Application of Rank Controlled Differential Quadrature Method for Solving an Infinite Steel Plate Cooling Problem

• Autorzy: Żak P. L. , Suchy J. S. , Lelito J. , Gracz B.
• Języki publikacji: EN

Abstrakty

EN

Rank Controlled Differential Quadrature method is a numerical method that allows to approximate the partial derivatives that appears in partial differential equations. Those equations with proper geometrical, physical, initial and boundary conditions make mathematical models of physical process. The heat transfer process is governed by Fourier–Kirchhoff equation, which is parabolic Partial Differential Equation. In this paper authors present the steel plate cooling problem. At the beginning of the process plate is heated up to 450 °C and is cooled to ambient temperature. The cooling of the plate is basic heat transfer problem. If the plates dimensions has proper proportions such problem may be described as one dimensional and solved exactly. The mathematical model and exact solution is given in the work. Authors apply the Rank Controlled Differential Quadrature to approximate derivatives in Fourier–Kirchhoff equation and in boundary conditions. After changing derivatives into quadrature formulation set of algebraic equations is obtained. Substituting thermo-physical parameters numerical model is obtained. The computer program was prepared to solve the problem numerically. Results of simulation are confronted with the exact ones. Error value at each time step as well as error value increase rate for examined numerical method is analyzed.

Słowa kluczowe

EN

information technology; foundry industry; solidification process; exact solution; rank controlled differential quadrature; numerical modelling

PL

technologia informatyczna; przemysł odlewniczy; proces krzepnięcia; dokładne rozwiązanie; kwadratura różniczkowa sterowanego rzędu; modelowanie numeryczne

• Wydawca: Komisja Odlewnictwa Polskiej Akademii Nauk Oddział w Katowicach
• Czasopismo: Archives of Foundry Engineering
• Rocznik: 2013
• Tom: Vol. 13, iss. 3 spec.
• Strony: 201--206
• Opis fizyczny: Bibliogr. 13 poz., wykr., tab.

Twórcy

autor

Żak P. L.

• AGH University of Science and Technology, Faculty of Foundry Engineering, 23 Reymonta Street, 30-059 Poland

autor

Suchy J. S.

• AGH University of Science and Technology, Faculty of Foundry Engineering, 23 Reymonta Street, 30-059 Poland

autor

Lelito J.

• AGH University of Science and Technology, Faculty of Foundry Engineering, 23 Reymonta Street, 30-059 Poland

autor

Gracz B.

• AGH University of Science and Technology, Faculty of Foundry Engineering, 23 Reymonta Street, 30-059 Poland

Bibliografia

• [1] Mochnacki, B., Suchy, J. S. (1995). Numerical method in computations of foundry processes. Krakow: Polish Foundrymen's Technical Association.
• [2] Taler, J., Duda, P. (2003). Rozwiązywanie prostych i odwrotnych zagadnień przewodzenia ciepła. Warszawa: WNT.
• [3] Majchrzak, E., Mochnacki, B., Suchy, J. S. (1999). Computer simulation of heat transfer between the particles and metal matrix during the solidification of a cast composite. International Journal of Cast Metals Research 12(4), 241-249.
• [4] Mochnacki, B., Suchy, J.S. (2006) Identification of alloy latent heat on the basis of mould temperature (part 1). Archives of Foundry Engineering 6(22), 324-330.
• [5] Majchrzak, E., Mochancki, B., Suchy, J.S. (2008) Journal of theoretical and applied mechanics. 46(2), 257-268.
• [6] Kalisz, D. (2012) Viscosity calculations of mould slag in continuous casting. Archives of Materials Science and Engineering. 58(2), 35-41.
• [7] Iwanciw, J., Kalisz (Podorska), D., Wypartowicz, J. (2011) Simulation of oxygen and nitrogen removal from steel by means of titanium and aluminium. Archives of Metallurghy and Materials. 56(3), 635-644.
• [8] Iwanciw, J., Kalisz (Podorska), D., Wypartowicz, J. (2011) Modelling of oxide precipitates chemical composition during steel deoxidization. Archives of Metallurgy and Materials. 56(4), 999-1005.
• [9] Żak, P.L., Suchy, J.S. (2012) Numerical Model for Dendrite Growth - Application of Rank Controlled Differential Quadrature Method. MaFE - Metallurgy and Foundry Engineering. 38, 55-65.
• [10] Żak, P. L. (2012). Zastosowanie metody kwadratur różniczkowych w komputerowej symulacji przewodzenia ciepła i krzepnięcia odlewów. Doctoral dissertation.AGH University of Science and Technology, Cracow, Poland.
• [11] Polyanin, A.D., Zaitsev, V.F. (2004) Handbook of nonlinear partial differential equations. Boca Raton, Florida, USA: Chapman & Hall.
• [12] Zong, Z., Zhang, Y. (2009) Advanced Differential Quadrature Methods. Boca Raton: CRC Press.
• [13] Shu, C. (2000) Differential Quadrature and Its Application In Engineering. London: Springer.