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. 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. 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 N_max. 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. 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.

The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ₁) and find proper method for the problem regularization. The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ₁ was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

This paper presents results of research on unevenness of cylinder heating in a furnace for thermo-chemical treatment. Experimental research was conducted with respect to nitriding. Various heating speeds and settings of the fan operation in the furnace were considered. Boundary conditions were calculated in the form of temperature and the heat transfer coefficient (HTC) on the cylinder boundary in four planes along the cylinder length. Calculations were performed with the use of the inverse problem for non-linear and unsteady heat conduction equations. Boundary conditions from individual planes were compared with the mean value of them all. The variability of the calculated boundary conditions (temperature and HTC) along the cylinder length was investigated based on values of the absolute and relative differences for temperature and HTC. Estimates: mean value, mean value from the absolute value and the maximum values for the absolute and the relative differences of temperature and HTC were also calculated. Estimates were the measurements of the unevenness of cylinder heating in a furnace for thermo-chemical treatment. Based on the results of our research, it was found that an increase of the fan rotational speed from 50% to 100%, with the same heating speed, resulted in a significant leveling of temperature in the analyzed planes. The difference in temperature along the cylinder length was reduced from 6.8 °C to 3.3 °C. The increase of the heating speed from 5 °C/min to 10 °C/min resulted in an increase of the unevenness of the cylinder heating. Values of the absolute differences of temperature in the analyzed planes with reference to the mean temperature changed from an interval from −2.7 °C to 2.3 °C to a range from −4 °C to 5 °C. In processes with a heating speed greater than 5 °C/min, more intensive heating in the end part of the cylinder (close to the cylinder) was achieved than it was in other planes. It was proven by temperature values, which were higher, even, by 5.4 °C, and by HTC values, higher by 11.4 W/m²K, when compared with mean values. Obtained results can form the basis for nitriding process optimization.

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.

Details var a2a_config = a2a_config || {}; a2a_config.onclick = 1; a2a_config.locale = "pl"; a2a_config.color_main = "D7E5ED"; a2a_config.color_border = "9D9D9D"; a2a_config.color_link_text = "ffffff"; a2a_config.color_link_text_hover = "ffffff"; a2a_config.color_bg = "d9d9d9"; a2a_config.icon_color = "#4e4e4e"; a2a_config.exclude_services = ["dihitt", "delicious", "whatsapp", "dzone", "sms", "stumpedia", "viber", "yahoo_messenger", "google_plus"]; Details PDF BIBTEX RIS Title Non-linear unsteady inverse boundary problem for heat conduction equation Journal title Archives of Thermodynamics Yearbook 2017 Issue No 2 Authors Joachimiak, Magda ; Ciałkowski, Michał Keywords inverse problem ; sensitivity of solution to inverse problem ; application of Chebyshev polynomials Divisions of PAS Nauki Techniczne Coverage 81-100 Publisher The Committee of Thermodynamics and Combustion of the Polish Academy of Sciences and The Institute of Fluid-Flow Machinery Polish Academy of Sciences Date 2017 Type Artykuły / Articles Identifier DOI: 10.1515/aoter-2017-0011 Source Archives of Thermodynamics; 2017; No 2; 81-100 References Ciałkowski (2010), A sequential and global method of solving an inverse problem of heat conduction equation, Mech Pol, 48, 111. ; Marois (2012), What is the most suitable fixed grid solidification method for handling time - varying inverse Stefan problems in high temperature industrial furnaces Heat Mass Trans, Int J, 55, 5471. ; Frąckowiak (2010), A fitting algorithm for solving inverse problems of heat conduction Heat Mass Trans, Int J, 53, 2123. ; Han (2008), Investigation of heat transfer coefficient in two - dimensional transient inverse heat conduction problems using the hybrid inverse scheme, Int J Numer Meth Engng, 1, 107. ; Duda (2000), Numerical method for the solution of non - linear two - dimen - sional inverse heat conduction problem using unstructured meshes, Int J Numer Meth Engng, 48, 881. ; Maciąg (2009), Trefftz functions for selected direct and inverse problems of mechanics Świętokrzyska Publishers in Polish, Polit. ; Trefftz (1926), Ein Gegenstük zum Ritz schenVerfahren nd of Applied Mechanics, Proc Int Cong, 131. ; Vakili (2013), Low cost surrogate model based evolutionary optimization solvers for inverse heat conduction problem Heat Mass Trans, Int J, 56, 263. ; Mierzwiczak (2011), The determination temperature - dependent thermal conductivity as inverse steady heat conduction problem Heat Mass Trans, Int J, 4, 790. ; Mierzwiczak (2011), The determination of heat sources in two dimensional inverse steady heat problems by means of the method of fundamental solutions Inverse Prob, Sci En, 19, 777. ; Joachimiak (2014), Optimal choice of integral parameter in a process of solving the inverse problem for heat equation, Arch Thermodyn, 35, 265. ; Mierzwiczak (2012), Application of the method of fundamental solu - tions with the Laplace transformation for the inverse transient heat source problem, Mech Pol, 4, 1011. ; Ciałkowski (1980), On a certain inverse problem of temperature and thermal stress fields, Acta Mechanica, 4169. ; Frąckowiak (2011), Solution of the inverse heat conduction problem described by the Poisson equation for a cooled gas - turbine blade Mass Trans, Int J Heat, 54, 1236. ; Grysa (2012), Solving nonlinear direct and inverse prob - lems of stationary heat transfer by using Trefftz functions Heat Mass Trans, Int J, 55, 7336. ; Taler (1999), Solution of inverse heat conduction problems using control volume approach Heat Mass Trans, Int J, 1123. Editorial Board International Advisory Board J. Bataille, Ecole Central de Lyon, Ecully, France A. Bejan, Duke University, Durham, USA W. Blasiak, Royal Institute of Technology, Stockholm, Sweden G. P. Celata, ENEA, Rome, Italy M. Colaco, Federal University of Rio de Janeiro, Brazil J. M. Delhaye, CEA, Grenoble, France M. Giot, Université Catholique de Louvain, Belgium K. Hooman, University of Queensland, Australia D. Jackson, University of Manchester, UK K. Kuwagi, Okayama University of Science, Japan J. P. Meyer, University of Pretoria, South Africa S. Michaelides, Texas Christian University, Fort Worth Texas, USA M. Moran, Ohio State University, Columbus, USA W. Muschik, Technische Universität Berlin, Germany I. Müller, Technische Universität Berlin, Germany H. Nakayama, Japanese Atomic Energy Agency, Japan A. Nenarokomov, Moscow Aviation Institute, Russia H. Orlande, Federal University of Rio de Janeiro, Brazil M. Podowski, Rensselaer Polytechnic Institute, Troy, USA M. R. von Spakovsky, Virginia Polytechnic Institute and State University, Blacksburg, USA A. Vallati, Sapienza University of Rome, Italy