Velocity Measurement Using Pitot-Static Tube
How can we determine the velocity of helium in a pipe using a Pitot-static tube?
Given the temperature and pressure as 44°F and 24 psia, along with a manometer reading of 3.0 in, how can we calculate the helium velocity and determine if the flow can be considered incompressible?
Answer:
(a) The helium velocity is approximately 2811.62 ft/s.
(b) No, it's not reasonable to consider the flow as incompressible due to the high Mach number (Ma ≈ 2.58), indicating compressibility effects.
Explanation:
To determine the helium velocity using a Pitot-static tube, you can use the Bernoulli's equation, assuming steady, incompressible flow:
P + 0.5 * ρ * V^2 + ρ * g * h = constant
Where: - P is the pressure in the pipe (psia) - ρ is the density of the fluid (helium in this case, lb/ft^3) - V is the velocity of the fluid (ft/s) - g is the acceleration due to gravity (32.2 ft/s^2) - h is the height difference in the manometer (inches)
Given: - Temperature (T) = 44°F = 44 + 460 = 504 Rankine (R) - Pressure (P) = 24 psia - Reading in manometer (h) = 3.0 inches
First, we need to find the density (ρ) of helium at the given conditions. You can use the ideal gas law: ρ = P / RT
Substitute the values to find ρ: ρ = 24 / (53.34 * 504) = 0.0008933 lb/ft^3
Now, we can calculate the velocity (V) using Bernoulli's equation: V = √(2 * (P - manometer correction) / ρ)
The manometer correction accounts for the density of the manometer fluid, which is typically water in this case. Since we're given that the manometer reading is in inches, we need to convert it to feet: Manometer Correction = 3.0 inches / 12 = 0.25 ft
Calculate the velocity: V = √(2 * (24 - 0.25 * 62.4) / 0.0008933) = 2811.62 ft/s
(a) The helium velocity is approximately 2811.62 ft/s.
(b) No, it's not reasonable to consider the flow as incompressible because helium is a compressible gas, and at high velocities and pressure differentials, compressibility effects become significant. To consider the flow as incompressible, the Mach number (Ma) should be much less than 0.3.
To calculate Ma:Ma = V / a Where a is the speed of sound in helium, which can be calculated using:a = √(γ * R * T) Calculate a and then Ma: a = √(1.66 * 53.34 * 504) = 1087.92 ft/s Ma = 2811.62 / 1087.92 ≈ 2.58 Since the Mach number is significantly greater than 0.3, the flow of helium in this case cannot be considered incompressible.