How do hydroelectric dams generate power?

[russ_watters]

But the piston is not moving if the fluid is incompressible.

Yes, that's my point: In order for the water to do work, the piston has to move.

And my question is, where exactly the error lies.

It lies in the fact that you're double-counting the gravitational potential energy. In the hyperphysics page on Bernoulli's equation, it lists 3 pressures:

-Static pressure
-Velocity pressure
-Hydrostatic pressure

The pressure at the bottom of a hydro dam is hydrostatic pressure, but you're making the mistake that if you can measure it with a pressure gauge, it must be static pressure, so you're double-counting it. For your example of putting a weight on a volume of water, you've substituted that weight for the extra water column, changing nothing: you're still double-counting gravitational potential energy by using the static pressure term when it doesn't apply.

I don't quite see how you can claim that pressure_energy and potential_energy represent the same quantity here. Why would you count the same thing twice?

You shouldn't: the equation isn't doing that, you are.

That is how I understand it too. And pressure energy is something different, that is accounted for separately.

Yes. And in both the case of the hydro dam and the case of your piston-cylinder-weight, the [static] pressure energy is zero (if you consider the weight to still have its gpe).

 

[rcgldr]

In the hyperphysics page on Bernoulli's equation, it lists 3 pressures:
-Static pressure
-Velocity pressure
-Hydrostatic pressure

At this web page, I see the description potential energy per unit volume for the term ρ g h, not the term "hydrostatic pressure":

http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html

Side note - the hyperphysics page also describes pressure as pressure energy, but pressure is energy per unit volume, the same as the other two terms in the equation.

hydrostatic pressure

Assuming I'm not misinterpreting this, hydrostatic pressure is the static pressure of a fluid due to gravity and is equal to ρ g h as described in this wiki article, but this is different than the ρ g h term used in Bernoulli's equation, which is a potential energy term:

http://en.wikipedia.org/wiki/Hydrostatic_pressure#Hydrostatic_pressure

Note that hydrostatic pressure increases with depth in a fluid, while the gravitational potential energy per unit volume term in Bernoulli's equation decreases with depth. Hydrostatic pressure corresponds to part or all of the static pressure term in Bernoulli's equation.