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  • Author: Michał Ciałkowski x
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Magda Joachimiak and Michał Ciałkowski

Abstract

Direct and inverse problems for unsteady heat conduction equation for a cylinder were solved in this paper. Changes of heat conduction coefficient and specific heat depending on the temperature were taken into consideration. To solve the non-linear problem, the Kirchhoff’s substitution was applied. Solution was written as a linear combination of Chebyshev polynomials. Sensitivity of the solution to the inverse problem with respect to the error in temperature measurement and thermocouple installation error was analysed. Temperature distribution on the boundary of the cylinder, being the numerical example presented in the paper, is similar to that obtained during heating in the nitrification process.

Open access

Magda Joachimiak, Michał Ciałkowski and Jarosław Bartoszewicz

Abstract

The paper presents the results of calculations related to determination of temperature distributions in a steel pipe of a heat exchanger taking into account inner mineral deposits. Calculations have been carried out for silicate-based scale being characterized by a low heat transfer coefficient. Deposits of the lowest values of heat conduction coefficient are particularly impactful on the strength of thermally loaded elements. In the analysis the location of the thermocouple and the imperfection of its installation were taken into account. The paper presents the influence of determination accuracy of the heat flux on the pipe external wall on temperature distribution. The influence of the heat flux disturbance value on the thickness of deposit has also been analyzed.

Open access

Magda Joachimiak, Andrzej Frąckowiak and Michał Ciałkowski

Abstract

A direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.