Bernoulli's principal and pressure difference due to altitude

[swaise]

Bernoulli's principal states that as velocity of a fluid increases, the static pressure decreases. I wonder if the reverse order can also happen. For example, if there is a pipe laid on the side of a mountain that is 3000 ft high. The bottom end of the pipe would be exposed to pressure at sea level, while the top would be exposed to a pressure much lower than sea level.

If we apply Bernoulli's equation, the indicated velocity is quite high. Yet intuitively, I can't believe this could be true.

How should I look at this concept?

 

[rcgldr]

Bernoulli's principal states that as velocity of a fluid increases, the static pressure decreases.

That's only true when there is no external work done during the transition when the fluid accelerates or declerates. The typical example is fluid in a pipe of varying diameters, with the assumption that the pipe does not perform any work on the fluid.

The pressure difference related to altitude is due to gravity, not speed. There's a third term in the Bernoulli equation, density x (gravitational acceleration) x height. Wiki article:

http://en.wikipedia.org/wiki/Bernoulli's_principle

 

[K^2]

Bernoulli equation is an energy balance equation, so yes, potential energy must be included. You can get a rough estimate by considering velocity you get from simple Bernoulli equation as what you need to get to the altitude in the v²=2gh sense, but this ignores the temperature changes, so it will only work for small altitude changes. In a more general form Bernoulli equation for atmosphere will include temperature gradients and gravitational potential.

 

[swaise]

Thank you for the replies. I have a second question.

See the attached picture for details. Let's assume we have a pipe that extends from sea level to 3000 meters. Let's also assume the the whole pipe is filled with air at sea level pressures and the bottom is open to the atmosphere while the top is sealed, as show in step 1 - left column.

If then, we go through the cycles of sealing and unsealing the segments as show in the picture. What would the final pressure be? At the location where the "?" is placed. Would it be P2 or something lower than P2 since gravity is lower at this level?

 

[K^2]

The air inside the pipe will have same gradient as atmosphere. If you close it at the top and leave it open at the bottom, the pressure will be P2 at the bottom and P1 at the top, with everything in between in the middle.

 

[swaise]

Really? I am so disappointed in nature...

Just to solidify my understanding of this concept. Imagine a bicycle air pump. That has a really good seal and is 3000 ft high. If we pull the pump up while the air is drawn from the bottom at sea level, would the pressure at the top of the cylinder at 3000 feet be ambient pressure or sea level pressure?

 

[K^2]

Give or take some corrections for weather phenomena, it will be ambient pressure at 3k feet.