Optimal choice of integral parameter in a process of solving the inverse problem for heat equation

• Autorzy: Joachimiak M. , Ciałkowski M.
• Języki publikacji: EN

Abstrakty

EN

Inverse boundary problem for cylindrical geometry and unsteady heat conduction equation was solved in this paper. This solution was presented in a convolution form. Integration of the convolution was made assuming the distribution of temperature T on the integration interval (t_i, t_i + Δt) in the form T (x, t) = T (x, t_i) Θ + T (z, t_i + Δt) (1 - Θ), where Θ ϵ (0,1). The influence of value of the parameter Θ on the sensitivity of the solution to the inverse problem was analysed. The sensitivity of the solution was examined using the SVD decomposition of the matrix A of the inverse problem and by analysing its singular values. An influence of the thermocouple installation error and stochastic error of temperature measurement as well as the parameter Θ on the error of temperature distribution on the edge of the cylinder was examined.

Słowa kluczowe

EN

inverse problem; sensitivity of solution; heat conduction

PL

zagadnienie odwrotne; przewodnictwo ciepła

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN ; Komitet Termodynamiki i Spalania PAN
• Czasopismo: Archives of Thermodynamics
• Rocznik: 2014
• Tom: Vol. 35, no. 3
• Strony: 265--280
• Opis fizyczny: Bibliogr. 17 poz., il.

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] CIAŁKOWSKI M.J., GRYSA K.W.: On a certain inverse problem of temperature and thermal stress fields. Acta Mech. 36(1980), 169-185.
• [2] CIALKOWSKI M., GRYSA K.: A sequential and global method of solving an inverse problem of heat conduction equation. J. Theor. Appl. Mech. 48(2010), 1, 111 134.
• [3] HAN-TAW CHEN XIN-Yi Wu: Investigation of heat transfer coefficient in two-dimensional transient inverse heat conduction problems using the hybrid inverse scheme. Int. J. Numer. Meth. Eng. 73(2008), 107-122.
• [4] CIAŁKOWSKI M.: A sequential and global method of solving an inverse problem of heat conduction equation. In: Proc. XIII Symposium on Heat and Mass Transfer, 2007 (in Polish).
• [5] TALER J., ZIMA W.: Solution of inverse heat conduction problems using control volume approach. Int. J. Heat Mass Tran. 42(1999), 1123 1140.
• [6] DUDA P., TALER J.: Numerical method for the solution of non-linear two-dimensional inverse heat conduction problem using unstructured meshes. Int. J. Numer. Meth. Eng. 48(2000), 881-899.
• [7] GROETSCH CH.W.: Inverse Problems in the Mathematical Sciences. Vieweg Mathematics for Scientist and Engineers, Vieweg, Wiesbaden 1993.
• [8] KRESS R.: Inverse Probleme. U. Goettingen, Goettingen 2009.
• [9] LOUIS A.K.: Inverse und schlecht gestellte Probleme. Teubner-Studienbücher Mathematic. Teubner, Stuttgart 1989.
• [10] GRĄDZIEL S.: Determination of temperature and thermal stresses distribution in power boiler elements with use inverse heat conduction method. Arch. Thermodyn. 32(2011), 3, 191-200.
• [11] 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.
• [12] GUZ E., KĄCKI E.: Temperature Field in Solids. PWN, Warsaw 1967 (in Polish).
• [13] KĄCKI E., SIEWIERSKI L.: Selected Sections of Higher Mathematics with Exercises. PWN, Warsaw 1985 (in Polish).
• [14] McLACHLAN N.W.: Bessel Functions for Engineers. PWN, Warsaw 1964 (in Polish).
• [15] BJÖRCK A., DAHLQUIST G.: Numerical Methods. PWN, Warsaw 1983 (in Polish).
• [16] CIAŁKOWSKI M.: Selected Methods and Algorithms for Solving the Inverse Problem of Heat Conduction Equation. Wydawnictwo Politechniki Poznańskiej, Poznań 1996 (in Polish).
• [17] MACKIEWICZ A.: Algorithms of Linear Algebra Direct Methods. Wydawnictwo Politechniki Poznańskiej, Poznań 2002 (in Polish).