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7/27/2019 Pre Measuring Instruments 2013

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Pressure above

atmospheric pressure

Ordinary pressure gage

reads difference between

absolute pressure andatmospheric pressure,

Pg

Pressure less than


Atmospheric pressure

Ordinary vacuum gage

reads difference between


and absolute pressure,

Pvac

Absolute pressure that is

less than atmospheric

pressure, Pabs

Absolute pressure

that is greater than

atmospheric

pressure, Pabs

Barometer reads

atmospheric pressure,

Patm

Zero pressure

Figure 1. Illustration of terms used in pressure measurements.

dead-weight pressure gages;

electrical-type pressure gages: piezoelectric pressure gages, capacitance pressure gages, strain-type pressure gages.

Here is the classification of instrumentation for pressure measurements with respect to the typeof the measured pressure:

pressure gages, for measurements of pressures above atmospheric pressure;

vacuum pressure gages, for measurements of pressures below atmospheric pressure;

vacuum manometers, for measurements of both pressures above and below atmosphericpressure;

barometers, for measurement of atmospheric pressure;

differential pressure and vacuum gages, for measurements of difference of pressures.

2. U-tube liquid filled manometers

These manometers are used for measurement of gauge pressures (up to 0.1 MPa), vacuumetricpressures (down to 0.1 MPa below atmospheric pressure) and for differential pressures of liquids

and gases. The principle is based on the static balance between the measured pressure and the

head of the liquid column. Fig. 2 shows the schematic of this manometer. A glass tube1, bended


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to the U-shape, is filled by one half of its volume with liquid 2 (water, mercury). This tube is

placed vertically, and a scale 3 (usually in mm) is attached along its height. Pressures 1P and

2P are supplied to legs of the tube, and levels of liquid in the legs change their position.

When static balance between a measuring pressure and the head of the liquid column is reached,this pressure can be evaluated according to the equation:

)( 2121 hhgHgPPP locloc +=== , (3)

where,

1P and 2P - pressures supplied to the legs of the manometer, Pa ;

1h and 2h - deviations of liquid levels from the zero point of the scale in two legs of the

manometer, m ;21 hhH += - total length of the liquid column corresponded to the measuring differential

pressure, m ;

- density of liquid filled the U-tube,3

m

kg;

locg - local gravitational acceleration, 2s

m.

We need always make two readings of the liquid level, namely, in each leg of the tube, because

in reality due to non-uniformity of the tube diameter along its length, values of 1h and 2h are

not equal. As the result of such reading the error introduced during pressure measurement will be

reduced. When this type of manometer is used for pressure measurements three cases may take

place:

1). 1P is above atmospheric pressure, atmPP =2 . In this case the manometer measures the

difference between absolute and atmospheric pressures: )( 211 hhgPP locg +== .

2). 2P is below atmospheric pressure, atmPP =1 . In this case manometer measures the

difference between atmospheric and absolute pressures: )( 212 hhgPP locvac +== .3). In this case the equation (4.3) refers to measurements of differential pressures.

Since the gravitational acceleration is used for the evaluation of pressure, then, when using U-

tube manometers, it is necessary to introduce correction which takes into account the difference

between gravitational acceleration in the place where this manometer was calibrated from thatwhere it is used.

Another source of the error is the deviation of liquid temperature in the tube from that

temperature when this manometer was calibrated. Due to thermal expansion of the liquid in thetube the volume of liquid will change and this inevitably introduces an error.

But the most common mistake is made by not correct reading the scale in respect to the meniscusof liquid in legs of the tube. Fig. 3 gives examples how operator should make readings when

using U-tube manometer with various liquids. We should always read a surface of the

meniscus in its centre. In the case with water - in the bottom, and in the case with mercury - inthe top of the meniscus. But in everyday industrial measurements the first two corrections

(gravitational and thermal) are not always used, whereas the last one (the meniscus correction)

must always be taken into account.


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0

P1P2

h2

h1

H

1

2

3

Figure 2. Liquid filled U-tube manometer.

When one measures low pressures several modifications of U-tube manometer are used,

namely, well or reservoir manometer, inclined manometer, absolute pressure gauge.

Water wets glass Mercury does not

wet glass

Figure 3. Correct reading of the U-tube manometer.

3. Bourdon and diaphragm gages3.1. Bourdon gauge

The most widely used in industry for pressure and vacuum measurements (from 20 kPa to 1000

MPa) is a pressure gauge with sensitive element made of a metallic (various stainless-steel

alloys, phosphor bronze, brass, beryllium copper, Monel, etc.) Bourdon tube 1 (see Figure 4).The tube was named after its inventor, E. Bourdon, who patented his invention in 1852. This

tube has an elliptical or oval cross-section AA and has the shape of a bended tube. When the

pressure inside the tube 1 increases, its cross-section dimension 1b also increases by the value of

1b , whereas the cross-section dimension a1 reduces its length by the value of 1a . Therefore,the tube tends to straighten (if pressure has increased) or twist (if pressure has decreased, forexample, during vacuum measurements), and the tip 2 of the tube moves linearly with applied

pressure. The movement of the tip is transmitted to the pointer3

through a mechanism4

. The


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tube tends to return to its original shape (the pointer returns to the starting position) after pressure

is removed. A relationship between the value of the tip movement x and the measuredpressure is linear, so the scale of this pressure gauge is uniform.

P

2

1

3

4

a1

a1- a1

b1

A - A

A

A

b1+ b1

Figure 4. Bourdon tube pressure gauge.

Some degree of hysteresis still exists during operation of these pressure gages, because metals

cannot fully restore their initial elastic properties. If we have two Bourdon tubes made of the

same metal, the tube with a bigger radius and a smaller thickness of the wall will have higher

sensitivity. An accuracy of a typical Bourdon-tube pressure gauge is equal to 1%, whereas a

specially designed gauge may have better accuracy which varies from 0.25 to 0.5%.

3.2. Diaphragm gauge

Another type of pressure gauge, which utilises elastic-element properties, is a diaphragmpressure gauge. These gages are used when very small pressures (from 125 Pa to 25 kPa) are to

be sensed. Fig. 5 shows a sensitive element for this type of pressure gauge.

A flexible disc 1 made of trumpet brass, or phosphor bronze, or beryllium copper, or titanium, or

tantalum, etc., is used to convert the measuring pressure to the deflection of the diaphragm.

Deflection vs pressurecharacteristic should be close to linear as much as possible. In reality for

a flat diaphragm this characteristic is non-linear. So, flat membranes are not used as sensitiveelements. To linearise this relationship special diaphragms with concentric corrugations 6 are

designed. Linearisation of a static characteristic of the membrane can be achieved by using a flat

spring 2, which is connected, to the diaphragm through the mechanism 3. The movement of themechanism 3 is transmitted by the link4 to a pointer of the gauge. The measuring pressure is

supplied to the pressure chamber5and causes the diaphragm to move upwards until the force

developed by this pressure on the diaphragm is balanced by the force acted from the spring. Toincrease the sensitivity of this type of pressure gauge, we may increase the diameter of the

diaphragm, to lengthen the spring, to change the material of the diaphragm and the spring to

more elastic, to increase the depth and the number of corrugations of the diaphragm.


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P

1

2

3

4

5

6

Figure 5. Sensitive element of a diaphragm pressure gauge.

When pressure is applied to both sides of the membrane, then the resultant reading isproportional to the differential pressure. The space above the diaphragm is connected to

atmosphere, so the diaphragm separates a measured media from the environment. In other words,

it serves as a fluid or gas barrier or as a seal assembly, thus preventing contact of corrosive andaggressive fluids with pressure elements.

Accuracy of diaphragm pressure gages varies from 1.0 to 1.5% of the span.

3.3. Bellows pressure gauge

These pressure sensitive elements are usually made of stainless steel; phosphor bronze, brass andare used for pressure measurements for pressures up to 6 MPa. Bellows sensors have large

displacement sensitivity. Figure 6 shows this type of sensor.

The effective area of a bellows can be calculated using the following formula:

4

)( 221 RRAef+

=

, (4)

When pressure is applied to the internal surface of a bellows the force is developed

according the formula: efef PAF = . (5)

4.4. Dead-weight pressure gagesThese are the most accurate pressure gages, so they are used mostly for calibration of other

pressure gages or for measurements, when high accuracy is required (for scientific purposes).The accuracy achieved may vary from 0.01 to 0.02% of the measuring pressure (these

instruments are used for calibration of dead-weight pressure gages), those with the accuracy of

0.05% are used for calibration of other types of pressure gages. The range of measured

pressures varies from 0.1 to 250 MPa. Figure 7 shows a schematic of a dead-weight pressuregauge.

A cylindrical piston 1 is placed inside a stainless-steel cylinder 2. The measuring pressure issupplied through the vent 8 to the fluid 4 of this gauge. To avoid contact of a measured media

with the fluid in the gauge, U-shape separating tubes (made of stainless steel with a thick wall)

are used. These tubes are filled by one half of their volume with mercury. The measuring

pressure spreads throughout the fluid in the dead-weight gauge system. Transformer mineral oil


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and castor oil are used for measurements of low (up to 6 MPa) and high (up to 250 MPa)

pressures, respectively. The measuring pressure by acting on the piston develops a force, whichtends to bring the piston upwards. The gravitational force developed by calibrated weights 3 can

balance this force and the piston itself. This force acts downwards. The balance should be

achieved for a certain position of the piston against a pointer 9 of the stainless-steel cylinder. A

manual piston pump 5is used to achieve approximate force balance (to increase pressure in the

system), whereas a wheel-type piston pump6

serves for accurate balancing. A Bourdon-typepressure gauge 7is used for visual reading of pressure, but not for pressure measurements in this

case.

2R2

2R1

P

a

r

Figure 6. Bellows pressure sensitive element.

1

2

3

4

5

6

7

8

2

9

Figure 7. Dead-weight pressure gauge.


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Thus measuring pressure can be evaluated using the formula:

locw

S

mmP

, (6)

where, P - measuring pressure, Pa ;

pm - mass of the piston, kg;

wm - mass of calibrated weights, kg;

locg - local gravitational acceleration, 2s

m;

pS - cross-section area of the piston, 2m .

In reality, this formula is more comprehensive in order to achieve high accuracy. Therefore,several corrections should be introduced, namely:

the correction for the variation of piston cross-section area with variation of its

temperature;

the correction which takes into account the difference between local

gravitational accelerations of the place where this dead-weight pressure gaugewas calibrated and where it is used for pressure measurements;

the buoyancy-type correction takes into account the weight of the air displaced

by the piston and calibrated weights;

we need to reduce friction of the piston inside the cylinder by spinning the

weight platform with the piston to keep the piston floating;

head of the oil should be constant in every measurement, this corresponds to a

certain position of the piston in the cylinder.

5. Piezoelectric pressure transducers

The principle of these pressure transducers is based on the well-known phenomenon, that whenan asymmetrical crystal is elastically deformed along its specific axes, an electrical charge isdeveloped on its sides. The value of this charge is proportional to the force applied to the crystal,

and, therefore, to the pressure under measurement.

Fig. 8 shows piezoelectric crystal circuit. An electrical charge developed on the sides of thecrystal is converted into a voltage-type signal using a capacitor. This voltage is proportional to

the electrical charge developed, and to the pressure to be measured. Piezoelectric sensors cannot

measure static pressures for more than a few seconds, but they have a very quick response whenmeasure dynamic pressures.

+

-

+

-

RC

P

U

Q

Figure 8. Piezoelectric pressure sensor with electrical circuit.


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Synthetically developed quartz crystals (barium titanate, lead zirconate) have similar propertiesas natural single crystal (quartz). But natural quartz still is the perfect material for manufacturing

piezoelectric sensitive elements, because it has perfect elasticity and stability, it is insensitive to

temperature variations and it has high insulation resistance.

These pressure transducers are used for measurements of hydraulic and pneumatic pulsations,flow instabilities, fuel injection, etc.

A variation of the electrical charge piezq , developed on the surfaces of a piezoelement for a

change in the input variable measured pressure, P. Now we should measure this electricalcharge. For this purpose two metal electrodes are attached to the opposite sides of a piezoelectric

crystal. Thus, a capacitor is formed. The value of capacitance of this capacitor can be evaluated

as follows:

d

AC xpiez

0= , (7)

where,

piezC - electrical capacitance of the piezoelement, F (Farad);

m

pF,85.80 = - the permittivity of vacuum, 1pF=10

-12 F;

- the relative permittivity of the material of the piezoelectric crystal,

this is the dimensionless parameter;

xA - cross-sectional area of the piezoelectric sensor in the direction,

perpendicular to the axis X, 2m ;

d - the thickness of the piezoelectric crystal in the direction,perpendicular to the axis X, m.

The relative permittivity, also called dielectric constant, for various piezoelectric materials isgiven below:

for quartz (natural piezoelectric material) 5.4= ; for tourmaline (natural piezoelectric material) 6.6= ; for lead-zirconate-titanate (man-made piezoelectric ceramic material) 1500= ; for lead metaniobate (man-made piezoelectric ceramic material) 250= .

It is also noted in the above mentioned reference, that natural piezoelectric materials have very

low charge to force sensitivity, and therefore man-made piezoelectric ceramic materials are usedas sensing elements:

charge sensitivity to force for quartzN

pC,3.2 ;

charge sensitivity to force for tourmalineN

pC,9.1 or

N

pC,4.2 ;

charge sensitivity to force for lead-zirconate-titanateN

pC,265 ;


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charge sensitivity to force for lead metaniobateN

pC,80 .

We need to develop an electrical circuit which will allow us to convert variations of the

capacitance of the piezoelectric sensor into the variation of an easy measurable electrical signal,voltage, for example. Such equivalent electrical circuit was developed, and is named after

Norton.

+

-

RloadCpiez

P

Ccable Vload

1 2 3

Ipiez

Figure 9. Norton equivalent electrical circuit for piezoelectric pressure/force measurements.

1- piezoelectric element, 2 connecting cable, 3 recorder.

The piezoelectric element can be represented as a current source (or a charge generator) which is

connected in parallel with a capacitance piezC . Then, this element is connected to a voltage

recorder via connecting cables, which have the capacitance cableC . A recorder has a resistive

load, loadR . The voltage measured across loadR is equal:

ZIV piezload *= , (8)

where,

loadCC RRRZ cablepiez

1111++= - the impedance of three resistances connected in

parallel, .Ohm

According to the definition, the capacitance is equal to the ratio of the charge to the voltageacross the capacitor plates, according to:

V

qC= , (9)

Lets consider capacitance

load

piez

piezV

qC = , (10)

After differentiating both sides of (10) we can get:


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load

piezpiez

Vdt

dq

dt

dC 1*= , (11) or

load

piez

piezV

I

dt

dC = , (12)

or, according to the Ohms Law,piez

piezRdt

dC

1= . (13)

Similar we can getcable

cableRdt

dC

1= . (14)

Substitution of (4-74) and (4-75) into (4-69) will give:

sCCR

RIV

cablepiezload

loadpiezload

)(*1*

++= , (15)

where,

dt

ds = - the Laplace operator.

Expressing variables loadV and piezI in deviation form and applying the Laplace transform to

(15) we can get:

sCCR

RsIsV

cablepiezload

loadpiezload

)(*1*)()( ''

++= , (16)

The transfer function for the Norton equivalent electrical circuit for piezoelectric pressure/force

measurement system (see Figure 9) is as follows:

sCCR

R

sI

sVsG

cablepiezload

load

piez

loadIV

)(*1)(

)()(

'

'

++==

, (17)

According to the definition:

dt

dqI

piez

piez =, (18). Expressing these variables in deviation form and applying the

Laplace transform to (18) we can get:

ssqsIpiezpiez

*)()( '' = . (19)

The transfer function relating current and charge of the piezoelectric sensor is as follows:

ssq

sIsG

piez

piez

qI ==)(

)()(

'

'

. (20)

The transfer function relating the voltage loadV and the measured pressure P can be determined

as follows:

)(*)(*)()()( sGsGsGsGsG IVqIPqPVoverall == . (21)


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After substitution of1

21)('

)()(

2

2

'

++

==

ss

K

sP

sqsG

nn

xpiez

Pq

, (17) and (19) into (21) we can

get an expression for an overall transfer function of the piezoelectric pressure/force measurement

system:

sCCR

Rs

ss

K

sI

sV

sq

sI

sP

sqsG

cablepiezload

load

nn

x

piez

load

piez

piezpiez

PV

*)(*1**

121

)(

)(*

)(

)(*

)('

)()(

2

2

''

''

++++

=

==

(22)

6. Capacitance pressure transducersFig. 10 presents a transducer for sensing and transmitting differential pressure. Pressures to be

measured act on isolating diaphragms 1 and 2 and are transmitted through a silicone oil 3, which

fills the system, to a sensing diaphragm 4. This sensing diaphragm is balanced by two forcesdeveloped by measured pressures and presents the sensitive element. Capacitor plates 5and 6detect the position of the sensing diaphragm, which moves to the left or to the right, and, thus,

the differential pressure applied to the sensitive element. The change in electric capacitance is

electronically amplified and converted to the standard electrical analog or digital output signal,which is directly proportional to the difference of pressures. In order the capacitance transducer

be able to measure comparatively low pressures, the device should produce about 25% change in

capacitance for a full-scale pressure change. These transducers have low mass and high

resolution. However, they are slightly dependent on temperature variation. Newly developed all-silicon capacitive pressure sensors have better thermal stability.

1 2

4

5

6

3

P2

P1

Figure 10. Variable capacitance differential pressure transducer.

Variable separation capacitance sensors have non-linear relationship between electrical

capacitance and the movement of the separating membrane according to the formula:


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ad

AC

+=

0 , (23)

where,

C - the electrical capacitance of the pressure sensor, F (Farad);

m

pF

,85.80=

- the permittivity of vacuum, 1pF=10

-12

F;

- the relative permittivity of the insulating material between plates of

the capacitor, this is the dimensionless parameter;

A - the cross-sectional area of the capacitor plate, 2m ;

d - the distance between the capacitor plates, m ;a - variation of the distance between the capacitor plates, m .

A three-plate differential version of the capacitive pressure sensor doesnt have such

disadvantage (see Figure 11).

Two fixed plates form two capacitances with the moving separating plate/membrane as follows:

ad

AC

+=

01 , (24) and

ad

AC

=

02 . (25)

2d

a

d-a

d

d+a

Figure 11. Three-plate differential pressure/displacement sensor(.

Figure 12 shows an a.c. deflection bridge for the detection of variations of capacitances.

Vcd

Z2

Z3

a b

d

c

Vab

Z4

Z1

I1I1

I2 I2

Figure 12. a.c. deflection bridge.

In this bridge:


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1

1

1

CjZ

= , (26)

2

2

1

CjZ

= , (27) RZZ == 43 ,(28)

where,

1Z and 2Z - reactive impedances, Ohm;

3Z and 4Z - resistive impedances, Ohm.

When 0=cdI , then cdV is called an open-circuit voltage of the bridge. According to the

Kirchoffs laws we have:

3141 ZIZIVab += , (29) 2212 ZIZIVab += . (30)

Let potential at 0=bV , then:

( ) ( )

d

Va

adad

adV

ad

A

ad

A

ad

A

V

CC

CV

RR

R

CjCj

CjV

ZZ

Z

ZZ

ZVZ

ZZ

VZ

ZZ

V

ZIZIZIVZIVVVV

ab

abab

abab

ababab

ababdccd

2*

2

1

11

1

*2

1*

2

1*

11

1

*

*

00

0

21

2

21

1

43

4

21

14

43

1

21

41121241

=

=

+

+

=

+

+

=

=

+=

+

+=

=

+

+=

+

+=

====

(31)

So, the relationship between cdV and a is linear.

pre measuring instruments 2013

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