Model for a Freezing Food Slab

This project solves for the moving boundary (MB) during freezing (thawing) of a one dimensional moist-food slab using the pseudo-steady state assumption (PSS), convective cooling at the exposed surface, and a temperature-dependent thermal conductivity. To solve this nonlinear problem, we compute first the surface temperature as function of the MB (s) using the Nest iterative function. From the energy differential equation, we back-calculate the time differential vector (time increments) and integrate it straightforwardly using the s List, the slab physical properties and cooling conditions as input parameters. Once the MB is computed, we calculate the surface heat flu, the temperature profile, and the effect of the (normalized) thermal conductivity parameter (Κ) perturbations on the MB location. This time-dependent effect is graphically presented as a dimensionless sensitivity σ. We compare our results to predictions given by the well-known Plank freezing solution. The interesting effects to notice are those due to the thermal conductivity parameter (Κ), the dimensionless latent heat number (λ), the Biot number (Bi), and the dimensionless cooling-medium temperature on the MB position and its sensitivity driven by Κ. This project thus allows the estimation of the extent to which deviations in thermal conductivity affects the MB location during freezing (thawing).

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DETAILS

The moving boundary (MB) 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, requires solving the heat conduction differential equation below 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: .
Other relevant 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 using Mathematica's iterative Nest function for in terms of a discretized List. We then back-calculate the time steps from the known increments and s List applying the simple forward Euler method. The slab temperature profile was solved applying the Nest function to Eq. (1) subject to the known temperature conditions at the surface and at the interface (s). As the MB penetrates the slab thickness, the (time-dependent) temperature profile develops in the MB penetration region only, while the unfrozen slab region is kept at the initial food freezing point (Ti).
We use Plank's solution to the freezing slab problem, Eq. (4) below, even though it assumes constant thermal properties [2] while the other assumptions are identical to our assumptions, because it still provides a simple approximations to this phase change problem:
s(τ) = + , (4)
where St is the Stefan number defined as St = .
The MB sensitivity to K perturbations is calculated with small but finite changes (=3.34%) in the thermal conductivity parameter Κ and is given by Eq. (5):
= (δ s(τ)/s(τ))/(δ , (5)
This sensitivity quantity represents a time-dependent response of the MB position to errors or perturbations in Κ; and thus provides a quantifiable measure of the influence of this relevant material thermal property on the interface movement. A sensitivity of σ = ±0.25 at a given point in time indicates that a ±10% perturbation or error in Κ, for instance, generates a ±2.5% variation in the MB position.
Results are plotted against dimensionless time , where ; here is the slab half 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 close to but not perfectly 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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