Bernoulli's equation confusion?

[JuanC97]

Some days ago I got confused trying to solve an exercise about fluid dynamics. Trying to simplify the problem here is a similar situation:

I have a cistern connected to a tube containing a fluid as shown in the picture below.

Assumming that the fluid is incompressible...
I know from the law of continuity that every single point in the same area has to move with the same speed.

However, Bernoulli's equation applied to the point B says that:

P atm + rho*g*h = Patm + (1/2)*rho*V²
This implies that v = ( 2gh )^1/2. but... in the point C it will be v = ( 2g(h+d) )^1/2.

That means that every single point has a different speed despite being in the same cross-sectional area.
What's the right equation?. Am I doing something wrong?. Where's the mistake?

PD: Both sides are open and it is supposed that the velocity of every "control volume" inside the cistern tends to zero.

 

[Simon Bridge]

The height parameter in the equation refers to the height of the pipe not of a point in the pipe.
Remember - B's equation is a model not reality.

 

[JuanC97]

The height parameter in the equation refers to the height of the pipe

I'm not sure about that.
I know it's possible to derive Bernoulli's eq using conservation of energy; however, you have to use the heights of the points because they are associated with the potential energy of each control volume. (It's not properly the height of the pipe)

 

[Simon Bridge]

I'm not sure about that.

Suit yourself - but the answer stands.
Note: your diagram is not the usual one used for the derivation ...

If you want to redo the derivation by taking smaller control volumes and the forces on all of them then be my guest.

 

[Chestermiller]

Just imagine a very long exit channel. Throughout most of the length of this channel, the velocity profile will be flat and horizontal, and the pressure variation will be hydrostatic vertically. Only very close to the outlet will the velocity profile begin to readjust to the flat pressure variation at the exit. This is strictly an exit effect, and will occur within about 2-3 channel heights of the exit. The fluid may even lose contact with the upper wall of the channel in this region.

Chet

 

[boneh3ad]

I know from the law of continuity that every sin
Usgle point in the same area has to move with the same speed.

Well, part of the problem is that Bernoulli's equation does not require this to be the case. Bernoulli's equation is simply an expression of a simplified energy balance between two points.

However, Bernoulli's equation applied to the point B says that:

P atm + rho*g*h = Patm + (1/2)*rho*V²
This implies that v = ( 2gh )^1/2. but... in the point C it will be v = ( 2g(h+d) )^1/2.

That means that every single point has a different speed despite being in the same cross-sectional area.
What's the right equation?. Am I doing something wrong?. Where's the mistake?

What you have done is correct. You are throwing yourself off, though, since usually ##d## is very small compared to ##h##, so the difference in height between the top of the pipe and the bottom are neglected. In that case, then Bernoulli's equation would predict a constant velocity at a given distance along the pipe.