E1C10 09/14/2010
13:4:37 Page 427
Then, the mean flow rate along each line of traverse is
Q 1 ¼ 0:252 m
3 /s Q 2 ¼ 0:252 m
3 /s Q 3 ¼ 0:254 m
3 /s
The average pipe flow rate Q is the pooled mean of the individual flow rates
Q ¼ hQi ¼
1
3
X 3
j¼1
Q j ¼ 0:253 m
3 /s
Example 10.2
Referring to Example 10.1, determine a value of the systematic standard uncertainty in mean flow
rate due to the measured spatial variation.
KNOWN Data of Example 10.1 over three (m ¼ 3) traverse sections.
SOLUTION The systematic standard uncertainty in mean flow rate due to error introduced by
the measured spatial variations is estimated by
b Q ¼
s hQi
ffiffiffiffi
m
p ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P 3
j¼1
Q j À hQi
À
Á 2 =2
s
ffiffi ffi
3
p
¼
0:0012
ffiffi ffi
3
p
¼ 0:0007 m
3 /s with n ¼ 2
10.5 PRESSURE DIFFERENTIAL METERS
The operating principle of a pressure differential flow rate meter is based on the relationship
between volume flow rate and the pressure drop Dp ¼ p 1 À p 2 , between two locations along the flow
path,
Q / p 1 À p 2
ð
Þ
n
ð10:4Þ
where n ¼ 1 for laminar flow occurring between the pressure measurement locations and n ¼ ½
for fully turbulent flow. Under steady flow conditions, an intentional reduction in flow area between
locations 1 and 2 causes a measurable local pressure drop across this flow path. This reduced flow
area leads to a concurrent local increase in velocity due to flow continuity (conservation of mass)
principles. The pressure drop is in part due to the so-called Bernoulli effect, the inverse relationship
between local velocity and pressure, but is also due to flow energy losses. Pressure differential flow
rate meters that use area reduction methods are commonly called obstruction meters.
Obstruction Meters
Three common obstruction meters are the orifice plate, the venturi, and the flow nozzle. Flow area
profiles of each are shown in Figure 10.2. These meters are usually inserted in-line with a pipe, such
as between pipe flanges. This class of meters as a whole operates using similar physical reasoning to
relate volume flow rate to pressure drop. Referring to Figure 10.3, consider an energy balance
between two control surfaces for an incompressible fluid flow through the arbitrary control volume
10.5 Pressure Differential Meters 427
13:4:37 Page 427
Then, the mean flow rate along each line of traverse is
Q 1 ¼ 0:252 m
3 /s Q 2 ¼ 0:252 m
3 /s Q 3 ¼ 0:254 m
3 /s
The average pipe flow rate Q is the pooled mean of the individual flow rates
Q ¼ hQi ¼
1
3
X 3
j¼1
Q j ¼ 0:253 m
3 /s
Example 10.2
Referring to Example 10.1, determine a value of the systematic standard uncertainty in mean flow
rate due to the measured spatial variation.
KNOWN Data of Example 10.1 over three (m ¼ 3) traverse sections.
SOLUTION The systematic standard uncertainty in mean flow rate due to error introduced by
the measured spatial variations is estimated by
b Q ¼
s hQi
ffiffiffiffi
m
p ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P 3
j¼1
Q j À hQi
À
Á 2 =2
s
ffiffi ffi
3
p
¼
0:0012
ffiffi ffi
3
p
¼ 0:0007 m
3 /s with n ¼ 2
10.5 PRESSURE DIFFERENTIAL METERS
The operating principle of a pressure differential flow rate meter is based on the relationship
between volume flow rate and the pressure drop Dp ¼ p 1 À p 2 , between two locations along the flow
path,
Q / p 1 À p 2
ð
Þ
n
ð10:4Þ
where n ¼ 1 for laminar flow occurring between the pressure measurement locations and n ¼ ½
for fully turbulent flow. Under steady flow conditions, an intentional reduction in flow area between
locations 1 and 2 causes a measurable local pressure drop across this flow path. This reduced flow
area leads to a concurrent local increase in velocity due to flow continuity (conservation of mass)
principles. The pressure drop is in part due to the so-called Bernoulli effect, the inverse relationship
between local velocity and pressure, but is also due to flow energy losses. Pressure differential flow
rate meters that use area reduction methods are commonly called obstruction meters.
Obstruction Meters
Three common obstruction meters are the orifice plate, the venturi, and the flow nozzle. Flow area
profiles of each are shown in Figure 10.2. These meters are usually inserted in-line with a pipe, such
as between pipe flanges. This class of meters as a whole operates using similar physical reasoning to
relate volume flow rate to pressure drop. Referring to Figure 10.3, consider an energy balance
between two control surfaces for an incompressible fluid flow through the arbitrary control volume
10.5 Pressure Differential Meters 427