The one-dimensional problem is treated in which a liquid metal column, of the original temperature ϑ0, is supplied with additional metal at the top, of temperature ϑb, at the rate v ft/min. At the bottom of the column a heat sink of temperature ϑa is maintained. The case in which ϑa exceeds the fusion temperature is solved for v = const, both rigorously and approximately; the case in which ϑa is below the fusion temperature so that a freezing front sweeps upward in the column is next treated by an approximate method. The foregoing analysis may represent, for instance, the filling stage of a casting process. The present analysis will be adapted to the more important continuous casting process in a succeeding paper.

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