measurementforcontrol v3_01

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Measurement for control

V3_01

C P Bodenstein

2009/10/19

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V3_01:Formulas for for flowrate corrected.

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1. Introduction

Modern civilizations cannot function without instrumentation since research and development and the consquential industrial activity ofany significance is impossible without measurement and control of process variables. Instrumentation and measurement are widely usedin the chemical process industry, transport, communications, power generation and supply, medicine, and many other activities inindustry and commerce. Without measurement meaningful research cannot be done and no useful industrial applications can beimplemented. Perhaps it can be said that instrumention makes research possible and that research in turn makes more and novelinstrumentation possible.

Industrial instrumentation covers a very wide field. This course will be limited to some of the most widely used instrumentation which isnecessary to operate control systems effectively and safely.

Widely used measurements are:Temperature in diverse industrial applications such as boiler plant, steam and gas turbines, heating, ventilation and air conditioning,furnace control, oil refineries, diverse chemical process plant.Pressure in boilers, turbines, pumping, transport of fuids in pipelines, compressors and chemical plant.Flowrate in steam plant, water treatment and pumping, chemical processes and transport of fluids in pipelines.

Levelof liquids and solids in diverse plant in electrical power generation and chemical process plant.Force. In metal processing, mass measurement, lifting and hoisting of material.Displacementin metal processing as thickness and manufacturing.Rotationalvelocity in turbines and compressors, mills for metal processing and in paper and pulp processing amongst others.These constitute the greater part of popular industrial measurements.

Other important industrial measurements are:pHin chemical plant and water treatment.Dissolved solids in boiler feedwater and water treatment.Components in gases by means ofgas chromatography.Oxygen in boiler and furnace flue gas for combustion control.Mass spectrometryin the analysis of products such as steel processing.

Many more examples are encountered in the daily life of any control engineer in a plant.

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Measurement and control is an important and diverse field of engineering. By itself, it forms a big industry worldwide with a large turnoverand which provides employment to many technical and non-technical staff. The instrumentation and control of a modern plant costsabout 15% - 20% of the total capital outlay. Employment is offered in research and development, project engineering, sales, construction

and in maintenance. Employers range from big international corporations to small local companies. The latter are frequently started byentrepreneurial engineers or technicians. A number of journals promote measurement and control and a number of institutes formeasurement and control actively promote this field of engineering.

Due to the wide variety of instruments that are available for industrial use, handbooks tend do be very bulky and expensive. Someclasses of instrumentation warrant specialist handbooks. This set of notes is an introduction to this fascinating field of engineering byconcentrating on a few of the most widely used instruments.

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2. Functional description of instrumentation

2.1. Objectives.

To compile functional descriptions of instrument systems for which the underlying physical effects are known.To incorporate interference and disturbance into the functional description.To compile functional descriptions in terms of the conversions between energy forms which form the basis of all instrumentation.

2.2. Introduction.

Transducers convert physical variables into more useful physical variables. In this set of notes sensors and transducers are synonyms.To convert a process variable such as temperature to a useful signal, one or more suitable physical effects are used to design aninstrument. In general, sensors employ inference to measure a variable. This generally means that the physical variable in theenvironment has some useful repeatable effect on the sensor. This change in the sensor can then be used to infer the value of thephysical variable that we want to measure. This will be explained here by means of an example.

Temperature has a formal definition as will be discussed later, but is hardly useful for practical temperature measurement. Nearly allmodern control systems use electronic signals for control, therefore physical effects which can deliver an electronic output are mostuseful. Temperature can be measured, amongst others, by using the effect whereby electrical resistance of a conductor is dependent ontemperature. This is the primary transducer in resistance temperature measurement. By passing an electrical current through the resistor,which is placed where the temperature is to be measured, the voltage drop over the resistor changes as the temperature changes whichforms the secondary transducer. A temperature change has been converted to a resistance change and then to a voltage change. Thatvoltage change can be converted into a standard signal for transmission to display and control. This is called manipulation of thevariables because the same signal type, namely voltage (or current), is processed. Note that the current source is also likely to beinfluenced by ambient temperature as are the conductors coupled to the measuring resistance. In these cases ambient temperature is aninterference input because it is not the process temperature which we want to measure. In addition, the measurement current also heatsup the sensor, changing its temperature and resistance. More examples will be covered in the notes.

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A generic layout of an instrument is shown in the figure:

Figure 1. Generic functional layout of instrument.

In all cases a primary transducer, based on an applicable physical effect, is required. If the output from this transducer is not in a desiredor useful form, it in turn has to be applied to secondary transducer. Transducers may be chained in series until the desired output signalcan be obtained. This comes at a price of more areas where intefering variables can act. It also increases the monetary price.

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2.3. Interference.

The transducers are generally also sensitive to other variables in the environment. These other variables give rise to outputs which we

may wrongly attribute to the variable which we actually want to measure. These are called interfering variables or inputs and requireeffort in the design of instruments to minimize their effect. Note that truly linear input-output relations are very seldom achieved.

Figure 2. Effects of interfering signals.

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Figure 3. Four sensor system effects.

In figure 3 all the possible inputs, interfering inputs or disturbances and possible outputs are shown. Note that the possibilities are limitedto a finite number of physical effects.

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2.5. Problems.

2.5.1. Find five physical effects which can be used in instrumentation. Give the defining equations as well as possible interfering inputs.

(An example: Linear expansion of a steel rod ( )( )00 1 TTLL TT += where the expansion can be used to measure temperature.Interfering input barometric pressure changes which change the dimensions of the steel rod this has to be quantified to determine if itreally qualifies as a disturbance or interfering input.)

2.5.2. Describe five instruments in terms of primary, secondary, (etc.) conversion between energy forms from environmental signal tomeasured to display.

2.5.3. List the instruments to be used in monitoring and control of turbo-compressor driven by an electric motor.

2.5.4. List the instruments to be used in monitoring and control of a HVAC plant in a shopping mall.

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3. Characteristics of transducers

3.1 Objectives

After studying this chapter the reader should be able to define and interpret the characteristics on the specification sheets ofcommercially available instruments.

3.2 Introduction

Transducers convert physical variables into more useful physical variables with the modern aim of producing an electrical output whichcan be readily used in control systems, stored and displayed. The devices used in this conversion are in general sensitive to otherphysical variables as well. This could cause errors if not compensated for. Calibration of instruments is the most important factor if thereadings are to be of value. Calibration is done with a standardized procedure to compare instrument response with a suitable standardreference.

3.3. Characteristics.

Accuracy

Accuracy is a measure of the transducer output representing the true value being measured. It can be defined as follows:

Ea = (Vt Vm)/Vt x 100%

or more generally as

Efs = (Vt Vm)/Vfs x 100%

Where Ea is the percentage error, Vt is the true value being measured, Vm is the transducer output, Efs is the full scale percentage errorand Vfs is the full scale output. Note that some reference standard or calibration source is required.

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Precision or random error

This is a measure of the deviation from a mean value of a set of readings obtained from an input of given value. Thus the repeatability of

a reading is defined by the precision as specified for the transducer.

Bias or systematic error

The difference between the mean value of the transducer output and the true value. Thus accuracy is the sum of random error andsystematic error.

Resolution

This denotes the smallest increment of the input signal that can be measured by the transducer.

Drift

The change in transducer output over time given a zero input. More specific the change in sensitivity over a period of time for a change intemperature, humidity or some other environmental factor.

Linearity

The deviation of a given calibration curve from a straight line within the full scale output of the transducer.

Conformance

For a non-linear transducer it is the tightness of fit to a specified curve, usually a least squares fit.

Span

This defined as the difference between the maximum and minimum outputs of the transducer.

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Hysteresis

This defined as the difference in output for a given input value with increasing input and the same input values with decreasing input.

Distortion

Defined by the difference of the actual output from an expected output as defined by a known input output relationship, whether linear ornon-linear.

Noise

A signal generated internally or externally that is added to the output signal.

3.4. Problems.

3.4.1. Find the specifications of two instruments which give numerical values for the characteristics.

3.4.2. Does it make sense to quote the pressure in an industrial boiler as 30,5231 MPa? Motivate.

3.4.3. Since the 1960s and 70s the distance to a spacecraft can be given to a few m. How is this possible? Do not answer the standardNASA coded pulse. Provide detail.

3.4.4. You buy a temperature sensor with seemingly impressive specifications with a resolution of 1 micro Kelvin. How can you be sure?

3.4.5. Research the effect of transducer characteristics on the claims for global warming.

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4. Instruments creating pressure difference as primary signal.

4.1. Objectives.

The student should be able to compile a block diagram for an instrument that converts a process variable to a pressure difference whichin turn can be converted to an electronic signal. The derivation of equations for simplified models of transducers that convert a physicalvariable to pressure difference should be accomplished. The transducers should be described in both qualitative(describe with words andsketches) and quantitative (derivation of equations, correct substitution of numerical values) terms.

4.2. Introduction.

Processes using fluids (liquids or gases) frequently employ pressure difference as a means to infer some behaviour of the process. In thecase of gases in pipes and pressure vessels, the gas pressure always needs to be known to control the process and also in the interestsof safety. Liquids in tanks and pipes exert a pressure proportional to the vertical distance below the liquid level in stationary systems. Thiscan be used to infer the height of liquid in tank.

A moving fluid in a pipe causes a pressure drop across a restriction which can be used to infer the flow velocity. Relative movementbetween a fluid and a suitably shaped object can also generate a pressure difference across parts of the object which is related to therelative velocity.

Pressure is always measured relative to reference which may in special cases be a vacuum. The primary signal, pressure, or moreaccurately, pressure difference, then has to be converted to something more useful. To explore this process, refer to figure 6. Thepressure difference creates a net force on an elastic membrane which is then deflected. Note that the deflection as shown in the figure isgreatly exaggerated. The membrane forms part of two capacitors which are created by adding two perforated plates. The capacitors areelectrically isolated from the metal parts of the transducer. The membrane deflects under pressure, causing one capacitance to increaseand the other to decrease. This can be processed in a bridge circuit to obtain a electrical voltage proportional to the pressure difference.The signal is further processed as shown in the block diagram. Note that inductive or resistive effects may also be used as will be shownlater.

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Figure 6. Conversion of pressure difference to an electrical signal.

4.3. Terms and definitions.

Gauge pressureis the amount of pressure relative to the prevailing atmospheric pressure, measured in Pascal (Pa).

1 Pa = 1 Newton/m2

= 1 N/m2.

Absolute pressureis measured relative to a perfect vacuum and can be expressed as the sum of the gauge pressure and atmospheric

pressure as measured by a barometer.

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Pa = Pg+ Pb PawherePa = absolute pressure in Pa.

Pg= gauge pressure in Pa andPb = prevailing atmospheric pressure in Pa.

In the case of fluids in motion, momentum has to be taken into account. The stagnation or impact pressure is the sum of the staticpressure and the dynamic pressure. The static pressure is that pressure whether the fluid is in motion or at rest. The dynamic pressure isdue to the kinetic energy when the fluid is decelerated to a standstill.

4.4. Gas pressure.

A gas expands to fill the entire volume. Gas laws use absolute pressure. In industrial applications, most pressure gauges andtransducers give gauge pressure, that is, relative to ambient or barometric pressure.

Figure 7. Gas pressure.

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4.5. Pressure due to height difference in fluids.

The pressure or pressure difference is due to static liquid level. Consider a liquid in a stationary vessel in the earths gravitational field

with the surface open to the atmosphere and with a pressure gauge fitted h m below the surface of the liquid as shown in the figure.

The pressure reading on the gauge is given by

Pg= gh Pa, ..(1)

where = density of liquid in kg/m3,

g= gravitational acceleration = 9,81 m/s

2

andh = height of liquid column above gauge

Note that the pressure is relative to the prevailing atmospheric pressure.

h m

Pressure gauge or transducer

Liquid in tank

Open to atmosphere

Figure 8. Pressure relative to atmosphere in open tank under gravitation.

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The height of the liquid above the reference point where the pressure transducer is located is given by:

h = Pg/(g) m ..(2)

When it is not feasible to have a pressure gauge at the height h below the surface, as for instance in a well, then compressed airmay be slowly bubbled through the liquid as shown in the figure. For moderate depth where the compressibility of the air is not great, thegauge pressure will also be given by equation (1), and the height by equation (2).

h m

Pressure gauge

or transducer

Liquid

Compressed air

supply

Bubbles

Figure 9. Depth measured by slowly bubbling compressed air.

Where the liquid is in a closed pressurized vessel and the level is to be measured by means of pressure, the arrangement shownin the next figure is required. A differential pressure transducer is required since the liquid is under pressure from the gas or vapour at thetop.

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h m

Differential pressure

transducer

Liquid

Gas

Figure 10. Level measurement by differential pressure in a closed vessel.

The height will be given by:

h = P/(g) m ..(3)

where Pis the pressure difference in Pa.

4.6. Flow rate measurement using differential pressure.

This method is widely used in various forms to measure flow rate in industrial processes. The pressure difference is generated by a fluid,gas or liquid in motion as opposed to the preceding sections where static fluids, in particular liquids, were considered.

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4.6.1. Measurement of flow rate with pressure drop over constant area in a Venturi.

A restriction is placed in a pipeline as diagrammatically depicted in the figure. Before and after the restriction the pressure is higher and

the velocity lower than in the restriction as will be derived. Although the restriction is shown with straight line sections in the sketch, thecurvature in practical systems has to comply with certain standards.

Q m3/s P1 Pa P2 Pa

D1 m D2 m

P=P1-P2 Pa

Q m3/s

P

Pressure loss

a, b pressure tap points

a b

Figure 11. Venturi tube.

Consider incompressible flow which would be applicable for liquids and gases with low pressure difference compared to the absolutepressure.

Mass balance:

kg/s222111 vAvA =

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where i = density at indicated position, kg/m3,

Ai = area at indicated position, m2,

vi = velocity at indicated position, m/s

With incompressible flow, 1 = 2 = kg/m3

yielding:

A1v1 =A2v2 m3/s

Energy rate or power balance for 1 m3:

(Potential energyat 1) + (kinetic energyat 1) = (Potential energyat 2) + (kinetic energyat 2)

where rates are considered for the dynamic system which is assumed lossless.

2

22

2

112

1

2

1vghvgh +=+ watt,

where hi m is the pressure height of the fluid under prevailing conditions and g= gravitational acceleration in m/s2. But Pi = ghi Pa as

shown before.

Watt2

1

2

1 222

2

11 vPvP +=+

From the mass balance or continuity of mass m/s21

21 v

A

Av = which is substituted into equation 9 to yield:

Pa12

121

2

1

22

2 PPA

Av =

( )m/s

1

22

1

2

212

=

A

A

PPv

and flow rate

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( )/sm

1

2 32

1

2

21222

==

A

A

PPAAvQ

.

In practice the energy rate or power balance is not satisfied by the simplified equation 6 and practical equations are of the form:

( )/sm

1

2 32

1

2

21222

==

A

A

PPACAvCQ dd

,

where Cd is a dimensionless flow coefficient.

4.6.2. Measurement of flow rate with pressure drop over constant area in an orifice.

Flow measurement can be done cheaper by using an orifice plate as shown in the figure.

Figure 12. Orifice plate.

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4.6.3. Velocity measurement with a Pitot tube.

This device, shown in the sketch, probes velocity over a very small area. Since Venturi tubes or orifice plate measurements areexpensive, the Pitot tube is sometimes used in pipes as a relatively cheap flow rate measurement device. It is also used in aviation. Theprinciple is also used in more some industrial flow meters where the velocity profile is probed at a number of points over the pipediameter.

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Figure 13. Velocity measurement with Pitot tube.

A simplified model for incompressible flow at low velocity is given here:

Power balance for 1 m3:

1

2

22

Pv

P =+

( )

PPPv

=

=

22 21 m/s

In practice

PCv d

=2

m/s

Note that the pressure drop is generally rather small.

4.6.4. Flow measurement in open channels.

Flow measurement in open channels can be found in water treatment plants, process industry and natural water courses where flow rateis to be measured.

A simplified model is derived for a rectangular weir.

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Figure 14. Rectangular weir.

Velocity at elementary area dA is ghvh 2= m/sLdhdA =But m2,

flow rate through element is dAvdQ h =

By integration 2/3223 HgLQideal = m

3/s.

Practical equations are of the form:

=2

32

0

23

2

0

221033,3

g

v

g

vH

nHLQ m

3/s

v0 = approach velocity,

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n = number of flow contractions in weir.

4.6.5. Flow measurement with constant pressure drop and variable flow area.

This flow meter has a float with higher density than the flow rate of the fluid to be measured. Consider the float that is placed in atruncated conical tube as shown in the figure. The pressure drop is determined by the weight minus the displaced fluid and is constant atsteady state operation. The float is kept suspended by the pressure drop as the flow area varies according to the flow rate. This flowmeter is available in a wide variety of forms from a simple see-through device to electronically operated versions.

The density of the fluid (gas or liquid) is given by 1 kg/m3.

The density of the fluid (gas or liquid) is given by 2 kg/m3.The velocity of the fluid through the annular area between float and tube is given by Bernoullis equation as:

1

2

Pv

= m/s.

The flow rate is given as1

22

2

PACvACQ DD

== m3/s,

whereA2 = area of annular opening, m2,CD = discharge coefficient,P= pressure drop, Pa.

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Figure 15. Conical flow meter.

LetAf= projected area of float, m2.

Vf= volume of float, m3,The weight of the float in the fluid = ( ) gVf 12 N,

whereg= gravitational acceleration, 9,81 m/s2,

1 = density of fluid, kg/m3,

2 = density of float, kg/m3.

Upward force due to fluid pressure = P1AfN.

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Downward force due to fluid pressure = P2AfN.

The total upward and downward forces are in balance at equilibrium:

( ) fff APgVAP += 2121 N.

( )gA

VPPP

f

f

1221 == Pa.

( )

=

= 12

2

1

2

2

1

12

2

f

f

D

f

f

DA

VgAC

gA

V

ACQ m3/s.

FindA2 as a function of the floating heightx:

( )2224

fx DDA =

m2.

From the figure:

( )2

tan2 xDD ix += m.

( ) ( )( )2222

2 2tan42tan44fii DxxDDA ++=

m

2

.

Let m.if DD =

( ) ( )( )2

tan42

tan44

22

2 xxDA i += m

2.

In practice the conical angle is small in order to get a great displacement of the float over the measuring range. The annular area canthen be approximated by:

( )( )2tan2 iDxA m

2

.Substitute into the equation for flow rate:

( ) xKA

VgDCxQ

f

f

iD =

= 12

2tan

1

2

m3/s.

In practical meters deviations from the ideal linear equation are due to the quadratic term and because CD is to some degree flow ratedependent. This is easily corrected by modern microprocessor-based meters.

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4.7. Exercises.

4.7.1. What should be added to enable maintenance to be carried out on the pressure gauges or pressure transducers without shutting

down the plant?

4.7.2. Refer to figure, Level measurement by differential pressure in a closed vessel. What would be the effect of gas bubbles in thelower tube from the liquid to the differential gauge? What would be the effect of liquid in the down-going tube the differential pressuregauge? What is the correct way to install the connecting tubes and differential pressure transducer where liquid is likely to enter thetubes?

4.7.3. A pressure gauge shows the pressure in a gas tank as 850 kPa. A barometer at the plant weather station shows that the prevailing

atmospheric pressure is 652 mm Hg. Find the absolute pressure of the gas in the tank.

4.7.4. A pressure gauge is situated 4,5 below an overhead pipeline. Over time, water has condensed in the pipe leading to the pressuregauge, filling it completely. The pressure gauge indicates 750 kPa. Find the actual pressure in the overhead pipeline as well as thepercentage error caused by the condensate.

4.7.5. The static pressure of a moving fluid in a pipeline is to be measured. Investigate the effect of the connecting hole or orifice. What

standards apply in the RSA?

4.7.6. A horizontal Venturi tube has a pipe diameter of 100 mm and a throat diameter of 75 mm. The fluid is water with a density of 1000kg/m

3. The discharge coefficient is 0,96. The mean flow velocity in the pipe is 3m/s.

a) Calculate the flow rate in m3/s at the given flow velocity.b) Calculate the pressure difference at the given flow velocity.

4.7.7. A triangular weir is shown in the sketch. This type of weir is particularly suitable for spot checks of flow rate in open channels.

a) Derive the equation for the flow rate Q in terms of flow height H.b) Calculate the height H if Q = 1,0 m3/s and = 90.

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Figure 16. Triangular weir.

4.7.8. A Pitot static tube with 10 mm diameter is placed concentrically inside a pipe with 20 mm diameter. The flow velocity far from thePitot tube is 3 m/s. What velocity would the Pitot tube indicate?

Figure 17. Pitot tube inside a pipe.

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4.7.9. The equations for the flow rate through the Venturi or orifice plate were derived for the device installed horizontally. What happenswhen the flow is vertically upwards (or any other angle)? How can this be compensated for?

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5. Converting force to an elastic displacement signal.

5.1. Objectives.

The student should be able to compile a block diagram for an instrument that converts a force or pressure difference to an electronicsignal. The derivation of equations for simplified models of transducers that convert a force variable to a deflection or displacement whichin turn can be converted to an electronic signal should be accomplished. The transducers should be described in bothqualitative(describe with words and sketches) and quantitative (derivation of equations, correct substitution of numerical values) terms.

5.2. Introduction.

In the previous section, a pressure difference was created in the measurement of pressure, liquid level and flow-rate. The pressuredifference creates a force which can, amongst others, be converted to an elastic displacement. In industry, force measurements are oftenrequired, and, by using gravity, to infer the mass of the contents of trucks, silos and tanks. It is used in conveyer belt systems togetherwith speed sensors to measure mass flow rate. The deflection or displacement can be converted to an electrical signal by using resistive,

capacitive, inductive, optical or piezoelectric transducers each within it own area of applicability.

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Figure 18. Conversion of force to an electrical signal.

Broadly speaking, strain is proportional to applied force provided the material remains elastic. The relation between strain and force isgiven by:

E

F=

L

L=

,

= strain, m/m,F= force, N,E= modulus of elasticity or Youngs modulus, Pa,

A = cross-sectional area, m2,L = length, m,

L = elastic displacement, m.

Temperature acts as an interference in two ways. Firstly, the material expands with increasing temperature and contracts when thetemperature drops. This change will be incorrectly interpreted as due to pressure or force and will cause an output even if no pressure orforce is applied. This manifests a zero offset. Secondly, the modulus of elasticity temperature dependent and changes the slope of strain

versus pressure or force.

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5.3. Converting force to strain with an elastic cylinder.

The cylindrical transducer is used to measure fairly large forces such as in weigh bridges and material in containers, as part of massmeasurement. These are called load cells.

Figure 19. Cylindrical load cell.

The strain in the axial direction is given by:

AE

F=

L

L m/m

L = axial deformation, m,L = length, m,

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F = applied force, N,A = cross-sectional area, m

2,

E = modulus of elasticity or Youngs modulus, Pa.

If the material is stretched in the axial direction, then it will shorten around the circumference, or transversally. Since the material iscompressed, the transversal deformation is less:

L

L= -

d

d ,

where = Poissons ratio.At an angle the strain is given by:

( ) ( )( )

2cos112

)( ++= L ,L

LL

=

By making use of the axial strain as well as the transversal strain, conversion to an electronic signal can be made which will have only avery small temperature effect.

5.4. Converting torque to strain with an elastic cylinder.

This transducer offers great challenges if it is mounted on a rotating shaft. Strain can be measured using a number physical effects.

Refer to the figure. The strain on the surface of the torsion bar at angle is given by:

( ) ( )

2sin3Gr

T= m/m

T= torque, Nm,

r= radius, m,

( )+=

12G

E= modulus of rigidity, Pa,

E= Youngs modulus, Pa,

= Poissons ratio.

Strain is a maximum at = 45 and is ( ) ( )Er

T3max

1245

+==o

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By measuring the strain along the 45 lines, positive and negative strain (see arrows) for the torque as shown is used to make conversionto an electrical signal less temperature sensitive.

Figure 20. Torque measurement.

5.5. Converting pressure differential to a deflection with a stretched membrane.

A pressure difference causes a taught membrane to bulge towards the lower pressure. The transducer described here, operates with avery small deflection which is only a small fraction of the thickness of the material. It is placed under tension by a conical ring which triesto expand a rim on the membrane circumference.The displacement can be converted to an electrical signal by using a suitable physical effect.

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Figure 21. Membrane to convert pressure difference to displacement.

Assume a thin membrane to which a radial force is applied around the circumference.Compile equations for the forces in balance when the force due to the pressure difference is balanced by the component of radialtension.

PaF = 21 N, the force due to the pressure difference.

r

aASaF 2cos22 ==

At equilibrium, F1 = F2 ,find radius of curvature R:

PaRSa 222 =

P

SR

2= m.

Find equation for membrane curve:From figure:

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S

rP=

S

aP =

2

cos

2cos

( )

)S

aP-(-)

S

rP-(

P

S=

-P

Sy =

- Ry = R

22

1

2 2

1

21

21

2

sinsin2

sinsin

Simplify with:

+...x!

)n(n-+nx+=)n+x( 2

2

111

and keep linear terms:

[ ]

m4

m4

m8

18

12

2

22

2

22

2

22

S

Pah =

r-aS

P=

S

Pa+-

S

Pr-

P

Sy

This deflection can be converted to a change in capacitance as will be shown later.

5.6. Exercises.

5.6.1. A steel cylinder is use as primary transducer in a load cell. Its length is 50 mm and its diameter is 20 mm. The linear temperaturecoefficient of expansion is = 1.1x10-7 K-1. Youngs modulus of elasticity is E = 210 GPa at 20C and temperature coefficient

5104.2

=E

E K-1. Poissons ratio = 0.3. It is stretched by a force of 10 kN and is subjected to temperature variations.

a) Find the change in length due to the force of 10 kN at constant temperature.b) Find the strain in the axial and transversal directions if a force of 10 kN is applied at constant temperature.c) Find the change in length if the temperature increases by 10C with zero force applied.

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d) Find the change in length due only to the change in Youngs modulus if the temperature increases by 10C with an applied force of 10kN.e) Compressive strength of the steel is 340 MPa. Assuming maximum design compression of 100 MPa, find the full scale load cell force.

5.6.2. The sketch shows a cantilever beam which is used as the elastic element in a load cell. The beam deflects if a force is applied asshown in the sketch. The relation between deflection ym of the end point of the beam and the force F N is given by:

3

3

4L

Ewt

y

F= ,

E= Youngs modulus Pa,w= width of beam, m,L = length of beam, m,

t= thickness of beam, m (and not t for time in this case).

Linear coefficient of expansion of steel: = 11,88 x 10-6/K;Temperature coefficient ofE: -239,4 x 10

-6/K.

Derive an equation for the fractional change in spring constant per unit temperature change and determine it for the given beam.

39


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Figure 22. Cantilever beam.

Additional information:

Strain atx: ( ) 26

Ewt

x

Fx = m/m.

Deflection at end point( ) 3

123

3

3

3

max

L

wtE

FL

EI

Fy == m.

5.6.3. The cantilever beam as shown in the sketch above is hardly practical as a load cell. How is the cantilever implemented in practicalload cells?

40


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6. Converting displacement or strain to electrical signals.

6.1. Objectives.

The student should be able to compile a block diagram for an instrument that converts displacement or strain to an electronic signal. Thederivation of equations for simplified models of transducers that convert a deflection or strain to a change in the parameters of anelectrical component which in turn can be converted to an electronic signal should be accomplished. The transducers should bedescribed in both qualitative(describe with words and sketches) and quantitative (derivation of equations, correct substitution of numericalvalues) terms.

6.2. Introduction.

The displacement or strain associated with force in load cells or pressure in differential pressure cells is generally very small. Thematerials that exhibit the physical effects to convert displacement and strain to electrical signals are mostly also temperature sensitive.The materials that are used for the elastic conversion of force or pressure to displacement, are also temperature sensitive. For thisreason the physical implementation of conversion to electrical signals can be made much less temperature sensitive if the properties ofboth elastic and electrical materials are matched.

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Figure 23. Conversion of displacement or strain to an electrical signal.

6.2. Resistive strain gauges.

6.2.1. Introduction and objectives.

The strain gauge is a resistor which can be mounted with adhesive on an elastic sensor. When subjected to strain, its resistance changesproportional to the strain. This may seem easy to use, but there are some physical effects which can render the transducer useless:

The resistance of the strain gauge changes with temperature. Both the strain gauge and the elastic member expand with increasing temperature, but at different rates. The modulus of elasticity of the elastic member changes with temperature.

42

This element is a very popular transducer in mechanical systems It is used in industrial instruments and research in mechanical


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This element is a very popular transducer in mechanical systems. It is used in industrial instruments and research in mechanicalengineering.

The reader should be able to define and interpret the characteristics of strain gauge transducers and describe their operation from a

phenomenological viewpoint and be able to describe their operation coupled to the mechanical environment.

6.2.2. The strain gauge factor.

Since we are dealing with resistive strain gauges, the theory of resistors applied to electric circuit theory can be found Appendix B.

A phenomenological theory (meaning we will not dwell on esoteric theory such as quantum physics) of resistance will be given here.

Refer to the figure. The resistance of a uniform conductor is given by=

A

lR

where

= specific resistivity, ml= length, m andA = cross-sectional area, m

2.

When subjected to deformation, the length of the conductor changes. Assume it lengthens as shown in the figure. The cross-sectionalarea will then decrease, but not such that the volume remains constant. The material is under pressure and Poissons ratio describes thiseffect. Three effects determine the change in resistance: Change in length, change in cross-sectional area and change due to the internalpressure in the material.

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Figure 24. Strain gauge and strain in an electrical conductor.

Find differential elements:

A

A

l

l

R

R

A

dA

l

dld

R

dR

lR

+=

+=

+=

lnlnlnln

But 2

4dA

= ,with ddiameter in m.

ddA = 24

m

2

From Poissons ratio : lateral strain = -x longitudinal strain44

ldA


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( )

++

=

=

21

22

l

l

R

R

l

l

d

d

A

A

Strain gauge factor

l

lR

R

l

lR

R

m

++=

=

=

21

Examples of strain gauge materials.

Material Strain gauge factor mConstantan 45% Ni, 55% Cu 2,1

Nichrome V 80% Ni, 20% Cr 2,5

Ni -12

Pt 4,8

p-Si 175

n-Si -133

6.2.2. Force measurement with a cylinder and strain gauges.

Strain gauges can be mounted on the cylinder in a number of configurations. In this example two strain gauges are placed in the axialdirection and two on the circumference, a total of four. These are then connected in a bridge. This configuration has a number ofadvantages.

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Figure 25. Strain gauges on cylinder connected in a bridge.

Strain in axial direction = 1, R1 = R2 = R(1-m1).Strain in transversal direction = -1, R3 = R4 = R(1+m1).Let m1 = .Find the output voltage of the bridge:

( )( )121

42

2

13

3

++

=++=

RR

R

RR

R

V

V

b

o

Note that the relation is not linear with strain, however, ( ) 21


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RT resistance at temperature TK in ohm,

R0= resistance at reference temperature T0K, ,= temperature coefficient of resistance, K

-1.

Since the strain gauges are mounted on the same steel cylinder which is a good thermal conductor, all will be at the same temperature.The placing of the load cell in industrial use should take this requirement into account.

The thermal expansion of the steel cylinder and the strain gauges can also create a strain:

( ) Testraingaugsteeltemp = , wheresteel= linear coefficient of expansion of steel, K

-1,

strain gauge = linear coefficient of expansion of strain gauge material, K-1,

T= temperature change, K orC.

Finally, the modulus of elasticity or Youngs modulus is also temperature sensitive, ( )( ) ( )TETTETEET +=+== 11)( 000 Pa.ET= modulus of elasticity at temperature TK in Pa,E0= modulus of elasticity at reference temperature T0K, Pa,

= temperature coefficient of modulus of elasticity, K-1

.This effect cannot be compensated for by the bridge arrangement.

6.2.3. Torque measurement.

Refer to the figure. Utilize the strain at the 45 lines:

( ) ( )Er

T345

1245

+==o .

The strain gauges are placed in a bridge as shown in the figure; the output is given by:

450 m

V

V

b

==

47


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Figure 26. Torque measurement.

6.3. Capacitance transducers for differential pressure.


Since we are dealing with capacitance transducers, the theory of capacitors will be briefly revisited in Appendix B.

Firstly, the simplified theory of a capacitor: Consider two equal parallel flat plates each with areaA m2 and distance D m apart. The

volume is filled with a dielectric material with relative permittivitypr(note that the symbol r is generally use for relative permittivity, but to

avoid confusion the symbolpr is used here). Since the material occurs in space with permittivityp0(or0) = 8,854x10-7

Farad/meter, the

permittivity of the region between the plates isprp0 . The capacitance is given byD

ppAC r 0

= Farad.

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Figure 27. Capacitor.

The capacitance equation can be used to design sensors for specific applications.Firstly, by varying the area A it can be used to measure displacement. This achieved by (say) moving the upper electrode in the figure tothe right since by doing so the area over which the field act is reduced.Secondly, the distance Dcan be changed in a displacement transducer.Thirdly, the area and distance can be kept constant, but the permittivity prcan be allowed to vary. By moving a dielectric in or out of thecommon volume, displacement can be converted changes in capacitance. This is widely used in level measurement of liquids and

granular solids.Fourthly, by keeping the area and distance as well as the dielectric volume constant, moisture constant of material (water has a highrelative permittivity, about 84 relative to air) can be converted to capacitance changes. It may also be used in thickness measurement ofplastic sheeting, etc.

The student should be able to derive simplified models for capacitance-based pressure transducers.

6.3.2. Capacitance differential pressure cell.

A pressure transducer is shown in the figure.

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Figure 28. Differential pressure to capacitance.

A pressure difference elastically displaces the central electrode relative to the fixed electrodes.

Let P= P1 P2 in Pa, the deflection from the zero position is d= KPm.

dD

ppAC r

+

= 01 F and

dD

ppAC r

= 02 F.

The bridge output is given by:

12

20

11

1

CjCj

Cj

RR

R

V

V

b

++= . The rest of the derivation is left as an exercise.

6.3.3. Stretched membrane capacitance cell.

A pressure difference causes a taught membrane to bulge towards the lower pressure. The membrane, which forms one electrode of acapacitor, bulges towards a fixed electrode (increasing capacitance) and away from another fixed electrode (decreasing capacitance) as

shown in the figure. The design whereby one capacitance increases and the other decreases inherently compensates for most of thetemperature effects. Note that some pressure transducers operate with a single fixed electrode only and design electronics tocompensate for temperature effects.

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Figure 29. Stretched membrane capacitance transducer.

For small deflections the form of the membrane is given by:

( )224

raS

Py

= m.

P= pressure difference, Pa,S= membrane tension, N/m,a = radius of membrane, m,

r= radius at ring element of capacitance, m.

The capacitance of a ring element relative to the upper fixed electrode is given by:

yd

rdrppdC r

=2

0 F.

For small deflections:

51

+=

y1

1111m-1


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+

dd

d

ydyd1

1

m .

From this approximation the capacitance has two parts, a fixed part and a variable part:

PSd

ppa

d

ppadrr

d

y

d

ppCC rr

ar +=

+=+ 204

0

2

0

00

81

2 F.

The sensitivitySd

Pa

C

C

8

2

0

=

. The change in capacitance is proportional to the pressure difference.

A similar derivation can be done for the lower electrode. As before, a bridge circuit is used to convert the differential capacitance to avoltage.

6.4. Inductive transducers for displacement.


The circuit theory of inductance will be briefly discussed in Appendix B.

A simplified model of an inductor is shown in the figure. In sensors the aim is generally to confine the magnetic flux as far as possible todefined and predictable paths. This aim can never be entirely achieved, but with modern materials the error is generally very small. Referto the figure. The self inductance of a network is given by:

elR

NL

2

= H,

whereN= number of windings,Rel= total reluctance or resistance to magnetic flux in Ampere/Weber.

The theory behind this sensor is stated here in short form:H= magnetic field strength in Ampere/m or A/m,i= current in A,for a path with n discrete members as shown in the figure:

52

iNlH kk = A,


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n

where kdenotes a path.

Figure 30. Inductor.

B = magnetic flux density in Tesla (T) or Weber/m2,

= r0= permeability in Henry/m, 0 = 4x10-7

H/m.

B = HT.

The magnetic flux is given by = BA Weber.Define F= N i, the magneto motive force or mmf given as Ampere or Ampere-windings to emphasize the role of windings.

53

iNA

lH kkk ==

A


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An kknkk

el

n kk

k R

A

lFNi===

, reluctance in A/Wb.

InductanceelR

N

i

NL

2

==

Henry or H.

6.4.2. Inductive displacement transducer for force and pressure.

Assume the iron sections (which may be ferrous or ferrite material) to have infinite permeability. The air gaps are therefore the only partswith reluctance.

Figure 31. Inductive displacement transducer.

54

The total reluctance is thus given by:x2


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A

xRel

0

2

= A/m.

Inductance = xAN 12

0

2

H.

Any of the previous membrane pressure transducers or the beam force transducer can be used with the inductive transducer. It may alsobe used in differential mode with two sets of windings.

6.5. Exercises.

6.5.1. A strain gauge has a nominal resistance of 100 . The strain gauge factorm = 2,5. Find the change in resistance if it is subjectedto a strain of 10-3 m/m.

6.5.2. A resistor of Constantan (45% Ni + 55% Cu) is used to measure hydrostatic pressure changes.

Nominal resistance: 100 .Pressure sensitivity: -7x10

-12m

2N

-1.

Temperature coefficient of resistance: 2x10-5 K-1.a) Find the change in resistance for a change in pressure of 10 MPa (~100 Atm) at constant temperature.

b) Find the change in resistance for a temperature change of 1C or 1K.

6.5.3. Refer to the strain gauges on the cylinder in 6.2.2. Let Poissons ratio = 0,3 and the strain gauge factorm = 2,1. Investigate thebridge output for a strain of 10-4 and 10-3 and comment on the linearity.

6.5.4. Verify 450 m

V

V

b

== for torque measurement, section 6.2.3.

6.5.5. How can the strain gauge bridge be applied to a rotating shaft?

6.5.6. Shown that the strain which increases linearly with distance, as measured by a strain gauge, is given by the strain at the centre ofthe strain gauge.

6.5.7. Show thatD

PK

V

V

b 2

0 = for 6.3.2.

55

6.5.8. Differentially connected inductors are used in a membrane differential pressure transducer. Show how you would connect the


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6.5.8. Differentially connected inductors are used in a membrane differential pressure transducer. Show how you would connect the

inductors in a bridge and derive the output ifx = KP.

Figure 32. Differential inductance transducer for differential pressure.

56

7. Converting temperature to electrical signals.


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7.1. Introduction and objectives.

The performance of some temperature transducers must be derived and quantitatively explained. The student should quantitativelyexplain temperature measurement using electrical resistance and thermocouples in terms of the relevant physics. Temperaturemeasurement using radiation should be explained in terms of the radiation laws.

Temperature measurement is important for the control of many industrial processes. Temperature is measured over a very wide range

with the result that it is not possible to use a single transducer from 0K to 3000K . Thermal power can be transmitted by conduction,convection and radiation, or a combination thereof to the transducer.

Electrical resistance is temperature dependent and is excellent for measuring temperature provided the materials are chosen forrepeatability and good sensitivity. Thermocouples, which generate a voltage related to temperature, are perhaps the most widely used.Again the choice of materials is essential for repeatability and stability. In some processes temperature is measured without directcontact by making use of radiation. In all cases thermal energy is exchanged until equilibrium is reached. The transient performance oftemperature measurement is important for control systems.

7.2. Heat transfer for thermal measurement systems.

Temperature measurements are generally not instantaneous. Heat transfer has to take place from the fluid being measured to thetransducer. In an environment where temperature changes occur rapidly, the indicated temperature of the sensor differs from the processtemperature. In order to appreciate this difference, consider the following simplified theory of heat transfer.

Consider flow of thermal power through a solid with cross sectional areaA m2 perpendicular to heat flow and with thickness d m. Refer tothe figure. The temperatures at the surfaces are T1 Kelvin and T2 K respectively. Since we are working with differences in temperature,temperature may also be expressed in Celsius.Definition: The thermal power flow is given by:

( )12 TTd

KAQ = Watt,

where

57

K= specific thermal conductance W/(mK).


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Figure 33. Thermal resistance of solid material.

Definition: Next consider thermal flow from a fluid to the surface of solid. Refer to the figure. Thermal power flows via a boundary layer.)af TThAQ = Watt,

whereh = heat transfer coefficient W/(m

2K),

A = area perpendicular to thermal power flow m2,Tf= bulk fluid temperature in K or Celsius,Ta = temperature on surface in K or Celsius.

58


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Figure 34. Thermal power flow from fluid to solid surface.

Definition: Thermal capacitance is obtained by considering a solid with uniform temperature as outlined in the figure, but with thetemperature relatively slowly time varying due to thermal power flow:

dt

dTmcQ ms= Watt,

whereTm = uniform temperature in Kelvin or Celsius since we will assume a reference temperature,cs = specific thermal capacitance in J/(Kg K).

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Figure 35. Thermal capacitance.

Consider thermal power flow through composite resistance, say boundary layer and solid as shown in the figure:

kA

d

hARRRR solidlayertotalT +=+==

1W/K

The thermal conductance is given by:

UA

k

d

h

A

k

d

hA

=+

=

+111

1K/W, where UW/(m

2K) is the total heat transfer coefficient. Similar relations could be derived for symmetric

cylindrical applications such as a thermocouple in a well.

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Figure 36. Composite layers.

Now consider heat transfer through a composite thermal conductance to an isothermal body:

(Thermal power inflow)-(thermal power outflow)=(rate of accumulation of thermal energy)( ) ( )( ) ( )refmmsrefmmrefff TT

dt

dmcTTTTUA = 0

Consider deviations around Tref:

( )dt

dTmcTTUA msmf = 0 , or

dt

dT

UA

mcTT msmf = 0

The time constantUA

mcs= seconds (Why?). The analogue electrical network is shown in the figure.

61


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Figure 37. Thermal RC network.

7.3. Temperature scales.

Formal definitions of temperature can be found in books on physics and thermodynamics, but these are not useful for constructingmeasurement sensors. Practical temperature scales can be defined by the triple point (gas, liquid and solid), the boiling point and thefreezing point of some standard substances. These scales are used for calibrating temperature transducers.

Defining fixed points C KTp hydrogen -259,34 13,81

Bp hydrogen 25/76 atm -256,108 17,042

Bp hydrogen -252.87 20,28

Bp neon -246,048 27,102

Tp oxygen -218,789 54,361

Bp oxygen -182,962 90,188

Tp water +0,01 273,16

Bp water 100 373,15

Fp zinc 419,58 692,73

62

Fp silver 961,93 1235,08

Fp gold 1064,43 1337,58

Tp = triple pointBp = boiling point


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Bp boiling pointFp = freezing point

Between these temperatures interpolating instruments are used:Pt resistance 13,81-273,15K 20

thorder polynomial

Pt resistance 0-630,74C Modified Callendar

Pt, 10%Rh&Pt thermocouple 630,74-1064,43C Parabola

Optical pyrometer Above 1064,43 Plancks law

7.4. Resistance temperature measurement.

7.4.1. Theory.

The temperature dependence of suitable materials can be used to construct temperature transducers. Platinum is the best candidate dueto the stability of its properties. For the purposes of control systems, a linear increase in resistance with increase in temperature ismodelled by:

( )( ) ( )TRTTRTRRT +=+== 11)( 000 , where

RT= resistance at temperature TK in ohm,R0= resistance at reference temperature T0K, ,= temperature coefficient of resistance, K-1.

Temperature coefficients over the range 0-100C.Platinum = +3.92*10

-3/K

Nickel = +6.80*10-3

/K

Copper = +4.30*10-3

/K

Tungsten = +4.80*10-3/KIron = (+2.00 - +6.00)*10

-3/K

Manganin (an alloy) = +2.00*10-5

/K

Constantan (an alloy) = +4.00*10-5

/K

Carbon = -7.00*10-4

/K

Electrolytes = (-2.00 - -9.00)*10-2

/K

Thermistors (components) = (-1.50 - -6.00)*10-2/K

63

Platinum must be of a certified purity which can be verified with 3925,1100 R

R, ratio of resistance at 100C and 0C.


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0R

7.4.2. Compensating for variations in connecting lead resistance.

When a resistance temperature detector (RTD) is used, the connecting leads are also temperature sensitive. Three configurations areshown in the figure.

Figure 38. RTD configurations.

64

7.5. Thermocouple temperature measurements.


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7.5.1. Theory.

When two metals are connected with the junctions at different temperatures, an emf is generated. This effect is known as the Seebeckeffect. This theory should be read in conjunction with the Peltier effect and the Thomson effect.

Figure 39. Thermocouple.

Modern measurements have shown that the emf versus temperature can be described by order polynomials. However, over a limitedrange, a second order approximation can be used.

e = ATc+0,5BT2

with T2 = 0; T1 = Tc:Metal A V/C B V/C2

Iron +16,7 -0,0297

Copper +2,7 +0.0079

Constantan -34,6 -0,0558

7.5.2. Rules.

When using thermocouples to measure temperature, three rules apply.

Rule of homogeneous metals: A thermo-electric current cannot be sustained by heat alone in a network of homogeneous metals. Non-homogeneous metals may generate spurious or parasitic emfs.

65

Rule of connecting metals: The algebraic sum of thermal emfs around a network consisting of metals equals zero, provided the wholenetwork is at the same temperature. This means that if one has a thermocouple of two metals, a connecting wire of a third metal may beused without affecting the emf, provided the extra two junctions are at the same temperature. It also implies that if the thermal emf e2,1 for


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g , p j p p 2,1metals 1 and as well as e3,1 for 3 and 1 are known, then e2,3 = e2,1+e3,1, provided all the junctions at temperatures T1 and T2 as shown in

the figure.

Figure 40. Rule of connecting metals.

Rule of intermediate temperatures: If the same two metal junctions at T1 and T2 generate ea, and at T2 and T3 generate eb, then withthe junctions at T1 and T3 the emf ec = ea + eb.

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Figure 41. Intermediate temperatures.

From the description thus far, it is obvious that one of the junctions be kept at a known temperature a thermocouple is to be used as atemperature transducer.

7.5.3. Thermocouple types.

Type Material Temperature range Slope

T Copper/Constantan 0-400C 4,80x10-2

mV/K

J Iron/Constantan 0-760C 5,76x10-2

mV/K

E Chromel/Constantan 0-1000C 7,50x10-2

mV/K

K Chromel/Alumel 0-1100C 3,96x10-2

mV/K

B Platinum/Platinum+30%Rhodium 0-1820C

R Platinum+13%Rhodium/Platinum 630-1665C 1,11x10-2mV/K

S Platinum+10%Rhodium/Platinum 630-1665C 1,02x10-2

mV/K

Notes:Slope with reference at 0CConstantan: ~ 40% Ni, 60% CuChromel: ~ 90% Ni, 10% CrAlumel: ~ 94% Al, 2% Ni, Si, etc

67

Some of these thermocouples may also be used below 0C

7 5 4 Th l f j ti


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7.5.4. Thermocouple reference junction.

The reference junction must be at a known temperature which is measured by a sensor (other than a thermocouple!) such as resistancetemperature sensor or the modern semiconductor pn-junction type.

7.6. Temperature measurement using radiation.

Where it is not feasible to use a contact sensor such as a thermocouple or RTD, radiation methods can be used. Radiation transducers

typically utilise radiation in the range 0,3m - 40m. Visible radiation fall within the spectral range of 0,3m 0,720m.

7.6.1. Theory.

The concept of black body radiation is used in radiation thermometry. A black body, best approximated by the entry hole into a cavity,radiates according to Plancks law for spectral hemispherical radiation intensity:

( )1

2

51

=

TC

Cw

W/(m2m), where

= wavelength, m,C1 = 3,743x10

8 Wm4/m2

C2 = 1,4387x104mK.

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Figure 42. Spectrum (w()) of black body radiation.

The positions of the peaks are given byT

p

2891= m.

The total power over the spectrum is given by 480

5

1 1067,5

12

TdCWtT

C

=

=

W/m2

Actual radiators differ from black body radiators. The hemispherical spectral emittance is defined by ( ) 1,


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Assume that Wta can be experimentally determined. Define the total hemispherical emittance: ( )t

tat

W

WT = . The total emitted power is

then modelled as ( )TTW tta = 481067,5 W/m

2. A body with constant ( )T, is defined as a grey emitter. In general radiation

thermometers operate over a limited band and

)d-(/

)d-(,T)/(

(b,T) =T/C

T)/C

b

a

b

a

11

1

)(5

(5

2

2

.

Emittance depends on the size, form and surface of the body as well as the angle through which observations are made.

Emittance = 1 r t , where r = coefficient of reflection, t = coefficient of transmission.In practice the part of the spectrum that is suited to measurements must be found. The absorption of the radiation by molecules in thetransmission path attenuates the radiation reaching the sensor. While absorption is a disturbance in the case of temperaturemeasurement, it is useful in the infra-red analysis of gas mixtures.

7.6.2. Detectors.

A number of detectors are available. These are designed to absorb the maximum amount of incoming radiation. The temperature of thedetector rises until equilibrium is used. Thermal detectors measure this temperature rise. Examples are thin film resistance temperaturetransducers, thin film thermocouples and thin film thermistors.

There are also detectors based on the piro-electric effect which operate over a large range of wavelengths, but the radiation needs to be

chopped since this type of detector delivers a current given by dt

dTKis = A, where Kis a constant depending on the material and Tis the

temperature.

Photon detectors do not depend on temperature rise and has a faster response. Types are photo conductive, photo voltaic and photoelectromagnetic which uses the Hall effect.

70

7.7. Exercises.

7.7.1. A platinum resistance temperature transducer has a first order lumped model.2 2


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Mass: 0.1 kg, specific heat is 250 J/kgK, surface area is 0,005 m2, total heat transfer coefficient is 50 W/m2K, Resistance at 0C is 25,5

, temperature coefficient of resistance is 3,85x10-3/K.Find the time constant.

7.7.2. Find the surface temperature whereby a black body would emit 1 MW/m2.

7.7.3. A resistance thermometer of nominally 100 carries a measuring current of 10 mA. Its surface area is 3,5x10-4 m2.a) In still air the total heat transfer coefficient is U = 8,5 W/m2K. Find the steady state error due to self heating.

b) In still water the total heat transfer coefficient is U = 567 W/m

2

K. Find the steady state error due to self heating.

7.7.4. Find the open circuit voltage of an iron-copper thermocouple with one junction at 200C and the other at 0C.

7.7.5. Analyse the transient response of a thermocouple in an oven where heat transfer is modelled by radiation only. The thermocoupleis insignificant in relation to the oven.

71

8. Diverse topics from measurement and control.


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8.1. Introduction and objectives.

This chapter concludes the course with a discussion of few short topics. The student should be able to describe some topics withsketches, equations, graphs and (few) words. Derivations should be performed. Numerical answers by substitution into the correctequations should also be done.

8.2. Control valves.

8.2.1. Introduction.

Approximately 15% of the cost of a plant in the process industry goes into measurement and control. Of this cost of control about twothirds go into control valves. Thus, in a chemical process plant the cost of control valves can run into hundreds of millions. It stands toreason that a few minutes lecture time can only be a very brief introduction to this topic. Interested students can download any number ofhandbooks from the Internet, the Control valve handbook, 4th ed, (Fisher) Emerson Process Management, 2005, is a good example.

A control valve forms a controllable flow resistance and has to be chosen correctly to achieve control of fluid flow. At rated flowapproximately 33% of the total pressure drop takes place over the control valve. This is best done with a computer program.

The flow rate through a valve is given by:

PKCq fv = where Kfdepends on the density of the fluid.

8.2.2. Characteristics.

The standard symbol for control valve usage is Cv. The value depends on the operating point as a function of valve stem position:

72

maxmax X

X

C

C

v

v = , linear characteristic.

X

X

CC


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max

min

max

min

X

v

v

v

v

C

C

C

C

= , equal percentage characteristic.

maxmin

max

min

maxmax 1

1

X

X

C

C

C

CC

C

v

v

v

vv

v

= , hyperbolic characteristic.

maxmax X

X

C

C

v

v = , square root characteristic.

73


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Figure 43. Control valve and pneumatic force motor.

8.2.3. Positioning of stem.

A valve with a pneumatic (compressed air) motor is shown in the figure. Only the valve motor is shown in the figure. The valve stemposition is controlled by feedback.

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Figure 44. Positioning servo for valve.

The positioner servo positions the valve stem against varying process pressure as well as friction of the valve stem seal packing.

In practical installations the valve has to investigated for sonic flow in the case of gases and for cavitation in the case of liquids. Very highgas velocity through a valve causes a lot of noise. In the case of liquids high velocity through the valve could cause the pressure to dropbelow the boiling point of the liquid at the prevailing temperature. As the liquid slows down when the area increases, the vapour bubblescollapse with very high pressure pulses and when this happens against the valve material, it is eroded. Note that bends, reducers, etc

play a role in this.

8.2.4. Sizing of control valves.

Many factors influence the choice of control valves.

For liquids: Sub critical flow occurs if the liquid never encounters a pressure lower than the vapour pressure at the prevailing temperatureof the liquid. Critical flow is encountered if the pressure at some stage through the valve falls below the boiling point. Cavitation occurs,which require a different design strategy since cavitation will cause serious damage to the valve. Turbulent flow occurs for Reynoldsnumbers > 2000. For laminar flow different design equations apply.

Note: The Reynolds number

vdRe = , dimensionless,

v= flow velocity, m/s,d= characteristic dimension, m, details of which can be found in any treatise on fluid mechanics,

75

= density of the fluid, kg/m3,

= dynamic viscosity, Pas.

An example of formulas for a valve installed full bore (no reducers) for liquid service, used in a computer program, are given here:


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p ( ) q , p p g , g

laminarPq0.072=C

critical

PP

P0.28-0.96-P

G

C

q=C

critical-sub

3

2

v

v

c

v1

f

f

v

PGq=C

f

v

The maximum value of Cvfrom the three formulas is used.

For these formulas the symbols have the following meanings:

q = flow rate m3/hCf = critical flow factor for the valve which depends on the manufacturers design, typically 0,4 to 0,96.P = pressure drop in bar.Gf = specific density of liquid relative to water at 15C.P1 = absolute upstream pressure, bar.

Pv = absolute vapour pressure of the liquid at the operating temperature, bar. For water it is 1,01 bar at 100C and 0,069 bar at 38C.Consult thermodynamic tables.

Pc = absolute pressure at the thermodynamic critical point for the liquid, bar. For water it is 221 bar at 374C. = viscosity, centipoise.

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8.4.3. Prevention of ignition.

Ignition can take place through electric arcing or by electrically heated surfaces. Note that personnel are to use spark-free tools as wellas intrinsically safe electrical instrumentation in classified zones in a plant


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y p

By experimentation ignition curves for combustible gas and air mixtures were obtained.For resistance: Minimum current for ignition against voltage.For inductance: Minimum current for ignition.For capacitance: Minimum voltage for ignition.

There is no correlation between ignition by hot surfaces and ignition by sparks or arcs because of totally different chemical mechanisms.

8.4.4. Safety barrier.

Figure 45. Intrinsically safe installations.

78

Intrinsically safe systems operate in three regions:

Apparatus in the safe area: Design may be uncertified, provided it does not involve the barrier.


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The safety barrier: This contains Zener diodes to withstand heavy current, resistors to withstand heavy current and fuses. May notcontain energy storage elements. This must be certified apparatus.

The danger area: Cables and devices. Energy storage limited. Intrinsically safe devices only. Certified by SABS in South Africa.

8.4.5. Zones.Classification of danger zones: shown here in the interests of safety. [ref]

Nature of hazard

Class I Class II Class III

Gases and vapors Dusts Flyinqs

Group A-Acetylene Group E-Metal dusts No group assigned:

Group B-Hydrogen or gases or Group F-Carbon black. coal, Typical materials are cotton,

similar hazards, such as man- coke dusts kapok, nylon, flax, wood chips-

ufactured gas, butadiene, Group G-Grain dust, flour, normally not in air suspension.

ethylene oxide, propylene- plastics, sugar. _

oxide.

Group C-Ethyl ether, ethyt-

ene, cyclopropane, unsym-

metrical dimethylhydrazine,acetaldehyde, isoprene.

Group D-Gasoline, hexane;

naphtha, benzine, butane,

propane, alcohol, acetone,

benzol, lacquer solvent, nat-

ural gas, acrylonitrile, ethyl-

ene dichloride, propylene,

styrene, vinyl acetate, vinyl

chloride, p-xylene.

Division 1

For heavier-than-air vapors, Cloud of flammable concentra- Areas where cotton, spanish moss,

below grade sumps, pits, et al, tion exists frequently, period- hemp, et al. are manufactured or

in Div. 2 locations. Areas ically, or intermittently-as processed.

around packing glands; areas near processing equipment.

where flammable liquids are Any location where conducting

handled or transferred; areas dust may accumulate.

adjacent to kettles, vats, mix-

ers, et al. Where equipment

failure releases gas or vapor

and damages electrical equip-

ment simultaneously.

Division 2

79

Areas adjacent to a Div. 1 area. Failure of processing equipment Areas where materials are stored

Pits, sumps containing piping may release cloud. or handled.

et al., in nonhazardous loca- Deposited dust layer on equip-

tion. Areas where flammable ment, floor, or other horizontal

liquids are stored or processed surface.

in completely closed piping

or containers. Div. I areas


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or containers. Div. I areas

rendered nonhazardous byforced ventilation.

80

8.4. Physical effects for use in transducers.

8 4 1 Introduction


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8.4.1. Introduction.

A viewpoint is that transducers operate in a three-dimensional matrix of the following types of energy: Chemical, Optical, Mechanical,Electrical, Thermal, Magnetic, Acoustic and Nuclear. It remains to find physical effects which can be used to construct transducers.Possible transducers are of three types:Self-generators such as the photovoltaic effect such as in photocells, piezo-electric effect such as in pressure transducers,accelerometers and ignition systems, electromagnetic induction such as in tachogenerators, etc. These generate a usable output.Modulators: Piezoresistive effect such as in strain-gauges, thermoresistive such as in resistance temperature transducers.

Modifiers: Radiation change such as in optical encoders, displacement due to pressure, etc.

8.4.2. Conductivity measurement.

Plant which uses water such as boilers or steam generators require that water quality be tightly monitored. Laboratory analyses of watersamples are accurate, but tedious. Instrumentation is required to monitor feed water quality between lab samples. By measuring the

electrical conductivity of water, the quality of the water can be inferred. Further applications: The condensate from condensing steamturbines can be re-used as feedwater, provided the condensate has not been contaminated by cooling tower water leaking in through thecondenser tubes. This can be detected by conductivity. Continuously monitoring the electrical conductivity in demineralization plant, itcan be determined when to regenerate the resin so that water with too high concentration of salts is not used as boiler feedwater. In thiscase resistance is modulated by ions in water. The electrical conductivity of water depends on the concentration of ions as well as themobility of ions present in the water.

The water to be analyzed is placed in a cell. The resistance of the liquid in the cell is given by:

GAllR 1===

whereG = conductance in Siemens,

= specific resistivity, m = specific conductivity, Siemens/m, where Ohm-1 = Siemens,l= length, m and

81

A = cross-sectional area, m2.

When constructing a cell, it is better to define a cell constant which can be calibrated with a standard solution, typically potassium

chloride at 25C.


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cCR = , Cc=cell constant m-1.

At low ion concentrations the specific conductivity is given by:

= iiC S/m whereCi = concentration in kgmol/m

3,

i = mobility factor [Sm2/kgmol].

The specific conductivity of an electrolyte is temperature dependent:( )( )25125 += TT S/m, where

T= specific conductivity at TC,

25 = specific conductivity at 25C,

= 0.02 0,025 K-1

for salts and bases,

= 0.01 0,016 K

-1

for acids.When measuring the resistance, direct current (DC) cannot be used due to electrolysis. Instead, alternating current is used, but electrodeeffects are still present, creating a circuit of the type shown in the figure.

Figure 48. Electrode effects.

R1, R2 model electrolysis effects,R3 models resistance of the electrolyte,

82

C1, C2 model film capacitance in cell between ions and electrodes, of the order of 0,1 to 1,0 F/m2.

When measuring conductivity, meaningful results can be obtained by compensating for temperature. Furthermore, in some applicationsconductivity measurements are interpreted in relation to laboratory analyses.


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8.4.3. Resistive displacement transducers.

The potentiometer has some form of slider over resistive material and can convert large displacements to resistance changes. Thepotentiometer does not use any exotic physical effects, but the resistive material and slider have exotic designs in order to withstand say106 + operations without failure.

Figure 49. Potentiometer displacement transducer.

A potentiometer designed for a linear characteristic is defined by:

m

x

x

x

R

R=

R= total resistance in ohm,Rx= resistance from starting point to slider,

83

x= displacement, linear in m or angular,xm = maximum design displacement, linear in m or angular.

When loaded with the input resistance of an amplifier, the relationship changes and is left as an exercise.


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8.4.4. Hot resistance anemometer.

The device is mainly used for measuring velocity fluctuations in gases. It also has many other applications, see later.

Figure 50. Hot-wire anemometer.At thermal equilibrium:

Electrical power input = Thermal power removed by convection.

( )fww TThARI =2 Watt,I= current, amp,Rw= resistance, ohm,

Tw= temperature of resistance, K orC,

84

Tf= temperature of fluid, K orC,A = heat transfer area, m2,

h = film coefficient of heat transfer, W/(m2K), by Kings law VCCh 10 += , V= velocity in m/s.

(Note: Do not confuse Vin Volt en Vin m/s).


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The resistance Ris temperature dependent.This device can be operated in two configurations:Constant current which could rapidly lead to destructive overheating and has a slow response time.Constant resistance, which is effectively constant temperature, by using a feedback amplifier. It cannot destructively overheat and has afast response time, 100-300x faster than the first method provided the feedback can handle the required current swings.

Other applications:

The heat transfer coefficient h depends on the gas provided the velocity is kept constant. This finds application in gas analysisinstrumentation such as gas chromatographs. In chromatography the gas components in a mixture travel at different rates through acolumn packed with silica and alumina, amongst others, with the aid of a carrier gas.

A variant is found in paramagnetic oxygen analyzers which is used in combustion instrumentation. Paramagnetic materials are weaklyattracted by a magnet. Oxygen, which is paramagnetic, is attracted into magnetic field, but the oxygen heated by the wire is then less

attracted. This creates a (feeble!) magnetic wind. The flow rate depends on the oxygen concentration.

8.4.5. Vortex-shedding flowmeters.

These flowmeters can be used for a wide variety of liquids, vapours and gases, but is not suitable for fluids with particles. The theory issomewhat complicated, but will be presented in a simplified manner. The physical effect of vortex shedding can be observed as aneveryday occurrence. A flag waving in the wind is caused by vortices peeling off the pole. A telephone lines makes a tone in the wind.Wind blowing around the corners of buildings making a howling noise.

A fluid (liquid or gas) which flows past an obstruction, causes a low velocity near surfaces. In obstructions that are not streamlined, thelow pressure behind the obstruction cause the boundary layer to wrap and cause vortices. The vortices are shed alternately from thesides of the blunt obstruction. By choosing the correct shape, the vortex shedding frequency is proportional to flow velocity over a largerange.

85

In order to derive a model for the vortex flowmeter, a few terms from fluid mechanics are revisited:

The Reynolds number

vdRe = , dimensionless,


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v= flow velocity, m/s,d= characteristic dimension, m, details of which can be found in any treatise on fluid mechanics,

= density of the fluid, kg/m3,

= dynamic viscosity, Pas.

The vortex-shedding frequency is stable forRe > 104

and is a function of the size of the obstruction. The Reynolds number is used forscaling the phenomena.

The Strouhal numberv

fdS= , dimensionless,

f= vortex-shedding frequency, Hz or s-1

,d= characteristic dimension, m,v= upstream velocity, m/s.

Figure 51. Vortex street downstream of obstruction.

In the case of flowmeters, the blunt obstruction is designed for maximum vortex strength. Note that the vortex street, named after VonKarmann, can have large effects some distance from the origin. The alternating peeling off of vortices can generate quite large forcesand have caused failure of towers, smoke stacks, tubes in heat exchangers, etc.

The Birkhoff model is derived here:

86


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Figure 52. Birkhoff oscillator.

The peeling off of vortices is viewed as an oscillating fish tail. In the Birkhoff model, the oscillator differential equation is given by:

02

2

=+ Kdt

dJ

= angle, radians,2

02

2

+= dd

dlJ kgm2, the moment of inertia as proposed by Birkhoff,

ddd

vddd

vK

+=

+= 02

0

2

2

24

22

1 N/m, the spring constant.

From the oscillator equation, the frequency of oscillation is given by:

J

K=0 rad/s,

+

==

0

0

0

2

42

dd

l

vf

Hz or s-1.

The vortex-shedding frequency is proportional to velocity.

87

8.5. Exercises.

8.5.1. In the section on intrinsic safety it as stated that a certain minimum energy is required to ignite a combustible mixture. Do a searchto find numerical values for some mixtures.


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8.5.2. The Strouhal number is given as 0.35 over the operating range of a vortex-shedding flowmeter in a pipeline. It is used in a gaspipeline with 150 mm (inner diameter). Gas velocity is 50 m/s. Find the vortex shedding frequency.

88

References

1. [http://uhavax.hartford.edu/~biomed/gateway/StaticCharacteristics.html]


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2. Joseph McGhee, Ian A. Henderson, Peter H. Sydenham. Sensor science essentials for instrumentation and measurementtechnology. Measurement 25 (1999) 89113.

3. Joseph McGhee, Ian A. Henderson, M. Jerzy Korczynski, Wlodzimierz Kulesza. The sensor effect tetrahedron: an extendedtransducer space. Measurement 24 (1998) 217236.

4. Bla G. Liptak, Kriszta Venczel. Process measurement instrument engineers handbook. 1982. Chilton book company.

5. Usher, M. J. Sensors and transducers. McMillan 1985.

89

Appendix A. Terms.

Gauge pressureIs the amount of pressure relative to the prevailing atmospheric pressure measured in Pascal (Pa)


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Is the amount of pressure relative to the prevailing atmospheric pressure, measured in Pascal (Pa).1 Pa = 1 Newton/m2 = 1 N/m2.

e.g. for example. It is an abbreviation for the Latin phrase exempli gratia.

Grouping of energy forms in transducers Grouping energy forms as Chemical, Optical, Mechanical, Electrical, Thermal, Magnetic,Acoustic and Nuclear, which is expressed by the acronym COMETMAN, has some benefits over other methods of classifying , as

pointed out by McGhee and others [24].

i.e. id estor, that is,

Instrument, measuring. A device for ascertaining the magnitude of a of a quantity or condition presented to it.Measuring device

Sensor. See transducer.

Transducers convert physical energy into more useful physical energy with the modern aim of producing an electrical output which canbe readily used in control systems, stored and displayed.

90

Appendix B. Basic electric circuit theory.

Resistance.


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Since we are dealing with resistive strain gauges, the theory of resistors will be briefly revisited. An electrical resistor is defined by Ohmslaw:

v(t) = i(t)RVolt, wherei(t) = current in Ampere, Amp or abbreviated A,

R= resistance in Ohm or abbreviated .

Figure B.1. Conventions for a resistor.

Capacitance.

An electrical capacitor is defined by:

( )dt

tdvCti

)(= A, where

v(t) = potential in Volt, V,i(t) = current in Ampere, Amp or abbreviated A,C= capacitance in Farad or abbreviated F.

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Figure B.2. Conventions for a capacitor.

When v(t) is a sinusoidal time function, v(t) = Vpsin(t) V, then i(t) = VpCcos(t) A in the steady state. By a long winded derivationmaking use of the Laplace transform, the so-calledjnot

measurementforcontrol v3_01

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