Finding analytical solutions to the problems of thermal conductivity with variable physical properties of the medium by classical analytical methods is very complicated mathematically. The known expressions repre-senting complex infinite series including two types of Bessel functions and gamma-functions are, in fact, numerical as they require a numerical solution to complex transcendental equations with eigenvalues of the boundary problem. Such solutions can hardly be used in engineering applications, especially in cases when a solution to a certain problem is only an intermediate stage in other problems (such as thermoelasticity and control problems, inverse problems, etc.) which can be solved effectively only by finding analytical solutions to the initial problems. Therefore, an urgent problem now is to develop new methods of obtaining analytical solutions to the abovementioned problems, at least approximate ones. The study employed methods of additional boundary conditions and additional unknown functions in the integral method of heat balance. High-precision approximate analytical solutions to the transient heat conduction problem with nonhomogeneous physical properties of the medium for an infinite plate under symmetric boundary conditions of the first type have been obtained. The initial problem for partial differential equations is reduced to two problems in which ordinary differential equations are integrated. Additional boundary conditions are defined in such a way that their fulfillment in accordance with the new method is equivalent to the result of solving the initial partial differential equation at the boundary points and at the temperature perturbation front (for the first stage of the process). By combining methods with finite and infinite heat propagation rate we have been able to obtain high-precision analytical solutions for the whole time range of the unsteady process including its small and ultra small values. The solutions look like simple algebraical polynomials not including special functions (Bessel, Legendre, gamma-functions and others). Since it is not necessary to directly integrate the initial equations by the space variable and to reduce them to ordinary differential equations with additional unknown functions, the considered method can be used for solving complex boundary problems in which differential equations do not allow distinguishing between the variables (into nonlinear, with linear boundary conditions and heat sources, etc.).
A mathematical and computer model of a district heating network fed by two heat sources located at significantly different elevation marks has been developed. The model is based on the electrohydraulic analogy of electric current spread in conductors and liquid pressure spread in pipelines, which are described by the same equations. In particular, the first and second Kirchhoff’s laws used in the calculation of electrical networks are applied to calculate the velocities and pressures in a complex multi-ring pipeline system. In order to maximize the approximation of the computer model to the real hydraulic network (in resistance to the process of heating agent flow), the method of automatic identification of the model is applied. This method is an iterative process of changing the hydraulic resistances in pipelines of the model in such a way that the results obtained from the calculations would have the least differences from the experimental data. The accuracy of identification depending on the number of points with known experimental data is 3 – 5%.
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.
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.
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