Introduction & Context

The analysis of surface heat transfer during freezing is a fundamental task in Process Engineering, particularly within the food processing, cryogenics, and pharmaceutical industries. Understanding the rate at which an object loses heat to its surroundings is critical for determining freezing times, ensuring product quality, and optimizing energy consumption in refrigeration systems.

This calculation evaluates the interplay between external convective heat transfer and internal thermal conduction. By determining the Biot number, engineers can identify whether the freezing process is limited by the surface heat transfer coefficient or by the internal thermal resistance of the material, allowing for the selection of appropriate cooling strategies and equipment sizing.

Methodology & Formulas

The calculation follows a systematic approach to determine the convective heat transfer coefficient and the subsequent dimensionless Biot number.

First, the Reynolds number (Re) is calculated to characterize the fluid flow regime around the object:

\[ Re = \frac{\rho_{\text{air}} \cdot V \cdot D}{\mu_{\text{air}}} \]

The Nusselt number (Nu) for a spherical geometry is determined using the empirical correlation for forced convection:

\[ Nu = 2 + \left( 0.4 \cdot Re^{0.5} + 0.06 \cdot Re^{2/3} \right) \cdot Pr_{\text{air}}^{0.4} \]

The convective heat transfer coefficient (h) is then derived from the Nusselt number:

\[ h = \frac{Nu \cdot k_{\text{air}}}{D} \]

The characteristic length (L_c) for a sphere is defined as:

\[ L_{c} = \frac{D}{6} \]

Finally, the Biot number (Bi), which represents the ratio of internal conductive resistance to external convective resistance, is calculated as:

\[ Bi = \frac{h \cdot L_{c}}{k_{s}} \]

Regime / Condition	Criteria	Physical Significance
Lumped Capacitance	\( Bi < 0.1 \)	Internal conduction is negligible; external convection dominates.
Intermediate Resistance	\( 0.1 \leq Bi \leq 10 \)	Both internal and external resistances influence the freezing rate.
Conduction Limited	\( Bi > 10 \)	Internal thermal resistance dominates; air velocity has minimal effect.
Empirical Validity	\( 1 \leq Re \leq 10^{5} \)	Correlation range for spherical Nusselt number.

The convective heat transfer coefficient (\(h\)) is a critical process parameter that dictates the rate of energy removal from the surface. Higher values of \(h\) accelerate the phase change process by:

Reducing the thermal resistance at the boundary layer.
Increasing the temperature gradient between the bulk fluid and the cooling medium.
Promoting faster crystal nucleation and growth rates at the interface.

Fouling introduces an additional conductive resistance layer on the heat transfer surface. For process engineers, this results in:

A significant decrease in the overall heat transfer coefficient (\(U\)).
Increased energy consumption to maintain the target freezing temperature.
Potential non-uniformity in the thickness of the frozen layer across the surface.

To minimize supercooling and ensure a controlled freezing front, consider the following engineering strategies:

Increasing surface turbulence to disrupt the stagnant boundary layer.
Utilizing surface treatments or coatings to reduce nucleation barriers.
Implementing precise control of the coolant flow rate to maintain a consistent heat flux.

Worked Example: Effect of Surface Heat Transfer on Freezing of a Small Spherical Food Particle

Scenario: A small spherical food particle (diameter \(D = 0.01\) m) is frozen in a blast freezer with air at \(-20^\circ\)C flowing at \(V = 1.0\) m/s. The solid thermal conductivity \(k_s = 0.5\) W/m·K. Determine whether the freezing process is limited by external convection or internal conduction by computing the Biot number.

Knowns (input parameters and constants):

Air properties: Prandtl number \(Pr = 0.7\), thermal conductivity \(k_{\text{air}} = 0.025\) W/m·K, dynamic viscosity \(\mu = 1.8 \times 10^{-5}\) kg/m·s, density \(\rho = 1.2\) kg/m³.
Sphere diameter: \(D = 0.01\) m.
Air velocity: \(V = 1.0\) m/s.
Solid thermal conductivity: \(k_s = 0.5\) W/m·K.

Step-by-step calculation:

Characteristic length for a sphere: \[ L_c = \frac{D}{6} = \frac{0.01}{6} = 0.001667\ \text{m} \]
Reynolds number: \[ Re = \frac{\rho \cdot V \cdot D}{\mu} = \frac{1.2 \times 1.0 \times 0.01}{1.8 \times 10^{-5}} = 666.667 \] Validity check: \(1 \le Re \le 10^5\) – within empirical bounds for the sphere Nusselt correlation.
Nusselt number (sphere correlation): \[ Nu = 2 + \left(0.4\,Re^{0.5} + 0.06\,Re^{2/3}\right) Pr^{0.4} \] Substituting \(Re = 666.667\) and \(Pr = 0.7\): \[ Nu = 2 + \left(0.4 \times (666.667)^{0.5} + 0.06 \times (666.667)^{2/3}\right) \times (0.7)^{0.4} = 14.925 \]
Convective heat transfer coefficient: \[ h = \frac{Nu \cdot k_{\text{air}}}{D} = \frac{14.925 \times 0.025}{0.01} = 37.312\ \text{W/m}^2\text{K} \]
Biot number: \[ Bi = \frac{h \cdot L_c}{k_s} = \frac{37.312 \times 0.001667}{0.5} = 0.124 \]

Final Answer: \(Bi = 0.124\). Since \(Bi > 0.1\), internal conduction resistance is not negligible; the freezing process is in the intermediate regime where both external convection and internal conduction contribute. Lumped capacitance analysis is not strictly valid. For a small particle at higher air velocity, external convection could dominate further, but at \(V = 1\) m/s, internal resistance already plays a role.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Effect of Surface Heat Transfer on Freezing

Introduction & Context

🚀 Skip the Manual Math!

Methodology & Formulas

Worked Example: Effect of Surface Heat Transfer on Freezing of a Small Spherical Food Particle