1
Pressure Measurements
2011
Pressure
Measurements
Lecture Notes
Systems & Biomedical Engineering Department Faculty of Engineering, Cairo
University
Prof. Bassel Tawfik
Biomedical Measurements
1/1/2011
Pressure Measurements
2
Lecture Outline
1.
2.
3.
4.
5.
6.
Definition of pressure
Terminology
Strain gages
Resonant wire
Piezo-electric transducers
Optical
1. Pressure
1.1 Definitions
Types of Pressure Measurements
There are three types (or definitions) of pressure: absolute, differential, and gauge.
Absolute pressure is measured relative to a perfect vacuum. An example is atmospheric
pressure. A common unit of measure is pounds per square inch absolute (psia).
Differential pressure is the difference in pressure between two points of measurement.
This is commonly measured in units of pounds per square inch differential (psid). DP
transducers are often used in flow measurement where they can measure the pressure
differential across a venturi, orifice, or any other type of flow resistances. The detected
pressure differential is related to fluid velocity and therefore to volumetric flow.
Last, gage pressure is measured relative to
ambient pressure. Blood pressure is one
example. Common measurement units are
pressure per square inch gauge (psig).
Intake manifold vacuum in an automobile
engine is an example of a vacuum gauge
measurement (vacuum is negative gauge
pressure).
Figure 7. The same sensor can be used for all three
types of pressure measurement; only the references
differ.
The three types of measurements are shown in Figure 7. Note that the same sensor may
be used for all three types; only the reference is different. Differential pressures may be
measured anywhere in the range—above, below, and around atmospheric pressure.
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Pressure Units
As previously noted, pressure is force per unit
area and historically a great variety of units
have been used, depending on their suitability
for the application. For example, blood
pressure is usually measured in mmHg because
mercury manometers were used originally.
Atmospheric pressure is usually expressed in
in.Hg for the same reason. Other units used for
atmospheric pressure are bar and atm. The
following conversion factors should help in
dealing with the various units:
1 psi = 51.714 mmHg = 2.0359 in.Hg =
27.680
1 in.H2O = 6.8946 kPa
1 bar = 14.504 psi
1 atm = 14.696 psi
Example: Convert 200 mmHg to psi:
200 mmHg • 1 psi/51.714 mmHg = 3.867
psi
Terminology
Often, the terms pressure gauge, sensor, transducer, and transmitter are used
interchangeably. The term pressure gage usually refers to a self-contained indicator that
converts the detected process pressure into a mechanical motion of a pointer. A
pressure transducer might combine the sensor element of a gage with a mechanical-toelectrical or mechanical-to-pneumatic converter and a power supply. The system may
also have a signal conditioner that converts the transducer signal into a standardized
output.
When transducers are directly interfaced with digital data acquisition systems and are
located at some distance from the data acquisition hardware, it is preferable that the
output voltage signal is high enough in order to be immune from electromagnetic
interference (EMI) during its travelling path to the receiver.
Accuracy: Transducer accuracy refers to the degree of conformity of the measured value
to an accepted standard. It is usually expressed as a percentage of either the full scale or
of the actual reading of the instrument. In case of percent-full-scale devices, error
increases as the absolute value of the measurement drops.
Repeatability refers to the closeness of agreement among a number of consecutive
measurements of the same variable.
Linearity is a measure of how well the transducer output increases linearly with
increasing pressure.
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Hysteresis error describes the phenomenon whereby the same process pressure results
in different output signals depending upon whether the current pressure is approached
(from a lower or higher pressure).
Practical Considerations
Strain gage transducers are available
for pressure ranges as low as 3 inches
of water to as high as 200,000 psig
(1400 MPa). Inaccuracy ranges from
0.1% of span to 0.25% of full scale.
Additional error sources can be a
0.25% of full-scale drift over six
months and a 0.25% full scale
temperature effect per 1000 F.
The way differential pressure transducers look like in
reality
Strain Gages
When a strain gage, as
described in detail in the
previous lecture, is used
to measure the deflection
of an elastic diaphragm or
Bourdon tube, it becomes
a component in a
pressure
transducer.
Strain-gage transducers
are used for narrow-span
pressure
and
for
differential
pressure
measurements.
Essentially, the strain gage
is used to measure the Figure 3-4: Strain-Gage Based Pressure Cell
displacement of an elastic
diaphragm due to a difference in pressure across the diaphragm. These devices can
detect gage pressure if the low pressure port is left open to the atmosphere. If the low
pressure side is a sealed vacuum reference, the transmitter will act as an absolute
pressure transmitter.
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Pressure Measurements
Capacitance Type
Capacitance
pressure
transducers were originally
developed for use in low
vacuum
research.
This
capacitance change results
from the movement of a
diaphragm element (Figure
3-5). The diaphragm is
usually metal or metalcoated quartz and is exposed
to the process pressure on
one side and to the
reference pressure on the
other. Depending on the
type of pressure, the
capacitive transducer can be
either an absolute, gauge, or
differential
pressure
transducer.
Figure 3-5: Capacitance-Based Pressure Cell
In
a
capacitance-type
pressure sensor, a high-frequency, high-voltage oscillator is used to charge the sensing
electrode elements. In a two-plate capacitor sensor design, the movement of the
diaphragm between the plates is detected as an indication of the changes in process
pressure.
As shown in Figure 3-5, the deflection of the diaphragm causes a change in capacitance
that is detected by a bridge circuit. This circuit can be operated in either a balanced or
unbalanced mode. In balanced mode, the output voltage is fed to a null detector and
the capacitor arms are varied to maintain the bridge at null. Therefore, in the balanced
mode, the null setting itself is a measure of process pressure. When operated in
unbalanced mode, the process pressure measurement is related to the ratio between
the output voltage and the excitation voltage.
Capacitance pressure transducers are widespread in part because of their wide
rangeability, from high vacuums in the micron range to 10,000 psig (70 MPa).
Differential pressures as low as 0.01 inches of water can readily be measured. And,
compared with strain gage transducers, they do not drift much. Better designs are
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Pressure Measurements
available that are accurate to within 0.1% of reading or 0.01% of full scale. A typical
temperature effect is 0.25% of full scale per 1000 F.
Capacitance-type sensors are often used as secondary standards, especially in lowdifferential and low-absolute pressure applications. They also are quite responsive,
because the distance the diaphragm must physically travel is only a few microns. Newer
capacitance pressure transducers are more resistant to corrosion and are less sensitive
to stray capacitance and vibration effects that used to cause "reading jitters" in older
designs.
Potentiometric
The potentiometric pressure sensor
provides a simple method for
obtaining an electronic output from
a mechanical pressure gauge. The
device consists of a precision
potentiometer, whose wiper arm is
mechanically linked to a Bourdon
or bellows element. The movement
of the wiper arm across the
potentiometer
converts
the
mechanically detected sensor
deflection into a resistance
measurement, using a Wheatstone
bridge circuit (Figure 3-6).
Figure 3-6: Potentiometric Pressure Transducer
The mechanical nature of the linkages connecting the wiper arm to the Bourdon tube,
bellows, or diaphragm element introduces unavoidable errors into this type of
measurement. Temperature effects cause additional errors because of the differences
in thermal expansion coefficients of the metallic components of the system. Errors also
will develop due to mechanical wear of the components and of the contacts.
Potentiometric transducers can be made extremely small and installed in very tight
quarters, such as inside the housing of a 4.5-in. dial pressure gauge. They also provide a
strong output that can be read without additional amplification. This permits them to be
used in low power applications. They are also inexpensive. Potentiometric transducers
can detect pressures between 5 and 10,000 psig (35 KPa to 70 MPa). Their accuracy is
between 0.5% and 1% of full scale, not including drift and the effects of temperature.
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Resonant Wire
The resonant-wire pressure
transducer was introduced in
the late 1970s. In this design
(Figure 3-7), a wire is gripped
by a static member at one
end, and by the sensing
diaphragm at the other. An
oscillator circuit causes the
wire to oscillate at its
resonant
frequency.
A
change in process pressure
changes the wire tension,
which in turn changes the
resonant frequency of the
wire. A digital counter circuit
detects the shift. Because
this change in frequency can
be detected quite precisely,
this type of transducer can
be used for low differential
pressure applications as well
as to detect absolute and
gauge pressures.
Figure 3-7: Resonant-Wire Pressure Transducer
The most significant advantage of the resonant wire pressure transducer is that it
generates an inherently digital signal, and therefore can be sent directly to a stable
crystal clock in a microprocessor. Limitations include sensitivity to temperature
variation, a nonlinear output signal, and some sensitivity to shock and vibration. These
limitations typically are minimized by using a microprocessor to compensate for
nonlinearities as well as ambient and process temperature variations.
Resonant wire transducers can detect absolute pressures from 10 mm Hg, differential
pressures up to 750 in. water, and gauge pressures up to 6,000 psig (42 MPa). Typical
accuracy is 0.1% of calibrated span, with six-month drift of 0.1% and a temperature
effect of 0.2% per 1000 F.
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Piezoelectric
When pressure, force or
acceleration is applied to a
quartz crystal, a charge is
developed across the crystal
that is proportional to the
force applied (Figure 3-8).
The fundamental difference
between
these
crystal
sensors and static-force
devices such as strain gages
is that the electric signal
generated by the crystal
decays
rapidly.
This
characteristic makes these Figure 3-8: Typical Piezoelectric Pressure Sensor
sensors unsuitable for the
measurement of static forces or pressures but useful for dynamic measurements.
Piezoelectric devices can further be classified according to the measured variable, being
either the crystal's electrostatic charge, its resistivity, or its resonant frequency.
Depending on which phenomenon is used, the crystal sensor can be called electrostatic,
piezo-resistive, or resonant.
When pressure is applied to a crystal, it is elastically deformed. This deformation results
in a flow of electric charge (which lasts for a period of a few seconds). The resulting
electric signal can be measured as an indication of the pressure which was applied to
the crystal. These sensors cannot detect static pressures, but are used to measure
rapidly changing pressures resulting from blasts, explosions, pressure pulsations (in
rocket motors, engines, compressors) or other sources of shock or vibration. Some of
these rugged sensors can detect pressure events having "rise times" on the order of a
millionth of a second.
The output of such dynamic pressure sensors is often expressed in "relative" pressure
units (such as psir instead of psig), thereby referencing the measurement to the initial
condition of the crystal. The maximum range of such sensors is 5,000 or 10,000 psir. The
desirable features of piezoelectric sensors include their rugged construction, small size,
high speed, and self-generated signal. On the other hand, they are sensitive to
temperature variations and require special cabling and amplification.
They also require special care during installation: One such consideration is that their
mounting torque should duplicate the torque at which they were calibrated (usually 30
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in.-lbs). Another factor that can harm their performance by slowing response speed is
the depth of the empty cavity below the cavity. The larger the cavity, the slower the
response. Therefore, it is recommended that the depth of the cavity be minimized and
not be deeper than the diameter of the probe (usually about 0.25-in.).
Electrostatic
pressure
transducers are small and
rugged. Force to the crystal
can be applied longitudinally
or in the transverse direction,
and in either case will cause a
high
voltage
output
proportional to the force
applied. The crystal's selfgenerated voltage signal is
useful
where
providing
power to the sensor is
impractical or impossible. Figure 3-9: Acceleration-Compensated Piezoelectric Sensor
These sensors also provide
high speed responses (30 kHz with peaks to 100 kHz), which makes them ideal for
measuring transient phenomena.
Figure 3-9 illustrates an acceleration-compensated pressure sensor. In this design, the
compensation is provided by the addition of a seismic mass and a separate
"compensation crystal" of reverse polarity. These components are scaled to exactly
cancel the inertial effect of the masses (the end piece and diaphragm) which act upon
the pressure-sensing crystal stack when accelerated.
Because quartz is a common and naturally occurring mineral, these transducers are
generally inexpensive. Tourmaline, a naturally occurring semi-precious form of quartz,
has sub-microsecond responsiveness and is useful in the measurement of very rapid
transients. By selecting the crystal properly, the designer can ensure both good linearity
and reduced temperature sensitivity.
Although piezoelectric transducers are not capable of measuring static pressures, they
are widely used to evaluate dynamic pressure phenomena associated with explosions,
pulsations, or dynamic pressure conditions in motors, rocket engines, compressors, and
other pressurized devices that experience rapid changes. They can detect pressures
between 0.1 and 10,000 psig (0.7 KPa to 70 MPa). Typical accuracy is 1% full scale with
an additional 1% full scale per 1000¡ temperature effect.
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Piezoresistive pressure sensors operate based on the resistivity dependence of silicon
under stress. Similar to a strain gage, a piezoresistive sensor consists of a diaphragm
onto which four pairs of silicon resistors are bonded. Unlike the construction of a strain
gage sensor, here the diaphragm itself is made of silicon and the resistors are diffused
into the silicon during the manufacturing process. The diaphragm is completed by
bonding the diaphragm to an unprocessed wafer of silicon.
If the sensor is to be used to measure absolute pressure, the bonding process is
performed under vacuum. If the sensor is to be referenced, the cavity behind the
diaphragm is ported either to the atmosphere or to the reference pressure source.
When used in a process sensor, the silicon diaphragm is shielded from direct contact
with the process materials by a fluid-filled protective diaphragm made of stainless steel
or some other alloy that meets the corrosion requirements of the service.
Piezoresistive pressure sensors are sensitive to changes in temperature and must be
temperature compensated. Piezoresistive pressure sensors can be used from about 3
psi to a maximum of about 14,000 psi (21 KPa to 100 MPa).
Resonant piezoelectric pressure sensors measure the variation in resonant frequency of
quartz crystals under an applied force. The sensor can consist of a suspended beam that
oscillates while isolated from all other forces. The beam is maintained in oscillation at its
resonant frequency. Changes in the applied force result in resonant frequency changes.
The relationship between the applied pressure P and the oscillation frequency is:
Where TO is the period of oscillation when the applied pressure is zero, T is the period
of oscillation when the applied pressure is P, and A and B are calibration constants for
the transducer.
These transducers can be used for absolute pressure measurements with spans from 015 psia to 0-900 psia (0-100 kPa to 0-6 MPa) or for differential pressure measurements
with spans from 0-6 psid to 0-40 psid (0-40 kPa to 0-275 kPa).
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Inductive/Reluctive
A number of early pressure transducer designs
were based on magnetic phenomena. These
included the use of inductance, reluctance, and
eddy currents. Inductance is that property of an
electric circuit that expresses the amount of
electromotive force (emf) induced by a given rate
of change of current flow in the circuit.
Reluctance is resistance to magnetic flow, the
opposition offered by a magnetic substance to
magnetic flux. In these sensors, a change in
pressure produces a movement, which in turn
changes the inductance or reluctance of an
electric circuit.
Flush-mount pressure sensor fits
1/4-in. NPT threads.
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Figure 3-10A illustrates
the use of a linear
variable
differential
transformer (LVDT) as
the working element of
a pressure transmitter.
The LVDT operates on
the inductance ratio
principle. In this design,
three coils are wired
onto an insulating tube
containing an iron core,
which is positioned
within the tube by the
pressure sensor.
Figure 3-10: Magnetic Pressure Transducer Designs
Alternating current is applied to the primary coil in the center, and if the core also is
centered, equal voltages will be induced in the secondary coils (#1 and #2). Because the
coils are wired in series, this condition will result in a zero output. As the process
pressure changes and the core moves, the differential in the voltages induced in the
secondary coils is proportional to the pressure causing the movement.
LVDT-type pressure transducers are available with 0.5% full scale accuracy and with
ranges from 0-30 psig (0-210 kPa) to 0-10,000 psig (0-70 MPa). They can detect
absolute, gauge, or differential pressures. Their main limitations are susceptibility to
mechanical wear and sensitivity to vibration and magnetic interference.
Reluctance is the equivalent of resistance in a magnetic circuit. If a change in pressure
changes the gaps in the magnetic flux paths of the two cores, the ratio of inductances
L1/L2 will be related to the change in process pressure (Figure 3-10B). Reluctance-based
pressure transducers have a very high output signal (on the order of 40 mV/volt of
excitation), but must be excited by ac voltage. They are susceptible to stray magnetic
fields and to temperature effects of about 2% per 1000¡ F. Because of their very high
output signals, they are often used in applications where high resolution over a
relatively small range is desired. They can cover pressure ranges from 1 in. water to
10,000 psig (250 Pa to 70 MPa). Typical accuracy is 0.5% full scale.
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Optical
Optical pressure transducers detect the effects of minute motions due to changes in
process pressure and generate a
corresponding electronic output
signal (Figure 3-11). A light emitting
diode (LED) is used as the light
source, and a vane blocks some of
the light as it is moved by the
diaphragm. As the process pressure
moves the vane between the source
diode and the measuring diode, the
amount of infrared light received
changes.
The
optical
transducer
must
compensate for aging of the LED light Figure 3-11: Optical Pressure Transducer
source by means of a reference
diode, which is never blocked by the vane. This reference diode also compensates the
signal for build-up of dirt or other coating materials on the optical surfaces. The optical
pressure transducer is immune to temperature effects, because the source,
measurement and reference diodes are affected equally by changes in temperature.
Moreover, because the amount of movement required to make the measurement is
very small (under 0.5 mm), hysteresis and repeatability errors are nearly zero.
Optical pressure transducers do not require much maintenance. They have excellent
stability and are designed for long-duration measurements. They are available with
ranges from 5 psig to 60,000 psig (35 kPa to 413 MPa) and with 0.1% full scale accuracy.
Practical Considerations
In industrial applications, good repeatability often is more important then absolute
accuracy. If process pressures vary over a wide range, transducers with good linearity
and low hysteresis are the preferred choice. Ambient and process temperature
variations also cause errors in pressure measurements, particularly in detecting low
pressures and small differential pressures. In such applications, temperature
compensators must be used.
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Thick-film silicon pressure sensor is available in ranges
from 10 to 30,000 psia.
Power supply variations also lower the performance of pressure transducers. The
sensitivity (S) of a transducer determines the amount of change that occurs in the
output voltage (VO) when the supply voltage (VS) changes, with the measured pressure
(Pm) and the rated pressure of the transducer (Pr) remaining constant:
In a pressure measurement system, the total error can be calculated using the rootsum-square method: the total error is equal to the square root of the sums of all the
individual errors squared.
Selection Criteria
Pressure transducers usually generate output signals in the mV range (spans of 100 mV
to 250 mV). When used in transmitters, these are often amplified to the voltage level (1
to 5 V) and converted to current loops, usually 4-20 mA dc.
The transducer housing should be selected to meet both the electrical area classification
and the corrosion requirements of the particular installation. Corrosion protection must
take into account both splashing of corrosive liquids or exposure to corrosive gases on
the outside of the housing, as well as exposure of the sensing element to corrosive
process materials. The corrosion requirements of the installation are met by selecting
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corrosion-resistant materials, coatings, and by the use of chemical seals, which are
discussed later in this chapter.
If the installation is in an area where explosive vapors may be present, the transducer or
transmitter and its power supply must be suitable for these environments. This is
usually achieved either by placing them inside purged or explosion-proof housings, or by
using intrinsically safe designs.
Probably the single most important decision in selecting a pressure transducer is the
range. One must keep in mind two conflicting considerations: the instrument's accuracy
and its protection from overpressure. From an accuracy point of view, the range of a
transmitter should be low (normal operating pressure at around the middle of the
range), so that error, usually a percentage of full scale, is minimized. On the other hand,
one must always consider the consequences of overpressure damage due to operating
errors, faulty design (waterhammer), or failure to isolate the instrument during
pressure-testing and start-up. Therefore, it is important to specify not only the required
range, but also the amount of overpressure protection needed.
Most pressure instruments are
provided with overpressure
protection of 50% to 200% of
range (Figure 3-12). These
protectors satisfy the majority
of applications. Where higher
overpressures are expected and
their nature is temporary
(pressure spikes of short
duration--seconds or less),
snubbers can be installed. These
Figure 3-12: Bourdon Tube Overpressure Protection
filter out spikes, but cause the
measurement to be less responsive. If excessive overpressure is expected to be of
longer duration, one can protect the sensor by installing a pressure relief valve.
However, this will result in a loss of measurement when the relief valve is open.
If the transmitter is to operate under high ambient temperatures, the housing can be
cooled electrically (Peltier effect) or by water, or it can be relocated in an airconditioned area. When freezing temperatures are expected, resistance heating or
steam tracing should be used in combination with thermal insulation.
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When high process temperatures are present, one can consider the use of various
methods of isolating the pressure instrument from the process. These include loop
seals, siphons, chemical seals with capillary tubing for remote mounting, and purging.
Maintenance
Without exception, pressure sensors
require
scheduled,
periodic
maintenance and/or recalibration. It is
necessary to periodically remove the
transducer from the process and to
make sure that this procedure does
not require shutting down the process
and does not cause injury or damage.
Because the process fluid may be
toxic, corrosive, or otherwise noxious
to personnel or the environment, it is
necessary to protect against the
release of such fluids during
maintenance.
Figure 3-13: Three-Valve Manifold for
Instrument Isolation
A three-way manifold (Figure 3-13) can provide such protection. In the illustration, valve
P is used to isolate the process and valve D serves to discharge the trapped process fluid
from the instrument into some safe containment. The purpose of valve T is to allow the
application of a known calibration or test pressure to the instrument. As all the
components of the manifold are pre-assembled into a compact package, space and field
assembly time are saved and chances for leaks are reduced.
Calibration
Pressure transducers can be recalibrated on-line or in a calibration laboratory.
Laboratory recalibration typically is preferred, but often is not possible or necessary. In
the laboratory, there usually are two types of calibration devices: deadweight testers
that provide primary, base-line standards, and "laboratory" or "field" standard
calibration devices that are periodically recalibrated against the primary. Of course,
these secondary standards are less accurate than the primary, but they provide a more
convenient means of testing other instruments.
A deadweight tester consists of a pumping piston with a screw that presses it into the
reservoir, a primary piston that carries the dead weight, and the gauge or transducer to
be tested (Figure 3-14). It works by loading the primary piston (of cross sectional area
A), with the amount of weight (W) that corresponds to the desired calibration pressure
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(P = W/A). The pumping piston then pressurizes the whole system by pressing more
fluid into the reservoir cylinder, until the dead weight lifts off its support.
Today's deadweight testers are
more accurate and more
complex than the instrument
in Figure 3-14, but the essential
operating principles are the
same. Sophisticated features
include
temperature
compensation and the means
to rotate the piston in its
cylinder to negate the effects
of friction.
Figure 3-14: Dead-Weight Tester Schematic
In the United States, the National Institute of Standards & Technology (NIST) provides
certified weights and calibrates laboratory piston gauges by measuring the diameter of
the piston. Deadweight testers can be used to calibrate at pressure levels as low as 5
psig (35 kPa) and as high as 100,000 psig (690 MPa). Tilting type, air-lubricated designs
can detect pressures in the mm Hg range. NIST calibrated deadweight testers can be
accurate to 5 parts in 100,000 at pressures below 40,000 psig (280 MPa). For an
industrial quality deadweight tester, error is typically 0.1% of span.
A typical secondary standard used for calibrating industrial pressure transducers
contains a precision power supply, an accurate digital readout, and a high-accuracy
resonant (quartz) pressure sensor. It is precise enough to be used to calibrate most
industrial pressure transducers, but must be NIST-traceable to be used as an official
calibration standard. The best accuracy claimed by the manufacturers is typically 0.05%
full scale.
Installation & Accessories
When possible, pressure instrumentation should be installed in visible, readily accessible
locations. Readouts should be located at eye elevation. Headroom should be provided
for instrument removal, as well as any space for tools and test equipment that might be
needed.
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In some applications, it is desirable to prevent the process fluid from coming in contact
with the sensing element. The process may be noxious, poisonous, corrosive, abrasive,
have the tendency to gel, freeze or decompose at ambient temperatures, or be hotter
or colder than the sensor can tolerate. Other reasons for inserting accessories between
the process and the pressure instrument are to filter out potentially plugging solids or to
remove potentially damaging pressure spikes or vibrations.
Snubbers & Pulsation Dampers
An unprotected pressure
sensor on the discharge of
a positive displacement
pump or compressor
would never come to rest,
and its pointer would
cycle continuously. To
filter out pressure spikes,
or to average out pressure
pulses, snubbers and
Figure 3-15: Pulsation Damper & Snubber Designs
pulsation dampers are
installed between the process and the instrument (Figure 3-15).
The first design shown in the illustration uses a corrosion-resistant porous metal filter
to delay the pressure reading by about 10 seconds. Other designs provide shorter delays
via fixed or variable pistons or restrictions. The advantage of an adjustable restriction is
that if, for example, a pressure gauge is placed on the discharge of a compressor, one
can see when the pointer cycling has stopped. Naturally, when one is interested in the
measurement of fast, transient pressures (such as to initiate safety interlocks on rising
pressures), snubbers must not be used, as they delay the response of the safety system.
Figure 3-16: Chemical Seal Alternatives
Chemical Seals
The chemical seal is also known as a "diaphragm protector." Its main components (the
upper and lower body and the clean-out ring) are shown in Figure 3-16A. The pressure
instrument is screwed into the upper body, which can be made of standard materials
because it contacts only the non-corrosive filling fluid, usually a silicone oil. The top
section with the filled diaphragm capsule can be removed with the pressure instrument
while the operator cleans out the material accumulated in the bottom housing. This
lower body is made of "pipe specification" (process compatible) materials and can be
continuously or periodically cleaned by purging.
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The seal shown in Figure 3-16A is an
off-line design; an in-line design is
shown in Figure 3-16B. In-line devices
are less likely to plug, but the process
has to be shut down if maintenance
is required. The ultimate in selfcleaning designs is shown in Figure 316C, in which all sharp edges and
dead-ended cavities (where solids
could accumulate) have been
eliminated. The flexible cylinder can
be made of a variety of plastics,
including PFA, and is available in
spool and wafer configurations.
Miniature pressure sensor fits in tight
spots.
As the process pressure changes, the amount of liquid displaced by the sealing
diaphragm is small, and is sometimes insufficient to fill and operate bellows-type
sensors. In that case, larger displacement "rolling" diaphragms are used. Volumetric seal
elements (Figure 3-17) also can eliminate cavities and sharp edges where material might
accumulate. They also are well suited for high pressure and high viscosity applications
such as extruders.
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Figure 3-17: Volumetric Seal Element Designs
Adding seals to a press measurement device can cause the following problems:
Long or large bore capillaries increase the volume of the filling fluid, increasing the
temperature error. Smaller diameter diaphragms are stiff and increase error,
particularly at low temperatures.
Filling fluid viscosity, acceptable at normal ambient temperatures, may be unacceptably
high
at
low
temperatures.
Long capillary lengths or smaller bores can cause slow response.
Uneven heating/cooling of seals and capillaries can cause errors.
Some fill fluids expand excessively with temperature and damage the instrument by
overextending
the
diaphragm.
High temperature and/or high vacuum may vaporize the fill fluid and damage the
instrument.
Fluid may contract excessively at low temperatures, bottoming the diaphragm and
preventing
operation.
Frozen
fill
fluid
also
will
prevent
operation.
For a successful seal installation, the following must also be considered:
Process
and
ambient
temperature
range.
Relative elevation of the seals and the instrument and the hydrostatic head of the fill
fluid. Instrument should be rezeroed after installation to correct for elevation.
Temperature, pressure, and physical damage potentials during cleaning and emptying.
Possible consequences of diaphragm rupture in terms of hazard and contamination.
Identical seals and capillary lengths for both sides of a differential pressure device.
Seal and instrument performance at maximum temperature/minimum pressure and
minimum pressure/temperature combinations.
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Pressure Measurements
Wet
Legs
&
Seal
Pots
When one or both impulse lines to a differential pressure device are filled with a stable,
process compatible fluid, the installation is called a "wet-leg" installation. The net effect
of the legs' height above the instrument and specific gravity of the fluid must be
considered in the calibration. Wet leg design must also allow for the filling and draining
of
the
leg(s).
Seal pots are used with wet legs when the instrument displaces a large volume of liquid
as the measurement changes. A seal pot is a small pressure vessel about one quart in
volume that is mounted at the top of the wet leg line. If two wet legs are used in a
differential application, the pots must be mounted at the same elevation. Each pot acts
as a reservoir in the impulse line where large volume changes will result in minimal
elevation change so that seal liquid is not dumped into the process line and elevation
shifts of the wet leg liquid do not cause measurement errors.
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Pressure Measurements
A check that the answer is correct is to verify that it is in the desired units and that mmHg
cancels.
Pressure Sensing
Pressure is sensed by mechanical elements such as
plates, shells, and tubes that are designed and
constructed to deflect when pressure is applied. This is
Figure 8. The typical pressure sensor has three
functional blocks.
the basic mechanism converting pressure to physical
movement. Next, this movement must be transduced
to obtain an electrical or other output. Finally, signal conditioning may be needed, depending on
the type of sensor and the application. Figure 8 illustrates the three functional blocks.
Sensing Elements
The main types of sensing elements are
Bourdon tubes, diaphragms, capsules, and
bellows (see Figure 9). The Bourdon tube is a
sealed tube that deflects in response to
applied pressure. All except diaphragms
provide a fairly large displacement that is
useful in mechanical gauges and for electrical
sensors that require a significant movement.
Mechanical Pressure Gauges
In mechanical gauges, the motion created by
the sensing element is read directly by a dial
or pointer. These devices are typically seen in
Figure 9. The basic pressure sensing element can be
low-performance applications, including
configured as a C-shaped Bourdon tube (A); a helical Bourdon
tube (B); flat diaphragm (C); a convoluted diaphragm (D); a
blood pressure measurement and automotive
capsule (E); or a set of bellows (F).
pressure gauges. The mechanical approach
used to couple the sensing element to the readout can introduce repeatability
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Pressure Measurements
Figure 10. Temperature has an effect on the offset of a pressure sensor.
errors, which will be discussed later. The mechanical mass of the gauges also limits the
frequency response and makes these sensors suitable only for slowly changing
measurements.
Electromechanical Pressure Sensors
Electromechanical pressure sensors convert the applied pressure to an electrical signal.
A wide variety of materials and technologies has been used in these devices, resulting in
performance vs. cost tradeoffs and suitability for applications. The electrical output
signal also provides a variety of choices for various applications.
Sensor Effects
A pressure sensor may be modeled as:
VOUT = kO + k1P
(13)
Where:
kO = offset
k1 = pressure sensitivity in V/pressure unit
Figure 11. The sensitivity of a pressure sensor is also affected by temperature.
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Pressure Measurements
A sensor will typically exhibit temperature coefficients of offset (also called null shift)
and sensitivity (see Figures 10 and 11). Linearity, on the other hand, refers to deviations
from the ideal straight line described by Equation (13). One way to measure linearity is
to use the least squares method, which gives a best fit straight line (see Figure 12).
Figure 12. The least squares method can be used to measure a pressure sensor's linearity.
Repeatability refers to the sensor's ability to produce the same output with consecutive
applications of the same pressure (see Figure 13). Hysteresis refers to the ability of the
sensor to give the same output when the same increasing and then decreasing
pressures are applied consecutively (see Figure 14).
Figure 14. Hysteresis is a sensor's ability to give the
same output at a given temperature before and after
a temperature cycle.
Figure 13. Repeatability refers to the ability of a
pressure sensor to provide the same output with
successive applications of the same pressure.
Temperature hysteresis refers to the ability of the sensor to give the same output at a
given temperature before and after a temperature cycle. Repeatability and hysteresis
effects are not easily compensated and are indicators of the basic stability of the device.
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Pressure Measurements
Gauge factor is a measure of the sensitivity of a sensor. It is defined as the ratio of the
change in an electrical transduction parameter over the full range of pressure to the
value of that parameter at zero pressure. Thus, the gauge factor of a resistive sensor is:
R/R
(14)
where:
R = base resistance
R = resistance change with full-scale pressure, for example
Pressure Sensor Technologies: Potentiometric Pressure Sensors
Potentiometric pressure sensors
use a Bourdon tube, capsule, or bellows
to drive a wiper arm on a resistive
element. For reliable operation the wiper
must bear on the element with some
force, which leads to repeatability and
hysteresis errors. These devices are very
low cost, however, and are used in lowperformance applications such as
dashboard oil pressure gauges. An
example is shown in Figure 15.
Figure 15. Potentiometric pressure sensors use a
Bourdon tube, capsule, or bellows to drive a wiper
arm on a resistive element. Such sensors tend to be
inexpensive, but subject to repeatability and
hysteresis errors.
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Pressure Measurements
Inductive Pressure Sensors
Several configurations based on varying
inductance or inductive coupling are used in
pressure sensors. They all require AC
excitation of the coil(s) and, if a DC output is
desired, subsequent demodulation and
filtering. The linear variable differential
transformer (LVDT) types have a fairly low
frequency response due to the necessity of
driving the moving core of the differential
transformer (see Figure 16). The LVDT uses
the moving core to vary the inductive
coupling between the transformer primary
and secondary.
Figure 16. An LVDT pressure sensor, one
configuration of inductive devices, drives a moving
core that varies the inductive coupling between
the transformer primary and secondary.
Capacitive Pressure Sensors
Capacitive pressure sensors typically use a
thin diaphragm as one plate of a capacitor.
Applied pressure causes the diaphragm to
deflect and the capacitance to change. This
change may or may not be linear and is
typically on the order of several picofarads
out of a total capacitance of 50-100 pF. The
change in capacitance may be used to
control the frequency of an oscillator or to
vary the coupling of an AC signal through a
network. The electronics for signal
conditioning should be located close to the
sensing element to prevent errors due to
stray capacitance.
Figure 17. The basic capacitive pressure sensor
consists of two plates with a vacuum between
them.
The capacitance of two parallel plates is given by:
C = µA/d
(15)
where:
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27
Pressure Measurements
µ = dielectric constant of the material between
the plates
A = area of the plates
D = spacing between the plates
If the dielectric constant of the material
between the plates isn't kept constant,
errors may result. Capacitive absolute
pressure sensors with a vacuum between
the plates are ideal in this respect. Because
the capacitance of this sensor depends only
Figure 18. A more complex capacitive pressure
on physical parameters, sensors with good
sensor can be built to detect differential pressure
performance can be constructed using
materials with low coefficients of thermal
expansion. Since the device has to be fairly large to obtain a usable signal, frequency
response may be a problem in some applications. Also, low-pressure capacitive sensors
exhibit acceleration and vibration sensitivity due to the necessity for a large, thin
diaphragm. A basic capacitive sensor is shown in Figure 17 and a more complex
differential pressure capsule is shown in Figure 18.
Piezoelectric Pressure Sensors
Piezoelectric elements are bi-directional transducers capable of converting stress into
an electric potential and vice versa. They consist of metallized quartz or ceramic
materials. One important factor to remember is that this is a dynamic effect, providing
an output only when the input is changing. This means that these sensors can be used
only for varying pressures. The piezoelectric element has a high-impedance output and
care must be taken to avoid loading the output by the interface electronics. Some
piezoelectric pressure sensors include an internal amplifier to provide an easy electrical
interface (see Figure 19).
Strain Gauge Pressure Sensors
Strain gauge sensors originally used a metal diaphragm with strain gauges bonded to it.
A strain gauge measures the strain in a material subjected to applied stress. Consider a
strip of metallic material (see Figure 20) with electrical resistance given by:
RO =
L/WT
(16)
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Pressure Measurements
where:
= resistivity
L, W, T = length, width, thickness
Figure 19. Piezoelectric sensors convert stress into an electric potential and vice versa. Sensors based on this technology are
used to measure varying pressures.
Figure 20. Strain gauge pressure sensors measure the strain in a material that is subjected to applied stress.
Metallic strain gauges depend only on dimensional changes to produce a change in
resistance. A stress applied to the strip causes it to become slightly longer, narrower,
and thinner, resulting in a resistance of:
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29
Pressure Measurements
R = (L + L) / (W R RO(1 + 3 )
W)(T -
T), or
(17)
As might be expected, the signal due to deformation of the material is small, on the
order of 0.1% of the base resistance.
Figure 21. A silicon bar can be bonded to a diaphragm to yield a strain gauge sensor with a relatively high output.
Semiconductor strain gauges are widely used, both bonded and integrated into a silicon
diaphragm, because the response to applied stress is an order of magnitude larger than
for a metallic strain gauge. When the crystal lattice structure of silicon is deformed by
applied stress, the resistance changes. This is called the piezoresistive effect. Following
are some of the types of strain gauges used in pressure sensors.
Photo 1. IC processing is used to form the piezoresistors on the surface of a silicon wafer to fabricate an integrated piezoresistive
pressure sensor.
Deposited strain gauge. Metallic strain gauges can be formed on a diaphragm by means
of thin film deposition. This construction minimizes the effects of repeatability and
hysteresis that bonded strain gauges exhibit. These sensors exhibit the relatively low
output of metallic strain gauges.
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Pressure Measurements
Bonded semiconductor strain gauge
A silicon bar may be bonded to a diaphragm to form a sensor with relatively high
output. Making the diaphragm from a chemically inert material allows this sensor to
interface with a wide variety of media (see Figure 21).
Piezoresistive Integrated Semiconductor. IC processing is used to form the piezoresistors
on the surface of a silicon wafer (see Photo 1). There are four piezoresistors within the
diaphragm area on the sensor. Two are subjected to tangential stress and two to radial
stress when the diaphragm is deflected.
They are connected in a four-element
bridge configuration (see Figure 22) and
provide the following output:
VOUT/VCC =
R/R
(18)
where:
VCC = supply voltage
R = base resistance of the piezoresistor
R = change with applied pressure and is
typically ~2.5% of the full R
Figure 22. Piezoresistive integrated semiconductor pressure
sensors incorporate four piezoresistors in the diaphragm.
When the diaphragm is deflected, two resistors are subjected
to tangential stress and two to radial stress. The four are
connected to a four-element bridge.
Etching of the back of the wafer is used
to form the diaphragm (see Figure 23).
The high output of the bonded strain
gauge is combined with the low
hysteresis of the deposited strain gauge
in this design, due to the integrated
construction and the nearly perfect
elasticity of single-crystal silicon. The Figure 23. The back of a wafer is etched out to form the
cost of the sensing element is low since diaphragm of a piezoresistive pressure sensor.
a large number of devices fit on a silicon
wafer. Typical die size is 0.1 in. square with a 50 mil square
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Pressure Measurements
diaphragm. The circuitry needed for amplification, temperature compensation, and
calibration may also be included on the same IC. Various pressure ranges are
accommodated by varying the diaphragm thickness and, for very low pressures,
by varying the diaphragm diameter. This device can be used to construct absolute, differential,
and gauge pressure sensors, depending on the
reference (see Figure 24).
Because the
sensing element is
so small (see Photo
2), the package can
have a great deal
Photo 2: This totally integrated silicon
of mounting and
pressure sensor measures 0.52 in. long
port interface
by 0.44 in. wide by 0.75 in. high,
including the port
flexibility. Also, the
small size means
that it has a wide
frequency response and may be used for dynamic
pressure measurements without concern about
errors. Mechanical vibration and acceleration have
a negligible effect.
Figure 24. Piezoresistive pressure sensors
can be configured to provide absolute,
differential, or gauge pressure readings,
depending on the reference. The diaphragm is
shown here as it deflects under applied
differential pressures.
Using a Flow Sensor to Measure Pressure
A recently developed micromachined flow
sensor (see Photo 3) represents a new
approach to low-pressure measurement in
applications where a small flow across the
sensing element may be tolerated. The sensor
consists of a thin film thermally isolated
bridge structure suspended over a cavity in
the silicon IC.
Photo 3. The recently developed microbridge mass
airflow sensor consists of a thin thermally isolated
bridge structure suspended over a cavity in the
silicon IC. The chip is 67 mil square
Because of its small size and excellent thermal
resistance, only a few mW of power are required to achieve high air flow sensitivity. By
calibrating the device with a flow restriction for a specified pressure drop, a sensor
capable of measuring pressures of a few inches of water results. This is ideal for airflow
measurement in HVAC applications, for example.
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Pressure Measurements
The sensor operates on
the principle of heat
transfer due to mass
airflow directed across the
surface of the sensing
element (see Figure 25).
Output voltage varies in
proportion to the mass air
(or other gas flow) through
the inlet and outlet ports
of the package (see Photo
4).
Figure 25. A silicon bar can be bonded to a diaphragm to yield a strain gauge
The sensor has a unique sensor with a relatively high output.
silicon chip based on
recent advances in microstructure technology.
It consists of a thin film, thermally isolated,
bridge structure containing a heater and
temperature-sensing elements. The bridge
structure provides a sensitive and fast response
to the flow of air over the chip. Dual sensing
elements flanking a central heating element
indicate direction as well as rate of flow. A
4. The output voltage of the microbridge
specially designed housing precisely directs and Photo
mass air flow sensor varies in proportion to the
controls the airflow across the sensing mass flow of air or other gas through the inlet and
ports of the package. The packaged sensor is
microstructure. Highly effective thermal outlet
1.20 in. wide by 1.24 in. long by 0.61 in. high
isolation for the heater and sensing resistors is
attained by the etched cavity air space beneath
the flow sensor bridge. The small size and thermal isolation of the microbridge mass
airflow sensor are responsible for the remarkably fast response and high sensitivity to
flows.
Pressure Switches
Pressure switches, combining a diaphragm or other pressure measuring means with a
precision snap switch, can provide precise single-point pressure sensing. Alternatively,
simple electronic switches may be combined with electrical sensors to construct a
pressure switch with an adjustable set point and hysteresis.
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Pressure Measurements
Electrical Interfacing
Care must be taken with the output from a pressure sensor to avoid corrupting the
signal by noise or 60 Hz AC pickup. If the signal must be run some distance to the
interface circuitry, twisted and/or shielded wire should be considered. A decoupling
capacitor located at the sensor and connected from the supply to ground will also filter
noise, as will a capacitor from output to ground.
For long runs, a current output sensor should be considered. These devices have a 2wire interface and modulate the supply current in response to applied pressure.
Obviously, wire resistance has no effect and noise must change the loop current, not
simply impress a voltage on the signal. The industry standard interface is:
PL = 4 mA
(19)
PH = 20 mA
(20)
Where:
PL = low pressure range limit
PH = high pressure range limit
Calibration
Manufacturers of pressure
sensors have elaborate
calibration facilities to verify
the accuracy of their
production test equipment.
These include dead weight
testers and temperaturecontrolled servo-rebalance
testers using Bourdon tubes
made from high-stability
quartz.
Dead-Weight Tester
A dead-weight tester uses
calibrated weights that
Figure 26. Dead-weight testers, used to calibrate pressure sensors,
incorporate calibrated weights that exert force on a piston which in
turn acts on a fluid to produce a test pressure. Oil-type testers like the
one shown here are commonly used for high pressures; pneumatic air
bearing devices are the usual choice for lower pressures.
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Pressure Measurements
exert force on a piston which then acts on a fluid to produce a test pressure. For high
pressures (>500 psi), oil is typically used (see Figure 26); for lower pressures, pneumatic
air bearing testers are available and are much more convenient as well as less messy to
use.
Manometer
A mercury manometer is a
simple pressure standard and
may be used for gauge,
differential, and absolute
measurements with a suitable
reference. It is useful mainly
for lower pressure work
because the height of the
column of mercury will
otherwise become unwieldy.
The difference in column
heights gives the pressure
reading (see Figure 27).
Figure 27. The mercury manometer, another calibration option for
pressure sensors, can be used on gauge, differential, and absolute
sensors with a suitable reference. The difference between column
heights gives the pressure reading. Manometers are used mainly to
calibrate sensors designed to measure in the lower pressure ranges
Low-Cost Calibration
Many of the higher
performance commercially
available pressure sensors
are furnished with
individual test data. A
sensor with excellent
repeatability and hysteresis
makes an excellent low-cost
in-house pressure
calibration reference when
combined with a pneumatic
pressure regulator and a
source of air pressure (see
Figure 28).
Figure 26. Dead-weight testers, used to calibrate pressure sensors,
incorporate calibrated weights that exert force on a piston which in
turn acts on a fluid to produce a test pressure. Oil-type testers like the
one shown
here are commonly
used for high
pressures;
pneumatic air
Biomedical
Measurements
| Bassel
Tawfik
bearing devices are the usual choice for lower pressures.
35
Pressure Measurements
Selection Considerations
Selection of a pressure sensor involves consideration of the medium for compatibility
with the materials used in the sensor, the type (gauge, absolute, differential) of
measurement, the range, the type of electrical output, and the accuracy required.
Manufacturer's specifications usually apply to
Figure 28. A sensor with excellent repeatability and hysteresis can be combined with a pneumatic pressure regulator and a
source of air pressure to yield an inexpensive in-house pressure calibration reference.
a particular temperature range. If the range of operation in a given application is
smaller, for example, the errors should ratio down. Total error can be computed by
adding the individual errors (worst-case) or by computing the geometric sum or root
sum of the squares (RSS). The latter is more realistic since it treats them as independent
errors that typically vary randomly. Following is a comparison of the two methods.
Given the following error terms:





Linearity = 1% F.S.
Null calibration = 1% F.S.
Sensitivity calibration = 1% F.S. Temperature errors are sometimes given as coefficients
per ºC referenced to 25ºC. Simply multiply the coefficient by the temperature range of
the application to obtain the total error.
Temperature error = 0.5% F.S.
Repeatability and hysteresis = 0.1% F.S.
(21)
Worst case error is equal to the sum of all the maximum errors:
Worst case error = 1 + 1 + 1 + 0.5 + 0.1 = 3.6%
(22)
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Pressure Measurements
Applications
Industrial. Fluid level in a tank: A gauge pressure sensor located to measure the
pressure at the bottom of a tank can be used for a remote indication of fluid level using
the relation:
h = P/ g
(23)
Fluid flow: An orifice plate placed in a pipe section creates a pressure drop. This
approach is widely used to measure flow because the pressure drop may be kept small
in comparison to some other types of flowmeters and because it is impervious to
clogging, which may otherwise be a problem when measuring flow of a viscous medium
or one containing particulate matter. The relation is:
(24)
In some cases, differential pressures of only a few inches of water are measured in the
presence of common-mode pressures of thousands of pounds per square inch. These
pressure sensors are built with elaborate mechanisms to prevent damage due to the
high common-mode pressures and also frequently have remotely controllable pressure
ranging.
Automotive. A wide variety of pressure applications exist in the modern electronically
controlled auto. Among the most important are:
Manifold absolute pressure (MAP). Many engine control systems use the speed-density
approach to intake air mass flow rate measurement. The mass flow rate must be known
so that the optimum amount of fuel can be injected. MAP is used in conjunction with
intake air temperature to compute the air density. This requires a 15 psia range or
higher (for supercharged or turbocharged engines). It is also desirable to include an
altitude correction in the control system, and this requires measurement of barometric
absolute pressure (BAP). Some systems use a separate sensor, but it is more common
for the MAP sensor to do double duty since it reads atmospheric pressure for two
conditions. One, before the engine begins cranking and two, whenever the throttle is
wide open.
Engine oil pressure. Engine lubrication requires pressures of 10-15 psig. The oil pump is
sized to achieve this pressure at idle and the pressure increases with engine speed. A
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Pressure Measurements
potentiometric gauge or pressure switch is used for this function since precision isn't
required.
Evaporative purge system leak detection. To reduce emissions, modern fuel systems are
not vented to the atmosphere. This means that fumes resulting from temperatureinduced pressure changes in the fuel tank are captured in a carbon canister and later
recycled through the engine. Government regulations require that leaks in this system
be detected by the onboard diagnostics system. One approach is to pressurize the
system and measure pressure decay over a fixed time interval. A 1 psig sensor is used
for this function.
Tire pressure. Recent development of the "run-flat" tire has prompted development of a
remote tire pressure measurement system. The reason is that a flat tire of this type is
difficult to detect visually and the distance over which it can be used without any
pressure is limited.
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Pressure Measurements
38
Appendix (A)
Basic Physics of Pressure
Basic Physics of Pressure Sensing1
This tutorial addresses the basic physics of pressure, three types of pressure measurement,
seven pressure sensor technologies, and three calibration techniques.
Static Pressure
Pressure, P, is defined as force, F, per unit area, A:
P = F/A
(1)
The measurement of pressure is generally associated with fluids, either liquids or gases.
A container filled with a liquid (see Figure 1) has a pressure (due to the weight of the
liquid) at a point in the liquid of:
P=
F/
A = hw
(2)
Where:
h = distance from the surface to the point
w = weight of the liquid (most liquids are nearly incompressible)
1
Source: Sensor Magazine (1998)Nov 1, 1998 - By: Robert E. Bicking
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Pressure Measurements
Figure 1. The pressure at any given point in a confined liquid is determined by the weight of the liquid and the distance from the
point to the surface.
The weight per unit volume, V, is given by:
w = mg/V
(3)
Where:
m = mass
g = gravitational acceleration
Note that this relation can be used to determine the height of the column of liquid in a
tank by measuring the pressure.
The density,
= m/V
, is given by:
(4)
Thus, the density of a liquid determines the pressure, P, exerted for a given height.
Mercury is 13.6 3 denser than water, so would exert a pressure 13.6 3 that of water for
a column of the same height. It should be noted that the pressure due to the height of a
column of liquid is in addition to the atmospheric pressure acting on the surface of the
liquid. The height of a column of liquid is:
h = P/ g
(5)
Archimedes's principle states that "A body wholly or partially submerged in a liquid is
buoyed up by a force equal to the weight of the liquid displaced." Given a block of
material submerged in a container of liquid (see Figure 2) with area A and length L, the
downward pressure exerted on the top face is:
PD = h g
(6)
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Figure 2. Archimedes' principle states that an object submerged in a confined liquid will be buoyed up by the weight of the liquid it
displaces.
and the upward pressure exerted on the bottom face is:
PU = (h + L)
g
(7)
giving a resultant pressure of:
PU - PD = L g
(8)
and the force is equal to the volume of the block multiplied by the weight of the liquid
displaced.
Figure 3. The pressure transmitted to a confined liquid is normal to the liquid's surface, as can be observed by breaching one of
the walls of the container.
A liquid exhibits nearly no shear stress, which leads to some interesting results. Pressure
is transmitted to the inside of a container normal to the surfaces, a fact that can be
most easily proved by punching a hole in a container of water and observing the stream
as it exits the hole (see Figure 3). This is important in the construction of dams, given
that they must resist the force of water. This pressure is called the static pressure.
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Pascal's law states that an increase in
pressure at any point in a liquid results in
a like increase at every other point in the
liquid. This principle is used in hydraulic
systems such as jacks and automobile
brakes and is the fluidic equivalent of the
principle of the lever, which allows large
forces to be generated easily by trading
large movement of a small piston for
small movement of a large piston (see
Figure 4).
Figure 4. Pascal's law states that an increase in
pressure at any point in a liquid causes a corresponding
increase at every other point in the liquid.
Pressure in Moving Fluids
The pressure in a moving fluid exerted parallel to the flow direction is called the impact
pressure, PI. This is due to the kinetic energy of the fluid:
PI =
VO²/2
(9)
where:
= fluid density
VO = fluid velocity
Bernoulli's theorem states that for horizontal flow the following relation holds:
PS = PO + PI
(10)
where:
PS = stagnation (or total) pressure
PO = static pressure
This relationship is useful for determining the flow velocity in a variety of applications.
Rearranging the equation gives:
(11)
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Pressure Measurements
Fluid flow velocity may be measured with the Pitot
tube assembly shown in Figure 5. The tube facing the
flow measures total pressure and the tube normal to
the flow measures static pressure. This approach is
used in HVAC applications and in aircraft to measure
flow velocity.
Figure 5. A Pitot tube assembly can be
used to measure the velocity of a moving
fluid.
Gases
Gases differ from liquids in two respects:
they are very compressible, and they
completely fill any closed vessel in which
they are placed. The nonlinear air pressure
variation with altitude shown in Figure 6 is
an example of the effect of the
compressibility of gases.
Figure 6. The compressibility of gases is illustrated by
nonlinear atmosphere pressure as a function of altitude
Dynamic Effects
Static pressure is measured under steady-state or equilibrium conditions, but most reallife applications deal with dynamic or changing pressure. For example, the
measurement of blood pressure usually gives the two steady-state values of systolic and
diastolic pressure. There is much additional information in the shape of the blood
pressure signal, however, which is the reason for the monitors used in critical-care
situations.
To measure changing pressures, the frequency response of the sensor must be
considered. As a rough approximation, the sensor frequency response should be 5-10 ×
the highest frequency component in the pressure signal. The frequency response is
defined as the highest frequency that the sensor will measure without distortion or
attenuation. Sometimes the response time is given instead. For a first-order system,
they are related as follows:
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Pressure Measurements
FB = 1/2
(12)
where:
FB = frequency where the response is reduced by 50%
= time constant where the output rises to 63% of its final value following a step input change
Another issue is the remote measurement of pressure where a liquid coupling medium
is used. Care must be taken to purge all air because its compressibility will corrupt the
waveform.
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