Question

The characteristic length (\(\text{L}_{\text{c}}\)) for the calculation of Biot number (Bi) for a sphere of radius 'R' subjected to Newtonian cooling is

• A. R
• B. \(\frac{\text{R}}{2}\)
• C. \(\frac{\text{R}}{3}\)
• D. 3R

Hint:

Standard Characteristic Lengths (\(L_c = V/A_s\)) to memorize: - Cube (side length $a$): \(L_c = \frac{a^3}{6a^2} = \frac{a}{6}\) - Long Solid Cylinder (radius $R$): \(L_c = \frac{\pi R^2 L}{2\pi R L} = \frac{R}{2}\) - Solid Sphere (radius $R$): \(L_c = \frac{\frac{4}{3}\pi R^3}{4\pi R^2} = \frac{R}{3}\)

Answer 1

The Correct Option is C

Step 1: Use a shortcut relation between area and volume for a sphere.
For a sphere of radius \(R\), the volume is \(V = \frac{4}{3}\pi R^3\) and the surface area is \(A_s = 4\pi R^2\). Multiplying \(V\) by \(3/R\) reproduces \(A_s\) exactly, so we get the handy identity: \[ A_s = \frac{3V}{R} \]

Step 2: Substitute this identity into the characteristic length formula.
The characteristic length is defined as \(L_c = V/A_s\). Replacing \(A_s\) with \(3V/R\) gives: \[ L_c = \frac{V}{\left(\dfrac{3V}{R}\right)} = \frac{R}{3} \] The volume terms cancel directly, leaving a clean result without ever writing out the full expressions for \(V\) and \(A_s\) separately. \[ \boxed{L_c = \frac{R}{3}} \]
This matches option 3.