Question

Saturated steam condenses on a vertical plate maintained at a constant wall temperature. If $x$ is the vertical distance from the top edge of the plate, then the local heat transfer coefficient $h(x) \propto \Gamma(x)^{-1/3}$, where $\Gamma(x)$ is the local mass flow rate of the condensate per unit plate width. The ratio of the average heat transfer coefficient over the entire plate to the heat transfer coefficient at the bottom of the plate is

• A. 4
• B. 4/3
• C. 3/4
• D. 3

Hint:

For vertical plate condensation, $h(x)$ decreases with height, and the average-to-bottom ratio always comes out to $4/3$.

Answer 1

The Correct Option is B

For laminar film condensation on a vertical plate, the local condensate mass flow rate varies as $\Gamma(x) \propto x^{3/4}$.
Given $h(x) \propto \Gamma(x)^{-1/3}$, we get:
\[ h(x) \propto x^{-1/4} \]
The average heat-transfer coefficient over the plate is:
\[ \bar{h} = \frac{1}{L} \int_0^L h(x)\, dx = \frac{1}{L} \int_0^L x^{-1/4}\, dx \]
Evaluate the integral:
\[ \int_0^L x^{-1/4} dx = \frac{4}{3} L^{3/4} \]
At the bottom of the plate ($x = L$):
\[ h(L) \propto L^{-1/4} \]
Thus, the required ratio is:
\[ \frac{\bar{h}}{h(L)} = \frac{\frac{4}{3} L^{3/4}}{L^{3/4}} = \frac{4}{3} \]
Final Answer: 4/3