Forecast of the temperature of self-heating of plant raw materials in a hearth of the plate shape


V.P. Olshanskiy* and O.V. Olshanskiy

An analytical solution of the differential equation of nonstationary thermal conductivity for a semi-infinite array of raw materials is constructed using the Fourier series and the integral cosine transformation. The array is bounded only by one of the spatial coordinates, in the direction of which a narrow self-heating hearth of the plate type is extended. The uniform distribution of thermal sources in the center which does not change over time is accepted. After calculating the improper integrals, the temperature increase in the center of the hearth is expressed by a single functional series. For further simplify the calculation, a compact approximation of the probability integral is introduced and an approximate analytical calculation of the sum of the trigonometric series is performed. As a result, the formula for the temperature increase is given by a series of fast convergence and it is shown that for large values of time in a series it can be limited to calculating only the first term. For predict the development of temperature over time, a combination of theory and experiment is provided. The experiment is required to measure the values of the temperature of the raw material at two points in time at the beginning of self-heating to identify the parameters of the hearth. Appropriate graphs have been constructed to simplify identification. The approximate results obtained using them can then be refined by numerical solutions of the corresponding transcendental equation. Examples of identification of parameters of internal thermal sources are given. After identification, the theoretical results become consistent with the experiment and suitable for forecasting the increase in self-heating temperature.

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