How do you work out the energy requirement for a submerged pump?

Question

I was wondering if you were to place a pump in a submerged container and then were to open a hatch allowing water to flood the previously vacant cavity how much energy would be required to discharge the water at the entry rate. What I would like to know also whether or not the depth would have any influence on the pump's output and efficiency. I understand submersible vehicles rely on the flooding and flushing of water from their hulls to dive and ascend but am curious if pumps must do more work when at depth. Please and thank you.

Answer 1

Suppose we have a pump submerged in a liquid of density $\rho$ at a depth $d$ which we want to use to raise the liquid to a height $h$. The pump has to do work to overcome the potential energy difference between the two heights of liquid. That is:

$$W(t)=\rho Qg (h-d)t$$

Where $g$ is the acceleration due to gravity (taken as $9.81m/s^2$). $Q$ is the volumetric flow rate (in $ m^3/sec$) of liquid being pumped and $ t $ is the time (in sec).

This would give us the minimum energy required. In practice, pumps have losses which are represented as 'percentage efficiency', $\eta$ so the actual mechanical energy required is:

$$E_M=W/ \eta$$

For example, the efficiency of a centrifugal pump might be $\eta=0.65$ (65%) due to losses in the pump such as turbulence. The actual efficiency of a pump will vary with flow rate $q$ and can be obtained from a manufacturers data sheet.