E1C10 09/14/2010
13:4:37 Page 428
shown. Under the assumptions of incompressible, steady and one-dimensional flow with no external
energy transfer, the energy equation is
p 1
g
þ
U
2
1
2g
¼
p 2
g
þ
U
2
2
2g
þ h L 1À2
ð10:5Þ
where h L 1À2 denotes the head losses occurring between control surfaces 1 and 2.
For incompressible flows, conservation of mass between cross-sectional areas 1 and 2 gives
U 1 ¼ U 2
A 2
A 1
ð10:6Þ
Then, substituting Equation 10.6 into Equation 10.5 and rearranging yields the incompressible
volume flow rate,
Q I ¼ U 2 A 2 ¼
A 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À A 2 =A 1
ð
Þ
2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 p 1 À p 2
ð
Þ
r
s
þ 2gh L 1À2
ð10:7Þ
Flow
Flow
Flow
Forward
View
Forward
View
(a)
(c)
(b)
Figure 10.2 Flow area profiles of common obstruction
meters. (a) Square-edged
orifice plate meter. (b)
American Society of
Mechanical Engineers
(ASME) long radius nozzle.
(c) ASME Herschel venturi
meter.
Eddy recirculation regions
Vena contracta
streamlines
Control
volume
p 1
p 2
1
d 1
d 2
d 0
2
Figure 10.3 Control volume concept as applied between two streamlines for flow through an obstruction meter.
428 Chapter 10 Flow Measurements
13:4:37 Page 428
shown. Under the assumptions of incompressible, steady and one-dimensional flow with no external
energy transfer, the energy equation is
p 1
g
þ
U
2
1
2g
¼
p 2
g
þ
U
2
2
2g
þ h L 1À2
ð10:5Þ
where h L 1À2 denotes the head losses occurring between control surfaces 1 and 2.
For incompressible flows, conservation of mass between cross-sectional areas 1 and 2 gives
U 1 ¼ U 2
A 2
A 1
ð10:6Þ
Then, substituting Equation 10.6 into Equation 10.5 and rearranging yields the incompressible
volume flow rate,
Q I ¼ U 2 A 2 ¼
A 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À A 2 =A 1
ð
Þ
2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 p 1 À p 2
ð
Þ
r
s
þ 2gh L 1À2
ð10:7Þ
Flow
Flow
Flow
Forward
View
Forward
View
(a)
(c)
(b)
Figure 10.2 Flow area profiles of common obstruction
meters. (a) Square-edged
orifice plate meter. (b)
American Society of
Mechanical Engineers
(ASME) long radius nozzle.
(c) ASME Herschel venturi
meter.
Eddy recirculation regions
Vena contracta
streamlines
Control
volume
p 1
p 2
1
d 1
d 2
d 0
2
Figure 10.3 Control volume concept as applied between two streamlines for flow through an obstruction meter.
428 Chapter 10 Flow Measurements