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

Concept: In heat transfer analysis, the characteristic length (\(L_c\)) of a solid body is defined as the ratio of its total internal volume (\(V\)) to its exterior geometric surface area (\(A_s\)) exposed to convection: \[ L_c = \frac{V}{A_s} \] This parameter represents the effective distance heat must travel from the interior core of the body to reach its outer cooling boundary surface. Let us compute this characteristic length for a solid sphere of radius \(R\):

Step 1: Identifying the geometric formulas for a sphere.
For a perfect solid sphere of radius \(R\):

• Total internal volume: \(V = \frac{4}{3}\pi R^3\)

• Total outer surface area: \(A_s = 4\pi R^2\)

Step 2: Substituting these formulas into the characteristic length definition.
\[ L_c = \frac{\frac{4}{3}\pi R^3}{4\pi R^2} \]

Step 3: Simplifying the fraction.
We can cancel common factors in the numerator and denominator:

• Cancel the factor of 4: \(\frac{\frac{1}{3}\pi R^3}{\pi R^2}\)

• Cancel the constant \(\pi\): \(\frac{\frac{1}{3}R^3}{R^2}\)

• Simplify the radius terms using exponent rules (\(\frac{R^3}{R^2} = R\)):
\[ L_c = \frac{1}{3} \cdot R = \frac{R}{3} \] Thus, the characteristic length for a sphere is exactly \(\frac{R}{3}\), which matches Option (3).