The GATE Grind

GATE 2022 CH – Question 42

Heat Transfer · Boiling, condensation and evaporation · 2 marks · Multiple choice

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

  1. 4
  2. 4/3
  3. 3/4
  4. 3

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Correct answer: (B) 4/3

Explanation

In Nusselt's film condensation the film thickness is $\delta \propto x^{1/4}$ and $\Gamma \propto \delta^3 \propto x^{3/4}$. So $h \propto \Gamma^{-1/3} \propto x^{-1/4}$. The average over the length $L$ is $\frac{1}{L}\int_0^LCx^{-1/4}dx = \frac{4}{3}CL^{-1/4}$, and the ratio to the value at the bottom, $CL^{-1/4}$, is $\frac{4}{3}$.