Question

A piezometer and a pitot tube measure the static and the total pressure of a fluid in a pipe flow respectively. The piezometer reads 100 kPa and the pitot tube shows 200 kPa. The density of the fluid is 1000 kg/m\(^3\). The velocity of the flow is .................... m/s (round off to one decimal place)

Hint:

The Pitot tube formula is a direct application of Bernoulli's equation. Remember that the difference between total and static pressure gives the dynamic pressure, \( \frac{1}{2}\rho v^2 \). Be sure to use pressures in Pascals (Pa) and density in kg/m\(^3\) to get the velocity in m/s.

Answer 1

Step 1: Understanding the Concept:
This problem uses the principle behind a Pitot tube to measure fluid velocity. A Pitot tube measures the total pressure (or stagnation pressure) at a point in the flow, while a piezometer (or a static pressure tap) measures the static pressure. The difference between these two pressures is the dynamic pressure, which is directly related to the fluid's velocity. This relationship is derived from Bernoulli's equation.
Step 2: Key Formula or Approach:
According to Bernoulli's principle for a horizontal streamline between a point in the free stream and the stagnation point at the tip of the Pitot tube: \[ P_{static} + \frac{1}{2}\rho v^2 = P_{total} \] The difference between the total and static pressure is the dynamic pressure: \[ P_{dynamic} = P_{total} - P_{static} = \frac{1}{2}\rho v^2 \] We can rearrange this formula to solve for the velocity, \(v\).
Step 3: Detailed Calculation:
Given values:
- Total pressure, \(P_{total} = 200\) kPa = \(200,000\) Pa
- Static pressure, \(P_{static} = 100\) kPa = \(100,000\) Pa
- Density of the fluid, \(\rho = 1000\) kg/m\(^3\)
First, calculate the dynamic pressure:
\[ P_{dynamic} = 200,000 \text{ Pa} - 100,000 \text{ Pa} = 100,000 \text{ Pa} \] Now, solve for the velocity \(v\): \[ 100,000 = \frac{1}{2} \times 1000 \times v^2 \] \[ 100,000 = 500 \times v^2 \] \[ v^2 = \frac{100,000}{500} = 200 \] \[ v = \sqrt{200} \approx 14.142 \text{ m/s} \] Rounding off to one decimal place: \[ v = 14.1 \text{ m/s} \] Step 4: Final Answer:
The velocity of the flow is 14.1 m/s.