# Model for a Freezing Food Slab

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This Demonstration shows a numerical solution for a nonlinear moving boundary model of a freezing food slab. We display the freezing or moving front (FF or MB), the convective surface temperature, the surface heat flux and the slab temperature profile (quantities labeled by PSS, for pseudo steady state). These results are compared graphically to the values obtained using the model based on Plank's equation, which assumes constant thermal properties. Due to the thermal properties (such as the thermal conductivity parameter and the dimensionless latent heat number ) on the output process quantities, the interesting effects to note are the nonlinear versus linear perturbations compared to the dominating nonlinear and effects in the above cases.

Contributed by: Victor M. Chavarria (March 2020)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The freezing front is the idealized infinitesimally thin interface separating the unfrozen region from the freezing region. To locate it, and assuming the pseudo steady state (PSS) condition, leads to the heat conduction differential equation that describes the freezing process of a one-dimensional slab [1]:

, (1)

where is the dimensionless slab temperature defined as

,

and are the (initial) freezing points of pure water and of the food material, respectively,

is the dimensionless freezing front (length) growing from to ,

is the normalized distance (spanning from the convective surface to the slab center),

and is the dimensionless latent heat number defined as

;

here is the heat of fusion of the moist food material adjusted by its unfreezable water content and is the volumetric specific heat capacity of the fully frozen slab [1].

The convective boundary condition (BC) is:

, (2)

where is the convective-surface slab temperature at . We adopt Schwartzberg's [1] temperature-dependent thermal conductivity model given by:

, (3)

where is the thermal conductivity parameter given by

and and are the unfrozen and fully frozen state thermal conductivities, respectively. governs the temperature response of thermal conductivity and influences the freezing process in a nonlinear fashion.

At the moving interface, the BC gives

.

In addition to these conditions, other key assumptions include the full release of the latent heat of fusion, the material temperature at time 0 and the phase change all take place at the initial freezing point of the material, and no sensible heat is released.

Equations (1) and (2) together with (3) were numerically solved; first for as a function of a discretized range and using the built-in Nest function. We then calculate the time steps from increments by applying Euler's method. The slab temperature profile was also solved using Euler's method, knowing both the fully developed freezing front for and the dimensionless surface temperature .

Results are plotted against dimensionless time , where ; here is the slab thickness and is dimensional time.

The interesting effects that can be controlled as parameters in the plots are:

1. thermal conductivity parameter

2. Biot number , where is the convective heat transfer coefficient,

3. dimensionless latent heat number

4. temperature driving force , based on , the medium-cooling temperature

5. slab temperature profile , which follows a nearly linear pattern

References

[1] H. G. Schwartzberg, "The Prediction of Freezing and Thawing Temperatures vs. Time through the Use of Effective Heat Capacity Equations," in *Proceedings of the Joint Meeting of Commissions C1 and C2 of the International Institute of Refrigeration*, 1977 pp. 311–317.

[2] V. M. Chavarria, "Modeling the Influence of Temperature-Dependent Thermal Properties on the Freezing Front," *Journal of Food Research*, 8(6), 2019 pp. 129–146. doi:10.5539/jfr.v8n6p129.

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