Unnecessary Pump in a Frictionless Pipe: Impact on Pressure and Work

[msanx2]

Imagine a pump is installed in a pipe where there would be no need for it (no friction, no height difference, etc.). In order for the energy balance to be correct, does this imply that the fluid would experience friction inside the pump the same as the head provided by the pump?

 

[BvU]

What energy balance ? Does it show anything with friction ?

 

[msanx2]

Suppose I would use Bernoulli equation to compare point 1 before the pump and point 2 after the pump:

$$ \frac{P_{1}}{\rho g} + \frac{v_{1}^2}{2g} + z_{1} + \Delta h_{p}= \frac{P_{2}}{\rho g} + \frac{v_{2}^2}{2g} + z_{2} + \Delta h_{f} $$

By the continuity equation, $$ v_{1} = v_{2} $$

Moreover, neglecting the pressure drop across the pump, $$ P_{1} = P_{2} $$

So, the previous equation becomes:
$$ \Delta h_{p}= \Delta h_{f} $$

If this analysis is correct, does it show that, when a pump is not needed (no energy input would be needed to overcome any fittings or friction in the pipe), the head that it produces is the same as the possible friction that it produces? I'm imagining this situation as if the pump stops the fluid and accelerates it again.

Thanks!

 

[BvU]

$P_{1} = P_{2}$ is a show stopper: it means the pump does no work. Conversely, if $P_{1} < P_{2}$ the pump does $pV$ work

I get the feeling I misunderstand your scenario: if there is a flow regulator further down the pipe and an unneeded pump is installed and turned on, the velocity stays the same but the pressure increases. Does that qualify ?