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 Eq. (1) 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 the dimensionless latent heat number defined
λ = ΔH/(C pf·(T0-Ti));
where
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).
Plank's solution serves as a simple but well known reference 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. This solution still provides a simple but useful 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.
[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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