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For each of the fluid types that are studied, the friction factor is presented graphically in terms of the appropriate Reynolds number, thedimensionless pipe diameter, the dimensionless

Introduction to Practical Fluid Flow

This book is dedicated to my

wife Ellen

An imprint of Elsevier Science

Linacre House, Jordan Hill, Oxford OX2 8DP

200 Wheeler Road, Burlington, MA 01803

First published 2002

Copyright#2002, R.P King All rights reserved

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2 Flow of fluids in piping systems

2.1 Pressure drop in pipes and channels 2.2 The friction factor

2.7 Symbols

2.8 Practice problems

3 Interaction between fluids and particles

3.1 Basic concepts

3.2 Terminal settling velocity

3.3 Isolated isometric particles of arbitrary shape

3.4 Symbols

3.5 Practice problems

5.6 Symbols used in this chapter

6 Sedimentation and thickening

6.1 Thickening

6.2 Concentration discontinuities in settling slurries

6.3 Useful models for the sedimentation velocity

6.4 Continuous cylindrical thickener

6.5 Simulation of the batch settling experiment

6.6 Thickening of compressible pulps

6.7 Continuous thickening of compressible pulps

6.8 Batch thickening of compressible pulps

6.9 Practice problems

6.10 Symbols

Index

This book deals with the transportation and handling of incompressiblefluids This topic is important to most process engineers, because large quan-tities of material are transported in the process engineering industries Theemphasis of this book is on suspensions of particulate solids although thebasic principles of simple Newtonian fluid flow form the basis of the devel-opment of models for the transportation of such material Both settlingslurries and dense suspensions are considered The latter invariably exhibitnon-Newtonian behavior Transportation of slurries and other non-Newtonianfluids is generally treated inadequately or perfunctorily in most of the textsdealing with fluid transportation This is a disservice to modern students inchemical, metallurgical, civil, and mining engineering, where problems relat-ing to the flow of slurries and other non-Newtonian fluids are commonlyencountered Although the topics of non-Newtonian fluid flow and slurrytransportation are comprehensively covered in specialized texts, this bookattempts to consolidate these topics into a consistent treatment that followsnaturally from the conventional treatment of the transportation of incompres-sible Newtonian fluids in pipelines In order to keep the book to a reasonablelength, solid±liquid systems that are of interest in the mineral processingindustries are emphasized at the expense of the many other fluid types thatare encountered in the process industries in general This reflects the particu-lar interests of the author However, the student should have no difficulty inadapting the methods that are described here to other application areas Thelevel is kept to that of undergraduate courses in the various process engineer-ing disciplines, and this book could form the basis of a one-semester coursefor students who have not necessarily had exposure to formal fluidmechanics This book could also usefully be adopted for students who have

or will take a course in fluid mechanics and who need to explore the typicalsituations that they will meet as practising process engineers The level ofmathematical analysis is consistent with that usually found in modern under-graduate engineering curricula and is consistent with the need to describe thesubject matter at the level that is used in modern engineering analysis.Modeling methods that are based on partial differential equations are used

in Chapter 6 because they are essential for the proper description of industrialsedimentation and thickening processes where the solid concentration fre-quently varies spatially and with time

An important novel feature of this book is the unified treatment of thefriction factor information that is used to calculate the flow of all types of fluid

in round pipes For each of the fluid types that are studied, the friction factor

is presented graphically in terms of the appropriate Reynolds number, thedimensionless pipe diameter, the dimensionless flowrate and the dimension-less flow velocity Each of these graphical representations leads to the most

convenient computational method for specific problems depending on whatinformation is specified and which variables must be computed The sameproblem-solving methods are used irrespective of the type of fluid be it

a simple Newtonian or a rheologically complex fluid such as those whosebehavior is described by the Sisko model This uniformity should assiststudents considerably in learning the basic principles and applying themacross a wide range of application areas

The presentation of material is somewhat different to that found in mosttextbooks in this field in that it is acknowledged that modern students ofengineering are computer literate These students are accustomed to usingspreadsheets and other well-organized computational aids to tackle technicalproblems They do not rely only on calculators and almost never plot graphsusing pencil and paper Few students submit handwritten reports Conse-quently, computer-oriented methods are emphasized throughout, and, whereappropriate, time-consuming or tedious computational processes are pre-programmed and made available in the computational toolbox that accompaniesthis text This toolbox has been designed with care to ensure that it does notprovide point-and-click solutions to problems Rather the student is encour-aged to formulate a solution method for every specific problem, but the tools

in the toolbox make it feasible to tackle realistic problems that would besimply too time consuming using manual computational methods or if thestudent were required to generate the appropriate computer code In any case,students of process engineering are becoming less fluent in the traditionalcomputational languages Fortran, C, Basic, and Pascal that almost all coulduse with some degree of proficiency during the last three decades of thetwentieth century Now, engineering students are far more likely to be fluent

in computer languages such as Java and HTML and are more likely to be able

to create a website on the Internet than to be able to quickly and correctlyintegrate a couple of differential equations numerically Nevertheless, theyare well-attuned to using solution methods that are preprogrammed andready to be used Students and instructors are encouraged to install the tool-box and to explore its constituent tools before tackling any material in thisbook No specific programming skills are required of the student or theinstructor The use of this modern problem-solving methodology makes itpossible to extend the treatment from a purely superficial level to a morein-depth treatment and so equip the student to tackle, and successfully solve,realistic engineering problems

The quantitative models that are described in this text will surely changeand evolve over the years ahead as a result of continuing research andinvestigational effort However, the basic approach should be sufficientlygeneral to accommodate these developments Because the computationaltoolbox has an open-ended design, new models can be inserted with ease atany time and it is intended that the toolbox should continue to expand wellinto the future

This book can be used as a reading text to support Internet-basedcourse delivery This method has been used with success at the University

of Utah, where such a course, supported by a fully equipped virtual tory, is now available At the time of writing this course can be previewed athttp://webct.tacc.utah.edu.

labora-Professor R.P Chabbra and labora-Professor Raj Rajamani made several usefulsuggestions for improving the first draft of this book These are gratefullyacknowledged

R.P KingSalt Lake City

Introduction

1.1 Fluid flow in process engineering

Process engineering deals with the processing of large quantities of ial In order to process materials, they must be transported to the process-ing plant, and they must be transported from one unit operation toanother within the processing environment Materials are usually trans-ported in a fluid phase, because this is generally much easier and morecost-effective than transportation as a solid Liquids can be easily movedthrough pipelines or open channels, and the energy that is required can beconveniently delivered to the fluid using a pump Gases too can be trans-ported economically by pipeline, but this book deals exclusively with thetransportation of incompressible fluids These include pure liquids, bothNewtonian and non-Newtonian, and suspensions of solid particles inliquids that form slurries or pastes Non-Newtonian fluids and suspensionsare commonly encountered by chemical, metallurgical, mining, and civilengineers

mater-This book does not start from the usual definition of the fluid as a tinuum from which the application of differential mass and momentumbalances leads to the equation of continuity and the Navier±Stokes equations.The approach taken is macroscopic with an emphasis of solving problems ofpractical engineering significance The accompanying computational toolboxprovides the tools that are necessary to solve these problems with conveni-ence, and the reader is expected to become familiar with the toolbox and itscontents

con-1.2 Dimensions, units, and physical quantities

A variety of physical quantities of both the fluids and the equipment will

be used throughout this book These quantities must be described tively, for which sets of dimensions and units are required For example,the density of a fluid is an important quantity that will influence thebehavior of the fluid in most situations The dimensions of density aremass per unit volume M/L3 To give the density a numerical value, a set

quantita-of units must be selected for all the dimensions that are to be used In thisbook all units will be specified in the SI (SysteÁme International) system.There is a good reason for this: the SI is the only practical set of units that

is coherent This means that no conversion factors are ever required whensolving problems This is in stark contrast to all other systems of units,

including the metric system, which require difficult-to-remember sion factors in almost every problem except perhaps only the most elemen-tary and trivial These older incoherent systems of units are now regarded

conver-as being obsolete for the purposes of scientific and technical calculations.The SI is based on a set of fundamental dimensions and units as shown inTable 1.1 The precise size of each of the fundamental dimensions is defined

by reference to a unique physical entity Because the size of the fundamentaldimensions that are used in the SI do not always conveniently match those

of the physical quantities that are encountered in practical problems, a set

of prefixes is defined which specify powers of 10 which multiply thefundamental units as required for convenient specifications of the numericalquantities These are given in Table 1.3

Clearly, the fundamental dimensions are not sufficient to describe all thephysical properties that are of interest, and a set of derived units that will be

of interest in this book is given in Table 1.2

For example, the unit of density in the SI system is kg/m3

The use of upper case letters in the unit abbreviations is restricted to thoseunits that are named for people In Table 1.2 these are the newton (N), hertz(Hz), pascal (Pa), joule (J), watt (W) and kelvin (K)

Some units that are outside the SI but which may be used with the SI aregiven in Table 1.4 These outside units are not coherent with the SI and shouldnever be used in calculations Convert any quantity in these units to the SIunit before calculations begin

The coherence of the SI system is demonstrated using the following simpleexample The energy that is required to transport a fluid from one location toanother can be calculated using the following equation, which is derived inChapter 2

Energy required ˆ Change in potential energy ‡ Change in kinetic energy

‡ specific volume of fluid  Change in pressure

‡ Energy dissipated by friction:

Table 1.1 Fundamental dimensions in the SI and their units

Such an energy balance is usually established for unit mass of fluid that flows.The energy required will now be calculated using obsolete units and SI units

to demonstrate the advantages that are gained through the coherence of thelatter system

Table 1.2 Some derived units in the SI

Table 1.5 Data for illustrative example

The data for this example is set out in Table 1.5 The standard method forsetting out this calculation in the old system of units, as taught in many highschools and universities in the United states, is as follows:

Energy required ˆ g…zfinal zinitial† ‡ 12…V2

1 lbf32:174 lbmft=s2

‡62:4 lbm=ft3

…0 65† lbf=inch2 122inch2=ft2

...

a flowing fluid is established in this section

2.4.1 Internal energy and energy dissipation in

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Flow of fluids in piping systems

2.1 Pressure drop in pipes and channels

When a fluid flows through a pipe, it transfers momentum... mobile fluid such as water The shearstress is related to the fluid velocity through an empirical equation

where V is the average velocity of the fluid in the pipe, fis the fluid