First-Order Solution to Glass-Ice Stefan Problem

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A Stefan problem is a type of boundary value problem in which a phase boundary can move with time. A first-order approximation of the Stefan problem is explored for a system consisting of a block of ice (0° C) attached to a 6 mm thick piece of glass pulled from boiling water at 100° C. Here the Stefan problem is simplified by assuming that the temperature of the glass decreases linearly across the glass from 100° C at the left edge to 0° C at the glass-ice interface, as indicated by the color gradient. As time progresses, heat flows from the glass into the water composite at a rate proportional to the temperature gradient of the glass. The heat entering the frozen ice melts a water layer. The thickness of the water is plotted as a function of time.

Contributed by: Sam Shames (May 2012)
(for Jeffrey C. Grossman; assisted by W. Craig Carter)
Open content licensed under CC BY-NC-SA


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Details

A first-order approximation is used, assuming the temperature of the glass decreases linearly with distance. This assumption reduces the problem to the following first-order differential equation

,

where is the width of the glass and is the temperature of the left edge of the glass at time . The constants depend on the heat capacity, thermal conductivity, and density of the glass and ice.



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