Power needed for turbines and pumps

[praondevou]

I have a friend who insists in an idea and I want to prove to him that it cannot work because it would violate basic physical laws.

But somehow I don't find the right words.

He says if he drilled a well of 2km (or any other depth) into the Earth and at the bottom there was a pump that pumped water through a pipe back into to the head of the well he could create more electric energy than the pump would be using.

He would install a few, not just one, turbines into the well at different heights, where the sum of energy generated by all turbines is greater than the energy needed to pump all the water up.

Now according to this:

A simple formula for approximating electric power production at a hydroelectric plant is:
where P is Power in kilowatts, h is height in meters, r is flow rate in cubic meters per second, g is acceleration due to gravity of 9.8 m/s2, and k is a coefficient of efficiency ranging from 0 to 1.

the only parameter that is different for all turbines is the height. Unless the flow rate is changing.

This formula is similar for pumps. Leaving acceleration (fixed) and water density and losses aside the only parameters ar flow rate and height.

It looks like very easy to explain why it cannot work, yet...

Does anyone have a calculation example?

 

[Jobrag]

The formula quoted is for the energy available from an uninterrupted flow, assuming that each turbine is 100% efficient then at the outlet of each turbine the velocity flow will be zero and you have to plug inthe numbers for the drop to the next turbine.

 

[berkeman]

I have a friend who insists in an idea and I want to prove to him that it cannot work because it would violate basic physical laws.

But somehow I don't find the right words.

He says if he drilled a well of 2km (or any other depth) into the Earth and at the bottom there was a pump that pumped water through a pipe back into to the head of the well he could create more electric energy than the pump would be using.

He would install a few, not just one, turbines into the well at different heights, where the sum of energy generated by all turbines is greater than the energy needed to pump all the water up.

Now according to this:

A simple formula for approximating electric power production at a hydroelectric plant is:
where P is Power in kilowatts, h is height in meters, r is flow rate in cubic meters per second, g is acceleration due to gravity of 9.8 m/s2, and k is a coefficient of efficiency ranging from 0 to 1.

the only parameter that is different for all turbines is the height. Unless the flow rate is changing.

This formula is similar for pumps. Leaving acceleration (fixed) and water density and losses aside the only parameters ar flow rate and height.

It looks like very easy to explain why it cannot work, yet...

Does anyone have a calculation example?

We do not waste people's time discussing Perpetual Motion Machines (PMMs) here at the PF. Here is a quote from the PF Rules link at the top of the page, giving some links to old locked PMM threads to help you understand why they do not work.

Perpetual motion and "free energy" discussions

Search PF and you will find many threads that have been closed in a number of forums. As for S&D, any claim of this nature would be reproducible and/or testable by the scientific community; hence there is no need for debate.
EDIT by berkeman -- here are some recent locked PMM threads:
https://www.physicsforums.com/showthread.php?t=522548
https://www.physicsforums.com/showthread.php?t=520290
https://www.physicsforums.com/showthread.php?t=7735
https://www.physicsforums.com/showthread.php?t=515402
https://www.physicsforums.com/showthread.php?t=403572