustrations are depicted.

Solution of Inverse Problem of Non-Stationary Heat Conduction Using a Laplace Transform

This paper presents some solutions to the one-dimensional Cauchy-type transient inverse heat conduction problem. Based on known courses of temperature and the heat flux on one of the boundaries of the region, courses of temperature and heat flux on the opposite boundary of the region were determined with the use of the Laplace’s transform. Solution of the inverse problem does not require determination of temperature distribution on subsequent time moments inside the region. Stability of the solution to the inverse problem of the Cauchy type was investigated by means of determining the minimal time step ensuring the stability of the solution. Two examples of one-dimensional transient inverse problems were solved. The first one is related to the reconstruction of shock temperature change on the boundary of the region based on known courses of temperature and heat flux on the opposite boundary. The second one concerned the reconstruction of the heat flux acting on one of the boundaries through a period of time. In both considered problems the reconstructed boundary conditions were discontinuous. Stable results were obtained for both investigated examples, however, the stability of the inverse problem depended on the length of time step.

Influence of Errors in Known Constants and Boundary Conditions on Solutions of Inverse Heat Conduction Problem

This work examines the effects of the known boundary conditions on the accuracy of the solution in one-dimensional inverse heat conduction problems. The failures in many applications of these problems are attributed to inaccuracy of the specified constants and boundary conditions. Since the boundary conditions and material properties in most thermal problems are imposed with uncertainty, the effects of their inaccuracy should be understood prior to the inverse analyses. The deviation from the exact solution has been examined for each case according to the errors in material properties, boundary location, and known boundary conditions. The results show that the effects of such errors are dramatic. Based on these results, the applicability and limitations of the inverse heat conduction analyses have been evaluated and discussed.

Implementation of total variation regularization-based approaches in the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes

In this work, we propose the extension of total variation regularization strategies in principle formulated to be implemented in image processing and signal analysis research fields, to the solution of one-dimensional linear inverse heat conduction problems concerning the estimation of surface heat fluxes. Three solution procedures are considered in the current study, and these include the lagged diffusivity fixed point iteration, the iteratively reweighted least-squares, and the split Bregman iteration methods. The performance of such procedures is tested using four cases, and the regularization parameter is selected via the L-curve criterion. The results are compared to the reconstruction obtained via a classical Tikhonov-like strategy. The main outcome is that the staircase effect dominated all the reconstructions. Despite it deteriorated the quality of the solutions in some specific cases, it did not prevent the appropriate fulfillment of the reconstruction task. The results achieved in this communication have been useful to demonstrate the suitability and reliability of extending total variation approaches to the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes, as an appropriate and novel alternative to standard procedures.

Experimental investigation on transient boiling heat transfer during quenching of fuel cladding surfaces

In case of postulated accidents in a nuclear reactor, the fuel rods uncovered from coolant can be overheated. Immediate quenching of overheated fuel rods is desired. Quenching experiment is performed at atmospheric pressure with rodlets of the FeCrAl alloy, one of the accident tolerant fuel cladding materials, and the Zircaloy-4, the conventional cladding material. In the experiment 600 °C rodlet specimens are plunged into the subcooled water bath. A high-speed camera is employed to record quenching phenomenon. With the embedded thermocouples the rodlet temperature is measured at the frequency of 50 Hz. One-dimensional inverse heat conduction problem is solved to obtain the surface temperature and heat flux. Influence of surface condition, solid thermal properties and liquid subcooling on transient boiling heat transfer is investigated. In the film boiling regime heat transfer is mainly affected by solid thermal properties and liquid subcooling. Small solid (kρCp)w and large liquid subcooling results in relatively thin vapor film and more efficient film boiling heat transfer. Roughness on the polished level shows no appreciable influence on the minimum film boiling temperature, TMFB, which increases with decreasing contact angle. Surface oxidation is slow on the FeCrAl surface and hence, does not significantly affect the boiling heat transfer. However, on the Zircaloy-4 surface the oxide layer with small (ρkCp)w increases TMFB and decreases the critical heat flux. Large (ρCp)w results in slow quench front propagation, long quench duration and weak subcooling effect. It is found that on each surface the vapor film collapses at the critical thickness which is irrelevant to liquid subcooling. Based on the critical thickness, a new approach for prediction of TMFB is proposed.

Application of repulsive particle swarm optimization for inverse heat conduction problem – Parameter estimations of unknown plane heat source

The major objective of the present study is to demonstrate the performance and efficiency of the Repulsive Particle Swarm Optimization (RPSO) method as an inverse analysis solver by extending its application to the inverse heat conduction problem. As the first outcome, an estimation of unknown parameters of a time-varying plane heat source in a one-dimensional inverse heat conduction problem was considered. The overall performance of the RPSO method was examined based on the effects of the forms of unknown heat source, the number of parameters, the measurement errors and the population sizes on the estimation accuracy. In addition, the final results were compared with those of the Levenberg-Marquardt Method (LMM), which is widely used for the inverse heat conduction problem, to investigate the advantages and disadvantages of the RPSO method.The present results prove that the RPSO method has more robust characteristics and yields rather confidential inverse estimations than the LMM for the inverse heat conduction problem although it requires more computational cost.

Revisiting the Temperature Gradient Method

Abstract Cooling curve analysis has emerged as a preferred method to characterize the cooling power of quenching media. This methodology is based on measuring the local cooling curve in a laboratory-scale probe using thermocouples. Several methods have been proposed to analyze cooling curves. Among them, the temperature gradient method (TGM) developed by Professor Liščić uses cooling curves measured at the surface and 1.5 mm below the surface of a cylindrical probe (the Liščić-Nanmac probe) to calculate the surface heat flux during quenching. In this work, we measured the thermal response at a location near the surface of cylindrical probes, fabricated with AISI 304 stainless steel, with two different geometries (conical- and hemispherical-end) during quenching from 850°C with water at 60°C, flowing parallel to the probe’s longitudinal axis. Together with the probe geometry, the experimental matrix included two values of water velocity (0.2 and 0.6 m/s). The data was used to estimate the surface heat flux and thermal response at the probe surface by solving a one-dimensional inverse heat conduction problem (IHCP) without phase change. The TGM was then applied to re-estimate the surface heat flux using the cooling curves at the subsurface (measured) and at the surface (estimated by solving the IHCP). The surface thermal gradient computed solving the IHCP is higher and, therefore, the surface heat flux estimated solving the IHCP is also higher than the value calculated with the TGM. The hemispherical-end probe delays the rupture of the vapor film, producing low values of the surface temperature at the time of maximum heat extraction; this results in extremely high values of the heat-transfer coefficient, which precludes the use of this geometry in conjunction with the TGM. The thermal profiles are parabolic, which restricts the maximum depth at which a subsurface thermocouple may be placed to use the TGM confidently.

Effect of Coating by a Carbon Nanostructure on Heat Transfer with Unsteady Film Boiling

We present the results of an experimental study of heat transfer regimes during cooling in water of high-temperature, heated steel balls with a technically smooth and modified surface. The modification consisted of the application of a finely dispersed carbon coating to the surface followed by electron beam treatment. When the samples are placed in subcooled water, an intensive heat transfer regime occurs on the both samples; it appeared upon film boiling with heat fluxes on the surface up to 6 MW/m2. The heat flux values based on the initial cooling thermograms were obtained from a one-dimensional inverse heat conduction problem. The carboncoating leads to a decrease in the surface temperature, which corresponds to the transition to the intensive cooling mode, whereas the intensive cooling mode itself is identical for samples with the different surface treatments. Experimental results confirm the approximate model proposed for the conditions for the occurrence of the intensive cooling regime during film boiling of a subcooled liquid.

FEATURES OF THE APPLICATION OF THE IQLAB PROGRAM FOR SOLVING THE INVERSE HEAT CONDUCTION PROBLEM FOR CHROMIUM-NICKEL CYLINDRICAL THERMOSONDES

The paper presents an experimental-computational study of the results of using the IQLab program to solve inverse heat conduction problem and restore the surface temperature of cylindrical thermosondes from heat-resistant chromium-nickel alloys while cooling them in liquid media. The purpose of this paper is to verify the correct operation of the IQLab program when restoring the surface temperature of thermosondes with 1-3 thermocouples. The IQLab program is also designed to solve one-dimensional nonlinear direct lines and inverse heat conduction problems with constant initial and boundary conditions specified as a function of time in a tabular form with a constant and variable time step. A finite-difference method is used to solve the heat equation. Experiments were carried out on samples D = 10-50 mm in liquids with different cooling capacities such as aqueous solutions of NaCl and Yukon-E polymer, rapeseed oil and I-20A mineral oil. For the calculation we used the readings of thermocouples installed at internal points of cylindrical thermosondes. The advantages of solving inverse heat conduction problems with the IQLab program include the possibility of restoring the surface temperature for cylindrical samples with a diameter of 10 mm to 50 mm with practical accuracy according to the indications of a single thermocouple located in the geometrical center of the thermosonde, which simplifies the manufacture of the probe. For larger dimensions with a diameter D ≥ 50 mm, it is necessary to install control intermediate thermocouples and perform additional tests. The solution of inverse heat conduction problems and restoration of the surface temperature of the sample makes it possible to calculate other important characteristics of the cooling process: the heat flux density and the heat transfer coefficient.

Two-probe calibration integral equation method for nonlinear inverse heat conduction problem of surface heat flux estimation

In this paper, a new two-probe surface heat flux calibration integral equation method (CIEM) is proposed for resolving nonlinear inverse heat conduction problem possessing temperature-dependent thermophysical properties and an unknown time-varying back surface boundary condition. Compared to the one-probe CIEM, the two-probe CIEM can be applied to a more general class of one-dimensional inverse heat conduction problems that can be used for accurately predicting the front surface heat flux without any restrictions on the back surface boundary condition. Temperature-dependent thermophysical properties and probe locations are not specified a priori but are implicitly accounted through a calibration campaign. The final mathematical formulation involves resolving a Volterra integral equation of the first kind for the unknown surface heat flux. A first kind Chebyshev expansion possessing undetermined coefficients, is used for approximating the introduced property transform function. Next, three well-defined calibration tests are used for estimating the undetermined coefficients associated with the Chebyshev expansion. A time sequential study of the coefficients is shown to yield the optimal truncation of the series expansion. A future-time method is applied for stabilizing the ill-posed first kind Volterra integral equation. Phase plane and cross-correlation analyses are employed for estimating the optimal regularization parameter (i.e., the future-time parameter) from a spectrum of chosen values. The effectiveness of the proposed approach is verified by numerical simulation using stainless steel as the host material. The calibration-based or “parameter free” inverse approaches significantly reduce time and costs in the resolution of the surface condition as there is no need to experimentally determine the thermophysical properties, probe locations, and probe characteristics. These system properties are implicitly included in the proposed calibration procedure. Many applications, especially in aerospace sciences where exotic materials are often used, do not have replicable properties among samples. Conventional inverse methods rely on accurately specifying the input parameters for forming the predictions. These “parameter required” methods are solely based on numerical algorithms and thus their predictions are strongly relied on accurate input of parameters and measured probe data.

A novel computational procedure based on league championship algorithm for solving an inverse heat conduction problem

Inverse heat conduction problems, which are one of the most important groups of problems, are often ill-posed and complicated problems, and their optimization process has lots of local extrema. This paper provides a novel computational procedure based on finite differences method and league championship algorithm to solve a one-dimensional inverse heat conduction problem. At the beginning, we use the Crank-Nicolson semi-implicit finite difference scheme to discretize the problem domain and solve the direct problem which is a second-order method in time and unconditionally stable. The consistency, stability and convergence of the method are investigated. Then we employ a new optimization method known as league championship algorithm to estimate the unknown boundary condition from some measured temperature on the line. League championship algorithm is a recently proposed probabilistic algorithm for optimization in continuous environments, which tries to simulate a championship environment wherein several teams with different abilities play in an artificial league for several weeks or iterations. To confirm the efficiency and accuracy of the proposed approach, we give some examples for the engineering applications. Results show an excellent agreement between the solution of the proposed numerical algorithm and the exact solution.

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.

Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.

Inverse analysis of heat conduction problems with relatively long heat treatment

This paper proposes a solution method for one-dimensional inverse heat conduction problems that require a relatively long time. A hybrid technique is applied to analyze laser surface heating and spray cooling on a hot surface. In the present study, the unknown temperature in half-range expansions form is assumed, and the shifting function method is used to obtain an analytic solution. The coefficients of the half-range expansions could be determined with the least-squares method in conjunction with the analytic solution and measured temperatures. Mathematical and experimental examples are given to illustrate the analyses.

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

Abstract In this paper, the meshless local radial point interpolation (MLRPI) method is applied to one-dimensional inverse heat conduction problems. The meshless LRPIM is one of the truly meshless methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. Numerical experiments are carried out and compared with implicit Finite difference method.

Calculation of Heat Flux Across the Hot Surface of Continuous Casting Mold Through Two-Dimensional Inverse Heat Conduction Problem

A novel method for the estimation of the mold hot surface heat flux based on the measured responding temperatures from two columns of thermocouples that embedded inside the mold during continuous casting has been developed. The method includes a Two-Dimensional Inverse transient Heat Conduction Problem (2D-IHCP) model that was solved by the conjugate gradient method with Adjoint Equation. The model was validated by comparing the results with those calculated by a robust One-Dimensional Inverse transient Heat Conduction Problem (1D-IHCP). The solution of a test problem indicated that the Mean Absolute Percentage Error of the estimated heat flux calculated by the new method is about 9 to 40 pct of those calculated by the 1D-IHCP. Then, the method is applied to compute the heat flux for a mold simulator experiment. The results indicated that the heat fluxes and temperatures across mold hot surface calculated by 2D-IHCP show the same variation tendency as those calculated by 1D-IHCP. However, the heat fluxes calculated by 2D-IHCP are about 1.2 to 2 times larger than those calculated by 1D-IHCP for the locations below the liquid mold flux surface and are about 50 to 90 pct of those calculated by 1D-IHCP for the locations above the liquid mold flux surface.

Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme

In this article, a one-dimensional inverse heat conduction problem with unknown nonlinear boundary conditions is studied. In many practical heat transfer situations, the heat transfer coefficient depends on the boundary temperature and the dependence has a complicated or unknown structure. For this reason highly nonlinear boundary conditions are imposed involving both the flux and the temperature. A numerical procedure based on the mollification method and the space marching scheme is developed to solve numerically the proposed inverse problem. The stability and convergence of numerical solutions are investigated and the numerical results are presented and discussed for some test problems.

Heat transfer in vertical casks engulfed in open pool fires

Characterization of the heat transfer to a cask engulfed in pool fire is extremely important. Experiments are carried out on diesel pool fires of diameters 0.5, 0.7, and 1.0 m with stainless steel 304L thermal casks of different sizes. Net surface heat flux on the cask is estimated using one-dimensional inverse heat conduction problem code. Velocities of the pool fires are measured using bidirectional probe to estimate the convective heat transfer coefficient ( h). The concept of adiabatic surface temperature, using plate thermometer, is applied to the pool fires and the thermal casks. By employing a mixed boundary condition (adiabatic surface temperature and h) in computational fluid dynamics package, transient temperature and heat flux of the cask are estimated. These predicted data are within 10% of the experimental results. This study demonstrates that the transient adiabatic surface temperature of the pool fire can be used to predict the behavior of the thermal cask engulfed in an open pool fire.

Regularization error analysis on a one-dimensional inverse heat conduction problem in multilayer domain

We investigate in this paper a non-characteristic Cauchy problem for a one-dimensional heat conduction equation in a multilayer domain, which is a well-known ill-posed problem. We first formulate the Cauchy problem in the frequency domain via Fourier transform technique. A modified version of the classical Tikhonov regularization technique together with the corresponding error estimates is presented. Several numerical examples are constructed to demonstrate the superior efficiency and accuracy of the proposed approach. Numerical results show that the modified Tikhonov regularization technique performs better than the classical Tikhonov regularization technique.