E1C09 09/14/2010
15:4:53 Page 384
ASSUMPTIONS Temperature and capillary effects in the manometer and gravity error in the
specific weights of the fluids are negligible. The degrees of freedom in the stated uncertainties are
large.
FIND u d
SOLUTION The relation between pressure and manometer deflection is given by Equation 9.5
with H ¼ L sin u:
Dp ¼ p 1 À p 2 ¼ Lðg m À gÞ sin u
where p 2 is the ambient pressure so that Dp is the nominal pressure relative to the ambient. For a
nominal Dp ¼ 100 N/m
2 , the nominal manometer rise L would be
L ¼
Dp
g m À g
ð
Þsin u
%
Dp
g m sin u
¼ 21 mm
where g m ) g and the value for g and its uncertainty are neglected. For the design stage analysis,
p ¼ f ðg m ; L; uÞ, so that the uncertainty in pressure, Dp, is estimated by
u d
ð Þ p ¼ Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qDp
qg m
u d
ð Þ g m
! 2
þ
qDp
qL
u d
ð Þ L
! 2
þ
qDp
qu
u d
ð Þ u
! 2
s
At assumed 95% confidence levels, the manometer specific weight uncertainty and angle
uncertainty are estimated from the problem as
u d
ð Þ g m ¼ 9770 N=m
3
ð
Þ0:005 ð
Þ%49 N=m
3
u d
ð Þ u ¼ 1 degree ¼ 0:0175 rad
The uncertainty in estimating the pressure from the indicated deflection is due both to the
manometer resolution, u o , and the zero point offset error, which we take as its instrument error, u c .
Using the uncertainties associated with these errors,
u d
ð Þ L ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
u 2
o þ u 2
c
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð0:5 mmÞ
2 þ ð0:5 mmÞ
2
q
¼ 0:7 mm
Evaluating the derivatives and substituting values gives a design-stage uncertainty interval of a
measured Dp of
u d
ð Þ Dp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ð0:26Þ
2 þ ð3:42Þ
2 þ ð3:10Þ
2
q
¼ Æ4:6N=m
2
ð95%Þ
COMMENT At a 30-degree inclination and for this pressure, the uncertainty in pressure is
affected almost equally by the instrument inclination and the deflection uncertainties. As the
manometer inclination is increased to a more vertical orientation, that is, toward the U-tube
manometer, inclination uncertainty becomes less important and is negligible near 90 degrees.
However, for a U-tube manometer, the deflection is reduced to less than 11 mm, a 50% reduction in
manometer sensitivity, with an associated design-stage uncertainty of 6.8 N/m
2 (95%).
Deadweight Testers
The deadweight tester makes direct use of the fundamental definition of pressure as a force per
unit area to create and to determine the pressure within a sealed chamber. These devices are a
384 Chapter 9 Pressure and Velocity Measurements
15:4:53 Page 384
ASSUMPTIONS Temperature and capillary effects in the manometer and gravity error in the
specific weights of the fluids are negligible. The degrees of freedom in the stated uncertainties are
large.
FIND u d
SOLUTION The relation between pressure and manometer deflection is given by Equation 9.5
with H ¼ L sin u:
Dp ¼ p 1 À p 2 ¼ Lðg m À gÞ sin u
where p 2 is the ambient pressure so that Dp is the nominal pressure relative to the ambient. For a
nominal Dp ¼ 100 N/m
2 , the nominal manometer rise L would be
L ¼
Dp
g m À g
ð
Þsin u
%
Dp
g m sin u
¼ 21 mm
where g m ) g and the value for g and its uncertainty are neglected. For the design stage analysis,
p ¼ f ðg m ; L; uÞ, so that the uncertainty in pressure, Dp, is estimated by
u d
ð Þ p ¼ Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qDp
qg m
u d
ð Þ g m
! 2
þ
qDp
qL
u d
ð Þ L
! 2
þ
qDp
qu
u d
ð Þ u
! 2
s
At assumed 95% confidence levels, the manometer specific weight uncertainty and angle
uncertainty are estimated from the problem as
u d
ð Þ g m ¼ 9770 N=m
3
ð
Þ0:005 ð
Þ%49 N=m
3
u d
ð Þ u ¼ 1 degree ¼ 0:0175 rad
The uncertainty in estimating the pressure from the indicated deflection is due both to the
manometer resolution, u o , and the zero point offset error, which we take as its instrument error, u c .
Using the uncertainties associated with these errors,
u d
ð Þ L ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
u 2
o þ u 2
c
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð0:5 mmÞ
2 þ ð0:5 mmÞ
2
q
¼ 0:7 mm
Evaluating the derivatives and substituting values gives a design-stage uncertainty interval of a
measured Dp of
u d
ð Þ Dp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ð0:26Þ
2 þ ð3:42Þ
2 þ ð3:10Þ
2
q
¼ Æ4:6N=m
2
ð95%Þ
COMMENT At a 30-degree inclination and for this pressure, the uncertainty in pressure is
affected almost equally by the instrument inclination and the deflection uncertainties. As the
manometer inclination is increased to a more vertical orientation, that is, toward the U-tube
manometer, inclination uncertainty becomes less important and is negligible near 90 degrees.
However, for a U-tube manometer, the deflection is reduced to less than 11 mm, a 50% reduction in
manometer sensitivity, with an associated design-stage uncertainty of 6.8 N/m
2 (95%).
Deadweight Testers
The deadweight tester makes direct use of the fundamental definition of pressure as a force per
unit area to create and to determine the pressure within a sealed chamber. These devices are a
384 Chapter 9 Pressure and Velocity Measurements