Introduction & Context

The calculation of convective freezing time is a fundamental task in food process engineering and industrial refrigeration. It allows engineers to predict the time required to reduce the temperature of a product to its freezing point and complete the phase change from liquid to solid. This analysis is critical for designing blast freezers, optimizing energy consumption, and ensuring food safety by controlling the rate of ice crystal formation, which directly impacts product texture and quality.

Plank’s Equation is the industry standard for estimating the freezing time of homogeneous food items, such as meat patties or slabs, where the removal of latent heat is the dominant thermal process. It is typically used in the preliminary design phase of cold chain logistics and food processing facilities.

Methodology & Formulas

The freezing time calculation follows a sequential approach, determining the driving thermal force, the convective heat transfer coefficient, and finally the total time required for phase change based on the slab geometry; understanding the freezing time versus thickness relationship enables engineers to predict how variations in product thickness influence the required freezing duration.

First, the driving temperature difference is calculated as:

\[ \Delta T = T_{fp} - T_{air} \]

The mass velocity of the cooling medium is determined by the product of air velocity and air density, a relationship that directly influences the freezing rate as described in the air velocity effect on freezing rate analysis.

\[ G = v \cdot \rho_{air} \]

The surface heat transfer coefficient is estimated using an empirical correlation for forced convection:

\[ h = 20 \cdot G^{0.8} \]

The Biot number is calculated to assess the relative importance of convective and conductive thermal resistances:

\[ Bi = \frac{h \cdot (\frac{d}{2})}{k} \]

Finally, the total freezing time is calculated using Plank’s Equation for an infinite slab:

\[ t = \left( \frac{\rho_{f} \cdot \lambda}{\Delta T} \right) \cdot \left( \frac{d}{2 \cdot h} + \frac{d^{2}}{8 \cdot k} \right) \]

Regime / Condition	Criteria	Engineering Implication
Convection Dominant	\( Bi < 0.1 \)	Conduction resistance is negligible.
Mixed Resistance	\( 0.1 \leq Bi \leq 40 \)	Both convection and conduction are significant.
Conduction Dominant	\( Bi > 40 \)	Convection resistance is negligible.
Mass Velocity Range	\( 3 \leq G \leq 30 \)	Empirical correlation for \( h \) is valid.
Freezer Temperature	\( T_{air} \leq -20 \)	Standard operating threshold for blast freezing.

To accurately calculate the convective heat transfer coefficient for non-standard shapes, process engineers should follow these steps:

Identify the characteristic length based on the primary axis of heat flow.
Utilize empirical correlations such as the Nusselt number relationship for the specific geometry.
Adjust for surface roughness factors that may influence the boundary layer thickness.
Validate the calculated coefficient against experimental temperature profile data from the core of the product.

The phase change introduces a latent heat component that significantly alters the thermal diffusivity of the material. When performing the analysis, consider the following:

The effective heat capacity increases drastically during the freezing plateau.
The thermal conductivity of the frozen layer is typically higher than the unfrozen core, creating a moving boundary problem.
The Stefan number must be calculated to determine the ratio of sensible heat to latent heat.

For successful scale-up, ensure that the following dimensionless groups are maintained or accounted for in your model:

Reynolds number to characterize the flow regime of the cooling medium.
Prandtl number to define the relationship between momentum and thermal diffusivity.
Biot number to assess the ratio of internal conductive resistance to external convective resistance.
Fourier number to track the dimensionless time required for the freezing front to reach the center.

Worked Example: Convective Freezing of a Meat Patty

A homogeneous meat patty of uniform thickness 0.04 m is to be frozen in a blast freezer. The freezer supplies forced air at –30.0°C and 5.0 m/s. The patty's initial temperature is at its freezing point (–2.0°C), so only latent heat removal is considered. Using Plank's equation for an infinite slab, compute the required freezing time.

Knowns

Slab thickness, \(d = 0.04\) m
Frozen density, \(\rho_f = 1050.0\) kg/m³
Latent heat, \(\lambda = 250000.0\) J/kg
Frozen thermal conductivity, \(k = 1.5\) W/(m·K)
Freezing point, \(T_{fp} = -2.0\)°C
Air temperature, \(T_{air} = -30.0\)°C
Air velocity, \(v = 5.0\) m/s
Air density, \(\rho_{air} = 1.3\) kg/m³

Step-by-Step Calculation

Driving temperature difference \(\Delta T = T_{fp} - T_{air} = 28.0\) K
Mass velocity \(G = v \cdot \rho_{air} = 6.5\) kg/(m²·s)
Validity check: \(G = 6.5\) is within the empirical range of 3–30 kg/(m²·s), so the HTC correlation is applicable.
Surface heat transfer coefficient \(h = 20 \cdot G^{0.8} = 89.405\) W/(m²·K)
Biot number \(\text{Bi} = \dfrac{h \cdot (d/2)}{k} = \dfrac{89.405 \cdot 0.02}{1.5} = 1.192\)
Since \(0.1 < \text{Bi} < 40\), both internal conduction and external convection resistances are significant.
Freezing time (Plank's equation for slab)
\[t = \frac{\rho_f \cdot \lambda}{\Delta T} \left[ \frac{d}{2h} + \frac{d^2}{8k} \right]\]
= \(\frac{1050.0 \cdot 250000.0}{28.0} \times \left[ \frac{0.04}{2 \cdot 89.405} + \frac{0.04^2}{8 \cdot 1.5} \right]\)
= \(9375000.0 \times \left[ 0.0002237 + 0.0001333 \right]\)
= \(9375000.0 \times 0.000357 = 3347.204\) s
Convert to hours \(t = \frac{3347.204}{3600} = 0.93\) h

Final Answer
The total freezing time for the meat patty under the specified blast-freezing conditions is 3347.204 seconds (approximately 0.93 hours).

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