The solidification of flat layers Stefan problem

A solidified body is maintained by cooling at x = 0 at the constant temperature i90, which is lower than the solidification temperature E, Fig. 2.35. Only onedimensional heat conduction in the x-direction will be assumed. At the phase boundary x = s, which is moving to the right, the solid is touching the liquid which has already been cooled to the solidification temperature. By advance of the phase boundary, or in other words by solidifying a layer of thickness ds, the enthalpy of fusion is released and must be conducted as heat to the cooled surface of the solid at x = 0. 

Neumann presented this solution in his lectures at the University of Konigsberg. The first time it was published was in a German book Die partiellen Differentialgleichungen der Physik, Editors B. Riemann and H. Weber, Vol.2, pp. 117-121. Braunschweig F. Vieweg 1912. 

A solution of the heat conduction equation (2.202) is the error function 

The argument of the error function must therefore be a constant 7 independent of t. The thickness of the solidified layer increases proportionally with /i, 

The still unknown constant 7 is yielded from the condition (2.206) for the solidification speed. It follows from (2.207) that 

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