Temperature distribution in molten metal is calculated by rapidly converging infinite series, with a casting plate having uniform thickness T and molten metal flowing at constant velocity inside the already solidified metal layers. In the range where the dimensionless Fourier number
τ is less than 0.2, the average temperature of the molten metal
θm is approximately expressed by the solution of the Fourier equation with error functions as follows :
θm−
θf=(
θp−
θf) {1−(2/√
π)√
τ}
where
θp is the initial temperature of the molten metal and
θf is the freezing point. In the range where
τ is greater than 0.2,
θm is approximately expressed by the solution of the equation with trigonometrical functions as follows :
θm−
θf=(
θp−
θf) exp {−(
π/2)
2(
τ+0.0851)}
Calculation errors of these expressions from the theoretical solutions remain less than 0.2%. The portion of heat energy over the freezing point is reduced to half during the flow from the entrance to the position which correspond to where
τ is 0.197. That portion of heat energy is reduced to half every 0.281 Fourier number in the range where
τ is greater than 0.197. A range where the once-formed metal layers have been remelted always exists near the entrance even though the duration of the molten metal flow may be short.
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