Solution of inverse heat conduction equation with the use of Chebyshev polynomials

2021

E3S Web Conf.

Self Cite

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.

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.

Self Cite

“…This leads to the inverse Cauchy problem, but the condition of energy conservation is ensured the stability of the inverse solution. Such problem is ill-posed in the Hadamard sense (Alifanov, 1994; Hadamard, 1902; Tikhonov and Arsenin, 1977) and generally needs a regularization (Frąckowiak and Ciałkowski, 2018; Joachimiak, 2020; Joachimiak et al , 2019a, 2016). However, the adopted method of calculation makes the problem under consideration possible to be solved without regularization.…”

Section: Introductionmentioning

confidence: 99%

Cauchy type nonlinear inverse problem in a two-layer area

Ciałkowski

Olejnik

et al. 2021

HFF

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Purpose To reduce the heat load of a gas turbine blade, its surface is covered with an outer layer of ceramics with high thermal resistance. The purpose of this paper is the selection of ceramics with such a low heat conduction coefficient and thickness, so that the permissible metal temperature is not exceeded on the metal-ceramics interface due to the loss ofmechanical properties. Design/methodology/approach Therefore, for given temperature changes over time on the metal-ceramics interface, temperature changes over time on the inner side of the blade and the assumed initial temperature, the temperature change over time on the outer surface of the ceramics should be determined. The problem presented in this way is a Cauchy type problem. When analyzing the problem, it is taken into account that thermophysical properties of metal and ceramics may depend on temperature. Due to the thin layer of ceramics in relation to the wall thickness, the problem is considered in the area in the flat layer. Thus, a one-dimensional non-stationary heat flow is considered. Findings The range of stability of the Cauchy problem as a function of time step, thickness of ceramics and thermophysical properties of metal and ceramics are examined. The numerical computations also involved the influence of disturbances in the temperature on metal-ceramics interface on the solution to the inverse problem. Practical implications The computational model can be used to analyze the heat flow in gas turbine blades with thermal barrier. Originality/value A number of inverse problems of the type considered in the paper are presented in the literature. Inverse problems, especially those Cauchy-type, are ill-conditioned numerically, which means that a small change in the inputs may result in significant errors of the solution. In such a case, regularization of the inverse problem is needed. However, the Cauchy problem presented in the paper does not require regularization.

“…In the paper of Liu and Wang (2018), the Cauchy problem for the Laplace’s equation was solved with the use of the method of fundamental solutions and the energy regularization technique to choose the source points. The Laplace’s equation was also solved with the use of iterative algorithms (Frąckowiak et al , 2015a, 2015b), of the Trefftz method (Ciałkowski and Frąckowiak, 2002; Ciałkowski and Grysa, 2010; Grysa et al , 2012; Hożejowski, 2016; Lin et al , 2018), of the method of fundamental solution (Kołodziej and Mierzwiczak, 2008; Mierzwiczak et al , 2015; Mierzwiczak and Kołodziej, 2011) and of the collocation method (Joachimiak et al , 2016). In many cases, the regularization of the inverse problem concerns the problem of choosing the regularization parameter.…”

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.