Bernoulli equation - pitot tubes

[yy205001]

Homework Statement

Homework Equations

P₁+pV₁²/2+pgh₁=P₂+pV₂²/2+pgh₂

The Attempt at a Solution

My thinking: since the pitot tubes measure the stagnation pressure (static + dynamics pressure) and the height of the tubes are the same. By Bernoulli's equation, the total pressure along a streamline is unchanged, therefore there is no pressure change between two tubes which gives h=0.

I just want to check is my thinking correct or not.
Thank you![/B]

 

[andrevdh]

The speed of the airflow at 2 will be higher than at 1 so that P₂ < P₁
so that h will be as indicated in the diagram - the water level higher
on the left than on the right.

 

[yy205001]

but isn't that P only the static pressure? But in this case we also have to include the dynamics pressure as well, so the total pressure
P+(p/2)V² stay the same everywhere on the streamline?

 

[andrevdh]

Bernoulli's equation describes fluid in motion so that P is the pressure in the fluid.
The pressure is transferred to the air and water in the manometer.
The 1/2 ρ v² term is the kinetic energy term of the equation
while the P term is the energy of the pressure (per unit volume of the fluid) and the
ρgh is the potential energy term of the equation.

 

[HalfLostAndSearching]

Your thinking is partially correct. The Bernoulli equation states that the total pressure along a streamline remains constant. This means that the sum of static pressure, dynamic pressure, and gravitational potential energy will be the same at any two points along a streamline. However, this does not necessarily mean that the pressure at two points will be the same.

In the case of pitot tubes, the height of the tubes does not necessarily have to be the same. The important factor is that the stagnation pressure is measured at the tip of the tube, where the velocity is zero. This means that the dynamic pressure will be zero at this point, and the total pressure will be equal to the static pressure.

The difference in height between the two tubes will affect the measured static pressure, as the pressure will decrease with increasing height. However, since the dynamic pressure is zero at both points, the difference in height will not affect the measured stagnation pressure.

In summary, the Bernoulli equation does not state that there is no pressure change between two points, but rather that the total pressure remains constant along a streamline. The height of the pitot tubes may affect the measured static pressure, but it will not affect the measured stagnation pressure.