Reference ID: MET-39A7 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations to determine the relationship between the conductive resistance within a solid body and the convective resistance at its surface. In process engineering, this parameter is critical for transient heat conduction analysis. It dictates whether a system can be modeled using the Lumped Capacitance Method (LCM), which assumes a uniform temperature distribution throughout the solid at any given time. This simplification is essential for the rapid design of quenching processes, heat treatment cycles, and thermal sensor response analysis.
Methodology & Formulas
The calculation of the Biot number requires the determination of the characteristic length (Lc) of the geometry, which represents the ratio of the volume of the body to its surface area. The governing equations are defined as follows:
The characteristic length is defined by the ratio of volume (V) to surface area (As):
\[ L_c = \frac{V}{A_s} \]
For a spherical geometry, the characteristic length simplifies to a function of the diameter (D):
\[ L_c = \frac{D}{6} \]
The Biot number is then calculated using the convection heat transfer coefficient (h), the characteristic length (Lc), and the thermal conductivity of the solid material (k):
\[ Bi = \frac{h \cdot L_c}{k} \]
Regime
Condition
Physical Interpretation
Lumped Capacitance Valid
Bi ≤ 0.1
Internal conductive resistance is negligible; temperature is spatially uniform.
Lumped Capacitance Invalid
Bi > 0.1
Internal temperature gradients are significant; transient conduction charts or numerical methods are required.
The Biot number represents the ratio of internal conductive resistance to external convective resistance of a solid body. It is a critical dimensionless parameter for process engineers because it determines whether a lumped capacitance model is valid for transient heat transfer analysis.
If Bi is less than 0.1, the internal temperature gradients are negligible, allowing for the use of the lumped capacitance method.
If Bi is greater than 0.1, internal resistance is significant, and spatial temperature variations must be accounted for using more complex analytical or numerical solutions.
The characteristic length is defined as the ratio of the volume of the solid to its surface area. Selecting the correct geometry is vital for accurate results.
For a plane wall of thickness 2L, the characteristic length is L.
For a long cylinder of radius r, the characteristic length is r/2.
For a sphere of radius r, the characteristic length is r/3.
While both numbers share a similar mathematical form, they describe fundamentally different physical phenomena.
The Biot number is based on the thermal conductivity of the solid, focusing on internal heat flow.
The Nusselt number is based on the thermal conductivity of the fluid, focusing on the efficiency of convective heat transfer at the surface.
Worked Example: Thermal Analysis of a Steel Sphere
In a process engineering application, a small steel ball bearing is being quenched in a cooling oil bath. To determine if the lumped capacitance method is appropriate for modeling the transient temperature profile, we must calculate the Biot number (Bi).
Knowns:
Diameter (D): 0.02 m
Radius (r): 0.01 m
Thermal conductivity of steel (k): 50.0 W/m·K
Convection heat transfer coefficient (h): 250.0 W/m²·K
Step-by-Step Calculation:
Calculate the characteristic length (Lc) for a sphere, defined as the ratio of volume (V) to surface area (As):
\[ L_c = \frac{V}{A_s} = \frac{\frac{4}{3}\pi r^3}{4\pi r^2} = \frac{r}{3} \]
\[ L_c = \frac{0.01}{3} = 0.003 \text{ m} \]
Calculate the Biot number using the formula:
\[ Bi = \frac{h \cdot L_c}{k} \]
\[ Bi = \frac{250.0 \cdot 0.003}{50.0} \]
Evaluate the result:
\[ Bi = 0.017 \]
Final Answer:
The calculated Biot number is 0.017. Since 0.017 is less than the threshold of 0.1, the lumped capacitance method is valid for this thermal analysis.
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