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On the convergence of the heat balance integral method. (English) Zbl 1151.80329
The authors apply the well-known Godmann’s approximate heat-balance method to the single-phase semi-infinite ice slab, whose exact solution is well-known and represents a special case of the Neumann general solution. The distance from the beginning of the slab \((x= 0)\) to the moving melt front \((x= s)\) is subdivided into \(n\) equal cells and the temperature is approximated at each node by a linear temperature profile. The further calculation is performed by the Godmann method. The authors demonstrated that the approximate solution obtained in such a way, is in good agreement with the exact solution. The study of the convergence of the approximate solution is accomplished in details.

80M25 Other numerical methods (thermodynamics) (MSC2010)
80A22 Stefan problems, phase changes, etc.
Full Text: DOI
[1] Goodman, T.R., Heat balance integral and its application to problems involving a change of phase, Trans. ASME J. heat transfer, 80, 335-342, (1958)
[2] Goodman, T.R., Application of integral methods in transient non-linear heat transfer, (), 51-122
[3] Wood, A.S., A new look at the heat balance integral method, Appl. math. mod., 25, 815-824, (2001) · Zbl 0992.80008
[4] Noble, B., Heat balance methods in melting problems, (), 208-209
[5] Bell, G.E., A refinement of the heat balance integral method applied to a melting problem, Int. J. heat mass transfer, 21, 1357-1362, (1978)
[6] Bell, G.E., Solidification of a liquid about a cylindrical pipe, Int. J. heat mass transfer, 22, 1681-1686, (1979)
[7] Bell, G.E., Accurate solution of one-dimensional melting problems by the heat balance integral method, (), 196-203
[8] Bell, G.E., The prediction of frost penetration, Int. J. numer. anal. meths. geomechanics, 6, 287-290, (1982)
[9] Bell, G.E.; Abbas, S.K., Convergence properties of the heat balance integral method, Numer. heat transfer, 8, 373-382, (1985) · Zbl 0568.73119
[10] Crank, J.; Phahle, R.D., Melting ice by the isotherm migration method, Bull. IMA, 9, 12-14, (1973)
[11] Stefan, J., Über einige probleme der theorie der Wärmleitung, Sber. akad. wiss. wien, 98, 473-484, (1889), (in German)
[12] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1965), Dover Press New York · Zbl 0515.33001
[13] Pearson, K., Tables of the incomplete gamma function, (1934), Cambridge University Press Cambridge
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