Question

A sphere of radius R is in motionless fluid (no forced or natural convection). The surface temperature of the sphere is T$_R$ and the bulk temperature T$_\infty$. What is the Nusselt number for this situation?

• A. 0
• B. 0.5
• C. 2
• D. \( \infty \)

Hint:

The expression \( Nu = 2 \) is the absolute minimum limit of heat transfer from a sphere.
Any fluid motion (convection) will increase the Nusselt number above 2, which is represented by empirical correlations like the Ranz-Marshall equation:
\[ Nu = 2 + 0.6 \cdot Re^{1/2} \cdot Pr^{1/3} \]

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the limiting value of the Nusselt number (\( Nu \)) for a sphere suspended in an infinite, motionless fluid, where heat is transferred solely via conduction.
This is a classical heat transfer problem.

Step 2: Key Formula or Approach:
The Nusselt number is defined as:
\[ Nu = \frac{h \cdot D}{k_f} \]
where \( h \) is the heat transfer coefficient, \( D \) is the diameter of the sphere (\( D = 2R \)), and \( k_f \) is the thermal conductivity of the fluid.
We can find \( h \) by analyzing steady-state, one-dimensional radial heat conduction in the surrounding stagnant fluid.

Step 3: Detailed Explanation:

• The heat conduction equation in spherical coordinates with radial symmetry is:
\[ \frac{d}{dr}\left(r^2 \cdot \frac{dT}{dr}\right) = 0 \]

• Integrate this equation once:
\[ r^2 \cdot \frac{dT}{dr} = C_1 \quad \implies \quad \frac{dT}{dr} = \frac{C_1}{r^2} \]

• Integrate a second time:
\[ T(r) = -\frac{C_1}{r} + C_2 \]

• Apply the boundary conditions:
1. At \( r = R \) (sphere surface), \( T = T_R \).
2. At \( r \to \infty \) (bulk fluid), \( T = T_{\infty} \).

• Solving for the constants:
\[ C_2 = T_{\infty} \]
\[ T_R = -\frac{C_1}{R} + T_{\infty} \quad \implies \quad C_1 = -R \cdot (T_R - T_{\infty}) \]

• Substitute the constants back into the temperature profile:
\[ T(r) = T_{\infty} + \frac{R}{r} \cdot (T_R - T_{\infty}) \]

• Calculate the heat flux at the surface (\( r = R \)) using Fourier's Law:
\[ q'' = -k_f \cdot \frac{dT}{dr}\Big|_{r=R} = -k_f \cdot \left[ -\frac{R}{R^2} \cdot (T_R - T_{\infty}) \right] = \frac{k_f}{R} \cdot (T_R - T_{\infty}) \]

• Equate this conduction heat flux to Newton's law of cooling:
\[ q'' = h \cdot (T_R - T_{\infty}) \quad \implies \quad h = \frac{k_f}{R} \]

• Substitute \( h \) into the Nusselt number equation, using \( D = 2R \):
\[ Nu = \frac{h \cdot D}{k_f} = \frac{\left(\frac{k_f}{R}\right) \cdot (2R)}{k_f} = 2 \]


Step 4: Final Answer:
The Nusselt number for a sphere in a stagnant fluid is exactly 2.