E1C09 09/14/2010
15:4:55 Page 405
This system is lightly damped. The magnitude frequency response, M(v), is shown in Figure 9.23.
The transmission line effects control the transducer response. In effect, the transmission line acts as
a mechanical low-pass filter for the transducer.
Heavily Damped Systems
In systems in which we estimate a damping ratio greater than 1.5, the frequency response model can
be simplified further. The behavior of the pressure-measuring system closely follows that of a firstorder system. Again, the typical pressure transducer–tubing system has a compliance C vp , which is
a measure of the transducer and tubing volume change relative to an applied pressure change.
The response of the first-order system is indicated through its time constant, which is estimated by
neglecting the second-order term in Equation 9.23 so that (12)
t ¼
128m‘C vp
pd
4
ð9:32Þ
Equation 9.32 shows that the time constant is proportional to ‘=d
4 . Long- and small-diameter
connecting tubes promote a more sluggish system response to changes in pressure.
9.9 FLUID VELOCITY MEASURING SYSTEMS
Velocity measuring systems are used to measure the local velocity in a moving fluid. Desirable
information can consist of the mean velocity, as well as any of the dynamic components of the
velocity. Dynamic components are found in pulsating, phasic, or oscillating flows, or in turbulent
flows. For most general engineering applications, information about the mean flow velocity is
usually sufficient. The dynamic velocity information is often sought during applied and basic fluid
mechanics research and development, such as in attempting to study airplane wing response to air
turbulence, a complex periodic waveform as the wing sees it. In general, the instantaneous velocity
can be written as
UðtÞ ¼ U þ u
ð9:33Þ
where U is the mean velocity and u is the time-dependent dynamic (fluctuating) component of the
velocity. The instantaneous velocity can also be expressed in terms of a Fourier series:
UðtÞ ¼ U þ
X
i
C i sin v i t þ f
0
i
ð9:34Þ
so that the mean velocity and the amplitude and frequency information concerning the dynamic
velocity component can be resolved with a Fourier analysis of the time-dependent velocity signal.
Pitot-Static Pressure Probe
For a steady, incompressible, isentropic flow, Equation 9.16 can be written at any arbitrary point x in
the flow field as
p t ¼ p x þ
1
2
rU
2
x
ð9:35Þ
9.9 Fluid Velocity Measuring Systems 405
15:4:55 Page 405
This system is lightly damped. The magnitude frequency response, M(v), is shown in Figure 9.23.
The transmission line effects control the transducer response. In effect, the transmission line acts as
a mechanical low-pass filter for the transducer.
Heavily Damped Systems
In systems in which we estimate a damping ratio greater than 1.5, the frequency response model can
be simplified further. The behavior of the pressure-measuring system closely follows that of a firstorder system. Again, the typical pressure transducer–tubing system has a compliance C vp , which is
a measure of the transducer and tubing volume change relative to an applied pressure change.
The response of the first-order system is indicated through its time constant, which is estimated by
neglecting the second-order term in Equation 9.23 so that (12)
t ¼
128m‘C vp
pd
4
ð9:32Þ
Equation 9.32 shows that the time constant is proportional to ‘=d
4 . Long- and small-diameter
connecting tubes promote a more sluggish system response to changes in pressure.
9.9 FLUID VELOCITY MEASURING SYSTEMS
Velocity measuring systems are used to measure the local velocity in a moving fluid. Desirable
information can consist of the mean velocity, as well as any of the dynamic components of the
velocity. Dynamic components are found in pulsating, phasic, or oscillating flows, or in turbulent
flows. For most general engineering applications, information about the mean flow velocity is
usually sufficient. The dynamic velocity information is often sought during applied and basic fluid
mechanics research and development, such as in attempting to study airplane wing response to air
turbulence, a complex periodic waveform as the wing sees it. In general, the instantaneous velocity
can be written as
UðtÞ ¼ U þ u
ð9:33Þ
where U is the mean velocity and u is the time-dependent dynamic (fluctuating) component of the
velocity. The instantaneous velocity can also be expressed in terms of a Fourier series:
UðtÞ ¼ U þ
X
i
C i sin v i t þ f
0
i
ð9:34Þ
so that the mean velocity and the amplitude and frequency information concerning the dynamic
velocity component can be resolved with a Fourier analysis of the time-dependent velocity signal.
Pitot-Static Pressure Probe
For a steady, incompressible, isentropic flow, Equation 9.16 can be written at any arbitrary point x in
the flow field as
p t ¼ p x þ
1
2
rU
2
x
ð9:35Þ
9.9 Fluid Velocity Measuring Systems 405