Pipe Flow E-Book

Donald C. Rennels

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 474

Veröffentlichungsjahr: 2012

Beschreibung

Pipe Flow provides the information required to design and analyze the piping systems needed to support a broad range of industrial operations, distribution systems, and power plants. Throughout the book, the authors demonstrate how to accurately predict and manage pressure loss while working with a variety of piping systems and piping components. The book draws together and reviews the growing body of experimental and theoretical research, including important loss coefficient data for a wide selection of piping components. Experimental test data and published formulas are examined, integrated and organized into broadly applicable equations. The results are also presented in straightforward tables and diagrams. Sample problems and their solution are provided throughout the book, demonstrating how core concepts are applied in practice. In addition, references and further reading sections enable the readers to explore all the topics in greater depth. With its clear explanations, Pipe Flow is recommended as a textbook for engineering students and as a reference for professional engineers who need to design, operate, and troubleshoot piping systems. The book employs the English gravitational system as well as the International System (or SI).

Leseprobe

Table of Contents

Cover

Title page

Copyright page

Epigraph

PREFACE

NOMENCLATURE

PART I: METHODOLOGY

1 FUNDAMENTALS

1.1 SYSTEMS OF UNITS

1.2 FLUID PROPERTIES

1.3 IMPORTANT DIMENSIONLESS RATIOS

1.4 EQUATIONS OF STATE

1.5 FLUID VELOCITY

1.6 FLOW REGIMES

2 CONSERVATION EQUATIONS

2.1 CONSERVATION OF MASS

2.2 CONSERVATION OF MOMENTUM

2.3 THE MOMENTUM FLUX CORRECTION FACTOR

2.4 CONSERVATION OF ENERGY

2.5 GENERAL ENERGY EQUATION

2.6 HEAD LOSS

2.7 THE KINETIC ENERGY CORRECTION FACTOR

2.8 CONVENTIONAL HEAD LOSS

2.9 GRADE LINES

3 INCOMPRESSIBLE FLOW

3.1 CONVENTIONAL HEAD LOSS

3.2 SOURCES OF HEAD LOSS

4 COMPRESSIBLE FLOW

4.1 PROBLEM SOLUTION METHODS

4.2 APPROXIMATE COMPRESSIBLE FLOW USING INCOMPRESSIBLE FLOW EQUATIONS

4.3 ADIABATIC COMPRESSIBLE FLOW WITH FRICTION: IDEAL EQUATION

4.4 ISOTHERMAL COMPRESSIBLE FLOW WITH FRICTION: IDEAL EQUATION

4.5 EXAMPLE PROBLEM: COMPRESSIBLE FLOW THROUGH PIPE

5 NETWORK ANALYSIS

5.1 COUPLING EFFECTS

5.2 SERIES FLOW

5.3 PARALLEL FLOW

5.4 BRANCHING FLOW

5.5 EXAMPLE PROBLEM: RING SPARGER

5.6 EXAMPLE PROBLEM: CORE SPRAY SYSTEM

6 TRANSIENT ANALYSIS

6.1 METHODOLOGY

6.2 EXAMPLE PROBLEM: VESSEL DRAIN TIMES

6.3 EXAMPLE PROBLEM: POSITIVE DISPLACEMENT PUMP

6.4 EXAMPLE PROBLEM: TIME-STEP INTEGRATION

7 UNCERTAINTY

7.1 ERROR SOURCES

7.2 PRESSURE DROP UNCERTAINTY

7.3 FLOW RATE UNCERTAINTY

7.4 EXAMPLE PROBLEM: PRESSURE DROP

7.5 EXAMPLE PROBLEM: FLOW RATE

PART II: LOSS COEFFICIENTS

8 SURFACE FRICTION

8.1 FRICTION FACTOR

8.2 THE COLEBROOK–WHITE EQUATION

8.3 THE MOODY CHART

8.4 EXPLICIT FRICTION FACTOR FORMULATIONS

8.5 ALL-REGIME FRICTION FACTOR FORMULAS

8.6 SURFACE ROUGHNESS

8.7 NONCIRCULAR PASSAGES

9 ENTRANCES

9.1 SHARP-EDGED ENTRANCE

9.2 ROUNDED ENTRANCE

9.3 BEVELED ENTRANCE

9.4 ENTRANCE THROUGH AN ORIFICE

10 CONTRACTIONS

10.1 FLOW MODEL

10.2 SHARP-EDGED CONTRACTION

10.3 ROUNDED CONTRACTION

10.4 CONICAL CONTRACTION

10.5 BEVELED CONTRACTION

10.6 SMOOTH CONTRACTION

10.7 PIPE REDUCER: CONTRACTING

11 EXPANSIONS

11.1 SUDDEN EXPANSION

11.2 STRAIGHT CONICAL DIFFUSER

11.3 MULTISTAGE CONICAL DIFFUSERS

11.4 CURVED WALL DIFFUSER

11.5 PIPE REDUCER: EXPANDING

12 EXITS

12.1 DISCHARGE FROM A STRAIGHT PIPE

12.2 DISCHARGE FROM A CONICAL DIFFUSER

12.3 DISCHARGE FROM AN ORIFICE

12.4 DISCHARGE FROM A SMOOTH NOZZLE

13 ORIFICES

13.1 GENERALIZED FLOW MODEL

13.2 SHARP-EDGED ORIFICE

13.3 ROUND-EDGED ORIFICE

13.4 BEVEL-EDGED ORIFICE

13.5 THICK-EDGED ORIFICE

13.6 MULTIHOLE ORIFICES

13.7 NONCIRCULAR ORIFICES

14 FLOW METERS

14.1 FLOW NOZZLE

14.2 VENTURI TUBE

14.3 NOZZLE/VENTURI

15 BENDS

15.1 ELBOWS AND PIPE BENDS

15.2 COILS

15.3 MITER BENDS

15.4 COUPLED BENDS

15.5 BEND ECONOMY

16 TEES

16.1 DIVERGING TEES

16.2 CONVERGING TEES

17 PIPE JOINTS

17.1 WELD PROTRUSION

17.2 BACKING RINGS

17.3 MISALIGNMENT

18 VALVES

18.1 MULTITURN VALVES

18.2 QUARTER-TURN VALVES

18.3 SELF-ACTUATED VALVES

18.4 CONTROL VALVES

18.5 VALVE LOSS COEFFICIENTS

19 THREADED FITTINGS

19.1 REDUCERS: CONTRACTING

19.2 REDUCERS: EXPANDING

19.3 ELBOWS

19.4 TEES

19.5 COUPLINGS

19.6 VALVES

PART III: FLOW PHENOMENA

20 CAVITATION

20.1 THE NATURE OF CAVITATION

20.2 PIPELINE DESIGN

20.3 NET POSITIVE SUCTION HEAD

20.4 EXAMPLE PROBLEM: CORE SPRAY PUMP

21 FLOW-INDUCED VIBRATION

21.1 STEADY INTERNAL FLOW

21.2 STEADY EXTERNAL FLOW

21.3 WATER HAMMER†

21.4 COLUMN SEPARATION

22 TEMPERATURE RISE

22.1 REACTOR HEAT BALANCE

22.2 VESSEL HEAT UP

22.3 PUMPING SYSTEM TEMPERATURE

23 FLOW TO RUN FULL

23.1 OPEN FLOW

23.2 FULL FLOW

23.3 SUBMERGED FLOW

23.4 REACTOR APPLICATION

APPENDIX A: PHYSICAL PROPERTIES OF WATER AT 1 ATMOSPHERE

APPENDIX B: PIPE SIZE DATA

B.1 COMMERCIAL PIPE DATA

APPENDIX C: PHYSICAL CONSTANTS AND UNIT CONVERSIONS

C.1 IMPORTANT PHYSICAL CONSTANTS

C.2 UNIT CONVERSIONS

APPENDIX D: COMPRESSIBILITY FACTOR EQUATIONS

D.1 THE REDLICH–KWONG EQUATION

D.2 THE LEE–KESLER EQUATION

D.3 IMPORTANT CONSTANTS FOR SELECTED GASES

APPENDIX E: ADIABATIC COMPRESSIBLE FLOW WITH FRICTION, USING MACH NUMBER AS A PARAMETER

E.1 SOLUTION WHEN STATIC PRESSURE AND STATIC TEMPERATURE ARE KNOWN

E.2 SOLUTION WHEN STATIC PRESSURE AND TOTAL TEMPERATURE ARE KNOWN

E.3 SOLUTION WHEN TOTAL PRESSURE AND TOTAL TEMPERATURE ARE KNOWN

E.4 SOLUTION WHEN TOTAL PRESSURE AND STATIC TEMPERATURE ARE KNOWN

APPENDIX F: VELOCITY PROFILE EQUATIONS

F.1 BENEDICT VELOCITY PROFILE DERIVATION

F.2 STREET, WATTERS, AND VENNARD VELOCITY PROFILE DERIVATION

Index

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Rennels, Donald C., 1937–

 Pipe flow : a practical and comprehensive guide / Donald C Rennels, Hobart M Hudson.

p. cm.

 Includes bibliographical references and index.

 ISBN 978-0-470-90102-1 (cloth)

 1. Pipe–Fluid dynamics. 2. Water-pipes–Hydrodynamics. 3. Fluid mechanics. I. Hudson, Hobart M., 1931– II. Title.

 TJ935.R46 2012

 620.1'064–dc23

2011043325

Knowledge shared is everything.

Knowledge kept is nothing.

—Richard Beere, Abbot of Glastonbury (1493–1524)

PREFACE

This book provides practical and comprehensive information on the subject of pressure drop and other phenomena in fluid flow in pipes. The importance of piping systems in distribution systems, in industrial operations, and in modern power plants justifies a book devoted exclusively to this subject. The emphasis is on flow in piping components and piping systems where greatest benefit will derive from accurate prediction of pressure loss.

A great deal of experimental and theoretical research on fluid flow in pipes and their components has been reported over the years. However, the basic methodology in fluid flow textbooks is usually fragmented, scattered throughout several chapters and paragraphs; and useful, practical information is difficult to sort out. Moreover, textbooks present very little loss coefficient data, and those that are given are desperately out of date. Elsewhere, experimental data and published formulas for loss coefficients have provided results that are in considerable disagreement. Into the bargain, researchers have not accounted for all possible flow configurations and their results are not always presented in a readily useful form. This book addresses and fixes these deficiencies.

Instead of having to search and read through various sources, this book provides the user with virtually all the information required to design and analyze piping systems. Example problems, their setups, and solutions are provided throughout the book. Most parts of the book will be easily understood by those who are not experts in the field.

Part I (Chapters 1 through 7) contains the essential methodology required to solve accurately pipe flow problems. Chapter 1 provides knowledge of the physical properties of fluids and the nature of fluid flow. Chapter 2 presents the basic principles of conservation of mass, momentum, and energy, and introduces the concepts of head loss and energy grade line. Chapter 3 presents the conventional head loss equation and characterizes the two sources of head loss—surface friction and induced turbulence. Several compressible flow calculation methods are presented in Chapter 4. The straightforward setup of series, parallel, and branching flow networks, including sample problems, is presented in Chapter 5. Chapter 6 introduces the basic methodology for solving transient flow problems, with specific examples. A method to assess the uncertainty associated with pipe flow calculations is presented in Chapter 7.

Part II (Chapters 8 through 19) presents consistent and reliable loss coefficient data on flow configurations most common to piping systems. Experimental test data and published formulas from worldwide sources are examined, integrated, and arranged into widely applicable equations—a valuable resource in this computer age. The results are also presented in straightforward tables and diagrams. The processes used to select and develop loss coefficient data for the various flow configurations are presented so the user can judge the merits of the results and the researcher can identify areas where further research is needed.

Friction factor, the main element of surface friction loss, is presented in Chapter 8 as an adjunct to quantifying the various features that contribute to head loss.

The flow configurations presented in Chapters 9 through 14 (entrances, contractions, expansions, exits, orifices, and flow meters) all exhibit some degree of flow contraction and/or expansion. As such, they have been treated as a family; where sufficient data for any one particular configuration were lacking, they were augmented by sufficient data in another.

Elbows, pipe bends, coils, and miter bends are presented in Chapter 15. The intricacies of converging and diverging flow through pipe junctions (tees) are presented in Chapter 16. Pipe joints are covered in Chapter 17, and valve information is offered in Chapter 18. The internal geometry of threaded (screwed) pipe fittings is discontinuous, creating additional pressure loss; and they are covered separately in Chapter 19.

Part III (Chapters 20 through 23) examines flow phenomena that can affect the performance of piping systems. Cavitation, when local pressure falls below the vapor pressure of a liquid, is studied in Chapter 20. Chapter 21 provides a brief depiction of flow-induced vibration in piping systems; water hammer and column separation are investigated. Situations where temperature rise in a flowing liquid may be of interest are presented in Chapter 22. Flow behavior in horizontal openings at low flow rates is evaluated in Chapter 23.

The book’s nomenclature was selected so that it would be familiar to engineers worldwide. The book employs two systems of units: the English gravitational system (often called the U.S. Customary System or USCS) and the International System (or SI for Système International). Conversions between and within the two systems are provided in the appendix.

This book represents industrial experience gained working together at Aerojet General Corporation, Liquid Rocket Engine Test Division, and later, working separately at General Electric Company, Nuclear Energy Division, and at Westinghouse Electric Cor­poration, Oceanic Division. We are indebted to the many engineering colleagues who helped shape our experience in the field of fluid flow. We especially appreciate Dr. Phillip G. Ellison’s helpful comments and suggestions.

We acknowledge the understanding and support of our wives, Bel and Joan.

DONALD C. RENNELSHOBART M. HUDSON

NOMENCLATURE

a See Section 1.1 in Chapter 1, “Fundamentals,” for the treatment of these units. There are instances identified in the text where lbm is used instead of lbf to simplify formulas for use with the English system and SI.

ABBREVIATION AND DEFINITION

PART I: METHODOLOGY

PROLOGUE

Part I of this work consists of Chapters 1 through 7. These chapters, with the exception of Chapters 5–7, establish the basic “rules of the road,” so to speak.

Chapter 1, “Fundamentals,” discloses the systems of units that are used throughout the book, nomenclature and meanings of fluid properties, important dimensionless ratios, equations of state, and expositions of flow velocity and flow regimes.

Chapter 2, “Conservation Equations,” elaborates on the conservation equations, that is, conservation of mass, of momentum and of energy. The general energy equation, head loss, and grade lines are treated under conservation of energy.

Chapter 3, “Incompressible Flow,” expounds on how the particulars of incompressible flow (i.e., flow of liquids) became known through the breakthroughs of Julius Weisbach (head loss formula, 1845), Osborne Reynolds (the Reynolds number, 1883), and Ludwig Prandtl (boundary layers and the smooth pipe friction factor formula, 1904–1929). Johann Nikuradse’s artificially roughened pipe experiments provided data (1933) to flesh out Prandtl’s smooth pipe friction factor formula and Theodor von Kármán’s complete turbulence formula (1930). Discrepancies between Nikuradse’s artificially roughened pipe data and data on commercial pipe were resolved by Cyril F. Colebrook and Cedric M. White (1937). Colebrook published a semiration­­al formula for random roughness (1939) that is still used today.

Chapter 4, “Compressible Flow,” gives several ways to calculate head loss in compressible flow in pipes using approximate formulas derived from incompressible flow formulas. It culminates in giving theoretical formulas for compressible flow using either the Mach number or absolute pressure. While the formulas are complicated enough to resist explicit solution, ways are given to solve them by trial-and-error methods.

Chapter 5, “Network Analysis,” gives methods to solve distribution of flow in networks. Chapter 6, “Transient Analysis,” provides methods for solving flow problems whose flow rates are not constant. Chapter 7, “Uncertainty Analysis,” gives methods for estimating the probable error or uncertainty in predicting pressure drop and flow rate.

1

FUNDAMENTALS

In this chapter we consider the fundamentals concerning fluid flow systems, such as the systems of units employed in this work, the physical properties of fluids, and the nature of fluid flow.

1.1 SYSTEMS OF UNITS

This book employs two systems of units: the U.S. Customary System (or USCS) and the International System (or SI, for Système International). The latter is based on the metric system, a system devised in France during the French Revolution in the late 1700s, but uses internationally standardized physical constants. Conversions between the systems may be found in Appendix C.

The USCS is virtually indistinguishable from the English gravitational system. There is some confusion in regard to the differences. Some authors imply that in USCS the slug is basic and the pound is derived, while others hold that the pound is basic and the slug is derived. In the English gravitational system the latter is assumed. For general engineering use it does not matter which is basic, because both systems agree that there is the slug for mass, the pound for force, the foot for length, and the second for time. This is all that need concern us in this work. The SI, derived from the metric system and having a much shorter pedigree, is consequently much more standardized.

Much confusion has resulted from the use in both English and metric systems of the same terms for the units of force and mass. To help eliminate the ambiguity owing to this double use the following treatment has been adopted.

The equation relating force, mass, and acceleration is

 (1.1)

where F, m, and a are defined in the nomenclature. In SI the unit of mass, the kilogram, is basic. The unit of force is derived by means of the equation above and is given a unique name, the newton. Mass is never referred to by force units and vice versa. In the English gravitational system (which predates the USCS) and the USCS, a similar set of units is available and familiar to engineers, but it is not uniformly used. The unit of force, the pound, can be considered to be basic and the unit of mass derived by means of the relation above. It is often called the slug. While the slug is not often used, its insertion here need not pose any inconvenience. Where mass units are called for they may be easily obtained from the pound-force unit by the use of Equation 1.1. By use of these conventions any fundamental equation given in this book may be used with either SI or English units.

It should be noted that Equation 1.1 returns, in the English gravitational system, a mass with units of lbf-s2/ft. This is not easily recognizable, so the engineering community has somewhat arbitrarily chosen the term “slug” to name the mass instead of lbf-s2/ft. Similarly, in SI, the force that comes out of the equation has units of kg-m/s2, and this force has been given the name “newton.” The equation does not contain a factor that transforms lbf-s2/ft to “slugs” or kg-m/s2 to “newtons.” We knowingly or unknowingly assume that there is an implicit conversion factor that changes the names of these units. This factor for SI is N/(kg-m/s2)/(kg), and in the English gravitational system it is lbf/slug-ft/s2. If you call these conversion factors “Cg,” Equation 1.1 becomes:

or

The numeric value of the conversion factor is 1.000, so it does not change the number obtained, but only the name of the number. This may be the reason many writers subscript the g with a c to obtain gc, when a is the acceleration of gravity.

Unfortunately the modern engineer must deal with mixed units and nomenclature used in some current practice and remaining from past practice. Conversions are offered in Appendix C that can help the user to work with mixed units. (Some secondary equations are given in which the units are mixed for the convenience of users of the English gravitational system. These equations and the units they require will be clearly indicated in the text.) Appendix C gives the important base units and derived units used here as well as the most frequently used conversions between systems.

1.2 FLUID PROPERTIES

Understanding the subject of pressure loss in fluid flow requires an understanding of the fluid properties that cause it. The principal concepts of interest in pressure loss due to flow are pressure, density, velocity, energy, and viscosity. Of secondary interest are temperature and heat.

1.2.1 Pressure

Pressure: The force per unit area exerted by a fluid on an arbitrarily defined boundary or surface, usually the walls of the conduit in which the fluid is flowing, or its cross section. Pressures are measured and quoted in different ways. A picture of pressure relationships can be gained from a diagram such as that of Figure 1.1, in which are shown two typical pressures, one above, and the other below, atmospheric pressure.*

Absolute Pressure: The pressure measured with respect to a datum of absolute zero pressure in which there are no fluid forces imposed on the boundary.†

Atmospheric Pressure: The absolute pressure of the local atmosphere.

Standard Atmospheric Pressure: The absolute pressure of the standard atmosphere at mean sea level. Standard atmospheric pressure, or one atmosphere, is 14.696 lb/in2, 760.0 mm of mercury, 1.01325 × 105 N/m2 (pascals), or 1.01325 bar.

Barometric Pressure: A barometer is an instrument used to measure atmospheric pressure by using water, air, or mercury. Thus atmospheric pressure is often called barometric pressure.

Critical Pressure: The pressure of a pure substance at its critical state; where the density of the saturated liquid is the same as the density of the saturated vapor. At pressures higher than the critical, a liquid may be heated from a low temperature to a very high one without any discontinuity indicating a change from the liquid to vapor phase. Values of critical pressure for selected gases are given in Appendix D.3.

Differential Pressure: The calculated or measured difference in pressure between any two points of interest.‡

Gauge Pressure: The pressure measured with respect to local atmospheric pressure. This is the pres­sure read by the common pressure gauge whose detecting element is a coil of flattened tube. (Sometimes this pressure is relative to standard atmospheric pressure. The reader is advised to determine which datum is used in other works or information sources.)§

Total Pressure: The pressure resulting from a moving fluid being brought to rest isentropically (without loss), as, for example, against a blunt object. (The kinetic energy of motion is converted to pressure when the fluid is brought to rest.) Total pressure is also known as stagnation pressure and pitot pressure (see Fig. 1.2).

Static Pressure: The pressure in a moving fluid before it is brought to rest. A pipe wall tap samples static pressure (see Fig. 1.2).

Vacuum: A pressure below local atmospheric pressure; often expressed as a negative pressure with respect to standard atmospheric pressure.

Vapor Pressure: The absolute pressure of a pure vapor in equilibrium with its liquid phase.

FIGURE 1.1. Pressure relationships.

FIGURE 1.2. Total and static pressure.

1.2.2 Density

Mass Density: The amount of material contained in a unit volume, measured in terms of its mass.

Weight Density: The amount of material contained in a unit volume, measured in terms of the force (weight) standard gravity exerts on the contained mass.

Specific Volume: The volume occupied by a unit mass or weight of material. (Which means it must be inferred from the units used. In the English system they will be ft3/lb[force]; in SI they will be m3/kg[mass].)

1.2.3 Velocity

Velocity: The speed of motion of a fluid with respect to a uniform datum. In pressure drop considerations it is usually used loosely with no direction implied. However, in impulse-momentum con­siderations, direction is an essential part of the measurement.

Average Velocity: A derived speed of a moving fluid whose various regions are not moving at the same speed but which accounts for the mass flux over the cross section of interest through which the fluid is moving.

Local Velocity: The actual speed of a moving fluid at a particular point of interest.

1.2.4 Energy

Energy (Work Energy): A measure of the ability of a substance to do or absorb work. It is usually measured in foot-pounds or newton-meters. (Newton-meters is also known as joules in SI.) Energy may exist in five forms: (1) potential, owing to a substance’s elevation above an arbitrary datum; (2) pressure, which is a measure of a fluid’s ability to lift some of itself to a level above an arbitrary datum or propel some of itself to a velocity; (3) kinetic, which resides in a substance’s speed or velocity; (4) heat, which ultimately is a measure of the kinetic energy of the molecules of a substance; and (5) work. Work, in the case of fluid flow, is actually an effect of pressure moving some resistance. The work may be added to or subtracted from a fluid to change the status of the other four forms of energy. Pressure energy is sometimes called flow work because of its role in transferring work from one end of a conduit to another. Heat is considered separately below.

1.2.5 Viscosity

Viscosity: The resistance offered by a fluid to relative motion, or shearing, between its parts.

Absolute Viscosity: The frictional or shearing force per unit area of relatively moving surfaces per unit velocity for a unit separation of the surfaces. It is also called coefficient of viscosity and dynamic viscosity.

Kinematic Viscosity: Absolute viscosity per unit mass per unit volume of the flowing fluid. (A fluid’s kinematic viscosity is its absolute viscosity divided by its mass density.)

1.2.6 Temperature

Temperature: In most fluid flow problems, temperature will refer simply to warmth (or lack of it), such as is perceived by our sense of touch and will be used to establish other fluid properties such as density and viscosity. It is usually measured on a somewhat arbitrary scale. The English system commonly uses the Fahrenheit scale, devised by the fifteenth-century German physicist Gabriel Fahrenheit [1]. It is based on the lowest temperature he could attain with a salt and ice mixture (assigned a value of 0°F) and human body temperature (to which he tried to assign a value of 96°F). This did not work out well and he ended up assigning 32°F to the melting point of ice and 212°F to the boiling point of water. The SI temperature scale (the Kelvin scale) is an absolute scale using the centigrade degree. The centigrade scale was devised by Swedish astronomer Anders Celsius in 1742 and incorporated into the metric system adopted in France at the close of the French Revolution [1]. On this scale—officially called Celsius since 1948—the melting point and boiling point of water at standard atmospheric pressure were assigned values of 0 and 100°C respectively.

Absolute Temperature: Temperature measured from absolute zero. It was noted in the late 1700s by the French physicist Jacques Charles (1746–1823) that gases expand and contract in direct proportion to their temperature changes. On a suitably chosen scale their volumes are thus directly proportional to their temperatures. The extrapolated temperature of zero volume according to the kinetic theory of gases is also the point at which molecular activity—and hence heat content—vanishes. No lower temperature is possible and so this temperature is called absolute zero. Two temperature scales based on this zero point are in common use. One, utilizing the Fahrenheit degree, is called the Rankine scale; temperatures on this scale are marked °R. The other, utilizing the Celsius degree, is called the Kelvin scale and its temperatures are marked K. The temperature 0°F corresponds to 459.67°R, and 0°C is identical to 273.15 K. Absolute zero is thus −459.67°F or −273.15°C.

Critical Temperature: The temperature of a pure substance at its critical state, above which its gas phase cannot be liquefied by the application of pressure, because at the critical temperature the latent heat of vaporization vanishes and the liquid cannot be distinguished from the gas. Values of critical temperature for selected gases are given in Appendix D.3.

1.2.7 Heat

Heat (Heat Energy): Heat is the measure of thermal energy contained in a substance. In fluid flow problems generally only sensible heat (i.e., heat obvious to the sense of touch or yielding a change in temperature) is of interest. It can be measured in the same units as work energy and indeed is interchangeable with energy. Usually heat is measured in units related to the heat–temperature relationship of water. In the English system the unit is the British thermal unit (usually abbreviated Btu). In SI it is the joule. The conversion to mechanical energy is the mechanical equivalent of heat. Its value is given in Appendix C.1.

Specific Heat: The measure of the change of heat capacity of a unit weight or mass of a substance for a unit change of temperature. It is almost always expressed in heat units, that is, Btu or joule. The units of specific heat are thus Btu/lb-R (or Btu/lb-F) and J/kg-K (or J/kg-C).

1.3 IMPORTANT DIMENSIONLESS RATIOS

Researchers have devised many dimensionless ratios in order to describe the behavior of physical processes. The most important to us in analyzing pressure drop in fluid systems are described in the succeeding sections.

1.3.1 Reynolds Number

Named for the British engineer Osborne Reynolds (1842–1912), the Reynolds number is the ratio of momentum forces to viscous forces. It is extremely important in quantifying pressure drop in fluids flowing in closed conduits. It is given by:

 (1.2a)

 (1.2b)

1.3.2 Relative Roughness

This quantity, as with the Reynolds number above, is extremely important in finding pressure drop in fluids flowing in pipes. It is rarely, if ever, assigned a symbol; but for illustration here let it be called RR. It is defined as:

where ε is the absolute roughness of the pipe inner wall and D is the pipe inside diameter. (In practice it is usually just called ε/D.)

1.3.3 Loss Coefficient

The loss coefficient, or resistance coefficient, is the measure of pressure drop in fluid systems. It is defined as:

 (1.3)

where:

K

 = 

loss coefficient measured in

velocity heads

,

f

  = 

Darcy friction factor,

L

  = 

length of pipe stretch for which the resistance coefficient applies, and

D

 = 

inside diameter of the pipe stretch.

More will be said about f and K in Chapters 3 and 8.

1.3.4 Mach Number

Named for the Czech physicist Ernst Mach (1838–1916), the Mach number is the ratio of the local fluid velocity u to the acoustic velocity A. It is very useful in describing compressible flow phenomena. It is given by:

 (1.4a)

The average velocity V is usually substituted when the flow is in a conduit and the velocity profile is fairly flat. With this convention, the equation becomes:

 (1.4b)

1.3.5 Froude Number

The Froude number NFr specifies the ratio of inertia force to gravity force on an element of fluid. It is named for William Froude, an English engineer and naval architect (1810–1879), who, in the later half of the nineteenth century, pioneered in the investigation of ship resistance by use of models. The Froude number is used in the investigation of similarity between ships and models of them. In this role, it is defined as the ratio of the velocity of a surface wave and the flow velocity. Our interest is in its application to pipe flow where the pipe is not flowing full. In this context it is expressed as:

 (1.5)

where V is the characteristic velocity, g is the acceleration of gravity, D is the pipe diameter, and R is the pipe radius. The Froude number, unlike the Reynolds number, is independent of viscosity and so it applies to inviscid flow analysis.

1.3.6 Reduced Pressure

Reduced pressure, along with reduced temperature (described below), is useful in quantifying departures from the ideal state in gases. Reduced pressure is given by:

where P is the pressure of interest and Pc is critical pressure.

1.3.7 Reduced Temperature

As with reduced pressure described above, reduced temperature helps to reduce the state point of most gases to a common base, making it possible to quantify departures of most gases from the ideal equation describing the relationship between pressure, temperature, volume, and quantity of substance (the equation of state, described below). Reduced temperature is given by:

where T is the temperature of interest and Tc is critical temperature.

1.4 EQUATIONS OF STATE

This section presents various equations which describe the physical properties of fluids—principally the fluid’s density as a function of pressure and temperature.

1.4.1 Equation of State of Liquids

An “equation of state of liquids” is not commonly expressed. This is because in usual engineering fluid-flow problems, the volume properties of the liquid are scarcely affected by changes in temperature or pressure in the flow path. Where their properties are significantly affected it is customary (because it is easiest and sufficiently accurate) to break the problem into small enough segments wherein the properties may be considered to be constant. Where this approach is not satisfactory, as, for instance, when dealing with liquids at pressures above the critical pressure, equations of state of liquids are available in the literature. Attention is directed to the works by Reid et al. [2] and Poling et al. [3], produced a quarter-century apart, which reflect the growth in information available in the literature on this subject.

1.4.2 Equation of State of Gases

Because gases exhibit large changes in volume, pressure, or temperature for comparable changes in one or both of the remaining of these three important variables, it has been necessary to formulate a workable expression relating them. The expression is called the equation of state. Two-variable relationships were discovered by Robert Boyle (1627–1691) and by Jacques Charles (1746–1823) and Joseph Gay-Lussac (1778–1850), which were soon combined into the perfect gas law:

 (1.6a)

where m = mass of the gas, V is the volume, and R is the individual gas constant; or

 (1.6b)

where n = number of mols of gas considered and is the universal gas constant. (In the English system Eq. 1.6a is usually written PV = wRT, where w = weight, lb, and the R used is expressed in weight units.)

Equation 1.6 adequately describes real gas behavior when pressure is low with respect to the critical pressure and temperature is high with respect to the critical temperature. However, with increasing pressure or decreasing temperature, or both, this relation departs increasingly from real gas behavior. A coefficient can be added to account for the departure, called the compressibility factor:

 (1.7)

where z is a function of the temperature and pressure of the gas. Dutch physicist Johannes van der Waals (1837–1923) noted that when z is plotted versus reduced pressure, that is, actual pressure divided by the critical pressure, for constant reduced temperature, that is, actual temperature divided by the critical temperature, the plotted points for any given reduced temperature for most gases fall into a narrow band [4]. If a line is faired through each band for each reduced temperature, a chart called a compressibility chart is obtained. A plot of this kind was published by L. C. Nelson and E. F. Obert in 1954 [5]. An example is shown in Figure 1.3 [3].* Many attempts have been made to find an analytic function, an equation of state, to describe this behavior, with varying success. Most of these “real gas” equations of state are limited in range of applicability. Two particularly attractive equations (solutions for z), suitable for wide ranges of pressure and temperature, the Redlich–Kwong equation and the Lee–Kesler equation, are described in Appendix D. Scores more are described by Poling et al. [3]. The utility of these equations is illustrated in Chapter 4, “Compressible Flow.”

FIGURE 1.3. (a) Generalized compressibility factor. (b) Generalized compressibility factor—subcritical range. (c) Generalized compressibility factor—pressure range to Pr = 10.

1.5 FLUID VELOCITY

As stated in Section 1.2, velocity (so called; more accurately it would be called speed) is usually considered to be uniform over the cross section of flow. In reality, it is not. The fluid in contact with the conduit wall must be at zero velocity, and velocity ordinarily increases toward the center. The assumption of uniform velocity immensely simplifies fluid flow calculations. There is an inaccuracy introduced by this assumption, but, fortunately, it usually does not affect the confidence level of fluid flow computations. The inaccuracies can be quantified and will be considered in the follow­ing chapter.

Another assumption that is usually made is that the velocity is one-dimensional, that is, that radial components of flow velocities are inconsequential. Inaccuracies introduced by this assumption are small and are absorbed by the loss coefficients.

1.6 FLOW REGIMES

In the study of fluid flow it has long been recognized that there are two distinct kinds of flow or flow regimes. The first is characterized by preservation of layers or laminae in the flow stream. This kind of flow is called laminar or streamline flow. In cylindrical conduits the layers are cylindrical, the local velocities are strictly parallel to the conduit axis, and they vary parabolically in velocity from zero at the wall to a maximum at the center. The second is characterized by destruction and mixing of the layers seen in laminar flow, and the local motions in the fluid are chaotic or turbulent. This kind of flow is thus appropriately called turbulent flow. In circular conduits the axial velocity distribution is more nearly uniform than it is in laminar flow, although local velocity at the pipe wall is still zero. Laminar and turbulent flow velocity profiles are illustrated in Figure 1.4. Because their effects will be treated in the following chapter you need to know that these two types of flow exist.

FIGURE 1.4. Velocity profiles.

Notes

* In the English system of units, pressure p is expressed in pounds per square inch, or psi.

† Absolute pressure is often expressed as psia in the English system.

‡ Differential pressure is often expressed as psid in the English system.

§ Gauge pressure is often expressed as psig in the English system.

* Large charts of the compressibility factor are available. One is reprinted by Poling et al [3]. Where more precision is desired, a computer program, called MIPROPS, which calculates many fluid properties, including density, viscosity, entropy, and acoustic velocity, was published by the National Bureau of Standards (now the National Institute of Standards and Technology) and is available from the Department of Commerce.

REFERENCES

1. Graham, L., Heat, thermometry, in Encyclopedia Britannica, Vol. 8, 15th ed., Encyclopedia Britannica, 1978, Macropedia, p. 706.

2. Reid, R. C., J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, 1977, p. 81.

3. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, Properties of Gases and Liquids, 5th ed., McGraw-Hill, 2001.

4. Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill, 1978, pp. 4–17.

5. Nelson, L. C. and E. F. Obert, Generalized pvT properties of gases, Transactions of the American Society of Mechanical Engineers, 76, 1954, 1057.

FURTHER READING

This list includes books and papers that may be helpful to those who wish to pursue further study.

Asimov, I., Understanding Physics, Vol. 1. Dorset Press, 1966.

Bedford, R. E., Thermometry, in Encyclopedia Britannica, Vol. 18, 15th ed., Encyclopedia Britannica, 1978, p. 322.

Fox, R. W., P. J. Pritchard, and A. T. McDonald, Introduction to Fluid Mechanics, 7th ed., John Wiley & Sons, 2008.

Streeter, V. L. and E. B. Wylie, Fluid Mechanics, 7th ed., McGraw-Hill, 1979.

Munson, B. R., D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics, 3rd ed., John Wiley & Sons, 1998.

2

CONSERVATION EQUATIONS

This chapter will consider the equations for conservation of mass, energy and momentum, velocity profiles, and correction factors for momentum and energy. In general, the English gravitational system uses weight flow rate (), and the International System of Units (SI) uses mass flow rate ().

2.1 CONSERVATION OF MASS

The continuity equation is simply a statement that there is as much fluid flowing out of a system under consideration as there is flowing into it. It assumes that mass is conserved and that fluid is not being stored or released from storage within the system. The equations for weight rate of flow and mass rate of flow are:

 (2.1a)

 (2.1b)

When the continuity equation holds, the inlet flow rate is equal to the outlet flow rate, so that

 (2.2a)

 (2.2b)

These equations are expressions of the continuity equation.

In these equations it is customary to assume that the velocity profile is flat, that is, the velocity in the fluid flowing in a conduit is the same everywhere in the cross section. The velocity that accounts for all the weight flux (or mass flux) across the cross section of the conduit is the average velocity.

The velocity profile is, of course, not flat across the cross section! Does this assumption therefore cause an error in the continuity equation? No, because we use the same relation to define the average velocity as to determine the weight flux through the cross section. The same cannot be said, however, for the momentum flux or the energy flux as we shall discover in the next sections.

2.2 CONSERVATION OF MOMENTUM

The momentum equation is a statement that a fluid stream, as it relates to fluid flow when acted upon by external forces whose sum is not zero, must acquire a change in velocity. The amount of this force may be found by use of the momentum equation. It is thus an application of Newton’s second law of motion (Eq. 1.1).

Consider an axisymmetric reducing flow passage as illustrated in Figure 2.1. Assume that velocity distribution is uniform at any cross section of the stream tube. P1A1 is the axial force acting on the fluid in the control volume owing to absolute pressure P1 acting over area A1; P2A2 is the axial force owing to absolute pressure P2 acting over area A2; and F is the apparent residual force owing to the diminishing stream pressure acting over the axial projection of the outer control volume boundary and to the frictional resistance on the surface of the stream tube. The terms and are the entering fluid momentum and exiting fluid momentum, respectively.

FIGURE 2.1. Axisymmetric reducing flow passage.

The sum of these axial forces is:

The sum of the forces is equal to the change in the momentum of the fluid between the inlet and outlet of the control volume:

Combining the axial force equation with the change in momentum equations gives:

In this derivation, an axisymmetric stream tube shape was chosen so that only axial forces need be considered. Because both force and velocity are vector quantities, that is, they include both quantity and direction, the momentum equation can be written for each of the three orthogonal directions:

Usually a nonaxisymmetric stream tube lies in a single plane so that an analysis in two directions is sufficient. For the stream tube shown in Figure 2.2 the momentum equations become:

FIGURE 2.2. Nonaxisymmetric reducing flow passage.

The angle ψ describing the orientation of F is:

2.3 THE MOMENTUM FLUX CORRECTION FACTOR

Up to this point it has been assumed that velocity distribution in the fluid has been uniform across a plane normal to the direction of flow, when in fact it never is (Section 1.5). An assessment of the error incurred by this assumption in the momentum equa­tion is in order. The total momentum at a given cross section of the stream tube is, assuming a flat velocity profile,

where V is the average fluid velocity. In an infinitely thin cylinder centered on the pipe center, this becomes the following differential equation,

where u is the local velocity. If we integrate this differential equation over the total cross sectional area A where the fluid velocity is not uniform throughout, we will arrive at a value that is not equal to . We need to introduce a correction factor:

 (2.3)

where θ is the momentum flux correction factor. For an axisymmetric velocity distribution the mass flow is:

 (2.4)

where r is the radius from the center of the pipe to the local velocity. The momentum flux is given by:

 (2.5)

Combining Equations 2.3–2.5, we obtain:

or

 (2.6)

Ludwig Prandtl, Johann Nikuradse, and Theodor von Kármán, during the period from 1926 to 1932, determined an equation for the velocity profile in pipe flow. From that equation Robert P. Benedict [1] shows that the velocity profile can be expressed as*:

 (2.7)

The plot of this equation is shown in Figure 2.3. It will be seen that the slope of the curve is not zero at the pipe centerline. About this, Hunter Rouse [2] says that “[these equations] do not give a zero slope of the velocity distribution curve at the center line. This is a defect in the formulas, which, from a practical viewpoint, is nevertheless of little significance. The equations actually portray the true velocity distribution in the central region of the flow very well, although they were derived for the region near the wall.”

FIGURE 2.3. Plot of fully turbulent velocity profile for f = 0.024.

Street et al. [3] give the following formulas for velocity profile and the resulting average velocity:

V* is the “friction velocity,” , where τ0 is the wall shear stress and ρm is the mass density (in either the English gravitational system or SI). By combining these three equations the following equation is obtained*:

 (2.8)

When Equation 2.8 is evaluated and compared with Equation 2.7, the difference is scarcely discernible. With either of these equations, performing the indicated integrations and ratio in Equation 2.6, the momentum flux correction factor is found to be:

A plot of this equation (for turbulent flow) is shown in Figure 2.4.

FIGURE 2.4. Momentum flux correction factor versus friction factor (for turbulent flow).

With a friction factor of 0.04, θ is about 1.038. Because most friction factors encountered in engineering work are less than 0.04, the error attendant to assuming a flat velocity profile is therefore usually negligible. Laminar flow, however, is an exception. Here the velocity profile is parabolic, and performing the indicated integrations and subsequent divisions yields θ = 1.333, a value which cannot be ignored. Other exceptions occur where the velocity profile is badly distorted, such as at the efflux of a conical expander.

2.4 CONSERVATION OF ENERGY

The energy equation is of paramount importance in our mathematical model of fluid flow losses. It accounts for the various energy changes within a flow system, or a portion of interest, and enables us to formulate a mathematical relationship that will provide consistently accurate predictions of pressure drop within it. The energy equation presents few difficulties once these energies have been identified.

As its name implies, the energy equation rests on the law of conservation of energy. This law, when applied to the steady flow of any real fluid, states that the rate of flow of energy entering a system is equal to that leaving the system. Figure 2.5 shows a hypothetical flow system with the fluid properties and circumstances and the energy fluxes affecting the energy balance.

FIGURE 2.5. Energy fluxes.

In order to relate the energy inflows and outflows in a system it is necessary to put them in common units. It is convenient for this discussion to express energy in work units such as foot-pounds or newton-meters, and unit energies in terms of foot-pounds per pound of fluid, or newton-meters per newton. From Figure 2.5 it is seen that five kinds of energy flux must be considered: potential, pressure, kinetic, heat, and work.

2.4.1 Potential Energy

Every unit of fluid lifted above an arbitrary datum required a certain amount of work to lift it there. If the unit of fluid quantity is pounds (or newtons), the work required (in a uniform gravity field) is its weight times the height it was lifted, ft-lb (or N-m). Thus the unit energy is ft-lb/lb or ft (or N-m/N or m), equal numerically and dimensionally to its elevation Z above the datum. This is called the elevation or poten­tial head.

2.4.2 Pressure Energy

Pressure is commonly expressed as force per unit area—for example, lb/in2, lb/ft2, or N/m2 (pascals). If the fluid’s pressure is divided by its weight density, its potential for doing work is expressed in potential energy terms. Consistent units will eliminate mixed unit problems. Thus lb/ft2 and N/m2 yield:

As an example, a fluid under pressure P can be lifted in a manometer to a height P/ρw or P/ρm. This is called the pressure head.

2.4.3 Kinetic Energy

The simple equations of motion show that in the absence of air or other resistance any body dropped from one elevation to another lower elevation acquires a velocity equal to the square root of twice the product of the elevation difference and the acceleration of gravity, that is,

Conversely, any body moving with velocity V can, if the velocity can be directed upward, attain a height of:

 (2.9)

A fluid’s energy of motion is thus V2/2g ft-lb/lb or simply ft (or N-m/N or m). This is called the velocity head. The symbol is HKE.

In the hypothetical flow system shown in Figure 2.5 we might assume that every molecule of fluid is passing through the conduit, at any one cross section, at the same velocity. In such a case the fluid’s average velocity would be the same as that of any particle of the flow, and its kinetic energy would be accurately described by Equation 2.9, where V is the fluid’s average velocity. A real fluid, however, never flows in quite this fashion. At the wall of the conduit its velocity always approaches zero and it increases to a maximum at the center of the conduit for fully developed flow. The kinetic energies of its parts vary depending on their locations in the cross section. Because the square of the average is not the same as the average of the squares, a correction factor ϕ must be included if the average velocity is used to calculate the kinetic energy of the flowing fluid:

The correction factor will be treated in more detail in a later section, but suffice it to say now that ϕ is required to measure precisely the kinetic energy of the fluid.

2.4.4 Heat Energy

The English physicist James Prescott Joule (1818–1889) showed conclusively in experiments conducted between 1843 and 1850 that heat is equivalent to work. The physical constant relating the two is denoted here by the symbol J. To convert common heat units (Btu/lb or kcal/kN) to specific work units (ft or m) the heat units are multiplied by J in the proper units. Because transferred heat flux Q is usually calculated in heat units and the energy equation is usually set up with work units, it is convenient to convert the heat units to work units:

The units in the foregoing expression, now in work units per unit time, must be further converted to potential energy units:

 (2.10a)

 (2.10b)

Internal heat energy, that is, heat energy possessed by the fluid upon entering the flow system or leaving it, like transferred heat, is usually expressed in heat units; but unlike transferred heat it is treated on a per-unit-weight basis or a per-unit-mass basis. (For this discussion let us continue to treat the individual terms of the general energy equation on a per-unit-weight basis.) Internal heat energy, or simply internal energy, denoted by the symbol U, is converted to potential energy units as follows:

2.4.5 Mechanical Work Energy

The mechanical work done on the fluid in the flow system by a pump and, as in the case of heat flux, the work done by the fluid in a turbine must be expressed in power units, or work per unit time, to maintain dimensional homogeneity in the energy equation. These units may be converted to potential energy units as they were in the case of heat flux (Eq. 2.10a and 2.10b):

The same conversion also applies to turbine work, ET.

The mechanical work energy is often called “flow work,” because without flow there is no work performed. In the case of the pump, flow work is added to the flow, and in the case of the turbine, flow work is subtracted from the flow.

2.5 GENERAL ENERGY EQUATION

Having defined the energy fluxes in the hypothetical flow system in common units, we may now write the energy balance:

 (2.11a)

Equation 2.11a is set up for weight units in either the English gravitational system (where lbf is basic) or SI (where the kilogram mass is basic) but using newtons as the force unit. For SI in mass units, the equation is:

 (2.11b)

As shown in Chapter 1, the units of ρm, m, and are changed to force units when multiplied by g, and this entity may not be easily recognized by the user. For this reason a conversion factor called Cg may be inserted into the conversion to change the name of the entity. This factor for SI is N/(m/s2)/(kg), and if you call it “Cg,” Equation 1.1 becomes:

With this convention, each term in the SI General Energy Equation has the units of meters.

Other forms of energy, such as chemical, electric, or atomic, may need reckoning in a particular flow problem. Their inclusion should present no difficulties if they are treated as the five forms shown here have been.

The first three terms on each side of Equation 2.11a and 2.11b are called the Bernoulli terms, after Swiss mathematician Daniel Bernoulli (1700–1782), and are referred to as heads—P/ρ is called the pressure head, ϕV2/2g is called the velocity head, and Z is called the elevation or potential head.

2.6 HEAD LOSS

The general energy equation as given above (Eq. 2.11a and 2.11b) is valid for any real fluid. There is, however, an observation that should be made here. Consider the most elementary flow system: a horizontal pipe of constant cross section, without pump or turbine, and without external heat transfer, carrying a fluid from one end to the other. Let us also assume that changes of fluid pressure or temperature do not affect the fluid density during its passage through the flow system. (This kind of flow is called incompressible flow and it is very closely approximated by the flow of most liquids.) By the continuity equation (Eq. 2.2a and 2.2b), the average velocity does not change; therefore the ϕV2/2g terms are equal on both sides of Equation 2.11 and may be dropped. The elevation does not change from one side of the equation to the other, so the Z terms may be dropped. Without pump or turbine work the terms may be dropped. Without external transferred heat the terms may be dropped. This leaves only the P/ρ terms and the JU terms. Collect the JU terms and lump them into one term called ΔJU; the resulting equation is:

Again, as in Equation 2.11, ρ is either ρw or ρm, depending on the units chosen. The pressure head change is equal to the thermal energy term, ΔJU! In this illustration, we could have included the other Bernoulli or head terms and shown that ΔJU is equal to the change in total head. Appropriately enough, the change is called head loss, or HL. In the general energy equation, where there is external heat transfer, only a portion of ΔJU is owing to head loss. But since we have observed that in incompressible flow the thermal terms usually do not affect the fluid density appreciably, we may drop the thermal terms altogether except for the portion that accounts for the loss of head, that is, HL. Then we may write a simplified energy equation:

 (2.12)

where ρ is either ρw or ρm, depending on the units chosen, as in Equation 2.11. Head loss is not a loss of total energy; it is a loss of useful mechanical energy by conversion of mechanical energy to heat energy. This energy is seldom recoverable, and, because in the study of pressure drop in liquid systems the heat energy is usually of no interest, the head loss term represents the loss of useful energy. (It would be an exceptional case indeed where this lost heat energy could be partially recovered, say, by a low tempera­ture, low pressure organic vapor turbine system, or a heating system.)

When a compressible fluid is flowing these generalizations cannot be made because there are significant conversions of heat energy to mechanical energy. Still, however, there are simplifications that can be made to make the general energy equation appear less formidable. These will be introduced in a later section (Section 2.8). Head loss will be treated in detail in the following chapters.

2.7 THE KINETIC ENERGY CORRECTION FACTOR

In Section 2.3 it was noted that the kinetic energy term requires a correction factor if the velocity profile is not flat and the energy is computed from the aver­age velocity V. The value of the correction factor is important if an accurate energy balance is to be obtained. The expression for the kinetic energy correction factor may be derived in very much the same fashion as the momentum correction factor was. The total kinetic energy flux at a given cross section of the stream tube is:

 (2.13)

where ϕ is the kinetic energy correction factor, V is the average velocity, and u is the local velocity. For an axisymmetric velocity distribution in a circular duct, the mass flow is given by Equation 2.4:

 (2.4, repeated)

The local kinetic energy flux is:

The total kinetic energy flux may be found by integrating along the radius:

 (2.14)

Combining Equations 2.13, 2.4, and 2.14 yields:

 (2.15)

Robert P. Benedict [1] gives the following equation for velocity profile:

 (2.7, repeated)

Using this equation, by performing the integrations indicated in Equation 2.15, Benedict obtains the following equation for the energy correction factor. It is (with coefficients rounded to four decimal places):

A plot of this equation (for turbulent flow) is given in Figure 2.6.

FIGURE 2.6. Kinetic energy correction factor versus friction factor (for turbulent flow).

In laminar flow, where the velocity profile is parabolic and is not a function of friction factor, the evaluation of Equation 2.15 may be accomplished analytically to show that ϕ = 2.000. When analyzing a laminar flow system, it is important therefore to include ϕ. Turbulent flow, however, is present throughout the operating range of most modern piping systems and consideration of the kinetic energy correction factor is much less important as will be seen in the following section.

2.8 CONVENTIONAL HEAD LOSS

By convention the kinetic energy correction factor ϕ is dropped in engineering computations because its value is close to 1. The head loss term in the incompressible general energy equation is defined by ignoring the ϕ coefficient so that the equation becomes:

 (2.16)

where, as in Equation 2.11, ρ is either ρw or ρm, depending on the units chosen. Notations (HL)C and (HL)E will be used momentarily to distinguish the conventional value from the exact value. By solving Equations 2.16 and 2.12 simultaneously, conventional head loss is seen to be:

It is evident that conventional head loss equals exact head loss when there is no change in flow area and thus, inherently, V2 = V1 and ϕ2 = ϕ1. When there is contraction of the flow passage as shown in Figure 2.7a, the contraction causes V2 to exceed V1 while flattening of the velocity profile causes ϕ2 to approach 1 more closely than ϕ1 does, so that (ϕ1