140 4 Perfect Flow
2
5
2
tan2
15
8
HCQ
d
¸
¹
·
¨
©
§
T
g
(10)
In engineering applications, the
range would be qddq 12022
where
60.|
d
C . For accurate measurement of discharge, meticulous calibration
is required for
d
C .
Exercise 4.1.4 Sudden Expansions, Contractions, Bend and Flow
In pipelines there are regions, where the pipe cross-sectional area changes
abruptly, as often seen at pipe joints. In long pipelines, total pressure loss
(head loss) is mostly caused by viscous loss along the pipelines, where
losses due to hydrodynamic effects, such as flow separations and secon-
dary circulations (vortices) in these abruptly changed cross-sectional area,
bends or other pluming (piping) parts, are normally small. Those losses
due to hydrodynamic effects are known as minor losses. However, in a
short pipe system, minor losses, are not to be neglected.
(i) Find an expression for the pressure loss (head loss) in a sudden expan-
sion in a pipeline, as shown in Fig. 4.12(a). Denoting that a pipe of cross-
sectional area
1
A of ٤
1
is connected to the sudden expansion part of cross-
sectional area
2
A of ٤
2
. The pressure at joint ٤
1
is
1
p and one at a down-
stream region of fully developed flow is
2
p . Assume uniform flow enters
the sudden expansion part
٤
1
with a velocity
1
u , and leaves the region at
٤
2
with
2
u . The fluid density is
.
(ii) Similar to (i), find an expression for the pressure loss (head loss) in a
sudden contraction in a pipeline, as shown in Fig. 4.12(b). Consider that in
the sudden contraction part the vena contract occurs so that the exit’s con-
dition is taken at region
٤
2
when the flow reaches fully developed uniform
velocity, where the pressure and velocity is
2
p and
2
u respectively.
(iii) There are many elements, which cause pressure loss (head loss) beside
frictional pressure loss (viscous loss) in pipelines. These are minor losses.
Explain a method to measure the minor loss in an experiment or to deter-
mine one from a numerical simulation result, taking an example of a
q90
bend in a flow channel as shown in Fig. 4.12(c).
Through Rotating Blades