Details

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 .

In summary:

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 [1] 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.

References

[1] B. S. Neumann, Conduction of Heat in Solids (H. S. Carslaw and J. C. Jaeger, eds.), 2nd ed., Oxford: Clarendon Press, 1959.

[2] M. N. Özisik, Heat Conduction, New York: Wiley, 1980.