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310. A bathtub is filled with water from a faucet at a flow rate of 10 liters per minute.
The
volume of the bathtub is 25 gallons when the depth of the water is one foot.
How long will it take
to fill the bathtub?
Suppose the bathtub plug has a hole and water drains at a rate of
0.1
leak
q
=
h
where the leak flow rate is in liters per second and the water height inside the tank
is in meters.
Assume that the cross section of the tub is constant. How long will it take to fill the
bathtub with a leak?
310. The macroscopic mass balance for a moving control volume
()
tt
d
dV
dA
dt
ρ+ ρ
−
=
⋅
∫∫
vw
n
VA
0
(1)
and with the assumption of constant density leads to
t
dt
dA
dt
+−
=
⋅
0
∫
n
A
V
(2)
The normal component of the
relative velocity
,
−
⋅
n
, is zero everywhere except at
the entrance and exit, thus Eq. 2 simplifies to
0
o
i
dV t
QQ
dt
+− =
(3)
where Q
o
is the outing volume flow rate and Q
i
is the entering volume flow rate.
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 Winter '10
 B.G.Higgins

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