The heat transfer problem is solved for the case of cooling, below the freezing temperature, an initially liquid material inside a spherical shell. The shell is limited by a fixed inner surface and by an outer surface, free to radially expand or contract. As boundary conditions it is imposed that the inner surface is kept constant below the freezing temperature of the liquid and the outer surface is maintained constant above it. The solution represents the formation of a solidified layer that expands outward, separated from the liquid by a spherical surface kept constant at the freezing temperature. The problem is solved in the form of two closed form solutions, written in non-dimensional variables: one for the heat conduction equation in the solid layer and the second for the heat conduction – advection equation in the liquid layer. The solutions formally depend of and are linked by the time dependent radius of the spherical solid–liquid interface and its time derivate, which are, at first, unknown. A differential equation describing the solid–liquid interface radius as function of time is obtained imposing the conservation of energy through the interface during the heat exchange process. This equation is non-linear and has to be numerically solved. Substitution of the interface radius and its time derivate for a particular instant in the heat transfer equations solutions furnishes the temperature distribution inside the spherical shell at that moment. The solution is illustrated with numerical examples.
The Solution of a Heat Transfer Problem With Change of State Inside a Spherical Shell
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de Oliveira, DS, & Ribeiro, FB. "The Solution of a Heat Transfer Problem With Change of State Inside a Spherical Shell." Proceedings of the ASME 2012 International Mechanical Engineering Congress and Exposition. Volume 7: Fluids and Heat Transfer, Parts A, B, C, and D. Houston, Texas, USA. November 9–15, 2012. pp. 2563-2576. ASME. https://doi.org/10.1115/IMECE2012-85241
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