(2.) flow measurement

FLOW MEASUREMENT I. FLOW-OBSTRUCTION METHODS: 1. Pitot tube: A pitot tube is a simple device used to measure fluid velocity at a point in a flow as shown in a sketch below: points 1 2 flow · · h pitot tube The pitot tube is based on the Bernoulli equation (with the assumption of constant density of the fluid between points 1 and 2) as:

SI units: 22

22

11

2

221 zgPVzgPV

++=++ρρ

(1)

British units: 22

22

11

2

221 z

ggP

gVz

ggP

gV

cccc++=++

ρρ (1.b)

Where V1 is the velocity of the flow and V2 = 0 at the stagnation point at the tip of the pitot tube. Points 1 and 2 are at the same elevation. The Bernoulli equation can be simplified as:

CHE 215 page of 20 H. Muhamad 1

( )ρ

122

21 PPg

V

c

−= (2)

( )ρ

121

2 PPgV c −= (3)

Therefore, ideally the fluid velocity can be determined from Equation (3) once the differential pressure (P1-P2) is measured by the pitot tube as: ( ) ( h

ggPP Mc

ρρ −=− 12 ) (4)

In practice, due to the non-ideal condition a calibration factor is added to Equation (4) for the determination of the fluid velocity at a given point in the flow as:

( )ρ

121

2 PPgCV c −= (5)

The calibration factor C is usually in the range of 0.98 to 1.0 Equation (5) applies to incompressible fluid (liquid). However, it can also be used for gas flow at moderate velocities and pressure change (P1-P2) less than 10% the total pressure. In gas flow application, an annubar is often used. The annubar is basically an integration of a number of pitot tubes on one bar that can installed across the diameter of a pipe. The annubar thus gives the reading of the average velocity of the flow in the pipe with a typical accuracy of ± 3% full scale.


2. Venturi flowmeter


P1 P2

flow

D1 D2

From the Bernoulli equation for incompressible fluid (with the assumption of constant fluid density between points 1 and 2):

22

22

11

2

221 z

ggP

gVz

ggP

gV

cccc++=++

ρρ

The relationship between the fluid velocity at points 1 and 2 can be written as:

( ) ( )cgVVPP

2

21

22

21−

=−ρ (6)

From the equation of continuity for isothermal condition (constant ρ):

1

221 A

AVV = (7)

Substitution of Equation (7) in to Equation (6):

( )cg

AAV

PP2

12

1

222

21

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎥⎦

⎤⎢⎣

⎡−

=−

ρ

(8)

The fluid volumetric flow rate is (ideal condition):


( ) 21

21c

21

2

1

2

2 PPg2

AA1

A=Q ⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

ρ (9)

Under practical conditions (some friction loss), a coefficient of discharge (or calibration coefficient), Co, which is a function of the ratio of the venturi throat diameter to that of the pipe and the Reynolds number, is added to the equation of the fluid volumetric flowrate as below:

( ) 21

21c

21

2

1

2

2o

PPg2

AA1

AC=Q ⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅ρ

(10)

Let M, velocity approach factor,

2

1

21

1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

AA

( ) 21

21c2o

PPg2C=Q ⎥

⎦

⎤⎢⎣

⎡ −⋅⋅⋅⋅

ρAM (11)

Advantages of a venturi flowmeter over an orifice plate:

• Low pressure loss • Less maintenance • More accurate: ± 0.5% full scale

However, a venturi flowmeter is more expensive than an orifice plate.


3. Orifice plate and Flow Nozzle orifice flow A1 A2 P1 P2

Orifice Plate


Flow nozzle

An orifice plate is a plate with an opening area A2 that is smaller the cross-sectional area A1 of the pipe. The fluid accelerates at the orifice opening. Therefore, the pressure at point 2 (P2) is lower than the pressure in the bulk stream P1 right in front of the orifice plate. Similarly, from the Bernoulli equation, the equation for the volumetric flowrate is identical to that for a venturi flowmeter as below:

( ) 21

21co

PPg2C=Q ⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅⋅

ρ2AM (12)

The orifice coefficient, Co, is a function of the ratio of the orifice diameter to that of the pipe and the Reynolds number.


The discharge coefficient for an orifice plate can be determined from the following equation: β (ratio of throat diameter to pipe diameter): 0.2 to 0.7

75.0D

5.241.2

oRe

71.91184.00312.05959.0C βββ +−+=

ReD: Reynolds number based on the pipe inside diameter A flow nozzle is basically an orifice plate with the opening extended to form a short tubing in the direction of the flow. The equation for the volumetric flowrate is exactly the same as that of an orifice plate. The orifice plate and the flow nozzle are suitable for clean and low viscosity fluid. The accuracy is usually in the range of ±0.75 to 2% full scale. The discharge coefficient can be estimated using the following equation: 5.06

Re1000653.09975.0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

DoC β

where the Reynolds number is based on the pipe diameter. Gas flow For a gas flow (compressible fluid) passing a venturi flowmeter or an orifice plate, the volumetric flowrate can be determined from the following equation:

( ) 21

21c2o

PPg2CY=Q ⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅⋅⋅

ρAM (13)

Where the gas expansion factor Y can be estimated from the following equations.


For an orifice plate:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

1

212

1

235.041.01P

PPAAY

γ (14)

Where γ = CP/CV, Cp is the gas specific heat at constant pressure and Cv is the gas specific heat at constant volume. For example, for air γ = CP/CV = 1.4 For a venturi flowmeter or a flow nozzle:

( ) 50

2

1

22

1

2

2

1

2

1

2

1

1

22

1

2

1

1

1

1

1

.

PP

AA

AA

PP

PP

PPY

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎠

⎞⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−

γ

γγ

γ

γγ (15)

In the graph for the values of the expansion coefficient Y, β is defined as D2/D1 (D1 diameter of the pipe, D2 the diameter of the throat)


II. FLOWMETER USING DRAG FORCE Rotameter (variable area flowmeter): d tapered tube float b angle ϴ y D


A rotameter is based on the force balance on a float inside a tapered tube. gbd FFF =+ (16) The drag force, Fd, can be expressed as:

c

2Fbdd g2

uACF ρ= (17)

Where Cd is the drag coefficient, Ab is the area of the float perpendicular to the flow (Ab = π d2/4), ρF is the density of the fluid and u is the fluid velocity at the annular space between the float and the tapered tube. The buoyant force, Fb is: b

cFb V

ggF ρ= (18)

where Vb is the volume of the float.


The gravitational force on the float, Fg is: b

cbg V

ggF ρ= (19)

By substitution of Equations (17), (18) and (19) into Equation (16) and isolation of u, the following equation is obtained:

5.0

12⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

F

b

bd

bACVgu

ρρ (20)

Equation (20) indicates that u is constant for a certain fluid and a specific float (Cd is relatively constant under turbulent flow). This can be applied to measure the stream flowrate. Initially, under no flow condition the float sits at the bottom of the tube where the gap between the float and the tube is almost negligible. As the fluid flows through the rotameter, the float is forced upward and moves to some position where all the forces applied on the float are balanced. The fluid velocity is then at the value of u accordingly to Equation (20). When a further increase of the flowrate occurs, the drag force is stronger. The total force applied on the float becomes unbalanced in the upward direction since the annular velocity is temporarily higher than the steady value given by Equation (20). As a result, the float moves up farther to open up the annular space so to regain the steady annular velocity u. Therefore, a scale can be calibrated and marked on the tapered tube to read the fluid flowrate. Precautions: • Rotameter is calibrated for a certain fluid. Therefore, it must be

used for the fluid it is calibrated for. Different fluids have different densities; hence, a density change due to a different fluid in use would cause some change in the value of the annular velocity u, leading to error in the flowrate reading.


Q = A· u A is the annular area between the float and the tube. At a given flowrate Q, when u changes, A will changes, i.e. the vertical position of the float in the rotameter changes.

• Use the rotameter at the same temperature at calibration since temperature changes cause changes in fluid viscosity. This in turn might change the Reynolds number, and hence, the drag coefficient, Cd.

• Low accuracy: up to ±5% full scale III. OTHER FLOWMETERS 1. Hot wire Anemometer


A hot wire anemometer is mainly a wire that is heated electrically and placed in a flow stream (usually gas). The heat transfer rate from the wire to the flow stream is: ( )( )∞−+= TTubaq W

5.0 (21) where a and b are calibration constants, u is the gas velocity, TW is the wire temperature and T∞ is the temperature of the stream. Also: q = i2 Rw

When the gas velocity, u, increases, more heat loss from the wire to the gas stream. The current, i, to wire is increased automatically to maintain TW. The change in the supplied current, i, with the gas velocity can be calibrated to read the gas velocity. Another arrangement involves a connection of a hot wire anemometer to a Wheatstone bridge, the change in the resistance


of the wire (due to the cooling of the wire by the gas flow) leads to a finite bridge output voltage that can be calibrated to measure the gas velocity. Hot wire anemometer is very sensitive. Some may have time constant of 0.001 second (such as a platinum wire of 0.0001 inch diameter). Therefore, it is suitable for transient measurements. 2. Magnetic flowmeter

A magnetic flowmeter is based on the principle of the Faraday law of electromagnetic induction that states a voltage (emf) will be induced in an electrical conductor moving through a magnetic field. In flow measuring application, an electrically conductive fluid flowing through a magnetic field has an “emf” induced in it at right angles to the direction of flow. The induced emf can be picked up by the electrodes, in contact with liquid, on the wall of the pipe, and sent to a read out unit. The electrodes are insulated from the pipe carrying the fluid. The induced emf, E, is directly


proportional to the velocity of the fluid, u, as shown in the following equation: E = B·L·u·10-8 (22) where E is the emf induced (volt), B is the magnetic flux density (gauss), L (cm) is the length of the conductor (it is the diameter of the pipe in the case of flow measurement of a stream in a pipe), u is the liquid velocity (cm/s) Two types of magnetic flowmeters are available. One type is integrated with a non-conductive pipe section. This type is suitable for liquids with low electrical conductivities such as water (with trace of ions such as the hardness of water in city water). The electrodes are installed in the nonconductive pipe section and they are flush with the inside wall of the non-conductive pipe. Another type is for liquid with high conductivities. For this case, the meter can be used with metal pipe. The electrodes can be attached directly to the outside of the pipe and diametrically opposed to each other. Magnetic flow meters are not suitable for non-conductive fluids such as several organic solvents, gases, steam and most hydrocarbons. The liquid should have a minimal conductivity of 1.0 μS/cm (conductivity of pure water is 0.04 μS/cm). Accuracy of a magnetic flowmeter is ±0.5-1.0%. 3. Ultrasonic flowmeter There are two types of ultrasonic flowmeters: transit-time meter and ultrasonic Doppler meter. a) Transit-time meter: For the transit-time meter, the time the ultrasound travels from a sound transmitter to a sound receiver on the opposite side of a pipe changes with the fluid velocity in the pipe. The change in the travel


time can be calibrated to measure the fluid velocity in the pipe. The transit-time in a quiescent liquid is used as a reference. This type of meter is often used with plastic pipes but not for concrete pipes since the sound wave can’t penetrate the concrete wall. The typical accuracy is ± 1% full scale. b) Utrasonic Doppler meter: The transmitter passes ultrasonic wave through the flow with suspended solid or bubbles (it requires at least 25 ppm particles or gas bubbles of 30 μm diameter or larger). The sound wave reflected from the particles in the flow has a frequency shift that is related to the velocity of the particles. The receiver receives the reflected waves and sends them to a microprocessor where signals are processed and converted to the flowrate output (accuracy of ± 2-5% full scale). transmitter ultrasound beam flow ultrasound receiver Transit-time meter transmitter ultrasound beam flow bubble or particle receiver Ultrasonic Doppler meter


4. Turbine flowmeter:

A turbine flowmeter mainly consists of a magnetized turbine in a flow path and a voltage pick-up coil on the outside wall of a pipe.


As fluid flows through the flowmeter, it makes the turbine to rotate and generate a voltage pulse every rotation. The voltage pick-up coil receives the voltage pulse and transfer it to a counter where the number of voltage pulses received per unit time can be translated to the flowrate of the fluid passing the turbine in the pipe as below:

KfQ = (23)

Where Q is the flowrate (gallon/s), f is the pulse frequency (cycle/s) and K is the flow coefficient of the meter (cycle/gallon). • Turbine flowmeter are quite accurate with the accuracy of

±0.5% over a large range of flowrates. • They are suitable for clean fluid, clean gas and steam. • They are not recommended for slurry, very viscous materials

and corrosive fluids.


(2.) flow measurement

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