Analysis of the heat transfer for processes of the cylinder heating in the heat-treating furnace on the basis of solving the inverse problem

2021

E3S Web Conf.

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In this paper a method for reducing leakage in labyrinth seals is presented. This method is based on CFD calculations and consists in the analysis of the phenomenon of gas kinetic energy carry-over in chambers of the seal between gaps. It belongs to the group of geometrical inverse problems and is designed for seals of given outside dimensions. For straight through labyrinth seals it enables determining the number of teeth and their optimal arrangement. This method was developed based on numerical and experimental tests. Examples of numerical calculations presented in this paper prove that this method is effective for straight through seals. We obtained the reduction of leakage ranging from 8.7 to 9.4% relative to the initial geometry with no change in the outside dimensions of the seal.

Section: Introductionmentioning

confidence: 99%

Method for the reduction of leakage in labyrinth seals by adapting the seal geometry to match the flow conditions

2021

E3S Web Conf.

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“…Monitoring of temperature of the edges of elements heated during processes of thermal and thermochemical treatment is very important due to material structures forming on the surface of the element being treated thermally [1][2][3]. Knowledge about the surface temperature allows the atmosphere in the furnace to be adjusted more precisely, which involves diffusion of relevant chemicals to heat treated steel and formation of a surface having desired properties [1].…”

Section: Introductionmentioning

confidence: 99%

Analysis of heating time and of temperature distributions for cylindrical geometry with the use of solution to the inverse problem

2021

E3S Web Conf.

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Changes in heating time of a cylinder in the furnace for thermal and thermochemical treatments depending on the given heating rate is analysed in this paper. Temperature distributions from the axis to the boundary of the cylinder were determined based on solving non-stationary and non-linear inverse problem for the heat equation. Differences between the temperature on the boundary and along the cylinder axis for processes with the given heating rates from 5 to 10ᵒC/min were calculated. Twofold increase in the heating rate allowed the heating time to be reduced significantly. Increase in the heating rate had no impact on the difference between the temperature on the boundary and on the axis of the cylinder and on the quantity of energy being consumed by heating elements.

“…Currently, research work focuses on finding new methods of regularization (Cheng and Feng, 2014;Zhuang and Chen, 2017) and on the modification of already known and used methods (Yang et al, 2015;Zheng and Zhang, 2018). Because of a wide application of inverse problems in engineering problems, such as the cooling of the blades in gas turbines (Frąckowiak et al, 2017;Frąckowiak et al, 2019b;Frąckowiak et al, 2011), analysis of the boiling heat transfer in minichannels (Ho_ zejowska et al, 2009;Maciejewska and Piasecka, 2017), analysis of thermal and thermo-chemical treatment (Joachimiak et al, 2019b) or monitoring of power boilers operation (Taler et al, 2016(Taler et al, , 2017, developing methods for regularization of these problems and investigating the process of choosing the regularization parameter are very significant.…”

Section: Introductionmentioning

confidence: 99%

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

2020

HFF

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Purpose In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy. Design/methodology/approach The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied. Findings Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm Nmax. Practical implications Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary. Originality/value The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.