Boundary-value Problem Of Heat Conduction Research Articles

The Mutual Influence of Thermal Contact Conductivity and Convective Cooling on the Temperature Field in a Tribosystem with a Functionally Graded Strip.

An analytical model to find the temperature field that has been developed for friction systems consists of a strip and semi-space. The strip is made of a two-component functionally graded material (FGM) with an exponentially changing coefficient of thermal conductivity. In contrast, the material of the semi-space is homogeneous. An appropriate boundary-value problem of heat conduction with constant specific friction power was formulated and solved using the Laplace integral transform method. The model takes into consideration the imperfect thermal friction contact between the strip and the semi-space, and also the convective cooling on the exposed surface of the strip. The appropriate asymptotic solutions to this problem for low and high values of Fourier number were obtained. It is shown how the determined exact solution can be generalized using Duhamel's formula for the case of a linearly reduction in time-specific friction power (a braking process with constant deceleration). Numerical analysis for selected materials of the friction pair was carried out in terms of examining the mutual impact on the temperature of the two Biot numbers, characterizing the intensity of the thermal contact conductivity and convective heat exchange on the exposed surface of the strip. The obtained results can be used to predict the temperature of friction systems containing elements made of FGM. In particular, such systems include modern disc braking systems.

Effect of Convective Cooling on the Temperature in a Friction System with Functionally Graded Strip.

An exact solution of the boundary-value problem of heat conduction was obtained with consideration of heat generation due to friction and convective cooling for the strip/semi-space system. Analytical solutions to this problem are known for the case with both friction elements made of homogeneous materials or a composite layer with a micro-periodic structure. However, in this study, the strip is made of a two-component functionally gradient material (FGM). In addition, the exact, asymptotic solutions were also determined at small and large values of the Fourier number. By means of Duhamel's theorem, it was shown that the developed solution for a constant friction power allows to obtain appropriate solutions with a changing time profile of this value during heating. Numerical analysis in dimensionless form was carried out for the FGM (ZrO2-Ti-6Al-4V) strip in combination with the cast iron semi-space. The influence of the convective cooling intensity (Biot number) on the temperature field in the considered friction system was investigated. The developed mathematical model allows for a quick estimation of the maximum temperature of systems, in which one of the elements (FGM strip) is heated on the friction surface and cooled by convection on the free surface.

Numerical Determination of the Nonsteady Thermal State of a Three-Layer Hollow Thermosensitive Cylinder Under the Conditions of Complex Heat Exchange

By using the straight-line method and conservative difference schemes, we reduce the nonlinear nonstationary boundary-value problem of heat conduction for a three-layer hollow cylinder to a Cauchy problem for a system of ordinary differential equations, which is solved numerically with the help of the formulas of backward differentiation. The conservative discretization of the heat conduction equation and the boundary conditions with respect to the space variable is performed by the integro-interpolation method. We also analyze the influence of temperature dependence of the thermal characteristics of the chosen materials on the temperature field formed in the layers of a three-layer cylinder.

Use of Functionally Graded Material to Decrease Maximum Temperature of a Coating–Substrate System

A mathematical model for determining the temperature distribution in the system consisting of a coating deposited on the surface of substrate was proposed. The foundation material is homogeneous, while the coating is made of a functionally gradient material (FGM) with thermal conductivity increasing exponentially along the thickness. Heating processes of the outer surface of the coating were considered with a constant and linearly decreasing in time intensity of the heat flux. Such thermal loads are common in thermal problems of friction, particularly regarding frictional heating during braking. An exact (in quadrature) solution of the corresponding boundary-value problems of parabolic heat conduction was obtained. Asymptotic solutions to these problems were also found for small and large values of the Fourier number. Calculations were performed for a coating made of two-component FGM ZrO2-Ti-6Al-4V, applied on a cast iron substrate. In order to explain the effect of FGM on temperature, corresponding analysis was carried out for the coating made of a homogeneous (ZrO2) material.

The Effect of Functionally Graded Materials on Temperature during Frictional Heating: Under Uniform Sliding.

The mathematical model of heating process for a friction system made of functionally graded materials (FGMs) was proposed. For this purpose, the boundary-value problem of heat conduction was formulated for two semi-spaces under uniform sliding taking into consideration heating due to friction. Assuming an exponential change in thermal conductivities of the materials, the exact, as well as asymptotic (for small values of time), solutions to this problem were obtained. A numerical analysis was performed for two elements made of ZrO2–Ti-6Al-4V and Al3O2–TiC composites. The influence of the gradient parameters of both materials on the evolution and spatial distributions of the temperature were investigated. The temperatures of the elements made of FGMs were compared with the temperatures found for the homogeneous ceramic materials.

Uncoupled Quasistatic Problem of Thermoelasticity for a Two-Layer Hollow Thermally Sensitive Cylinder Under the Conditions of Convective Heat Exchange

We determine the unsteady temperature field depending on the radial coordinate and the thermoelastic state caused in a two-layer hollow thermally sensitive cylinder both by this field and the action of external force loads. The nonlinear boundary-value problem of heat conduction with discontinuous coefficients is reduced by the integro-interpolation method to the Cauchy problem for a system of ordinary differential equations, which is solved numerically. The stressed state is determined from the integral equations of the second kind and integral conditions obtained as a result of direct integration of the problem of quasistatic thermoelasticity in stresses. We study the influence of the temperature dependence of the thermal and mechanical characteristics of the layers of materials on the values and character of distributions of temperature and stresses caused by this dependence in a two-layer cylinder.

Solution of the Boundary-Value Problem of Heat Conduction with Periodic Boundary Conditions

We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions. Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution, and its continuous dependence on the data by using the generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of the problem.

Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions

A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented.

Temperature Field in the Contact Zone in the Course of Rotary Friction Welding of Metals

We propose a mathematical model for the investigation of the temperature field caused by the friction welding of metals. An axisymmetric nonlinear boundary-value problem of heat conduction is formulated with regard for the friction heating of two cylindrical specimens of finite length made of AISI 1040 steel. It is taken into account that the thermal properties of this steel, its yield strength, and the friction coefficient change as functions of temperature. The numerical solution of the problem is obtained by the finite-element method. We study the influence of two mechanisms of heat generation (caused by friction on the contact surface and by the plastic deformation) on the temperature field of the specimens. It is shown that the obtained numerical results are in good agreement with the corresponding experimental data.

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.

Effect of the temporal profile of the friction power on temperature of a pad-disc brake system

A mathematical model to analyse the influence of change of friction power over time on the temperature of a pad-disc tribosystem has been proposed. For this purpose a boundary-value problem of heat conduction for two semi-infinite bodies with taking into account a heat generation due to the friction on the contact surface was formulated. Exact solutions of this problem were obtained for seven temporal profiles of the specific heat generation power, which were established experimentally. For selected friction materials, numerical analysis of the spatiotemporal temperature distributions and heat fluxes intensities were executed. The obtained results were compared with a corresponding data which were found by means of known approximate solution.

Stable difference scheme for a nonlocal boundary value heat conduction problem

Abstract In this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.

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

Thermal Stresses Due to Frictional Heating with Time-Dependent Specific Power of Friction

Abstract In this paper influence of temporal profile of the specific friction power (i.e. the product of the coefficient of friction, sliding velocity and contact pressure) on thermal stresses in a friction element during braking was investigated. Spatio-temporal distributions of thermal stresses were analytically determined for a subsurface layer of the friction element, based on the model of thermal bending of a thick plate with unfixed edges (Timoshenko and Goodier, 1970). To conduct calculations, the fields of dimensionless temperature were used. These fields were received in the article (Topczewska, 2017) as solutions to a one-dimensional boundary-value problem of heat conduction for a semi-space heated on its outer surface by fictional heat flux with three, different time profiles of the friction power.

Integral methods of solving boundary-value problems of nonstationary heat conduction and their comparative analysis

The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.

Influence of the Friction Power on Temperature in the Process of Braking

For three experimental dependences used to describe the changes in the specific power of friction as a function of the braking time, we construct the exact solutions of the corresponding thermal problems of friction. For this purpose, we use the Duhamel formula and the well-known analytic solution of a one-dimensional boundary-value problem of heat conduction for two semiinfinite bodies and a constant power of friction on the contact surface. We illustrate the application of obtained solutions to modeling the process of friction heating of a pad–disk tribosystem. We also study the influence of the chosen time profiles of friction power on the temperature distribution in a cast-iron (ChNMKh) disk and in a pad made of FM–16L retinax.

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.