При совместном использовании ортогональных методов Л. В. Канторовича, Бубнова-Галеркина и интегрального метода теплового баланса получено точное аналитическое решение нестационарной задачи теплопроводности для бесконечной пластины при симметричных граничных условиях первого рода. Нахождение точного решения при использовании приближенных методов оказалось возможным вследствие использования тригонометрических координатных функций, обладающих свойством ортогональности. Их применение позволяет находить собственные числа не через решение краевой задачи Штурма-Лиувилля, в котором интегрированию подлежит дифференциальное уравнение второго порядка, а через решение дифференциального уравнения относительно неизвестной функций времени, являющегося уравнением первого порядка. Благодаря этому же свойству координатных функций при нахождении из начальных условий констант интегрирования удается избежать решения больших систем алгебраических линейных уравнений с плохо обусловленными матрицами коэффициентов. В связи с чем значительно упрощается как процесс получения решения, так и окончательная формула для него при возможности нахождения не только приближенного, но и точного аналитического решения в форме бесконечного ряда.
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).
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.
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 Bi2 = 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.
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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