# Pump Power

There’s been some discussion of pump power. It is extremely easy to calculate it when using SI units.

A flow front in a tube has a power:

$W=Fv$

W power, F force, v speed.

The force is

$F=AP$

Where A is cross sectional area and P the pressure.

Also,

$\dot{V}=Av$

Where V is volume so its time derivative is the volume flow.

Thus we can combine these and get

$W=Fv=APv=P\dot{V}$.

1 kg/s at water density is a volume flow of 1E-3 m^3/s. If pressure is 10 bars, that’s 1 MPa (Megapascal, 1 Pascal is 1 N/m^2) in SI, and thus the power is 1000 W or 1 kW.

1 kW for 90 seconds means 25 Watt hours (I know, I know, this is a horrible unit of energy). With modern LiFe batteries that have 100 Wh/kg energy densities (J/kg could often be more useful measure), a quarter of a kg battery could power the rocket for 90 seconds. With an exhaust velocity of 2 km/s, 1 kg/s gives a thrust of 2 kN. Thus the rocket could be in the 200 kg mass category. The battery mass seems negligible.

Of course, in real life the pump efficiency is a small number and thus plays a big factor, and the pump times are longer too.

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