Head loss in turbulent flow in a pipe

In turbulent flow, the fluid moves in an irregular and chaotic manner, characterized by the formation of eddies and swirls. This type of flow is commonly observed at high velocities or when the fluid encounters obstacles or rough surfaces. The head loss in turbulent flow refers to the energy loss or pressure drop that occurs due to the friction between the fluid and the pipe wall.

Relationship between head loss and velocity

The head loss in turbulent flow is approximately proportional to the square of the velocity. This relationship is expressed by the Darcy-Weisbach equation:



- Hf: Head loss
- f: Darcy friction factor
- L: Length of the pipe
- D: Diameter of the pipe
- V: Velocity of the fluid
- g: Acceleration due to gravity

Explanation

1. Directly as velocity: If the head loss were directly proportional to the velocity, it would mean that as the velocity increases, the head loss would also increase. However, this is not the case in turbulent flow.

2. Inversely as the square of velocity: If the head loss were inversely proportional to the square of the velocity, it would mean that as the velocity increases, the head loss would decrease. This is not the observed behavior in turbulent flow.

3. Inversely as velocity: If the head loss were inversely proportional to the velocity, it would mean that as the velocity increases, the head loss would decrease. This is also not the observed behavior in turbulent flow.

4. Approximately as the square of velocity: The most accurate relationship between head loss and velocity in turbulent flow is that the head loss is approximately proportional to the square of the velocity. This means that as the velocity increases, the head loss increases significantly.

The square of the velocity term in the Darcy-Weisbach equation represents the kinetic energy of the fluid, which is converted into frictional losses as the fluid interacts with the pipe wall. The formation of eddies and swirls in turbulent flow results in a higher energy loss compared to laminar flow, hence the need for a squared velocity term in the equation.

In conclusion, the head loss in turbulent flow in a pipe is approximately proportional to the square of the velocity. This relationship highlights the significant impact of velocity on the energy loss due to friction in turbulent flow conditions.