Introduction & Context
Plate freezer contact heat transfer optimization is a critical process engineering task used to determine the efficiency of industrial food freezing systems. By utilizing direct contact between refrigerated metal plates and the product, this method maximizes heat extraction rates compared to air-blast freezing. Accurate modeling of this process is essential for maintaining product quality, ensuring structural integrity, and optimizing throughput in high-volume production lines. This calculation is typically employed during the design phase of freezing equipment or when adjusting operational parameters for different product geometries and packaging materials.
Methodology & Formulas
The calculation follows a systematic approach to determine the thermal resistance at the interface, the resulting heat transfer coefficient, and the total time required for the phase change of the product.
First, the composite surface roughness (σ) and effective thermal conductivity (keff) are determined to calculate the contact resistance (Rc) using the Yovanovich correlation:
\[ \sigma = \sqrt{\sigma_{1}^{2} + \sigma_{2}^{2}} \]
\[ k_{eff} = \frac{2 \cdot k_{frozen} \cdot k_{steel}}{k_{frozen} + k_{steel}} \]
\[ R_{c} = 0.4 \cdot \frac{\sigma}{k_{eff}} \cdot \left( \frac{P}{H} \right)^{-0.95} \]
The total surface resistance (Rsurf) accounts for both the contact resistance and the conductive resistance of the packaging material, which then defines the convective heat transfer coefficient (h):
\[ R_{surf} = R_{c} + \frac{t_{pack}}{k_{pack}} \]
\[ h = \frac{1}{R_{surf}} \]
The thermal regime is validated using the Biot number (Bi). If the system satisfies the conduction-dominated regime, Plank's equation is applied to calculate the total freezing time (t):
\[ Bi = \frac{h \cdot d}{k_{frozen}} \]
\[ t = \left( \frac{\rho \cdot L_{f}}{T_{f} - T_{s}} \right) \cdot \left( \frac{d}{2 \cdot h} + \frac{d^{2}}{8 \cdot k_{frozen}} \right) \]
| Parameter |
Condition/Regime |
Threshold |
| Pressure Ratio |
Empirical Validity |
\(0.01 \leq P/H \leq 0.3\) |
| Biot Number |
Conduction-Limited |
\(Bi > 10\) |
| Product Thickness |
Geometric Constraint |
\(40 \text{ mm} \leq d \leq 80 \text{ mm}\) |
| Freezing Time |
Quality Limit |
\(t \leq 90 \text{ min}\) |
| Applied Pressure |
Deformation Limit |
\(P \leq P_{yield}\) |
The refrigerant flow rate directly influences the convective heat transfer coefficient within the plate channels. Increasing the flow rate reduces the boundary layer thickness, which enhances the overall heat transfer rate. However, engineers must balance this against:
- The pressure drop across the plate circuit.
- The energy consumption of the circulation pumps.
- The potential for flow-induced vibration or mechanical fatigue in the plate assembly.
Worked Example: Plate Freezer Contact Heat Transfer Optimization
A fish block is frozen in a plate freezer. The design engineer must verify that the contact pressure, surface resistance, and freezing time meet product quality and throughput constraints.
Knowns
- Fish block thickness, \(d = 0.06\) m (60 mm)
- Density, \(\rho = 1000\) kg/m³
- Latent heat of fusion, \(L_f = 280000\) J/kg (280 kJ/kg)
- Freezing point, \(T_f = -2°C\)
- Frozen fish thermal conductivity, \(k_{\text{frozen}} = 1.5\) W/m·K
- Plate temperature, \(T_s = -40°C\)
- Plate roughness, \(\sigma_1 = 0.5\) µm
- Fish surface roughness, \(\sigma_2 = 4.97\) µm
- Steel thermal conductivity, \(k_{\text{steel}} = 15\) W/m·K
- Package thickness, \(t_{\text{pack}} = 0.0001\) m (0.1 mm)
- Package thermal conductivity, \(k_{\text{pack}} = 0.3\) W/m·K
- Contact pressure, \(P = 30\) kPa
- Compressive yield limit, \(\sigma_y = 50\) kPa
- Fish hardness, \(H = 2.5\) MPa
Step-by-Step Calculation
- Compute composite surface roughness:
\[
\sigma = \sqrt{\sigma_1^2 + \sigma_2^2} = \sqrt{(0.5)^2 + (4.97)^2} = 4.995 \; \mu\text{m}
\]
- Compute effective thermal conductivity at the contact:
\[
k_{\text{eff}} = \frac{2 k_{\text{frozen}} k_{\text{steel}}}{k_{\text{frozen}} + k_{\text{steel}}}
= \frac{2 \cdot 1.5 \cdot 15}{1.5 + 15} = 2.727 \; \text{W/m·K}
\]
- Compute pressure ratio and check empirical bounds:
\[
\frac{P}{H} = \frac{0.03}{2.5} = 0.012
\]
\(0.01 \leq 0.012 \leq 0.3\) ✓ Valid range for Yovanovich correlation.
- Calculate contact resistance (Yovanovich):
\[
R_c = 0.4 \frac{\sigma}{k_{\text{eff}}} \left( \frac{P}{H} \right)^{-0.95}
= 0.4 \cdot \frac{4.995 \times 10^{-6}}{2.727} \cdot (0.012)^{-0.95}
= 4.894 \times 10^{-5} \; \text{m}^2 \cdot \text{K/W}
\]
- Calculate package film resistance:
\[
R_{\text{pack}} = \frac{t_{\text{pack}}}{k_{\text{pack}}}
= \frac{0.0001}{0.3} = 3.333 \times 10^{-4} \; \text{m}^2 \cdot \text{K/W}
\]
- Determine total surface resistance:
\[
R_{\text{surf}} = R_c + R_{\text{pack}}
= 4.894 \times 10^{-5} + 3.333 \times 10^{-4}
= 3.823 \times 10^{-4} \; \text{m}^2 \cdot \text{K/W}
\]
- Compute effective surface heat transfer coefficient:
\[
h = \frac{1}{R_{\text{surf}}}
= \frac{1}{3.823 \times 10^{-4}}
= 2615.938 \; \text{W/m}^2 \cdot \text{K}
\]
- Calculate Biot number to confirm conduction regime:
\[
\text{Bi} = \frac{h d}{k_{\text{frozen}}}
= \frac{2615.938 \cdot 0.06}{1.5}
= 104.638
\]
\(Bi > 10\) \(\checkmark\) Conduction-dominated regime.
- Compute freezing time using Plank's equation (conduction-limited form):
\[
t = \frac{\rho L_f}{T_f - T_s} \left( \frac{d}{2h} + \frac{d^2}{8 k_{\text{frozen}}} \right)
= \frac{1000 \cdot 280000}{(-2) - (-40)} \left( \frac{0.06}{2 \cdot 2615.938} + \frac{0.06^2}{8 \cdot 1.5} \right)
= 38.250 \; \text{minutes}
\]
- Validate constraints:
- Contact pressure: \(P = 30\) kPa < 50 kPa ✓ Within compressive yield limit.
- Freezing time: \(t = 38.250\) min < 90 min ✓ Meets quality limit.
Final Answer
The freezing time is \(t = 38.250\) minutes. The Biot number of 104.638 confirms conduction-limited heat transfer. All pressure and time constraints are satisfied.
