Question

For condensation of a pure vapor on a vertical surface, the Nusselt theory predicts:

• A. Constant film thickness
• B. Film thickness increasing from top to bottom
• C. Turbulent flow throughout
• D. Film thinning from top to bottom

Hint:

Remember that \( \delta(x) \propto x^{1/4} \) and \( h(x) \propto x^{-1/4} \).
These proportionalities are frequently tested in competitive examinations and help solve multi-part questions quickly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks what the Nusselt theory of film condensation predicts regarding the thickness of the liquid condensate film as it flows down a vertical surface.
This is a standard topic in phase-change heat transfer.

Step 2: Key Formula or Approach:
Nusselt's analysis of laminar film condensation on a vertical plate assumes that a continuous film of liquid flows downward under the influence of gravity.
The local film thickness, \( \delta(x) \), at a distance \( x \) from the top of the plate is given by:
\[ \delta(x) = \left[ \frac{4 \cdot \mu_l \cdot k_l \cdot (T_{\text{sat}} - T_w) \cdot x}{g \cdot \rho_l \cdot (\rho_l - \rho_v) \cdot h'_{fg}} \right]^{1/4} \]

Step 3: Detailed Explanation:

• Mechanism of Film Growth: As vapor condenses on the cold vertical wall, a liquid film is formed at the top of the plate (\( x = 0 \)).
As this liquid film flows downward under the action of gravity, additional vapor condenses onto the liquid-vapor interface.
This continuous condensation adds mass to the falling liquid film.

• Proportionality: From Nusselt's analytical equation, the local film thickness \( \delta(x) \) is proportional to the fourth root of the distance from the top:
\[ \delta(x) \propto x^{1/4} \]
This means the film thickness starts at zero at the very top of the plate and continuously increases as it flows toward the bottom.

• Thermal Resistance Impact: Because the liquid film acts as a thermal barrier, the local heat transfer coefficient \( h(x) \) is inversely proportional to the film thickness:
\[ h(x) = \frac{k_l}{\delta(x)} \propto x^{-1/4} \]
Consequently, the local heat transfer coefficient is highest at the top (where the film is thinnest) and decreases toward the bottom.


Step 4: Final Answer:
Nusselt's theory of condensation predicts that the liquid film thickness increases from top to bottom on a vertical surface.