Boundary-value Heat Conduction Research Articles

Approximate analytical solution to the stationary two-dimensional heat conduction problem on infinite bar with the source of heat

An approximate analytical solution to the boundary-value heat conduction problem for an infinite bar with a heat source was obtained with the use of the integral method of heat balance, by introducing a complementary required function and complementary boundary conditions. The boundary - value problem for a partial differential equation is reduced to an ordinary differential equation with respect to this function due to the complementary required function that characterizes the change in temperature along the axis of symmetry in the cross-section of the bar. The complementary boundary conditions determined by the initial differential equation and the given boundary conditions are found so that their satisfaction is equivalent to the solution of the initial equation of the boundary value problem at the boundary points. The fulfilment of the equation at the boundary points as well as the heat balance integral results in the fulfilment of the initial equation inside the domain. The approximate analytical solution obtained can be used to identify the amount of internal heat generated by various production processes (vibration and deformation loads, electromagnetic fields effects, etc.) in thermal and nuclear power plants, in the rocket and space industry and other industrial facilities.

INTEGRAL METHOD OF SOLVING HEAT-CONDUCTION PROBLEMS WITH THE SECOND-KIND BOUNDARY CONDITION. 1. BASIC STATEMENTS

On the basis of systems of identical equalities and integral boundary characteristics, a new algorithm of solving a boundary-value problem on the nonstationary heat conduction in a canonical body with boundary condition of the second kind has been developed. The scheme proposed for finding approximate analytical solutions of boundary-value problems on nonstationary heat conduction with boundary conditions of the second kind involves the introduction into consideration of a temperature-disturbance front and separation of the whole heating process into two stages. For the first stage of this process, on the basis of the differentiation of the heat-conduction equation over a space variable and the application of symmetric integral and differential operators to the expressions obtained, two sequences of integral and differential identical equalities have been constructed. Each of these sequences includes integral or differential limiting characteristics for a definite boundary condition of the second kind. For the second stage, by way of introduction of a boundary function, differentiation of the heat-conduction equation with respect to a spatial coordinate, and application of integral operators to the expression obtained, a sequence of integral identical equalities involving integral boundary characteristics for the second-kind boundary condition has been constructed. On the basis of the integral and differential identical equalities obtained, closed systems of equations, allowing one to find polynomial coefficients of the temperature profile for the first and second stages of the heating process, have been constructed. A general scheme of determining approximate eigenvalues of boundary-value problems with boundary conditions of the second kind on the basis of construction of an ordinary differential equation and transformation of it into the characteristic equation is proposed. For each of the two stages of the heating process, special integral operators, reducing the boundary-value heat-conduction problem to the ordinary differential equation, are proposed.

Modeling of the temperature regime and stress state in the thermal sensitive pad-disk brake system

An analytical–numerical nonlinear model to investigate temperature fields and thermal stresses in a pad and a disk for a single braking with a constant deceleration has been proposed. For this purpose, the boundary-value heat conduction problem for a tribosystem strip–semi-space has been formulated, which takes into account the temperature dependence of the thermophysical properties of materials. The solution to this problem has been obtained by a partial linearization through the Kirchhoff substitution, and next with a subsequent use of the method of linearizing parameters. Knowing the distribution of temperature fields in the elements of a friction pair, the thermal stresses have been established within the theory of thermal bending of plates. In this case, the temperature dependence of the Young’s modulus, Poisson’s ratio, and linear thermal expansion of the pad and disk materials have been additionally taken into account. The numerical analysis has been performed for a friction pair consisting of a titanium alloy pad and a steel disk.

Frictional heating of the brake disc with essential nonlinearity thermal barrier coating

A mathematical model to investigate temperature field distributions in a brake disc with thermal barrier coating (TBC) was proposed. For this purpose, a one-dimensional boundary-value heat conduction problem was formulated for the coating (TBC) - substrate (disc) system. It was assumed that the materials of both the coating and the substrate are thermally sensitive and characterized by essential thermal nonlinearity. The surface of the coating was heated by a heat flux with intensity proportional to the friction power during single braking. Thermal contact between the coating and substrate is perfect. The solution to the problem was obtained by means of an integro-interpolation variant of the method of lines and the DIFSUB package for numerical solving of systems of ordinary differential equations. Quasi-static thermal stresses in the strip with taking into account change in temperature mechanical properties, were determined, too. Numerical analysis was conducted for a coating made from Yttria-Stabilized Zirconium ZrO2–3Y2O3 (YSZ) deposited on an AISI 1040 steel disc.

К решению нестационарных нелинейных граничных обратных задач теплопроводности

К решению нестационарных нелинейных граничных обратных задач теплопроводности

Influence of thermal sensitivity of the materials on temperature and thermal stresses of the brake disc with thermal barrier coating

The time-dependent frictional heating of a disc with applied thermal barrier coating (TBC) on its working surface was investigated. To determine the temperature fields in the coating and the disc a one-dimensional friction heat problem during braking was formulated, with taking into account the dependence of thermal properties of materials from temperature. A model was adopted for materials with a simple non-linearity, i.e. materials whose thermal conductivity and specific heat are temperature dependent, and their ratio – thermal diffusivity is constant. The linearization of the corresponding boundary-value heat conduction problem was made by the Kirchhoff transformation and the linearizing multipliers method. A numerical-analytical solution to the obtained problem was found by Laplace transform method. Knowing the temperature distributions, quasi-static thermal stresses in the strip (TBC) with taking into account change in temperature mechanical properties, were determined. The distribution of temperature and thermal stresses in the strip made from ZrO2 deposited on the UNS G51400 steel disc, was investigated.

Determination of the maximum temperature at single braking from the FE solution of heat dynamics of friction and wear system of equations

ABSTRACTA system of equations of heat dynamics of friction and wear (HDFW) for a pad–disc tribosystem during a single braking has been formulated. It takes into account the coefficient of friction dependent on the maximum temperature, which is the sum of the mean temperature of the nominal contact region (the macro contact) and the flash temperature on the real contact area (the micro contact), temperature-dependent properties of materials and wear. Numerical solution of the initial problem of motion and the nonlinear boundary-value heat conduction problem was obtained. Computations were performed using the axisymmetric FE contact model for the metal ceramic pad and the cast-iron disc.

К решению нелинейных обратных граничных задач теплопроводности

К решению нелинейных обратных граничных задач теплопроводности

Refined Model of Heat Transfer in Composite Bodies Reinforced with Tubes with a Liquid Heat-Transfer Agent Moving in a Developed Turbulent Regime

The author has obtained equations describing thermal conductivity of composite bodies spatially reinforced with a system of smooth tubes in which an incompressible liquid heat-transfer agent is pumped in a developed turbulent regime. The corresponding boundary-value heat-conduction problem was formulated and its qualitative analysis was made. Specific calculations were performed for steady-state temperature fields in cylindrical concrete shells spirally reinforced with steel tubes through which a heat-transfer agent (air) is pumped. A study has been made of the influence of the reinforcement parameters and of the velocity and direction of the heat-transfer agent in the tubes and the dimensions of their cross sections on the temperature field. It has been established that variation of these characteristics enables one to substantially change the intensity of heat removal from the shells, opening up wide opportunities for efficient control of the heat transfer in them.

Method of lines in nonlinear frictional heating during braking

The boundary-value heat conduction problem for two semi-infinite bodies compressed by constant pressure and sliding with a constant deceleration has been formulated. Due to friction at the contact surface the heat is generated. It is assumed that the materials of the bodies are thermal sensitive, i.e. their thermophysical properties are temperature dependent. This problem allows determining temperature fields in such elements of brakes, as a pad and a disk. At the first stage, the Kirchhoff transformation has been used for partial linearization of the nonlinear problem. Next, the Kirchhoff functions have been found by the method of lines. The transition has been made from the Kirchhoff functions to temperatures on the basis of formulas obtained for the friction couple aluminum alloy series (a disk) — the metal-ceramic (a pad). For the case of braking with a constant deceleration and at constant pressure the comparison of the results obtained with the proposed approach, and on the basis of an iterative procedure has been executed. The influence of duration of pressure increase on the temperature has been investigated.

Solving Nonlinear Thermal Problems of Friction by Using Method of Lines

Abstract One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.

Some Methods for Calculating Temperature During the Friction of Thermosensitive Materials

The thermal problem of friction for two semi-infinite bodies, with the dependence of their thermal properties on the temperature taken into account, is considered. It is assumed that a specific power of friction is constant and that a thermal contact between semi-spaces is imperfect. As a consequence of the latter assumption, the linearization of a corresponding boundary-value heat conduction problem using the Kirchhoff transformation is incomplete. Depending on the type of nonlinearity of friction body material—simple or arbitrary—the various methods of linearizing the final task are suggested. The effectiveness of these methods is illustrated by the numerical analysis of the friction materials for couples with a linear or nonlinear temperature dependence on coefficient of thermal conductivity and specific heat.

Temperature in thermally nonlinear pad–disk brake system

The influence of the thermal sensitivity of pad and disk materials on temperature at braking is under investigation. A mathematical model of process of frictional heating in a pad–disk brake system, which takes into account the temperature-sensitive materials, is proposed. The basic element of this model is the thermal problem of friction—a one-dimensional boundary-value heat conduction problem with temperature-dependent thermal conductivity and specific heat. Contrary to the prior studies of authors, where a simple nonlinearity was considered, in this article the arbitrary nonlinearity of the thermophysical properties of materials is studied. The solution of a nonlinear boundary-value heat conduction problem is obtained by the method of successive approximations. The numerical analysis of temperature is executed for some materials of a pad and a disk with and without taking into account their thermal sensitivity.

Mutual influence of the velocity and temperature in the axisymmetric FE model of a disc brake

High energy dissipation systems with friction such as disc brakes require high and constant coefficient of friction. Its value, however, depends beside the materials of the contacting components, on the sliding velocity, contact pressure, temperature, etc. Generally the relationship of these parameters is nonlinear. In this paper an axisymmetric FE contact model was developed to study an influence of the coefficient of friction on the braking time, braking distance, maximum temperature of sliding components of a disc brake and the evolution of the sliding velocity. Computations were carried out based on the equation of motion and the boundary-value heat conduction problem of friction. The main purpose of the study was to compare the transient temperature fields of the pad and the disc, braking time, and braking distance at constant and temperature-dependent coefficient of friction. Dependencies of the coefficient of friction on the temperature and the contact pressure were adopted from experimental measurements and implemented to the FE model of a brake. Temperature and wear evolutions were presented and relevant correlations were discussed.

Influence of thermal sensitivity of the pad and disk materials on the temperature during braking

The transient frictional heating of pad–disk tribosystem at single braking is under consideration. To determine the average friction surface temperature, the one-dimensional thermal problem of friction at braking has been formulated. The linear dependence of the thermophysical properties of the disk and pad materials on the temperature has been taken into account. Model of materials with a simple nonlinearity has been adopted, i.e. materials in which coefficients of heat conduction and specific heat depend on the temperature, and their ratio – coefficient of thermal diffusivity – is constant. Linearization of the corresponding boundary-value heat conduction problem by the Kirchhoff transformation and linearizing parameter method has been performed. The numerical–analytical solution to the problem has been found by using the integral Laplace transform and the Newton–Raphson methods. The influence of the thermosensitive materials of titanium pad, sliding over the surface of the disk made of steel, aluminum alloy or gray cast iron, on the temperature has been studied.

Numerical methods for solving a boundary-value inverse heat conduction problem

This paper considers the boundary-value inverse heat conduction problem with steady boundary. To solve this problem, different approaches based on the Laplace and Fourier transforms are proposed. Application of the Laplace transform makes it possible to obtain an operator equation describing the explicit dependence of the desired boundary-value function on the initial data at the other boundary. This approach to solving the problem is used for the first time. The method based on the direct and inverse Fourier transforms with respect to the time variable provides stable solutions. The estimation of error of these solutions is the best with respect to the order. The range of application of each of the proposed methods and the stability of the solutions obtained by these methods was evaluated by a computational experiment.

Meshless method to solve nonstationary heat conduction problems using atomic radial basis functions

The authors consider a meshless method to solve 3D nonstationary boundary-value heat conduction problems. It is implemented through an iterative scheme based on a combination of the double substitution method and the method of fundamental solutions with the use of atomic radial basis functions. The approaches to the visualization of the desired solution are considered.

Friction-induced formation of heat with account for heat transfer through the contact surface between homogeneous and piecewise-homogeneous semispaces

An analytical solution of the boundary-value heat conduction problem is obtained for a tribosystem consisting of a homogeneous semispace sliding with a constant velocity along the surface of a plane-parallel strip applied to a semi-infinite foundation. The tribosystem is heated as a result of the frictional heat formation on the sliding surface. It is assumed that the thermal contact of the semispace with the strip is not full. With the aid of the Duhamel theorem, a solution for the considered tribosystem, with sliding at a constant deceleration, is also constructed that models heat formation from friction in disk brakes. For the materials of the friction pair "pig iron semispace (disk)–metal-ceramic strip (lining)–steel foundation (frame)," the influence of the coefficient of thermal conductivity of the contact (Biot number) on the temperature distribution was investigated.

Account for heat transfer between elements of a plane parallel strip–foundation friction unit

An analytical solution of the boundary-value heat conduction problem for a tribosystem consisting of a semiinfinite foundation and a plane-parallel strip sliding at a constant velocity along the foundation surface is obtained. The thermal contact between the strip and the foundation is partial. The asymptotics of the solution for low and high values of time have been found. For the materials of the friction pair “metal ceramics strip-pig iron foundation” the influence of the coefficient of thermal conductivity of the contact on temperature distribution was studied.

Temperature in a Frictionally-Heated Ceramic-Metal Patch and Cast Iron Disc during Braking

Analytical solution of a boundary-value heat conduction problem of friction for the friction couple consisting of a semi-infinite foundation and a plane-parallel strip sliding at a linear velocity over its surface, for two thermal boundary conditions on the upper surface of the strip (zero temperature and thermally insulated) is obtained. The results of numerical evaluation of the temperature fields in the strip and the foundation and the maximum contact temperature are presented in graphic form.