The rate at which a fluid moves through a conduit is measured in terms of the quantityknown as the flow rate. Some of the most common and accepted methods for measuringflow rate are discussed here. Flow rate can be expressed in terms of a flow volume perunit time, known as the volume flow rate, or as a mass flow per unit time, known as themass flow rate. Flow rate devices, called flow meters, are used to quantify, totalize, ormonitor flowing processes. Type, accuracy, size, pressure drop, pressure losses, capitaland operating costs, and compatibility with the fluid are important engineering designconsiderations for choosing a flow metering device. All methods have both desirable andundesirable features that necessitate compromise in the selection of the best method forthe particular application.
The objective is to present both an overview of basic flow metering techniques forproper meter selection, as well as those design considerations important in the integrationof a flow rate device with the process system it will meter.
1.1 Brief Historical background
The importance to engineered systems give flow measurement methods their rich history.The history of measurement systems in India begins in early Indus Valley Civilizationwith the earliest surviving samples dated to the 5th millennium BC. A complex systemof weights and measures was adopted by the Maurya empire (322–185 BCE), which alsoformulated regulations for the usage of this system. The Arthashastra offers a wealth ofevidence for the wide varieties of standardized weights and measures of the time.
Accounts of flow metering were recorded by Hero of Alexandria (ca. 150 B.C.) whoproposed a scheme to regulate water flow using a siphon pipe attached to a constant headreservoir. The early Romans developed elaborate water systems to supply public bathsand private homes. Evidence suggests that Roman designers understood correlationbetween volume flow rate and pipe flow area.
Following a number of experiments conducted using olive oil and water, Leonardoda Vinci (1452–1519) first formally proposed the modern continuity principle: that ductarea and fluid velocity were related to flow rate. Isaac Newton (1642– 1727), DanielBernoulli (1700–1782), and Leonhard Euler (1707–1783) built the mathematical andphysical bases on which modern flow meters would later be developed. By the nineteenthcentury, the concepts of continuity, energy, and momentum were sufficiently understoodfor practical exploitation. Relations between flow rate and pressure losses were devel-oped that would permit the tabulation of the hydraulic coefficients necessary for thequantitative engineering design of many modern flow meters.
2 Flow Rate and Velocity measurement
The amount of mass flowing through a cross section per unit time is called mass flowrate and is denoted by m. The volume of the fluid flowing through a cross section perunit time is called the volume flow rate denoted by v.
2
Some flowmeters measure the flow rate directly by discharging and recharging ameasuring a chamber of known volume continuosly and keeping track of number ofdischarges per unit time. But most flowmeters measure the flow rate indirectly—theymeasure the average velocity V or a quantity that is related to average velocity such aspressure and drag, and determine the volume flow rate v from
v = V Ac (1)
where Ac is the cross-sectional area of flow. Therefore, measuring the flow rate is usuallydone by measuring flow velocity, and most flowmeters are simply velocimeters used forthe purpose of metering flow.
The devices that are commonly used to measure velocity and flow rate are broadlyclassified as follows:
1. Pitot and Pitot static probes
2. Pressure Differential meters
3. Insertion Volume Flowmeters
4. Mass flow meters
The general features of these devices are discussed in brief in the following sections.
2.1 Pitot and Pitot static probes
Pitot probes (also called Pitot tubes) and Pitot-static probes, named after the Frenchengineer Henri de Pitot (1695–1771), are widely used for flow rate measurement.
The Pitot-static probe measures local velocity by measuring the pressure differencein conjunction with the Bernoulli equation. It consists of a slender double-tube alignedwith the flow and connected to a differential pressure meter. The inner tube is fullyopen to flow at the nose, and thus it measures the stagnation pressure at that location(point 1). The outer tube is sealed at the nose, but it has holes on the side of theouter wall (point 2) and thus it measures the static pressure. For incompressible flowwith sufficiently high velocities (so that the frictional effects between points 1 and 2 arenegligible), the Bernoulli equation is applicable and can be expressed as
P1
ρg+V 21
2g+ z1 =
P2
ρg+V 22
2g+ z2 (2)
3
Figure 1: Close-up of a Pitot-static probe, showing the stagnation pressure hole and twoof the five static circumferential pressure holes.
Noting that z1 ≈ z2 since the static pressure holes of the Pitot-static probe arearranged circumferentially around the tube and V1 = 0 because of the stagnation condi-tions, the flow velocity V = V2 becomes
V =
√2(P1 − P2)
ρ(3)
The Pitot-static probe is a simple, inexpensive, and highly reliable device since it hasno moving parts. It also causes very small pressure drop and usually does not disturbthe flow appreciably.
2.2 Pressure Differential meters
The operating principle of a pressure differential flow rate meter is based on the rela-tionship between volume flow rate and the pressure drop ∆p = p1 − p2 , between twolocations along the flow path. The different types of Pressure differential meters areexplained briefly in the following sections.
4
2.2.1 Obstruction Flowmeters: Orifice, Venturi, and Nozzle Meters
Consider incompressible steady flow of a fluid in a horizontal pipe of diameter Dthat is constricted to a flow area of diameter d, as shown. A simple analysis using thecontinuity and the Bernoulli equations yields the following equation:
V2 =
√2(P1 − P2)
ρ(1 − β)4(4)
where β = D/d, the diameter ratio.This analysis shows that the flow rate through a pipe can be determined by con-
stricting the flow and measuring the decrease in pressure due to the increase in velocityat the constriction site. Noting that the pressure drop between two points along theflow can be measured easily by a differential pressure transducer or manometer, it ap-pears that a simple flow rate measurement device can be built by obstructing the flow.Flowmeters based on this principle are called obstruction flowmeters and are widelyused to measure flow rates of gases and liquids.
The velocity in Eq. 4 is obtained by assuming no loss, and thus it is the maximumvelocity that can occur at the constriction site. All the losses that occurs in practice canbe accounted for by incorporating a correction factor called the discharge coefficientCd whose value (which is less than 1) is determined experimentally. The volume flowrate v is finally determined from Eq. 1.
Of the numerous types of obstruction meters available, those most widely used areorifice meters, flow nozzles, and Venturi meters. The orifice meter has the simplestdesign and it occupies minimal space as it consists of a plate with a hole in the middle, butthere are considerable variations in design. The Venturi meter, invented by the Americanengineer Clemans Herschel (1842–1930) and named by him after the Italian GiovanniVenturi (1746– 1822) for his pioneering work on conical flow sections, is the most accurateflowmeter in this group, but it is also the most expensive. Its gradual contraction andexpansion prevent flow separation and swirling, and it suffers only frictional losses onthe inner wall surfaces. The common types of obstruction meters are shown.
5
2.3 Insertion Volume Flowmeters
Dozens of volume flow meter types based on a number of different principles have beenproposed, developed, and sold commercially. A large group of meters is based on somephenomenon that is actually sensitive to the average velocity across a control surfaceof known area. Another common group, called positive displacement meters, actuallymeasure parcels of a volume of fluid per unit time. Some of these designs are includedin the discussion below.
2.3.1 Electromagnetic Flowmeters
It has been known since Faraday’s experiments in the 1830s that when a conductor ismoved in a magnetic field, an electromotive force develops across that conductor as aresult of magnetic induction. Faraday’s law states that the voltage induced across anyconductor as it moves at right angles through a magnetic field is proportional to thevelocity of that conductor. This suggests that we may be able to determine flow velocityby replacing the solid conductor by a conducting fluid, and electromagnetic flowmetersdo just that. Electromagnetic flowmeters have been in use since the mid- 1950s, andthey come in various designs such as full-flow and insertion types.
6
A full-flow electromagnetic flowmeter is a nonintrusive device that consists of a mag-netic coil that encircles the pipe, and two electrodes drilled into the pipe along a diameterflush with the inner surface of the pipe so that the electrodes are in contact with thefluid but do not interfere with the flow and thus do not cause any head loss. The elec-trodes are connected to a voltmeter. The coils generate a magnetic field when subjectedto electric current, and the voltmeter measures the electric potential difference betweenthe electrodes. This potential difference is proportional to the flow velocity of the con-ducting fluid, and thus the flow velocity can be calculated by relating it to the voltagegenerated.
Insertion electromagnetic flowmeters operate similarly, but the magnetic field is con-fined within a flow channel at the tip of a rod inserted into the flow, as shown.
2.3.2 Vortex Flowmeters
Most flows encountered in practice are turbulent, and a disk or a short cylinder placed inthe flow coaxially sheds vortices. It is observed that these vortices are shed periodically,and the shedding frequency is proportional to the average flow velocity. This suggeststhat the flow rate can be determined by generating vortices in the flow by placing anobstruction along the flow and measuring the shedding frequency. The flow measurementdevices that work on this principle are called vortex flowmeters.
A vortex flowmeter consists of a sharp-edged bluff body (strut) placed in the flow thatserves as the vortex generator, and a detector (such as a pressure transducer that recordsthe oscillation in pressure) placed a short distance downstream on the inner surface of thecasing to measure the shedding frequency. The detector can be an ultrasonic, electronic,or fiber-optic sensor that monitors the changes in the vortex pattern and transmitsa pulsating output signal. A microprocessor then uses the frequency information tocalculate and display the flow velocity or flow rate.
7
2.3.3 Variable-Area Flowmeters (Rotameters)
A simple, reliable, inexpensive, and easy-to-install flowmeter with low pressure drop andno electrical connections that gives a direct reading of flow rate for a wide range of liquidsand gases is the variable-area flowmeter, also called a rotameter or floatmeter.
A variable-area flowmeter consists of a vertical tapered conical transparent tubemade of glass or plastic with a float inside that is free to move, as shown. As fluid flowsthrough the tapered tube, the float rises within the tube to a location where the floatweight, drag force, and buoyancy force balance each other and the net force acting onthe float is zero. The flow rate is determined by simply matching the position of thefloat against the graduated flow scale outside the tapered transparent tube.
The drag force acting on the fluid increases with flow velocity. The weight andthe buoyancy force acting on the float are constant, but the drag force changes withflow velocity. Also, the velocity along the tapered tube decreases in the flow directionbecause of the increase in the cross-sectional area. There is a certain velocity thatgenerates enough drag to balance the float weight and the buoyancy force, and thelocation at which this velocity occurs around the float is the location where the floatsettles. The degree of tapering of the tube can be made such that the vertical risechanges linearly with flow rate, and thus the tube can be calibrated linearly for flowrates. The transparent tube also allows the fluid to be seen during flow.
2.3.4 Turbine Flowmeters
We all know from experience that a propeller held against the wind rotates, and the rateof rotation increases as the wind velocity increases.These observations suggest that theflow velocity in a pipe can be measured by placing a freely rotating propeller inside apipe section and doing the necessary calibration. Flow measurement devices that workon this principle are called turbine flowmeters or sometimes propeller flowmeters,although the latter is a misnomer since, by definition, propellers add energy to a fluid,while turbines extract energy from a fluid.
Turbine flowmeters give highly accurate results (as accurate as 0.25 percent) over awide range of flow rates when calibrated properly for the anticipated flow conditions.Turbine flowmeters have very few blades (sometimes just two blades) when used to
8
Figure 2: Two types of variable-area flowmeters: (a) an ordinary gravity-based meterand (b) a spring-opposed meter.
9
measure liquid flow, but several blades when used to measure gas flow to ensure adequatetorque generation. The head loss caused by the turbine is very small.
Turbine flowmeters have been used extensively for flow measurement since the 1940sbecause of their simplicity, low cost, and accuracy over a wide range of flow condi-tions. textbfPaddlewheel flowmeters are low-cost alternatives to turbine flowmetersfor flows where very high accuracy is not required. A sensor detects the passageof each of the paddlewheel blades and transmits a signal. A microprocessor thenconverts this rotational speed information to flow rate or integrated flow quantity.
2.3.5 Ultrasonic Flowmeters
It is a common observation that when a stone is dropped into calm water, the waves thatare generated spread out as concentric circles uniformly in all directions. But when astone is thrown into flowing water such as a river, the waves move much faster in the flowdirection (the wave and flow velocities are added since they are in the same direction)compared to the waves moving in the upstream direction (the wave and flow velocities aresubtracted since they are in opposite directions). As a result, the waves appear spreadout downstream while they appear tightly packed upstream. The difference between thenumber of waves in the upstream and downstream parts of the flow per unit length isproportional to the flow velocity, and this suggests that flow velocity can be measuredby comparing the propagation of waves in the forward and backward directions to flow.Ultrasonic flowmeters operate on this principle, using sound waves in the ultrasonicrange.
Ultrasonic (or acoustic) flowmeters operate by generating sound waves with a trans-ducer and measuring the propagation of those waves through a flowing fluid. There aretwo basic kinds of ultrasonic flowmeters: transit time and Doppler-effect (or frequencyshift) flowmeters.
The transit time flowmeter transmits sound waves in the upstream and downstreamdirections and measures the difference in travel time. A typical transit time ultrasonicmeter is shown schematically It involves two transducers that alternately transmit andreceive ultrasonic waves, one in the direction of flow and the other in the opposite direc-
10
tion. The travel time for each direction can be measured accurately, and the difference inthe travel time can be calculated. The average flow velocity V in the pipe is proportionalto this travel time difference ∆t, and can be determined from
V = KL∆t (5)
where L is the distance between the transducers and K is a constant.
Doppler effect forms the basis for the operation of most ultrasonic flowmeters.Doppler-effect ultrasonic flowmeters measure the average flow velocity along the sonicpath. This is done by clamping a piezoelectric transducer on the outside surface of apipe (or pressing the transducer against the pipe for handheld units). The transducertransmits a sound wave at a fixed frequency through the pipe wall and into the flowingliquid. The waves reflected by impurities, such as suspended solid particles or entrainedgas bubbles, are relayed to a receiving transducer. The change in the frequency of thereflected waves is proportional to the flow velocity, and a microprocessor determines theflow velocity by comparing the frequency shift between the transmitted and reflectedsignals.
2.3.6 Positive Displacement Flowmeters
When the quantity of interest is the total amount of mass or volume of a fluid thatpasses through a cross section of a pipe over a certain period of time rather than theinstantaneous value of flow rate, positive displacement flowmeters are well suitedfor such applications. There are numerous types of displacement meters, and they arebased on continuous filling and discharging of the measuring chamber. They operateby trapping a certain amount of incoming fluid, displacing it to the discharge side ofthe meter, and counting the number of such discharge– recharge cycles to determine thetotal amount of fluid displaced. The clearance between the impeller and its casing mustbe controlled carefully to prevent leakage and thus to avoid error.
11
Figure 3: A positive displacement flowmeter with double helical three-lobe impellerdesign
2.4 Mass flow meters
There are many situations in which the mass flow rate m is the quantity of interest. Massflow meters are used in such cases. A brief description of Thermal Anemometer whichis a type of Mass flow meter is given below.
2.4.1 Thermal (Hot-Wire and Hot-Film) Anemometers
Thermal anemometers were introduced in the late 1950s and have been in commonuse since then in fluid research facilities and labs. Thermal anemometers involve anelectrically heated sensor, as shown, and utilize a thermal effect to measure flow veloc-ity. Thermal anemometers have extremely small sensors, and thus they can be used tomeasure the instantaneous velocity at any point in the flow without appreciably dis-turbing the flow. They can take thousands of velocity measurements per second withexcellent spatial and temporal resolution, and thus they can be used to study the detailsof fluctuations in turbulent flow. They can measure velocities in liquids and gases accu-rately over a wide range—from a few centimeters to over a hundred meters per second.
12
A thermal anemometer is called a hot-wire anemometer if the sensing element is awire, and a hot-film anemometer if the sensor is a thin metallic film (less than 0.1 µmthick) mounted usually on a relatively thick ceramic support having a diameter of about50 µm. The hot-wire anemometer is characterized by its very small sensor wire—usuallya few microns in diameter and a couple of millimeters in length. The sensor is usuallymade of platinum, tungsten, or platinum–iridium alloys, and it is attached to the probethrough holders.
The operating principle of a constant-temperature anemometer (CTA), which is themost common type and is shown schematically, is as follows: the sensor is electricallyheated to a specified temperature (typically about 200 ◦C). The sensor tends to coolas it loses heat to the surrounding flowing fluid, but electronic controls maintain thesensor at a constant temperature by varying the electric current (which is done by vary-ing the voltage) as needed. The higher the flow velocity, the higher the rate of heattransfer from the sensor, and thus the larger the voltage that needs to be applied acrossthe sensor to maintain it at constant temperature. There is a close correlation betweenthe flow velocity and voltage, and the flow velocity can be determined by measuringthe voltage applied by an amplifier or the electric current passing through the sensor.
Using proper relations for forced convection, the energy balance can be expressed by
13
Figure 4: Thermal anemometer probes with single, double, and triple sensors to measure(a) one-, (b) two-, and (c) three-dimensional velocity components simultaneously.
King’s law asE2 = a+ bV n (6)
where E is the voltage, and the values of the constants a, b, and n are calibrated fora given probe. Once the voltage is measured, this relation gives the flow velocity Vdirectly.
Thermal anemometers can be used to measure two or three-dimensional velocity com-ponents simultaneously by using probes with two or three sensors, respectively. Whenselecting probes, consideration should be given to the type and the contamination levelof the fluid, the number of velocity components to be measured, the required spatial andtemporal resolution, and the location of measurement.
14
References
[1] Richard S. Figliola, Donald E. Beasley , Theory and Design for Mechanical Measure-ments, John Wiley and Sons, 5nd Edition, 2011.
[2] Yunus A. Cengel, John M. Cimbala , Fluid Mechanics: Fundamentals And Applica-tions, Tata McGraw Hill, 2nd Edition, 2012.
[3] Iwata, Shigeo, Weights and Measures in the Indus Valley, Encyclopaedia of the His-tory of Science, Technology, and Medicine in Non-Western Cultures, 2nd Edition,2008.
15
Questions
1. Which of these terms best describes the amount of fluid that’s passing some pointat any given time?
(a) Flow rate X
(b) Total flow
(c) Laminar flow
(d) Turbulent flow
2. What is a meter that indicates gallons per minute most likely measuring?
(a) Flow rate X
(b) Total flow
(c) Laminar flow
(d) Turbulent flow
3. Select the term that best describes the amount of fluid that has passed a designatedpoint.
(a) Flow rate
(b) Gallons per minute
(c) Laminar flow X
(d) Flow velocity
4. Select the term that best describes the principle on which this nutating disc meteroperates.
(a) Laminar turbulence
(b) Equalization of level
(c) Positive displacement X
(d) Differential viscosity
5. The D/P cell in a flow measurement device relies on differential pressure createdby which component?
(a) Plummet
i
(b) Float
(c) Rotor
(d) Orifice plate X
6. What is the purpose of the orifice plate in this flow measuring arrangement?
(a) To create a differential pressure X
(b) To change the level of the fluid
(c) To change the temperature of the fluid
(d) To stop the flow of the fluid
7. When a weir is placed in an open channel, which of these factors is used to deter-mine fluid flow rate?
(a) Pressure
(b) Weight
(c) Level X
(d) Clarity
8. A meter uses the velocity of a fluid to determine flow rate. What is the metercalled?
(a) Nutating disc meter
(b) Oval gear meter
(c) Orifice plate meter
(d) Turbine flow meter X
9. Select the choice that best describes what flow rate is.
(a) The amount of fluid that’s passing some point at any given time X
(b) The total amount of fluid that has passed a designated point
(c) The amount of air needed to make bubbles rise in a bubbler system
(d) The ratio of a fluid’s turbidity to its opacity
10. Which unit of measurement is commonly associated with flow rate?
(a) Gallons per minute X
(b) Pounds per square inch
(c) Feet per second
(d) Inches of water
11. If a meter measures the total amount of fluid that has passed a designated point,then what is it most likely measuring?
ii
(a) Flow rate
(b) Total flow X
(c) Laminar flow
(d) Turbulent flow
12. Select all of the statements that can be applied to a nutating disc meter.
(a) It traps fluid above and below the disc. X
(b) It contains a plummet.
(c) It operates on the principle of positive displacement. X
(d) has turbine blades mounted in the fluid flow path.
13. What is the main industrial application of orifice meter?
Ans: The orifice meter is widely used in industry for the measurement of single-phase streams.
14. What is the difference between notch and orifice?
Ans: Notch is a device used for measurement of rate of flow of liquids through smallchannels in tanks. Orifice is a device used to measure the rate of flow through pipes.
15. Why is a V notch preferred over a rectangular notch?
• Expression for discharge through right-angled notch is very simple
• V notch gives more accurate values than rectangular notch
• Ventilation of V notch is not necessary
16. Pitot static tubes are based on the principle of Bernoulli’s Equation
17. What are the primary considerations when selecting a flowmeter to measure theflow rate of a fluid?
Ans: The engineering decision involving the selection of a particular meter dependson a number of constraining factors. In general, flow rate can be determined towithin about 0.25% of actual flow rate with the best of present technology butpractical values for industrial installations are more nearly 3–6% for obstructionmeters and 1–3% for insertion meters. However, new methods may push theselower limits even further, provided that calibration standards can be developed todocument method uncertainties and the effects of installation.
