The Engineer’s Guide to DP Flow Measurement Rosemount Documen… · 1 – DP Flow Measurement 1...

The Engineer’s Guide to DP Flow Measurement

Table of Contents

1. Differential Pressure Flow Measurement ..................1

2. Fluids Basics and Concepts ......................................13

3. Theory of DP Flow ..................................................23

1 DP Flow Measurement

TO PIC PAGE

1.1 Introduction ............................................... 2

1.2 Objectives ................................................... 2

1.3 History of DP Flow ..................................... 2

1.4 Pressure ..................................................... 3

1.5 DP Flow 101 ............................................... 4

1.6 DP Flow Measurement Applications ......... 7

1.7 Flow Meter Installations ............................. 7

1.8 Alternate Flow Technologies ...................... 9

1.9 Summary .................................................... 9

1.3.2 Reynolds

Osborne Reynolds (1842-1912) is a second key researcher who contributed significantly to the theoretical development of DP Flow technologies.

Reynolds was not a physicist but a student of mechanics. His work began with the practical steamfitting of ships, though he progressed to an astonishing array of studies. Among them are the mechanism of the drag of ships in water, condensation of steam, propeller design, turbine propulsion design, and hydraulic brakes.

1 – DP Flow Measurement

2

1.1 INTRODUCTION

1.1.1 Differential Pressure Flow Measurement

Differential pressure flow measurement (DP Flow) is one of the most common technologies for measuring flow in a closed pipe. Flow rate of the fluid in the pipe is derived from the pressure differential between the upstream (high) side and downstream (low) side of an engineered restriction in the pipe.

There are many reasons for the wide usage of DP Flow technology:

• Its technology is based on well-known laws of physics, particularly around fluid dynamics and mass transport phenomena• Its long history of use has also led to the development of standards for manufacture and use of DP flow meters• Manufacturers offer a large catalog of both general and application-specific instrumentation and installation choices• Finally, DP Flow technologies can achieve high accuracy and repeatability

1.2 OBJECTIVES OF THIS BOOKThis handbook is intended to help engineers and process technicians bring all the elements of DP Flow together in a comprehensive reference. The book will present enough theory and technical background to provide a solid context for engineering, procurement, and configuration of DP Flow technologies. This includes the following:

• The necessary equations and calculations commonly needed for developing DP Flow systems• A full discussion of components commonly found in DP Flow systems in gas, liquid, and steam applications• Discussions of DP Flow instrumentation technologies found in primary elements and transmitters• Common uses including the challenges and points to consider with specific applications• Installation guidelines• Maintenance and calibration procedures


This book focuses on practical engineering problems and challenges. That said, the elements of theory found here are crucial to the successful engineering of most DP Flow solutions.

1.3 HISTORY OF DP FLOWFlow measurement in general began thousands of years ago. Ancient Egyptians made approximate predictions of harvests based on the relative level of spring floods of the Nile. Centuries later, as Romans engineered aqueducts to convey water into their cities for sustenance, baths and sanitation, the need to monitor steady flow became important. Operators used flow through an orifice or the welling of water over obstructions to roughly gauge flow rates. Marks on the walls of the flow stream, strength of the stream through the orifice or other methods gave a rough idea of flow rates.

Advancements toward measured, repeatable flow metering increased after Newton’s discovery of the law of gravitation in 1687. This concept enabled physicists and mathematicians to begin to formulate a broad range of theories and hypotheses around motion and force. These in turn helped develop a range of instruments that quantified flow volumes and rates.

1.3.1 Bernoulli

Swiss mathematician Daniel Bernoulli (1700-1782), whose study of hydrodynamics centered on the principle of conservation of energy, provided the first key breakthrough in the development of technologies for flow measurement.

Bernoulli strove to discover as much as possible about the flow of fluids and his work led to the development of what is known as Bernoulli’s Principle. This states that for a hypothetical fluid with no viscosity, an increase in the speed of the fluid creates a simultaneous decrease in the fluid’s potential energy.

Conservation of energy is the basis for DP Flow measurements. The Bernoulli principle says that the sum of all energy in the fluid flow—remains constant regardless of conditions. When the speed of the fluid increases, its static pressure and potential energy decrease, while its dynamic pressure and kinetic energy increase.

3

Bernoulli’s principle dictates that the total pressure within a system is equivalent to the summation of its dynamic and static pressures.

Mathematically in the simplest terms, this is expressed in Bernoulli’s equation:

Where:

q = Dynamic pressure

p = Static pressure

p0 = Total pressure

Much more detail on Bernoulli’s equation can be found in the next chapter. Bernoulli’s equation describes the conservation of hydraulic energy across a constriction in a pipe. It states that the sum of the static energy (pressure), kinetic energy (velocity), and potential energy (elevation) upstream and downstream of the constriction are equal.

In the years since Bernoulli, many follow-on expressions of Bernoulli-based equations have been developed. These capture the behaviors of a broad range of compressible and incompressible liquids in many types of applications.

Much has been developed from Bernoulli’s work. For example, flow over an airfoil, the mechanism of lift, harnesses Bernoulli’s principle in aircraft. Flow through a restriction, while named for another researcher, Giovanni Battista Venturi (1746-1822), exhibits Bernoulli’s principle in carbureted internal combustion engines, where the pressure drop across a venturi sucks gasoline into the air stream entering the engine.

(1.1)

Most famously, he studied the flow of fluids in pipes, and more specifically, the conditions under which the flow transitions from laminar flow to turbulent flow. Out of this was created the dimensionless Reynolds number (Re).

The Reynolds number quantifies the relation of inertialforces to viscous forces, thus:

Reynolds number quantifies the relative importance of these two types of flow forces in a given flow condition.

Because Reynolds number describes the flow regime of a fluid, under certain conditions it is central to designing and operating DP Flow meters. Specifically, Reynolds number can be applied as a constraint on the range of a flow meter’s applicability. Operating a flow meter outside of its Reynolds number range constraints can degrade accuracy.

1.4 PRESSUREThe most critical background concept in the domain of DP Flow is pressure.

Accurate measurement of liquid, gas, and steam pressure is basic to many industrial processes. A typical plant will use more pressure measurement and pressure control devices than all other types of measurement and control instruments combined.

1.4.1 What Is Pressure?

Pressure is the amount of force applied over a defined area.

The Pressure EquationThe relationship between pressure, force, and area isrepresented in the following formula:

(1.2)

(1.3)

Where:

P = Pressure

F = Force

A = Area

1 – DP Flow Measurement 1 – DP Flow Measurement

1.5 DP FLOW 101–THE ROOTS (SQUARED)1.5.1 What Is Flow?Flow theory is the study of fluids in motion. A fluid is any substance that can flow, and thus the term applies to both liquids and gases. Precise measurement and control of fluid flow through pipes requires in-depth technical understanding, and is extremely important in almost all process industries.

1.5.2 Key Factors of Flow Through Pipes

There are 5 factors that are key to flow:

1. Physical piping configuration2. Fluid velocity3. Friction of the fluid along the walls of the pipe4. Fluid density5. Fluid viscosity

Piping Configuration: The diameter and cross- sectional area of the pipe enables both the determination of fluid volume for any given length of pipe and is included in the determination of the Reynolds number for a given application.

Velocity: Depends on the pressure or vacuum that forces fluid through the pipe.

Friction: Because no pipe is perfectly smooth, fluid in contact with a pipe encounters friction, resulting in a slower flow rate near the walls of the pipe compared to at the center. The larger, smoother, or cleaner a pipe, the less effect on the flow rate.

Density: Density affects flow rates because the more dense a fluid, the higher the pressure required to obtain a given flow rate. Because liquids are (for all practical purposes) incompressible and gases are compressible, different methodologies are required to measure their respective flow rates.

Viscosity: Defined as the molecular friction of a fluid, viscosity affects flow rates because in general, the higher the viscosity more work is needed to achieve the desired flow rates. Temperature affects viscosity, but not always intuitively. For example, while higher temperatures reduce most fluid viscosities, some fluids actually increase in viscosity above a certain tempera-ture.

If a force is applied over an area, pressure is being applied. Pressure increases if the force increases, or the size of the area over which the force is being applied decreases.

Why Measure Pressure?Four of the most common reasons that process industries measure pressure are:

• Safety• Process efficiency• Cost savings• Measurement of other process variables

Safety: Pressure measurement helps prevent overpressurization of pipes, tanks, valves, flanges, and other equipment; minimizes equipment damage; controls levels and flows; and helps prevent unplanned pressure or process release or personal injury.

Process Efficiency: In most cases, process efficiency is highest when pressures (and other process variables) are maintained at specific values or within a narrow range of values.

Cost Savings: Pressure or vacuum equipment (e.g., pumps and compressors) uses considerable energy. Pressure optimization can save money by reducing energy costs.

Measurement of Other Process Variables: Pressure is used to measure numerous processes. Pressure transmitters are frequently used in a number of applications, including:

• Flow rates through a pipe• Level of fluid in a tank• Density of a substance• Liquid interface measurement

The square root of the differential pressure across a restriction in a pipe is proportional to flow. This is expressed mathematically as:

(1.4)

4

Where:

Q = Flow rate

∝ (signifies proportionality)

ΔP = Differential pressure

Reynolds Number: By factoring in the relationships between the various factors in a given system, Reynolds number can be calculated to describe the type of flow profile. This becomes important when choosing how to measure the flow within the system.

There are three different flow profiles that are defined by different Reynolds number regimes. Laminar flow, Reynolds number below 2000, is a smooth flow in which a fluid flows in parallel layers. It is usually characterized with low velocities, very little mixing, and sometimes high fluid viscosity. When a fluid’s flow profile has a Reynolds number between 2000 and 4000, it is in a transitional zone. A Reynolds number above 4000 is called turbulent flow. This is characterized by high velocity, low viscosity, and rapid and complete fluid mixing.

Best accuracy in DP flow metering occurs with turbulent flow, where Reynolds number is greater than 4000 (varies primary element to element). This is because in turbulent flow, the point at which the fluid separates from the edge of the flow restriction is more predictable and consistent. This separation of the fluid creates the low pressure zone on the downstream side of the restriction, thus allowing that restriction to function as the primary element of a DP meter. Depending on the type of restriction and design of the flow meter, the minimum pipe Reynolds number at which a specific meter should be operated can be considerably higher than 4000.

1.5.3 Flow Continuity

When liquid flows through a pipe of varying diameter, the same volume flows at all cross sectional slices. This means that the velocity of flow must increase as the diameter decreases and, conversely, velocity decreases when the diameter increases.

Volumetric flow equates to the volume of fluid dividedby time:

Where:

Q = Volumetric flow rate

V = Volume

t = Time

5

(1.5)

Figure 1.5.3.a - Graphical representation of the simplest representation of the flow law where Q1 = Q2.

Volume can be broken down to area multiplied by length:

Where:

A = Areas = length

Flow can thus be expressed as:

(1.6)

(1.7)

(1.8)

This can be further simplified, since length divided bytime yields velocity:

Substituting velocity for s/t:

This yields the simplest representation of equation.

(1.9)

Hence,

(1.10)

(1.11)

(1.12)

(1.13)


The derivation of flow continuity above describes the basic principle of energy conservation. The Bernoulli equation, which will be covered in more detail in Chapter 3, builds on this principle to define the energy conservation appropriate for flowing fluid.

1.5.4 The DP Flow meterDifferential pressure is the most common flow measurement methodology today. There are three important elements that are combined to create a differential pressure flow meter.

The primary element creates a pressure drop across the flow meter by introducing a restriction in the pipe. This pressure drop is measured by the secondary element, a differential pressure transmitter. The tertiary element consists of everything else within the system needed to make it work, including impulse piping and connectors that route the upstream and downstream pressures to the transmitter.

By creating an engineered restriction in a pipe, Bernoulli’s equation can be used to calculate flow rate because the square root of the pressure drop across the restriction is proportional to the flow rate.

There are some important cautions around DP flow metering, including (1) ensuring that impulse lines do not clog with particles or sludge; (2) orienting impulse lines correctly—they have to be sloped to prevent gas accumulation in liquid applications and liquid accumulations in gas applications; and (3) ensuring that periodic calibration does not degrade accuracy—avoided by the use of highly accurate calibration equipment.


P High

Flow

P Low

Figure 1.5.5.a - Pressure flow diagram showing how DP Flow works. As fluids pass the restriction from the high side, the restriction induces a pressure drop. Flow is then calculated from the pressure drop (DP) across the restriction.

Figure 1.5.4.a - An integrated DP flow meter.

1.5.5 Primary Element Types

There are many kinds of primary elements:

• Single hole and conditioning orifice plates• Single and multiple-port pitot tubes• Venturi tubes• Flow nozzles• Cones• Segmental wedges

1.5.6 Transmitter Options

There are two main types of pressure transmitters used to calculate flow using differential pressure. The first is the traditional differential pressure type, which only measures differential pressure, with no ancillary functionality. The second is the multivariable transmitter.

A multivariable transmitter is a differential pressure transmitter that is capable of measuring a number of independent process variables, including differential pressure, static pressure, and temperature. When used as a mass flow transmitter, these independent values can be used to compensate for changes in density, viscosity, and other flow parameters.

Although multivariable transmitters can be more expensive than traditional differential pressure transmitters, they eliminate the need for multiple devices at a single measurement point. This means fewer transmitters, less wiring, fewer process penetrations, and lower overall installed cost.

Multivariable transmitters, unlike traditional differential pressure transmitters, are capable of calculating mass flow, energy flow, volumetric flow, and totalized flow.

1.6 DP FLOW MEASUREMENT–APPLICATIONSDP Flow measurement allows the optimization of many different aspects of a process including the following:

• Product consistency• Production efficiency• Process variable control• Safety• Internal billing / allocation• Custody transfer

Product Consistency: Batch-based products depend on accurate proportions of ingredients—DP Flow helps ensure the accurate of delivery of liquids and gases.

Production Efficiency: Metering and measurement of flow are part of a broad range of process control variables related to efficiency, from batch control, to by-product scavenging, to emissions monitoring.

6 7

Figure 1.6.a - Utility monitoring is a major component of production efficiency.

1

23

Figure 1.7.1.a - The traditional DP Flow installation has separate primary element, top left, tertiary elements (impulse lines, valves, connectors, and manifold), and secondary element, the transmitter, center right).

Process Variable Control: Processes often include multiple variable inputs. Control over these variables, including flow rates, is key to quality production.

Safety: DP Flow helps prevent a broad range of threats to safety including overfilling, reactor control, and others.

Internal Billing & Resource Allocation: Tighter control over inventories and process rates contributes directly to profitability. For many sophisticated producers, internal billing around process costs directly impacts the bottom line.

Custody Transfer: Flow metering is the cash register for products sold by volume or weight. An accurate measurement on the dispensing side accounts for every drop and on the receiving side minimizes over-charging.

1.7 Flow meter INSTALLATIONS: TRADITIONAL VERSUS INTEGRATEDSensor and process instrumentation in general has seen a great deal of integration of both form and function over the last two decades with DP Flow being no exception.

At this point in time, there are two broad types of DP flow meters installations: traditional and integrated.

1.7.1 Traditional

The Traditional Installation Method calls for threeseparate component categories.

1. Primary element (differential pressure producer)2. Secondary element (transmitter)3. Tertiary elements (impulse lines, connecting hardware, tubing, fittings, valves, etc.)


2 3

1

5

6

4

8

9

10

7

2

1 5

36

4

8

9

10

7

Figure 1.7.3.a - Traditional DP Flow structure. See page 17 for call outs.

Figure 1.7.3.b - Integrated multivariable instrumentation.

In Figures 1.7.3.a and 1.7.3.b:

1. Flow Computer2. Primary Element3. Thermowell4. Temperature Sensor5. Temperature Transmitter6. Sensor Wiring7. Pressure Transmitter8. DP Transmitter9. Manifold

10. Connection Hardware

1.8 ALTERNATE FLOWTECHNOLOGIESFlow measurement can be performed with a broad range of technologies other than pressure-based. These include open channel, mechanical, ultrasonic, electromagnetic, Coriolis, optical, thermal mass, and vortex types.

Electromagnetic flow meters, which require an electrically-conductive fluid and a means for inducing magnetic energy to the flow, use electrodes to sense current induction from the magnetic flux.

Coriolis flow meters, as the name implies, use the Coriolis effect, which induces distortion in a vibrating tube.

8 9

The traditional form enables component-by-component engineering to meet a wide variety of applications and can be engineered to meet custody transfer standards.

Owing to its long history, many traditions around DP-based flow measurement have emerged. Some of these traditions, however, have led to inherent limitations or problems. These include multiple potential leak points at connectors, separate /incorrect piping and manifolding; accuracy problems traceable to long impulse lines. In addition, installation is complex, requiring long straight runs (dependent on the primary element used) and careful configuration of components.

Much work has been done over the years to correct forsome of these issues, and thus extend the usefulnessand value of DP Flow installations.

1.7.2 IntegratedThe integrated flow meter integrates the primary element and the transmitter into a single flow meter assembly. It was in large part developed to minimize the issues around installations of the older-style traditional flow meter. As a result, its installation calls for components and less labor than traditional flow meters installations.

Figure 1.7.2.a - This integrated DP flow meter combines both the primary element and the transmitter into a single flow meter assembly, reducing potential leak points during installation and use..

The integrated flow meter works much the sameway as that of the traditional flow meter. It uses thesame equations, works largely with the same primaryelements, and is available with the same transmitters(both differential pressure and multivariable).

1.7.3 Benefits of the IntegratedFlow meterAn integrated flow meter design eliminates the need for fittings, tubing, valves, adapters, manifolds, and mounting brackets.

Compared to traditional installation, the benefits of integrated flow meters include:

• Fewer potential leak points (factory leak- checked)• Fewer flow measurement error sources• Simplified ordering and installation• Decreased susceptibility to freezing and plugging• More compact footprint

Rosemount integrated flow meters combine industry leading transmitters with innovative primary element technologies and connection systems. There are in effect 10 devices in one flow meter, simplifying engineering, procurement, and installation.

Optical flow meters use photodetectors to gauge the movement of particles in an illuminated fluid stream.

Vortex flow meters use electrical pulse generators—commonly a piezoelectric crystal—to measure flowdisturbances (vortices) around a calibrated obstruction.

Each of the various flow measurement technologies in existence today has its ideal range of applications. However, thanks to its long history, its ease of use, and its immense range of applicability, DP Flow remains the most commonly used form of flow measurement in industry.

1.9 SUMMARYDifferential pressure flow metering is the mostcommon technology for measuring flow in a closed pipe.

• The technology is based on well-known laws of physics, fluid dynamics and mass transport• Its long history of use has led to the creation of basic and practical engineering solutions for a broad range of applications• Manufacturers offer a large catalog of general and application-specific DP Flow instrumentation

1.9.1 History

Over the past centuries great strides have been made in the advancement of flow measurement. Two major of the major players in these advancements were Daniel Bernoulli and Osborne Reynolds.

• Decreased susceptibility to freezing and plugging• More compact footprint

Rosemount integrated flow meters combine industry leading transmitters with innovative primary element technologies and connection systems. There are in effect 10 devices in one flow meter, simplifying engineering, procurement, and installation.


Figure 1.9.3.a - The modern DP Flow meter which integrates the primary and secondary elements.

10 11

1.9.2 The Phenomenon of Pressure

The most crucial background concept in the domain of DP flow is pressure, the physical phenomenon that is harnessed to derive measurements. Accurate measurement of liquid, gas, and steam pressure is basic to many industrial processes—and, of course, specific to DP flow measurement.

1.9.3 DP Flow 101—The Basics

A DP flow meter consists of two major elements, a primary element, a restrictor in a pipe; and a secondary element, the differential pressure transmitter.

There are many kinds of primary elements:

• Orifice plates• Venturi tubes• Elbows• Flow nozzles• Single- and multiple-port pitot tubes• Cones• Segmental wedges

1.9.4 Applications of DP Flow Measurement

Process engineering and cost engineering are the two primary disciplines that exploit DP flow. The primary objectives include engineering for:

• Product consistency• Production efficiency• Process variable control• Safety• Internal billing / allocation• Custody transfer

1.9.5 Instrument Form Factors: Traditional Versus Integrated

There are two broad types of DP flow meters available, traditional and integrated.

The traditional form consists of three component categories.

1. Primary element (Differential pressure producer)2. Secondary element (transmitter)3. Tertiary elements (impulse lines, connecting hardware, tubing, fittings, valves, etc.)

The integrated form integrates the primary element and the transmitter into a single entity. See figure 1.9.3.a.

1.9.6 Alternate Flow Technologies

Flow measurement can be performed with a broad range of technologies other than pressure-based.These include:

• Open channel• Mechanical• Ultrasonic• Electromagnetic• Coriolis• Optical• Thermal mass• Vortex

Thanks to its range of usability and its critical mass of knowledge, DP flow remains the most-used form of flow measurement in industry.

2 Fluids Basics and Concepts

TO PIC PAGE

2.1 Introduction ................................................ 14

2.2 Force, Weight , and Mass ............................ 14

2.3 Density ........................................................ 15

2.4 Specific Gravity ............................................ 15

2.5 Pressure ....................................................... 15

2.6 Temperature ................................................ 17

2.7 Viscosity ...................................................... 18

2.8 Fluid Velocity ............................................... 19

2.9 Mass and Volumetric Flow .......................... 19

2.10 Isentropic Exponent .................................... 19

12

2 – Fluids Basics and Concepts

14


15

2.1 INTRODUCTIONThis chapter presents an overview of the physical properties of fluids, a review of basic concepts, and covers the fundamentals required to understand the theory of measuring flow, which is presented in Chapter 3.

A fluid is a substance that continues to deform when subjected to a shear stress. Fluids can be liquids, vapors, or gases. For most fluids, some of the fluid properties can be calculated by knowing other properties.

There are five key fluid properties that must be known to properly size and use a DP flow meter: density or specific weight, static pressure, temperature, isentropic exponent and viscosity. These properties factor into DP Flow calculations.

2.2. FORCE, MASS AND WEIGHT

Property Symbol Units

Mass m lbm , kg

Force F lbf , N

When working with fluids in motion, it is important to review the physics behind the concept of force, mass, and weight. The DP flow meter uses the concept of energy conversion to determine the rate of flow in a pipe by measuring a physical difference in pressures.

The US Customary unit of mass is the lbm and the SI unit is the kilogram. They are related by:

Force

Acceleration, a

Mass,m

Figure 2.2.a - Newton’s Second Law states force is equal to mass times its acceleration.

The SI unit of force is the Newton, N, and is the amount of force needed to move a 1kg mass at an acceleration of 1m/s2. So:

(2.1)

(2.2)

In SI units, gc = 1kg-m/(N-s2). Since the value of gc is 1, this factor is often left out of force calculations using the metric system.

In imperial US units, the unit of force is the pound force, lbf, and is the amount of force needed to move 1 pound mass, lbm, one foot at an acceleration of 32.174 ft/s2. Thus:

Where:

(2.3)

(2.4)

Newton’s Second Law of Motion describes the relationship between force, mass and acceleration as:

(2.1)

Where:

F = The force applied to or by an object

m = The mass of the object

a = The resulting acceleration of the object

To account for the force of gravity, a gravity conversion constant gc is needed. Thus, Newton’s second law becomes:

Weight of an object can be defined as:

Where:

W = The weight in force units

gl = The local acceleration of gravity

The value of earth’s gravity is slightly higher at the poles and slightly lower at the equator due to the earth’s rotation. At 45° latitude, gl = gc, and the force exerted by a Lbm is exactly a Lbf .

This concept is used to account for the weight of fluids in the energy equation (such as the elevation or “z” term in the Bernoulli flow equation), and in fluid statics when converting gravitational head to pressure, as is done with the manometer.

2.3 DENSITY


Density ρ lbm /ft2, kg/m2

The density of a fluid is its mass per unit volume. Note that for the same quantity of mass, the volume occupied by that mass will vary with temperature and pressure (see 2.9 Mass and Volumetric Flow). Fluctuations in density are typically small for liquids but is much greater for gases. Fluids whose densities change slightly with moderate temperature and pressure fluctuations are considered incompressible. If the density changes significantly with varying pressures and temperatures, it is considered a compressible fluid.

For industrial gas measurement applications it is common to use the Real Gas Law or the Ideal Gas Law. The Ideal Gas Law is applicable for moderate temperatures and low pressures. It fails to account for the interaction between gas molecules. When assumptions regarding the Ideal Gas Law do not apply the Real Gas Law is applicable. See Chapter 4 for more information on density and compressibility considerations.

2.4 SPECIFIC GRAVITYSpecific gravity (SG) is the ratio of density of one substance to the density of a second, or reference, substance. The reference substance for liquids is generally water at 68°F (20°C). The density of distilled water at 68°F is 62.316 lbm /ft3, or at 20°C, 998 kg/m3. Specific gravity thus provides a simple number that indicates whether a liquid is lighter or heavier than water.

Specific gravity of liquids is generally obtained with hydrometers, instruments whose scales read in specific gravities, degrees Baume (°B), or degrees API (American Petroleum Institute).

The reference fluid for specific gravity of gases is air. Specific gravity of a gas is defined as the ratio of the molecular weight of the gas of interest to the molecular weight of air (defined as 28.9644). This method avoids the difficulty of calculating the gas density based on temperature and pressure, as well as accounting for the non-linear behavior of gases. As long as the composition of a gas does not change, the ratio of molecular weight against that of a reference gas will remain the same regardless of temperature, pressure, or location.

2.5 PRESSURE


Pressure P psi, Pa

Pressure is the force acting on a surface in the normal direction (i.e., perpendicularly) per unit area (figure 2.2.4.a). The US unit of pressure is related by:

where the lb /in is force and not mass.

The SI unit of pressure is the Pascal and is equal to:

Where:

N = Newton and is in units of kg-m/s2


16


17

Force Area

Figure 2.5.1.a - Pressure is force acting perpendicularly on an area.

This is a difference of 0.08% between the two reference temperatures, thus it is important to know the reference temperature.

Differential pressure can be calculated with a simple relation:

The simplest of instruments used for the measurement of a small difference in pressures is the manometer A manometer uses a U-shaped tube or two vertical tubes connected at the bottom, with a liquid filling the tubes part-way (figure 2.5.2.b). When the two pressures are applied at the top of each tube, the liquid in the two tubes change height, and a scale fixed to the manometer is used to measure the height or elevation difference, h. The differential pressure is then calculated using the relation:

(2.6)

(2.7)

(2.5)

The Pascal is a very small unit of pressure and is generally expressed in kPa (kiloPascals) or MPa (megaPascals). Another common SI unit of pressure is the bar which is equal to 100 kPa.

2.5.1 Absolute and Gage Pressure

The absolute pressure is the pressure relative to a perfect vacuum. The gage pressure is the pressure relative to atmospheric pressure. Thus:

The absolute pressure is used to compute the density of gases. The gage pressure, since it is relative to atmospheric pressure, is used to ensure that pressure retaining parts (ie the pipe or parts of flow meters that retain the pressure when installed in the pipe) will remain within safe working limits.

Standard atmospheric pressure is typically defined as 1atm, or 14.69595 psi or101.325 kPa. Actual atmospheric pressure at any given location depends on that location’s elevation above sea level and day-to-day weather conditions. Typically, changes in the atmospheric pressure due to weather are not used when calculating absolute pressure. However local standard atmospheric pressure–adjusted for elevation –is used to determine the atmospheric pressure for purposes of measuring the flow rate.

The value of pressure for a flow meter application is used to provide information for two separate but important engineering tasks:

• For the calculation of fluid parameters– especially gas or vapor density and gas expansion factor• To check the compatibility and safety margins for the mounting hardware

2.5.2 Differential Pressure

When the difference in two pressures is needed, as called for in DP Flow calculations, it is called the differential pressure or DP (figure 2.5.2.a).

The SI unit for DP is Pa or kPa.

The US unit for DP is psi or inches of water (inH2O) at a specified temperature. The inches of water unit is a carryover from the past where manometers were used to measure flow rate and indicates the pressure at the bottom of a column of water of the specified height when the water is at a specific temperature. As an example a DP of 25 inches H2O at 68°F means that this is the pressure at the bottom of a column of water that is 25 inches high when the temperature of the water is a uniform 68°F. There are two commonly used versions of this unit inches H2O at 68°F (used in the US process control industry) and inches H2O at 60°F (used in the US natural gas industry). The conversion factors in psi for each of these is:


Differential Pressure ΔP or DP inH20 @ 68°F, inHg*, kPa, mbar

*inches of mercury; can also be inches of water, alcohol, oil or other fluid

Figure 2.5.2.a - Differential pressure. P1 has higher pressure than P2.

Where:

ΔP = Differential pressure, in appropriate units

gl = The local acceleration of gravity

gc = Units conversion constant from mass to

force

pm = The density of the manometer fluid

pf = The density of the fluid conveying the

pressure

h = Elevation difference or height of the fluid

at the points of measure

The standard manometer fluid for gas flow DP measurement is water, with the height indicated in inches, and the DP expressed as “Inches of Water Column.” Of course, if water or other liquids are being measured, a heavier manometer fluid is needed. Typically, mercury (S.G. = 13.5), or bromide-based fluids are used (S.G. = 2.5 to 3.0). Any fluid that is heavier than water could be used, but it must be “immiscible,” or unable to mix with the fluid in contact with the manometer.

P

P0Unknown Pressure

Ambient Fluid(Atmosphericpressure inmost cases)

Reference FluidDensity p (Liquid,e.g., water or mercury)

hFluid of Interest(Gas in most cases)

Figure 2.5.2.b - Example of how a manometer works.

Note that manometers and mechanical pressure gages are old technology. Current best practices exploit electronic differential pressure and static pressure transmitters (figure 2.5.2.c), which provide extremely accurate readings over very large ranges of pressure or DP, and can operate over a wide range of ambient temperatures without external correction. The electronic signal output is easily fed into microprocessors for calculating the flow rate or logging data.

Figure 2.5.2.c - Modern differential pressure transmitter.

2.6 TEMPERATURE


Temperature T °C, °F, °K, °R

1 More in-depth information can be found in The Engineer’s Guide to Industrial Temperature Measurement, Rosemount literature reference number 00805-0100- 1036 or go to Rosemount.com/temperature.

Standard orifice plate

Rosemount ConditioningOrifice Plate

Nozzle

Venturi

P1

P2

Outside of the engineering world, measurements are generally done on the relative Fahrenheit or Celsius scales, which were originally devised to measure the earth’s temperate range. However, flow engineering problems require a different temperature scale, one that represents an absolute temperature. Absolute temperature has units of Kelvin in SI units and Rankine in US customary units. The relationships between absolute temperature and the conventionally used units of °F and °C are:

The absolute temperature is required in the calculation of the fluid properties (i.e., density, viscosity and insentropic exponent). The calculation of thermal expansion effects involves temperature differences so it is common to use °F or °C.

Absolute zero Rankine and absolute zero Kelvin are equivalent. The Rankine scale increments by degrees Fahrenheit, while the Kelvin scale increments by degrees Celsius.


18


2.7 VISCOSITY

(2.10)

Every real fluid has viscosity, which changes primarily with temperature. For this reason, fluid viscosity is usually plotted against temperature, and equations are developed that allow the calculation of viscosity once the temperature is known. For a liquid, the viscosity decreases with temperature; for a gas, the viscosity increases with temperature.

Kinematic viscosity is the absolute viscosity divided by the density of the fluid at the same temperature, or:

Fluids are classified by the relationship between fluid stress (the force needed to overcome viscosity) and the strain (the fluid velocity). Figure 2.7.b shows the plot for different types of fluids based on the behavior of μ. DP flow meters are restricted to “Newtonian” type fluids, or those where the slope of the fluid stress/strain curve (μ) is constant. Shear-thinning fluid viscosities decrease with increasing shear stress; examples include ketchup, lava, or polymer solutions and molten polymers. Shear thickening (with viscosity increasing as shear stress increases) include suspensions—corn starch in water for example. Bingham plastics do not flow until a critical stress yield is exceeded; examples of this type of fluid include toothpaste.

(2.11)

boundary plate (2D, stationary)

boundary plate(2D, moving)

shear stress,τ

velocity,ν

dimensionγ

δυδγ

gradient,�uid

Figure 2.7.a - Viscosity defines the resistance to movement of a fluid.

Shea

ring

stre

ss,τ

δυ

μ

δγRate of shearing strain,

Shear thickening

Shear thinning

Bingham plasticNewtonian

1

Figure 2.7.b - Fluid classification based on the behavior of viscosity, m.

Absolute viscosity defines the resistance to movement of a fluid (figure 2.7.a). It is a measure of the resistance of fluid molecules to velocity change due to shear stress. Stated differently, viscosity tends to resist one particle from moving faster than an adjacent particle. Absolute viscosity is defined as:

Velocity is not a fluid property, but can be used to predict the behavior of fluids in motion and will frame the application of DP flow meters. In general, velocity is the rate of change of an object’s position relative to a reference, and is equivalent to the specification of speed and direction of an object. As applied to fluid dynamics, velocity defines the speed of a particle of fluid with respect to a stationary reference such as a pipe. When a fluid flows around an object or through a pipe, the viscosity of the fluid creates a velocity profile. If there were no viscosity, the velocity of a flowing fluid in a pipe would be uniform across a pipe section. With the slightest of viscosity, however, shearing between adjacent fluid particles produces a non-uniform velocity profile in the pipe, with a velocity of zero at the pipe wall and maximum at the pipe centerline for developed flow.

Fluid flow in a pipe also defines a velocity field. The flow of fluids through pipes has been studied extensively and velocity fields can be predicted when the rate of flow and fluid properties are known.

459.67

(2.8)

(2.9)


Absolute Viscosity μ centipoise (μcp), lbm/ft-s, kg/m-s

Dynamic Viscosity ν centistokes

Where:

τ = The shearing stress in the fluid, or the

force required to move the fluid against a

surface per unit area

= The change in velocity or “strain” between

the wall or surface and the free-stream

velocity

2.8 FLUID VELOCITY


Velocity v ft/s, m/s

2.9 MASS AND VOLUMETRIC FLOW

Volumetric flow is measured in terms of volume, as the name implies, yielding how much volume is passing through a given area, as follows:


Volumetric Flow Rate Qv

1/hr, m3/hr

Mass Flow Rate Qm

lb/hr, kg/hr

(2.14)

(2.13)

(2.12)

Mass flow is dependent on density and the volumetric flow rate, as follows:

Volumetric flow and mass flow can be related by the following:

19

Industrial methods of measuring temperature are based on substances that change electrical resistance with temperature, such as RTDs (resistance temperature detectors), or thermocouples that generates a voltage at the junction of dis-similar metals that is based on temperature.

In other words, when the mass flow rate (units of mass/unit time) is divided by the density at reference conditions, the flow rate is equivalent to the volume the fluid would occupy if its pressure and temperature were adjusted from the flowing conditions to reference conditions. So if the flowing pressure is 146.96 psia, the temperature is 68 °F, and the reference conditions are 14.696 psia and 68°F when the mass flow rate is converted to standard volume flow rate (by dividing by the density of the fluid at reference conditions) the numerical value will increase by a factor of 10. The mass will remain the same, but in order for the pressure to be at 14.696 psia it must occupy 10 times the volume as it did under flowing conditions.


For a given quantity of liquid, mass does not change, but volume can with changes in pressure and temperature (figure 2.9.a).

The measurement of mass flow is preferred for most gases and liquids, while volumetric flow can be acceptable for stable liquids. Flow meters suitable for mass flow include multivariable DP flow meters or Coriolis meters. Volumetric flow meters include DP flow meters, turbine, vortex, magmeter, or variable area meters.

2.10 ISENTROPIC EXPONENTAs gases flow through a restriction in a pipe the density changes due to pressure changes. The expansion of the gas is assumed to be an isentropic process and the effect of the density changes on the flow rate can be determined theoretically for some obstructions or empirically for others. The relevant fluid property is the isentropic exponent of the gas, designated by k (sometimes designated by g) and primarily is a function of temperature. It is common practice to determine k at a nominal temperature and use this value for all flow rates. Typical values of k can range from 1.0 to 1.4.

40.9 gal

342 lb 342 lb

20̊ F 60̊ F

42.0 gal2.7%

0.0%

Figure 2.9.a - Mass does not change with fluctuations in pressure and temperature, but volume can. This illustration depicts an example of how much volume can change with temperature.

20 21

3 Theory of DP Flow

TO PIC PAGE

3.1 Introduction ............................................... 24

3.2 The Physics and Engineering of Fluids and Flow ..................................................... 24

3.3 Developed and Undeveloped Flow ............ 24

3.4 Reynolds Number ....................................... 25

3.5 The Bernoulli Principle................................ 27

3.6 The DP Flow Equation ................................. 32

3.7 Types of Area Meters .................................. 33

3.8 Averaging Pitot Tubes................................. 35

3.9 Things to Consider ...................................... 40

3.10 Summary .................................................... 40

22

3 – Theory of DP Flow 3 – Theory of DP Flow

3.1 INTRODUCTIONChapter 3 covers the theoretical and computational details for DP Flow. Its purpose is two-fold:

• Introduce industry users to some of the general aspects of fluid flow and specifically DP Flow technologies • Explain the underlying assumptions and approaches behind the engineering of Rosemount DP Flow products

Note that an in-depth understanding of the physical relationships that affect DP Flow are useful in helping technical personnel cover all the aspects in the engineering of a specific application, but is not required for the installation and daily operation of DP flow meters. What is important to understand is that there is a great deal of complexity that exists under the surface of even the simplest application.

3.1.1 Available Resources

There are many readily available resources that allow engineers to resolve complexities. Among these resources are the following:

• Application and sales engineering resources available from the vendor of a given product • Industry training and discussion by both experts and peers at user group and formal workshop sessions • Software toolboxes and utilities usually developed by vendors and designed to streamline the engineering of a given application • A large body of technical articles and books on the subject • Standards documents from standards organizations (ISO, ASME, AGA, etc)

The following chapters discuss the practical side of DP Flow — which technologies best serve a given class of application (gases, liquids, steam), insight into the hardware and software of available products (transmitters, primary elements), and considerations for installation and use.

(3.1)

3.2 THE PHYSICS AND ENGINEERING OF FLUIDS AND FLOWThe concepts used in DP Flow theory and calculations originate mainly in two areas of fluid mechanics: fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effects of forces due to fluid motion. The DP Flow equation is based on the conservation of energy, and applies to the measurement of almost every type of fluid found in industrial or commercial use.

The advantages of using DP devices to measure flow rate include the simplicity of the sensing system, the availability of many types of primary devices, the ability to verify the measurement, and the wide range of applications that are suitable for DP Flow.

The challenges for using DP flow meters are overcome by becoming familiar with the theory and operation of the devices used for DP Flow measurement.

3.3 DEVELOPED AND UNDEVELOPED FLOWWhen evaluating the performance of flow meters and in assessing their use in a potential application, the condition of the velocity profile at the plane of measurement should be considered. A flow rate is considered “developed” when the velocity profile does not change significantly as it travels downstream. DP Flow calculations assume there is a developed flow. Achieving developed flow requires either a sufficient length of straight piping, or devices installed upstream that remove excessive turbulence or “straighten the flow.” Since most flow meters are primarily tested in developed flows, the potential effects on the performance of a meter must be considered separately if the flow at the measuring plane is not developed.

Different types of flow devices are affected differently by underdeveloped flows. Underdeveloped flow can result from additional turbulence in a pipe caused by piping fittings and types of valves installed upstream of the measurement location. Because of this, manufacturers usually provide a chart that shows how the flow meter device should be installed to achieve the stated performance.

3.4 REYNOLDS NUMBERReynolds Number, is an important non-dimensional parameter used in fluid mechanics. It is defined as the ratio of the inertial force of a fluid to the viscous force. The Reynolds number allows modeling of a fluid flow so that specific operational characteristics can be indexed to a common value. Aerodynamicists use the Reynolds number to allow the determination of how an aircraft will perform based on much smaller models used in wind tunnels. For flow metering, the Reynolds number is used to define a universal measuring range for all types of fluids. This ability greatly simplifies the evaluation, sizing, and use of flow meters.

For flow through a pipe the Reynolds number is given by:

Where:

ρ = Fluid density

D = Pipe diameter

= Average fluid velocity

μ = Absolute (dynamic) velocity of the fluid

The flow through a pipe is characterized by ranges of the Reynolds Number. The identification of these ranges, or regimes, is the result of extensive studies by scientists and engineers researching the theory that fluids flowing in pipes go through a transition between low and high velocities. This transition causes a change in the velocity profile in a pipe, which greatly affects the dynamics of the fluid and the ability accurately to measure the flow rate.

The regime at very low Reynolds Numbers is referred to as “Laminar” flow, where the fluid remains in layers. The velocity increases consistently from the pipe wall to the pipe axis. The velocity profile for laminar flow is represented by a parabola. In this case, fluid viscosity plays a major role in driving the flow pattern to remain in steady layers. See Figure 3.4.a.

As the velocity increases, this laminar condition begins to change and the flow transitions. The layers breakdown into smaller eddies as the parabolic shape of the velocity profile begins to flatten. At higher velocities, the laminar region exists only at the wall and is very thin. The fluid throughout the rest of the pipe becomes turbulent. Although, the velocity profile flattens, the highest velocity is still at the center.

Figure 3.4.a shows the profiles for the two types of flow. In Reynolds number values, the regimes are as follows:

• Re <2000 = laminar flow • Re 2000 ≤ 4000 = transition flow • Re >4000 = turbulent flow

The turbulent regime covers the majority of the velocities seen for fluids used in industrial and commercial flow. It is rare that piping is sized such that flows to be read are in the laminar regime unless the fluid has a high viscosity.

Figure 3.4.a - Flow profiles of laminar and turbulent flow.

24 25

TurbulentRe > 4000

Laminar flowRe < 2000

averageV


The poise is the unit of measurement for absolute viscosity, also known as dynamic viscosity and is commonly measured using centipoise (cP) in US units and Pa·s in the SI system of measurement. To convert to the viscosity units shown above use:

(3.2)

Table 3.4.1.b - Units conversions.

While the density and viscosity can usually be found, the velocity is not typically on the specification sheet for a flow meter. Instead the desired minimum and maximum flow rates are commonly given. It is possible to calculate the Reynolds number using the flow rate rather than the velocity. We start with the area for a circular pipe or duct:

(3.3)

(3.4)

(3.5)

Standard Volume Flow:

(3.7)

(3.8)

(3.9)

Figure 3.4.2.a - Rosemount Annubar™ primary element remote-mounted in a non-circular duct.

1 For introduction, see Chapter 1, “Bernoulli.”2 Swiss physicist and mathematician, 1707-1783. He created much of the mathematical terminology and notation used today in addition to his work in mechanics, fluid dynamics, astronomy, and optics.

3.5 THE BERNOULLI PRINCIPLE1 In fluid dynamics, the Bernoulli Principle and the equations derived from it are a special form of the conservation of energy equation first described mathematically by Leonhard Euler2 in 1757. This principle is a collection of related equations whose forms can differ for different kinds of flow. The basic Bernoulli Equation for steady, incompressible flow is:

(3.10)

(3.11)

3.4.1 Calculating the Pipe Reynolds Number

The basic Reynolds number equation is described by a velocity, pipe ID, the fluid density, and viscosity. Since the Reynolds number is dimensionless, the units must be given in a consistent mass, volume/length, and time basis. The following base units are commonly used to calculate the pipe Reynolds number for US and SI units:

Table 3.4.1.a - Reynolds Number Base units.

Parameter US-CU Base Units SI Base Units

Velocity ft/s m/s

Density lbm /ft3 kg/m3

Diameter ft m

Viscosity lbm/ft•s kg/m•s

Volumetric Flow ft3/s m3/s

Mass Flow lbm/min kg/min

To convert from to Multiply by

cP kg/m•s 1 x 10-3

cP lbm/ft•s 6.7197 x 10-4

26 27

For units other than the base units, a conversion factor is needed. The following are the equations for converting to the pipe average velocity, and calculating the Reynolds Number:

Actual Volume Flow:

Where:

Qv = Actual volumetric flow rate

ρ = Fluid density

μ = Dynamic viscosity of the fluid

D = Pipe ID

Mass Flow:

Knowing

substitute 3.4 for V then simplify to base units:

Where:

Qm = Actual volumetric flow rate

ρf = Fluid density

(3.6)

Knowing


Knowing


Where:

DH = Hydraulic diameter

A = Duct wetted area

P = Duct wetted perimeter

H = Duct height (span)

W = Duct width

Where:

Qs = Standard volume flow rate

ρb = Density at standard conditions

3.4.2 Special Case: Non-Circular Ducts

For non-circular ducts (Figure 3.5.2.a), the hydraulic diameter is used in place of pipe diameter. This is defined as 4 times the cross-sectional area divided by the wetted perimeter. The equation is:

Where:

ρ = Density of the fluid

p = Pressure of the fluid

g = Local gravitational constant

z = Height above a datum

Vs = Velocity of the fluid in the streamline

This equation applies to a fluid that is moving along a “streamline” (denoted as “s”), a continuous path that is followed. All changes in the fluid will only occur along the streamline, and no fluid will flow out of or into the streamline. For the application of this concept to fluid meters, where the fluid is flowing in a conduit or pipe, the pipe is the streamline. For steady-state conditions with developed flow, this one-dimensional model is sufficient to describe the flow field in a pipe.

or


(3.11)

(3.12)

Figure 3.5.a - Typical energy and flow diagram for a restriction in a pipe.

To determine the flow rate in mass/time or volume/time the continuity equation must be used which assures that mass is conserved (Eqn 3.13):

For an incompressible fluid, For an incompressible fluid, p1 = p2, and the continuity equation becomes:

To reach the above form of the continuity or Bernoulli’s equation, some assumptions were made:

• Steady flow – the equations represent constant velocity flow • Negligible viscous effects – represents a fluid with perfect uniformity while flowing and a consistent flow profile. • No work is added to the system – Bernoulli’s equation was derived from an energy balance on a system boundary around the meter. • Incompressible fluid – it is assumed that density remains constant across the streamline. • Negligible heat transfer effects – the simplified Bernoulli energy balance excludes frictional effects which create local energy transfer in the form of heat.

(3.13)

(3.14)

3.5.1 Deriving the DP Flow Equation

Beginning with Bernoulli’s equation:

And re-writing where z = h for height, the equation becomes:

(3.12)

(3.15)

Figure 3.5.1.a - Categorization of the energy terms in the Bernoulli equation..

(3.16)

(3.17)

(3.14)

(3.18)

(3.19)

Equations 3.22 and 3.23 are the theoretical mass and volumetric flow equations based on the assumptions listed in section 3.5. These are not representative of real-world fluid interactions. As a result two correction factors were developed, the discharge coefficient Cd and the gas expansion factor Y1.

The discharge coefficient, Cd corrects for the following assumptions:

• No viscous effects • No heat transfer • Pressure taps at ideal locations

(3.20)

(3.21)

(3.22)

(3.23)

The Bernoulli Equation acts as the operating equation for DP Flow. It is the transfer function between the input: the flow rate and fluid condition, and the output: the differential pressure. The benefit of the Bernoulli Equation is that it is simple, well defined, and accepted in the engineering community as a viable method for measuring fluid flow.

For flow metering, the flow of fluid must be considered “steady-state,” meaning that there is no appreciable change in the rate or conditions while measurements are made. While these conditions might seem restrictive, in reality most fluid systems are designed to operate at a steady state with changes occurring slowly to prevent excessive pressure transients or vibration in the system.

For an energy balance, the assumption is that no heat is added to the system and no work is done to or by the system. Of course work is done to the fluid by pumps and fans. However when the system boundary is drawn around the meter, it is a good approximation to eliminate the system energy terms. In the applied form for fluid flow, the Bernoulli Equation represents an energy transformation at two points in the fluid flow stream. In this description, Point 1 records the higher pressure; point 2 records the lower pressure (Figure 3.5.a), due to a transformation of energy from potential (pressure) to kinetic (velocity) energy.

If Bernoulli’s Equation is applied to two points along the same stream line:

When an energy balance is applied around the area change or restriction within the pipe, the equation becomes:

The equation shows the sum of the energy terms going into the restriction at point 1 must equal the sum of the energy terms after the restriction at point 2.

28 29

When it’s assumed the fluid is flowing horizontally and there is no change in height, the potential energy terms are equal:

Combined with the continuity equation (Equation 3.15) and rearranged:

Assuming density is constant:

And with a circular pipe and a circular restriction (for example, an orifice plate),

To calculate a volumetric flow rate multiply both sides of the equation by the area of the restriction:

To calculate a mass flow rate, multiply both sides of the velocity equation by the density:

Re-writing Equation 3.14 and substituting Equation 3.18 and 3.19 for A1 and A2,

Combining with 3.17 and substituting in Equation 3.20, the resulting equation relates the velocity at the restriction to a differential pressure:

21 p, V , A

Z

1

1 Z2

P2

P1

(P P )1 2

1

p, V , A2 2

Constant Energy Line

Arbitrary Datum Plane

P1

P2

V1

+ + + +=12

12p pgh

1pgh

2

2 V2p2

Pressure Energy

Kinetic Energy

Potential Energy

Where:

D = Inside diameter of the pipe

d = Inside diameter of the restriction


30 31

The Gas Expansion Factor corrects for gas density changes as a gas flows through a restriction.

3.5.2 Beta Ratio

Instead of a “restriction,” this type of meter can be called an “area” meter, as the meter is based on a change in area. For convenience, the ratio “d/D” is called “Beta”, or “β.” The term “d4/D4” would then be β4. Area meters such as orifice plates or venturis are defined by Beta, and the result of calibrations is classified by the type of area meter. To further simplify the flow equation, the term for Beta replaces the diameter ratio:

(3.24)

(3.25)

The parameter: is defined as “E,” sometimes

called the velocity of approach, so that the equation simplifies to:

This is still the theoretical equation for incompressible flow, as it does not account for energy losses for a real fluid. When the discharge coefficient is added to the equation, it is called the “Actual Mass Flow Equation for an incompressible flow” and is:

(3.26)

3.5.3 The Discharge Coefficient (Cd)

The discharge coefficient is dependent on the Reynolds Number, and the value approaches a constant as the Reynolds number approaches infinity. A given meter type, Beta, and Reynolds number value will generate a unique discharge coefficient. Discharge coefficients are determined in the flow laboratory where an actual flow rate and the fluid conditions are known for a defined flow field. The following example illustrates how this factor is determined.

Figure 3.5.3.a - Flow lab set-up for one discharge coefficient data point.

However, during the same period, the weigh tank actually collected 607 pounds of water (Figure 3.5.3.a). This means that the discharge coefficient (Equation 3.31) for this orifice plate was 0.607 at the steady flow rate that was observed. This discharge coefficient represents just one data point on the graph in Figure 3.5.3.b.

(3.27)

Since the discharge coefficient for most primary elements varies with Reynolds number, this test is done over a range of Reynolds numbers to determine the Cd vs. Re curve, or the meter characteristic. For area meters the same curve is also determined for various beta ratios. This body of data characterizes the discharge coefficient over a wide range of possible flow conditions for a range of area ratios, or Beta. This will result in hundreds or thousands of data points depending on the extent of the parameters to be tested.

Once all of the data is collected, an equation can be developed to fit the curve of the data, as represented by the black line in Figure 3.5.3.b. This equation can then be used to predict the discharge coefficient of any geometrically similar primary element. In this way, the equation serves as a calibration constant so that primary elements of similar construction do not need to each be calibrated in a laboratory. The uncertainty for this variable can then be determined as shown by the orange dotted line in Figure 3.5.3.b. The uncertainty of a curve that is fitted to data is done using the Standard Estimate of the Error (SEE), which is the standard deviation of the data sample referenced to the calculated (curve) values.

Figure 3.5.3.b - The curve fit and uncertainty of the discharge coefficient data collected.

3.5.4 The Gas Expansion Factor (Y1)

The gas expansion factor is also derived from laboratory testing where a gaseous fluid (typically air) can be used to generate a known flow rate. The reason the incompressible fluid assumption does not hold true in actual flowing conditions is because as a gas flows through a restriction, there is a decrease in pressure which results in the expansion of the gas and a decreased density, so ρ1 ≠ρ2. With a lowered density, the velocity will be slightly higher than predicted by the theoretical flow equation.

Since:

(3.28)

This is shown in the below lab test case (Figure 3.5.4.a) using the discharge coefficient from Figure 3.5.3.b:

Figure 3.5.4.a - Flow lab set-up for one gas expansion data point using air flow.

In this example, the lab testing for the gas expansion factor determines that 90 lbs should be collected according to the theoretical equation and 54.1 lbs was actually collected. The 54.1 lbs represents the mass flow including the discharge coefficient and the gas expansion factor.

Thus:

Since the same pipe and beta ratio was used for this test, Cd= 0.607. Solving for Y1 is now possible.

For gas or vapor flows, the density is determined at the upstream tap. For liquid flows, Y1 = 1.000. Figure 3.5.4.b below shows a plot of the expansion factor vs. the ratio ∆P : Pabs for gases with a ratio of specific heats = 1.4 for the typical concentric, square-edged orifice plate. The gas expansion factor is plotted this way because the slight change in density is proportional to the percent change in line pressure.

(3.30)

54.1 lb

90.0 lb

=Cd Y1 =54.190.0

0.601

607 lb

1000.0 lb

Assume that an orifice plate is installed in a flow lab so that a steady flow of water can be collected in a weigh tank. A flow calculation based on the theoretical equation shows that a total 1000 pounds of water flowed through the orifice plate during the test period and was collected in the tank.

(3.29)c


32 33

Figure 3.5.4.b. - After data is collected a line can be fit and the uncertainty of the gas expansion factor determined.

Because the calculated flow rate depends on Cd and Y1 and these factors depend on the flow rate, it is necessary to re-calculate the flow rate, and then calculate new values for Cd and Y1, or iterate, until the difference between subsequent calculations is small.

3.6 The DP Flow EquationBy adding the discharge coefficient and gas expansion factor, the flow equation now can accurately calculate flow applications. Recall the theoretical mass flow equation:

Substituting:

And

Where:

Qm = Mass flow

N = Units conversion factor

CD = Discharge coefficient

Y1 = Gas Expansion Factor

E = Velocity of approach

d2 = Bore diameter of differential producer


ρ = Density

Based on Bernoulli’s equation and an energy balance (Continuity Equation), this is the basic flow equation that uses the physical fluid properties mentioned above. d2, E, are geometrical terms and are determined by the primary element geometry. Cd, Y1 are empirical terms and are calculated using the test-derived equations either for a fixed set of flow equation parameters, or when a microprocessor-based flow computer is used, are continuously calculated DP, and ρ vary with changing process conditions such as flow, temperature, and pressure.

3.6.1 Difference Between Empirical and Geometric Terms

As mentioned above d2 and E are geometric terms that change depending on the primary element geometry. Orifice plates, venturis, and flow nozzles are considered area-change primary elements. Averaging pitot tubes use a velocity calculated by measuring the stagnation pressure (see section 3.8 for details).

The Bernoulli equation is used to calculate the velocity at an area change in a pipe or conduit. This form is used for DP meters based on an area change, also called “area meters,” including orifice plates, flow nozzles, and venturi meters.

Figure 3.6.1.a - Types of area meters, also known as throated meters, include Conditioning and standard orifice plates, nozzles, and venturis.

Each meter will have different levels of energy loss, so the values of the discharge coefficient will be different. Figure 3.6.1.b shows the values of Cd for three throated primary elements plotted against the pipe Reynolds Number. When the primary meter calibration factor is plotted over the operating range, it is called the “signature curve” of the meter. Note that the venturi shown in the figure approximates the path taken by the streamlines of the flow. For this reason, there is little energy loss, so the value of the discharge coefficient is nearly 1.00. The nozzle has more energy loss as the streamlines separate from the walls, but the orifice has the most because it is an abrupt change in area that creates more turbulence in the fluid. (3.31)

(3.32)

The flow equation is simplified to become:

(3.33)

(3.34)

Figure 3.6.1.b - Discharge coefficient curves for three types of DP flow meters.

For a DP flow meter, there are two primary design drivers:

1. The geometry of the meter – including the pipe, the location and size of the openings to measure the DP signal (called the “taps”), and the condition of the components that make up the meter.

2. The discharge coefficients assigned to the appropriate meter geometry.

Primary elements such as the orifice plate or venturi shown in Figure 3.6.1.a have been tested and many standards have been created to establish the value of the discharge coefficients and design requirements for fabricating and installing an orifice plate. These efforts resulted in equations that were derived from series of calibrations for a range of pipe sizes and beta ratios to allow the calculation of the discharge coefficient. Different types of Cd prediction equations have been developed with varying degrees of success. Because the orifice plate is the simplest, least expensive and easiest to retrofit and maintain, it is the most widely used. Each primary element type has a slightly different flow coefficient, but the equation used to calculate the flow rate is the same.

3.7 Types of Area MetersThere are many design variations of the primary elements shown above. These variations allow the application of DP Flow to fluid and process conditions that would not be possible with standard designs. In every case, the modification to the standard design will use the same basic Bernoulli equation form, but with a modified discharge coefficient and expansion factor (for compressible flows). These types of primary elements include Rosemount Conditioning Orifice Plates, standard orifice plates, venturis, and nozzles. Please see Chapters 8 and 9 for more information on the types of primary elements available.

Standard orifice plate

Rosemount ConditioningOrifice Plate

Nozzle

Venturi

Gas

Expan

sion F

acto

r

0

0.95

0.96

0.97

0.98

0.99

10.025 0.05 0.075 0.1 0.125 0.15

PΔPabs

Uncertainty of theGas Expansion Factor

Dis

char

ge

Coef

fici

ent

Pipe Reynolds Number, RD

Area Meter Discharge CoefficientFor Venturi, Orifice Plate and Flow Nozzle

1,000 10,000 100,000 1,000,000 10,000,0000.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Venturi Tube

Orifice Plate

Flow NozzleOrifice Plate

Flow Nozzle

Venturi Tube


34 35

3.7.1 Standard Orifice Plates

As discussed in Chapter 1, the type of flow meter most often specified for industrial flow is the orifice plate (Figure 3.7.1.a). The diameter of the orifice bore or throat is less than the diameter of the pipe, creating pressure differential as it restricts flow. As with all differential pressure producing flow meters, the underlying theoretical principle for orifice flow meters is Bernoulli’s Equation, while the calculation of actual flow rates depend on the addition Cd and Y1 terms to the equation.

Note that the discussion in the next section is generic and applicable to square-edge concentric orifice plates. The discussion of other orifice geometries is in Chapter 8.

Figure 3.7.1.a - Standard orifice plate.

3.7.2 ISO, ASME, and AGA Standards Provide a Basis for Calculating Discharge Coefficients

Three major standards have been written to define the Cd and Y1 coefficients, as well as the detailed construction, installation requirements and uncertainty factors. They are ISO 5167 Parts 1-4 from the International Organization for Standardization; ASME MFC-3M from the American Society of Mechanical Engineers; and AGA Report No. 3 for Natural Gas and Hydrocarbon fluids from the American Gas Association.

Many independent test laboratories both public and private have contributed to the test data to correlate Reynolds Number, discharge coefficient, and gas expansion factor.

Each standards organization continuously examines the equation structure to be used for their standard, usually when more data becomes available or a new analysis of the original data has been completed. For example, the ASME MFC-3M committee in 2004 updated its equation structure to one virtually identical to ISO 5167.

3.7.3 Rosemount Conditioning Orifice Plates

The Rosemount Conditioning Orifice Plate design is an orifice plate with four bores (Figure 3.7.3.a.) The primary purpose of this type of orifice is to condition the flow being measured. This self-conditioning eliminates the need to install separate flow conditioners or the need for, in some cases, 40+ diameters of straight run after a flow disturbance dictated by the standards mentioned above.

Figure 3.7.3.a - Rosemount Conditioning Orifice Plate showing its characteristic four holes orthogonally arranged around the center.

The Rosemount Conditioning Orifice Plate (COP) technology is based on the same Bernoulli equation as is used for standard orifice plates. As a result, the Conditioning Orifice Plate follows the same general discharge coefficient versus Reynolds number relationship as ISO 5167 designed orifice plates with a slight shift in value depicted by the calibration factor (Fc).

The four holes in the plate are placed equally around the plate center. When a kinetic energy balance is done at the conditioning orifice plate and the continuity equation is applied, the result requires the rate of the flow through the four holes to be the same. This pattern forces a distribution of the flow through the holes, creating a consistent downstream dynamic even when the upstream fluid velocity distribution is highly asymmetric or cyclonic. Since most of the orifice DP signal is created downstream, the COP provides equivalent results when installed in very close proximity to typical piping components or long runs of straight pipe. This removes the requirement for a flow conditioner to provide high performance in short straight pipe runs.

Figure 3.7.3.b - Illustration of how the four holes in the Rosemount Conditioning Orifice Plate conditions irregular flow profiles to provide accurate flow measurement with little straight run.

Total Straight Pipe Run Diameters Upstream (in Pipe Diameters) Downstream (in Pipe Diameters)

1595 and 405C 2 2

ASME MFC 3M Up to 54 Up to 5

AGA Report Number 3 Up to 95 Up to 4.2

ISO 5167 Up to 60

Up to 7

Flow Conditioners

Pressure Taps Flange Taps Corner Taps D and D/2

Not Reqired. All three standards sometimes require flow conditioners to shorten required straight pipe run.

Complies with all three standards Complies with ASME and ISO. Corner taps not included in AGA Report Number 3 In development

O-Plate Thickness 2’’ to 4’’ 6’’ 8’’ to 20’’

Complies with all three standards Complies with ASME and ISO. Thicker than AGA Report Number 3 Complies with all three standards

Beta Area of 4 holes = Area of same β for standard oriface of all three standards.1

All other plate dimensions (including angle of bevel, bore thickness (e), etc.) Complies with all three standards

Surface Finish Complies with all three standards

Discharge Coefficient Uncertainty Follows ISO 5167.2

Expansion Factor Follows ISO 5167.

1 At Schedule Standard 2 Follows ISO 5167 with a bias shift – the bias is determined from previous test or is determined by lab calibration by request

Category 1595 and 405C Conditioning Orifice Plate Technology

Rosemount Conditioning Orifice flow meters follows the intent of three main standards, ISO 5167/ASME MFC 3M and AGA Report Number 3. See Table 3.8.3.I for details of compliance and deviations from the standards.

3.8 Averaging Pitot TubesPitot tubes calculate velocity by measuring the pressure created by the fluid impact of the fluid on the pitot tubes. If the pressure at the low static pressure tap is considered to be the pipe or conduit static pressure, the DP is called the dynamic pressure of the fluid. This form is used for the pitot tube, invented and first used by Henri de Pitot in 1784. The modern incarnation of the pitot tube is the Averaging Pitot Tube, or APT. The purpose of the APT is to measure the flow rate in a pipe or duct by measuring the velocity pressure over the diameter of the pipe and average the results.

Table 3.7.3.1 – Comparison of the Rosemount Conditioning Orifice Plate to single-hole concentric orifice plates.

Even Profile

Even Profile


36 37

Figure 3.8.a - Pressure points for a single point pitot tube.

The average velocity can be calculated from these sample values. The sampling locations provide positions in a pipe or duct so that the average velocity can be obtained.

The averaging pitot tube (APT) was developed to provide a faster method to obtain a velocity average. Figure 3.8.b depicts a Rosemount 485 Annubar averaging pitot tube. The main benefits of an APT over a traditional area meter such as a nozzle or an orifice plate are:

1. The APT can be installed through a pipe coupling which requires less welding and expense. 2. The APT can be “hot-tapped,” or installed while the pipeline is under pressure. 3. The APT creates a much lower permanent pressure loss than a typical area meter.

Primary element selection is covered in more detail in Chapters 8 and 9.

Figure 3.8.b - The Rosemount Annubar 485 averaging pitot tube design.

When compared to a single-point pitot tube, the following are the important distinctions for the Rosemount Annubar APT:

1. The velocity profile is sampled at the slots or holes in the front of the tube, which is installed across the pipe. This is equivalent to a “continuous pitot-traverse.” 2. The fluid that comes to rest (or stagnates) in front of each slot or hole creates a pressure that represents the velocity at that point in the velocity field. In addition, the opening at the front of the pitot tube must be perpendicular to the fluid velocity vector to achieve a proper stagnation pressure. 3. The pressure sensed at the top of the Annubar APT front chamber is the averaged stagnation pressure for the sampling slots or holes. 4. The rear chamber measures the pressure at the rear of the tube, or the suction pressure. This pressure will be below the pipe static pressure due to the fact that for real fluids in the turbulent flow regime, separation of the fluid from the element has occurred. This is advantageous because the DP signal is higher than that obtained with a standard pitot.

The pressure sensed at the top of the rear chamber is the average suction pressure. However in most cases, the pressure created behind the tube is nearly the same across the pipe diameter due to the span-wise vortex-shedding or separation of the flowing fluid from the element.

The sensor shape design of averaging pitot tubes varies greatly from manufacturer to manufacturer and can have a great deal of impact. Generally, sensor shapes such as the bullet shape, round, scalloped or ellipse shapes (Figure 3.8.d) will perform more poorly over a flow range, especially at lower Re numbers because the signal strength of the DP signal is weaker with no fixed separation point.

Figure 3.8.c - Depiction of how vortices are shed off the Rose-mount Annubar sensor. The Rosemount Annubar T-shape design has a flat upstream surface which creates a fixed separation point. This improves the performance over a wider flow range over other APT sensor designs as well as stabilize the low pressure measurement.

A pitot tube measures only a point velocity. Unless the velocity profile is known or the flow is considered well developed, a single velocity measurement will not represent an average velocity needed to calculate an accurate flow rate. A pitot traverse, a procedure which involves moving the pitot tube across the pipe or duct while taking samples, can be done to improve the accuracy of the measurement.

Figure 3.8.d - Various averaging pitot tube sensor shapes. The above shapes can result in weaker DP signal strengths due to lack of a separation point.

The Rosemount 485 Annubar T-shape has a flat upstream surface which creates a fixed separation point (Figure 3.8.e), resulting in a strong DP signal. Additionally, the T-shape design includes frontal slots (Figure 3.8.f) which capture more of flow profile for a more comprehensive averaging and higher accuracy. The fixed separation point also creates a stagnation zone (Figure 3.8.c) in the back of the T-shape, which stabilizes the low pressure measurement for overall less signal noise.

Figure 3.8.e - Cutaway of the Rosemount Annubar T-shaped sen-sor. Holes in the backside of the Rosemount 485 Annubar T-Shape average the low pressure measurement.

Low Pressure Ports

Bullet Shape Round Shape

Scallop Shape Scallop Shape

StagnationZone

StagnationZone

FLOW

Impact Pressure

Fluid Flow

Low (Static) Pressure Tap

High (Impact)Pressure Tap

Static PressureSensing Ports


38 39

Figure 3.8.f - The 485 Annubar T-Shape APT design includes fron-tal slots that average the high pressure side measurements

For more information on averaging pitot tubes please see Chapter 8, DP Flow Primary elements, or Chapter 9 for the Rosemount Annubar averaging pitot tube offering.

3.8.1 Averaging Pitot Tube Flow Equation

Recall Bernoulli’s equation assuming a horizontal pipe:

(3.35)

For pitot style DP meters, the velocity at the sensing port is stagnated, meaning the velocity V2, is actually zero simplifying the equation to:

And solving for velocity, V1:

(3.36)

(3.37)

Mulitplying the velocity by the cross-sectional area of the pipe the theoretical volumetric flow equation is obtained:

(3.38)

And multiplying by flowing density to obtain the theoretical mass flow equation:

(3.39)

Simplifying to:

(3.40)

Again the theoretical equations are based on the following assumptions:

• No viscous effects • No heat transfer • Incompressible fluid

For averaging pitot tubes the flow coefficient (K) corrects for the following assumptions:

• Negligible viscous effects • Negligible heat transfer • Pressure taps at ideal locations

The gas expansion factor corrects for the incompressible fluid assumption.

So the full flow equation for averaging pitot tubes becomes:

(3.41)

Where:

N = Units conversion factor and

K = Averaging pitot tube flow coefficient

Y1 = Averaging pitot tube gas expansion factor

D = Pipe diameter


ρ = Density

3.8.2 Flow Coefficient, K, for Averaging Pitot Tubes

The K factor has to be determined by extensive laboratory testing, similar to that of the discharge coefficient for orifice plates. Empirical equations have been created to calculate the K factor based on the test data. To calculate the K factor for an averaging pitot tube, it is common to start from a function of blockage. Blockage is the ratio of the area of the averaging pitot tube to the area of the pipe.

And substituting terms shown in figure 3.8.2.a (B is the blockage factor and is unitless)

(3.43)

(3.42)

Figure 3.8.2.a - Cross section of pipe with averaging pitot tube installed, showing terms of the blockage equation.

Frontal Slot

D

d

A A

Aπ

a

= x

4

a d D

= D2

Sensor Size Probe Width

1 0.59 in

2 1.06 in

3 1.935 in

Table 3.8.2.1 - Rosemount 485 Probe Widths for each available sensor size. Probe width is dimension d in Figure 3.8.2.a

Once the blockage is known, the K factor can be calculated.

For a blockage, B, ≤0.25 use the following K factor equation and C1 and C2 values from Table 3.8.2.2:

For a blockage, B >0.25 use equation 3.45 below and 3.8.2.2:

(3.45)

(3.44)

Table 3.8.2.2 - Constants for determining the flow coefficient for the Rosemount 485 Annubar primary element. Where C1, C2, and C3 are constants that are determined empirically based on the sensor width and shape. The values shown are applicable to the Rosemount 485 Annubar primary element.

Sensor Size C3

1 5.3955

2 —

3 —

C1

-1.515

-1.492

-1.5856

C2

1.4229

1.4179

1.3318


40 41

3.8.3 Gas Expansion Factor for Averaging Pitot Tubes

The gas expansion factor for averaging pitot tubes is calculated slightly differently than area meters such as orifice plates. It is a function of blockage, DP, static line pressure, and the ratio of specific heats. Again, this factor is determined by laboratory testing.

The equation for the gas expansion factor of the Rosemount Annubar primary elements is as follows – note this form requires the pressure and differential pressure to be in the same units:

(3.46)

Where:

Ya = Gas expansion factor for an averaging

pitot tube

Y1 = Adiabatic gas expansion factor

(0.3142329)

Y2 = Pressure ratio factor (0.09483556)

B = Blockage factor of averaging pitot tube


Pf = Static line pressure

γ = Ratio of specific heats

Note ∆P and Pf must be in the same units of measure for pressure so that Ya will be unitless.

3.9 THINGS TO CONSIDER 3.9.1 Computational Software

Flow computers are often used to calculate flow utilizing the variables from the DP Flow installation or other measurement points. Flow computers are configured to calculate the flow based on the fluid properties and installation specifics such as line size and process variables either from individual pressure and temperature measurements or a multivariable transmitter such as the Rosemount 4088 (see chapter 7 for details).

The other option is to utilize multivariable transmitters with the ability to calculate flow specifically the Rosemount 3051SMV. Rosemount Engineering Assistant is a PC-based software program used for configuring Rosemount MultiVariable™ devices with mass flow output. In addition to being able to configure and calibrate the device, Engineering Assistant also performs configuration of the mass flow equation inside the transmitter. This software makes setting up a compensated flow equation simpler than manually setting up the flow equation in the control system. This is because the configuration of the flow equation all happens within Engineering Assistant and the flow calculation is done with the transmitter. The user only needs to enter their basic flow meter and process information to configure their transmitter for fully compensated mass or energy flow.

Engineering Assistant can be used as a “Stand-Alone” Windows based program, or as a SNAP-ON to AMS (covered in detail in Chapter 7). The SNAP-ON version runs within AMS, while the stand-alone version can be run without an AMS installation.

A common error in DP Flow installations is performing a double square root, or taking the square root of the Differential Pressure in the flow equation in both the transmitter and in the control system. The square root should only be taken once, either in the control system or in the transmitter.

3.10 SUMMARY Chapter 3 has focused on the theoretical and computational details for DP Flow, with a two-fold purpose:

1. Introduce industry users to some of the many aspects of fluid flow in general and of DP Flow technologies specifically 2. Explain the underlying assumptions and approaches behind the engineering of Rosemount DP Flow products

Note that an in-depth understanding of the physical relationships that affect DP Flow are useful to help technical personnel to cover all the bases in the engineering of a flow measurement, but is not required for the installation and daily operation of DP flow meters. There are many readily available resources that allow engineers in charge of DP Flow projects to resolve complexities, including the following:

• Engineering expertise available from the vendor of a given product • Industry training and discussion by both experts and peers at user group and formal workshop sessions • Software toolboxes and utilities usually developed by vendors and designed to streamline the engineering of a given flow measurement • A large body of technical articles and books on the subject

3.10.1 The Physics and Engineering of Fluids and Flow

The concepts used in DP Flow theory and calculations originate mainly in two divisions of fluid mechanics: fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effects of forces due to fluid motion. The basic DP Flow equation is based on the conservation of energy.

3.10.2 Developed and Undeveloped Flow

The condition of the velocity profile at the plane of measurement is necessary for the evaluation of flow meters and their applications. A flow rate is considered “developed” when the velocity profile does not change significantly as it travels downstream. Achieving developed flow requires either a sufficient length of straight piping, or devices installed upstream that remove excessive turbulence or “straighten the flow.” Since most flow meters are primarily tested in developed flows, the potential effects on the performance of a meter must be considered separately if the flow at the measuring plane is not developed.

3.10.3 Reynolds Number

Reynolds number is an important non-dimensional parameter used in fluid mechanics. It is defined as the ratio of the inertial force of a fluid to the viscous force. The Reynolds number allows modeling of a fluid flow so that specific operational characteristics can be indexed to a common value.

3.10.4 The Bernoulli Principle

In fluid dynamics, the Bernoulli Principle and the equations derived from it is a form of the conservation of energy. It is in actuality a collection of related equations whose forms can differ for different kinds of flow.

The Bernoulli Equation acts as the operating equation for DP Flow — that is, the transfer function between the input: the flow rate and fluid condition, and the output: the differential pressure. The benefit of the Bernoulli Equation is that it is simple, well defined, and accepted in the engineering community as a viable method for measuring fluid flow.

3.10.5 Beta Ratio

Some DP flow meters are called “area” meters, because the flow calculation is based on a change in area. For convenience, the ratio “d/D” is called “Beta,” or “β.” Area meters such as orifice plates or venturis are defined by Beta.

3.10.6 Discharge Coefficient

The discharge coefficient is dependent on the Reynolds number. A given primary element, Beta, and Reynolds number value will generate a unique discharge coefficient. Discharge coefficients are determined in the flow laboratory where an actual flow rate and the fluid conditions are known. Once all of the data is collected, an equation can be developed to fit the curve of the data. This equation can then be used to predict the discharge coefficient of any geometrically similar primary element.


42 43

3.10.7 Gas Expansion Factor

The gas expansion factor is also derived from laboratory testing where a gaseous fluid (typically air) can be used to generate a known flow rate. The reason the incompressible fluid assumption does not hold true in actual flowing conditions is because as a gas flows through a restriction, there is a decrease in pressure which results in the expansion of the gas and a decreased density, so ρ1 ≠ ρ2. With a lowered density, this means the velocity will be slightly higher than predicted by the theoretical flow equation.

3.10.8 The DP Flow Equation

By adding the discharge coefficient and gas expansion factor, the flow equation now can accurately calculate flow rate. Refer to section 3.7 above to see how this equation is simplified to become the following:

(3.34)

There have been standards created to establish the value of the discharge coefficients as well as the design requirements for fabricating and installing each type of meter.

3.10.9 Types of Area Meters

There are many variations of the basic types of DP area meters, venturi, orifice plate, and flow nozzle. These different designs offer flexibility to allow the use of DP flow meters to various applications. In every case area meters will use the same basic Bernoulli equation form, but with a modified discharge coefficient, and expansion factor (for compressible flows). These types of primary elements include Rosemount Conditioning Orifice Plates, standard orifice plates, venturis, and nozzles. Please see Chapter 8 for more information on the available types of primary elements.

3.10.10 Standard Orifice Plates

The type of primary element most often specified in industry is the orifice plate. The diameter of the orifice bore is less than the diameter of the pipe, creating differential pressure as it restricts flow. As with all Bernoulli Principle-based differential pressure producing primary elements, the calculation of actual flow rates depends on Cd and Y1.

3.10.11 ISO, ASME and AGA Standards Provide the Means for Calculating Discharge Coefficients

Three major standards have been written to detail the Cd and Y1 coefficients, as well as the detailed construction, installation requirements and uncertainty factors. They are ISO 5167 Parts 1-4 from the International Organization for Standardization; ASME MFC-3M from the American Society of Mechanical Engineers; and AGA Report No. 3 for Natural Gas and Hydrocarbon fluids from the American Gas Association. Each standards organization continuously examines the equation structure to be used for their standard, usually when more data becomes available or a new analysis of the original data has been completed. For example, the ASME MFC-3M committee in 2004 updated its equation structure to one virtually identical to ISO 5167.

3.10.12 Rosemount Conditioning Orifice Plates

The Rosemount Conditioning Orifice Plate design is an orifice plate with four bores (refer to Figure 3.8.3.a). The primary purpose of this type of orifice is to condition the flow being measured within the area. This self-conditioning eliminates the need to install separate flow conditioners or the need for 40+ diameters of straight run after a flow disturbance. The Rosemount Conditioning Orifice Plate (COP) technology is based on the same Bernoulli equation as is used for standard orifice plates. As a result, the conditioning orifice plate follows the same general discharge coefficient versus Reynolds number relationship as standard orifice plates with a slight shift in value, depending on the Beta ratio.

3.10.13 Averaging Pitot Tubes

The purpose of the averaging pitot tube is to measure the flow rate in a pipe or duct by measuring the average differential pressure over the entire flow profile.

The Averaging Pitot Tube (APT) was developed to provide a faster method to obtain a velocity average. Figure 3.9.b depicts a Rosemount 485 averaging pitot tube. The main benefits of an APT are:

• The APT can be installed through a pipe coupling which requires less welding and material expense • The APT can be “hot-tapped,” or installed while the pipeline is under pressure • The APT creates a much lower permanent pressure loss than a typical area meter

3.10.14 Flow Coefficient K for Averaging Pitot Tubes

The K factor has been determined by extensive laboratory testing, similar to that of the discharge coefficient for orifice plates. Empirical equations have been created to calculate the K factor based on the test data. The K factor for an averaging pitot tube is a function of blockage and differs with differing types of meters. See Section 3.8.2 for full details.

3.10.15 Gas Expansion Factor for Averaging Pitot Tubes

The gas expansion factor for averaging pitot tubes is calculated slightly differently than area meters such as orifice plates. It is a function of blockage, DP, static line pressure, and the ratio of specific heats. Again, this factor is determined by laboratory testing. See Section 3.8.3 for details.

Standard Terms and Conditions of Sale can be found at: www.rosemount.com/terms_of_sale.The Emerson logo is a trademark and service mark of Emerson Electric Co.Rosemount and Rosemount logotype are registered trademarks of Rosemount Inc.All other marks are the property of their respective owners.

© 2015 Rosemount Inc. All rights reserved.

www.rosemount.com

Literature reference number: 00805-0100-1041 Rev AA March 2015

http://www.rosemount.com/terms_of_sale

http://www.rosemount.com

The Engineer’s Guide to DP Flow Measurement Rosemount Documen… · 1 – DP Flow Measurement 1...

Documents

Transcript of The Engineer’s Guide to DP Flow Measurement Rosemount Documen… · 1 – DP Flow Measurement 1...