Measuring velocity in a pitot-static tube and manometer

[JJBladester]

Homework Statement

A Pitot-static probe connected to a water manometer is used to measure the velocity of air. If the deflection (the vertical distance between the fluid levels in the two arms) is 7.3 cm, determine the air velocity. Take the density of air to be 1.25 kg/m³.

Homework Equations

http://composmentisconsulting.com/12-36.jpg

Pressure in a manometer = ρ_fluidgh

Dynamic pressure from the Bernoulli equation = $\rho \frac{V^2}{2}$

The Attempt at a Solution

I reason that the height change in the water of the manometer is due only to the dynamic pressure entering the device. Therefore,

$$\rho_{air} \frac{V^2}{2}=\rho_{H_{2}o}gh$$

$$V=\sqrt{2\frac{\rho_{H_{2}O}}{\rho_{air}}gh}$$

$$V=\sqrt{2\left (\frac{1000}{1.25} \right )\left (9.81m/s^2 \right )(.073m)}=33.8m/s$$

I think I'm doing the math correctly. I just want to be sure my original assumption (that the pressure in the manometer = dynamic fluid pressure) is correct.

 

[Adam Nur]

But, how can I relate this to the Bernoulli's equation?

 

[rude man]

But, how can I relate this to the Bernoulli's equation?

Well, if you don't get any other replies, my take is that you can't. Bernoulli applies to a streamline only, and the pitot and static pressure sensors are not in the same streamline.
Other takes solicited!

 

[JohnMoeDoe]

Assume your manometer is water. If you look at the manometer on both ends, then you'll see that
P₀₁ = P₀₂
P_s1 + 1/2*rho_water*V₁² + rho_water*g*h₁ = P_s2 + 1/2*rho_water*V₁² + rho_water*g*h₂
assuming that the change in height is only dependent on the static pressure,
dP_s,water = P_s2 - P_s1 = rho_water*g*(h₁ - h₂)

Now if you assume that the wind streamline starts at zero (1), and compare to the wind at the inlet to the manometer (2)
P₀₁ = P₀₂
P_s1 + 1/2*rho_air*V₁² + rho*g*h₁ = P_s2 + 1/2*rho_air*V₁² + rho*g*h₂
assume that the height is negligible
dP_s,air = P_s2 - P_s1 = 1/2*rho_air*(V₁² - V₂²)

Since the change is static pressure of the wind is directly affecting the change in static pressure at the manometer,
dP_s,air = dPs_,water
rho_water*g*(h₁ - h₂) = 1/2*rho_air*V₂²
If you measure the manometer's change in height, then you can find the wind velocity. So yes, your assumption is correct.