Mechanics of Fluids

286 Flow and losses in pipes and fittings
Fig. 7.25
where a represents the cross-sectional area of the orifice and h the head above
it. Again A is much greater than a. The orifice size and the Reynolds number
of the flow are both usually large enough for Cd to be assumed constant.
For the special case of a reservoir with constant cross-section and with no
inflow
t =
� t
0
dt =
� h2
h1
−Adh
C da � (2g)h =
2A
Cda � (2g) (h1/2
1
− h1/2
2 ) (7.35)
If two reservoirs are joined by a pipe, both ends of which are submerged
(Fig. 7.25), the rate of flow between them is determined by the difference in
the reservoir surface levels. As one surface falls, the other rises, so
A1
�
− dz1
dt
�
dz2
= Q = A2
dt
The rate at which the difference of levels changes is
dh d
=
dt dt (z1 − z2) = dz1
�
A1
− −
dt A2
dz1
�
=
dt
dz1
�
1 +
dt
A1
�
A2
The continuity equation gives
�
A1 − dz1
�
= au = anh
dt
1/2
(from eqn 7.33)
Substitution from eqn 7.36 gives
−A1 dh
= anh1/2
1 + (A1/A2) dt
(7.36)
The relation now has a single head variable h and may be integrated in the
usual manner.
In any problem containing more than one head variable, care should be
taken to distinguish between them, and integration should not be attempted
until the principle of continuity has been used to express all heads (and
differentials of heads) in terms of a single variable.
Example 7.6 Two vertical cylindrical water tanks, each open to
atmosphere and of diameters 3 m and 2 m respectively, are connected
by two pipes in parallel, each 50 mm diameter and 75 m long. Initially
the water level in the larger tank is 1.8 m above that in the smaller.