Pressure measurement using a Piezometer

[fog37]

Hello everyone,

I understand the hydrostatic fluid pressure in static fluids: static pressure varies linearly with depth and the infinitesimal fluid parcels located at the same depth in the fluid experience the same isotropic hydrostatic pressure. By Pascal principle, the atmospheric pressure at the liquid's free surface must added to the hydrostatic pressure.

Let's now consider a liquid flowing with speed v inside a horizontal pipe of diameter D with piezometers (small vertical tubes open to the atmosphere) connected to the pipe. The height reached by the fluid inside the piezometric tubes correlates with the pressure at the base of the tubes. But what pressure does the height indicate? Does it indicate the total pressure, the static pressure or the dynamic pressure? I know that, physically, there is only one type of pressure but it is sometimes convenient to separate it in different contributions.
Since the piezometric tubes are vertical (perpendicular to the flow direction), I think the height reached by the fluid inside the piezometric tubes only indirectly measures the static pressure without the kinetic pressure contribution coming from the fluid motion. Is that correct? If that is the case, would this measured static pressure be the same as the pressure the fluid was not moving inside the pipe?

To measure the total pressure we would need to a stagnation point where the moving fluid impacts and exerts the total pressure...

Thanks!
Fog37

 

[Useful nucleus]

In the case of inviscid and irrotational flow, you can apply Bernoulli equation between a point at the base of the Piezometer and a point at the center of the pipe. They both have same velocity (inviscid flow results in uniform velocity distribution). The only difference between them is the height difference which is the radius of the pipe in this case. Thus you end up with
P_{pipe center} = P_base + ρgr
Thus, what you measure is the static (thermodynamic) pressure at the center of the pipe. As you said, if you want the so-called total pressure, you need a stagnation tube that obstructs the flow in the pipe and allows you to convert the kinetic energy carried by the flow to a static fluid column.

 

[fog37]

Thanks Useful nucleus.

Glad we agree, I think, that the height reached by the fluid inside the vertical piezometric tubes correlates with the static pressure in the inviscid fluid itself and it is the same as the static pressure that would exist if the fluid had zero velocity...

 

[fog37]

Now that I understand that better, I have a related dilemma:

In the Torricelli experiment, fluid flows out of an orifice near the bottom of a container full of water. Considering an ideal, inviscid fluid, let's connect a long horizontal pipe to the orifice. The fluid will flow inside that pipe and eventually exit in the open air. If piezometric tubes were placed along this long horizontal tube, the fluid would reach NO height inside the piezometric tubes. But why? How do we justify that using Bernoulli's equation? The piezometric measure the gauge pressure so zero height would imply zero gauge pressure and the static pressure inside the fluid flowing along the pipe would be equal to atmospheric pressure... That seems strange. I know that an ideal fluid has no viscosity so it there would be no pressure drop (change in fluid height in the piezometric tubes) but I am surprised that the height is zero!

I the fluid had nonzero viscosity instead, the piezometric tube would have fluid reaching a nonzero height with the piezometric tubs closer to the container orifice having liquid reach a higher height (viscosity causes a pressure drop, hence a progressive decrease in fluid height in the tubes).

Thanks for any helpful hint.
Fog37

 

[Useful nucleus]

I think in the ideal flow there will be pressure inside the pipe that you connect to the container and the pressure will remain constant along the pipe. However, at the exit a jet will form, and the pressure at the last point in the jet will be the atmospheric pressure. So the jet starts compressed and then expands until it reaches to the atmospheric pressure.

 

[fog37]

Hi useful nucleus,

I agree with you but apparently the pressure inside the entire tube is also atmospheric and that can be proven using Bernoulli's equation. Not sure why though. The liquid would rise to zero height inside the piezometers apparently...

 

[Useful nucleus]

Hi fog37,

Please can you elaborate how did you apply Bernoulli eqn in this case and obtained zero pressure inside the pipe?

 

[fog37]

Sure.

Since the pipe cross-section is uniform and pipe' height is the same at every point, the fluid is inviscid and the pressure at the exit point of the pipe is the atmospheric pressure, the velocity is the same at every point inside the pipe and the pressure is also constant and equal to the atmospheric pressure. That is why the piezometers register zero fluid height (zero gauge pressure)...Does that make sense? But real fluids always have some minimal viscosity...

 

[Useful nucleus]

But if you apply exactly the same analysis to the side of the pipe that intersects with tank that contains water, then you will conclude that the pressure in the pipe is ρgh where h is the distance between the surface of water in the tank and the center-line of the pipe. Water will continue to flow in the pipe under constant velocity ~ √2gh with constant pressure. Once it reaches to the exit that opens at the atmosphere, a jet will come out. The jet pressure decreases gradually until it reaches to the atmospheric pressure it the end of the jet. This sound more logical to me compared to zero gage pressure inside the pipe. Notice, that I did not need to invoke any arguments related to losses or friction inside the pipe. It is all based on ideal flow.

 

[fog37]

Well, I see you point very clearly but my result, although strange, seem correct even (found it in several textbooks)...I see if I can provide more solid arguments. Maybe someone more knowledgeable will chime in...