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In hydraulic analogy one compares electrical circuits with water circuits. For the electric case the formula $P = U \cdot I$ for the electric power holds. The analogous formula for water flow would be $P = \Delta p \cdot I_W$ where $\Delta p$ ist the pressure difference and $I_W$ the flow rate of the water through the pipe. I have some questions about this:

  • under what circumstances/assumptions does this analogous formula hold
  • $P$ in the electric case can be interpreted as the energy per second which is dissipated for example in a resistor. Is there a similar interpretation in the water case and why does it hold?
  • with the assumptions from above, how can one derive the formula from first principles (e.g. from Bernoulli-equation or even from Navier-Stokes)?
  • with the assumptions from above, is there a nice conceptual argument, why the formula holds in the water case?
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Power is the rate of transfer of energy i.e. it's the rate of doing work. When you write $Power = VI$ you are assuming that there is some device consuming energy, and $V$ is the voltage drop across this device. The device could be a resistor, that just turns the energy into heat, or it could be something like an electric motor that uses the energy to do work on something else.

So in the hydraulic analogy you just have to work out the rate that work is done. If you have some pressure $P$ and the area of your pipe is $A$, then the force is $F = PA$. If the linear flow rate of the water is $L$ m/sec the work done per second (force times distance) is just $W = PAL$. $AL$ is just the volume flow rate, $I_w$. So the power, i.e. work per second is just:

$$ Power = PI_w $$

hence the analogy with the electric circuit. As with the electric circuit the power could be used, e.g. in a water wheel, to do work on something else or it could be converted to heat e.g. by putting in something like a filter that obstructs the water flow.

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