Reference ID: MET-8CCE | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations to determine the ratio of the internal thermal resistance of a solid to the external convective thermal resistance at its surface. In the context of Process Engineering, specifically for the thermal processing of canned foods, the Biot number is critical for determining whether the temperature distribution within the food can be assumed to be uniform at any given time.
This calculation is typically used during the design of sterilization or pasteurization cycles. By evaluating the Biot number, engineers can decide if the lumped capacitance model is valid—which simplifies the analysis to a single ordinary differential equation—or if a more complex distributed thermal analysis (such as transient conduction in a cylinder) is required to ensure food safety and quality.
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The calculation follows a systematic approach to determine the characteristic length and the resulting dimensionless Biot number based on the geometry of the cylindrical container and the thermal properties of the food product.
First, the radius R is derived from the diameter D, and all dimensions are converted to meters:
\[ R = \frac{D}{2} \]
The characteristic length Lc for a cylindrical geometry is defined as the ratio of the volume to the surface area:
\[ L_c = \frac{R \cdot H}{2 \cdot (R + H)} \]
The Biot number is then calculated using the convective heat transfer coefficient h, the characteristic length Lc, and the thermal conductivity of the food kfood:
Uniform temperature assumption is valid; simple exponential decay model.
Distributed Thermal Analysis
\(\mathrm{Bi} \geq 0.1\)
Internal conduction resistance is significant; requires transient conduction solutions (e.g., Heisler charts or numerical methods).
The Biot number is a dimensionless quantity that determines the ratio of internal conductive resistance to external convective resistance. For process engineers, it is essential because:
It dictates whether the internal temperature of the food product is uniform during heating.
It helps determine if the lumped capacitance model is valid for your specific container geometry.
It identifies if the heat transfer rate is limited by the surface film coefficient or the thermal conductivity of the food matrix.
The characteristic length, denoted as Lc, is defined as the ratio of the volume of the body to its surface area. For a standard cylindrical can, use the following approach:
Calculate the volume using \(V = \pi r^2 h\).
Calculate the total surface area using \(A_s = 2\pi r^2 + 2\pi r h\).
Divide the volume by the surface area to obtain Lc.
Ensure that the units for radius and height are consistent to avoid calculation errors in the Biot number formula.
A high Biot number, typically greater than 0.1, indicates that the internal thermal resistance of the canned food is significant compared to the external convective resistance. In this scenario:
Temperature gradients within the product will be substantial.
The surface of the food will reach the retort temperature much faster than the geometric center.
You must utilize the Heisler charts or analytical solutions for transient heat conduction rather than the lumped capacitance method to ensure accurate lethality calculations.
Worked Example: Biot Number for Canned Food in a Rotary Retort
A food processing plant uses a rotary retort to sterilize cans of cream-style corn. The cans are standard #2 cans (307×409: diameter 87.3 mm, height 116 mm). The retort uses saturated steam at 121°C, providing a high heat transfer coefficient. The thermal conductivity of the corn product is typical for high-moisture food. Determine the Biot number to assess if the lumped capacitance method is valid for modeling the heating time.
Knowns:
Diameter: \(D = 87.3\) mm
Height: \(H = 116.0\) mm
Thermal conductivity of food: \(k_{\text{food}} = 0.5\) W/(m·K)
Convective heat transfer coefficient: \(h = 2000.0\) W/(m²·K)
Step-by-step calculation:
Convert dimensions to meters: Radius \(R = D/2 = 87.3/2000 = 0.04365\) m; Height \(H = 116.0/1000 = 0.116\) m.
Compute characteristic length for a cylinder:
\[
L_c = \frac{R H}{2(R+H)} = \frac{0.04365 \times 0.116}{2(0.04365 + 0.116)} = 0.015858 \text{ m}
\]
(rounded to 6 decimal places from exact value 0.01585781396805512).
Use given thermal conductivity: \(k_{\text{food}} = 0.5\) W/(m·K).
Use given heat transfer coefficient: \(h = 2000.0\) W/(m²·K).
Compute Biot number:
\[
\mathrm{Bi} = \frac{h L_c}{k_{\text{food}}} = \frac{2000.0 \times 0.01585781396805512}{0.5} = 63.431
\]
(rounded to three decimal places).
Validity check: \(\mathrm{Bi} = 63.431 > 0.1\). Therefore, the lumped capacitance model is invalid; internal conduction resistance dominates. A full transient conduction analysis (e.g., Heisler charts or FEM) is required.
Final Answer: \(\mathrm{Bi} = 63.431\). Lumped capacitance valid: No.
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