Meassurement and Instrumentation

8/9/2019 Meassurement and Instrumentation

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WELLCOME ! I am:

Andrey Elenkov Assoc. Prof. Dr.

Technical University of SofiaRoom 2448

Contacts / Councils :Tuesday 12:00 - 14:00hThursday 12:00 - 14:00h


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Measurement and

Instrumentation 4-th semester – Part 2 Lectures – part 2

15 hours/ 7 weeks – tutor Assoc.Prof.Dr. Andrey Elenkov

Laboratory works – start 3-th week

Exam – end of the semester

part 1 + part 2 – R. Dinov, (A. Elenkov)


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Sources:Main:

Measurement and Instrumentation – Part 2, edited by N.Kolev,Technical University of Sofia, 2006

Measurement and Instrumentation – Part 1, edited by N.Kolev, Technical University ofSofia, 2006

John Bentley, Principles of measurement Systems, Longman Scientific @ Technical 1992 Doebelin E.O. Measurement Systems, Application and Design, IV edition, McGraw-Hill

Publishing Company, 1990 Galeyer J.F.W., C.R. Shotbolt, Metrology for Engineers, Cassel Publishers Limited, London,

1990 Anthony D.M. Engineering metrology. Pergamon Press, oxford, 1992

Auxiliary: George Barney, Intelligent Instrumentation – microprocessor applications in measurement and

control, Prentice Hall International, University Press, Cambridge, 1990. John Fulcher, Microcomputer Systems – Architecture and interfacing, Addison- Wesley Publishing

company, 1991 Optical Methods in Engineering Metrology, Edited by D.C.Williams, Chapman&Hall, 1993 Dakin J., Culshaw B., Optical Fiber Sensors: Principles and Components, vol.1, Artech House, Inc.,

Norwood, 1988


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Part 2

Andrey Elenkov


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8. DISPLACEMENT AND MOTION

MEASUREMENTS 8.1. BASIC DEFINITIONS AND UNITS

Length is probably the most measuredphysical parameter. This parameter isknown under many alternative names:

displacement, movement, motion

The displacement may determinethe extent of a physical specimen,or it may establish the extent of amovement. It is characterized by

the determination of a componentof space .

The position of a point P can bedefined in various coordinatesystems. Cartesian coordinatesystem is usually employed inmanufacturing engineering .


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In each instance the

general position of a point Pwill need threemeasurement numbers,each being measured byseparate sensing element.

Two measurement numbers

are needed to determinethe position of a point in aplane (P1 ) and only onesensor is required for lengthmeasurement in single axis(P").


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The basic equations related to lineardisplacement, velocity andacceleration are as follows:

- displacement

- velocity

- acceleration

The corresponding relationships for

angular motion are:

- displacement

- velocity

- acceleration

])[( mt f S ]/[ sm

dt

dS v

]/[ 22

2

smdt

S d

dt

dva

])[( rad t f ]/[ srad

dt

d

]/[ 2

2

2

srad dt

d

dt

d a


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MEASUREMENTS 8.2. INDUSTRIAL RANGE OF LENGTH MEASUREMENTS

8.2.1. General Length Measurements

A large proportion of industrial range of length measurements can beperformed quite adequately using simple sensors in conjunction withadvanced electronic interrogating and measurement circuitries. A number ofdevices that are basically linear and angular displacement – sensitive islisted:Resistance potentiometers; Resistance strain gauges;

Inductive sensors; Variable inductance sensors;Linear variable differential transformers; Variable reluctance sensors;Electromagnetic generating se nsors; Capacitance sensors;Differential capacitance sensors; Hall - effect sensors;Magnetoresistive sensors; Photoelectric sensors;Optical grating sensors, both incremental and absolute;

Fiber - optic sensors


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Some sort of displacement measurements are covered in Chapter 9 formicro - displacements and in Chapters 10 and 11 and 12 for comparativelysmall - scale displacements.

One field of interest for manufacturing engineers is the measurement ofdisplacement and positioning of numerically controlled and computerized

numerically controlled (NC and CNC) machine tools , robots and flexiblemanufacturing systems (FMS).

The fundamental measurement task related to the positioning of NC andCNC machine tools, robots and FMS is the measurement of linear andangular displacements .


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The measurement of angular displacements basically are accomplished byrotary encoders. Such optical grating sensors are commercially availablewith up to 20 000 pulses per revolution. Both incremental and absolutesensors are available with built - in advanced electronics providinginformation for position, displacement, velocity and direction. Their dynamicresponse is up to several meters per second and meets successfully theentire requirements of manufacturing engineering.

Linear displacement measurement at NC and CNC machine toolscan be accomplished in two ways: indirectly by using rotary sensors inconjunction with the corresponding lead screw or directly by using lineardisplacement sensors.


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The principle of measurement of linear displacement by rotary

sensor is depicted in a simplified form in the figure.


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MEASUREMENTS8.2.1. General Length Measurements

The driving motor rotates the leadscrew. The slide has a nut workingwith the leadscrew. When the leadscrew rotates, the nut (and theslide itself) is forced to move linearly in a direction corresponding tothe direction of rotation of the leadscrew.


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MEASUREMENTS8.2.1. General Length Measurements

The displacement of the slide is proportional to the product of the number of the leadscrewangular displacement and of the step - constant of the leadscrew thread:

- ∆l is the displacement of the slide [m] - k

tis the step-constant of the leadscrew thread

- θ is the angular displacement of the leadscrew ][mk l t


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The angular displacement is measured by a rotary displacement sensor.Thus, by measuring the angular displacement we can obtain the relevantinformation for the linear displacement. The fundamental disadvantage ofthis approach is caused by the backlash which normally exists in the" leadscrew and nut" junction. Any leadscrew error due to the pitch,backslash, torsional wind-up or end float will result in a loss of accuracywhen rotary sensors are used. These hindrances can be drastically

minimized by using recirculating -ball leadscrew - and - nut arrangement.

The load, the friction and the backslash are reduced by introducing ballingfriction in place of sliding friction in the leadscrew nut. Such devices arecommercially available in broad range of sizes and driving power,designated basically for CNC machine tools, robots and photoplotters.


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8. DISPLACEMENT AND MOTIONMEASUREMENTS

8.2. INDUSTRIAL RANGE OF LENGTH MEASUREMENTS

8.2.2. The inductosyn

The Inductosyn is ineffect, a linear transformer.The philosophy of themetamorphosis of the ordinarytransformer leading to aninductosyn-type linear

displacement sensor isillustrated in the Figure. As shown in Fig. (b) the

windings of the ordinarytransformer are modified inrectangles.


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The primary and secondarycoils can be placed in twoparallel plains, the primary coil(called scale) being fixed andthe secondary coil (calledslider) being movable – (c).


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Both coils are produced by theprinted - circuit – board techniques(d) with one important difference:metal plates are utilized instead ofinsulating plates. The printed coilsare electrically isolated from themetal plate base.

Fig. (e) illustrates the assembly ofthe slider and the scale withspacing δ . One plate is fitted tothe machine bed and the other tothe machine slide, with an air gapδ maintained between them at aconstant value of about 0.2 mm.


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The principle of operation of the linearinductosyn is depicted in Figure. Thevoltage induced in the secondary coil foran ordinary transformer is (8.1):

k is the transformation ratio of thetransformer

v s = V sp sin ωt is the transformerexcitation voltage

V sp is the peak value of the transformersupply voltageω=2πf is the frequency of the excitationvoltage . For various reasons f is in the

frame of 5 to 10 kHz.

t kV kvv sp s sin0


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From equation (8.1):

If the secondary coil is displaced from its"optimal" position (shown in Fig. a) thefactor V op also will change due to thechanging of the coupling effect betweensecondary and primary coils (say due tothe change of the mutual inductancebetween the two coils).

The physical phenomenon of this effect is

shown successively in Fig. (b to h)

spop kV V

t V vop sin0


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The change of the secondary voltagepeak value with the displacement isdepicted in Fig. (h)

Taking into account the relationshipf*T *= 1 we get

l V v op sin0Ω is the angular velocity

∆l is the displacement

l T

V v opop *2

sin

l s

V v opop 2sin l

st V v opo

2sinsin


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The draw back of the above inductosyndesign is that is still not capable ofsensing the direction of displacement.

To detect direction of motion the slider isdesigned to include a second coil,displaced s/4=π/2 from the first coil.

With both a sine and a cosine outputavailable the inductosyn electronics cansense the direction of motion.

l

s

t V v opo

2sinsin1

l s

t V s

l s

t V v opopo

2

cossin4

2sinsin2


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8.2.3. Laser Interferometer

The laser interferometer is,without doubt, the superiorlength measuring instrumentfor general purpose industrialwork with a range (parts ofmicron up to more than 10 m)not provided by any other

length sensor. The interferometer is

illustrated in Figure assuggested by Albert Michelson(1852 -1931).


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8.2.3. Laser Interferometer

One reflector can be traversed alonga length to be measured . When themotion is a distance ∆ l, the path of itslight beam increases by 2∆ l . Thenumber of successive dark fringesthat occur at the photodetectorduring this motion is equal to thenumber of wavelengths in the pathchange:

∆l - is the moved distance [m] λ - is the wavelength [m] N - is the number of wavelengths

N l 2


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8.3 SUMMARY

Length is the most measured physical parameter, known also asdisplacement, movement, motion. Some times the length is affiliated

with the direction of the motion, as well as with the velocity and theacceleration of motion. The industrial range of length measurement islimited from 10-8 to 102 m.

A large proportion of industrial range of length measurements can beaccomplished using the sensors described in Chapter 7 – III-thSemester.

Angular displacements are measured by rotary sensors. Lineardisplacements are measured by rotary sensors in conjunction with theleadscrew or directly by linear - sensitive sensors.

The inductosyn is the most sensitive and widely used linear sensor forcomputerized numerically controlled CNC machine tools.

The laser interferometer is the superior length - measuring

instrument for generalpurpose industrial works.


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9. STRAIN, STRESS AND FORCEMEASUREMENT

9.1 INTRODUCTION

9.1.1 Strain, Stress and Force

Consider a free solid body (simply arod) in uniaxial tension stretched by a

force as shown on Figure. Due to the force applied to the rod,

normal to the area s , the rodincreases in length by ∆l . The ratio ofthe change in length of the rod(which results from applying the loadF) to the original length l is the axial

strain defined as:

ε is the axial strain [-] ∆l is the change of length [m] l is the original unloaded length [m]

l

l


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9.1 INTRODUCTION

9.1.1 Strain, Stress and Force

In many handbooks strain is reportedin units of 10-6 m/m. These units are

equivalent to a dimensionless unitcalled a microstrain ( μ s).

Typically, ε = 10-6 + 10-3 (i.e. 1 to1000 μ s ). When the force is appliedto the body as shown in Figure thematerial is subject to a stress σ (i.e.the internal force per unit area):

σ is the stress [N/m2]

F is the force applied [N] s is the cross-sectional area of the

rod [m2]

s

F


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9.1 INTRODUCTION

9.1.1 The Hooke’s Law

The stress - strain relationships arevery important for understanding the

properties as well as the behaviour of amaterial under load.

The general relationship is shown inFigure. For the linear region thestress/strain ratio is a constant figure(the Hook's law):

σ is the stress of the material [N/m2]ε is the strain of the material [-] or[μs]

E is the modulus of elasticity or Young's modulus. Typically, E= 1011 [N/m2] for carbon steel.

const l l

E


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9.2 STRAIN MEASUREMENT

The measurement of strain is of particular importance for a broad range ofengineering design, control, measurement evaluation and maintenanceactivities.

As mentioned above the strain in the linear region (which is the usual case)is comparatively very small (10-6 to 10-3). This is in the range of 1μm to1000 μm change for an 1 m original length. Direct measurements of suchsmall relative changes are both extremely difficult as well as non-accurate.

The application of strain gauges offers a good solution for this problem sincethe bridge measurement circuits offer an easy and accurate measurement ofthe strain gauge resistance change associated with the strain change.

Once we can transfer the strain of the body under measurement to equalstrain of a strain gauge we can measure the strain of the strain gauge by aWheatstone bridge, i.e. the strain of the body under measurement itself.

This strain transfer is possible if we can fix properly the strain gauge to thebody. The strain gauge fixed adequately to a solid body is known asbonded strain gauge.


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9.2 STRAIN MEASUREMENT 9.2.1 Installation of the Strain Gauge

The surface where the strain gauge is to be fixed must be thoroughly cleaned,probably best by abrasion followed by chemical degreasing.

The next step is the fixation of the gauge by a special cement. Cementscommonly used are cellulose nitrate (up to 100° C), epoxy (up to 200° C) andceramic (above 200° C), where special techniques must be used.

Gauge manufacturers usually provide a particular cement for their strain gaugesaccompanied with the relevant techniques descriptions.

After the gauge is fixed down, its leads should be fastened in position and

accordingly connected. It is most important for leads to be mounted securely (bysoldering or spot-welding) in order to withstand the vibrations they may besubjected.

In practice there are more failures of leads than in strain gauges themselves.Unless the installation is in a friendly environment it must be protected bycovering with wax, rubber or similar protective materials. Moisture could causecorrosion and electrical leakage conductance.


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9.2.1 Example 9.1 Assume that a 10 MΩ leakage resistance is introduced in parallel

with a 100 Ω strain gauge (G=2) due to the moisture. Calculate

the false strain indication due to the above leakage resistance. Solution

The changed equivalent resistance of the strain gauge will become (10 M Ω in parallel to 100 Ω ):

The absolute change of the resistance is

The relative resistance change is

R

R

R

R

Gl

l

2

11

10010

10010

7

7

I s R

10010

10010100

7

7

I s R R R

5

7

77

7

1010010

101

100

10010

10010100

R

R

s R

R 510.510

2

1

2

1 65


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9.2 STRAIN MEASUREMENT 9.2.2 . Measurement Circuits for Strain Gauges

For measurement of its small resistancechanges a strain gauge is generally

connected in a Wheatstone bridge(balanced for laboratory measurements orunbalanced for industrial multitaskapplications) - Figure.

∆V 0 is the bridge output voltage [V]

Vs is the bridge supply voltage [V]

∆R SG is the absolute change of theresistance of the strain gauge dueto the strain of the strain gauge (i.e. thestrain of the body undermeasurement) [Ω]

R SG is the original resistance of the strain

gauge [Ω]


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The bridge circuit may be energised withDC voltage or current or AC voltage or

current. The last offers some advantagesof avoiding errors from thermocouplepotentials that can arise in the leads whenthe junctions of dissimilar metals are atdifferent temperatures. The outputvoltage from the above bridge withcommon metallic strain gauge is quitesmall (a few μ V to a few mV), therefore

an amplification is needed, usually anoperational amplifier with very high inputimpedance. The fixed stable resistors arenormally with equal value

which for various reasons is a commonpractice.

SG R R R R R 0432


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The output voltage V 0 for a balanced bridge is zero. The out-of-balanced voltageV 0 (assume high input resistance operational amplifier or high input digital

voltmeter) is:

In order to calculate the output voltage change, relevant to the strain gaugeresistance change we first take the natural logarithm of both parts of theequation:

oSG

SG s

oSG

SG

s

s

SG

SG

s

s

SG

SG

s

bcacab

R R

R RV

R R

RV R

R R

V R

R R

V

R R R

V R

R R

V V V V V

0

0

000

4

432

0

22

1

000 lnln2

lnln R R R RV

V SGSG s


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Next we take the partial derivatives:

which gives (V s =const; R 0=const):

Finally

In case R 0=R SG we get:

Since the changes are very small we can take the finite increments:

000

0

lnln2

ln R R R

R R R

V

V V

V SGSGSGSG

s

s

SGSGSGSG

SG

SG

SG

dR R R R R

RdR

R RdR

R RV

dV

00

0

000

0 2110

SGSGSGoSGSG s dR

R R R R R

R R R RV dV

00

000

22

SG

SG s

R

dRV dV

40

SG

SG s

R

RV V

40


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9.2.2 Example 9.2 Calculate the strain of a solid body ε

SB with properly bonded strain gaugeunder the following conditions: the bridge supply voltage V S=10V; thegage factor of the strain gauge is G =2. The bridge output voltage change

(due to the strain) measured by a digital voltmeter is ∆V o =500 μ V.

Solution The strain of the solid body ε SB is equal to the strain ε SG of the bonded on its surface strain gauge:

The strain of the strain gauge is

The strain gauge relative resistance change is

And finally, the strain of the solid body is

SGSB

sSG

SG

V

V

R

R 04

SG

SGSG

R

R

G

1

00 441 V

GV GV

V

R

R

G s sSG

SGSG

48

0 10

10.2

10.500.44

s

SGSB

GV

V


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9.2 STRAIN MEASUREMENT 9.2.3. Temperature Effect

Since the strain gauge is a metal wire, its resistance will vary with thetemperature:

This temperature resistance change will be sensed by the measuring circuitand will be indicated wrongly as a strain

The above indication is a temperature error. In order to avoid such an error atemperature compensation is required.

T f RSGt

SG

SGt t

R

R

G

1


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9.2 STRAIN MEASUREMENT 9.2.3. Temperature Effect

For temperature compensation a DUMMYstrain gauge (identical to the active one)

is used in the adjacent leg of theWheatstone bridge.

This dummy strain gauge is in"temperature contact" with theactive strain gauge, i.e. thetemperature of both gauges is equal.

If we repeat the calculations for thecircuit shown in Figure we shall see thatthe temperature resistance change of thelive strain gauge will be completelycompensated by the correspondingtemperature resistance change of thedummy strain gauge, thus thetemperature error will be eliminated.


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9.2 STRAIN MEASUREMENT 9.2.3. Lead Wire Effect

Gauges are often mounted somedistance from their associated

interrogating circuitries - figure.

Therefore care must be taken thatthe long leads involved do notintroduce error - these leads aremetal wires and are subject totemperature resistance change.


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When the live strain gauge is located at a remote placethe resistance r+r=2r of itsconnecting leads are part of the gauge bridge leg and any changes in their resistance

will be indistinguishable from the live gauge resistance change, thus introducing leadwire effect error. This can be avoided by so called three leads configuration as shown inFigure.


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As shown in Figure the connecting leads resistances (as well as their temperaturechanges) are neutralised since they are in the adjacent legs of the bridge circuit (r 1 and

r 2). The third lead going to the power source is not critical. The above circuit isidentical in function to the three-wire RTD circuit (shown in Fig. 7.21).


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9.2 STRAIN MEASUREMENT 9.2.3. The Bridge Power Supply

We have called the power source "bridgesupply voltage", implying that the supply

is at constant potential. It canalternatively come as a constant currentsource and this has some advantages forlinearity. As shown in Figure the constantcurrent I S, flows into two parallelresistance legs R SG+R 2 and R 3+R 4

Applying the Ohm's law we get

In case R SG=R 2=R 3=R 4

and yields

432 R R R R I V SGS s

SGS SGSGS S R I R R I V 22

SG

S R I

V 4

0


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9.2 STRAIN MEASUREMENT 9.2.3. The Bridge Power Supply

The sensitivity of the strain gauge bridgecircuit is expressed by the relationship

between the input strains ε and theoutput voltage of the bridge ∆V 0.

As follows from equations the larger thevoltage or current applied to the straingauge bridge, the higher will its sensitivitybe. The practical limit is set by self-heating in the gauge. Self-heating variesa lot with the details of an installation, butwith metal substrates, can generally beignored below 1 mW/mm2 of gauge area.

GV

V s

40

4

00 GV

l l

V V S sV

SGS

I RG I V

S .4

0


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9.2 STRAIN MEASUREMENT 9.2.3. Half Bridge Configuration

The considerations in theabove paragraphs have beenbased on single strain gauge(quarter bridge)arrangement as shown inFigure (a).

Let us consider now thedouble strain gauge (halfbridge) arrangement asshown in Figure (b).


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9.2 STRAIN MEASUREMENT 9.2.3. Half Bridge Configuration

Following the procedure described inparagraph 9.2.2 we get:

Assuming R SG1=R SG2=R s and

R 2=R 4=R 0 and in case R 2=R 4=R s weget:

Corresponding sensitivity is:

Half bridge configuration provides a two

times better sensitivity

4

42

1

21

0 R R R

V R

R R

V V V V V V

SG

sSG

SG

sbcacba

s

s s s

s

s s

R

dRV dR

R

RV dV

24

220

s

s s

R

RV V

2

0

2

0 GV

l l

V S sV


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9.3 STRESS MEASUREMENT

The measurement of the stress is based on the Hooke's law (refer to paragraph 9.1.2:

The Young's modulus E is given in manufacturer's data sheets for a specific type ofmaterial. The strain ε is proportional to the relative change of the bonded straingauge(s):

The Gauge factor G is given by the manufacturer of the strain gauges or can be defined

by special laboratory experiments. The relative resistance change of the bonded strain gauge is measured by bridge

configurations (refer to paragraph (9.2). For instance for a quarter Wheatstone bridgeconfiguration the relative resistance change is:

After substitution:

E

SG

SG

SG R

R

G

1

sSG

SG

V

V

R

R 04

0

4V

GV

E

s


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9.3 STRESS MEASUREMENT

9.3 Example 9.3

Calculate the stress for the case given in example9.2. if the Young's modulus E = 1011N/m2.

Solution

27611

1010.50010.2

104

m N


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9.4 FORCE MEASUREMENT

The unit of force is a derived unit - N [kgm/s2]. The measurement of

force is intimately related to strain and (or) stress measurements asfar as the engineering constructions are concerned. The stress andstrain of a solid body varies with the force acting on the solid body, aswell as with the mode of acting of the force applied. The force can bea product of acceleration of the mass of the solid body.

We shall limit the presentation of force measurement concerned with

strain gauge application for some typical cases only.


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9.4 FORCE MEASUREMENT Case No1

F – force

l – length S – cross –sectional area

ε= ∆l/l – strain

σ=E.ε – stress

E – Young’s modulus

Quarter-bridge strain gaugeconfiguration

Half bridge configuration

0

.4 V GV

s E F

s

sE

F

s

F

0

.2 V

GV

s E F

s


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F – force

l – length I – Moment of inertia


ε – strain

σ – stress



V I GV

l E F

s

2

.192

El

FI x

2

48

1

l

sI

x

2

3

1

V

I GV

l E F

s

2

.96


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F – force

l – length I – Moment of inertia


ε – strain

σ– stress



V I GV

l E F

s

2

.12

l

sI x

2

3

1

El

FI x

2

3

1

V

I GV

l E F

s

2

.6


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9.4 STRESS MEASUREMENT 9.4 Example 9.4 Calculate the force applied to the solid body, case 1 under the following

conditions:the output voltage of the quarter bridge configuration of the bonded strain

gauge ∆V = 100μ V DCthe cross - sectional area of the solid body s = 3.14 cm2.the bridge supply voltage V s = 10 V.the gauge factor of the strain gauge G = 2the Young's modulus of the material of the solid body E = 1011 N / m2.

Solution

N x x x

x x x F 62810100

102

1014.3104 6

411


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9.5 SUMMRY

The application of strain gauges for strain / stress

measurements can be made an analogue for essentially any ofthe various mechanical - type variables of interest to theengineer: force, torque, displacement, pressure, temperature,motion, acceleration, etc .

For this reason strain gauges are very widely used in measuringsystems. Their response characteristics are excellent and they

are reliable, relatively linear and inexpensive. It is important, therefore, that the engineer concerned with

experimental and measurement works be well versed in thetechniques of their basic use and application.


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10. PRESSURE MEASUREMENT

10.1 INTRODUCTION

The pressure exists both in solids and fluids.Pressure on the surface of solids ismeasured by force transducers (refer toChapter 9), and the force is divided by the

appropriate area.

This chapter will be devoted only to the fluid pressure measurements.


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10.1 INTRODUCTION 10.1.1 Units

Pressure is a derived quantity and is formulated as force per unit area that afluid exerts on its surroundings:

F is the force [N] ; s is the area [m2] The units of the pressure are named the Pascal [Pa] so that

Pa = 1 [N/m2] Other commonly used units to be mentioned are: 1 atmosphere (at) = 101 325 Pa = 101.325 kPa 1 bar = 100 000 Pa = 100 kPa 1 psi = 1 Ib / in2 * ≈ 6 895 Pa = 6.895 kPa 1 mm Hg = 133.322 Pa

s

F p


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10.1 INTRODUCTION 10.1.2 Gauge pressure, absolute pressure, differential pressure

The earth's atmosphere exerts a

pressure (because of its weight) atthe surface of the earth ofapproximately 1 at = 101.325 kPa.Therefore if a closed tank at theearth's surface contained a gas atan absolute pressure of 101.325 kPathen it would exert no effective

pressure on the walls of the tankbecause the atmospheric gas exertsthe same pressure from the outside- Figure.


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The illustration in Figure shows,

that the reaction of the gaugesensing element, i.e. the gaugereading will be:

where:

p g is the gauge reading p abs is the absolute gas pressure in

the tank

p at is the atmospheric pressure

at abs g p p p


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Practically, the absolute

pressure is not the quantity ofmajor interest in describing thepressure since it is based on theassumption that themeasurement is accomplishedin absolute vacuumenvironment - Figure.


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In ordinary pressureenvironment theatmospheric pressure is

always present – Figure.


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A special case is shown in

Figure, when the reaction ofthe pressure gauge is due tothe action of two pressures.The reading of the pressuresensor now will be relatednot to the absolute pressuresunder measurement but to

their difference.Therefore the last measurementis referred to as differentialpressure.


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10.1 INTRODUCTION 10.1.3 Head pressure

For liquids the expression head pressure

or pressure head is often used todescribe the pressure of the liquid in acontainer. This refers to the staticpressure, (i.e. the liquid is at rest)produced by the weight of the liquidabove the point at which the pressureis described - Figure.

p is the pressure [Pa] ρ is the density of the fluid [kg/m3], i.e.

the units are mass per volume.g =9,8 is the acceleration due to gravity

[m/s2]h is the depth (height) of the liquid [m]

][ Pa gh p


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10.1 INTRODUCTION

Example 10.1

A tank holds mercury with a depth of 760 mm.Define the pressure at the tank bottom

(The density of mercury is p Hg =13604 kg/m3).

Solution The pressure is given by equation

kPa Pa x x gh p Hg Hg Hg 322.101101322760.08.913600


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10.2 U-TUBE MANOMETERS

The principle of the U- tube manometer isbased on the balance between theunknown pressure against the pressureproduced by a column of liquid of knowndensity - usually mercury. Thus thederived quantity pressure is traced tothe two fundamental quantities massand length.

The level difference of the liquid in Figure(a) is a result of the interaction of twoopposite forces acting in the two legs

s is the cross-section area of the U-tube

II

x

II sp F


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The force in the left-hand leg is a sum oftwo force components. The firstcomponent is due to the pressure p x

I:

The second component is due to the weightof the liquid column with height h :

m = ρ V is the mass of the liquid column

ρ the density of the liquid

V = sh is the volume of the column

s is the cross-section area of U-tube

g is the acceleration due to gravity

I

x

I ap F 1

shg mg F I 2


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At equilibrium

The above equation shows that the scale maybe calibrated directly in differential pressureunits.

0 F

x

I

x

II

x p g

p ph

shg ap F F F I x I I I 21

II x I x sp gh s sp

II I F F


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If for the case shown in Figure (b)we substitute p' x with p at ,respectively p" x with p' xabs we get:

The above equation shows thatthe scale may be marked-off ingauge pressure units.

g at xabsat xabs p p p

g

p ph


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If for the case shown in Figure ( c) wesubstitute in p‘ x with zero (i.e.vacuum), respectively p" x with p' xgbswe get:

The above equation shows that thescale may be marked-off in absolutepressure units.

U-tube manometers although verysimple and reliable have poordynamic response and are not systemoriented.

xabs xabs p

g g

ph

10


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Example 10. 2. Define the liquid height h for a U-tube manometer with p' x =102 kPa and

p" x =122 kPa if the liquid substance of the manometer is: Water (p = 1000 kg/m3)

Mercury (p = 13604 kg/m3)

Solution This is a differential pressure U-tube manometer. The substitution of the above

data in equation yields:

a) For water

b) For mercury

m x

x x

g

p ph

I

x

II

x 28.91000

1010210122 33

m x

h 15.08.913604

10102122 3


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10.3 ELASTIC SENSNG ELEMENT CONCEPT

Elastic pressure sensors operate on

the principle that the deflection (ordeformation) accompanying abalance of pressure and elasticforces (Hooke's law!) may be usedas a measure for pressure. Thisphenomenon may bedemonstrated by objects generallyreferred to as springs.

A traditional spring of coiled steelwire serves as good example -Figure.


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If an external force is applied to the

spring, this causes the spring toextend from l to a new lengthl + ∆l , where the system againbecomes stationary - equilibriumtakes place due to the interactionbetween the internal elastic F and

the external force F ext :

ext F F


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The Hook's law states that the

equilibrium force of a springcompression or extension is givenby:

F is the equilibrium force [N]

k is the spring constant [N/m] ∆l is the change in length [m]

The negative sign indicates thatthe spring force is opposite theapplied force so that they balance.

l k F


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The term spring is used as a

general term to describe anyconfigurations that relatedisplacement to force.

Selected such configurations,transducing the pressure to

displacement are shown in nextFigures.

The Bourdon tube , normally ofoval cross-section is coiled into acircular arc as shown in Figure (a).


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As a pressure is applied the ovalsection tends to round out (equalpressure on the tube walls !),becoming more circular in shape. Theinner and outer arc lengths will remainapproximately equal to their originallengths, and hence the only recourse

is for the tube to uncoil, thusproducing a deflection ∆l which is

proportional to the gauge pressureapplied. Spiral, helical and twistedtube versions are also used for various

applications.


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The bellows sensing element is a

thin-walled, flexible metal tubeformed into deep convolutions andsealed at one end as shown inFigure (b).

The pressure applied will cause the

bellows to change in length.The bellows can be arranged to react

to absolute, to gauge and todifferential pressure.


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The diaphragm is a thin elasticcircular plate supported about itscircumference. The diaphragmsmay be flat (Figure c) orcorrugated (Figure d). Thedeformation of the diaphragm isproportional to the pressureapplied. By using relevantchambers the diaphragms may

provide deflection ∆l proportionalto absolute, to gauge and todifferential pressure.

The flat diaphragms provide smalldeflections, while the corrugateddiaphragms are useful when

larger deflections are required.


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10.4. INTELLIGENT PRESSURE SENSORS

The achievements of microelectronic semiconductor technology hasled to development of a variety of very fast, very small, highly

sensitive pressure gauges. Supported by advanced microelectronicssuch pressure sensors offer flexible interface with the computerizedprocess control systems, both analog and digital output signals,temperature compensation, autocalibration, self diagnostic, highreliability operation in hazardous environment and many otherprocess oriented advantages.

Two specific examples that illustrate features we feel to beparticularly significant and interesting.

- Strain-Gauge Pressure Sensors

- Capacitance Based Pressure Sensors


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10.4. INTELLIGENT PRESSURE SENSORS 10.4.1 Strain-Gauge Pressure Sensors

The advanced strain-gauge-based

pressure sensors make use of aflat diaphragm with bonded straingauges fabricated on a single chip.

The principle is illustrated in Figure.The pressure stresses the diaphragm

as shown in Figure (c).

Four strain gauges are bonded on theappropriate places in order toachieve the best sensitivity – Figure (a),(b),(d).


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The design shown in Figure (b) offers

separation of the strain gaugesfrom the fluid under measurement,which for some processes is ofvital importance.

The full bridge arrangement providestheoretically complete temperature

compensation in terms of thestrain gauge temperature

variations


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The intelligent pressure sensor is

based on the above concept.The pressure chamber, the

diaphragm and the strain gaugeare fabricated on a single chip.

The sensing device is supported bypowerful microprocessor circuitry

which is in the sensorencapsulation or even on thesensor's chip, i.e. sensingelement and electronics on asingle chip \\\

A thin-film polysillicon sensor isdepicted in Figure.


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When the pressure is applied to

the isolating diaphragm, it istransmitted to thepolycillicon sensingdiaphragm by means of asilicon fill fluid (or Neobee fill

fluid for sanitary standardprocesses) - the thin straighttube shown in Figure.


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The silicon fill fluid (proportional to the

process pressure) creates a very smalldeflection of the sensing diaphragm,which in turn applies strain to aWheatstone bridge circuit containingstrain gauges.

The change in resistance is sensed andconverted to a digital signal for

processing by the microprocessor.This digital signal is then converted to a 4- 20 mA DC signal to be used by thecustomer in parallel or independentlywith the digital output, if required.


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The sensor contains a build-in

temperature sensor.The microprocessor-based circuitry usesthis temperature measurement tomake corrections to the pressuremeasurement and minimises theeffects of temperature changes.

This greatly reduces the errors caused by

temperature changes.The accuracy offered is 0,25% with a

range up to 30 000 kPa. Absolute, gauge and differential pressure

sensors are readily available.


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10.4. INTELLIGENT PRESSURE SENSORS 10.4.2 Capacitance Based Pressure Sensors

The principle of operation of this type

of pressure sensors is based onthe differential capacitor

described in paragraph 7.7.1. The two parts of the differential

capacitor are incorporated in ainductive bridge circuit. The sensor

Module Cross Section is shown inFigure.The sensor module also has a built-in

temperature sensor that is used tocorrect for temperature effects


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10.4. INTELLIGENT PRESSURE SENSORS 10.4.2 Capacitance Based Pressure Sensors

The analog pressure and temperature signals from the sensing module enter

the analog-to-digital converter (ADC) and are converted to a digital formatfor the microprocessor.

The module memory EPROM stores data from the factory characterizationprocess. The microcomputer controls operation of the sensor.

In addition, it performs calculations for sensor linearization, reranging,

engineering units conversion, self-diagnostic, and digital communication.The digital - to - analog converter (DAC) changes the corrected digital signalfrom the microcomputer into 4-20 mA analog signal.

The digital communications circuitry provides an interface between thesensor and external devices of the network


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10.5. MEASUREMENT OF VACUUM

The pressure range below 1O7 Pa is very frequently referred to as theultrahigh or extra-high vacuum region. The interest in measurement of

these pressures is usually confined to systems where the mainobjective is to have as minimal an amount of gas as possible in aprescribed volume.

Limitations

The techniques for the measurement of these pressures are limitedusually by the difficulties in achieving stable, reproducible lowpressures, and the fact that these pressures are so small that themechanical force exerted (i.e. momentum transferred) by the gasmolecules is extremely small and hence virtually impossible tomeasure directly.


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Difficulties

The main difficulties in achieving low, stable pressures are due to thedependence of the rate of vacuum degradation from the walls andother materials of the vacuum enclosure on the cleanliness, leak-tightness, and temperature of the system. Other difficulties aresometimes related to the capacity and stability of the vacuum pumpingsystem. The difficulty in measuring the mechanical forces exerteddirectly by the gas molecules in this pressure regime has not beenovercome. Gauges designed for use in the ultrahigh vacuum regiontypically detect gas density and not gas pressure directly.


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Some kinetics backgrounds

While primary techniques based on the kinetic theory of gases have been developedthat could, in principle, be used to calculate the absolute gas pressure in ultrahighvacuum systems, in practice it is found that these techniques for measuring pressuresbelow 10-7 Pa are inadequate. What has been done instead is to use the primarytechniques at higher pressures, where the techniques work well, to measure pressure orflow ratios, and then to use these directly measured ratios to indirectly determine the

lower pressures. It is reasonably assumed that the measured ratios are valid at thelower pressures since the mean free paths of the gas molecules are so large thatmolecular collisions are a negligible factor even at the relatively higher pressures wherethe ratios are measured.

For pressure measurement in the vacuum range (less than atmospheric

pressure) only the more specialized instruments are applicable.


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10.5. MEASUREMENT OF VACUUM 10.5.1 Thermocouple Gauge

This gauge works on the thermalconductivity principle. There exists a

linear relationship between pressure andthermal conductivity. Operation of thegauge (Figure) depends on the thermalconduction of heat between a thin hotmetal strip in the center and the coldouter surface of a glass tube.

The metal strip is heated by passing a

current through it (at a temperature ofabout 320 K) and its temperature ismeasured by a thermocouple. Thetemperature measured depends on thethermal conductivity of the gas in thetube and hence on its pressure. Themeasurement range is from 500 Pa to

0.1 Pa.


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10.5. MEASUREMENT OF VACUUM 10.5.2 Pirani Gauge

The Pirani gauge is similar to a

thermocouple gauge but has aheated element which consists offour tungsten filaments connectedin parallel - Figure.


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10.5. MEASUREMENT OF VACUUM 10.5.2 Pirani Gauge

Two identical elements are normally

used connected in a bridge circuit.

The resistance of the elementchanges with temperature andcauses an imbalance of the

measurement bridge.

The measurement range is from10 Pa to 0.1 Pa.


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10.5. MEASUREMENT OF VACUUM 10.5.3 The McLeod Gauge

The principle of operation is based on measuring the force exerted by a sample

of the gas of known volume after a known degree of compression. The low-pressure fluid is compressed to a higher pressure which is then

measured by manometer techniques. This gauge can be visualized as a U-tube manometer, sealed at one end, and where the bottom of the U can beblocked at will. To operate the gauge, the piston is first withdrawn, causing thelevel of mercury in the lower part of the gauge to fall below the level of the junction J between the two tubes marked Y and Z. Fluid at unknownpressures p x is then introduced via the tube marked Z from where it also flowsinto the tube marked Y , of cross sectional-area S . Next, the piston is pushedin, moving the mercury level up to block the junction J. Thus, the underpressure p x is trapped in S tube volume.


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Measurement of the height (h)

above the mercury column in tubeY then allows calculation of thecompressed volume of the fluid V c:

V C is the compressed volume[m3], h is the height [m], s is the area [m2].

shV C .


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Then, by Boyle's law:

P x is the unknown pressure [Pa] p c is the compressed pressure [Pa] V 0 = const is the volume of Y tube

from the top up to junction J [m3] V c is the compressed gas volume [m3]

C C x V pV p .. 0


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Applying the normal manometer

equation:

p x is the unknown pressure [Pa],

p c

is the compressed pressure [Pa],

h is the height [m],

ρ is the density of mercury[kg/m3],

typically p Hg = 13604 [kg/m3]

g is the gravity acceleration [m/s2].

g h p p xC .


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Hence:

or

because p x is the unknown pressure [Pa], p c is the compressed pressure [Pa],

h is the height [m], ρ is the density of mercury[kg/m3],typically p Hg = 13604 [kg/m

3]s is the area [m2].V 0 = const

g is the gravity acceleration [m/s2].

shV

g sh p

x

0

2

0

2

V

g sh p x

0V sh


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10.5. MEASUREMENT OF VACUUM 10.5. Example 10.3 Calculate the vacuum measured by McLeod gauge under the

following conditions:

- the measured height h = 10 mm- the standard volume of the gauge V 0 = 22 500 [cm

3]- ρ Hg = 13604[kg/m

3]- the cross sectional area of the top sections of Y and Z tubes s = 1[cm2]- the local gravity acceleration g = 9.8 [m/s2]

Solution After relevant substitution we get:

Pa x

x x x x x p x 059.0

1022500

8.91360410101016

234


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10.5. SUMMARY

An attempt has been made to introduce the student to some of theproblems attending measurement of pressure.

Absolute pressure, differential pressure and gauge pressure have beenexplained and illustrated by U-tube manometer concept.

Blasting sensing element concept has been described and illustrated bydiaphragm strain gauge pressure sensors and by diaphragmdifferential capacitor - based Intelligent pressure sensor.

Measurement of vacuum has been briefly covered.We realize that many approaches to the problem have been omitted and

that in certain respects the coverage has been brief and somewhatsuperficial

Such is the penalty that must be paid in assembling a textbook of thisgeneral nature.


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11. FLOW MEASUREMENT

11.1. INTRODUCTION 11.1.1 Historical background

Fluid flow has been measured since the dawn of civilization. The

Egyptians measured river Nile flow to control the crop irrigation. Flowmeasurement helped the Romans to develop heating systems,aqueducts, baths, fountains.

The history of flow measurement is closely related with theestablishment of the principle of conservation of mass, namely that ofa steady state mass m entering a system over a unit time ∆t equals the

mass leaving the system over the same period:

ou t in t

m

t

m


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11.1. INTRODUCTION 11.1.2 General consideration

Flow measurement is mainly concerned with fluids, defined as liquids, gases, vapoursand slurries. The flow measurement methods considered here are for use in closedconduit systems. Besides there are methods for use in open ducts. The basic quantitiesmeasured are:

- mean flow velocity in a pipeline v or point velocity in [m/s] using a velocitymeter as a sensing element, not concerned here

- flow rates of two types - volumetric flow rate G v in [m3/s], when the volume of a

fluid flow per unit of time is measured, or mass flow rate G m = ρ G v in [kg/s], when themeasurement of the weight of a fluid of density ρ flowing past a point is considered, thesensor used in both cases is a ratemeter ,

- quantity of fluid passed in a given time in [m3], the sensing element being aquantity meter


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The important fluid characteristics relatedto flow measurement are :

* density

where

m is the fluid mass, [m]

V is the fluid volume, [m3]

(Typically pair =1.184 [kg/m3] and ρwater =1000 [kg/m3]);

3, mkg V

m


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* viscosity μ in [N s/m2] or [Pa s], defined as thetangential force on a unit area of either of two horizontalplanes separated by unit distance, one plane fixed and theother moving with unit velocity, the space between the

planes being filled with the viscous substance μ air =18.2x10

-6 [Pa s]

μ water =10-3[Pa s]


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- Reynolds number

that specifies the flow form laminar for Re 4000, where l is the flow

length

and l =0 for round pipes of diameter D .

Vl

Re

tion FtheFLOW perimeterO

SECTION AofTHEflow surfaceARE xl

sec

4


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11.2. DIFFERENTIAL PRESSURE FLOWMETERS 11.2.1 Principle of operation

Differential pressure flowmeters or else pressure difference flowmeters

measure the pressure drop over an introduced constriction to the flowin a pipe. The constriction is the reason for the velocity v of the fluid toincrease until it reaches a maximum at the area of minimumcontraction. The increase of velocity causes the static pressure p todecrease. Thus the pressure drop is correlated with the rate of the fluidflow. The maximum pressure is located slightly downstream from the

restriction where the stream is the narrowest and is called venacontracta. Beyond this point the pressure rises due to the partialtransformation of the kinetic energy of the fluid back into pressure butdoes not restore its upstream value.


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This part of the kinetic energy that has been spent to

overcome the friction and vortex shedding due to theobstruction was turned irreversibly into heat. The heatleads to an increase of the enthalpy E

where u is internal energy, [J] ν is the specific volume (ν = V/m - volume per unit of

mass), [m3 /kg] p is the pressure, [N/m2].

pu E


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Hence, as a result of the introduced extra resistance in

the flow system a permanent loss of pressure is observedthat depends on the type of restriction and the ratio ofthe diameter of the constriction d to the diameter of thepipe D , called diameter ratio (d/D = 0.2+0.6).

The smaller the diameter ratio the more considerable thelosses are but the more accurate the flow ratemeasurement is.


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11.2. DIFFERENTIAL PRESSURE FLOWMETERS 11.2.2 Derivation of the equation for mass flow rate of a

compressible fluid through a constriction

The subsonic flow of a perfect gasflowing horizontally through aconstriction is considered in Figure.

The gas flows from conditions ( p 1 , T 1 ,v 1 , ρ 1) at cross-sectional area of fluid

flow s 1 to conditions ( p 2 , T 2 , v 2 , ρ 2) at the area of maximum contractions 2, where by p is denoted theabsolute pressure, by T - the absolutetemperature, by v - the fluid velocityand by ρ -the fluid density.


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From the conservation of energyequation

it follows that the change of the kineticenergy is equal to the change in theenthalpy per unit mass

22

2

22

2

11

v E

v E

21

1

1

2

2

2 E E

vv


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Since the processes are isentropic (the student is referred to subject Physics, 1st year), itholds

or or

K is a constant

y = C p / C V is the ratio of specific heat at constant pressure

and at constant volume and the gas constant is R = c p - c v.

Besides, dE = vdp , which after integration yields

const p

p

dt

duc

const V

V dt

duc

K p

2

2

1

1 p p

1

2

1

2

p

p

1

2

2

1

1.21

p

p

p

p

dpdp E E


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Expressing from

and substituting in, it is obtained

Where

The final result is reached after substituting K

1

1

1 11

p

11

1

11

21 11

1

2

r p K dp K E E

p

p

1

2

p

pr

1

1

1

2

1

2

2 112

r pvv


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From mass continuity equation the theoretical fluid flow rate G T = ρ sv

at the inlet G T1 is equal to the theoretical fluid flow rate at the outletG TZ , i.e. ρ 1s 1 v 1 = ρ 2s 2v 2 . Hence:

After substituting into for v 2 is obtained

Considering that in p 1 can be expressed as p 1=( p 1- p 2 ) / (1- r) and( p 2/ p 1)

2 as ( p 2/ p 1)2 =r 2/γ, for the mass flow rate is finally

obtained:

12

1

2

1

2

12

211

1

2

r

s

sr

p

pv

2

1

2

1

1

2

2

1

2

1

2

1 v s

s

p

pv

s

sv

212222

2. p p sv sGT


110/223


111/223




After substituting and expressing v2, for G T is obtained

∆ p = p 1 - p 2 is the pressure difference or drop over the restriction

K is a constant for constant areas s 1 and s 2 and constant density ρ .The actual flow rate G is always less than the theoretical G T mainlydue to fluid friction effects and density gradients which is reflected bythe introduction of a discharge coefficient c d. As a result the actualflow rate is:

p K p p s s

s sGT

21

2

2

2

1

21 2

T d GcG


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The above equations are derived after some simplifications in order toenhance understanding.

In reality the fluid pressure and temperature are changingduring process continuity.

The density changes with temperature as well.This could lead to unacceptable increase of measurement errors.

Therefore, more advanced flowmeters are manufactured which besidesthe pressure difference account for the fluid pressure andtemperature.

These data are used by the advanced microprocessor circuitry tointroduce the relevant error corrections.


113/223



compressible fluid through a constrictionThe differential pressure sensor is based on the differential capacitance concept -

refer to para 10.4.2. The absolute pressure sensor consists of a Wheatstonebridge circuit made of polysilicon resistors deposited on a silicon substrate -refer to para10.4.1. The absolute pressure sensor is hydraulically connected tothe high pressure side of the transmitter. Process pressure is transmittedthrough the fill fluid to the sensing element, creating a very small deflection ofthe silicon substrate, acting as a diaphragm. The resulting strain of thesubstrate changes the bridge resistance in proportion to the pressure applied,and the resistance change produces an "out-of-balance" voltage as describedin para 7.5. and Chapter 9. The differential pressure signal, the absolutepressure (high) signal and the temperature signal from a 4-wire RTDconfiguration (refer to para 7.10.1.) are fed to a microprocessor basedelectronics for relevant calculations, corrections, calibrations, communicationsand diagnostic procedures as illustrated on the block diagram shown in.

The above intelligent sensor provides high accuracy flow measurement bydynamically compensating flow equation variables like discharge coefficients,velocity factor, thermal expansion effects and density.


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11.2. DIFFERENTIAL PRESSURE FLOWMETERS 11.2.3 Types of restrictions

The main types of restrictions used

are:orifice plate - Fig.a),

nozzle - Fig.b)

Venturi tube – Fig. c).

The discharge coefficient depends

on the type of the flow,obstruction device and Reynoldsnumber and is the least for Venturi meter, where thepressure loss is the least.


115/223



The Venturi meters Fig.c) howeverare expensive and occupysubstantial space.

Orifice meters Fig.a) are the leastexpensive, the easiest to install andrequire the smallest possible spacebut at the same time suffer from

head losses and the accuracy ishighly affected by the dirt in thefluids, their edge tends to wear.

The common disadvantage is thesquare-root relationship betweenpressure drop and flow rate.


116/223



For a nozzle Fig. b) the area ofmaximum contraction occurs atthe throat while for an sharp-edged orifice the minimumcontraction is slightly

downstream the plate.The isentropic assumptions are not

valid for orifice plates and ε is

determined experimentally.


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11.2. DIFFERENTIAL PRESSURE FLOWMETERS Example 11.1

A nozzle is fitted in a horizontal pipe of diameterD =15 [cm], carrying a gas of density ρ =1,15 [kg/m3] for the purpose of flow measurement.The differential pressure head indicated by a U- tube manometer containing oil of specific gravity0.8 is ∆h oil=10 [cm]. If the discharge coefficientand the nozzle diameter are c d=0.8 and d=5[cm] respectively, determine the flow of gasthrough the nozzle flowmeter .


118/223



Solutionsince

The specific gravity of oil is 0.8, so:

gasd actual h g s s

s scG

22

2

2

1

21 g h p gas gas oil oil gas hh

34

80010008.0

,1000

,

,8.0

mkg x

const V V

V

m

m

oil

C aterAT distilledW

ater distilledW

oil

ater distilledW

oil

o


119/223



SolutionHence,

sl scm x x x xGactual /348.58/21.58348

62.1971.176

52.695681.9262.1971.176 222

ofGascmhh oil oil gas 52.695615.1

80010

22

22

1

62.194/2514.3

71.1764/22514.34

cm x s

cm x D

s


120/223


121/223


122/223


11.3. VARIABLE AREA FLOWMETERS 11.3.1. Principle of operation

The height of the float is therefore ameasure of the flow velocity or flowrate provided that the fluid densityand the pressure difference across thefloat remain constant.

For measurements at pressures andtemperatures close to ambient, thetapered tube is made of transparentglass and the float position can be

observed directly against a scale,scribed on the tube wall andgraduated in flow rate units.

Otherwise for high pressure the tube ismade of metal and a magneticcoupling indicates the flow position.


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11.3. VARIABLE AREA FLOWMETERS 11.3.2. Derivation of the theoretical flow equation for

compressible fluids

It is assumed that the metered fluid is an ideal gas, the taper angle of the tube is

negligibly small and the potential energy differences between plane 1 and plane 2 can be neglected. The mass flow rate is then expressed as:

where

a is the annular area between tube and float at plane 2, [m2] a=[Stube(x)-Sfloat] S float is the float area, [m

2] S float = π d2 /4

s is the area corresponding to diameter D , [m2] γ is the isentropic exponent r is the pressure ratio p 2 / p 1.

12

2

12

21

22 111

2

r

s

ar r p paGT


124/223



compressible fluids

Denoting by a 2

=s /S 1 and a /s =(a 2

-1)/a2

and putting p 1=( p 1 - p 2 )/(1-r ),

it is obtained

The force balance at the float if the pressure difference due toelevation is neglected, yields

where F is the Buoyed weight of the float.

1

222

4

1

121

22 1

11

1

12

r a

a

r

r p p sGT

)( 1221 vvG F p p s T


125/223



compressible fluids

The velocities v-i and vz are expressed from mass flow continuityequation:

after inserting

The last equation is used to eliminate ( p 1 - p 2 )

1

1

22

2

2

2

1

1

1 r sa

Ga

a

Gv

s

Gv

T T

T

1

11

2

2

1

2

21 r a

a

s

G F p p s T


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compressible fluids

where

For incompressible fluids Y =1

1

2

21 sF Y aGT

211

111

11

11

1

12

2

12

22

r

r

r a

a

r a

aa

Y


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11.3. VARIABLE AREA FLOWMETERS 11.3.2. Area of application

The variable area meters for not very high flows are less

expensive than the pressure difference flowmeters.The accuracy without special calibration is about ±2% of

full scale deflection.The range of flow rates covered is 0+0.5 [m3/s] for gases

and 0+0.1 [m3/s] for liquids.

Pressures and temperatures are generally close to ambientbut there are special instruments suitable for pressuresup to 3.5 [MPa] and temperatures up to 350 [°C].

Rangeability of 10:1 can be achieved.


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11.3. VARIABLE AREA FLOWMETERS Example 11.2.

A rotameter is calibrated for metering a liquid of density 1000

[kg/m3

] and has a scale ranging from 1 to 100 [l/min]. Itis intended to use this meter for metering the flow of gas ofdensity 1.25 [kg/m3] with a flow range between 20 to2000 [l/min].

Determine the density of the new float, if the original one has a

density of 2000 [kg/m 3

]. The shape and the volume of both floats is assumed to be thesame.


129/223



Solution

Let the subscripts 1 and 2 refer to the liquid flow and the gas flowrespectively through the rotameter. Consider the equation:

where ρ f and ρ ff are the densities of the float and the flowing fluid respectively

V f is the float volume S t and S f are the tube area at the float level and the area of the

float respectively c d is the discharge coefficient which is slight if (s t - s f ) / s t «1.

f f

ff f t

t

f t

f t d

actual s

V g

s

s s

s scG

2

12

2


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The simplified equations are:

for liquid flow

for gas flow

where k is the rotameter constant.

2

22

2

2

2

1

11

1

2

1

ff

ff f

f

f

t

f t

f t d

ff

ff f

k G

s

V g

s

s s

s sck

k G


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The scale ratio between gas flow and liquid flow is 20:1=2000:100=20.

Therefore G 2=20G 1 or G 2/G 1=20 and considering the above equationsfor G 1 and G 2 it is obtained:

which solved with respect to ρ f2 yields ρ f2 =501. 25 [kg/m3].

211122

1

2 20 ff ff f

ff ff f

G

G


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11.4. TURBINE FLOWMETERS 11.4.1. Principle of operation

A typical turbine flowmeteris depicted in Figure.

The rotary speed of aturbine wheel placed in apipe depends on the flowrate of the fluid and can besimply measured bycounting the rate at whichthe turbine blades pass agiven point using amagnetic proximity pickupto produce voltage pulses.


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11.4. TURBINE FLOWMETERS 11.4.1. Principle of operation

By feeding these pulses toan electronic pulse-ratemeter a flow is measured orby accumulating the totalnumber of pulses during atime interval, the total flowis obtained.

Thus the turbine meter can

serve both as a ratemeterand a quantity meter. Ifanalogue voltage signal isdesired the pulses can feeda frequency-to-

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Meassurement and Instrumentation