Introduction & Context

Estimating the cooling time of packaged frozen foods is essential for designing blast freezers, cold storage facilities, and industrial packaging lines. The method below treats the product as a slab subjected to transient heat conduction on both sides. Note: This calculation applies to sensible cooling only (product already frozen, i.e., entirely below its freezing point). For the complete freezing process involving latent heat and phase change, refer to Plank’s equation or other appropriate phase‑change models.

Methodology & Formulas

Heat transfer is modeled by one‑dimensional transient conduction in an infinite slab. The key parameters are the thermal diffusivity, Biot number, dimensionless temperature ratio, and Fourier number.

Characteristic length for a slab of total thickness δ (symmetric cooling from both sides):

\[ L = \frac{\delta}{2} \]

Thermal diffusivity α from thermal conductivity k, density ρ, and specific heat c_{p}:

\[ \alpha = \frac{k}{\rho \cdot c_{p}} \]

Biot number Bi (ratio of convective to conductive resistance):

\[ Bi = \frac{h \cdot L}{k} \]

Dimensionless temperature ratio θ, based on initial temperature T_{\text{initial}}, ambient temperature T_{\text{ambient}}, and target core temperature T_{\text{target}}:

\[ \theta = \frac{T_{\text{target}} - T_{\text{ambient}}}{T_{\text{initial}} - T_{\text{ambient}}} \]

For a slab, the one‑term approximation of the infinite series solution yields the Fourier number Fo using the first eigenvalue \zeta_{1} and coefficient C_{1} associated with the Biot number:

\[ Fo = \frac{-\ln\left(\dfrac{\theta}{C_{1}}\right)}{\zeta_{1}^{2}} \]

The required cooling time t is:

\[ t = \frac{Fo \cdot L^{2}}{\alpha} \]

Parameter	Condition/Constraint	Engineering Significance
Thickness	δ > 0	Physical dimension must be positive for a valid geometry.
Biot Number	Bi ≥ 0.1	Indicates that internal thermal gradients are non‑negligible; the one‑term approximation method is appropriate.
Temperature Ratio	0 < θ < 1	Target temperature must lie between initial and ambient temperatures.

To ensure product integrity during sub‑zero storage, materials must provide high resistance to moisture vapor transmission and maintain structural stability. Key requirements include:

High puncture resistance to prevent damage from sharp food edges at low temperatures.
Excellent seal integrity to prevent freezer burn and dehydration.
Compatibility with high‑speed automated filling lines.
Low‑temperature flexibility to prevent brittleness and cracking.

The sealant layer is critical for maintaining a hermetic seal under thermal stress. Process engineers should consider:

The use of metallocene‑based resins for superior seal strength and hot tack.
The impact of seal contamination from food particles on the integrity of the seal.
The necessity of adjusting dwell time and temperature settings to account for the thermal conductivity of the specific packaging substrate.

Packaging failures often stem from environmental stressors and mechanical handling. Common issues include:

Thermal shock causing micro‑fractures in the barrier layer.
Inadequate headspace management leading to package bursting during pressure changes.
Poor moisture barrier properties resulting in ice crystal formation inside the package.
Material fatigue caused by repeated freeze‑thaw cycles during distribution.

Worked Example: Sensible Cooling Time for a Packaged Frozen Food Slab

Scenario: A process engineer needs to estimate the time required to cool a 5 cm thick slab of already‑frozen food from –5.0 °C to –15.0 °C in a blast freezer. The ambient air temperature is –20.0 °C. Cooling occurs symmetrically from both sides. The answer is derived using the one‑term approximation of the Heisler chart for an infinite slab.

Knowns:

Total slab thickness, \( \delta = 0.05 \; \text{m} \)
Thermal conductivity of frozen food, \( k = 1.25 \; \text{W/(m·K)} \)
Density of frozen food, \( \rho = 950.0 \; \text{kg/m}^3 \)
Specific heat of frozen food, \( c_{p} = 1900.0 \; \text{J/(kg·K)} \)
Convective heat transfer coefficient, \( h = 25.0 \; \text{W/(m}^2\text{·K)} \)
Initial temperature, \( T_{\text{initial}} = -5.0 \; \text{°C} \)
Ambient temperature, \( T_{\text{ambient}} = -20.0 \; \text{°C} \)
Target core temperature, \( T_{\text{target}} = -15.0 \; \text{°C} \)

Step-by-Step Calculation:

Characteristic length (half‑thickness): \( L = \dfrac{\delta}{2} = \dfrac{0.05}{2} = 0.025 \; \text{m} \).
Biot number: \( \text{Bi} = \dfrac{h \cdot L}{k} = \dfrac{25.0 \cdot 0.025}{1.25} = 0.5 \). This confirms that internal temperature gradients are significant and the Heisler method is appropriate.
Dimensionless temperature ratio: \[ \theta = \frac{T_{\text{target}} - T_{\text{ambient}}}{T_{\text{initial}} - T_{\text{ambient}}} = \frac{-15.0 - (-20.0)}{-5.0 - (-20.0)} = \frac{5.0}{15.0} = 0.3333. \]
One‑term approximation: For Bi = 0.5, the first eigenvalue and coefficient are \( \zeta_{1} = 0.6533 \) and \( C_{1} = 1.0701 \). The dimensionless core temperature is given by \( \theta = C_{1} \exp\left(-\zeta_{1}^{2} \cdot \text{Fo}\right) \). Rearranging for the Fourier number: \[ \text{Fo} = \frac{-\ln\left(\dfrac{\theta}{C_{1}}\right)}{\zeta_{1}^{2}} = \frac{-\ln\left(\dfrac{0.3333}{1.0701}\right)}{(0.6533)^{2}} = \frac{-\ln(0.3115)}{0.4268} = \frac{1.1663}{0.4268} = 2.733. \]
Thermal diffusivity: \( \alpha = \dfrac{k}{\rho \cdot c_{p}} = \dfrac{1.25}{950.0 \cdot 1900.0} = 6.925 \times 10^{-7} \; \text{m}^2/\text{s} \).
Required cooling time in seconds: \( t = \dfrac{\text{Fo} \cdot L^{2}}{\alpha} = \dfrac{2.733 \cdot (0.025)^{2}}{6.925 \times 10^{-7}} = \dfrac{2.733 \cdot 6.25 \times 10^{-4}}{6.925 \times 10^{-7}} \approx 2466 \; \text{s} \).
Convert to minutes: \( t_{\text{min}} = \dfrac{t_{\text{s}}}{60} = \dfrac{2466}{60} \approx 41.1 \; \text{min} \).

Final Answer: The time required to cool the frozen slab from –5.0 °C to a core temperature of –15.0 °C is approximately 41.1 minutes.

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