Stable solution to nonstationary inverse heat conduction equation

• Autorzy: Joachimiak M. , Ciałkowski M.

Identyfikatory

DOI

10.1515/aoter-2018-0002

• Języki publikacji: EN

Abstrakty

EN

The paper presents investigations related to solving of a direct and inverse problem of a non-stationary heat conduction equation for a cylinder. The solution of the inverse problem in the form of temperature distributions has been obtained through minimization of a functional being the measure of the difference between the values of measured and calculated temperatures in M points of the heated cylinder. The solution of the conduction equation was presented in the convolutional form and then numerically integrated approximating one of the integrand with a step function described with parameter Θ ∈ (0, 1]. The influence of the integration parameter Θ on the obtained solution of the inverse problem (including a number of temperature measurement points inside the heated body) has been analyzed. The influence of the parameter Θ on the sensitivity of the obtained temperature distributions has been investigated.

Słowa kluczowe

EN

inverse problem; nonstationary heat conduction equations; solution sensitivity; integral parameter

PL

zagadnienie odwrotne; parametr; całka; wrażliwość; roztwór

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN ; Komitet Termodynamiki i Spalania PAN
• Czasopismo: Archives of Thermodynamics
• Rocznik: 2018
• Tom: Vol. 39, no 1
• Strony: 25--37
• Opis fizyczny: Bibliogr. 22 poz., rys., wz.

Twórcy

autor

Joachimiak M.

• Poznan University of Technology, Chair of Thermal Engineering, Piotrowo 3, 60-965 Poznań, Poland

autor

Ciałkowski M.

• Poznan University of Technology, Chair of Thermal Engineering, Piotrowo 3, 60-965 Poznań, Poland

Bibliografia

• [1] Buczek A., Cebo-Rudnicka A., Malinowski Z., Telejko T.: Influence of phase transformations on the inverse boundary problem solution during steel quenching. Thermodynamics in Science and Technology (L. Bogusławski, Ed.). Poznań 2011, 551–558 (in Polish).
• [2] Ciałkowski M.: Selected methods and algorithms of solving inverse problems of heat conduction. Wydawnictwo Politechniki Poznańskiej, Poznań 1996 (in Polish).
• [3] Ciałkowski M., Grysa K.: A sequential and global method of solving an inverse problem of heat conduction equation. J. Theor. App. Mech-Pol 48(2010), 1, 111–134.
• [4] Ciałkowski M.J., Grysa K.W.: On a certain inverse problem of temperature and thermal stress fields. Acta Mech. 36(1980), 169–185.
• [5] Frąckowiak A., Botkin N.D., Ciałkowski M., Hoffmann K.H.: A fitting algorithm for solving inverse problems of heat conduction. Int. J. Heat Mass Tran. 53(2010), 2123–2127.
• [6] Frąckowiak A., Wolfersdorf J.V., Ciałkowski M.: Solution of the inverse heat conduction problem described by the Poisson equation for a cooled gas-turbine blade. Int. J. Heat Mass Tran. 54(2011), 1236–1243.
• [7] Guz E., Kącki E.: Temperature field in solids. PWN, Warszawa 1967 (in Polish).
• [8] Groetsch Ch.W.: Inverse Problems in the Mathematical Sciences. Vieweg Mathematics for Scientist and Engineers., Vieweg, Wiesbaden 1993.
• [9] Jaremkiewicz M.: Determination of the transient fluid temperature using inverse method – experimental verification. In: Modern Energy Technologies, Systems and Units, (J. Taler, Ed.), Wydawnictwo Politechniki Krakowskiej, Kraków 2013, 83–99.
• [10] Joachimiak M.: Analysis of heating process based on the solution of inverse problem for heat conduction equation. Politechnika Poznańska, Poznań 2014 (in Polish).
• [11] Joachimiak M., Ciałkowski M.: Optimal choice of integral parameter in a process of solving the inverse problem for heat equation. Arch. Thermodyn. 35(2014), 3, 265–280.
• [12] Joachimiak M., Ciałkowski M.: Stable method of solving the inverse boundary problem for transient heat conduction equation. In: Analysis of systems (B. Węglowski, P Duda, Eds.), Wydawnictwo Politechniki Krakowskiej, Kraków 2013, 135–151 (in Polish).
• [13] Joachimiak M., Ciałkowski M., Bartoszewicz J.: Analysis of temperature distribution in a pipe with inner mineral deposit. Arch. Thermodyn. 35(2014), 2, 37–49.
• [14] Kress R.: Inverse Probleme. Institut für Numerische und Angewandte Mathematik, Universität Göttingen 2009.
• [15] Louis A.K.: Inverse und Schlecht Gestellte Probleme. Teubner Studienbücher, Stuttgart 1989.
• [16] Marois M.A., M. Désilets M., Lacroix M.: What is the most suitable fixed grid solidification method for handling time-varying inverse Stefan problems in high temperature industrial furnaces? Int. J. Heat Mass Tran. 55(2012), 5471–5478.
• [17] Mierzwiczak M., Kołodziej J.A.: The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem. Int. J. Heat Mass Tran. 54(2011), 790–796.
• [18] Özişik M.N.: Heat Conduction. 2nd Edn. Wiley,1993.
• [19] Taler J.: Solving direct and inverse problems of heat conduction. Ossolineum, Wrocław Warszawa Kraków 1995 (in Polish).
• [20] Taler J., Duda P.: Rozwiązywanie prostych i odwrotnych zagadnień przewodzenia ciepła. WNT, Warszawa 2003 (in Polish).
• [21] Taler D., Sury A.: Inverse heat transfer problem in digital temperature control in plate fin and tube heat exchangers. Arch. Thermodyn. 32(2011), 4, 17–31.
• [22] Taler J., Zima W.: Solution of inverse heat conduction problems using control volume approach. Int. J. Heat Mass Tran. 42(1999), 1123–1140.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).