fluid flow for the practicing chemical engineer - essential engineering calculations series

FLUID FLOW FOR THE PRACTICING CHEMICAL ENGINEER... It is no secret that the teaching of Unit Operations-fluid flow, heat transfer, and mass tmsfer-is now required in any chemical enginee

FLUID FLOW FOR THE PRACTICING CHEMICAL

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FLUID FLOW FOR THE PRACTICING CHEMICAL

ENGINEER

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FLUID FLOW FOR THE PRACTICING CHEMICAL

Copyright 0 2009 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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(L.T.)

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CONTENTS

PREFACE

INTRODUCTION

1 History of Chemical Engineering-Fluid Flow

Units and Dimensional Consistency / 9

3 Key Terms and Definitions

3.1 Introduction / 19

3.2 Definitions 3.1.1 Fluids / 20 / 19

3.2.1 Temperature / 20

xvii xix

Classification of Non-Newtonian Fluids / 50

6.2.1 Non-Newtonian Fluids: Shear Stress / 5 1

6.3.1 Flow in Tubes / 54

6.3.2 Flow Between Parallel Plates / 55

6.3.3 Other Flow Geometries / 57

7.2 Conservation of Mass / 61

8.3 Total Energy Balance Equation / 75

8.3.1 The Mechanical Energy Balance Equation / 79

8.3.2 The Bernoulli Equation / 79

10.2.1 Buoyancy Effects; Archimedes’ Law / 102

11 Ideal Gas Law

11.1 Introduction / 109

11.2 Boyle’s and Charles’ Laws / 110

11.3 The Ideal Gas Law / 110

1 1.4 Non-Ideal Gas Behavior / 1 16

References / 119

12 Flow Mechanisms

12.1 Introduction / 123

12.2 The Reynolds Number / 124

12.3 Strain Rate, Shear Rate, and Velocity Profile / 126

97

109

121

1 23

14.3 Relative Roughness in Pipes / 150

14.4 Friction Factor Equations / 151

14.5 Other Considerations / 153

14.6 Flow Through Several Pipes / 154

14.7 General Predictive and Design Approaches / 155

16.4 Gas (Turbulent) Flow-Liquid (Viscous) Flow / 184

16.5 Gas (Viscous) Flow-Liquid (Viscous) Flow / 186

16.6 Gas-Solid Flow / 188

16.6.1 Introduction / 188

16.6.2 Solids Motion / 189

16.6.3 Pressure Drop / 190

CONTENTS Xi

16.6.4 Design Procedure / 190

16.6.5

References / 192

Pressure Drop Reduction in Gas Flow / 191

18.3 Expansion and Contraction Effects / 222

18.4 Calculating Losses of Valves and Fittings / 223

18.5 Fluid Flow Experiment: Data and Calculations / 234

20.2 IndoorAirQuality / 264

20.3 Indoor Air/Ambient Air Comparison / 264

20.4 Industrial Ventilation Systems / 266

References / 278

23.4 Particle Force Balance / 330

23.5 Cunningham Correction Factor / 335

Hydrostatic Equilibrium in Centrifugation / 355

25 Porous Media and Packed Beds

26.2 Fixed Beds / 378

26.3 Permeability / 382

26.4 Minimum Fluidization Velocity / 385

Bed Height, Pressure Drop and Porosity / 390

Fluidization Experiment Data and Calculations / 396

29.2.1 Comprehensive Environmental Response, Compensation,

and Liability Act (CERCLA) / 446 29.2.2 Superfund Amendments and Reauthorization Act

of 1986 (SARA) / 447 29.3 Health Risk Assessment / 448

29.3.1 Risk Evaluation Process for Health / 450

29.4 Hazard Risk Assessment / 451

29.4.1 Risk Evaluation Process for Accidents / 452

29.5 Illustrative Examples / 454

References / 462

3 1.3 Simultaneous Linear Algebraic Equations / 484

3 1.3.1 Gauss- Jordan Reduction / 485

32.3.6 Fabricated Equipment Cost Index / 503

32.3.7 Capital Recovery Factor / 504

32.3.8 Present Net Worth / 504

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PREFACE

Persons attempting to find a motive in this narrative will be prosecuted;

Persons attempting to find a moral in it will be banished;

Persons attempting to find a plot in it will be shot

By order of the Author, Mark Twain (Samuel Langhome Clemens, 1835-1910),

It is becoming more and more apparent that engineering education must provide courses that will include material the engineering student will need and use both professionally and socially later in life It is no secret that the teaching of Unit Operations-fluid flow, heat transfer, and mass tmsfer-is now required in any chemical engineering curriculum and is generally accepted as one of the key courses in applied engineering In addition, this course, or its equivalent, is now slowly and justifiably finding its way into other engineering curricula

Chemical engineering has traditionally been defined as a synthesis of chemistry, physics and mathematics, tempered with a concern for the dollar sign and applied

in the service of humanity During the 120 years (since 1888) that the profession has been in existence as a separate branch of engineering, humanity’s needs have changed tremendously and so has chemical engineering Thus it is that today, this changing profession faces a challenge and an opportunity to put to better use the advances that have occurred since its birth

The teaching of Unit Operations at the undergraduate level has remained relatively static since the publication of several early to mid-1900 texts At this time, however, these and some of the more recent texts in this field are considered by many to be too advanced and of questionable value for the undergraduate engineering student The present text is the first of three texts to treat the three aforementioned unit operations-fluid flow, heat transfer, and mass transfer This initial treatise has been written in order to offer the reader the fundamentals of fluid flow with appropri- ate practical applications, and to possibly serve as an introduction to the specialized and more sophisticated texts in this area

It is no secret that the teaching of both stoichiometry (material and energy bal- ances) and the three unit operations, including fluid flow, has been a major factor

xvii

xviii PREFACE

in the success of chemical engineers and chemical engineering since the early 1900s The authors believe that the approach presented here is a logical step in the continual evolution of this subject that has come to be defined as a unit operation This “new” treatment of fluid flow is offered in the belief that it will be more effective in training engineers for successful careers in and/or out of the chemical process industry

The present book has primarily evolved from notes, illustrative examples, problems and exams prepared by the authors for a required three semester fluid flow course given

to chemical engineering students at Manhattan College The course is also offered as an elective to other engineering disciplines in the school and has occasionally been attended by students outside the Department It is assumed the student has already taken basic college physics and chemistry, and should have as a minimum background

in mathematics courses through differential equations

The course at Manhattan roughly places equal emphasis on principles and appli- cations However, depending on the needs and desires of the lecturer, either area may

be emphasized, and the material in this text is presented in a manner to permit this Further, no engineering tool is complete without information on how to use it By the same token, no engineering text is complete without illustrative examples that serve the important purpose of demonstrating the use of the procedures, equations, tables, graphs, etc., presented in the text There are many such examples There are also prac- tice problems (available at a website) at the end of each chapter It is believed that most, if not all, of the illustrative examples and practice problems are “original”; some have been drawn from National Science Foundation (NSF) workshops/semi- nars conducted at Manhattan College, and some have been employed for over such

a long period of time that the original authors can no longer be identified and properly recognized If that be the case, please accept the authors’ apologies and be assured that appropriate credit (where applicable) will be given in the next printing

In constructing this text, topics of interest to all practicing engineers have been included The organization and contents of the text can be found in the table of con- tents The table consists of six main parts-Introduction to Fluid Flow, Basic Laws, Fluid Transport Classification, Fluid Flow Applications, Fluid-Particle Applications, and Special Topics

It is hoped that this writing will place in the hands of teachers and students of engineering, plus practicing engineers, a text covering the fundamental principles and applications of fluid flow in a thorough and clear manner Upon completion of the course, the reader should have acquired not only a working knowledge of the prin- ciples of fluid flow, but also experience in their application; and, readers should find themselves approaching advanced texts and the engineering literature with more confidence

Finally, the authors are particularly indebted to Shannon O’Brien for her extra set

of eyes when it came time to proofreading the manuscript

J PATRICK ABULENCIA LOUIS THEODORE

INTRODUCTION

thought is viscous

The history of unit operations is interesting Chemical engineering courses were orig- inally (late 1800 and early 1900s) based on the study of unit processes and/or indus- trial technologies However, it soon became apparent that the changes produced in equipment from different industries where similar in nature, i.e., there was a common- ality in the fluid flow, heat transfer, and mass transfer operations in the petroleum industry as with the utility industry These similar operations became known as unit operations

This book-“Fluid Flow”-was prepared as both a professional book and as an undergraduate text for the study of the principles and fundamentals of the first of the three aforementioned unit operations Some of the introductory material is pre- sented in the first two parts of the book Understandably, more extensive coverage

is given in the remainder of the book to applications and design Furthermore, seven additional topics were included in the last part of the book-special topics These topics are now all required by ABET (Accreditation Board for Engineering and Technology) to be emphasized in course offerings: each of these seven topics

is briefly discussed below

The first chapter in Part VI addresses environmental concerns; nearly one third of undergraduates chose environmental careers The second topic is health, safety, and accident prevention; new and existing processes today require ongoing analyze in these areas To better acquaint the student with human relations, engineering and environmental ethics is the third topic Numerical methods are the next topic encoun- tered since computers are not only used to design multi-component distillation columns but also routinely used in the work force The success or failure of any business related activity is tied to economics and finance, and this too receives treat- ment The “hot” topic-Biomedical Applications-receives treatment in Chapter 33 Finally, open-ended problems (problems that can have more than one solution), are

xix

XX INTRODUCTION

treated in the last chapter This final chapter requires the reader to ask questions, not always accept things at face value, and select a methodology that will yield the most effective and efficient solution Illustrative examples on each of these topics are included within each chapter

Although not a complete treatment of the subject, the text has attempted to present theory, principles, and applications of unit operation in a manner that will benefit the reader and/or prospective engineer in their career as a practicing engineer Those desiring more information on these topics should proceed to specialized texts in these areas

This book is the result of several years of effort by the Chemical Engineering Department at Manhattan College The first rough draft was prepared during the

2001 -2002 academic year and underwent peripheral classroom testing during the ensuing years; the manuscript underwent significant revisions during this past year, some of it based on the experiences gained from class testing

In the final analysis, the problem of what to include and what to omit was particu- larly difficult However, every attempt was made to offer engineering course material

to individuals at a level that should enable them to better cope with some of the pro- blems they will later encounter in practice As such, the book was not written for the student planning to pursue advanced degrees; rather, it was primarily written for those individuals who are currently working as practicing engineers or plan to work as engineers in the future solving real world problems

The entire book can be covered in a three-credit course At Manhattan, Fluid Flow

is taught in the second semester of the sophomore year (Heat and Mass Transfer are taught in the junior year) Finally, it should be again noted that the Manhattan approach is to place more emphasis on the macroscopic approach; however, some microscopic material is included

INTRODUCTION TO

FLUID FLOW

This first part of the book provides an introduction to fluid flow It contains six chap- ters and each serves a unique purpose in an attempt to treat important introductory aspects of fluid flow From a practical point-of-view, systems and plants move liquids and gases from one point to another; hence, the student and/or practicing engineer is concerned with several key topics in this area These receive some measure of treatment in the six chapters contained in this part A brief discussion

of each chapter follows

Chapter 1 provides an overview of the History of Chemical Engineering-Fluid Flow Chapter 2 is concerned with Units and Dimensional Analysis Chapter 3 intro- duces Key Terms and Definitions Chapter 4 provides a discussion of Transport Phenomena versus Unit Operations The final two chapters introduce the reader to Newtonian Fluids (Chapter 5 ) and Non-Newtonian Flow (Chapter 6) These subjects

are important in developing an understanding of the various fluid flow equipment and operations plus their design, which is discussed later in the text

Fluid Flow for the Practicing Chemical Engineer By J Patrick Abulencia and Louis Theodore Copyright 0 2009 John Wiley & Sons, Inc

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HISTORY OF CHEMICAL ENGINEERING-FLUID FLOW

1.1 INTRODUCTION

Although the chemical engineering profession is usually thought to have originated shortly before 1900, many of the processes associated with this discipline were devel- oped in antiquity For example, filtration operations (see Chapter 27) were carried out

5000 years ago by the Egyptians During this period, chemical engineering evolved from a mixture of craft, mysticism, incorrect theories, and empirical guesses

In a very real sense, the chemical industry dates back to prehistoric times when people first attempted to control and modify their environment The chemical industry developed as any other trade or craft With little knowledge of chemical science and

no means of chemical analysis, the earliest “chemical engineers” had to rely on pre- vious art and superstition As one would imagine, progress was slow This changed with time The chemical industry in the world today is a sprawling complex of raw- material sources, manufacturing plants, and distribution facilities which supplies society with thousands of chemical products, most of which were unknown over a century ago In the latter half of the nineteenth century, an increased demand arose for engineers trained in the fundamentals of chemical processes This demand was ultimately met by chemical engineers

Fluid Flowfor the Practicing Chemical Engineer By J Patrick Abulencia and Louis Theodore Copyright 0 2009 John Wiley & Sons, Inc

3

4 HISTORY OF CHEMICAL ENGINEERING-FLUID FLOW

1.2 FLUID FLOW

With respect to fluid flow, the history of pipes and fittings dates back to the Roman Empire The ingenious “engineers” of that time came up with a solution for supplying the never-ending demand for fresh water to a city and then disposing of the waste- water produced by the Romans Their system was based on pipes made out of wood and stone and the driving force of the water was gravity.“) Over time, many improvements have been made to the piping system These improvements have included the material choice, shape and size of the pipes; pipes are now made from different metals, plastic, and even glass, with different diameters and wall thick- nesses The next challenge was the connection of the pipes and that was accomplished with fittings Changes in piping design ultimately resulted from the evolving indus- trial demands for specific requirements and the properties of fluids that needed to be transported

The first pump can be traced back to 3000 B.C in Mesopotamia It was used to supply water to the crops in the Nile River valley.‘2’ The pump was a long lever with a weight on one side and a bucket on the other The use of this first pump became popular in the Middle East and this technology was used for the next 2000 years Sometimes, a series of pumps would be put in place to provide a constant flow of water to the crops far from the source Another ancient pump was the bucket chain, a continuous loop of buckets that passed over a pulley-wheel; it is believed that this pump was used to imgate the Hanging Gardens of Babylon around 600 B.c.‘~) The most famous of these early pumps is the Archimedean screw The pump was invented by the famous Greek mathematician and inventor Archimedes (287-212 B.c.) The pump was made of a metal pipe in which a helix- shaped screw was used to draw water upward as the screw turned Modem force pumps were adapted from an ancient pump that featured a cylinder with a piston “at the top that create[d] a vacuum and [drew] water upward.”‘2’ The first force pump was designed by Ctesibus of Alexandria, Egypt Leonard0 Da Vinci (1452-1519) was the first to come up with the idea of lifting water by means of centrifugal force; however, the operation of the centrifugal pump was first described scientifically by the French physicist Denis Papin (1647-1714) in 1687.’3’ In 1754, Leonhard Euler further developed the principles on which centrifugal pumps operate and today the ideal pump performance term, “Euler head,” is named after him.(4) In the United States, the first centrifugal pump to be manufactured was by the Massachusetts Pump Factory James Stuart built the first multi-stage centrifugal pump in 1 849.‘3’

1.3 CHEMICAL ENGINEERING

The first attempt to organize the principles of chemical processing and to clarify the professional area of chemical engineering was made in England by George E Davis

In 1880, he organized a Society of Chemical Engineers and gave a series of lectures

in 1887, which were later expanded and published in 1901 as “A Handbook of Chemical Engineering.” In 1888, the first course in chemical engineering in the

6 HISTORY OF CHEMICAL ENGINEERING-FLUID FLOW

United States was organized at the Massachusetts Institute of Technology by Lewis

M Norton, a professor of industrial chemistry The course applied aspects of chem- istry and mechanical engineering to chemical proces~es.‘~’

Chemical engineering began to gain professional acceptance in the early years of the twentieth century The American Chemical Society was founded in 1876 and, in

1908, it organized a Division of Industrial Chemists and Chemical Engineers while authorizing the publication of the Journal of Zndustrial and Engineering Chemistry Also in 1908, a group of prominent chemical engineers met in Philadelphia and founded the American Institute of Chemical Engineers.‘”

The mold for what is now called chemical engineering was fashioned at the 1922 meeting of the American Institute of Chemical Engineers when A D Little’s com- mittee presented its report on chemical engineering education The 1922 meeting marked the official endorsement of the unit operations concept and saw the approval

of a “declaration of independence” for the profe~sion.(~’ A key component of this report included the following:

nated series of what may be termed ‘unit operations,’ as pulverizing, mixing, heating, roasting, absorbing, precipitation, crystallizing, filtering, dissolving, and so on The number of these basic unit operations is not very large and relatively few of them are involved in any particular process An ability to cope broadly and adequately with the demands of this (the chemical engineer’s) profession can be attained only through the analysis of processes into the unit actions as they are carried out on the commercial scale under the conditions imposed by practice.”

The key unit operations were ultimately reduced to three: Fluid Flow (the subject title of this text), Heat Transfer, and Mass Transfer The Little report also went on to state that:

“Chemical Engineering, as distinguished from the aggregate number of subjects com- prised in courses of that name, is not a composite of chemistry and mechanical and

civil engineering, but is itself a branch of engineering, .”

A time line diagram of the history of chemical engineering between the pro- fession’s founding to the present day is shown in Fig 1.1 .(5) As can be seen from the time line, the profession has reached a crossroads regarding the future edu- cation/curriculum for chemical engineers This is highlighted by the differences of Transport Phenomena and Unit Operations, a topic that is discussed in Chapter 4

REFERENCES

2 hnp://www.bookrags.corn/sciences/sciencehistory/water-pump-woi.hhl

REFERENCES 7

4 R D Flack, “Fundamentals of Jet Propulsion with Applications,” Cambridge University

5 N Serino, “2005 Chemical Engineering 125th Year Anniversary Calendar,” term project,

Press, New York, 2005

submitted to L Theodore 2004

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UNITS AND DIMENSIONAL

ANALYSIS

2.1 INTRODUCTION

This chapter is primarily concerned with units The units used in the text are consist- ent with those adopted by the engineering profession in the United States One usually refers to them as the English or engineering units Since engineers are often concerned with units and conversion of units, both the English and SI system of units are used throughout the book All the quantities and the physical and chemical properties are expressed using these two systems

2.1.1 Units and Dimensional Consistency

Equations are generally dimensional and involve several terms For the equality to hold, each term in the equation must have the same dimensions (i.e., the equation must be dimensionally homogeneous or consistent) This condition can be easily proved Throughout the text, great care is exercised in maintaining the dimensional formulas of all terms and the dimensional consistency of each equation The approach employed will often develop equations and terms in equations by first examining each

in specific units (feet rather than length), primarily for the English system Hopefully, this approach will aid the reader and will attach more physical significance to each term and equation

Consider now the example of calculating the perimeter, P, of a rectangle with length, L, and height, H Mathematically, this may be expressed as P = 2L + 2H

Fluid Flowfor the Practicing Chemical Engineer By J Patrick Abulencia and Louis Theodore Copyright 0 2009 John Wiley & Sons, Inc

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10 UNITS AND DIMENSIONAL ANALYSIS

This is about as simple as a mathematical equation can be However, it only applies when P, L, and H are expressed in the same units

A conversion constant/factor is a term that is used to obtain units in a more convenient form All conversion constants have magnitude and units in the term, but can also be shown to be equal to 1 O (unity) with no units An often used conversion constant is

12 inches/foot This term is obtained from the following defining equation:

12in = 1 ft

If both sides of this equation are divided by 1 ft one obtains

12in/ft = 1.0 Note that this conversion constant, like all others, is also equal to unity without any units Another defining equation is

454 g

Terms in equations must also be constructed from a “magnitude” viewpoint Differential terms cannot be equated with finite or integral terms Care should also

be exercised in solving differential equations In order to solve differential equations

to obtain a description of the pressure, temperature, composition, etc., of a system, it

is necessary to specify boundary and/or initial conditions for the system This infor- mation arises from a description of the problem or the physical situation The number

of boundary conditions (BC) that must be specified is the sum of the highest-order derivative for each independent differential term A value of the solution on the boundary of the system is one type of boundary condition The number of initial conditions (IC) that must be specified is the highest-order time derivative appearing

in the differential equation The value for the solution at time equal to zero constitutes

an initial condition For example, the equation

requires 2 BCs (in terms of z) The equation

=O; t=time

dT

-

dt requires 1 IC And finally, the equation

- dCA = D%; D = diffusivity

at i3y2 requires 1 IC and 2 BCs (in terms of y )

12 UNITS AND DIMENSIONAL ANALYSIS

2.2 DIMENSIONAL ANALYSIS

Problems are frequently encountered in fluid flow and other engineering work that involve several variables Engineers are generally interested in developing functional relationships (equations) between these variables When these variables can be grouped together in such a manner that they can be used to predict the performance

of similar pieces of equipment, independent of the scale or size of the operations, something very valuable has been accomplished

Consider, for example, the problem of establishing a method of calculating the power requirements for mixing liquids in open tanks The obvious variables would

be the depth of liquid in the tank, the density and viscosity of the liquid, the speed

of the agitator, the geometry of the agitator, and the diameter of the tank There are therefore six variables that affect the power, or a total of seven terms that must

be considered To generate a general equation to describe power variation with these variables, a series of tanks having different diameters would have to be set

up in order to gather data for various values of each variable Assuming that ten different values of each of six variables were imposed on the process, lo6 runs would be required Obviously, a mathematical method for handling several variables that requires considerably less than one million runs to establish a design method must be available In fact, such a method is available and it is defined as dimensional analysis ( I )

Dimensional analysis is a powerful tool that is employed in planning experiments, presenting data compactly, and making practical predictions from models without detailed mathematical analysis The first step in an analysis of this nature is to write down the units of each variable The end result of a dimensional analysis is a list of pertinent dimensionless numbers A partial list of common dimensionless numbers used in fluid flow analyses is given in Table 2.1

Dimensional analysis is a relatively “compact” technique for reducing the number and the complexity of the variables affecting a given phenomenon, process or calcu- lation It can help obtain not only the most out of experimental data but also scale-up data from a model to a prototype To do this, one must achieve similarity between the prototype and the model This similarity may be achieved through dimensional analy- sis by determining the important dimensionless numbers, and then designing the model and prototype such that the important dimensionless numbers are the same

in both

There are three steps in dimensional analysis These are:

1 List all parameters and their primary units

2 Formulate dimensionless numbers (or ratios)

3 Develop the relation between the dimensionless numbers experimentally

Further details on this approach are provided in the next section

2.3 BUCKINGHAM Pi (m) THEOREM 13

Table 2.1 Dimensionless numbers

Pressure Inertia Kinetic energy Inertia Pressure Inertia Inertia Gravity

st = -

Surface tension

U Note: p’ = vapor pressure, C,, = heat capacity

2.3 BUCKINGHAM Pi (m) THEOREM

This theorem provides a simple method to obtain dimensionless numbers (or ratios) termed T parameters The steps employed in obtaining the dimensionless T par- ameters are given below‘*):

1 List all parameters Define the number of parameters as n

2 Select a set of primary dimensions, e.g., kg, m, s, K (English units may also be employed) Let r = the number of primary dimensions

3 List the units of all parameters in terms of the primary dimensions, e.g., L [=I

m, where “[=I” means “has the units of.” This is a critical step and often requires some creativity and ingenuity on the part of the individual performing the analysis

14 UNITS AND DIMENSIONAL ANALYSIS

I

4 Select a number of variables from the list of parameters (equal to r) These are called repeating variables The selected repeating parameters must include all r independent primary dimensions The remaining parameters are called “non- repeating” variables

5 Set up dimensional equations by combing the repeating parameters with each

of the other non-repeating parameters in turn to form the dimensionless par- ameters, T There will be (n - r) dimensionless groups of (m)

6 Check that each resulting T group is in fact dimensionless

D

b

Note that it is permissible to form a different T group from the product or division

of other m, e.g.,

Note, however, that a dimensional analysis approach will fail if the fundamental vari-

ables are not correctly chosen The Buckingham Pi theorem approach to dimension- less numbers is given in the Illustrative Example that follows

Illustrative Example 2.2 When a fluid flows through a horizontal circular pipe, it undergoes a pressure drop, AP = (P2 - P I ) For a rough pipe, A P will be higher than

a smooth pipe The extent of non-smoothness of a material is expressed in terms of the roughness, k For steady state incompressible Newtonian (see Chapter 5 ) fluid flow, the pressure drop is believed to be a function of the fluid average velocity u, viscosity p, density p, pipe diameter D, length L, and roughness k (discussed in more detail in Chapter 14), and the speed of sound in fluid (an important variable

if the flow is compressible) c, i.e.,

Determine the dimensionless numbers of importance for this flow system

Solution A pictorial representation of the system in question is provided in Fig 2.1

r

Figure 2.1 Pipe

The non-repeating parameters are then AP, p, k, c, and L

Determine the number of m:

n - r = 8 - 3 = 5 Formulate the first T, r l , employing A P as the non-repeating parameter

Tl = A P V ~ ~ ~ D ~ Determine a, b, and f by comparing the units on both sides of the following equation:

0 [=I (kg m-I s-*)(m s-')O(kg m-3)b(m)f

16 UNITS AND DIMENSIONAL ANALYSIS

This represents the Euler number (see Table 2.1) Formulate the second 72, n2 as

722 = pvapbDf Determine a, b, and f by comparing the units on both sides:

0 [=I (kg m-'s-')(m s-')a(kg m-3)b(m)f Compare kg:

O = 1 +b Thereforeb= -1 Compare s:

0 = -1 -a Therefore a = -1 Compare m:

Substituting back into 72 yields:

Replace 72. by its reciprocal:

where Re = Reynolds number (see Chapter 12)

Similarly, the remaining non-repeating variables lead to

k

723 = kvapbDf t -

D

2.4 SCALE-UP AND SlMllARllY 17

L

m5 = -

D Combine the m into an equation, expressing mI as a function of m2, m3, m4, and m5:

AP

Eu = - = f (Re, = the Euler number

PV2/2 Consider the case of incompressible flow

The result indicates that to achieve similarity between a model (m) and a prototype (p), one must have the following:

Since Eu =f(Re, k/D, LID), then it follows that Eu, = Eu, (see Table 2.1)

2.4 SCALE-UP AND SIMILARITY

To scale-up (or scale-down) a process, it is necessary to establish geometric and dynamic similarities between the model and the prototype These two similarities a~ discussed below

Geometric similarity implies using the same geometry of equipment A circular pipe prototype should be modeled by a tube in the model Geometric similarity estab- lishes the scale of the model/prototype design A l/lOth scale model means that the characteristic dimension of the model is 1 / 10th that of the prototype

Dynamic similarity implies that the important dimensionless numbers must be the same in the model and the prototype For a particle settling in a fluid, it has been shown (see Chapter 23) that the drag coefficient, CD, is a function of the