Introduction & Context
The Individual Quick Freezing (IQF) process is a critical unit operation in food processing, designed to freeze small, discrete food items rapidly to preserve quality, texture, and nutritional value. In process engineering, calculating the freezing time is essential for sizing equipment, determining conveyor belt speeds, and ensuring that the product core reaches the target temperature to meet food safety standards.
Plank's Equation is the industry-standard analytical model for estimating the time required to freeze a product. It is particularly useful for fluidized bed freezers, where high convective heat transfer coefficients are achieved by suspending particles in a cold air stream. This calculation focuses on the latent heat removal phase, assuming the product is already at its initial freezing point.
Methodology & Formulas
The freezing time is determined by evaluating the resistance to heat transfer from both the external convective environment and the internal conductive properties of the product. The total freezing time \(t_{f}\) is calculated using the following relationship:
\[ t_{f} = \frac{\rho \cdot L_{f}}{\Delta T} \left( \frac{P \cdot d}{h} + \frac{R \cdot d^{2}}{k} \right) \]
Where the temperature difference is defined as the absolute difference between the freezing point of the product and the ambient air temperature:
\[ \Delta T = |T_{\text{frz}} - T_{\infty}| \]
To determine the validity of using Plank's Equation, the Biot number (\(Bi\)) is calculated to assess the ratio of internal conductive resistance to external convective resistance:
\[ Bi = \frac{h \cdot d}{k} \]
The following table outlines the empirical constraints and validity regimes for this calculation:
| Parameter |
Condition / Threshold |
Engineering Significance |
| Convective Coefficient (h) |
\(50 \leq h \leq 150\) W/m²·K |
Typical range for fluidized bed air velocity. |
| Biot Number (Bi) |
\(Bi < 10\) |
Ensures Plank's Equation remains a valid approximation. |
| Geometry (P, R) |
\(P = 1/6,\; R = 1/24\) |
Shape factors specific to spherical geometry. |
Worked Example: IQF Freezing Time for Small Vegetable Spheres
Scenario: A fluidized bed individual quick freezer (IQF) is used to freeze small spherical vegetable pieces (approximated as spheres of diameter 1 cm). The product enters at its freezing point, so only latent heat removal is considered. Cold air at –35°C is the freezing medium.
Knowns:
- Density of product, ρ = 1000.0 kg/m³
- Latent heat of fusion, Lf = 335,000.0 J/kg
- Thermal conductivity of frozen product, k = 1.5 W/(m·K)
- Freezing point, Tfrz = –1.0°C
- Air temperature, T∞ = –35.0°C
- Characteristic diameter, d = 0.01 m
- Convective heat transfer coefficient, h = 80.0 W/(m²·K)
- Sphere shape factor P = 1/6 ≈ 0.1667
- Sphere shape factor R = 1/24 ≈ 0.04167
Step-by-Step Calculation Using Plank's Equation:
- Calculate the temperature difference (absolute):
\[
\Delta T = |T_{\text{frz}} - T_{\infty}| = |-1.0 - (-35.0)| = 34.0\ \text{K}
\]
- Compute the pre-factor:
\[
\frac{\rho \cdot L_{f}}{\Delta T} = \frac{1000.0 \times 335{,}000.0}{34.0} = 9{,}852{,}941.176\ \text{J/(m}^3\!\cdot\!\text{K)}
\]
- Compute the external convective resistance term:
\[
\frac{P \cdot d}{h} = \frac{(1/6) \times 0.01}{80.0} = \frac{0.001667}{80.0} = 2.083 \times 10^{-5}\ \text{m}^3\!\cdot\!\text{K/W}
\]
- Compute the internal conductive resistance term:
\[
\frac{R \cdot d^{2}}{k} = \frac{(1/24) \times (0.01)^{2}}{1.5} = \frac{4.167 \times 10^{-6}}{1.5} = 2.778 \times 10^{-6}\ \text{m}^3\!\cdot\!\text{K/W}
\]
- Sum the resistance terms:
\[
\text{Sum} = 2.083 \times 10^{-5} + 2.778 \times 10^{-6} = 2.361 \times 10^{-5}\ \text{m}^3\!\cdot\!\text{K/W}
\]
- Calculate freezing time in seconds:
\[
t_{f} = 9{,}852{,}941.176 \times 2.361 \times 10^{-5} = 232.639\ \text{s}
\]
- Convert to minutes:
\[
t_{f} = 232.639\ \text{s} \div 60 = 3.877\ \text{min}
\]
- Validity check – Biot number:
\[
Bi = \frac{h \cdot d}{k} = \frac{80.0 \times 0.01}{1.5} = 0.533
\]
Since Bi < 10, Plank's equation is applicable.
Final Answer: The predicted freezing time is 232.6 seconds (approximately 3.88 minutes). This result falls within the typical 3–5 minute range for IQF freezing of small vegetable pieces, confirming the approach is sound.
