Simple analysis and working equations for the solidification of cylinders and spheres
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unidirectional heat flow,7 analytic results for the s o l i dification t i m e s of cylinders and spheres are presented which are easy to handle and much more accurate than previous results of equally simple structure. 1. MODEL AND MATHEMATICAL FORMULATION The model to be discussed and evaluated here is formally the same as the one for slab-shaped bodies,v However, with cylindrical or spherical systems the physical significance of the m o d e l as well as the quality of the results are expected to be quite different and w e have t o draw on previous conclusions in o r d e r to justify the present procedure.
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J. KERN and G. L. WELLS are Senior Lecturerand Post-Graduate Research Student, respectively, Department of Chemical Engineering, University of Witwatersrand, Johannesburg, South Africa. Manuscript submitted June I0, 1976. METALLURGICAL TRANSACTIONS B
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Fig. 1--(a) Assumed temperature distribution during solidification-slab analysis. (b) Assumed temperature distribution-cylindrical and spherical solidification. VOLUME 8B,MARCH 1977 99
1.1. Conclusions From the Slab Analysis For the p u r p o s e of summarizing the slab analysis we r e f e r t o Fig. l(a) and assume that t h e r e is a l i n e a r temperature profile in the solidified l a y e r at any t i m e . Neglecting the heat capacity of the solid one obtains a simple analytic result for the solidification time as a function of the l a y e r thickness 5. This result, known as the quasistationary solution, clearly provides a lower bound for the solidification time bec a u s e in reality not only latent heat but also some sensible heat has to be removed. This is easily seen by following the temperature-time history of some volume element at 0 < x < 5. On the other hand we know that the true temperature profile is always curved so that the temperature at any point x < 5 is l a r g e r than the one specified by a l i n e a r temperature gradient. Therefore, if w e a s s u m e that a f t e r an incremental increase of 5 internal energy is removed until this steady profile is r e a c h e d then less capacity is available for latent heat removal and an upper bound for the solidification time should result. Again, the result is obtained from an energy balance and consists of an extremely simple relationship between t, 5 and the system parameters. It is pointed out that the bounding character of the second result has not yet been proved mathematically but was verified by comparison with all the n u m e r i cally and analytically e x a c t results which are a v a i l able from the literature. Indeed, for l a r g e temperature changes over the solid layer and l a r g e solidification r a t e s (typically encountered in metallurgical applications) heavy overpredictions for the solidification time occur; the temperature profile is curved markedly and far less internal energy is removed in the true process than is anticipated in the model. This suggested the formula
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