Objective . The purpose of the study is to develop a technique for obtaining an analytical solution to the thermal conductivity problem for a 2-layer plate under boundary conditions of the 1st kind. Method . The research method is based on the integral method. In this case, an additional function (DF), additional boundary conditions (ABC) and local coordinate systems are introduced. The DF describes the temperature over time at one of the points of the twolayer system. Its use reduces the partial differential equation to an ordinary equation. DKUs allow you to perform equations on boundaries. Result . It is shown that satisfying the equations on the boundaries leads to their fulfillment inside the domain. Note that additional boundary conditions are satisfied for any other method of obtaining analytical solutions. The only difference is that they are not accepted as conditions subject to separate consideration. An additional function is also a quantity that is determined by any other method of obtaining a solution. The only difference is that it is not singled out for separate consideration. Conclusion . It can be stated that the introduction of an additional function and additional boundary conditions does not distort the original formulation of the problem and is only a means to significantly simplify the process of obtaining an approximate analytical solution and the final expression for it.

Based on the definition of an additional function and additional boundary conditions in the integral heat balance method, an approximate analytical solution is obtained for a nonstationary two-dimensional heat conduction problem for an infinite hollow cylinder under boundary conditions of the third kind with variable heat-transfer coefficients in the circumferential direction. An additional function describes the change in temperature over time at one particular point in the spatial variable. Using it, one can reduce the solution of the original partial differential equation to the integration of an ordinary differential equation, from which the eigenvalues of the boundary value problem are found. That is, a different concept of determining eigenvalues based on the time equation with respect to an additional function is presented, in contrast to classical methods, where eigenvalues are found when solving the Sturm–Liouville boundary value problem for a region of a spatial variable. The assignment of additional boundary conditions is the solution of the original equation at the boundary points. It is shown that solving the equation at the boundary points leads to its solution inside the considered region as well. Additional boundary conditions are derived using the original differential equation and basic boundary conditions. Repeatedly differentiating the equation with respect to the spatial variable, and the boundary conditions with respect to time, by comparing the obtained relations, one can find any number of additional boundary conditions. The accuracy of solving the equation inside the considered region depends on the number of approximations, and, consequently, on the number of additional boundary conditions used. The approximate analytical solution obtained in this way is characterized by a simple design that is convenient for use in engineering applications.

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution. THE PURPOSE . Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed. METHODS . The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased. RESULTS . The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function). CONCLUSION . Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

An analytical solution of the heat conduction problem for a two-layer plate based on the Galerkin method and usage of additional boundary conditions (ABC) is obtained. The ABCs are found in such a way that their implementation is to be adequate to the implementation of the original equations at the boundary points. Execution of equations at boundary points leads to their execution over the entire range of spatial coordinates. The use of ABC allows us to obtain chain systems of algebraic equations for unknown solution coefficients. The equations of these systems have highly sparse well-conditioned square matrices. In this connection, their solutions are so simplified that with a large number of approximations, in general form, a system of only two algebraic equations should be solved. A high accuracy of finding the eigenvalues is observed that is explained by the use of a special ABC design.

This paper depicts an approximate analytical solution of the non-stationary heat conduction problem for an infinite plate under asymmetric boundary conditions of the third kind according to the heat balance integral method. The solution is a simple, engineering-friendly product of exponential and coordinate functions. The coordinate functions are found by the method of undetermined coefficients so that the asymmetric boundary conditions of the third kind meet the main criteria primarily. Satisfactory accuracy of the obtained solution for engineering use is provided by residual orthogonality condition abidance of the differential equation to the coordinate function obtained in this research. This method allows for errors by no more than 1% in the second approximation, in the range 1.0 < Fo < ∞ for Bi = 0 and Bi 2 = 0.1. The heat transfer coefficient was determined a plate surface using the named solution along with data on the temperature change at a fixed plate spot, thus solving the inverse heat conductivity problem.

This paper presents an approximate analytical solution of the heat conduction problem for a two-layer plate under symmetric boundary conditions of the first kind. The solution was determined on the basis of the property of the parabolic heat transfer equation associated with an infinite velocity of heat propagation, by determining the accessory unknown function and accessory boundary conditions in the integral heat balance method. Local coordinate systems are given in order to obtain the simplest possible coordinate system satisfying the matching conditions and boundary conditions for each separate layer. An accessory unknown function is the temperature change over time in the center of symmetry. The use of this function in the heat balance integral method allows for reduction in the solution of the initial partial differential equation to the integration of an ordinary differential equation with respect to the additional unknown function. Further boundary conditions are defined in such a way that their satisfaction by an unknown solution is equivalent to the satisfaction of the equations at the boundary points. Studies have shown that the equation fulfillment at the boundaries leads to their fulfillment within the regions.

The paper presents an analytical solution of the heat transfer boundary value problem for a plate with a time-varying boundary condition of the 1st kind using the Kantorovich method. The solution involving a power-law algebraic polynomial with coefficients that are exponentially stabilized in time is characterized by high accuracy. So, already in third approximation in the range from 0 to 4, the discrepancy from the exact solution does not exceed 1%. The time-varying boundary condition of the 1st kind is recovered by solving the inverse heat transfer problem using experimental data on the temperature in the center of the plate.

Abstract Based on thermal profiling of the cylinder of a high pressure cylinder (HPC) of T–100–130 stream turbine detailed researched has been performed to study its temperature conditions during startup. Using experimental data about the temperature condition of an external surface of the cylinder, by way of solving the inverse problem of heat conductivity the average heat transfer coefficients have been determined during the period of startup on its internal surface (on the steam side). At the same time, approximated analytical solution of the heat conductivity problem has been applied for the cylinder’s two-layered insulation (thermal insulation – metal wall). Using the data of experimental and theoretical research thermal stresses have been identified in the cylinder wall, as well as the stresses due to the effect of steam pressure forces. It has been shown that in certain profiles of the turbine cylinder stresses are able to exceed the ultimate stress limit for that material. Using experimental data about the changes in certain parameters (temperature differential for the top- and bottom sections of the cylinder, differential in the shaft- and cylinder extension, vibration indicators, etc.) a theoretical method has been developed for forecasting their changes during a certain time interval as measured from the time of current measurement.

Abstract In order to induce residual compressive stresses in the boundary layer of a steel plate heated to temperature T 0 = 600 °C, a method of jet water cooling with time-varying heat transfer coefficients is considered. By solving the corresponding heat conduction problem, it is shown that the maximum temperature difference between the surface and the center of a 2 mm thick plate is 105 °C. It is observed within 0.02 s since the start of cooling, the time of its complete cooling is equal to 0,32 s. Calculations of temperature stresses using the finite element method showed that temperature tensile stresses occur on the outer surface of the plate. They may reach 26 kg/mm 2 which exceeds the yield strength of this material. As a result of plastic deformation, the thermal tensile stress relieving occurs in the course of cooling of the plate surface layer; when it is completely cooled, the residual compressive stresses 21.8kg/mm 2. (on the surface of the plate) are induced in this layer (0,25 mm depth).

An approximate analytical solution to a heat transfer problem for a moving fluid in a cylindrical channel is obtained using an additional new function and additional boundary conditions in the heat balance integral method and taking into account energy dissipation under a first-order boundary condition that varies along the longitudinal coordinate. The use of an additional new function that determines the temperature change along the longitudinal variable in the center of the channel makes it possible to reduce the solution of the partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are found in such a way that their satisfaction for the new solution is equivalent to the satisfaction of the differential equation at boundary points.

Using additional required functions and additional boundary conditions in an integral method of a heat balance, high-precision approximate analytical solutions of the task of heat conduction for the infinite plate with variable physical properties of the environment in case of the symmetric boundary conditions of the first kind are received. For finding of the decision into areas 0,05 ≤ < ∞ the additional required function characterizing change of temperature in center of a plate which in view of the infinite speed of distribution of the warmth put in the parabolic equation of heat conduction begins to change right after application of a boundary condition of the first kind is entered. Therefore, the range of its change includes all range of time of nonstationary process and all range of change of temperature. For obtaining the decision in case of small and midget values of time the model with a final speed of distribution of warmth based on determination of the front of temperature perturbation and additional boundary conditions is used. The combination of these two models (with the infinite and a final speed warmth distribution) allowed to gain rather simple look approximate analytical solutions of the complex non-linear challenge (with nonlinearity of the second kind) in all range of time of nonstationary process, practically with the given accuracy rating. Reviewing in both models of additional required functions allows to consolidate the solution of partial equations to integration of ordinary differential equations.

Through the interplay of orthogonal methods by L. V. Kantorovich, Bubnov-Galerkin and a heat balance integral method there have been obtained an exact analytical solution of a nonstationary heat conduction problem for an infinite plate under the symmetrical first-type boundary conditions. It was possible to obtain an exact solution through the employment of approximate methods due to the appliance of trigonometric coordinate functions, possessing the property of orthogonality. They enable us to determine eigenvalues not through the solution of the Sturm-Liouville boundary value problem, which supposes the second-order differential equation integration, but through the solution of a differential equation for an unknown function on time, which is the first-order equation. Due to the property of coordinate functions mentioned above, while determining constants of integration out of initial conditions it is possible to avoid solving large systems of algebraic linear equations with ill-conditioned matrix of coefficients. Thus, it simplifies both the process of obtaining a solution and its final formula and provides an opportunity to find not only an approximate, but also an exact analytical solution, represented by an infinite series.