process instrumentation i module code: eipin1b study...

2/10

VUT Vaal University of Technology

PROCESS INSTRUMENTATION I

MODULE CODE: EIPIN1B

STUDY PROGRAM: UNIT 1

EIPINI Chapter 1: Introduction to Industrial Instrumentation Page 1-1

1. INTRODUCTION TO INDUSTRIAL INSTRUMENTATION

"……… when you can measure what you are speaking about, and express it in numbers, you know something about it;....." Lord Kelvin (1824-1907), Institute of Civil Engineers, London, 3rd May 1883

1.1 MEASUREMENT

Measurement is defined as the determination of the existence or magnitude of a variable for monitoring and controlling purposes.

1.2 UNITS AND STANDARDS

A measurement is done with an instrument in terms of standard units. The system of units, which is most widely used, is the SI (Systems International d'Unites). The seven so called base units of the system, are the following:

meter (length) kilogram (mass) second (time) Kelvin (temperature) ampere (current) candela (luminous intensity) Mole (amount of substance) Standards for these units are classified as follows:

International standards

International standards are defined by international agreement, representing units of measurements to the best possible accuracy allowed by measurement technology.

Primary standards

Primary standards are maintained at institutions in various countries. The main function is to check the accuracy of secondary standards.

Secondary standards

Secondary standards are employed in industry as reference for calibrating high-accuracy equipment and components. Calibration and comparison are done periodically by the involved industries against the primary standards maintained in the national standards labs. The main function of the secondary standards is to verify the accuracy of working standards.

Working standards

Workplace standards are used to calibrate instruments used in industrial applications and instruments used in the field, for accuracy and performance. Working standards are checked against secondary standards for accuracy.


1.3 FUNCTIONAL ELEMENTS OF INSTRUMENTS 1.3.1 Functions of instruments Instruments may be classified according to the functions they perform. Indicating function An instrument may provide the information about the value of a quantity under

measurement, in the form commonly known as an indicating function.

For example, the pointer and scale on a speedometer, indicates the speed of an automobile at that instant. Recording function An instrument may provide the information of the value of a quantity under

measurement against time or some other variable, in the form of a written record, usually on paper.

For example, an instrument may record the room temperature every second, as a graph on a strip chart. Controlling function This is one of the most important functions of an instrument, especially in the

field of industrial control processes. In this case, the information provided by the instrument is used by the control system to control the original measured quantity.

For example, the temperature measured in a room may be used to switch the cooling system on or off, in order to keep the room temperature within preset values. 1.3.2 Elements of instruments When examining different instruments, one soon recognizes a recurring pattern of similarity with regard to function. This leads to the concept of breaking down instruments into a number of elements according to the function of each element. Consider for example, the liquid filled pressure type thermometer in Figure 1-1.

A temperature change results in a pressure build-up within the bulb because of the constrained thermal expansion of the filling fluid. This pressure is transmitted through the tube to a Bourdon type pressure gauge, which converts pressure to displacement. This displacement is manipulated by the linkage and gearing to give a larger pointer movement. We can now recognise the following basic functional instrument elements, using this liquid filled thermometer as an example:

Bulb

Tube

Bourdontube

Scale and pointer

Link andgears Figure 1-1


Primary element: The primary element is that part of the instrument that first utilizes energy from

the measured medium and produces an output depending in some way on the measured quantity.

Note: For the liquid filled thermometer example, the bulb is the element in contact with the measured medium. The energy it extracts from the medium in this case is heat energy. The variable conversion from temperature to pressure is accomplished when the heat energy absorbed by the liquid in the bulb, causes an increase in pressure energy within the volume constrained liquid.

Data transmission element: The data transmission element transmits data from one element to another.

Note: When the elements of an instrument are physically separated, it becomes necessary to transmit the data from one to the other. It may be as simple as the tube in the liquid filled thermometer example, that transmits the pressure information from the bulb to the Bourdon tube, or as complicated as the telemetry system between a ship and the cruiser missile it has launched.

Secondary element: The secondary element converts the output of the primary element, to another

more suitable variable for the instrument to perform the desired function.

Note: In the thermometer example, the Bourdon tube is the secondary element (or the variable conversion element, as it is often called). It responds with a movement when receiving a pressure input. Every instrument need not include a second variable conversion element, while some require several.

Manipulation element: The manipulation element processes the information received from the primary or

secondary element and transforms the data into a more useful form.

Note: By manipulation we mean specifically a change in the numerical value of the variable according to some definite rule, while preserving the original character of the variable. In the thermometer example, a small movement of the Bourdon tube is amplified by the gears to produce a large circular movement of the pointer. A variable manipulation element does not necessarily follow a variable conversion element; it may precede it, appear elsewhere in the chain, or not appear at all.

Functioning element: The functioning element is that part of the instrument that is used for indicating,

recording or controlling of the measured quantity.

Note: The presentation of measured information may assume many different forms. It could include the simple indication of a pointer moving over a scale or the recording of a pen moving over a chart. It may also be in the form of a digital readout or even in a form not directly detectable by human senses as in the case of a digital computer used to perform a control function according to the value of the measured variable.


In summary then, the interconnection between the various functional elements for this particular thermometer instrument, is shown in Figure 1-2. It must be stressed though, that different instruments are not necessarily composed of all these elements or may not adhere to the same order of interconnection, as depicted in Figure 1-2.

Figure 1-2

1.4 RANGE AND SPAN OF AN INSTRUMENT

Range:

The range of an instrument is the minimum and maximum values of the measured variable that the instrument is capable of measuring.

Span:

The span of an instrument is the arithmetic difference between the minimum and maximum range values, used to describe both the input and the output.

Example: A thermometer can measure temperature between -20 ºC and 90 ºC. The range of the instrument is from -20 ºC to 90 ºC. The span of the instrument is 90 – (-20) = 110 ºC.

Bulb Primary element (Variable conversion element: temperature to pressure)

Temperature Measured medium

Tube Data transmission element (Pressure to pressure)

Bourdon tube Secondary element (Variable conversion element: pressure to motion)

Linkage and gear Variable manipulation element (Motion to motion)

Scale and pointer Functioning element (Data presentation: indicating function performed by moving pointer over scale)

Observer


1.5 STATIC CHARACTERISTICS OF INSTRUMENTS Information about the static performance or static characteristics of an instrument, is obtained by a process called static calibration. Static calibration refers to a situation where the input is varied over some range of constant values, causing the output to vary over some range of constant values. Each reading is taken when the output has settled to a steady value. The input-output relations developed in this way comprise what is known as a static calibration. The characteristics for an instrument with ideal static calibration, is shown in Figure 1-3. Because of instrument errors, the actual static calibration of an instrument will deviate from the expected or ideal input-output relationship Figure 1-3 (Ideal static calibration curve)

As different instruments measure different variables, the input and output values may sometimes be conveniently expressed in percentage values. Of course the ideal input-output characteristic does not necessarily have to be a straight line but most instruments are designed to produce a linear input-output relationship.

The following concepts; error of measurement, accuracy, precision, repeatability, reproducibility, resolution and sensitivity, are associated with the static characteristics of an instrument, and will subsequently be defined.

Error of measurement

The error of measurement is the difference between the measured value and the true value.

Note: The value of the measurement error can only be evaluated when the instrument is used to measure a standard value as the true value.

Accuracy

The accuracy of a measurement is the closeness with which the reading approaches the true value of a variable.

Output y (%)

yMAX (100 %)

yMIN (0 %)

xMIN (0 %)

xMAX (100 %)

Input x (%) Input span

Output span


Precision

Precision is the closeness with which repeated measurements of the same quantity agree with each other.

Students often confuse the terms precision and accuracy but a precise instrument may not be accurate. Precision simply means that if the measuring device is subjected to the same input for several times and the indicated results are tightly grouped together around some mean value (though not necessarily the true value), then the instrument is said to be of high precision. See Figure 1-4 for an interpretation of accuracy and precision. Figure 1-4

Two concepts related to precision, are repeatability and reproducibility. Repeatability is basically a measure of the instrument precision when the same operator in the same laboratory or the same environment, measures a constant input repeatedly, over a short time. Reproducibility is a measure of the instrument precision when a constant input is measured repeatedly, but these experiments are performed in different laboratories or locations with different ambient conditions and over a longer time span.

Repeatability

Repeatability is the closeness of the instrument readings when the same input is applied repeatedly under the same conditions over a short period of time.

Reproducibility

Reproducibility is the closeness of the instrument readings when the same input is applied under different conditions over a long period of time.

Resolution

Resolution is the smallest variation in the measured variable that can still be measured.

For example, the resolution of an ordinary digital wristwatch is normally 1 second as it can measure the flow of time to a maximum “fineness” of 1 second.

Accurate and precise

Inaccurate but precise

Accurate on average but imprecise

Inaccurate and imprecise


Sensitivity

Sensitivity is the rate of change of the output of a system with respect to input changes.

For a linear calibration curve, the sensitivity or gain K of an instrument is constant but will vary for a non-linear curve. The sensitivity at any particular input x, may be expressed as the slope of the line tangent to the calibration curve at that point.

K = xy

ΔΔ . Equation 1-1

Example 1-1: What is the sensitivity of a linear instrument that records the following values? 0 ºC = 12.3 V and 45 ºC = 24.3 V Answer: From Equation 1-1:

K = xy

ΔΔ = 045

3.123.24−− = 45

12

= 0.2667 volt per ºC

1.6 INSTRUMENT ERRORS 1.6.1 Classification of errors No measuring instrument is entirely free from errors. We can broadly classify instrument errors into three main groups; gross errors, systematic (bias) errors and random (precision) errors. Gross errors:

Gross errors are mistakes made, for instance, by the operator in gross misreading of a scale. These errors can be minimized by care and self-discipline.

Systematic errors:

Systematic errors affect all readings in such a way that the error of measurement has a fixed sign throughout the whole range of the instrument. These errors are usually caused by an error in the instrument, poor calibration, improper technique of the operator or loading of the instrument. Normally systematic errors are corrected by careful recalibration of the instrument.

Random errors:

Random errors occur because of unknown and unpredictable variations that exist in all measurement situations. This results in slightly different values obtained for each repeated measurement (scattered evenly about the mean value) of the same input. The influence of random errors on the integrity of measurements can be reduced with statistical methods and refined experimental techniques.


1.6.2 Typical instrument errors Some errors that may be encountered while using an instrument, are errors because of non-linearity, drift, hysteresis and dead band. Non linearity

Non-linearity is the maximum deviation from a straight line connecting the zero and full-scale calibration points.

Note: A straight line connecting the minimum and maximum input-output operating points, would represent perfect linear operation of the instrument. The actual static calibration of the instrument will normally deviate from this line. Non- linearity can be expressed in a variety of ways but a widely used method is to determine the maximum deviation of the output from this line, as shown in Figure 1-5. Non-linearity is then expressed as a percentage of the maximum output value.

Drift

Drift is the change in instrument indication over time while the input and ambient conditions are constant.

Note: A typical error because of drift is a change in sensitivity. This will cause an error across the whole range of the instrument as indicated in Figure 1-6. An error because of drift is an example of a systematic error. As was mentioned before, systematic errors may normally be corrected with routine calibration of the instrument.

yMIN

yMAX

xMIN xMAX

Output y

Input x

Desired linear input- output relationship

Actual static calibration

Maximum non-linearity

Figure 1-5

yMIN

yMAX

xMIN xMAX

Output y

Input x

Range error

Figure 1-6


Hysteresis

Hysteresis is the difference between the readings obtained when a given value of the measured variable is approached from below and when the same value is approached from above.

Note: It is possible to find that when performing a static calibration for an instrument starting from the minimum input value to the maximum input value (also called the upscale direction), that the calibration curve obtained in this way, may differ from the static calibration obtained when the input variable is allowed to vary from the maximum value down to the minimum value (also called the downscale direction). This phenomenon, illustrated in Figure 1-7, is called hysteresis. This is usually caused by friction or backlash in the gearing of the instrument.

Dead band

Dead band is the largest change of input to which the instrument does not respond due to friction or backlash effects

Note: Dead band error is normally associated with hysteresis. Dead band operation is sometimes intentionally built into the instrument for instance in a room temperature regulator to prevent excessive on-off switching. As an example of dead band behaviour in an instrument, Figure 1-8 illustrates instrument insensitivity near zero input, typically because of friction.

yMIN

yMAX

xMIN xMAX

Output y

Input x

Figure 1-7

x0

Downscale static calibration

Upscale static calibration

Hysteresis error for input x0

yMIN

yMAX

xMIN xMAX

Output y

Input x

Insensitivitynear zero

input

Figure 1-8


Example 1-3

A displacement sensor has an input range of 0.0 to 3.0 cm. Using the calibration results given in the table, calculate:

a) the input and output span. b) the maximum non-linearity as a percentage of f.s.d. (full scale deflection). c) the sensitivity of the instrument at an input of 1.0 cm.

Displacement (cm) 0 .5 1.0 1.5 2.0 2.5 3.0

Output Voltage (mV) 0 16.5 32.0 44.0 51.5 55.5 58.0

a) Input span = 3-0 = 3 cm and output span = 58-0 = 58 mV

b) The maximum deviation from the straight line connecting the range values

appears to occur when the displacement is 1.7 cm. The non-linearity at this point is approximately 48 – 33 = 15 mV. Non-linearity expressed as percentage of full scale is (15/58)×100 ≈ 26 %.

c) Sensitivity at x = 1 cm, is equal to the slope of the line tangent to the curve

at x = 1 cm. ∴K = xy

ΔΔ = 09.1

460−− = 29.5 mV/cm

0 0.5 1.0 1.5 2.0 2.5 3.0

y - output voltage in mV

x – displacement in cm 0

10

20

30

40

50

60

Maximum non- linearity ≈ 47-33 = 15 mV


1.7 INDUSTRIAL INSTRUMENTATION STANDARDS AND SCHEMATICS

1.7.1 Instrument identification lettering

Letter First letter Second / third letter A Analysis Alarm B Burner or combustion User’s choice* C User’s choice Control D User’s choice E Voltage Primary element F Flow rate G User's choice Glass (sight tube) H Hand (manually initiated) High I Current Indicate J Power K Time schedule Control station L Level Light / Low M Moisture or humidity Middle N User’s choice User’s choice O User’s choice Orifice, restriction P Pressure or vacuum Point (test connection) Q Quantity R Radiation Record or print S Speed or frequency Switch T Temperature Transmit U Multivariable Multifunction V Vibration or viscosity Valve, damper or louver W Weight or force Well X Unclassified Unclassified Y Event, state, or presence Relay or compute Z Position, dimension Driver, actuator, final control

Table 1-1


* The user’s choice entry in the table may be used to denote a particular meaning, and the user must describe the particular meaning(s) in the legend accompanying his drawing. The letter Y in the second position has an extended meaning of variable manipulation, and some of this instrument functions are given in table 1-2.

Symbol Function Σ Δ × ÷ Add, subtract, multiply and divide Xn ± Raise to power, square root, bias K -K Proportional reverse proportional > < > < High select, low select, high limit, low limit ∫ D or d/dt Integral, derivative

X/Y Convert X to Y with X and Y selected from: P=Pressure, E=Voltage, I=Current, H=Hydraulic O=Electromagnetic or sonic, A=Analog, D=Digital

Table 1-2

1.7.2 Instrument signals and connections

1.7.3 Standard methods to transmit pneumatic and electrical signals The standard industrial range for pneumatic signals is 20 to 100 kPa above atmospheric pressure, which corresponds to a 0% to 100% process condition. Note that the transmitter output signal starts at 20 kPA and not 0 kPa. This 20 kPa output is called a live zero. A live zero allows control room staff to distinguish between a valid process condition of 0% (a 20 kPa reading) and a disabled transmitter or interrupted pressure line (a 0 kPa reading) – providing a rough rationality check.

The accepted industrial electronic standard is a 4 mA to 20 mA current signal or a 1 V to 5 V voltage signal to represent a 0 % to 100 % process condition. Again, a live zero is used to distinguish between 0% process variable (4 mA or 1 V) and an interrupted or faulted signal loop (0 mA or 0 V).

Instrument supply or connection to process

Pneumatic signal

Electrical signal

Hydraulic signal

Electromagnetic, sonicor radioactive signal

Primary process flow


Example 1-4 A 20 - 100 kPa output pneumatic transmitter is used to monitor the water level inside a tank. The calibrated range is 100 to 200 cm. of water above the base of the tank. Calculate the output of the transmitter when the water level is at 175 cm. above the base of the tank.

Span (difference between the upper and lower limit) of the transmitter output = 100 kPa - 20 kPa = 80 kPa Fraction of measurement = (175 – 100)/(200 – 100) = 0.75 Output Signal = (Fraction of Measurement) × (Signal Span) + Live Zero = 0.75×80 + 20 = 80 kPa

Example 1-5 An electronic transmitter with an output of 4 - 20 mA is calibrated for a pressure range of 70 - 150 kPa. What pressure is represented by a 12 mA signal?

Span of transmitter = 20 mA - 4 mA = 16 mA Fraction of Measurement Change = (Output Signal - Live Zero)/Signal Span = (12 – 4)/16 = 0.5 Actual Process Change = (Fractional Change) × (Process Span) = 0.5×(150 - 70 kPa) = 40 kPa Actual Process Value = Base Point + Process Change = 70 + 40 kPa = 110 kPa.

Note: One advantage of a pneumatic system is that sparks will not be produced if a transmitter malfunction occurs, making it much safer when used in an explosive environment. The biggest problem with pneumatic systems is that air is compressible. This means that a pressure transient representing a process change will only travel in the air line at sonic velocity (approximately 300 m/sec.). Long signal lines will cause substantial time delays, which is a serious drawback. Electronic signals on the other hand, travel at speeds which approach the speed of light and can therefore be transmitted over long distances without the introduction of unnecessary time delays.

1.7.4 Power supply abbreviations AS Air supply ES Electric supply GS Gas supply WS Water supply SS Steam supply HS Hydraulic supply NS Nitrogen supply

100%

Process

75%

50%

25%

0%

Pneumatic transmitter

100kPa

80kPa

60kPa

40kPa

20kPa

100%

Process

75%

50%

25%

0%

Electronic transmitter

20mA (5V)

16mA (4V)

12mA (3V)

8mA (2V)

4mA (1V)

output output


1.7.5 Instrument symbols

Orifice plate flowmeter

Venturi flowmeter

M Electric motor

Instrument mounted locally (field mounted)

Instrument mountedon board (panel mounted in control room)

Instrument mounted behind board (mounted behind panel in control room, not accessible to operator)

Instruments sharing common housing (measures two variables or single variable with two functions)

Valve with diaphragm actuator Valve

Butterfly valve Valve with hand actuator

Rotameter flowmeter


1.7.6 Schematics

The key to instrument identification, is given in Figure 1-9 Example 1-6

Identify the following instruments:

a) b) Exercise Identify the instrumentation blocks in the heat exchanger below

Steam supply

1 – Component function (table 1-1) 2 – Component sequence number 3 – Instrument function (table 1-2) 4 – Vendor designation 5 – Panel number 6 – Set point(s) 7 – Application notes

2

3 4 5 1

7

6

TRC Answer: Temperature (1’st letter) recording (2’nd letter) controller (3’rd letter also from second column table 1-1)mounted on board

FY

Answer: Flow compute instrument mounted behind board (or rack mounted). The instrument receives a pneumatic signal and converts this signal into a pneumatic output signal representing thesquare root of the input signal.

√

Figure 1-9

4b

4a

PIC 3

M

TRC

TSH

Product stream to be heated

4c TAH

FIT 2a

PIT 1b

FY 2b

PR 1a

FR 2c

√

EIPINI Chapter 2: Pressure Measurement Page 2-1

2. PRESSURE MEASUREMENT The purpose of this chapter is to introduce students to the definitions and units of pressure related quantities and to discuss typical methods to measure pressure.

2.1 PRESSURE CONCEPTS AND DEFINITIONS

2.1.1 Pressure Pressure is defined as the force exerted over a unit area. The SI unit is newton per

square meter (N/m2) or pascal (Pa).

AFP = Equation 2-1

2.1.2 Density Density of a substance is defined as the mass of a unit volume of a substance. The

SI unit is kilogram per cubic meter (kg/m3).

VMρ = Equation 2-2

2.1.3 Relative density (Specific gravity) Relative density of a substance is defined as the ratio of the density of the

substance to the density of water.

δsubstance = waterρ

substanceρ

∴ρsubstance = 1000×δsubstance Equation 2-3

(Note: If the substance is a gas, the relative density is defined as the ratio of the density of the gas to the density of air at the same temperature, pressure and dryness.)

Weight 100 N

Area 0.1 m2

Area 0.01 m2 P = 1000 Pa P = 10000 Pa

Same force, different area different pressure

ρwater = 1000 kg/m3 ρmercury = 13600 kg/m3 ρtransformer oil = 864 kg/m3 ρair = 1.2 kg/m3


2.1.4 Absolute zero of pressure The absolute zero of pressure (or perfect vacuum), is the pressure that would

exist in a chamber, if all molecules were removed from the chamber, so that no pressure forces could be exerted on the chamber walls.

2.1.5 Absolute pressure Absolute pressure (or total pressure), is the pressure measured from absolute zero

pressure. 2.1.6 Atmospheric pressure Atmospheric pressure is the absolute pressure caused by the weight of the earth’s

atmosphere.

(Notes: Atmospheric pressure is often called barometric pressure. Local atmospheric pressure depends on the height above sea level. Standard atmospheric pressure at sea level is 101.326 kPa. or 760 mm. mercury. 2.1.7 Gauge pressure Gauge pressure, is the difference between the absolute pressure in a medium and

local atmospheric pressure, when the pressure in the medium is higher than atmospheric pressure.

Pgauge = Pabs – Patm

2.1.8 Vacuum pressure Vacuum pressure, is the difference between local atmospheric pressure and the

absolute pressure in a medium, when the pressure in the medium is lower than atmospheric pressure.

Pvacuum = Patm – Pabs

2.1.9 Differential pressure Differential pressure is the difference between two pressures. Summary: Absolute zero pressure (0 Pa)

Atmosphericpressure

Absolute pressure

Pabs

Patm Pvacuum

Pgauge

A comparison of absolute pressure, atmospheric pressure, gauge pressure, vacuum pressure and differential pressure.

Differentialpressure


2.2 PRESSURE IN A LIQUID The cylinder in Figure 2-1, with a cross sectional area A meter2, is filled with a liquid of density ρ kilogram/meter3, to a height of h meter. The weight of the liquid will exert a pressure P pascal on the bottom of the container. We will now obtain an expression for P.

Volume of the liquid = V = cross-sectional area×height = Ah. ∴Mass of the liquid = m = volume×density = V×ρ = (Ah)×ρ. ∴Weight of the liquid = w = mg = (Ahρ)×g. ∴Pressure on the bottom of container due to weight of the liquid = w÷A = Ahρg/A = ρhg

We conclude therefore that the pressure (pascal) caused by a liquid column h meter

high and with density ρ kilogram/meter3, is given by:

P = ρhg Equation 2-4 where g is the gravitational acceleration. We will always use g = 9.81 m/s2 in pressure calculations.

Note: If the absolute atmospheric pressure, exerted on the surface of the liquid, is P0 pascal, the total pressure acting on the bottom of the container is Ptotal = P0 + ρhg Example 2-1 a) Convert a pressure of 150 cm. water, to a pressure expressed in pascal. P = ρhg = 1000×(150×10-2)×9.81 = 14715 Pa. b) Convert a pressure of 10 kilo pascal to a pressure expressed as meter water. P = ρhg ⇒ 10000 = 1000×h×9.81 ⇒ h = 1.019 meter Therefore 10 kPa = 1.019 meter H2O. c) Convert a pressure of 760 millimeter mercury to a pressure expressed in pascal. P = ρhg = 13600×(760×10-3)×9.81 = 101400 Pa = 101.4 kPa. d) Convert a pressure of 50 kPa to a pressure expressed as millimeter mercury. P = ρhg ⇒ 50000 = 13600×h×9.81 ⇒ h = 0.3748 meter Therefore 50 kPa = 374.8 mm Hg.

A

h P

Figure 2-1


2.3 PRESSURE MEASUREMENT WITH MANOMETERS

2.3.1 The U tube manometer A simple U tube manometer is formed, when a glass tube, in the form of a U, is half filled with a liquid (for example mercury), as shown in Figure 2-2. If the pressures in both legs of the manometer are the same, for instance, if both legs are open to atmospheric pressure, the manometer liquid level will lie in the same horizontal plane. This is called the zero level or zero line. Example 2-2

A U tube manometer is half filled with mercury. A pressure of 200 kPa is applied to the left hand leg and a pressure of 100 kPa is applied to the right hand leg. Calculate the reading h on the manometer.

It is important to remember that mercury cannot be compressed by typical pressures. Therefore, if a pressure differential causes a movement of the mercury away from the zero level, the downward movement in the one leg, will be equal to the upward movement in the other leg. Secondly, the density of air is very small in comparison with the density of the manometer liquid. The pressure contribution of the air in the tubes may therefore be neglected. Thirdly, when we compare the pressures in the two legs of the manometer, we need to remember the important theorem from hydrostatics that states:

The pressure at two points, in the same horizontal plane, in a liquid at rest, is the same, if a curve can be drawn from the one point to the other point, without leaving the liquid.

It is now clear that we can equate the pressures in the XY plane, as this plane cuts the mercury in the same horizontal plane, and the points of intersection, may be joined via the mercury.

200×103 = 100×103 + 13600×h×9.81 ∴h = 0.7495 m

If the reading were taken from the zero line upwards, it would be 0.7495/2=0.3748 m

Zero level

Patm Patm

Figure 2-2

Zero level

100 kPa200 kPa

h

X Y


2.3.2 Using a U tube manometer to measure differential pressure, gauge pressure and absolute pressure Differential pressure: To measure the difference between two unknown pressures, the one pressure is applied to one leg and the other to the second leg, as shown in Figure 2-3 (a). The reading h is directly proportional to the pressure difference P1-P2.

Comparing pressures in the XY plane: P1 = P2 + ρhg

∴P1-P2 = ρhg Gauge pressure. The arrangement to measure gauge pressure, is shown in Figure 2-3 (b). A pressure P1, larger than atmospheric pressure, is applied to one leg, and atmospheric pressure to the other. The reading h will be indicative of the pressure difference P1 – Patm or the gauge pressure.

Comparing pressures in the XY plane: P1 = Patm + ρhg ∴P1 – Patm = ρhg

∴Pgauge = ρhg Absolute pressure. In order to measure absolute pressure, it is necessary to compare the unknown pressure with zero pascal, as shown in Figure 2-3 (c). For that purpose, all the air must be removed from one leg, to form a perfect vacuum. That leg is then sealed. The two mercury levels will take on their zero line position, only if zero pascal is applied to the open leg.

Equating pressures in the XY plane: Pabs = 0 + ρhg

∴Pabs = ρhg

P1

h

P2

Zero level

Figure 2-3 (a)

X Y

P1

h

Patm

Zero level

Figure 2-3 (b)

X Y

Vacuum (0 Pa)

Pabs

hZero level

Figure 2-3 (c)

X Y


Example 2-3 A u-tube manometer is filled with two liquids, one liquid with a relative density of 1 and the other with a relative density of 13.6. Calculate the pressure difference, P1 – P2 , applied across the manometer.

Comparing pressures on the XY line: P1+1000×1×9.81=P2+1000×0.5×9.81+13600×0.5×9.81 ∴P1 + 9810 = P2 + 4905 + 66708 ∴P1 – P2 = 61.803×103 = 61.80 kPa.

Example 2-4 You are requested to design a scale plate for a U-tube manometer that uses zeal oil, with relative density of 0.88, as manometer liquid. You are told that the maximum differential pressure to be measured, will be 10 kPa. From the zero line upward, the following values must be marked off on the scale plate: 2.5 kPa, 5 kPa, 7.5 kPa and 10 kPa. Calculate the distances between the markings, and sketch the designed plate.

P1 – P2 = ρhg, so for P1-P2=10 kPa: 10×103=880×h×9.81 ⇒ h = 1158 mm. ∴Distance from zero line to 10 kPa marking=579 mm. Intervals=144.75 mm.

Example 2-5 The distance from the zero level to the top of a mercury manometer is 1 meter, when both tubes are open to an atmospheric pressure of 100 kPa. The right hand tube is now sealed off and a pressure of 200 kPa is applied to the left hand tube. Calculate the manometer reading h.

When 200 kPa is applied to the left hand tube, the pressure in the sealed tube, will rise to a new, higher than 100 kPa, pressure which we will call Px. If the cross sectional area of the

tube is A, we may use Boyle’s law to obtain an expression for Px. The volume of the air in the right hand tube is 1×A when open to 100 kPa and sealed, with 200 kPa applied to the left hand tube, it is (1-h/2)×A. Using Boyle’s law, P1V1=P2V2: 100×103×[1×A] = Px×[(1-h/2)×A] ∴Px = 100×103/[1-h/2] …………. (1) Comparing pressures on the XY line: 200×103 = Px + 13600×h×9.81 ……. (2) (1) in (2): 200×103 = 100×103/[1-h/2] + 13.6×103×h×9.81 ∴200 = 100/[1-h/2] + 13.6×9.81×h ⇒ 200 = 100/[1-h/2] + 133.4h ∴200×(1-h/2) = 100 + 133.4h×(1-h/2) ⇒ 200-100h=100+133.4h – 66.71h2 ∴66.71h2 –233.4h + 100 = 0 {ax2+bx+c=0 ⇒ x=[-b±√(b2-4ac)]/2a} ∴h = [233.4±√(233.42-4×66.71×100)]/2×66.71 = 0.5 m or h=3m (unacceptable)

Y X

1 m

P1 P2

0.5 m

δ=13.6

Zero line 0 kPa

2.5 kPa

5 kPa

7.5 kPa

10 kPa

289.5mm

434.25mm

579mm

144.75mm

h/2

1-(h/2)

X

200 kPa100 kPa

1 m

Px

h Y

δ=1


2.3.3 The well type manometer (cistern type manometer) The well type manometer, is essentially a manometer with one of the limbs (the well or reservoir) having a large cross sectional area of A1, and the second limb, a glass tube, with much smaller cross sectional area, of A2. Figure 2-4 (a) Figure 2-4 (b)

When the two limbs are open, as shown in Figure 2-4 (a), the manometer liquid meniscuses will fall on the zero line. If a pressure differential, P1 - P2 (P1 > P2), is applied to the instrument, in Figure 2-4 (b), the rise and fall of the manometer liquid in the two limbs will be different (h > d). The level h, in the glass tube, to which the manometer liquid rises above the zero line, can be measured, while the fall in the liquid level d, in the well, can not be observed, and as such, will be eliminated from our equations below.

Comparing the pressures in the two limbs, on level XY, in figure 2-4 (b): P1 = P2 + ρ(h+d)g …………………………………………… (1) Also, the volume of manometer liquid, leaving the well, is equal to the volume of manometer liquid, entering the tube:

A1d = A2h

∴d = 1A2A

h ………………………………………………….. (2)

(2) in (1): P1 = P2 + ρ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+ hh

1A2A

g

∴P1 – P2 = ρhg⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

1A2A

1 Equation 2-5

P2

d

Zero level

h

Cross sectional Area of well = A1

Cross sectional Area of tube = A2

P1

Manometer liquid density = ρ

X Y

High

Low

A1 A2


2.3.4 The inclined limb manometer The inclined manometer is a variation of the well type in that the tube is not vertical, but leaning to one side. Referring to Figure 2-5, the movement L, of the manometer liquid along the tube, is amplified with respect to its vertical height h. This facilitates the detection of small changes in applied differential pressure. Figure 2-5

Deriving the relationship between the applied pressure differential P1-P2, and the manometer reading L, is very similar to that of the well type. The only difference is that the tube and horizontal does not form an angle of 90°, but an angle α.

Comparing the pressures on level XY, in figure 2-5: P1 = P2 + ρ(h+d)g …………………………………………… (1) Equating rise and fall of manometer liquid: A1d = A2L

∴d = 1A2A

L ………………………………………………….. (2)

And from figure 2-5: h = Lsinα ……………………………………………………. (3)

(2) and (3) in (1): P1 = P2 + ρ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+α LLsin

1A2A

g

∴P1 – P2 = ρLg⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+α

1A2A

sin Equation 2-6

L

Cross sectional Area of tube = A2

Zero level

Cross sectional Area of well = A1

X Y

h

d

α

P1

P2

High

Low


Example 2-6

An inclined limb manometer is used for the measurement of pressure. The inclined limb forms an angle of 30 degrees with the horizontal plane. The relative density of the manometer fluid is 0.8 . The internal diameter of the well is 3 cm and the internal diameter of the inclined limb is 12 mm. a) Calculate the maximum applied pressure (in pascal), for a maximum scale reading

(L) of 100 cm on the scale attached to the inclined limb. b) The range of the above inclined manometer must be extended so that the

maximum pressure that can be applied to the manometer is increased by 1000 pascal, by using a different manometer fluid, without changing the construction of the manometer. Calculate the relative density of the manometer fluid that is required.

a) A1= 4

21Dπ

=4

2)2103( −×π =706.9×10-6 and A2= 4

22Dπ

=4

2)31012( −×π =113.1×10-6

∴A2/A1 = 113.1×10-6/706.9×10-6 = 0.16 {or A2/A1=(D2/D1)2=(12/30)2 =0.16} From equation 2-6: P1 – P2 = ρLg(sinα+A2/A1) = 800×1×9.81×[sin30° + 0.16] = 7848×(0.5 + 0.16) = 7848×0.66 = 5180 Pa. b) (P1 – P2)new = 5180 + 1000 = 6180 Pa. ∴P1 – P2 = ρLg(sinα+A2/A1) ⇒ 6180 = ρ×1×9.81×0.66 ∴ρ = 6180/6.475 = 954.4 kg/m3 ⇒ δnew = 0.9544 Example 2-7

The reading h on a well type mercury manometer, is 73 cm when measuring a pressure of 100 kPa. a) Calculate the ratio of well diameter to the diameter of the tube. b) Determine the change in level that the well mercury experiences.

a) From equation 2-5:

P1 – P2 = ρhg⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

1A2A

1

∴100×103 = 13600×(73×10-2)×9.81×[1 + (A2/A1)] ∴1 + (A2/A1) = 100×103/[13600×(73×10-2)×9.81] = 1.027 ∴(A2/A1) = 0.027 ∴(D2/D1)2 = 0.027 ∴D2/D1 = 0.1643 ∴D1/D2 = 6.086 (ratio of well to tube diameter)

b) d = 1

2

AA h

∴d = 0.027×73×10-2 = 19.71 mm.


2.3.5 Liquids used in manometers

2.3.5.1 Transformer oil Relative density: 0.864 Applications: Useful when measuring small pressure differences. Suitable for pressure

measurement in ammonia gas installations. Advantages: Low density for measuring small pressure differences. Unaffected by

ammonia. Can be easily seen. Does not readily evaporate. Disadvantages: Tends to cling to inside of tubes. Density of transformer oil varies.

2.3.5.2 Aniline Relative density: 1.025 Applications: Suitable for pressure measurement in low pressure gas or air installations,

with the exception of ammonia and chlorine. Advantages: Low density for measuring small pressure differences. Evaporates slowly.

Does not mix with water. Can be easily seen. Disadvantages: Attacks paint. Poisonous, penetrates the skin and causes blood poisoning.

Aniline darkens on contact with air.

2.3.5.3 Dibutylphathalate Relative density: 1.047 Applications: Suitable for pressure measurement in ammonia gas installations. Advantages: Does not mix with water. Disadvantages:

2.3.5.4 Carbon Tetrachloride Relative density: 1.605 Applications: Useful when measuring higher pressure differences. Suitable for measuring

pressure in chlorine gas installations. Advantages: Not attacked by chlorine. Disadvantages: Not easily seen. Readily evaporates.

2.3.5.5 Tetrabromoethane Relative density: 2.964 Applications: Useful when measuring higher pressure differences. Suitable for pressure

measurement in ammonia gas installations. Advantages: Evaporates slowly. High density. Disadvantages:

2.3.5.6 Bromoform Relative density: 2.9 Applications: Useful where pressure measurement demands manometer liquid with density

between water and mercury. Advantages: Density that falls between water and mercury. Disadvantages: Density uncertain. Poisonous. Freezes easily. Subject to attack. Attacks rubber.

2.3.5.7 Mercury

Relative density: 13.6 Applications: Pressure measurements in compressed gas, and in water and steam

applications. Advantages: High density. Can be easily seen. Mercury does not: i) evaporate, ii) mix

with other liquids, iii) wet sides of tubes. Disadvantages: Expensive. Mobility and density are affected by contamination.


2.4 ELASTIC PRESSURE SENSORS

2.4.1 The C type bourdon tube gauge

Bourdon tube pressure gauges are usually used where relatively large static pressures are to be measured. A typical bourdon tube pressure gauge is shown in Figure 2-6. The Bourdon tube pressure gauge consists of a C-shaped tube with one end sealed. The sealed end is connected by a mechanical link to a pointer on the dial of the gauge. The other end of the tube is fixed and open to the pressure being measured. The inside of the Bourdon tube experiences the measured pressure, while the outside of the tube is exposed to atmospheric pressure. Therefore, the tube responds to changes in Pmeasured – Patm. Increasing this pressure will tend to straighten out the tube and move the pointer to a higher scale position.

Figure 2-6

Pivot point

Bourdon tube

Pinion gear

HairspringAdjustable link

Sector gear

Pointer and scale

Pressure connection Tube cross section

Range adjust

0 30

25

2015

10

5


2.4.2 Bellows pressure sensor

The bellows element is basically a flexible metallic cylinder with a ripple profile, which can expand when a pressure differential exists between the interior pressure of the bellows and the pressure surrounding the bellows. In Figure 2-7(a), a bellows pressure sensor is used to measure a differential pressure P1 – P2.

Figure 2-7 (a) Differential pressure

The bellows element may also be used to measure gauge pressure if P2 is equal to atmospheric pressure, as depicted in Figure 2-7 (b). Absolute pressure may be measured, see Figure 2-7 (c), if all air is removed from the bellows enclosure, so that the pressure in the bellows, acts against a vacuum (0 Pa).

Figure 2-7 (b) Gauge pressure Figure 2-7 (c) Absolute pressure

Pressure indication

P1

P2 Bellows element

Spring

Low pressure

High pressure

Moving end

P1

Atmospheric pressure

Pressure indication

P1

Vacuum (0 Pa)

Pressure indication


2.4.3 Diaphragm pressure sensors

Diaphragms are round flexible disks, formed from thin metallic sheets with concentric corrugations. Two diaphragms may be used together to form a diaphragm capsule. Figure 2-8 (a) shows the structure of a single diaphragm while Figure 2-8 (b) and Figure 2-8 (c), indicate the design of convex and nested diaphragm capsules, respectively.

Figure 2-8 (a) (single) Figure 2-8 (b) (Convex) Figure 2-8 (c) (Nested)

Figure 2-9 (a), shows a diaphragm used to measure a pressure difference, P1 - P2, while in Figure 2-8 (b), the same function is fulfilled with a diaphragm capsule. Figure 2-9 (a) (Diaphragm) Figure 2-9 (b) (Capsule)

Capsules are sometimes filled with silicone oil and a solid plate mounted in the center of the capsule to protect against over-pressure. Pressure is then applied to both side of the diaphragm (Figure 2-10) and it will deflect towards the lower pressure. Most pneumatic differential pressure trans-mitters (discussed in section 2.6) are built around the pressure capsule concept.

Pressure indication

P1

P2 Diaphragm

Pressure indication

P1

P2 Capsule

(High pressure) (Low pressure)P1

Silicone oil

Pressure indication

P2

Capsule

Force bar

Seal and pivot

Figure 2-10

Backup plate


2.5 FORCE-BALANCE GAUGE CALIBRATOR

This instrument is also known as the piston type gauge or the dead weight tester. Its main purpose is to calibrate other pressure gauges.

The deadweight tester consists of a pumping piston that screws into the oil filled reservoir, a primary piston that carries the dead weight, and the gauge under test (Figure 2-11). The primary piston (of cross sectional area A), is loaded with the amount of weight (W) that corresponds to the desired calibration pressure (P = W/A). When the screw is rotated, the pumping piston pressurizes the whole system by pressing more oil into the reservoir cylinder, until the dead weight lifts off its support. The gauge under test is also exposed to the oil pressure that at this stage is equal to the calibration pressure.

Figure 2-11 Example 2-8 A dead weight tester has a primary piston with a diameter of 1.5 cm. The mass of the platform and primary piston together, is 300 gram. Calculate the mass m, of the mass pieces, that must be placed on the platform to check a gauge at 150 kPa.

Pressure = pistonprimary of Area

pistonprimary andplatformofweight masspieces ofWeight +

∴150×103 =

⎥⎦

⎤⎢⎣

⎡ ×

⎥⎦⎤

⎢⎣⎡ ××+×

4

2)2-10(1.5π

9.81)3-10(300 9.81m

∴(150×103)×(176.7×10-6) = 9.81m + 2.943 ⇒ 9.81m = 26.51 – 2.943 ∴9.81m = 23.57 ⇒ m = 2.403 kg. The total mass of the mass pieces to be placed on the platform is therefore 2.403 kg.

Gauge under test Primary piston

Secondary (pumped)

piston Screw

Platform

Oil

Mass pieces


2.6 The pneumatic differential pressure transmitter (DP cell) The purpose of this instrument is to measure a differential pressure Phigh – Plow, and convert the measured value into a standard output pressure that varies between 20 kPa and 100 kPa. The measured value may then be transmitted as a pressure variable, to a station some distance away. A simplified schematic of a pneumatic pressure transmitter is given in Figure 2-12. Figure 2-12

The operation of the differential transmitter is governed by the flapper and nozzle feedback mechanism, which keeps the range bar, pivoted by the range wheel, in balance. The upper part of the range bar and force bar is connected by a flexible plate. When the input pressure differential, P1-P2, increases, the force bar will pivot in a clockwise direction, and that will in turn cause the range bar to pivot clockwise. The flapper will therefore move towards the nozzle and airflow from the supply, through the nozzle, will consequently be reduced (blocked by flapper). This will result in a lower pressure drop across the restriction in the supply line and thus a higher pressure will be presented to the feedback bellows via the pilot relay that serves as a pneumatic buffer amplifier between the nozzle and feedback bellows (for clarity, a direct connection via the pilot relay, is shown in Figure. 2-12).

Pilot relay

A

B

P0

Low pressure (P2)

Pivot and seal

Nozzle Flapper

High pressure (P1)

Liquid filleddiaphragm

capsule

Range bar

Zero adjustment(20 kPa)

Output pressure

Feedbackbellows

Pivot point (range wheel adjust)

Regulatedair supply

Restriction

Force bar

Cross flexure

L2

L1

Capsule flexure


The feedback bellows will now push the range bar in an anti-clockwise direction, thereby restoring balance of both the range bar and the force bar, but at a higher output pressure value of P0, indicative of an increased value of P1-P2.

Similarly, when P1-P2 decreases, the flapper will be pushed away from the nozzle, thereby increasing the airflow through the nozzle resulting in a higher pressure drop across the restriction and a lower pressure transmitted to the feedback bellows. Balance will thus be restored, but at a lower value of P0.

The zero adjustment represents a pressure of 20 kPa in opposition to P0, so that when the pressure differential, P1 – P2, is zero, the output must still be 20 kPa. To simplify the discussion, let us assume that the effective clockwise moment at point A is (P1 - P2)L1 while the anti-clockwise moment at point B is (P0 – 20)L2. Equating these moments around the range wheel:

(P1 – P2)L1 = (P0 - 20)L2

∴P0 = 2

L1

L(P1 – P2) + 20 kilopascal ….……..….(1)

The ratio L1/L2 is adjusted during calibration, by changing the position of the range wheel, to ensure that P0 equals 100 kPa when (P1-P2) reaches it’s maximum value. Setting this ratio equal to m, we can rewrite equation (1) as: P0 = m×(P1 – P2) + 20 kilopascal Equation 2-7

In Equation 2-7, the variables P0, P1 and P2, must be expressed in kilopascal. A graphical representation of Equation 2-7 is given in Figure 2-13 Example 2-9 A differential pressure transmitter is correctly calibrated for a process variable that varies from 0 kPa to 170 kPa. Determine the output of the DP transmitter when the process variable reaches 90 kPa.

Output P0 [kPa]

Input (P1-P2) [kPa]

0

20

(P1-P2)MAX

100

P0 = m×(P1 – P2) + 20 where m = 80/(P1-P2)MAX

80

(P1-P2)MAX

Figure 2-13


From Equation 2-7, the output of the transmitter is given by: P0 = m×(P1 – P2) + 20 When (P1-P2) = 170 kPa, the output is P0 = 100. ∴100 = m×170 + 20 ⇒ m = 0.4706 ∴P0 = 0.4706(P1 – P2) + 20 kilopascal If (P1-P2) = 90 kPa.: P0 = 0.4706×90 + 20 = 62.35 kPa. The Pilot Relay In Figure 2-12, the pressure developed by the nozzle, is enhanced by a pilot relay. Theoretically, without a pilot relay, as shown in Figure 2-14, the restriction, flapper, nozzle and feedback bellows mechanism, would still function properly and respond to the force applied to the flapper, with an output pressure related to the force. The practical problem however, is that an increase in output pressure, must be accompanied by an increase in air flow through the very narrow restriction, while a decrease in pressure, must be accompanied by air bleeding away through the small nozzle opening. The response of the output pressure to changes in flapper movement, will inevitably be slow.

The pilot relay alleviates this problem by allowing the nozzle pressure to operate a small diaphragm which in turn controls the output pressure of the pilot relay in such a way, that it will follow the nozzle pressure, but this time, the output pressure is derived directly from the more powerful air supply. In Figure 2-15, the arrangement of the flapper and nozzle, assisted by a pilot relay, is shown.

When the force F moves the flapper towards the nozzle, the airflow through the nozzle will be reduced, thereby causing a smaller pressure drop across the restriction, so that more of the supply pressure will arrive at the diaphragm

Output

Output

Flapper and nozzle

Pivot

Restriction

Feedback bellows

Air supply

Figure 2-14

Pivot

Flapper and nozzle Restriction

Feedback bellows

Figure 2-15

Air supply

Vent

Supply valve(ball)

Spring

Exhaust valve(cone)

Diaphragm

Valve stem

F

F


chamber of the pilot relay, pushing the valve stem to the left. Moving the valve stem to the left, will have a dual effect. Firstly the supply valve will allow more of the air supply to reach the output (increasing the output pressure), and secondly, the exhaust valve will close a bit more, making it more difficult for the newly established higher output pressure, to relax itself through the vent. Balance will again be restored by the higher pressure in the feedback bellows, that opposes the disturbing force.

Similarly, when the external force pulls the flapper away from the nozzle, the air flow through the nozzle will increase. The increased air flow will cause more of the available supply pressure to fall across the restriction, making less pressure available on the nozzle side of the restriction. The diaphragm will slacken, as it is now exposed to a lower pressure and the valve stem will move to the right. The supply valve will begin to close, thereby restricting the flow of air from the supply to the output (thereby decreasing the output pressure) and at the same time, the exhaust valve will open more, thus providing a wider escape route for the original high output pressure, facilitating in this way with the rapid change in output pressure from a higher value to a lower value. As always, the feedback bellows, now receiving a lower pressure, will oppose the external force and bring the flapper back into balance.

The flapper/nozzle, pilot relay arrangement, is an important pneumatic mechanism and is also used in other instruments, such as the pneumatic control valve discussed in Chapter 6, in addition to the differential pressure transmitter.

2.7 Strain gauges Many pressure instruments such as an electronic differential pressure transmitter, may need to develop a standard electrical signal of 4 to 20 mA or 1 to 5 V, instead of the standard 20 to 100 kPa pressure signal. The strain gauge is one of the devices used to convert a pressure or force into an electrical signal. The majority of strain gauges are foil types, shown in Figure 2-16. They consist of a pattern of resistive foil which is mounted on a backing material and operate on the principle that as the foil is subjected to stress, the resistance of the foil changes in a defined way.

Figure 2-16

Backing material

Alignment marks

Solder tabs Grid


For a metal wire, the electrical resistance is given by:

R = aρl , where R is the resistance of the wire (Ω), ρ is the metal’s resistivity

(Ω-m), ℓ the length of the wire (m) and a the cross sectional area of the wire (m2).

The resistance will increase with increasing length of the wire or as the cross sectional area decreases. When force is applied, as indicated in Figure 2-17, the overall length of the wire tends to increase while the cross-sectional area decreases. This increase in resistance is proportional to the force that produced the change in length and area. The gauge factor (GF) of the strain gauge is defined as:

GF = ll/Δ

ΔR/R ,

where ΔR is the change in resistance, corresponding to a change in length, Δℓ.

Figure 2-17

The fractional change in length Δℓ/ℓ is called the strain ε, so that the gauge factor may be expressed as:

GF = εΔR/R , Equation 2-8

where ε = llΔ . Equation 2-9

The value of GF for a metallic strain gauge is 2.

The strain gauge pattern can be bonded to the surface of a pressure capsule or embedded inside the capsule. The change in the process pressure will cause a resistive change in the strain gauge, which can be used to produce a 4 to 20 mA or 1 to 5 V signal.

To facilitate with converting a change in resistance to a corresponding voltage change, a Wheatstone bridge, shown in Figure 2-18, is used. The Wheatstone bridge is excited with a stabilised DC supply and the bridge can be zeroed at the null point of measurement. As stress is applied to the bonded strain gauge, a resistive change takes place and unbalances the Wheatstone bridge. This results in a signal output, related to the stress value.

Force Force

Cross sectional area decreases Length increases

Wire without tension

Wire under tension (stress)


Figure 2-18

Using the voltage division rule, the output voltage of the bridge is easily obtained as:

V0 = ⎟⎟⎠

⎞⎜⎜⎝

⎛+−+ 3R1R3R

4R2R4R

E.

From this equation it is apparent that when 4R2R

4R+

= 3R1R

3R+ (which implies

that R1R4 = R2R3 or 3R4R

= 1R2R

), the voltage output V0 will be zero. Under these

conditions, the bridge is said to be balanced. Any change in resistance in any arm of the bridge will now result in a nonzero output voltage. Therefore, if we replace R4 in Figure 2-16 with an active strain gauge, any change in the strain gauge resistance will unbalance the bridge and produce a nonzero output voltage, related to the stress. Let us assume that when the bridge is in balance, the nominal values of the bridge arms are R1 = R, R2 = R, R3 = R and R4 = R. Now if R4 is put under tension (stress), R4 will change its value to R + ΔR and the bridge output will become:

V0 = ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

Δ++Δ+

RRR

R)R(RR)(R E = ⎟

⎠⎞⎜

⎝⎛ −Δ+

Δ+21

R2RRR E

∴V0 = ⎟⎟⎠

⎞⎜⎜⎝

⎛Δ+

Δ+−Δ+R)2(2R

R)(2RR)2(R E

∴V0 = R24RRΔ+

Δ E

From Equation 2-8:

ΔR = (GF)Rε.

Using this expression for ΔR in the expression for V0 above:

V0 = ε+ε

R)GF(24RR)GF( E = /2](GF)4R[1

R(GF)ε+ε E

+ –

Gauge in tension

(R + ΔR)

F

V0

R1 R2

R4

E

Strain gauge

R3


∴V0 = ⎥⎦⎤

⎢⎣⎡ ε+

ε

2(GF)14(GF) E Equation 2-10

Equation 2-10 is the bridge equation for one strain gauge in the bridge or what is

known as a quarter bridge. Other structures are possible, such as one active and one dummy strain gauge or two active strain gauges (half-bridge) or four active strain gauges (full bridge).

The bridge output voltage is typically very small and additional electronic circuitry is needed to amplify the signal and condition it for a 4 to 20 mA or a 1 to 5 V signal. Example 2-10 A strain gauge, imbedded in a silicone filled pressure capsule, is used to measure a differential pressure P1 – P2. The strain gauge is connected to a quarter Wheatstone bridge arrangement shown in the figure below. Each of the resistors in the three fixed arms, has a resistance of 120 Ω. The strain gauge has a nominal resistance of 120 Ω and the bridge is therefore in balance if the capsule experiences no stress. The gauge factor of the strain gauge is two (GF = 2). The pressure cell is put under stress by applying a differential pressure P1 – P2 = 100 kPa which results in a strain of ε = 0.005 in the strain gauge. Calculate the amplifier gain required to produce an output of 1 volt from the Wheatstone output voltage V0, when P1 - P2 = 100 kPa.

From Equation 2-10,

V0 = ⎥⎦⎤

⎢⎣⎡ ε+

ε

2(GF)14(GF) E =

⎥⎦⎤

⎢⎣⎡ ×+××

20.005214

0.0052 ×10 = 24.88×10-3 V (= 24.88 mV)

The amplifier gain A must therefore be 1/(24.88×10-3) = 40.19

P2Pressure capsule

Output

A

120 Ω

120 Ω

Straingauge

+ – V0 10 V

120 Ω

P1

EIPINI Chapter 3: Flow Measurement Page 3-1

3. FLOW MEASUREMENT

The purpose of this chapter is to introduce students to the definitions and units of flow related quantities and to discuss typical methods to measure volumetric flow and flowrate.

3.1 VOLUMETRIC FLOW AND FLOWRATE

Volumetric flow

Volumetric flow is the total volume of a liquid or gas passing a given point over a certain period of time, and is measured in cubic meter (m3).

Note: An example of volumetric flow measurement is municipal water meters that measures the total volume of water used by the customer over a month period. Another example is measuring the total volume of petrol at a gas station, when filling up a car’s tank.

Flowrate Flowrate is the volume of a liquid or gas passing a given point per unit time, and is

measured in cubic meter per second (m3/s).

Note: The flowrate (q) may also be expressed as the product of the velocity (v) of the flow and the cross sectional area (A) of the pipe through which the flow occurs:

q = Av Equation 3-1

3.2 VISCOSITY

Viscosity is a measure of a fluid's resistance to flow and is measured in poiseuille (PI).

Note: Not all liquids are the same. Some are thin and flow easily. Others are thick and sticky. Honey or syrup will pour more slowly than water. A liquid's resistance to flow is called its viscosity. Imagine two layers of a liquid at a distance y from each other and with layer area A, as shown in Figure 3-1. If we assume that the bottom plate is the layer of stationary liquid molecules, clinging to the wall of the pipeline, then the force F that we must apply to move the top plate at a constant velocity v relative to the bottom plate, will be indicative of the fluid’s flow resistance.

v

v Distance cylinder travels in 1 second

A Total volume transported in 1 second = q = Av


The quantity

AF , is called the shear stress in the fluid and the ratio

yv is called the

velocity gradient (or shear rate). For typical liquids (Newtonian liquids), the shear stress is proportional to the velocity gradient and the constant of proportionality is called the viscosity η of the liquid. η =

v/yF/A Equation 3-2

The SI units for viscosity are the poiseuille (PI) or pascal-second (Pa-s) or newton-

second per square meter (N-s/m2). Another common (cgs) unit used to express viscosity is the poise (1 poise = 0.1 PI).

Some examples of viscosity of liquids (at 20 ºC): ηair = 0.00002 PI ηwater = 0.001 PI ηmercury = 0.0015 PI ηoil = 1 PI (typical) ηhoney = 100 PI ηpeanut butter = 2500 PI

Notes: i) Pressure has very little effect on viscosity. ii) Viscosity is not related to density. iii) The viscosity of liquids decrease while the viscosity of gasses increase

with increasing temperature. iv) The viscosity of a liquid divided by the density of the liquid is called the

kinematic viscosity of the liquid.

3.3 STREAMLINED FLOW AND TURBULENT FLOW Streamlined flow

In a streamlined flow (also called laminar flow), all the particles in the liquid, flow in the same direction and parallel to the walls of the pipe, and the streamlines are smooth (Fig 3-2a).

Turbulent flow

In a turbulent flow, the particles in the stream, flow axially as well as radially, and the streamlines are in a chaotic pattern of ever changing swirls and eddies (Fig 3-2b).

y

F

v

v

Figure 3-1

Figure 3-2 (a) Laminar flow

Figure 3-2 (b) Turbulent flow


The Reynolds number

The Reynolds number for a flowstream is given by:

Re = ηDvρ Equation 3-3

where D is the pipe diameter (m), v is the flow speed of the fluid (m/s), ρ is the

density of the fluid (kg/m3) and η is the viscosity of the fluid (PI).

At low Reynolds numbers (generally below Re = 2000) the flow is streamlined while at high Reynolds numbers (above Re =3000) the flow becomes fully turbulent.

Flow straighteners (straightening vanes)

When flow is measured and the flow is not streamlined, errors may arise in the readings obtained. This problem can be prevented by installing flow straighteners or straightening vanes, inside the pipe as shown in Figure 3-3.

Example 3-1 The average velocity of water at room temperature in a tube of radius 0.1 m is 0.2 ms-1. Is the flow laminar or turbulent?

Re = (0.2×0.2×1000)/0.001 = 40000 > 3000 hence turbulent.

3.4 POISEUILLE’S LAW

The flowrate of a streamlined liquid in a horizontal pipe is given by:

q = )2p1(pL8

4πR −η

Equation 3-4

where q is the liquid’s flowrate (m3/s), R is the radius of the pipe (m), η is the

viscosity of the fluid (PI), L is the length of the pipe (m) and p1–p2 is the pressure differential across the pipe (Pa).

The variables used in Equation 3-4, are illustrated in Figure 3-4, and the typical parabolic velocity profile associated with laminar flow in a pipe, is also shown. L

Flow Flow

Figure 3-3

Figure 3-4

p1 p2 q R


Example 3-2:

Calculate the flowrate of water in a pipe with diameter of 0.15 m and length 100 m that discharges into air (p2=100 kPa) while the pump at the other end maintains a

pressure of 102 kPa. Ans: q = )310100310(102100(0.001)8

4π(0.075)×−×

×× = 0.2485 m3/s.

3.5 ENERGY OF A LIQUID IN MOTION

Pressure energy

Pressure energy is the energy which a liquid has by virtue of its internal pressure. A body of liquid with volume V meter3 under pressure p newton/meter2, possesses pressure energy equal to pV joule. Pressure energy per unit volume (when V equals 1 cubic meter), equals p joule.

Kinetic energy

Kinetic energy is the energy a liquid has by virtue of its motion. A body of liquid with mass m kilogram, moving at velocity v meter/second, possesses kinetic energy equal to ½mv2 joule. If the density of the liquid is ρ kilogram/meter3, then the kinetic energy per unit volume (m = ρV; if V = 1, then m = ρ), equals ½ρv2 joule.

Potential energy

Potential energy is the energy that a liquid has by virtue of its height above a given plane. A body of liquid with mass m kilogram and a height h meter above a reference plane, possesses potential energy equal to mgh, where g is the gravitational acceleration constant. If the density of the liquid is ρ kilogram/meter3, then the potential energy per unit volume, equals ρgh joule.

3.6 BERNOULLI’S LAW

If an incompressible fluid is in a streamlined flow with no friction, the sum of the pressure energy, the kinetic energy and potential energy per unit volume, is constant at every point in the flow.

ρgh2ρv21p ++ = constant. Equation 3-5 (a)

Or alternatively, at point 1 and 2 in the stream:

2ρgh22ρv2

12p1ρgh2

1ρv21

1p ++=++ . Equation 3-5 (b)


Static pressure

Static pressure is the pressure that would be measured by a pressure gauge moving with the flow.

Dynamic pressure

Dynamic pressure is the pressure exerted by a flow because of the flow velocity.

Stagnation pressure

Stagnation pressure is the sum of the static and dynamic pressure in a flow.

Note: Referring to equation 3-5a, the energy per unit volume, p, is the static pressure and the kinetic energy per unit volume ½ρv2, is the dynamic pressure (energy per unit volume may be associate with pressure as [Joule/meter3] is equivalent to [Pascal] and interested students are encouraged to verify this dimensional equivalence). The static and dynamic pressures taken together, is called the stagnation pressure (or impact pressure), which is the pressure realized when a flowing fluid is brought to rest.

pstag = p + ½ρv2

∴pstag = pstat + pdyn ……………………………………..……… Equation 3-6

3.7 THE PITOT TUBE One of the earliest flow meters , operation based on Equation 3-6, is the Pitot tube.

The L shaped tube, has the pitot opening directly facing the oncoming flow stream, as shown in Figure 3-5. At the Pitot opening, the stream is brought to rest, and the impact or stagnation pressure is measured. The static pressure is measured

at right angles to the flow direction. The flow velocity and therefore the flowrate q = Av, may be determined from the difference between the stagnation pressure and the static pressure:

v = ρ

)statpstag2(p −. Equation 3-7

A disadvantage of the Pitot tube is that it measures the flowrate only at one point. An annubar flowmeter overcomes this problem by positioning several Pitot tubes across the pipe diameter, providing a better average.

Note: The Pitot tube measures flowrate, by making direct use of Equation 3-6, pdyn = pstag – pstat ⇒ ½ρv2 = pstag – pstat ⇒ v = √[2(pstag – pstat)/ρ].

Stagnation (impact) pressure pstag

v

Static pressure pstat

Impact hole

Figure 3-5


3.8 THE FLOW EQUATION In this section, we will derive the flow equation for a horizontal flow stream. Given that the flow stream is horizontal, gravitational forces and potential energy of the flow, will be neglected. A flow restriction will be formed by allowing a section of the pipe to become narrower than the rest of the pipe. We will then show that the flowrate q, may be determined from the square root of the pressure difference across the restriction.

Referring to Figure 3-6, consider a unit volume in the flow stream with mass ρ (given that the density of the stream is ρ). Let us now follow this unit volume in the flow stream, as it passes the point X (flow area A1), travelling with velocity v1 and under pressure p1, and then later passing the point Y (with smaller flow area A2) at a higher velocity v2 and under the influence of a smaller pressure p2.

According to Bernoulli’s theorem for a steady stream, the total energy content (pressure energy plus kinetic energy) of the unit volume should stay constant.

2p22ρv

21

1p21ρv

21 +=+ ………………………………….. Equation 3-8 (a)

And flow continuity demands that the flow rate q must be the same at X and Y:

q = A1v1 = A2v2, therefore v1 = 1A2A

v2 ……….…………… Equation 3-8 (b)

Using Equation 3-8 (b), v1 can be eliminated from Equation 3-8 (a), which will allow us to obtain an expression for v2 and to determine the flowrate from q = A2v2.

2p 22ρv

21

1p22)v1/A2(Aρ

21 ][ +=+ ⇒ 2p22

2ρv1p222v2)1A/2ρ(A +=+

∴ )2p1p(22)1A/2A( 122ρv −=− ⎥⎦

⎤⎢⎣⎡ ⇒

⎥⎦⎤

⎢⎣⎡

−=

2)1/A2(A-1ρ

)2p12(p22v

v2 v1 p1 p2 Flowdirection

Flowrate = q

h

ρ ρ

X Y

h1

A2 A1

Figure 3-6


∴v2 =

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −

−2)1/A2(A1ρ

)2p12(p ………….………...…… Equation 3-8 (c)

The flow rate q is given by:

q = A2v2 …………………………….…………..…………... Equation 3-8 (d)

From Equations 3-8 (c) and 3-8 (d) it then follows that,

q =

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −

−2)1/A2(A1ρ

)2p12(p2A ……...……...……… Equation 3-8 (e)

Equation 3-8 (e), expresses essentially what we wanted to show, namely, the flow rate q varies with the square root of the pressure difference across the restriction. Equation 3-8 (e) may be simplified further, if we choose to specifically measure the pressure difference p1 – p2 by allowing the liquid in the stream to rise up in the two vertical tubes, as shown in Figure 3-6, and then take the reading h.

p1 = Patm + ρ(h1 + h)g and p2 = Patm + ρh1g. ∴p1 – p2 = ρhg ………………………….……….…….……. Equation 3-8 (f)

Using p1 – p2 from Equation 3-8 (f) in Equation 3-8 (e):

q = ⎥⎦⎤

⎢⎣⎡ −

ρ2)1/A2(A1ρ

gh22A = 2)1/A2(A1

gh22A

− …….…… Equation 3-8 (g)

We now define a calibration constant:

k = A2 ]2)1A/2A(2g/[1− ………………...………. Equation 3-8 (h) From Equations 3-8 (f) and 3-8 (g), we may conclude that,

q = k h Equation 3-9

Note: Although we assumed a horizontal flow stream, it can be shown that Equation 3-9 is equally valid for inclined flow streams or even vertical flow streams. Please remember that h in Eq. 3-9 still implies the pressure difference p1 – p2 (pascal).


Example 3-3 A flow rate meter, uses a restriction in the flow stream, to measure the flow rate of a liquid in a horizontal pipe. The pressure difference across the restriction is determined by allowing the liquid into two vertical tubes installed on top of the pipe and on both sides of the restriction. When the flow rate is 0.1 cubic meter per second, the level difference of the liquid in the tubes is 0.3 meter. Calculate the flow rate when the level difference of the liquid in the two tubes is 0.6 meter.

Answer: Using Equation 3-9: q1 = k 1h ⇒ 0.1 = k 0.3 ⇒ k = 0.1/ 0.3 = 0.1826

∴q2 = k 2h = 0.1826 0.6 = 0.1414 m3/s

Example 3-4 A cylindrical object with volume Vf = 100×10-6 m3, density ρf = 2000 kg/m3 and cross sectional area Af = 4×10-3 m2, is suspended in the centre of a vertical tapered tube by water with density ρℓ = 1000 kg/m3, rushing upwards with a velocity v1 meter/sec. driven forward by pressure p1 pascal when the tube’s cross-sectional area is A1 = 5×10-3 m2 and speeding up to a velocity v2 meter/sec. when it reaches the restricted cross-sectional area A2 = 1×10-3 m2, around the object, where the pressure has diminished to p2 pascal. Calculate the flow rate of the water.

Answer: There are two forces operating on the object, a gravitational force Fw pulling it downwards and a drag force Fd caused by the water stream, pulling it upwards.

Fw = Weight of the object – Weight loss of the object in the water = ρfVfg – ρℓVfg (Weight loss, according to Archimedes’s law, equals weight of water displaced by object) = 2000×(100×10-6)×9.81 – 1000×(100×10-6)×9.81 = 0.981 N

Fd = Force caused by pressure p1 (up) – Force caused by pressure p2 (down) = p1×Cross-sectional area of object – p2×Cross-sectional area of object = p1×Af – p2×Af = p1×(4×10-3) – p2×(4×10-3) = 4×10-3(p1 – p2) newton

For this object to be stationary and stay suspended at one position in the water, the water drag force acting on the object, must equal the gravitational force. ∴Fd = Fw ⇒ 4×10-3(p1-p2) = 0.981 ∴p1 – p2 = 245.3 Pa

From Equation 3-8 (e): q = A2√{2(p1-p2)/ρℓ[1-(A2/A1)2]} = (1×10-3)×√{[2×245.3]/[1000×(1 – (1×10-3/5×10-3)2)]} = (1×10-3)×√{(490.6)/[1000×(1-0.04)]}=(1×10-3)√(490.6/960) = 1×10-3×√0.511 = 714.8×10-6 m3/sec.

p2 p1

0.001m2

A2

A1

v2 v2

v1

q

p2

p1

q 0.005m2 0.004m2

Fd

Fw


Example 3-5 A flow rate meter, uses a restriction in the flow stream, to measure the flow rate of a liquid in a horizontal pipe. When the flow rate of the stream reaches its maximum value (100%), the differential pressure meter also registers its maximum reading (100%). Assuming that the pressure meter will show a zero reading when the flow rate is zero, calculate the flow rate when the pressure meter indicates 20%, 40%, 60% and 80% of its full scale reading. Draw a graph of percentage flow rate versus percentage differential pressure.

It is given that the flow rate q is 100% when the differential pressure h is 100%. Using Equation 3-9, q = k h , 100 = k 100 ⇒ k = 10 For h = 20%, q = 10 20 = 44.72%. For h = 40%, q = 10 40 = 63.25% For h = 60%, q = 10 60 = 77.46%. For h = 80%, q = 10 80 = 89.44%

20 kPa

36 kPa

52 kPa

68 kPa

84 kPa

100 kPa

If we did use a differential pressure transmitter to measure the pressure difference, then 0% input pressure would correspond to 20 kPa, 20% input pressure to 36 kPa, 40% input pressure to 52 kPa, 60% input pressure to 68 kPa, 80% input pressure to 84 kPa and 100% input pressure to 100 kPa. These values are also shown on the differential pressure axis of the graph.

Example 3-6: Use Equation 3-8 (c) to derive the simplified Bernoulli’s equation, p1 – p2 ≈ 4v2

2, that medical doctors use when they examine a patient’s blood circulation. Assume δblood = 1, p1 – p2 is measured in mm. Hg and (A2/A1)2 << 1.

v2 = √{[2(p1-p2)pascal]/[ρblood(1-(A2/A1)2]} ≈ √[2(p1-p2)pascal/1000] because A22 << A1

2 and ρblood ≈ 1000 kg/m3. Also (p1-p2)pascal = [13600×(p1-p2)mmHg×9.81]/1000 ∴v2 = √[2×13600(p1-p2)mmHg×9.81/10002] = √0.27(p1-p2) ⇒ p1-p2 ≈ 4v2

2

Differential pressure h(percent of full scale)

0 40 60 80 10020

20

40

60

80

100

Flowrate q(percent of full scale)

q = 10 h or

h = 0.001q2 (y = ax2)


Allowing a pressure differential to develop across a restriction in a flow stream, is a popular method to measure the flowrate in a liquid as well as a gaseous flow. Two important meters that utilise this method are the venturi tube and the orifice plate.

3.9 THE VENTURI TUBE 3.9.1 Operation and construction

The venturi tube, illustrated in Figure 3-7, has a converging conical inlet, a cylindrical throat, and a diverging conical outlet cone. It has no projections into the fluid, no sharp corners, and no sudden changes in contour. The inlet section decreases the area of the fluid stream causing the velocity to increase and the pressure to decrease. The low pressure is measured in the centre of the cylindrical throat since the pressure will be at its lowest value, and neither the pressure nor the velocity is changing. The outlet or recovery cone allows for the recovery of pressure such that total pressure loss is only 10% to 25%. The high pressure is measured upstream of the entrance cone.

Figure 3-7

3.9.2 Advantages and disadvantages of the venturi tube Advantages

• Pressure loss is small • Operation is simple and reliable

Disadvantages

• Highly expensive • Occupies considerable space.

D/2

d

High pressure tap (upstream tap)

Low pressure tap(downstream tap)

Throat

Flow Inlet cone (19º – 23º)

Outlet cone (5º – 15º)

d/2

D d


3.9.3 Pressure profile across the venturi tube In Figure 3-8 an experimental setup is shown to measure the pressure near the venturi tube wall, and thereby obtaining some feeling about the way the pressure varies with the flow along the venturi tube. The pressure in the flow experiences a dramatic drop in value at the entrance of the throat, accompanied by two pressure spikes and then remains essentially constant in the throat. As the flow enters the output cone, the pressure slowly recovers but never reaches the original pressure. For a venturi tube however, the total pressure loss is only 10% to 25%. Figure 3-8

3.10 THE ORIFICE PLATE 3.10.1 Positioning of an orifice plate between pipe flanges

The orifice meter, is a common and simple method to measure flow rate. It consists of a thin steel plate with a hole, through which the flow passes. The plate is held between two flanges as shown in Figure 3-9. Some orifice plates have a tag with identification information that is still visible after the orifice plate has been clamped between the two pipe flanges.

Non-recoverable pressure loss

Flow

Pipe flangePacking Orifice Orifice plate

Tag

Packing Pipe flange

Flow

Figure 3-9


3.10.2 Drain and vent holes Drain holes:

Drain holes (Figure 3-10a) are provided to prevent solid particles in liquids and condensate in gasses to accumulate at the bottom of the pipe on the upstream side of the orifice plate. Drain holes provide a path for the particles and condensate to be transported further down the pipe with the flow.

Vent holes:

Vent holes (Figure 3-10b) are provided to prevent gasses to accumulate at the top the pipe on the upstream side of the orifice plate, when transporting liquids. Vent holes provide a path for the trapped gasses to be carried downstream with the flow.

3.10.3 Concentric, eccentric and segmental orifice plates Concentric orifice plate

The concentric orifice plate (Figure 3-11a) has its opening located exactly in the centre of the plate. This plate is used for gasses and clean liquids. It is unsuitable for liquids with poor flow characteristics.

Eccentric orifice plate

The eccentric orifice plate (Figure 3-11b) is used in gaseous installations in which condensates are present or where solid particles are present in liquid mediums.

Segmental orifice plate

The segmental orifice plate (Figure 3-11c) is used in systems where solid particles are present in the liquid medium or if the medium is in pulpy form.

Flow Orifice plate

Drain hole

Figure 3-10 (a)

Flow

Orifice plate

Vent hole

Figure 3-10 (b)

Figure 3-11 (a)

Figure 3-11 (b)

Figure 3-11 (c)


3.10.4 Advantages and disadvantages of orifice plates Advantages

• Orifice plates are cheap and easy to install. (A home made orifice is often entirely satisfactory, whereas a venturi meter is practically always purchased from an instrument dealer.)

• Orifice plates are reliable and require a minimum amount of maintenance. • The orifice plate can easily be changed to accommodate widely different flow

rates, whereas the throat diameter of a venturi is fixed.

Disadvantages

• The orifice meter has a large permanent loss of pressure because of the presence of eddies on the downstream side of the orifice-plate; the shape of the venturi meter, however, prevents the formation of these eddies and greatly reduces the permanent pressure loss.

• The higher pressure loss may be associated with higher cost and when an orifice is inserted in a line carrying fluid continuously over long periods of time, the cost of the power loss may be out of all proportion to the saving in first cost.

3.10.5 The vena contracta position in a flow stream

The point where the stream is at its smallest cross-sectional area is called the vena contracta. At this point, flow velocity is a maximum and pressure is a minimum.

The position of the vena contracta point, indicated in Figure 3-12, is of course not stationary but varies with flow velocity. (It is useful here to think about a hosepipe nozzle.) 3.10.6 Positions of the various orifice plate tap points. Depending on the application to which the orifice plate is put, tap points can be located at various positions. The following tap points are usually used:

Corner taps

Corner taps are mounted directly against the sides of the orifice plate, as shown in Figure 3-13a. Corner taps are very sensitive and calibration is very easily affected if the d/D ratio changes as a result of wear. These tap points are therefore not recommended for liquids which contain solid particles.

Flow

Vena contracta Figure 3-12


Flange taps

Flange taps are mounted directly in the pipe flange 25 mm. upstream and 25 mm. downstream, as shown in Figure 3-13b. They are usually accurately built into the flange by the manufacturer.

Radius taps

Radius taps are the most common type in use and are usually called D and D/2 taps or throat taps. The high pressure tap is mounted one pipe diameter upstream and the low pressure tap, one half pipe diameter downstream. These taps give a very high differential pressure.

Flow D

High pressure tap Low pressure tap

d

Figure 3-13 (a)

Flow D


d

25mm

25 mm

Figure 3-13 (b)

½D

Flow D

High pressuretap Low pressure tap

d

D

Figure 3-13 (c)


Vena contracta taps

These taps, shown in Figure 3-13d, are much like radius taps. The high pressure tap is also situated one pipe diameter upstream and the low pressure tap is situated at the vena contracta point. A disadvantage of this system is that the vena contracta point does not stay static at one point, its position varies with flow velocity. This system does however provide the highest differential pressure of all other tap positions. It is not recommended for conditions where there is a large variation in flow rate.

Pipe taps

Pipe taps are mounted two and a half pipe diameters upstream and eight pipe diameters downstream, as shown in Figure 3-13e. The differential pressure developed is not very high however, but it is not affected very much by the rate of flow or wear of the pipe walls or the orifice hole.

Other flow meters The remaining flow meters that we will discuss are not specifically based on the flow equation although the next two, the target meter and rotameter may still be considered to be similar to the orifice plate in the sense that they cause a flow restriction with associated pressure difference, but in the case of the target meter, the pressure difference translates into a force (Δp×area) that varies with the flow rate, while the rotameter attempts to keep the pressure difference constant by varying its float position with the flow rate.

8D

Flow


2½D

Figure 3-13 (e)

d D

Flow D

High pressuretap Low pressure tap

d

D

Figure 3-13 (c)

Vena contracta


3.11 TARGET METERS The target meter (also called a drag plate meter), uses a flat disk or target positioned at right angle to the fluid flow, as shown in Figure 3-14. The drag force exerted on the target by the approaching stream, is transmitted via a force bar to a bonded strain gauge bridge (or differential pressure arrangement for pneumatic output). The strain gauge converts the mechanical

stress caused by the target, into an electrical signal.

In Appendix 3-1 at the end of Chapter 3, it is shown that the flow rate q may be determined from the measured force F exerted on the target, with the expression:

q = 2ρπd8F

4)2d2π(D − Equation 3-10

where D is the pipe diameter (meter), d the target diameter (meter), F the drag force on the target (newton) and ρ the fluid density (kg/m3)

3.12 ROTAMETERS The rotameter (also called a variable area flow meter) consists of a gradually tapered transparent tube, mounted vertically in a frame with the large end up, as shown in Figure 3-15. The fluid flows upward through the tube and a metal displacer or float, is suspended in the fluid. The float is the indicating element and the reading is taken on the scale in line with the top of the float. The position in the tube where the float reaches equilibrium, depends on the flow rate of the fluid. The greater the flow rate, the further up the tube the float rises. The tube is often made of high strength glass to allow for direct observation of the float position, but if greater strength is required or if the liquid is

very dark or dirty, a metal tube is used and the float position detected externally.

The operation of the rotameter is discussed in Appendix 3-2. With the rotameter the flow rate q may be determined from:

q = k(At – Af). Equation 3-11

where At is the tube area at the current float position (meter2), Af the float area (meter2) and k a calibration constant (meter/second).

d D Target

Force barPivot and sealFlow

Strain gauge Electronics

housing

Figure 3-14

Flow

Tapered tube

Float

Fd Scale

Fw

Figure 3-15


3.13 VORTEX FLOW METERS

When an object (also called a bluff body or shredder bar) is located in a flow stream, it causes an alternating series of vortices and whirls to be formed (or shedded) downstream in the flow (called a von Karman vortex street), as shown in Figure 3-16. The number of vortices passing downstream over a given interval of time is proportional to the mean flow velocity. Vortex flow meters utilize this phenomenon by counting the number of vortices, using different techniques such as pressure sensors, capacitance sensors or thermistor temperature sensors built into the bluff body. A popular technique is to use ultrasonic sensors placed outside the pipe just after the shedder bar, which will send an ultrasonic beam across the pipe. The vortices will modulate the frequency of the ultrasonic signal and the vortices are then counted by electronic circuitry.

Note: A flag is a good example of vortex shedding. The flagpole acts as the shredder bar that sheds the wind into vortices that makes the flag wave. In spite of all the complex fluid dynamics involved inside a flow stream, the formula for calculating the frequency of the vortexes, is surprisingly simple and is given by, f = d

vtS , where v is the flow

velocity, d is the width of the vortex generator and St is a proportional constant called the Strouhal number. The Strouhal number is a function of the shape of the vortex generator but it is constant over a broad range of Reynolds numbers and flow velocities. The measured vortex frequency may thus be utilised to calculate the flow velocity v = fd/St, and from the flow velocity, the flow rate q = Av may be obtained.

The final expression for the flowrate q is:

q = A×tS

fd Equation 3-12

where A is the unblocked flow area (meter2), f is the measured vortex frequency (hertz), d is the width of the bluff body (meter) and St the Strouhal number (dimensionless).

Figure 3-16

Flow

Vortex generator (bluff body)

d

Ultrasonic sensors

v

Vortices (swirls)


3.14 MAGNETIC FLOW METERS

Magnetic flow meters (magmeters) can measure the flow rate of any conductive liquid while offering no obstructions to the flow stream. A simplified schematic diagram of a magnetic flow meter is shown in Figure 3-17. Magnetic flow meters are based on Faraday’s law of electromagnetic induction (e = Bℓv), which states that when a conductor is moved through a magnetic field, an emf e (volt) will be generated that is proportional to the velocity v (m/s) of the conductor, the length ℓ (m) of the conductor, and the strength B (tesla) of the magnetic field. The section of pipe that is part of the flow meter, contains the coils through which current is passed to produce the magnetic field as well as the electrodes that produce the voltage that is proportional to the flow rate. This section must be made of a material that is non-magnetic so as not to distort the magnetic field and also a material that is non-conductive so that the electrodes are not short circuited. To ensure that the electrodes make contact with the liquid at all times, they should, preferably lie in a horizontal plane.

Note: Faraday’s law can be applied to a flowmeter, if one imagines the liquid to consist of a series of liquid tubes moving through the magnetic field and cutting through the field lines. The velocity of the conductor v, becomes the velocity of the fluid flowing through the flow tube and the length of

the conductor ℓ, is now the distance D between the electrodes. The flux density B, is the strength of the magnetic field generated by the coils and e is the voltage produced between the electrodes. For a flowmeter, Faraday’s law reduces to e = k1BDv, where k1 is a proportionality constant. From the expression for e, the flow velocity can be calculated as v = e/k1BD. With the flow area given by A = πD2/4, the flow rate may now be determined as q = Av = (πD2/4)×(e/k1BD) = (π/4k1)×(De/B) = kDe/B, where k is a second constant, determined by calibration.

The expression for the flow rate q is therefore:

q = k eBD . Equation 3-13

where D is the distance between electrodes or pipe diameter (meter), B the magnetic flux density (tesla), e the measured emf (volt) and k a calibration constant (dimensionless).

B

Flow

v Electrodes

Magnet coils

D

Figure 3-17

B

vℓ

e = Bℓv


3.15 DOPPLER FLOWMETERS

Doppler flow meters use the well known Doppler frequency shift effect, to determine the flow velocity of a stream. In order to operate, the liquid must contain some small particles or bubbles. Ultrasonic sound waves are transmitted at an angle into the flow, by an

ultrasonic transmitter, and reflected back by the moving bubbles or particles in the stream, as shown in Figure 3-18. The receiver picks up the reflected waves at a higher frequency than the transmission frequency and the flow velocity is a function of the frequency difference between the received and transmitted frequencies.

The flow velocity v is given by:

v = cθcos2f

ff

T

TR − Equation 3-14

The flow rate is then obtained from q = Av, where A is the pipe area.

3.16 TRANSIT TIME FLOWMETERS

The transit time flowmeter (also called transmissivity, time of flight or time of travel flowmeter) uses two ultrasonic transducers to beam a high frequency sound wave (ultrasonic wave), alternatively upstream and downstream at an angle θ, across the flow, as shown in Figure 3-19. The difference in times required for the sound waves to travel upstream (t12) and

downstream (t21), can be used to calculate both the sound speed and the mean fluid velocity along the path followed by the sound. This meter gives accurate results but is only applicable to clean liquids and gasses. It is however not easy to accurately measure the extremely short time intervals that are involved.

The flow velocity is given by:

v = ( )θcost2t

t-tL2112

2112 Equation 3-15

The flow rate is then obtained from q = Av, where A is the pipe area.

Ultrasonic transceiver 2 (Piezoelectric crystals)

Ultrasonic transceiver 1(Piezoelectric crystals)

θ

Bubbles or solid particles

Piezoelectric crystals

Figure 3-18

v

Receiver Transmitter

L t21 θ

Flow t12

Figure 3-19

v

where L is the distance between the sensors (meter), t12 is the travel time from sensor 1 to 2 (second), t21 is the travel time from sensor 2 to 1 (second) and θ the incidence angle. (Refer also to Appendix 3-4)

where c is the speed of sound in the liquid (meter/sec.), fR is the received (echo) frequency (hertz), fT is the transmitted frequency (hertz) and θ is the incidence angle. (Refer also to Appendix 3-3)


3.17 CORIOLIS FLOW METERS Mass flow meters can measure the mass flow (kg/s) of a flow stream. This is in contrast with flow meters that measure volumetric flow rate (m3/s). Principle examples of mass flow meters are thermal mass flow meters and Coriolis mass flow meters. We will discuss the Coriolis flow meter.

Coriolis flow meters use the Coriolis effect to measure the mass flow rate through the meter. The substance to be measured runs through one or two U-shaped tubes (or other complex geometric shape that is proprietary to the manufacturer) that is driven to vibrate back and forth, perpendicular to the direction of the flow. The fluid flowing through the tube causes the tube to twist in synchronism with the vibration. The angle of twist is measured by a pick-up

coil and the greater the angle of the twist, the greater the flow. The density of the fluid is proportional to the damping of the tube oscillations which is measured, so that the volumetric flow rate of the fluid may also be made available. A schematic diagram of the meter is shown in Figure 3-20.

The fascinating principle of operation of the extremely accurate Coriolis mass flow meter, is discussed in Appendix 3-5 at the end of Chapter 3 VOLUMETRIC FLOW METERS It is possible to use flow rate meters for the purpose of measuring the total volume of a fluid passing a certain point, as it is only necessary to accumulate the flow rate over the relevant time interval. The principal purpose, however, of volumetric flow meters is to measure volume of flow and they are completely engineered towards this goal. The next two meters that we will discuss; the turbine flow meter and reciprocating piston flow meter, are examples of volumetric flow meters.

3.18 TURBINE FLOW METERS

When properly installed and calibrated, the modern axial turbine flow meter, shown in Figure 3-21, is a reliable device capable of providing the highest accuracies attainable by any currently available flow sensor for both liquid and gas volumetric flow measurement. The meter consists of a turbine rotor mounted in-line with the flow, by a shaft and bearings supported on the upstream and downstream side, by aerodynamic structures called cones. The angular speed of the rotor, is determined by the flow rate of the stream. Permanent magnets are embedded in the rotor blades

Pickupcoil

Exciter

Flow

Figure 3-20

U-tube


and an alternating voltage is induced in the coil as the blades pass the coil. Each voltage pulse represents a discrete volume of liquid that passed through the meter. The total volume is obtained by counting the number of pulses generated.

POSITIVE DISPLACEMENT FLOW METERS A positive displacement flow meter, commonly called a PD meter, measures the volumetric flow rate of a continuous flow stream, by momentarily entrapping a segment of the fluid into a chamber of known volume and releasing that fluid back into the flow stream on the discharge side of the meter. The total volume is obtained by counting the number of these quantities during a certain period of time.

3.19 RECIPROCATING PISTON PD METER

A reciprocating piston flow meter, fills one piston chamber with fluid, while the fluid is discharged from the opposite piston chamber. In Figure 3-22(a), the piston is shown as it is pushed upwards by the incoming fluid. When it reaches the top of its stroke, a slide valve opens the top piston chamber to the inlet port while the outlet is connected to the bottom chamber. The piston is now forced downwards, as shown in Figure 3-22(b), and when it reaches the bottom of its stroke, the cycle repeats.

Outlet Inlet Outlet Inlet

Pivot Pivot

Slide valve Slide valve

Figure 3-22

(a) (b)

Flow

Permanent magnet Magnetic pickup coil

Rotor blade

Bearing

Support and flow straightener

Rotor shaft

Cone

Figure 3-21


Appendix 3-1 (Target meter) Referring to Figure 3-14, if the force F acting on the target is measured, the pressure difference across the target may be determined from p1 – p2 = 4F/πd2. The restricted flow area A2 is given by A2 = π(D2/4) - π(d2/4) – that is the area not blocked by the target. Assuming that (A2/A1)2 ≈ 0, we can now use Equation 3-8 (e) to determine the flow rate q:

q = ][ 2)/A-(A1ρ

)p-2(p2A

12

21 = ρ

)2d(4F24

)2d2π(D /π×− = 2ρπd8F

4)2d2π(D −

Appendix 3-2 (Rotameter) Referring to Figure 3-15, the downward gravitational force Fw, acting on the float, is the weight of the float minus its weight loss in the fluid. Therefore, for a given fluid density, Fw is constant. The drag force Fd acting upwards on the float, is given by (p1 – p2)Af where p1 is the pressure acting on the bottom of the float, p2 is the pressure acting on the top of the float and Af is the effective cross sectional area of the float. It is now clear that wherever the float finds itself in the tube, in order to be stationary, Fd must be equal to Fw. This means that the drag force acting on the float, Fd, must be constant and because the float area Af is fixed, the pressure differential p1 – p2 across the float, must stay constant, independent of the flowrate. Careful inspection of Equation 3-8(e), will reveal that if p1 - p2 is to remain constant, the value of q can only increase if the restricted flow area A2 increases with respect to the unrestricted flow area A1. For the rotameter this implies that the restricted flow area, At - Af, must increase. The float can accomplish this by shifting its position to a higher level in the tube where the tube area At is bigger and it can be shown that the flow rate is proportional to the difference between the tube area and the float area, that is q = k(At – Af).

Appendix 3-3 (Doppler meter) Consider a transmitter that beams high frequency sound waves at an object. If the frequency of the sound waves is fT hertz and the speed of sound is c meter per second, then the wavelength (distance between two wave crests) is λT = c/fT meter. If the object is stationary with respect to the transmitter, it will reflect part of each wave back to the transmitter with the same wavelength and frequency as the transmitted signal. If, however, the object is moving straight towards the transmitter at a speed of v meter per second, the object will meet each wave crest sooner than it would have if it had stayed motionless. Let us consider the moment when the first wave crest (I) has just collided with the object and is echoed back, as depicted in the drawing to the right. Let us further consider the moment, some time Δt later, when the object encounter the second wave crest (II) and echoes it back. In the time interval Δt, wave crest II must have moved a distance cΔt while the object has moved a distance vΔt.

III

λR 0

cΔt cΔt

vΔt

λT c

0

cI II

c

c II c

I

I c c II Object

v

F

p1

Target area=πd2/4 p2

F1=p1×πd2/4 F2=p2×πd2/4

F = F1-F2 = p1×πd2/4 – p2×πd2/4∴p1 – p2 = F/(πd2/4) = 4F/πd2


The total distance the two have moved before they collided is clearly λT and therefore vΔt + cΔt = λT or Δt = λT/(v+c). Furthermore, during the same time interval Δt, wave crest I has already moved a distance cΔt from the origin where it was reflected, towards the receiver. Wave crest II is now also on its way back to the receiver and it is again clear from the sketch that the distance between I and II (which is the wavelength λR, of the reflected sound wave with corresponding received frequency, fR = c/λR) is given by: λR = cΔt - vΔt = (c – v)Δt = (c – v)λT/(v + c) ⇒ λR(c + v) = λT(c - v) ∴(c/fR)(c + v) = (c/fT)(c – v) ⇒ fT(c + v) = fR(c – v) ⇒ vfT + vfR = cfR –cfT ∴v = c(fR – fT)/(fT + fR).

Because fT ≈ fR, it follows that fT + fR ≈ 2fT, and we conclude that the speed of the object is given by, v = (fR - fT)c/2fT. When we refer back to Figure 3-18, we see that the sound waves are transmitted at an angle θ into the stream. This means that we are not measuring the flow velocity v, but rather the magnitude of the velocity vector, vcosθ, that is directly in line with the sound beam (see sketch). We therefore replace v with its component vcosθ in v = (fR - fT)c/2fT, to obtain the flow velocity v = c(fR – fT)/2fTcosθ.

Appendix 3-4 (Transit time meter) Referring to Figure 3-19, if the flow velocity is v meter/second, then the component of v that is parallel to the beam path is vcosθ. If the speed of sound is c meter/second, the upstream speed of the sound beam is c-vcosθ, as it is opposed by the flow. The downstream speed of the sound pulse is c+vcosθ as it is assisted by the flow. The time it takes a sound pressure pulse to move upstream from transducer 1 to 2 is then t12=L/(c-vcosθ), where L is the distance between the transceivers, and the time it takes a sound pressure pulse to move downstream from transducer 2 to 1 is t21=L/(c+vcosθ). The difference of the inverses of the times is then, 1/t21-1/t12=(c+vcosθ)/L-(c-vcosθ)/L and it follows easily that the flow velocity v = L[(1/t21)-(1/t12)]/2cosθ, from which the flowrate q = Av may be determined.

Appendix 3-5 (Coriolis meter) It is easy to demonstrate the tube twisting phenomenon. If water is flowing in an elastic hosepipe and a section of the hosepipe, formed in a loop, is swung slowly forward and backward, the twisting of the lower section of the hosepipe will immediately be observed.

The Coriolis force was first described by the French civil engineer Gaspard Coriolis in 1843, who observed that the wind, the ocean currents and even airborne artillery shells will always drift sideways because of the earth’s rotation. If one could imagine a person standing in the centre of a rotating disk revolving in a clockwise direction, as depicted in Figure A3-1 (a), and he rolls a ball straight from A to B, he would observe that as far as he and his rotating disk is concerned, the ball is not moving in a straight line but rather it is curling to the left. This person would come to the conclusion that there is apparently a force present that is pushing the ball away.

v

vcosθ

θ

θ c


This apparent force is known as the Coriolis force and is given by Fc = 2mωv where m is the mass of the body moving in the rotational frame, ω is the angular speed of rotational frame and v is the outward speed of the mass body. If the person would repeat the experiment but this time rolling the ball through an elastic rubber pipe fixed at A and B, he would observe that the pipe will be bent by the ball as it is trying, but prevented by the pipe, to follow its natural path as illustrated in Figure A3-1 (b). Similarly, if the

person would stand on the edge of the disk and roll the ball from A to B towards the centre of the disk, he would observe that the ball, because of the disk’s rotation, would again be veered off the intended path, as shown in Figure A3-2 (a). If he would roll the ball through a rubber pipe, fixed at A and B, the ball will bend the pipe once more, but this time in the opposite direction, as illustrated in Figure A3-2 (b). With this in mind, we can now turn our attention to the Coriolis flowmeter tubes, depicted in Figure A3-3.

The driver element (exciter) is at this moment rotating the tubes downward with an angular velocity ω. The fluid moving into the entrance tube, now acts in the same way as the ball rolling out from the centre of the disk and is bending the tube upwards. The fluid moving in the exit tube behaves the same way as the ball rolling in from the edge of the spinning disk and is bending the tube downwards. The total effect is a clockwise twist of the tubes. When the driver element starts to push the tubes upwards (the driver operates near the resonance frequency of the tubes), the whole scenario changes around and the tubes will twist in an anti-clockwise direction.

Figure A3-3

ω

Driver Tube Flow

End view of flow tube

showing twist

ω ω

(a) (b)

AB

AB

Figure A3-2

(b) (a)

BA BAω ω

Figure A3-1

process instrumentation i module code: eipin1b study...

Documents