Nonlinear Heat Conduction Research Articles

A modified sequential gradient-based method for the inverse estimation of transient heat transfer coefficients in non-linear one-dimensional heat conduction problems

In this communication, we introduce an online sequential implementation of a gradient-based method for reconstructing transient heat transfer coefficients in the context of non-linear one-dimensional heat conduction problems. Such a method employs a quasi-Newton updating strategy for computing the descent direction, in contrast with the traditional approach based on the conjugate gradient method. We denote the resulting procedure as the sequential quasi-Newton method (SQNM). The performance of the proposed algorithm was tested in the reconstruction of triangle-, sine-, and square-wave functions that models different transient heat transfer coefficients and compared with the results obtained using a standard sequential function specification method. The SQNM was capable of properly reconstruct the aforementioned exact functions independently of the location of the temperature sensors within the body. The proposed strategy is fast, robust, and reliable, which demonstrates the suitability of employing the sequential gradient-based implementation, together with the quasi-Newton updating strategy, for reconstructing transient heat transfer coefficients in the context of one- and two-dimensional non-linear heat conduction problems. Thus, the proposed method is a novel alternative strategy to other online inverse estimation procedures.

ANALITYCAL SOLUTION OF THE ONE DIMENSIONAL NONLINEAR TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN’S FUNCTIONS

Analytical solutions showed to be an important and strong tool for understand thermal problems using mathematic tools. In this work we propose an approach about one dimensional analytical solution for a nonlinear transient heat conduction problem, were used mathematical elements such as Kirchhoff transformation, Green’s functions and the combination of them. The combination of this two methods showed that was possible to determinate an analytical solution for the nonlinear thermal problem, and showed a good approximation when compared with results from numerical methods.

AN UNCONDITIONALLY STABLE METHOD FOR TRANSIENT HEAT CONDUCTION

Time-dependent transient heat conduction problems are widely encountered in aerospace, civil engineering, metallurgical engineering, etc., and for such problems, accurate and fast numerical approaches have always attracted attention in the past decades. To achieve this goal, this paper proposes an unconditionally stable single-step time integration method for general transient heat conduction systems. In the proposed method, the temperature vector and its time derivative are formulated independently by the Langrage interpolation function, and then the relation between the temperature vector and its time derivative is defined with the weighted residual method. Theoretical analysis, including convergence rate and amplification factor, illustrates that the proposed method is strictly second-order accurate for the temperature vector and its time derivative, and it has the strong algorithmic dissipation (L-dissipation), meaning that it can quickly filter out the unwanted numerical oscillations in the high-frequency range. At present, most existing time integration methods, such as the Crank-Nicolson method and the Galerkin method, are unconditionally stable for linear transient heat conduction systems, but they are conditionally stable for nonlinear ones. To this end, this work improved the stability analysis theory for nonlinear transient heat conduction systems proposed by Hughes and used the improved stability analysis theory to design the free parameters of the proposed method. Because of this reason, the proposed method is unconditionally stable for both linear and nonlinear transient heat conduction problems. Due to the desirable algorithmic stability, the proposed method can still provide accurate and stable predictions for nonlinear transient heat conduction problems where the excellent Crank-Nicolson method fails. Some linear and nonlinear transient heat conduction problems are solved in this paper, and the results of these problems show that compared to the currently popular time integration methods, such as the Crank-Nicolson method and the backward difference formula, the proposed method enjoys noticeable advantages in accuracy, dissipation and stability.

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.

Heat Source Analysis on Ultrasonic Welding of Plastic Structural Components Based on Numerical Simulation

Compared with traditional arc welding, ultrasonic welding of plastic structural components has such advantages as fast welding speed, high weld strength, small surface damages, simple equipment operation and cleanness and no pollution, thus having a good application prospect. However, the existing literatures have discussed little on the thermal stability analysis of the welding process and the influences of the parameters affecting the fixation effect of welding on the heat transfer performance of the welding process. For this reason, this paper analyzes the heat source in the ultrasonic welding of plastic structural components based on numerical simulation. First, numerical simulation was performed on the temperature field of the heat source in the ultrasonic welding of plastic structural components, the basic heat transfer forms in the welding process were elaborated in detail, the corresponding governing equations and boundary conditions were given and the governing equation of nonlinear transient heat conduction were solved by the finite element method. Then, ABAQUS thermal analysis and heat source parameter calibration were carried out on the welding heat source. In the experiment section, the heat source analysis results were given regarding the ultrasonic welding of plastic structural components.

Global strong solution to a thermodynamic compressible diffuse interface model with temperature‐dependent heat conductivity in 1D

In this paper, we investigate the wellposedness of the nonisentropic compressible Navier–Stokes/Allen–Cahn system with the heat conductivity proportional to a positive power of the temperature. This system describes the flow of a two‐phase immiscible heat‐conducting viscous compressible mixture. The phases are allowed to shrink or grow due to changes of density in the fluid and incorporates their transport with the current. We established the global existence and uniqueness of strong solutions for this system in 1D, which means no phase separation, vacuum, shock wave, mass, or heat or phase concentration will be developed in finite time, although the motion of the two‐phase immiscible flow has large oscillations and the interaction between the hydrodynamic and phase‐field effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov–Shelukhin's result (Kazhikhov and Shelukhin, 1977) for the compressible single‐phase flow with constant heat conductivity to the non‐isentropic compressible immiscible two‐phase flow with degenerate and nonlinear heat conductivity.

Temperature-constrained topology optimization of nonlinear heat conduction problems

Abstract This paper presents topology optimization of nonlinear heat conduction problems with multiple domains and multiple constraints, including regional temperature and material volume for reducing temperature. Maximum approximation temperatures in the constraint regions are accurately and dynamically calculated, though temperature and temperature-dependent thermal conductivity change with the update of material distribution. A temperature measure with constant error to approximate regional maximum temperature is adaptive to different temperature ranges. A strategy of hole nucleation generation combined with the regional temperature constraints is presented for the level set-based topology optimization. The solid isotropic material with penalization (SIMP) and parametrized level set methods are compared for the temperature-constrained topology optimization. Finally, several numerical examples are solved by the SIMP and parametrized level set methods. The results demonstrate that the proposed approach can obtain intricate topological details and reduce regional temperatures for the nonlinear heat conduction problems.

Numerical investigations on model order reduction to SEM based on POD-DEIM to linear/nonlinear heat transfer problems

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

Inverse estimation of hot-wall heat flux using nonlinear artificial neural networks

Compared with cold-wall heat flux measurement, hot-wall heat flux indicates more information from dynamic heat transfer between the aerothermodynamic boundary layer and the true model’s surface; thus, hot-wall heat flux measurement has become predominant in the aerothermodynamic measurement field. In harsh aerothermodynamic in-flight or ground tests, the hot-wall heat flux must be determined from time history temperature measurements at one or more interior locations. Therefore, accounting for the temperature dependence of the thermophysical properties, hot-wall heat flux measurement essentially results in the solution of the nonlinear inverse heat conduction problem (IHCP). In this paper, a novel inverse estimation of hot-wall heat flux using nonlinear artificial neural networks (ANN) is presented. First, motivated by the hybrid method proposed by Clayton A. Pullins, David O. Hubble, and Tom E. Diller et al., [2010], a modified hybrid heat flux measurement method using two in-depth thermocouples is proposed, which avoid to directly measure surface temperature of gauge or model under harsh aerodynamic heating environments; accounting for the unknown temperature dependent thermophysical properties, a new heat flux inverse estimation model of a nonlinear ANN is proposed and identified to approximate the modified hybrid measurement method through calibration experiment. This heat flux inverse estimation method does not need to solve a first kind Volterra integral equation and to obtain the information about the thermophysical properties of heat conduction body, and the thermal inertia, locations of thermocouples. This paper presents xenon lamp calibration and arc-heated wind tunnel experiments that validate the new inverse estimation method combined with the fabricated hot-wall heat flux sensor and its probe. These experimental results show that, in general, the dynamical hot-wall heat flux estimated based on the proposed method agree well with the known calibrated values and the stagnation heat fluxes of the classic slug calorimeter.

Analytical thermal model of orthogonal cutting process for predicting the temperature of the cutting tool with temperature-dependent thermal conductivity

High temperatures generated in cutting processes significantly affect the surface integrity of machined parts and tool wear, leading to workpiece thermal damage, tensile residual stresses in the workpiece and a reduction in tool life. In recent years, different analytical thermal models to predict cutting temperatures have been developed in literature based on 2D modeling of the cutting process and the assumption that thermal conductivities of workpiece and chip are not dependent on temperature. However, this dependence of conductivity on temperature may have a significant influence on predicted temperatures and must be taken into account. In this paper, a thermal model of the orthogonal cutting process that considers thermal conductivity of materials (chip and tool) to be dependent on temperature is developed. A linear variation of thermal conductivity with temperature is assumed for chip (workpiece) and tool materials. The model is based on application of: (1) the Kirchhoff transformation in order to convert the nonlinear heat conduction problem into a linear one, (2) the theory of moving and stationary heat sources in semi-infinite and infinite mediums in order to model primary and secondary deformation zones and (3) imaginary heat sources to meet adiabatic boundary conditions in the chip and tool. Imaginary heat sources were defined in the thermal model proposed in this paper in such a way that the effect of the tool-chip interface dimensions and of cutting tool width on the tool temperature could be taken into account. This allows the temperature on the rake face and lateral faces of the tool to be predicted. To this end, a new methodology that considers the temperature-dependent thermal conductivity of materials was developed in order to estimate heat partition ratio along the secondary heat source (tool-chip interface), which is assumed to be non-uniform. Orthogonal cutting tests were also performed in order to verify model predictions by comparing them to tool temperature distributions measured using an IR camera.

Analysis of Thermodynamic Parameter Variability in a Chamber of a Furnace for Thermo-Chemical Treatment

This paper presents results of research on unevenness of cylinder heating in a furnace for thermo-chemical treatment. Experimental research was conducted with respect to nitriding. Various heating speeds and settings of the fan operation in the furnace were considered. Boundary conditions were calculated in the form of temperature and the heat transfer coefficient (HTC) on the cylinder boundary in four planes along the cylinder length. Calculations were performed with the use of the inverse problem for non-linear and unsteady heat conduction equations. Boundary conditions from individual planes were compared with the mean value of them all. The variability of the calculated boundary conditions (temperature and HTC) along the cylinder length was investigated based on values of the absolute and relative differences for temperature and HTC. Estimates: mean value, mean value from the absolute value and the maximum values for the absolute and the relative differences of temperature and HTC were also calculated. Estimates were the measurements of the unevenness of cylinder heating in a furnace for thermo-chemical treatment. Based on the results of our research, it was found that an increase of the fan rotational speed from 50% to 100%, with the same heating speed, resulted in a significant leveling of temperature in the analyzed planes. The difference in temperature along the cylinder length was reduced from 6.8 °C to 3.3 °C. The increase of the heating speed from 5 °C/min to 10 °C/min resulted in an increase of the unevenness of the cylinder heating. Values of the absolute differences of temperature in the analyzed planes with reference to the mean temperature changed from an interval from −2.7 °C to 2.3 °C to a range from −4 °C to 5 °C. In processes with a heating speed greater than 5 °C/min, more intensive heating in the end part of the cylinder (close to the cylinder) was achieved than it was in other planes. It was proven by temperature values, which were higher, even, by 5.4 °C, and by HTC values, higher by 11.4 W/m2K, when compared with mean values. Obtained results can form the basis for nitriding process optimization.

Quench and Stability Modelling of a Metal-Insulation Multi-Double-Pancake High-Temperature-Superconducting Coil

The most detailed, first-principle simulation model of a quench in a high-temperature superconducting metal-insulation multi-module/multi-double-pancake coil, wound with a REBCO coated superconductor/tape and normal-metal tape (co-wind), was developed. It enables one to simulate a normal zone propagation, or non-propagation, in the subtle multi-turn structure of the coil's pancakes due to the inductive coupling and transverse electric contact between all the REBCO turns and co-wind turns along with the external circuit and active and passive quench protection systems. In doing so, the multi-module coil's structure is represented as a “rectilinear equivalent” electric circuit, a branched multi-decker multi-rectangle planar grid, which facilitates the mathematical formulation, using the standard electric circuit differential equations and the Kirchhoff's rules. The transient non-linear comprehensive heat conduction equations are employed to model the thermal part of the phenomenon at the same level of detail. All this turns the model into an effective design tool for metal-insulated, -stabilized and/or -reinforced multi-module coils. A parametric study of quench in a 6-module no-insulation (NI) test coil, using a stainless-steel thin co-wind, was conducted to identify the possibilities of the model. The profound effect of the transverse contact resistivity on an NI-coil quench behavior is confirmed and detailed.

Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.

Reduction of plane-radial freezing - thawing problems to plane-parallel ones

Construction in northern conditions is complicated by the processes of seasonal freezing and thawing of soils. This factor is of particular importance in the area of permafrost distribution. The paper deals with the problems that arise when laying communication networks. Any engineering solution must be justified by a heat engineering calculation. In one case, it is necessary to exclude the thawing of water or sewer networks, in the other case – to prevent the destructive effect of the heating main on the permafrost soil. The most common mathematical model of the heat transfer process in the presence of phase transitions is the Stefan problem. When calculating the processes of freezing-thawing near the pipes of communications, it is rational to set the problem in cylindrical coordinates, using the property of plane-radial symmetry. But the Stefan problem has an exact solution only in Cartesian coordinates, for the case of plane-parallel symmetry. For this case, many approximate dependencies are also obtained. The paper presents a method that allows using the results of solving the plane-parallel Stefan problem to solve the plane-radial problem with the same values of the input parameters. Approximate values of the coordinate of the phase transition front as a function of time for the plane-parallel and plane-radial areas are obtained using the method of sequential change of stationary states. A one-to-one relationship is established between the obtained values. If there is a solution obtained by any method for a plane-parallel area, and then using this dependence, it can be applied to a plane-radial area. This technique is suitable not only for the Stefan problem, but also for many nonlinear heat conduction problems. The results given in the paper are of practical importance: using the above method, almost any solution obtained for one type of symmetry directly extends to other types of symmetry.

Time fractional heat transfer analysis in thermally sensitive functionally graded thick hollow cylinder with internal heat source and its thermal stresses

The aim of this work is to investigate the time fractional thermoelastic state of a thermally sensitive functionally graded thick hollow cylinder subjected to internal heat source. Convection boundary conditions are applied on the curved surface of cylinder also sectional heating is assumed on the lower surface. The material properties except density and Poisson’s ratio are assumed to be dependent on temperature. Kirchhoff’s variable transformation is used to reduce the nonlinear heat conduction with temperature dependent thermal conductivity and specific heat capacity to linear form. Further the solution of linear form is obtained by using integral transform technique in the form of Bessel’s and trigonometric functions. All physical and mechanical quantities are taken as dimensional for the theoretical analysis whereas for numerical computations non-dimensional parameters are assumed. Numerical results of temperature change and stress distribution are illustrated graphically for ceramic-metal-based functionally graded materials and shown in figures with the help of Mathematica software.

Combustion Model of a Dual‐Fuel Diesel Engine

AbstractA mathematical model is presented for the volumetric combustion of a homogeneous fuel mixture in compression‐ignition and forced‐ignition engines. With careful consideration of all combustion parameters, the profile of the burning rate has a two‐peak structure. A simple nonlinear heat conduction problem was solved to illustrate the importance of including the isobaric process and heat conduction as functions of the temperature. The study shows that, when the pre‐exponential parameter was chosen as a calibrated parameter, the ignition delay time significantly depends on the start of combustion and the pressure in the common rail, while the effect of the engine speed seems less noticeable. The results can be extended to quasi‐sized combustion models.

Effect of quantum correction of electron–ion collision frequency on ion temperature and expansion energy in laser-driven fusion

The nonlinear heat conduction equation can be solved by quantum modified heat transfer coefficient in deuterium–tritium (DT) plasma driven by laser above the critical electron temperature. As a result, the electron profile temperature and the heat flux solutions are improved. In this paper, we have studied the subsequent modification of ion temperature due to ion heating by energy transfer between electrons and ions through electron–ion collisions. The differential equation between Te and Ti, expansion energy of plasma as well as the threshold energy of the laser beam for plasma ignition are renewed by considering quantum modification in electron–ion collision frequency. The achieved results improve the values of the ion temperature and the threshold energy for plasma ignition compared with the values obtained from the calculations using the classical electron–ion collision frequency.

An experimental and straightforward approach to simultaneously estimate temperature-dependent thermophysical properties of metallic materials

This article describes an experimental and straightforward technique towards the simultaneous estimation of temperature-dependent thermal conductivity and specific heat of metals. The thermal model is based on transient nonlinear one-dimensional heat conduction across a metallic sample, which is subject to a constant heat flux on the upper side, and insulated on the bottom side. The thermal analyses are performed in plates of 304 stainless steel and WC10Co cemented carbide. The Levenberg–Marquardt method is employed to provide the solution to an inverse heat conduction problem capable of simultaneously evaluating the temperature-dependent thermophysical properties using transient temperature measurements. Through sensitivity analysis, it is possible to obtain prior information about estimation feasibility and establish all experimental aspects. The imperfect contact at the heater-plate interface causes contact resistance effect, which is considered a reducing agent on heat flux. Beck's nonlinear function minimization technique is used to confirm the reliability of the inverse estimation technique by using the achieved outcomes to recover the heat flux imposed on the test plate. Furthermore, the statistical study into confidence bounds and comparison with literature reveal the robustness of the results. Finally, the accuracy of the developed approach is investigated through the analysis of the errors deriving from experimental and numerical procedures.

Анализ живучести магистрального нефтепровода в зоне стыкового сварного соединения

The article describes the development of a technique for analyzing the survivability of the trunk pipeline in the zone of the transverse weld during operation, taking into account the residual welding stresses. The calculation of residual welding stresses was performed by solving the problem of thermoelasto-viscoplasticity for a material with a non-stationary structure by the finite element method. Solving the nonlinear nonstationary heat conduction problem was carried out by the finite difference method using boundary conditions of the third kind. Modeling the kinetics of transformation of austenite into ferrite-pearlite and bainite under non-isothermal conditions during welding was carried out on the basis of the theory of isokinetic reactions. The calculation of survivability is based on the Irwin criterion and the Paris formula.

Non-Linear Steady State Heat Conduction using Element-Free Galerkin Method

This study aims to interpret the non-linear steady-state heat conduction for temperature-dependent thermal conductivity ($k$ ($T$)) using Element-Free Galerkin (EFG) method. In this present study, a one-dimensional heat conduction problem with uniform heat generation was explicated. Moving Least Squares (MLS) approximants were applied to estimate the unknown function of temperature $T$ ($x$) with $T^h$ ($x$) using linear basis and weight functions. The variational method has been used to develop discrete equations. Essential boundary conditions are enforced by using the Penalty method. The results have been obtained for the one-dimensional model using essential MATLAB codes. The results obtained by the EFG method are compared with the analytical and finite-element method results. The results are also studied by increasing the number of nodes to study the convergence which indicated that EFG has good convergence behavior. The results have also been obtained for different values of the scaling parameter ($α_s$) and any values of αs between 1.8 and 2.0 were found suitable for providing better results in the EFG method.