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Optimal exponent heat balance and refined integral methods applied to Stefan problems. (English) Zbl 1183.80091
Summary: When using a polynomial approximating function the most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term. In this paper we employ a method recently developed for thermal problems, where the exponent is determined during the solution process, to analyse Stefan problems. This is achieved by minimising an error function. The solution requires no knowledge of an exact solution and generally produces significantly better results than all previous HBI models. The method is illustrated by first applying it to standard thermal problems. A Stefan problem with an analytical solution is then discussed and results compared to the approximate solution. An ablation problem is also analysed and results compared against a numerical solution. In both examples the agreement is excellent. A Stefan problem where the boundary temperature increases exponentially is analysed. This highlights the difficulties that can be encountered with a time dependent boundary condition. Finally, melting with a time-dependent flux is briefly analysed without applying analytical or numerical results to assess the accuracy.
MSC:
80A22Stefan problems, phase changes, etc.
80A20Heat and mass transfer, heat flow
80M25Other numerical methods (thermodynamics)
References:
[1]Alexiades, V.; Solomon, A. D.: Mathematical modeling of melting and freezing processes, (1993)
[2]Antic, A.; Hill, J. M.: The double-diffusivity heat transfer model for grain stores incorporating microwave heating, Appl. math. Modelling 27, 629-647 (2003) · Zbl 1043.80001 · doi:10.1016/S0307-904X(03)00072-6
[3]W.F. Braga, M.B.H. Mantelli, J.L.F. Azevedo, Approximate analytical solution for one-dimensional ablation problem with time-variable heat flux, in: AIAA Thermophys. Conference, June 2003.
[4]W.F. Braga, M.B.H. Mantelli, J.L.F. Azevedo, Approximate analytical solution for one-dimensional finite ablation problem with constant time heat flux, in: AIAA Thermophys. Conference, June 2004.
[5]Caldwell, J.; Kwan, Y. Y.: Numerical methods for one-dimensional Stefan problems, Comm. num. Meth. eng. 20, 535-545 (2004) · Zbl 1048.65095 · doi:10.1002/cnm.691
[6]Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids, (1959)
[7]Goodman, T. R.: The heat-balance integral and its application to problems involving a change of phase, Trans. ASME 80, 335-342 (1958)
[8]Hristov, J.: Research note on a parabolic heat-balance integral method with unspecified exponent: an entropy generation approach in optimal profile determination, Therm. sci. 13, No. 2, 49-59 (2009)
[9]Hristov, J.: The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and exercises, Therm. sci. 13, No. 2, 27-48 (2009)
[10]Kutluay, S.; Wood, A. S.; Esen, A.: A heat balance integral solution of the thermistor problem with a modified electrical conductivity, Appl. math. Modelling 30, 386-394 (2006) · Zbl 1096.78005 · doi:10.1016/j.apm.2005.05.002
[11]Langford, D.: The heat balance integral method, Int. J. Heat mass transfer 16, 2424-2428 (1973)
[12]S.L. Mitchell, T.G. Myers, The application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Review, in press. · Zbl 1188.80004 · doi:10.1137/080733036
[13]Mitchell, S. L.; Myers, T. G.: A heat balance integral method for one-dimensional finite ablation, AIAA J. Thermophys. heat transfer 22, No. 3, 508-514 (2008)
[14]Mitchell, S. L.; Vynnycky, M.: Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. math. Comp. 215, No. 4, 1609-1621 (2009) · Zbl 1177.80078 · doi:10.1016/j.amc.2009.07.054
[15]Myers, T. G.: Extension to the messinger model for aircraft icing, Aiaa j. 39, No. 2 (2001)
[16]Myers, T. G.: Optimizing the exponent in the heat balance and refined integral methods, Int. comm. Heat mass transfer 44 (2008)
[17]Myers, T. G.; Charpin, J. P. F.: A mathematical model for atmospheric ice accretion and water flow on a cold surface, Int. J. Heat mass transfer 47 (2004) · Zbl 1078.76074 · doi:10.1016/j.ijheatmasstransfer.2004.06.037
[18]Myers, T. G.; Mitchell, S. L.: Application of the heat balance and refined integral methods to the Korteweg-de Vries equation, Therm. sci. 13, No. 2, 113-119 (2009)
[19]Myers, T. G.; Mitchell, S. L.; Muchatibaya, G.: Unsteady contact melting of a rectangular cross-section material on a flat plate, Phys. fluids 20 (2008) · Zbl 1182.76548 · doi:10.1063/1.2990751
[20]Myers, T. G.; Mitchell, S. L.; Muchatibaya, G.; Myers, M. Y.: A cubic heat balance integral method for one-dimensional melting of a finite thickness layer, Int. J. Heat mass transfer 50, 5305-5317 (2007) · Zbl 1140.80389 · doi:10.1016/j.ijheatmasstransfer.2007.06.014
[21]Ockendon, J. R.; Howison, S. D.; Lacey, A. A.; Movchan, A.: Applied partial differential equations, (1999) · Zbl 0927.35001
[22]Sadoun, N.; Si-Ahmed, E-K.; Colinet, P.: On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions, Appl. math. Modelling 30 (2006) · Zbl 1183.80093 · doi:10.1016/j.apm.2005.06.003
[23]Sadoun, N.; Si-Ahmed, E. K.: A new analytical expression of the freezing constant in the Stefan problem with initial superheat, Numer. meth. Therm. probl. 9, 843854 (1995)
[24]Schlichting, H.: Boundary layer theory, (2000)
[25]Spearpoint, M. J.; Quintiere, J. G.: Predicting the piloted ignition of wood in the cone calorimeter using an integral model effect of species grain orientation and heat flux, Fire safety J. 36, No. 4, 391-415 (2001)
[26]Wood, A. S.: A new look at the heat balance integral method, Appl. math. Modelling 25, 815-824 (2001) · Zbl 0992.80008 · doi:10.1016/S0307-904X(01)00016-6 · doi:http://www.elsevier.com/gej-ng/10/10/8/47/42/28/abstract.html