Abstract
The analysis of solidification processes is complicated by a nonlinear boundary condition at the moving solid-liquid interface, and exact solutions are rare. Various attempts to predict the rate of solidification are available in the literature but most of the results seem to be of limited use for operation and design studies on metallurgical processes. In this article we present a physical model which can be solved analytically for the most commonly encountered boundary conditions; that is constant temperature at the cooling wall or finite heat transfer to the cooling fluid. The model is based on the assumption of a linear temperature profile in the solidified shell and a corresponding differential removal of internal energy. As a result one obtains a very simple expression for the soli-dification time as a function of the space variable and the pertinent system parameters. By comparison with numerical results the prediction error is shown to be less than 10 pet over a wide range of parameter combinations. In extreme situations, where a larger error may occur, equally accurate working equations can be generated by slightly modifying the basic results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Planck:Z. Gesamte Kalte Ind. Beih, 1932, vol. 39, pp. 56–58.
A. L. London and R. A. Seban:Trans. Am. Soc. Mech. Eng., 1943, vol. 65, pp. 771–78.
H. S. Carslaw and J. C. Jaeger:Conduction of Heat in Solids, pp. 282–96, Oxford University Press, London, 1959.
C. Lapadula and W. K. Mueller:Int. J. Heat Mass Transfer, 1966, vol. 9, pp. 702–04.
R. H. Tien:Trans. TMS-AIME, 1965, vol. 233, pp. 1887–91.
K. Stephan:Kaeltetech. Klim., 1971, vol. 23, pp. 42–46.
J. Kern:Int. J. Heat Mass Transfer, in press.
R. T. Beaubouef and A. J. Chapman:Int. J. Heat Mass Transfer, 1967, vol. 10, p. 1581–87.
K. Stephan and B. Holzknecht:Waerme Stoffübertragung, 1974, vol. 7, pp. 200–07.
A. W. D. Hills:J. Iron Steel Inst., 1965, vol. 203, pp. 18–26.
J. Savage and W. H. Pritchard:J. Iron Steel Inst., 1954, vol. 178, pp. 269–77.
J. K. Brimacombe and F. Weinberg:J. Iron Steel Inst., 1973, vol. 211, pp. 24-33.
F. Megerlin:Forsch Geb. Ingenieurw., 1968, vol. 34, pp. 40–46.
D. Langford:Int. J. Heat Mass Transfer, 1966, vol. 9, pp. 827–28.
D. S. Riley, F. T. Smith, and G. Poots:Int. J. Heat Mass Transfer, 1974, vol. 17, pp. 1507–16.
K. Stewartson and R. T. Waechter:Proc. Roy. Soc. London, Ser. A, 1976, vol. 348, pp. 415–26.
K. Stephan and B. Holzknecht:Int. J. Heat Mass Transfer, 1976, vol. 19, pp. 597–602.
L. C. Tao:AIChEJ., 1967, vol. 13, pp. 165–69.
K. Nesselmann:Kaeltetech., 1949, vol. 1, no. 7, pp. 169–72.
A. W. D. Hills, S. L. Malhotra, and M. R. Moore:Met. Trans.B, 1975, vol. 6B, pp. 131–42.
W. A. Stein:Verfahrenstechnik, 1971, vol. 5, pp. 453–55.
S. Lin:Dr.-Ing. Thesis, T. H. Karlsruhe, 1964.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kern, J., Wells, G.L. Simple analysis and working equations for the solidification of cylinders and spheres. Metall Trans B 8, 99–105 (1977). https://doi.org/10.1007/BF02656357
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02656357