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Difficulty: MediumFluid Dynamics and Bernoulli's Principle

A steady stream of water flows through a horizontal pipeline of varying diameter. Four vertical piezometer tubes (standpipes open to the atmosphere) are installed along the pipe to measure the local static pressure. The height of the water column in each tube is observed as follows:

- Station 1: 1.21.2 meters
- Station 2: 2.52.5 meters
- Station 3: 0.80.8 meters
- Station 4: 1.81.8 meters

Based on Bernoulli's principle, arrange the stations in order from the fastest fluid velocity to the slowest fluid velocity.

  1. 1Station 3 (0.80.8-meter water column)
  2. 2Station 1 (1.21.2-meter water column)
  3. 3Station 4 (1.81.8-meter water column)
  4. 4Station 2 (2.52.5-meter water column)

Answer

The correct order from fastest to slowest velocity is: Station 3, Station 1, Station 4, Station 2.
According to Bernoulli's principle, for an incompressible fluid flowing through a horizontal pipe, an increase in fluid velocity occurs simultaneously with a decrease in static pressure. The height of the water in a vertical piezometer tube is a direct measure of the static pressure at that location. Therefore, the station with the lowest water column (Station 3, 0.80.8 m) has the lowest static pressure and the fastest velocity. The station with the highest water column (Station 2, 2.52.5 m) has the highest static pressure and the slowest velocity. Sorting from lowest to highest water column gives the correct sequence from fastest to slowest velocity.

Step-by-Step Solution

1
Understand the relationship between water column height and static pressure.
The height of a fluid column in a piezometer tube directly indicates the static pressure at that point (P=ρghP = \rho g h).
Establishing what the given measurements represent.
2
Apply Bernoulli's principle to relate static pressure and fluid velocity.
In a horizontal pipe, regions with higher static pressure correspond to slower fluid velocities, and regions with lower static pressure correspond to faster fluid velocities.
This inverse relationship is the core concept of Bernoulli's principle for steady, horizontal flow.
3
Rank the static pressures based on the water column heights.
Lowest pressure to highest pressure: Station 3 (0.80.8 m), Station 1 (1.21.2 m), Station 4 (1.81.8 m), Station 2 (2.52.5 m).
To find the order of fluid velocities, we first need to order the static pressures.
4
Determine the final order of fluid velocities.
Since the fastest velocity occurs at the lowest pressure, the order from fastest to slowest is: Station 3, Station 1, Station 4, Station 2.
Translating the pressure ranking into the final velocity ranking required by the question.

Key Concept

Fluid Dynamics and Bernoulli's Principle
Estimated Time:1m 30s
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Topics

Fluid Dynamics and Bernoulli's Principle

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