Boundary Inverse Heat Conduction Problems Research Articles

METHOD OF LINES IN DISTRIBUTED PROBLEMS OF EXPERIMENTAL DATA PROCESSING

In many cases, the mathematical support of non-stationary thermal experiments is based on methods for solving the inverse heat conduction problem (IHCP), which include boundary thermal conditions determination, identification of heat and mass transfer processes, restoration of external and internal temperature fields, etc. However, at present, the main field of the IHCP application remains the processing and interpretation of the results of the thermal experiments. It was here where the most considerable theoretical and applied successes were achieved in methods' effectiveness and the breadth of their practical use. This paper highlights the issues of mathematical modeling of multidimensional non-stationary problems of metallurgical thermophysics.The primary research purpose aims at solving problems associated with identifying parallel structures of algorithms and programs and their reflection in the computers’ architecture in solving a wide range of applied problems. Supercomputers are currently inaccessible due to the enormous cost and service price. In this regard, a real alternative is cluster-type computing systems by which the simulation results are covered in this paper.Being a relatively new technology, cluster-type parallel computing systems are useful in solving a large class of non-stationary multidimensional problems, while allowing to increase the productivity and quality of computations. The software developed in this paper can be used to plan and process the results of a thermophysical experiment. The algorithms developed in the application program package are simply reconstructed to solve other coefficient and boundary problems of thermal conductivity. The developed algorithms for solving thermophysical problems are highly accurate and efficient: the test solution for IHCP with accurate input data coincides with the thermophysical features of the sample material. The developed software for processing the results of a thermophysical experiment is self-regulating. Moreover, it is quite merely tuned to the solution of others and, in particular, of boundary IHCP.

Thermal Regulation of Space Platform Panels by Using Electric Heaters

In orbital flight, the temperature state of spacecraft and their constituent space platforms complexly changes in time and space. The inhomogeneous temperature field of the space platform can cause shape distortion, which negatively affects the performance of targets. To equalize the temperature field of the space platform, it is proposed to use electric heaters with differentiated power distributions. The formulation of the boundary inverse heat conduction problem is presented and the specific power of the heaters necessary for uniform distribution of the temperature field of a panel of composite material is determined.

Effect of Ice on the Heat-Moisture Regime of Soil Foundation of Gas Pipeline

Operation of long-haul trunk pipelines in permafrost areas has shown that they undergo various negative processes, such as sweating, thermokarst, ice formation, etc. Especially in the mountainous areas, crossing the water barrier, in the wintertime ground is covered with ice. The interaction of pipelines with ice is not well researched. The purpose of this work is the numerical simulation of the heat-moisture regime of the soil base of the gas pipeline during the formation of ice. A mathematical model of heat and moisture transfer taking into account the actual process of freezing-thawing of the pore solution of soil is given. For mathematical modelling, a numerical experiment was performed to restore the heat flow of the icy valley using the method of solving boundary inverse heat conduction problems. As a result of a numerical experiment, it was established that the permafrost groundwater increases the average annual temperature of the soil around the pipeline and has a warming effect. During long-term operation of a gas pipeline with a positive temperature, the temperature of the soil base rises, but the thawing halo around the gas pipeline is small, which depends on the depth and thickness of pipe insulation.

Algorithm for restoration of friction power in drum brake device with a layer from the compositional material on temperature data

A quasilinear two-dimensional mathematical model of the thermal process in a drum braking device is considered, taking into account the rotational motion of the brake cylinder. The basic heat conduction equations and the algorithm for solving by method of conjugate gradients of the boundary inverse heat conduction problem to restore the friction power between the brake pad and the cylinder on temperature data are given.

An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.

Estimation of spatially varying thermal contact resistance from finite element solutions of boundary inverse heat conduction problems split along material interface

This paper presents a method for estimating spatially varying thermal contact resistance from a computational point of view. The method starts by splitting the computational domain along material interface to yield two boundary inverse heat conduction problems. Temperatures computed from analytical solutions are specified at only three interior points of each material instead of temperatures available from experiments. A number of equations constructed from modified cubic spline specified along the interface are incorporated into finite element equations during the solution process. After extracting temperatures and heat transfer rates from both finite element solutions, which are solved separately, thermal contact resistance at each pair of coincident nodes on the interface are calculated.Constant, sinusoidal, and triangle functions are selected as spatial functions of thermal contact resistance along material interface to test the method. Constant thermal contact resistance is estimated accurately. Shapes of sinusoidal and triangle functions are fairly captured, locations of their maximums are correctly predicted. Thermal load is suggested to be applied on the boundary of material that has lower thermal conductivity to let the method works properly. Quantifying the uncertainty of estimated thermal contact resistance is demonstrated by adding bias error based on accuracy of sensors to the temperatures specified on the interior points of both materials.

Time-Dependent Heat Flux Estimation in Multi-Layer Systems by Inverse Method

In this paper, the conjugate gradient method, coupled with the adjoint problem, is used to solve the inverse heat conduction problem and to estimate the time-dependent heat flux, using temperature distribution at a point in a three-layer system. Also, the effect of noisy data on the final solution is studied. The numerical solution of the governing equations is obtained by employing a finite difference technique. For solving this problem, the general coordinate method and body-fitted coordinate method are used. The irregular region in the physical domain (, ) is transformed into a rectangle in the computational domain (, ). 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 show the good accuracy of the presented method. Also, the solutions have good stability even if the input data includes noise. The problem is solved in an axisymmetric case. Applications of this model are in the thermal protection systems and heat shield systems.

A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems

This paper presents a finite element method in predicting unknown boundary conditions of homogeneous and two-layered materials subjected to steady-state heat conduction. Firstly, a finite element formulation is introduced to solve a steady-state boundary inverse heat conduction problem of homogeneous material. Effects of bias error of temperature specified on interior nodes and locations of such nodes on accuracy of predicted temperature distribution are examined. Then, modified cubic spline is specified on material boundary to stabilize predicted temperature distribution. Cubic spline functions using different numbers of control points are used in examining their effects on accuracy of predicted temperature distribution and computing time when specifying no bias temperatures. Finally, the formulation with cubic spline function specification is employed in predicting unknown boundary conditions of two-layered materials with thermal conductivity ratio of 0.1, 1, and 10. Concept of coincident nodes is applied in handling physical condition characterized by thermal contact resistance and heat source strength at layer interface. Effect of bias error of temperatures specified on nodes within thicker layer is examined under three interface conditions. Cubic spline function with five control points can predict temperature distributions accurately for all interface conditions when specifying no bias temperatures. RMS errors vary linearly with bias errors for interface conditions with no heat source but are drastically affected by bias error when heat source exists at the interface.

Investigation of heat transfer coefficient during quenching in various cooling agents

Heat transfer coefficients, HTCs, at the surface of a metal sample during immersion quenching were measured and evaluated using numerical procedures. The boundary inverse heat conduction problem has been defined and solved. A FEM self-developed computer code has been used to obtain a solution to the direct problem. The sensitivity of the method enabled us to examine the effect of various quenching parameters on the heat transfer for two mineral oils and a polymer quenchant. At 800°C the HTC values were equal to ∼0.5kW/(m2K) and ∼3.2kW/(m2K), for mineral oils and a polymer coolant, respectively. They increased to ∼4.7kW/(m2K) – oil A, ∼6.0kW/(m2K) – oil B and ∼7.4kW/(m2K) – polymer, respectively. The peak of HTC was sharp and occurred at a narrow temperature interval between 520 and 550°C for the oils, whereas for the polymer, the peak was lower by approx. 100K and flat over 100–120K interval. Subsequently HTC decreased, and at ∼150°C the values were ∼0.5kW/(m2K) and ∼2.0kW/(m2K), for mineral oils and a water polymer coolant, respectively.

Influence of the Finite Element Model on the Inverse Determination of the Heat Transfer Coefficient Distribution over the Hot Plate Cooled by the Laminar Water Jets / Wpływ Modelu Metody Elementów Skonczonych Na Współczynnika Wymiany Ciepła Wyznaczany Z Rozwiazania Odwrotnego Procesu Laminarnego Chłodzenia Płyty Metalowej

The industrial hot rolling mills are equipped with systems for controlled cooling of hot steel products. In the case of strip rolling mills the main cooling system is situated at run-out table to ensure the required strip temperature before coiling. One of the most important system is laminar jets cooling. In this system water is falling down on the upper strip surface. The proper cooling rate affects the final mechanical properties of steel which strongly dependent on microstructure evolution processes. Numerical simulations can be used to determine the water flux which should be applied in order to control strip temperature. The heat transfer boundary condition in case of laminar jets cooling is defined by the heat transfer coefficient, cooling water temperature and strip surface temperature. Due to the complex nature of the cooling process the existing heat transfer models are not accurate enough. The heat transfer coefficient cannot be measured directly and the boundary inverse heat conduction problem should be formulated in order to determine the heat transfer coefficient as a function of cooling parameters and strip surface temperature. In inverse algorithm various heat conduction models and boundary condition models can be implemented. In the present study two three dimensional finite element models based on linear and non-linear shape functions have been tested in the inverse algorithm. Further, two heat transfer boundary condition models have been employed in order to determine the heat transfer coefficient distribution at the hot plate cooled by laminar jets. In the first model heat transfer coefficient distribution over the cooled surface has been approximated by the witch of Agnesi type function with the expansion in time of the approximation parameters. In the second model heat transfer coefficient distribution over the cooled plate surface has been approximated by the surface elements serendipity family with parabolic shape functions. The heat transfer coefficient values at surface element nodes have been expanded in time by the cubic-spline functions. The numerical tests have shown that in the case of heat conduction model based on linear shape functions inverse solution differs significantly from the searched boundary condition. The dedicated finite element heat conduction model based on non-linear shape functions has been developed to ensure inverse determination of heat transfer coefficient distribution over the cooled surface in the time of cooling. The heat transfer coefficient model based on surface elements serendipity family is not limited to a particular form of the heat flux distribution. The solution has been achieved for measured temperatures of the steel plate cooled by 9 laminar jets.

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.

Estimation of time-dependent heat flux using temperature distribution at a point in a two layer system

In this paper, the conjugate gradient method, coupled with the adjoint problem, is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux, using temperature distribution at a point in a two layer system. Also, the effect of noisy data on the final solution is 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. 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. The problem is solved in an axisymmetric case. Applications of this model are in the thermal protect systems (t.p.s.) and heat shield systems.

A method of determining nonstationary heat flux and heat conduction using parametric identification

The formulation and solution of the combined (boundary and coefficient) inverse heat-conduction problem of the simultaneous determination of the thermal characteristics of a material and the heat flux density entering the heat receiver are considered. The solution is based on a parameterization of this problem and the use of an extended digital Kalman filter. The results of a simulation of the simultaneous reestablishment of the heat flux and the heat conduction in a single experiment to measure the temperature of a surface are presented.

Effectiveness and Efficiency of Particle Swarm Optimization Technique in Inverse Heat Conduction Analysis

Three variations of the particle swarm optimization (PSO) method are used to solve the boundary inverse heat conduction problem, in one, two, and three dimensions. Both steady and transient problems are studied. It is shown that PSO can be successfully applied to inverse heat conduction problems, and can alleviate some of the stability problems of the classical approaches. The computational costs of the three variations of PSO (basic, repulsive, and complete repulsive) are compared with each other, and with an implementation of the genetic algorithm. For these problems, some variants of PSO are proven to be more efficient than other algorithms. Also, the effectiveness of PSO in dealing with noisy domains is investigated.

Effortless Application of the Method of Lines for the Inverse Estimation of Temperatures in a Large Slab With Two Different Surface Heating Waveforms

The boundary inverse heat conduction problem (BIHCP) deals with the determination of the surface heat flux or the surface temperature from measured transient temperatures inside a conducting body where the initial temperature is known. This work addresses a BIHCP related to the spatiotemporal heat conduction in a large slab when a time-variable heat flux is prescribed at an exposed surface and the other surface is thermally insulated. Two different heating waveforms are studied: a constant heat flux and a time-dependent triangular heat flux. The numerical temperature-time history at the insulated surface of the large slab provides the “temperature-time measurement” with one temperature sensor. Framed in the theory of the method of lines (MOL) first and employing rudimentary concepts of numerical differentiation later, the main objective of this paper is to develop a simple computational methodology to estimate the temporal evolution of temperature at the exposed surface of the large slab receiving the two distinct heat fluxes. In the end, it is confirmed that excellent predictions of the surface temperatures versus time are achievable for the two cases tested while employing the smallest possible system of two heat conduction differential equations of first-order.

Boundary inverse heat conduction problem: algorithm and error analysis

Methods based on solving boundary inverse heat conduction problems are widely used at present in experimental investigations of thermal processes between solids and the environment. An iterative regularization method is used in the presented paper to solve ill-posed inverse problems. The method is based on minimizing the residual functional by means of gradient methods of the first kind and spline-approximation of unknown functions. An optimal number of approximation parameters using the residual principle is chosen. The accuracy of the solution of inverse problems obtained by the suggested algorithms is discussed.

Three-dimensional boundary inverse heat conduction problem for regular coordinate systems

Computational methods for the solution of boundary inverse heat conduction problems are widely used at present in experimental investigations of thermal processes between the solids and the environment. To solve three-dimensional ill-posed boundary inverse problems, an iterative regularization method is suggested in this study. The method is based on gradient methods of the first kind, and on the principle of the residual. The accuracy of the solution of inverse problems obtained by the suggested algorithms is discussed. The application of the considered method is demonstrated as the reconstruction of spatial-time heat flux distribution over the thermal protection of the spacecraft equipment.

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.

On the solution of parabolic and hyperbolic inverse heat conduction problems

A parabolic and hyperbolic numerical formulation of the boundary inverse heat conduction problems is considered in this paper. The control volume algorithm is combined with a digital filter method to estimate temperature and heat flux values on a surface of a body based on the temperature measurement inside the body. Numerical experiments were carried out to obtain the best value of the non negative coefficient of the hyperbolic equation using noisy and smoothed input data. The accuracy of the method is verified by comparison with a direct (analytical) solution of the problem. The influence of the relatively high noise into measurement data is studied.

A GENERAL SPACE MARCHING ALGORITHM FOR THE SOLUTION OF TWO-DIMENSIONAL BOUNDARY INVERSE HEAT CONDUCTION PROBLEMS

A numerical procedure for solving two-dimensional boundary inverse heat conduction problems is presented in this article. A boundary condition (surface temperatures and surface heat fluxes)of a body is estimated. The major advantage of this work is that the solution can be obtained using limited information about the problem. Another advantage is that the formulation of the problem is completely general. The proposed procedure does not need any stabilization method when exact data are used for solving the problem. A digital filter method is used to stabilize the inverse algorithm by smoothing the real (noisy) data. Then, we apply the control-volume formulation for the filtered data. Accuracy and stability of the solution method is verified by utilizing a solution of a direct problem. The influence on the numerical solution of the time step, digital filter coefficients, and measurement errors is also considered.