Consider two identical pipes. One pipe is to be filled with water by suddenly raising the liquid…

Consider two identical pipes. One pipe is to be filled with
water by suddenly raising the liquid pressure upstream of the pipe to Po.
The other pipe is already filled with water. We would like to expel the water
from the pipe by suddenly increasing the upstream pressure of the gas to Po.
Find the time to fill and to drain each pipe and compare the results. The atmospheric
pressure for both cases is shown by Pf. Pipe area is A.

[Hint: For the filling the pipe example, write the momentum
equation for the control volume shown in the figure as _F = d(mV)/dt +
_(momentum flux). Since no
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Consider two identical pipes. One pipe is to be filled with
water by suddenly raising the liquid pressure upstream of the pipe to Po.
The other pipe is already filled with water. We would like to expel the water
from the pipe by suddenly increasing the upstream pressure of the gas to Po.
Find the time to fill and to drain each pipe and compare the results. The atmospheric
pressure for both cases is shown by Pf. Pipe area is A.

[Hint: For the filling the pipe example, write the momentum
equation for the control volume shown in the figure as _F = d(mV)/dt +
_(momentum flux). Since no mass is leaving, momentum flux at the exit is
zero and momentum flux at the inlet is the flow rate into the control volume.
The momentum equation then becomes:

(Po – Pf)A – Ffriction =
d(mV/dt) + (0 – mi V)

Relate mi to m from the continuity equation and substitute
for m from _Ax and for V from V = dx/dt].

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