Introduction & Context
The freezing rate of food products is a critical parameter in Process Engineering, directly influencing the microstructural integrity and final quality of the product. During the transition from liquid water to ice, the rate of heat extraction dictates the nucleation and growth kinetics of ice crystals. Rapid freezing promotes a high number of small, uniformly distributed crystals, whereas slow freezing allows for the growth of large, dendritic ice structures. These large crystals can cause mechanical damage to cellular membranes, leading to significant drip loss, texture degradation, and nutrient leaching upon thawing.
This calculation is typically employed in the design and optimization of Individual Quick Freezing (IQF) tunnels and blast freezers to ensure that the freezing process remains within a regime that preserves product quality.
Methodology & Formulas
The freezing process is modeled using a conduction-controlled approach. The total freezing time is estimated using a modified version of Plank’s equation, which accounts for both convective resistance at the surface and conductive resistance within the product slab.
The freezing time \( t_{f} \) is calculated as:
\[
t_{f} = \frac{\rho \cdot L_{h}}{T_{m} - T_{air}} \cdot \left( \frac{L}{h} + \frac{L^{2}}{2 \cdot k_{frozen}} \right)
\]
The average freezing rate \( v_{avg} \), defined as the velocity of the ice front moving through the product half-thickness, is derived as:
\[
v_{avg} = \frac{L}{t_{f}}
\]
The resulting ice crystal diameter \( D_{cryst} \) is estimated using an empirical power-law correlation, which relates the crystal size to the freezing rate:
\[
D_{cryst} = a \cdot (v_{avg})^{-b}
\]
| Parameter |
Description |
Regime/Condition |
| Biot Number (Bi) |
\( Bi = \frac{h \cdot L}{k_{frozen}} \) |
If \( Bi > 50 \), internal resistance dominates; neglect external convection term. |
| Freezing Rate (vavg) |
Empirical Validity Range |
Valid for \( 0.02 \leq v_{avg} \leq 5.0 \) mm/min. |
| Water Content |
Product Composition |
Model assumes high-moisture content (\( > 70\% \)). |
Worked Example: Effect of Freezing Rate on Ice Crystal Size in a Spinach Leaf
Scenario: A 10 mm thick spinach leaf (half-thickness \(L = 5\) mm) is frozen under two distinct conditions to compare the resulting ice crystal size. The leaf is high-moisture (>70% water) and is modeled as a slab with pure conduction-controlled freezing.
Knowns:
- Half-thickness: \(L = 0.005\) m
- Density: \(\rho = 900.0\) kg/m³
- Latent heat of fusion: \(L_h = 280000.0\) J/kg
- Thermal conductivity (frozen): \(k_f = 1.5\) W/m·K
- Freezing point of leaf: \(T_m = -1.0\) °C
- Fast freezing (IQF): Air temperature \(T_{air,fast} = -35.0\) °C, heat transfer coefficient \(h_{fast} = 50.0\) W/m²·K
- Slow freezing (block): Air temperature \(T_{air,slow} = -18.0\) °C, heat transfer coefficient \(h_{slow} = 10.0\) W/m²·K
- Empirical correlation constants: \(a = 300.0\) μm·(mm/min)0.75, \(b = 0.75\)
Step-by-Step Calculation:
- Biot Number Check:
\[
Bi = \frac{h L}{k_f}
\]
- Fast freezing: \(Bi = \frac{50.0 \cdot 0.005}{1.5} = 0.167\)
- Slow freezing: \(Bi = \frac{10.0 \cdot 0.005}{1.5} = 0.033\)
Both values are well below 50, so the Plank equation (including both surface and internal resistances) is applicable.
- Freezing Time (Plank Equation for slab):
\[
t_f = \frac{\rho L_h}{(T_m - T_{air})} \left( \frac{L}{h} + \frac{L^2}{2 k_f} \right)
\]
- Fast freezing: Temperature difference \(\Delta T = T_m - T_{air,fast} = -1.0 - (-35.0) = 34.0\) K.
\(t_f = \frac{900.0 \cdot 280000.0}{34.0} \left( \frac{0.005}{50.0} + \frac{0.005^2}{2 \cdot 1.5} \right) = 802.941\) s = 13.382 min.
- Slow freezing: \(\Delta T = -1.0 - (-18.0) = 17.0\) K.
\(t_f = \frac{900.0 \cdot 280000.0}{17.0} \left( \frac{0.005}{10.0} + \frac{0.005^2}{2 \cdot 1.5} \right) = 7535.294\) s = 125.588 min.
- Average Freezing Rate:
\[
v_{avg} = \frac{L}{t_f} \quad \text{(converted to mm/min)}
\]
- Fast: \(v_{avg,fast} = \frac{0.005 \text{ m}}{802.941 \text{ s}} \times 1000 \times 60 = 0.374 mm/min\)
- Slow: \(v_{avg,slow} = \frac{0.005 \text{ m}}{7535.294 \text{ s}} \times 1000 \times 60 = 0.04 mm/min\)
Both rates lie within the valid empirical range (0.02–5 mm/min).
- Estimate Ice Crystal Diameter:
\[
D_{cryst} = a \cdot (v_{avg})^{-b}
\]
- Fast: \(D_{cryst,fast} = 300.0 \cdot (0.374)^{-0.75} = 627.759 μm\)
- Slow: \(D_{cryst,slow} = 300.0 \cdot (0.04)^{-0.75} = 3365.933 μm\)
Final Answer: Under fast freezing (IQF) with an average rate of 0.374 mm/min, the estimated ice crystal diameter is 627.759 μm. Under slow freezing (block) at 0.04 mm/min, the estimated diameter is 3365.933 μm. The slower freezing rate produces crystals over five times larger, which would cause significantly more cellular damage and loss of texture.
