Nonlinear Heat Conduction Problem Research Articles

Derivation and application of the adjoint method for estimation of both spatially and temporally varying convective heat transfer coefficient

In this study, a numerical algorithm is developed to determine the heat transfer coefficient distribution of mixed convection on a vertical plate with a two dimensional inverse method. Cooling procedures of hot plate include forced convection with buoyancy effect is simulated by solving two dimensional, transient heat conduction equation using finite difference method while the Grashof number is varied through natural convection in combination with forced convection resulting in a variation of the Richardson number from 0.1 to 1. The nonlinear inverse heat conduction problem is then implemented to directly predict the time-space-varying convective heat transfer coefficient of cooling in the mixed convection regime. The sum of squared differences between calculated and measured temperature data at thermocouples’ locations is the objective functional. The adjoint method is employed to optimize the functional using conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems in a whole time-domain optimization process. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by the adjoint method at four cooling strategies for ten minutes of data acquisition in the presence of noise with standard deviations of σ=0.01 °C and σ=0.1 °C. Although results are affected by noisy simulated temperatures, a satisfactory agreement between exact and estimated heat transfer coefficients is achieved. Furthermore, increasing data acquisition time to sixty minutes reveals that the inverse scheme is able to predict 1200 convective heat transfer coefficient components efficiently employing the rapidly convergent adjoint method.

Similarity solution for one-dimensional heat equation in spherical coordinates and its applications

A similarity type of general solution for the one-dimensional heat equation in spherical coordinates is developed, and the solution is expressed by the Kummer functions. After introducing some useful properties of the general solution, the solution is then applied to establish analytical solutions of finite line source problems and moving boundary problems. For the finite line source problems, a continuous point source problem with a power-type source intensity is solved first, and then analytical solutions for a vertical finite line source problem and an inclined finite line source problem are established using the mirror image method and the solution of the continuous point source problem. For the moving boundary problems, a one-phase Stefan problem with power-type latent heat in spherical coordinates and a nonlinear heat conduction problem induced by a continuous point source are investigated, and analytical solutions are developed using the general solution. Computational examples are presented. For the finite line source problems, computational error caused by neglecting vertical heat transfer under the situation of a time-varying source intensity is investigated. For the moving boundary problems, an illustrative example of the one-phase Stefan problem is presented, and the computational results can be used to calibrate numerical solutions of Stefan problems.

Variational estimates of the parameters of a thermal explosion of a stationary medium in an arbitrary domain

The variational formulation of the nonlinear problem of steady heat conduction in an arbitrary configuration domain is applied to investigate the conditions for the existence of a steady-state temperature distribution in a stationary medium with the intensity of volumetric energy release rising with increasing temperature. Based on the relations of the time-independent theory of thermal explosion, a variational model of this phenomenon is constructed, which makes it possible to obtain estimates of the critical values of the parameters that determine the temperature state of the medium preceding the thermal explosion. The examples of a comparative analysis of such values for a solid and a hollow cylinder of finite length are given.

Estimation of thermal conductivity of low thermal conductive solid materials using the jet flush method

A novel and practical transient-type technique named as “Jet Flush” is proposed for measuring temperature-dependent thermal conductivity of solid materials with low thermal conductivity. In order to estimate thermal conductivity of the sample, its surface is exposed to an impinging air jet flow with higher (or lower) temperature than the sample for a short period of time, while an infrared thermometer records the heated (or cooled) surface temperature history. A nonlinear 1D inverse heat conduction problem with constant convective heat transfer coefficient on the boundary condition is then employed for determining the temperature-dependent thermal conductivity using the measured temperatures. As a requirement for solving the inverse problem, the convective coefficient on the surface of the sample is measured in a separate experiment with the same flow condition using a standard sample. Sensitivity and error analyses are performed to discuss the effects of the estimation points (the temperatures in which the thermal conductivity is to be estimated), and the simplification assumption of constant convective heat transfer coefficient (with respect to time) on the solution of the inverse problem and the accuracy of the estimated results. In order to demonstrate the applicability of the proposed method, the thermal conductivities of three polymer-type materials (HDPE, PMMA and PA6) are measured over the temperature range of ~22–62 °C, and the results are found to be in good agreement with literature data. Moreover, the uncertainty analysis performed in this study showed that the accuracy of the proposed method can be better than 5% for an appreciable temperature range.

Analysis of Crack Resistance of an Oil Trunk Pipeline Considering the Varying Failure Viscosity in the Neighborhood of a Welded Joint

This article presents a development of the method of analysis of crack resistance of an oil trunk pipeline in the zone of the transverse welded joint during operation. The nonlinear nonstationary heat conductivity problem was solved by the finite difference method using the boundary conditions of third kind. Modeling of the kinetics of conversion of austenite to ferrite and bainite under non-isothermal conditions during welding was carried out based on the theory of isokinetic reactions. The calculation of the residual welding stress is performed by solving the problem of thermoelastic plasticity using the finite element method for material with a nonstationary structure. The calculation of the crack resistance is based on Irwin’s force failure criterion, and the dependence of the failure viscosity on the structural composition is considered.

Визначення квазістатичного термопружного стану шаруватих термочутливих плит

The technique of determining the quasistatic thermoelastic state of the layered thermosensitive plates free of load is illustrated. Much attention is paid to finding analytical-numerical solutions of one-dimensional non-stationary heat conduction problems taking into account the temperature dependences of the thermal and temperature conductivity coefficients. Their finding involves use of the Kirchhoff transformation, generalized functions, Green's functions of the corresponding linear heat conduction problem, exact sums of the series, in particular those for which the Gibbs effect takes place, linear splines and solving the received recurrent systems of nonlinear algebraic equations relative to the values in the nodes of the spline of the Kirchhoff variable on the layer division surfaces and the derivative in time on inner flat-parallel surfaces of layers. The results of numerical calculations of temperature fields in two-layer plates with different thicknesses of layers and the external surface heated by a constant heat flux are presented. The accuracy of the found solution is investigated. The comparison of the temperature fields, which are determined assuming simple nonlinearity, stable thermophysical characteristics with the ones based on the exact solution of the corresponding nonlinear stationary heat conduction problem is fulfilled.

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACTDiffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.

INVESTIGATION OF TEMPERATURE MODES IN THERMOSENSITIVE NON-UNIFORM ELEMENTS OF RADIOELECTRONIC DEVICES

Context. The non-linear boundary value problem of heat conduction for a thermosensitive non-homogeneous strip-shaped element of a radio-electronic system with a through inclusion has been solved whose analytical-numerical solution enables us to analyze temperature regimes in the element. Objective. Is to develop such a method of linearization of mathematical model of heat conduction which enables us to obtain analytical numerical solution of the corresponding non-linear boundary value problem for determination of temperature field in elements of radio electronic devices, which are geometrically represented by a thermosensitive plate with a through inclusion. Method. A linearizing function which enables us to partially linearize the initial non-linear mathematical model of heat conduction for a thermosensitive non-homogeneous element of a radio electronic system in the form of “plate-inclusion” structure has been suggested. The introduced piece-wise linear approximation of temperature on plate-inclusion interfaces has enabled us to completely linearize the corresponding partially linearized boundary value problem relative to the linearizing function. After this, it became possible to apply Fourier’s integral transformation to the obtained linear problem with respect to one of the spatial coordinates, as well as to determine the linearizing function. The linear dependence of the coefficient of heat conductivity on temperature for structure materials with the use of the linearizing function has been considered. By solving the boundary value problem, the formulae for determination of temperature field in the “plate-inclusion” thermosensetive structure have been obtained. Results. The obtained formulae for determination of temperature field in a thermosensitive non-homogeneous element of radio electronic system were used to create the software which enables us to obtain distribution of value of temperature and to analyze temperature regimes. Conclusions. A mathematical model for the calculation for the temperature field in a “plate-inclusion” thermosensitive structure is adequate to the actual physical process, because no jump of temperature at “plate-inclusion” interfaces is observed. The numerical results for the chosen materials under linear dependence of the coefficient of thermoconductivity on temperature differ by 7% from the results which are obtained for constant coefficient of heat conductivity. Prospect of further investigation will consider more complicated geometric representation of elements of radio electronic systems.

A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.

Nonlinear Numerical Model of Friction Heating during Rotary Friction Welding

A calculation procedure to determine the temperature field during rotary friction welding of metal is proposed. An axisymmetric nonlinear boundary value problem of heat conduction taking into account the frictional heating of two cylinders (specimens) of finite length is formulated. It is assumed that the materials of specimens are thermally sensitive, and the friction coefficient depends on temperature. The solution of the problem is obtained by the finite element method. The numerical analysis was carried out for two identical specimens made of AISI 1040 grade steel.

Reduced order model analysis method via proper orthogonal decomposition for nonlinear transient heat conductionproblems

A reduced order analysis method is proposed to solve the nonlinear transient heat conduction problems with temperature-dependent conductivity of material by using the proper orthogonal decomposition (POD) model reduction method and the finite element method. Firstly, the POD basis vectors are developed by calculating eigenvectors of an autocorrelation matrix composed of snapshots, which clustered by the given results of obtained from experiments or numerical methods for the linear transient heat transfer problem. Secondly, the finite element discrete scheme of the nonlinear transient heat conduction problem is reduced the order by using the POD basis vectors of the linear problem, and a lower order ordinary differential equation system of the reduced order model of the nonlinear transient heat conduction problem is obtained. Finally, the value of temperature field at each time is obtained by solving the reduced-order ordinary differential equations. Several numerical examples are given to verify the effectiveness of the proposed method. The results show that the proposed method has a good accuracy and the computational efficiency can be improved by 1–2 orders of magnitude.

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.

An extended approach of a Kalman smoothing technique applied to a transient nonlinear two-dimensional inverse heat conduction problem

An extended Kalman smoother (EKS) introducing the concept of future measurement information is developed to handle nonlinear inverse heat conduction problems. In the present study, the formulation of the EKS algorithm is generalized in the case of a two-dimensional problem in order to reconstruct a non-homogeneous heating condition on the front surface of a cylindrical sample. This leads to apply the new method in a multiple sensor case for the simultaneous reconstruction of several parameters at each time. The analyzed inverse problem is nonlinear due to the variability of the thermophysical properties with temperature and to the presence of radiation boundary conditions. Inverse estimation is successfully performed where temporal and spatial variations of the front surface heat flux are recovered based on non intrusive transient temperature measurements made on the back surface. Numerical experiments show that the use of an optimal number of future data greatly improves the solution of the EKS compared to the extended Kalman filter (EKF) estimates, where a noticeable reduction of time lag and sensitivity to measurement errors is observed. Inversion results show that the optimal number of future data, chosen on the basis of a minimum measure of the heat flux bias, depends on the modeling error, the measurement time step, the distance between the sensors and the front heated surface and the measurement noise level. The proposed algorithm is robust in recovering different heat flux profiles and provides for all the examined patterns symmetric and stable solutions that superposes well with the exact functions. The EKS formulation based on an augmented state vector allows to efficiently recover the front surface temperature simultaneously with the heat flux.

Solution of an inverse transient heat conduction problem in a part of a complex domain

The purpose of this work is to formulate two simple methods which can be used to solve nonlinear inverse heat conduction problems in a part of a complex-shaped component in the on-line mode. The proposed methods can be useful if temperature measurements are carried out only in a selected part of an outsize element or if the whole element cannot be analysed due to the inverse heat conduction problem being ill-conditioned. It is assumed that the conductive heat transfer occurs through the surfaces separating the domain from the rest of the component. It is shown that the simplifying assumption of thermal insulation on these surfaces, which is often presented in literature, can cause significant errors. Compared to works published previously, if two additional unknown boundary conditions are introduced on the separating surfaces, the inverse problem conditioning deteriorates substantially. Despite this, stable solutions are achieved for “noisy measured data”.The presented methods can be used to optimize the power unit start-up and shutdown operation. They may also enable a reduction in the heat loss arising during the process and extend the power unit life. The methods presented herein can be applied in monitoring systems working both in conventional and nuclear power plants.

Numerical solution of multi-dimensional transient nonlinear heat conduction problems with heat sources by an extended element differential method

In this paper, the element differential method is extended to solve a transient nonlinear heat conduction problem with a heat source and temperature-dependent thermophysical properties for the first time. The transient term is discretized by employing a finite difference scheme. An iterative methodology is developed to deal with the nonlinearity caused by temperature-dependent thermophysical properties. Examples of two-dimensional (2D) and three-dimensional (3D) problems are given to validate the present method for solving multi-dimensional transient nonlinear heat conduction problems. The results show that the present EDM provides a promising way that is effective and with high accuracy for solving multi-dimensional transient nonlinear heat conduction problems.

An isogeometric independent coefficients (IGA-IC) reduced order method for accurate and efficient transient nonlinear heat conduction analysis

This article develops an isogeometric independent coefficients (IGA-IC) reduced order method for transient nonlinear heat conduction analysis. Herein, we first exactly represent the geometric model via isogeometric analysis (IGA), and therein provide an accurate solution for the semi-discretized equations. Next, our proposed GSSSS-1 time-stepping framework is employed to solve the transient nonlinear temperature in space and time domains. We advance our independent coefficients (IC) reduced order method to efficiently solve IGA-based transient nonlinear heat conduction problems. We extend the IC method to significantly reduce the original full IGA-discretized formulations and calculate the reduced equilibrium formulations in each Newton–Raphson iteration. Thereby, hugely improving the efficiency and guaranteeing the accuracy simultaneously. Illustrative numerical examples validate this proposed IGA-IC method is reliable, accurate, and efficient; especially, the larger the scale of the problem, the more advantages the proposed IGA-IC will inherit.

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.

Nonlinear inverse heat conduction problem of surface temperature estimation by calibration integral equation method

A calibration integral equation method is proposed for estimating the surface temperature in the context of a nonlinear inverse heat conduction problem. The temperature-dependent thermophysical properties and probe positioning are implicitly accounted in the integral equation formulation through calibration tests. A first kind Chebyshev expansion is applied to represent the temperature-dependent property transform function. The undetermined expansion coefficients associated with the Chebyshev expansion are then estimated through two calibration tests. Regularization of the ill-posed problem is achieved by the future-time method. The optimal regularization parameter is estimated using a phase plane and cross-correlation phase plane analyses. Numerical simulation for stainless steel yields highly favorable surface temperature prediction.

Solution of a Nonlinear Heat Conduction Equation for a Curvilinear Region with Dirichlet Conditions by the Fast-Expansion Method

The analytical solution of the nonlinear heat conduction problem for a curvilinear region is obtained with the use of the fast-expansion method together with the method of extension of boundaries and pointwise technique of computing Fourier coefficients.

Notes on using temperature-dependent thermal diffusivity—forgotten rules

New materials and structures are applied in order to maintain the engineering products more and more efficient. Different types of thermal analysis/modelling of such materials and structures are also needed for the proper planning or design. The thermal analysis/modelling processes are supported by many commercial codes but in some cases, the experts might have applied a self-developed method also. The success of the analysis/modelling depends on many factors, but the basis is the correct application of the heat conduction theory which is more important in case of transient thermal processes. As the used tools are more and more powerful, the users pay less attention to the theory background, according to the author experience. It seems that the long tradition of the inconsistent application of the temperature-dependent thermal diffusivity is continuing, even strengthened by the increased availability of data of the temperature-dependent thermal properties. Regarding the origination of the thermal diffusivity, this might lead to the misinterpretation or misunderstanding of the nonlinear heat conduction problem solution procedures. This paper calls attention to avoid this, shows what kind of mistakes might occur related to this widely observable practice, summarizes the proper application and interprets where and how one could find room for the temperature-dependent thermal diffusivity.