1. THEORY

1.1. Introduction

Measuring the flow of liquids is a critical need in many industrial plants. In some operations,

the ability to conduct accurate flow measurements is so important that it can make the

difference between making a profit or taking a loss. In other cases, inaccurate flow

measurements or failure to take measurements can cause serious (or even disastrous) results.

1.2.Definitions Of Flow Meter

Flowmeter is a device that meters movement of fluid in a conduit or an open space. This fluid could be water,chemicals, air, gas , steam or solids.

1.3.Selection of flowmeters

The basis of good flowmeter selection is a clear understanding of the requirements of the

particular application. Therefore, time should be invested in fully evaluating the nature of the

process fluid and of the overall installation. Here are some key questions which need to

answered before selecting a flowmeter:

What is the fluid being measured by the flowmeter or flowmeters (air,water,etc…)?

Do you require rate measurement and/or totalization from the flow meter?

If the liquid is not water, what viscosity is the liquid?

Is the fluid clean?

Do you require a local display on the flow meter or do you need an electronic signal

output?

What is the minimum and maximum flowrate for the flow meter?

What is the minimum and maximum process pressure?

What is the minimum and maximum process temperature?

Is the fluid chemically compatible with the flowmeter wetted parts?

If this is a process application, what is the size of the pipe?

Summary characteristics of common devices are shown in Appendix 1.

1.4.Flowmeter Types

Numerous types of flowmeters are available for closed-piping systems. In general, the

equipment can be classified as differential pressure, positive displacement, velocity, and mass

meters.

1.4.1.Differential Pressure Meters

The use of differential pressure as an inferred measurement of a liquid's rate of flow is well

known. Differential pressure flowmeters are, by far, the most common units in use today.

Estimates are that over 50 percent of all liquid flow measurement applications use this type of

unit.

The basic operating principle of differential pressure flowmeters is based on the premise that

the pressure drop across the meter is proportional to the square of the flow rate. The flow rate

is obtained by measuring the pressure differential and extracting the square root.

Differential pressure flowmeters, like most flowmeters, have a primary and secondary

element. The primary element causes a change in kinetic energy, which creates the differential

pressure in the pipe. The unit must be properly matched to the pipe size, flow conditions, and

the liquid's properties. And, the measurement accuracy of the element must be good over a

reasonable range. The secondary element measures the differential pressure and provides the

signal or read-out that is converted to the actual flow value.

1.4.1.a.Orifices 

Orifices are the most popular liquid flowmeters in use today. An orifice is simply a flat piece

of metal with a specific-sized hole bored in it. Most orifices are of the concentric type, but

eccentric, conical (quadrant), and segmental designs are also

available.

In practice, the orifice plate is installed in the pipe between two flanges. Acting as the primary

device, the orifice constricts the flow of liquid to produce a differential pressure across the

plate. Pressure taps on either side of the plate are used to detect the difference. Major

advantages of orifices are that they have no moving parts and their cost does not increase

significantly with pipe size.

1.4.1.b.Venturi

Venturi tubes have the advantage of being able to handle large flow volumes at low pressure

drops. A venturi tube is essentially a section of pipe with a tapered entrance and a straight

throat. As liquid passes through the throat, its velocity increases, causing a pressure

differential between the inlet and outlet regions.

The flowmeters have no moving parts. They can be installed in large diameter pipes using

flanged, welded or threaded-end fittings. Four or more pressure taps are usually installed with

the unit to average the measured pressure. Venturi tubes can be used with most liquids,

including those having a high solids content.

1.4.1.c.Pitot tubes

Sense two pressures simultaneously, impact and static. The impact unit consists of a tube with

one end bent at right angles toward the flow direction. The static tube's end is closed, but a

small slot is located in the side of the unit. The tubes can be mounted separately in a pipe or

combined in a single casing.

Pitot tubes are generally installed by welding a coupling on a pipe and inserting the probe

through the coupling. Use of most pitot tubes is limited to single point measurements. The

units are susceptible to plugging by foreign material in the liquid. Advantages of pitot tubes

are low cost, absence of moving parts, easy

installation, and minimum pressure drop.

1.4.1.d.Variable-area meters

Often called rotameters, consist essentially of a

tapered tube and a float, Fig. 3. Although

classified as differential pressure units, they are, in reality, constant differential pressure

devices. Flanged-end fittings provide an easy means for

installing them in pipes. When there is no liquid flow, the float

rests freely at the bottom of the tube. As liquid enters the

bottom of the tube, the float begins to rise. The position of the

float varies directly with the flow rate. Its exact position is at

the point where the differential pressure between the upper

and lower surfaces balance the weight of the float.

Because the flow rate can be read directly on a scale mounted next to the tube, no secondary

flow-reading devices are necessary. However, if desired, automatic sensing devices can be

used to sense the float's level and transmit a flow signal. Rotameter tubes are manufactured

from glass, metal, or plastic. Tube diameters vary from 1/4 to greater than 6 in

Some other differential pressure meters are:

Flow tubes

Flow Nozzles

Elbow meters

Target meters

1.4.2.Positive-Displacement Meters

Operation of these units consists of separating liquids into accurately measured increments

and moving them on. Each segment is counted by a connecting register. Because every

increment represents a discrete volume, positive-displacement units are popular for automatic

batching and accounting applications. Positive-displacement meters are good candidates for

measuring the flows of viscous liquids or for use where a simple mechanical meter system is

needed.

Some positive-displacement meters are:

Reciprocating piston

Oval-gear meters

Nutating-disk meters

Rotary-vane meters

1.4.3.Velocity Meters

These instruments operate linearly with respect to the volume flow rate. Because there is no

square-root relationship (as with differential pressure devices), their rangeability is greater.

Velocity meters have minimum sensitivity to viscosity changes when used at Reynolds

numbers above 10,000. Most velocity-type meter housings are equipped with flanges or

fittings to permit them to be connected directly into pipelines.

Some velocity meters are:

Turbine meters

Vortex meters

Electromagnetic meters

Ultrasonic flowmeters

Mass flowmeters

Coriolis meters

Thermal-type mass flowmeters

1.4.4.Open Channel Meters

The "open channel" refers to any conduit in which liquid flows with a free surface. Included

are tunnels, nonpressurized sewers, partially filled pipes, canals, streams, and rivers. Of the

many techniques available for monitoring open-channel flows, depth-related methods are the

most common. These techniques presume that the instantaneous flow rate may be determined

from a measurement of the water depth, or head. Weirs and flumes are the oldest and most

widely used primary devices for measuring open-channel flows.

Some open channel meters are:

Weirs

Flumes

1.5. Measurement of Flowing Fluids

1.5.1.Bernoulli Equation

The Bernoulli equation applies to steady, incompressible flow along a streamline with no heat or work interaction. One form of the Bernoulli equation is

p1

ρ+ρ v1

2

2+gh1=

p2

ρ+ρ v2

2

2+gh2

wherep is the pressure,ρ is the density,v is the fluid velocity,g is the acceleration of gravity, and h is the elevation measured from an arbitrary datum. The subscripts 1 and 2 denote two positions along the streamline.

Deriving Bernoulli’s Principle Using Conservation of Energy

Bernoulli's Equation can be derived using the conservation of energy. There are two approaches to show how the fluid pressure is equivalent (Equals) to an energy per volume. In both approaches, we consider a rectangular volume with seven rigid sides and one movable side. The movable side is then allowed to move a little, but the pressure throughout this experiment is kept constant.

1.5.1.a.Algebra Approach

Pressure is the distance traveled determining the amount of work involved, and the change in volume. The work equals the amount of energy added to maintain the constant pressure.

P= FA

P= F .dA .d

P=WV

P= EV

1.5.1.b.Calculus Approach

Work is derived from the definitions of pressure and volume that are used to set up a definite integral equation, which is then integrated.

∆W=F .∆ d

∆ E=F .∆ d

∆ E=P . A . ∆d

∆ E=P .∆V

∫0

E

∆ E=∫0

V

P .∆V

E=P.V

Where P is static pressure in newtons per meter2, F is force in newtons, A is area in meter2, d is distance in meter, W is work in joules and E is energy in joules.

Since energy is conserved, the energy for a mass of fluid at point a equals the energy fort he same mass off fluid at point b. We therefore write the energy equation as

PaV +12mva

2+mgha=PbV + 12vb

2+mghb

where m is mass of fluid in kilograms, v is velocity of fluid in meters per second, g is gravitational acceleration in meters per second2 and h is heigt in meters. Dividing both sides by volume, and using a single density at both points, gives the Bernoulli Equation:

Pa+12ρ v a

2+ ρgha=Pb+12ρ v b

2+ ρghb

where ρ is density of fluid in kilograms per meter3.

1.5.2.Flow Measurements of Incompressible Fluids

1.5.2.a.Equations for Pitot-tube:

Bernoulli equation between point 1, where the velocity v1 is un disturbed before the fluid decelerates, and point 2 , where the velocity v2 is zero:

v12

2−v2

2

2+P1−P2

ρ=0(1)

Setting v2=0 and solving for v1,

v=Cp√ 2 (P2−P1 )ρ

(2)

Where v is the velocity v1 in the tube at point 1in m/s, P2 is the stagnation pressure, ρ is the density of the flowing fluid at the static pressure P1, and Cp is a dimensionless coefficient to take into account deviations that generally varies between about 0.98 and 1.0.

The value of pressure drop, ∆ P or (P2-P1) in Pa is related to ∆ h, the on the manometer, as follows;

∆ P=∆h (ρA−ρ ) g(3)

Where ρA is the density of the fluid in the manometer in kg/m3 and ∆ h is the manometer reading in m.

1.5.2.b.Equations for Venturi meter:

To derive the equation for the venturi meter, firction is neglected and the pipe is assumed horizontal. Assuming turbulent flow and the writing the Bernoulli equaion between points 1 and 2 for an incompressible fluid,

v12

2+P1

ρ=v2

2

2+P2

ρ(4)

The continuity equation for constant ρ is

v1π D12

4=v2π D2

2

4(5)

Combining equations (4) and (5) and eliminating v1,

v2=1

√1−(D2

D1)

4 √ 2 (P1−P2 )ρ

(6)

To account for the small friction loss, an experimental coefficient Cv is introduced to give,

v2=C v√ 2(P1−P2)

(1−(D2

D1)

4

)ρ

(7)

For many meters and a Reynoulds number >104 at point 1, Cv is about 0.98 for pipe diameters below 0.2m and 0.99 for larger size.

To calculate the volumetric flow rate, the velocity v2 is multiplied by the area A2:

flow rate=v2π D2

2

4(8)

1.5.2.c.Equations for Orifice meter:

The equation fort he orifice meter is smiliar to equation (7).

vo=Co

√1−¿¿¿¿

Where v0 is the velocity in the orifice in m/s. Do is the orifice diameter in m, and Co is the dimensionless orifice coefficent. The orifice coefficient Co is always determined experimentally.

1.5.2.d.Equations for Rotameter:

For a given flow rate , the equilibrium position of the float in a rotameter is established by a balance of three forces. First one is the weight of the float, second one is the buoyant force of the fluid on the float and the third one is the drag force on the float. The weight of the float acts downward, the buoyant force and the drag force upward.

For equilibrium;

−∆ P A f=FD=V f ρf g−V f ρg(10)

where; ∆ P : Pressure difference acting on float, FD: Drag force, g: acceleration of gravity, Vf: Volume of float, ρ f : Density of float, ρ: Density of fluid.

Applying the Bernoulli Equation,

m=CRa

√1−( aA )2 √ 2 gV f (ρ f−ρ )

A f(11)

where; A: cross-sectional area of rotameter, Af: cross-sectional area of float, a: flowing area of fluid(=A-Af).

1.5.3.Flow Measurements of Compressible Fluids

Strictly speaking, most of the questions that have been presented in the preceding part

of this section only to incompressible fluids, but practically, that may be used for all liquids

and even for gases and vapors where the pressure differential is small relative to the total

pressure. As this is the condition usually encountered in the metering of all fluids, even

compressible ones, the preceding treatment has extensive application. However, there are

conditions in metering fluids where compressibility must be considered.

As in the case of incompressible fluids, equations may be derived for ideal frictionless

flow and then a coefficient introduced to obtain a correct result. The ideal condition that will

be imposed on the compressible fluid is that the flow be isentropic, i.e, frictionless adiabatic

process (no transfer of heat). The latter is practically true for metering devices, as the time for

the fluid pass through is so short that very little heat transfer can take place. An expression

applicable to pitot tubes for subsonic flow of compressible fluids can be derived by

introducing the conditions at the upstream tip of the tube (that is, V2=0 and V1=0) in equation

below:

(eq.1)

Substituting the first expression for R and p in case of V2=0 from equation (2):

C’p = kR / (k - 1) ; C’v = R / (k - 1) and = P/ RT (eq. 2)

Where k is the ratio of specific heats at constant pressure and volume,

Doing so gives the pitot tubes:

(eq.3)

The static pressure P1 may be obtained from the side openings of the pitot tube from

regular piezometer, and the stagnation pressure Ps (= P2) is indicated by pitot tube itself. A

coefficient must be applied if the side openings do not measure the true static pressure.

Equation (3) does not apply to supersonic conditions because a shockwave would from

upstream of the stagnation point. In such a case a special analysis considering the effect of the

shock wave is required. To develop an expression applicable to compressible flow through

venturi tubes equation (1) is taken and combined with continuity to get:

(eq. 4)

This equation can be transformed into an equation for the actual weight rate of flow

through venturi tubes by introducing by introducing the discharge Cv and an expansion factor

Y. The resulting equation is:

(eq. 5)

Y is a dimensionless expansion factor, which is the function of P2/P1 and β that is equal to D2/D1

ratio.

In equation (5), Cv has the same value as for an incompressible fluid at the same Reynolds

number and 1 may be replaced by P1/RT1 if desired. Also the equation can be used for flow nozzles

and orifice meters, though for flow nozzles Cv should be replaced. Values of Y for k=1.4 is plotted in

the Figure 1:

Figure 1- Expansion factors