Question

For heat transfer calculation in fins, if the Biot number is very small (\(\text{Bi} \lt 0.1\)) it implies

• A. Higher convective resistance
• B. High temperature gradient in the fin
• C. Fin is infinitely long
• D. Negligible internal conductive resistance

Hint:

Biot Number physical interpretation: - \(Bi \lt 0.1\): Conductive resistance is negligible. Temperature inside the cross-section is uniform. This justifies using the Lumped Parameter Capacity analysis for transient problems. - \(Bi \gt 0.1\): Internal temperature gradients are significant and cannot be ignored; spatial variations must be accounted for using Fourier equations.

Answer 1

The Correct Option is D

Concept: The Biot number (\(Bi\)) is a dimensionless parameter used in transient conduction and extended surface (fin) thermal analysis. It relates the internal conduction resistance within a solid body to the external convection resistance at the body's surface. The Biot number is defined mathematically as: \[ Bi = \frac{\text{Internal Conductive Resistance to Heat Transfer}}{\text{External Convective Resistance to Heat Transfer}} = \frac{\left(\frac{L_c}{k \cdot A}\right)}{\left(\frac{1}{h \cdot A}\right)} = \frac{h \cdot L_c}{k} \] Where:

• \(h\) = Convective heat transfer coefficient of the surrounding fluid.

• \(L_c\) = Characteristic length of the solid body (defined as volume divided by surface area, \(L_c = \frac{V}{A_s}\)).

• \(k\) = Thermal conductivity of the solid material.
Let us evaluate the physical meaning of a small Biot number:

Step 1: Analyzing the mathematical limit \(Bi \lt 0.1\).
When the Biot number is less than 0.1, the numerator in our resistance ratio is much smaller than the denominator: \[ \frac{\text{Internal Conductive Resistance}}{\text{External Convective Resistance}} \ll 0.1 \quad \Rightarrow \quad \text{Internal Conductive Resistance} \to 0 \] This indicates that the material's internal resistance to heat conduction is negligible compared to the resistance to heat convection at the surface.

Step 2: Deducing the spatial temperature distribution.
Because the internal conductive resistance is negligible, heat distributes rapidly within the solid cross-section compared to the rate of heat transfer across the fluid boundary. As a result, the temperature gradient within the solid cross-section is virtually flat. For a thin fin or pin, a small Biot number (\(Bi = \frac{h t}{k} \lt 0.1\)) justifies the fundamental assumption that temperature varies only along the length of the fin (\(x\)-direction) and remains uniform across its cross-section.

Step 3: Matching the given options.
A Biot number of \(Bi \lt 0.1\) directly implies that the system has a negligible internal conductive resistance, which corresponds to Option (4).