The Neumann Solution for Phase Changes in Foods
The Neumann solution (1860) gives an exact analytical solution to the phase change in a solid material initially at a temperature above its freezing point. The analysis involves heat conduction in two regions, with and without latent heat removal, in a semi-infinite geometry, assuming constant thermal properties and Dirichlet boundary conditions. This Demonstration implements this solution for varying physical and geometric properties, including the depth on the moving boundary during freezing or thawing of moist food. The results are shown in a three-dimensional plot of temperature as a function of depth and time. The moving boundary is shown as a purple line.
The partial differential equation describing the energy balance involves heat transport in the freezing (thawing) and unfrozen (frozen) regions of a semi-infinite body, as represented in Eq. (1.1):
Here, refers to the phase-change region for with its exposed surface at . Cooling (thawing) occurs in the medium at above or below the food material with an initial freezing point . Similarly, refers to the frozen or unfrozen pure-heat conduction region for .
Food material thermal properties and are assumed constant but different for each region. The specific heat and latent heat are expressed as volumetric energy quantities [1, 2].
The problem requires solving for both temperature fields in each region and for the position of moving boundary or interface between the phase-change and pure-heat conduction regions. Using similarity transformations, Neumann  derived the exact solution for the temperature fields in each region, given by Eqs. (2.1) and (2.2):
where the thermal diffusivity is .
The interface energy-balance equation leads to the transcendental Eq. (3.1) where the parameter is obtained from Eq. (3.1):
The Stefan number is a key dimensionless parameter for the phase change phenomena. Eq. (3.1) is solved using the Wolfram Language function FindRoot. The moving boundary has an explicit classic power law solution proportional to , given by Eq. (3.2):
is shown on the plotted 3D surface as the thick purple isotherm line. You can compare the effects of each of the phase thermal properties and , latent heat , initial freezing point , the process parameters and , and the scales of time and depth. The Stefan number captures the key physical property and process quantities and is left to the reader to calculate.
A word of caution: although you can vary the controls at will, take care to restrict values to real-world food properties, that is, and , where the subscripts and stand for fully frozen and fully thawed material properties, respectively.
 B. S. Neumann, Conduction of Heat in Solids (H. S. Carslaw and J. C. Jaeger, eds.), 2nd ed., Oxford: Clarendon Press, 1959.
 M. N. Özisik, Heat Conduction, New York: Wiley, 1980.