To calculate the air velocity in a duct using a Pitot-static tube, we apply Bernoulli’s principle and account for the manometer reading, which gives the pressure difference (dynamic pressure) between the total and static pressures.
The Correct Answer and Explanation is:
Given:
- Manometer reading = 8 cm of water = 0.08 m of water
- Static pressure = 9 kN/m² = 9000 N/m²
- Air temperature = 320 K
- Barometric pressure = 740 mm Hg = $740 \times 133.322 = 98657 \, \text{Pa}$
- Gas constant for air = 287 J/kg·K
- Coefficient of velocity (Cv) = 0.98
- Density of water ≈ 1000 kg/m³
- Acceleration due to gravity = 9.81 m/s²
Step 1: Calculate differential pressure from manometer
Dynamic pressure (ΔP) = ρ_water × g × h
$$
ΔP = 1000 \times 9.81 \times 0.08 = 784.8 \, \text{Pa}
$$
Step 2: Find air density (ρ_air) using ideal gas law:
$$
ρ = \frac{P}{RT} = \frac{9000}{287 \times 320} = \frac{9000}{91840} \approx 0.098 \, \text{kg/m}^3
$$
Step 3: Calculate theoretical velocity (V) from dynamic pressure:
$$
V = \sqrt{\frac{2 \times ΔP}{ρ}} = \sqrt{\frac{2 \times 784.8}{0.098}} = \sqrt{16016.3} \approx 126.5 \, \text{m/s}
$$
Step 4: Apply velocity coefficient (Cv):
$$
V_{\text{actual}} = C_v \times V = 0.98 \times 126.5 \approx \boxed{124 \, \text{m/s}}
$$
Explanation (300 words):
A Pitot-static tube measures the fluid velocity based on the difference between the stagnation (total) pressure and the static pressure. When air flows into the Pitot tube, it is brought to rest, causing the pressure to rise — this rise is captured by the manometer in terms of the height of a liquid column, which here is 8 cm of water.
To determine the velocity, the pressure difference (ΔP) is first calculated using the hydrostatic pressure formula:
$$
ΔP = \rho_{\text{water}} \cdot g \cdot h
$$
This gives the dynamic pressure caused by the velocity of air in the duct.
Next, we calculate the air density using the ideal gas law, which is essential because air density changes with pressure and temperature. The static pressure (9000 Pa) and temperature (320 K) allow us to compute this density accurately using the gas constant for air (287 J/kg·K).
Then, we apply Bernoulli’s equation to solve for velocity based on dynamic pressure:
$$
V = \sqrt{\frac{2 \cdot ΔP}{ρ_{\text{air}}}}
$$
This gives the theoretical velocity. However, real-world readings are affected by non-idealities like tube friction and turbulence. Thus, we multiply the result by a velocity coefficient (Cv), typically slightly less than 1, to obtain the actual velocity.
In conclusion, after incorporating all these parameters, the air velocity in the duct is approximately 124 m/s, providing a reliable estimate of flow speed using practical fluid mechanics principles.