Subtopic: Preservation of food by cooling and freezing
Question Statement:
A contact plate freezer extracts heat from a 24 mm thick slab of boneless meat (85% w/w water). Initial slab temperature = freezing point = 272.5 K; latent heat of freezing = 335 kJ/kg water. Plate temperature = 247.5 K (steady). Bulk density = 750 kg/m³; thermal conductivity of frozen meat = 1.5 W/m·K. Find the minimum duration for complete freezing (s).
Official Answer Key:
409 to 411
Step-by-Step Solution & Explanation
• **Governing Equation (Plank\'s Equation for Food Freezing):**
$t_F = \frac{\rho_m \lambda}{\Delta T} \left( \frac{P \cdot a}{h} + \frac{R \cdot a^2}{k} \right)$
• **Given Data:**
- Meat density $\rho_m = 750\text{ kg/m}^3$, water fraction $w = 0.85$, slab thickness $a = 24\text{ mm} = 0.024\text{ m}$.
- Latent heat of freezing of water $\lambda_w = 335\text{ kJ/kg} = 335,000\text{ J/kg}$.
- Effective volumetric latent heat $\rho_m \lambda = 750 \times 0.85 \times 335,000 = 2.135625 \times 10^8\text{ J/m}^3$.
- Temperature driving force $\Delta T = T_f - T_p = 272.5 - 247.5 = 25\text{ K}$.
- Frozen meat thermal conductivity $k = 1.5\text{ W/m}\cdot\text{K}$.
- For infinite slab frozen symmetrically from both contact plates: $P = \frac{1}{2}$, $R = \frac{1}{8}$.
- In ideal contact plate freezing, surface convective resistance is negligible ($h \to \infty$, so $\frac{P a}{h} \approx 0$):
• **Freezing Time Calculation:**
$t_F = \frac{2.135625 \times 10^8}{25} \times \left( \frac{1}{8} \times \frac{(0.024)^2}{1.5} \right) = 8,542,500 \times \left( \frac{0.000576}{12} \right) = 8,542,500 \times 0.000048 = 410.04\text{ s}$
• **Conclusion:** Minimum freezing time $\approx 410\text{ seconds}$ (Official range: 409 to 411).
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