Mechanics of Fluids

So
For two pipes in parallel, and similar assumptions,
h f = 32fQ2 1 l1
π 2 gd 5 1
�
π
Q1 + Q2 =
2ghf 32f
�
π
=
2ghf 32f
� ⎧
1/2 �
⎨ d5 1
⎩ l1
�1/2 �
= 32fQ2 2 l2
π 2 gd 5 2
� 1/2
d5 �1/2 l
� � ⎫
d5 1/2⎬ 2
+
⎭
for the equivalent pipe.
For a first approximation, pipes in which the flow rate is obviously small
may be supposed closed.
7.8.5 Pipe with side tappings
Fluid may be withdrawn from a pipe by side tappings (or laterals) along
its length. Thus, for a constant diameter, the velocity, and hence the slope
of the total head line, varies along the length. If the side tappings are very
close together the loss of head over a given length of the main pipe may be
obtained by integration of the equation
dhf = 4f dl u
d
2
2g
l2
(7.27)
between appropriate limits. In the general case, integration might require,
for example, a graphical or numerical method in which values of 4fu2 /2gd
are expressed as a function of l. However, if the tappings are uniformly
and closely spaced and are assumed to remove fluid at a uniform rate q with
respect to the distance along the main pipe, the volume flow rate at a distance
l from the inlet is Q0 − ql, where Q0 denotes the initial value. For a uniform
cross-sectional are A:
Q0
= A = Q0 − ql
u
so
u0
�
u = 1 − ql
�
u0
Q0
Substitution for u in eqn 7.27 and integration from l = 0tol = l gives
h f = 4f
d
where f is assumed constant.
u2 0 Q0
2g 3q
�
1 −
�
1 − ql
Q0
� 3 �
(7.28)
Pipes in combination 281