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Question:

Is a perpetual energy water pump possible?

If you use a water pump to move water through a circular trench, essentially making a little river, and then place some kind of turbine or water wheel or something to generate electricity from the moving water, couldn't you use the electricity generated by the turbine to power the pump, thus requiring no outside power?

Answer:

Perpetual Water Pump
All the energy that you can get from the stream you have created will always be less than the one you are using to drive the pump that feeds the stream. In engineering field you must get into your brain that Perpetual. motion machines just does not exist. you can use it to understant thermodynamics processes when using in simulated problems but it is only ideal situations.
No, every energy conversion process is less than 100% efficient. In other words, when the pump and motor raise the water energy is lost. Now when the water flows more energy is lost. When the water turn the turbine more energy is lost. Finally, when the turbine drives the motor to produce electricity more energy is lost.
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