tank of volume (
V
) 100 m
3
is ?tted with two pumps, one at the entrance and the other
at the exit of the tank. Each pump is a positive displacement pump, i.e., it pumps at a
?xed volume ?ow rate of the ?uid entering the pump. The in?ow pump sucks air from
the atmosphere and delivers it to the tank, while the out?ow pump sucks air from the
tank and delivers it to a pipeline.
Initially, neither pump is working and the vessel is ?lled with air at atmospheric density
?
a
=
1.2 kg/m
3
. At t=0 both pumps are switched on, instantaneously drawing a volume
?ow rate
Q
in
=
2 m
3
/s into the in?ow pump and
Q
out
=
1 m
3
/s into the out?ow pump.
Because the mass in?ow and out?ow rates are different initially, the density
?
(
t
)
of the air
in the tank changes with time. After a long time, however the air density inside the tank
reaches a steady value.
(a) Calculate the value of the air density
?
s
in the tank after the steady state has been
reached, (b) Derive and solve a differential equation for
?
(
t
)
for any time
t
?
0.
2 of 2
