Problem 37 solution
According to the problem statement, the volume of the horizontal cylindrical tank is given by
(1)
V
= p
L R
2

L R
2
ÅÅÅÅÅÅÅÅÅ
2
H
a 
sin
a
L
where
a
is a function of time. From the geometry we can use trigonometry to relate h to
a
:
(2)
h
=
R
+
R cos
I
a
ÅÅÅÅ
2
M
It follows then that
h=0 when
a
ê
2
=
180
° . and h=2R when
a
/2=0°
We select a moving control volume
(t) that encloses the liquid in the tank at any given instant in time with a cut
where the liquid enters the tank. The macroscopic mass balance for this moving control volume is
(3)
„
ÅÅÅÅÅÅÅ
„
t
‡
H
t
L
r „
V
+
‡
H
t
L
r
H
v

w
L
ÿ
n
„
A
=
0
and the assumption that the density is constant leads to
(4)
„
ÅÅÅÅÅÅÅÅ
„
t
+
‡
H
t
L
H
v

w
L
ÿ
n
„
A
=
0
The normal component of the relative velocity
H
v

w
L
is zero everywhere except at the entrance cut where the
liquid flow into the control volume. Thus we have
(5)
„
ÅÅÅÅÅÅÅÅ
„
t
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 Winter '10
 B.G.Higgins
 Expression

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