Question

Consider film condensation on a vertical plate. Will the heat flux be higher at the top or at the bottom of the plate? Why?

Short answer

Please explain your answer.

Step-by-step solution

Understanding Film Condensation

Film condensation occurs when a vapor cools and turns into a liquid, forming a thin film on a cold surface like a vertical plate. The heat is conducted through the liquid film to the cold surface. The heat flux depends on the film thickness and temperature difference between the film and the cold surface.

Effect of Gravity

Gravity plays a significant role in film condensation on a vertical plate. The liquid film flows downward due to gravitational force, causing the film to become thicker as it moves down the plate. This increase in thickness leads to a reduction in heat flux at the bottom of the plate as compared to the top.

Nusselt's Theory and Heat Flux

According to Nusselt's theory of film condensation, for laminar flow on a vertical plate, the film thickness (δ) is given by the equation: \[ \delta = \left( \frac{4 \nu_{L} x}{g \Delta T} \right)^{1/4} \] where: - \( \delta \) is the film thickness - \( \nu_{L} \) is the kinematic viscosity of the liquid - x is the distance from the top of the plate - g is the acceleration due to gravity - \( \Delta T \) is the temperature difference between the film and the cold surface Since the film thickness (δ) increases with x (distance from the top of the plate), the heat flux decreases as we move from the top to the bottom of the plate.

Conclusion

In conclusion, the heat flux will be higher at the top of the vertical plate compared to the bottom. This is because the liquid film becomes thicker as it moves downward due to gravity, and the thickness of the film plays a major role in determining the heat flux.

Key concepts

Heat Flux

Heat flux is a term used to describe the rate of heat energy transfer through a surface. In the context of film condensation on a vertical plate, it is crucial to understand how this transfer occurs. The rate at which heat moves depends on several factors, such as the temperature difference across the film, the thermal conductivity of the material, and the thickness of the liquid film.

The thinner the film, the higher the heat flux, since a thinner film means heat can be conducted more efficiently to the cold surface. Now, imagine the process of condensation at the top of the plate. Here, the film starts thin, allowing a larger heat flux. As it travels down the plate, gravity causes the film to thicken, reducing the rate of heat transfer.

Vertical Plate

The vertical plate in our discussion serves as a surface for condensation. In heat transfer studies, vertical plates are often used to observe how fluids behave under the influence of gravity.

In film condensation scenarios, a vertical plate allows us to see how gravity affects the thickness of a liquid film. At the top of the plate, the film is at its thinnest. As the liquid film makes its way down, it gains thickness due to the accumulation of the liquid above it. This change in thickness is crucial for understanding how heat transfer rates alter from the top to the bottom of the plate.

Nusselt's Theory

Nusselt's theory provides us with a mathematical approach to understand film condensation over a surface. It helps in predicting the thickness of the film, assisting in determining the heat transfer efficiency. According to Nusselt's theory, the film thickness (\( \delta \)), in this case, increases with distance along the vertical plate due to gravitational forces.

The formula \[ \delta = \left( \frac{4 u_{L} x}{g \Delta T} \right)^{1/4} \]demonstrates that the film gets thicker as you move down the plate (where \( x \)is larger). Consequently, with greater film thickness, less heat flows through, impacting the overall heat flux.

Gravity Effect on Thermal Processes

Gravity is a silent yet powerful force influencing thermal processes like film condensation on a vertical plate. When a vapor condenses, it forms a liquid that is pulled downward by gravity. This creates a flow that increases the film thickness as it descends down the plate.

This thickness affects thermal conductivity, as a thicker film acts as an insulating layer, slowing down heat transfer. The interplay between gravity, plate orientation, and liquid properties defines how efficiently heat can be evacuated from the system. Understanding gravity's impact helps engineers optimize designs for better thermal management across various applications.