Unit Operations of Chemical Engineering - Fifth Edition

Unit Operations of Chemical Engineering Fifth Edition UNIT OPERATIONS OF CHEMICAL ENGINEERING McGrawHill Chemical Engineering Series Editorial Advisory Board James J Carberry Professor of Chemical Engineering University of Notre Dame James R Fair Professor of Chemical Engineering University of Texas Austin William P Schowalter Dean School of Engineering University of Illinois Matthew Tirrell Professor of Chemical Engineering University of Minnesota James Wei Dean School of Engineering Princeton University Max S Peters Emeritus Professor of Chemical Engineering University of Colorado Building the Literature of a Profession Fifteen prominent chemical engineers first met in New York more than 60 years ago to plan a continuing literature for their rapidly growing profession From Industry came such pioneer practitioners as Leo H Baekeland Arthur D Little Charles L Reese John V N Dorr M C Whitaker and R S McBride From the universities came such eminent educators as William H Walker Alfred H White D D Jackson J H James Warren K Lewis and Harry A Curtis H C Parmelee then editor of Chemical and Metallurgical Engineering served as chairman and was joined subsequently by S D Kirkpatrick as consulting editor After several meetings this committee submitted its report to the McGrawHill Book Company in September 1925 In the report were detailed specifications for a correlated series of more than a dozen texts and reference books which have since become the McGrawHill Series in Chemical Engineering and which became the cornerstone of the chemical engineering curriculum From this beginning there has evolved a series of texts surpassing by far the scope and longevity envisioned by the founding Editorial Board The McGrawHill Series in Chemical Engineering stands as a unique historical record of the development of chemical engineering education and practice In the series one finds the milestones of the subjects evolution industrial chemistry stoichiometry unit operations and processes thermodynamics kinetics and transfer operations Chemical engineering is a dynamic profession and its literature continues to evolve McGrawHill with its editor B J Clark and its consulting editors remains committed to a publishing policy that will serve and indeed lead the needs of the chemical engineering profession during the years to come The Series Bailey and Ollis Biochemical Engineering Fundamentals Bennett and Myers Momentum Heat and Mass Transfer Brodkey and Hershey Transport Phenomena A Unified Approach Carberry Chemical and Catalytic Reaction Engineering Constantinides Applied Numerical Methods with Personal Computers Coughnowr Process Systems Analysis and Control de Nevers Fluid Mechanics for Chemical Engineers Douglas Conceptual Design of Chemical Processes Edgar and Himmelblau Optimization of Chemical Processes Gates Katzer and Schuit Chemistry of Catalytic Processes Holland Fundamentals of Multicomponent Distillation Holland and Liapis Computer Methods for Solving Dynamic Separation Problems Katz and Lee Natural Gas Engineering Production and Storage King Separation Processes Lee Fundamentals of Microelectronics Processing Luyben Process Modeling Simulation and Control for Chemical Engineers McCabe Smith and Harriott Unit Operations of Chemical Engineering Mickley Sherwood and Reed Applied Mathematics in Chemical Engineering Middleman and Hochberg Process Engineering Analysis in Semiconductor Device Fabrication Nelson Petroleum Refinery Engineering Perry and Chilton Editors Perrys Chemical Engineers Handbook Peters Elementary Chemical Engineering Peters and Timmerhaus Plant Design and Economics for Chemical Engineers Reid Prausnitz and Rolling Properties of Gases and Liquids Smith Chemical Engineering Kinetics Smith and Van Ness Introduction to Chemical Engineering Thermodynamics Treybal Mass Transfer Operations ValleRiestra Project Evaluation in the Chemical Process Industries Wei Russell and Swartzlander The Structure of the Chemical Processing Industries Wentz Hazardous Waste Management UNIT OPERATIONS OF CHEMICAL ENGINEERING Fifth Edition Warren L McCabe Late R J Reynolds Professor in Chemical Engineering North Carolina State University Julian C Smith Emeritus Professor of Chemical Engineering Cornell University Peter Harriott Fred H Rhodes Professor of Chemical Engineering Cornell University McGrawHill Inc New York St Louis San Francisco Auckland Bogotá Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto UNIT OPERATIONS OF CHEMICAL ENGINEERING International Editions 1993 Exclusive rights by McGrawHill Book Co Singapore for manufacture and export This book cannot be reexported from the country to which it is consigned by McGrawHill Copyright 1993 1985 1976 1967 1956 by McGrawHill Inc All rights reserved Except as permitted under the United States Copyright Act of 1976 no part of this publication may be reproduced or distributed in any form or by any means or stored in a data base or retrieval system without the prior written permission of the publisher 1 2 3 4 5 6 7 8 9 0 CWP PMP 9 8 7 6 5 4 3 This book was set in Times Roman The editors were BJ Clark and Eleanor Castellano the production supervisor was Louise Karam The cover was designed by Joseph Gilians Library of Congress CataloginginPublication Data McCabe Warren L Warren Lee date Unit operations of chemical engineering Warren L McCabe Julian C Smith Peter Harriott 5th ed p cm McGrawHill chemical engineering series Includes index ISBN 0070448442 1 Chemical processes I Smith Julian C Julian Cleveland date II Harriott Peter III Title IV Series TP1557 M393 1993 6602842dc20 9236218 When ordering this title use ISBN 0071127380 Printed in Singapore ABOUT THE AUTHORS Julian C Smith BChem ChemE Cornell University is Professor Emeritus of Chemical Engineering at Cornell University where he joined the faculty in 1946 He was Director of Continuing Engineering Education at Cornell from 1965 to 1971 and Director of the School of Chemical Engineering from 1975 to 1983 He retired from active teaching in 1986 Before joining the faculty at Cornell he was employed as a chemical engineer by EI duPont de Nemours and Co He has served as a consultant on process development to Du Pont American Cyanamid and many other companies as well as government agencies He is a member of the American Chemical Society and a Fellow of the American Institute of Chemical Engineers Peter Harriott B ChemE Cornell University ScD Massachusetts Institute of Technology is the Fred H Rhodes Professor of Chemical Engineering at Cornell University Before joining the Cornell faculty in 1953 he worked as a chemical engineer for the EI duPont de Nemours and Co and the General Electric Co In 1966 he was awarded an NSF Senior Postdoctoral Fellowship for study at the Institute for Catalysis in Lyon France and in 1988 he received a DOE fellowship for work at the Pittsburgh Energy Technology Center Professor Harriott is the author of Process Control and a member of the American Chemical Society and the American Institute of Chemical Engineers He has been a consultant to the US Department of Energy and several industrial firms on problems of mass transfer reactor design and air pollution control Agitation and Mixing of Liquids Heat Transfer by Conduction Principles of Heat Flow in Fluids Heat Transfer to Fluids without Phase Change Heat Transfer to Fluids with Phase Change 374 Radiation Heat Transfer 397 HeatExchange Equipment 427 Evaporation 463 Membrane Separation Processes 838 Separation of Gases 838 Separation of Liquids 859 Dialysis 860 Membranes for LiquidLiquid Extraction 862 Pervaporation 864 Reverse Osmosis 871 Symbols 878 Problems 879 References 881 EquilibriumStage Operations 501 Crystallization 882 Crystal Geometry 883 Principles of Crystallization 884 Equilibria and Yields 884 Nucleation 892 Crystal Growth 899 Crystallization Equipment 902 Applications of Principles to Design 909 MSMPCR Crystallizer 909 Crystallization from Melts 918 Symbols 920 Problems 921 References 923 Distillation 521 Properties Handling and Mixing of Particulate Solids 927 Characterization of Solid Particles 927 Properties of Particulate Masses 936 Storage of Solids 939 Mixing of Solids 941 Types of Mixers 942 Mixers for Cohesive Solids 943 Mixers for FreeFlowing Solids 952 Symbols 957 Problems 958 References 959 Introduction to Multicomponent Distillation 588 Leaching and Extraction 614 This revised edition of the text on the unit operations of chemical engineering contains much updated and new material reflecting in part the broadening of the chemical engineering profession into new areas such as food processing electronics and biochemical applications Its basic structure and general level of treatment however remain unchanged from previous editions It is a beginning text written for undergraduate students in the junior or senior years who have completed the usual courses in mathematics physics chemistry and an introduction to chemical engineering An elementary knowledge of material and energy balances and of thermodynamic principles is assumed Separate chapters are devoted to each of the principal operations which are grouped in four main sections fluid mechanics heat transfer equilibrium stages and mass transfer and operations involving particulate solids Onesemester or onequarter courses may be based on any of these sections or combinations of them In this edition SI units are emphasized much more than in previous editions but the older cgs and fps systems have not been completely eliminated Chemical engineers must still be able to use all three systems of units The great majority of the equations and correlations it should be noted are dimensionless and may be used with any set of consistent units A new chapter on membrane separations has been added and the order of the chapters on multicomponent distillation extraction drying and crystallization has been made more logical The discussion of particulate solids has been shortened and two former chapters on properties and handling of solids and of solids mixing have been combined into one New material has been added on flow measurement dispersion operations supercritical extraction pressureswing adsorption crystallization techniques crossflow filtration sedimentation and many other topics The treatment of dimensional analysis has been condensed and moved from the appendices to Chapter 1 Principles of Diffusion and Mass Transfer between Phases 647 About twothirds of the problems at the ends of the chapters are new or revised with a large majority of them expressed in SI units Nearly all the problems can be solved with the aid of a pocket calculator although a computer solution may be preferred in some cases McGrawHill and the authors would like to thank Edward Cussler University of Minnesota and Robert Kabel Pennsylvania State University for their helpful reviews of the manuscript The senior author Dr Warren L McCabe died in August 1982 This book is dedicated to his memory Julian C Smith Peter Harriott Gas Absorption 686 SECTION I Humidification Operations 738 INTRODUCTION Drying of Solids 767 CHAPTER 1 DEFINITIONS AND PRINCIPLES Chemical engineering has to do with industrial processes in which raw materials are changed or separated into useful products The chemical engineer must develop design and engineer both the complete process and the equipment used choose the proper raw materials operate the plants efficiently safely and economically and see to it that products meet the requirements set by the customers Chemical engineering is both an art and a science Whenever science helps the engineer to solve a problem science should be used When as is usually the case science does not give a complete answer it is necessary to use experience and judgment The professional stature of an engineer depends on skill in utilizing all sources of information to reach practical solutions to processing problems The variety of processes and industries that call for the services of chemical engineers is enormous Products of concern to chemical engineers range from commodity chemicals like sulfuric acid and chlorine to hightechnology items like polymeric lithographic supports for the electronics industry highstrength composite materials and genetically modified biochemical agents The processes described in standard treatises on chemical technology and the process industries give a good idea of the field of chemical engineering as does the 1988 report on the profession by the National Research Council Adsorption 810 Because of the variety and complexity of modern processes it is not practicable to cover the entire subject matter of chemical engineering under a single head The field is divided into convenient but arbitrary sectors This text covers that portion of chemical engineering known as the unit operations UNIT SYSTEMS The official international system of units is the SI system Système International dUnités Strong efforts are underway for its universal adoption as the exclusive system for all engineering and science but older systems particularly the centimetergramsecond cgs and footpoundsecond fps engineering gravitational systems are still in use and probably will be around for some time BASIC EQUATIONS The basic proportionalities each written as an equation with its own proportionality factor are F k1 dm u dt 11 F k2 m a m b r 2 12 Q c k3 W c 13 T k4 lim p V m p0 14 where F force t time m mass u velocity r distance W c work Q c heat p pressure V volume T thermodynamic absolute temperature k1 k2 k3 k4 proportionality factors STANDARDS By international agreement standards are fixed arbitrarily for the quantities mass length time temperature and the mole These are five of the base units of the SI system Currently the standards are as follows The standard of mass is the kilogram kg defined as the mass of the international kilogram a platinum cylinder preserved at Sèvres France The standard of length is the meter m defined since 1983 as the length of the path traveled by light in vacuum during a time interval of 1299792458 of a second The standard of time is the second s defined as 9192631770 frequency cycles of a certain quantum transition in an atom of 133Ce The standard of temperature is the kelvin K defined by assigning the value 27316 K to the temperature of pure water at its triple point the unique temperature at which liquid water ice and steam can exist at equilibrium The mole abbreviated mol is defined as the amount of a substance comprising as many elementary units as there are atoms in 12 g of 12C The definition of the mole is equivalent to the statement that the mass of one mole of a pure substance in grams is numerically equal to its molecular weight calculated from the standard table of atomic weights in which the atomic weight of carbon is given as 1201115 This number differs from 12 because it applies to the natural isotopic mixture of carbon rather than to pure 12C EVALUATION OF CONSTANTS From the basic standards values of m ma and mb in Eqs 11 and 12 are measured in kilograms r in meters and u in meters per second Constants k1 and k2 are not independent but are related by eliminating F from Eqs 11 and 12 This gives k1k2 dmudtmambr2 Either k1 or k2 may be fixed arbitrarily Then the other constant must be found by experiments in which inertial forces calculated by Eq 11 are compared with gravitational forces calculated by Eq 12 In the SI system k1 is fixed at unity and k2 found experimentally Equation 11 then becomes F ddt mu The force defined by Eq 15 and also used in Eq 12 is called the newton N From Eq 15 1 N 1 kgms² Constant k2 is denoted by G and called the gravitational constant Its recommended value is G 66726 1011 Nm²kg² 1 J 1 Nm 1 kgm²s² Power is measured in joules per second a unit called the watt W HEAT The constant k3 in Eq 13 may be fixed arbitrarily In the SI system it like k1 is set at unity Equation 13 becomes Qc Wc Heat like work is measured in joules TEMPERATURE The quantity pVm in Eq 14 may be measured in Nm²m³kg or Jkg With an arbitrarily chosen gas this quantity can be determined by measuring p and V of m kg of gas while it is immersed in a thermostat In this experiment only constancy of temperature not magnitude is needed Values of pVm at various pressures and at constant temperature can then be extrapolated to zero pressure to obtain the limiting value required in Eq 14 at the temperature of the thermostat For the special situation when the thermostat contains water at its triple point the limiting value is designated by pVm0 For this experiment Eq 14 gives 27316 k4 lim p0 pVm0 For an experiment at temperature T K Eq 14 can be used to eliminate k4 from Eq 110 giving T 27316 lim p0 pVmTlim p0 pVm0 Equation 111 is the definition of the Kelvin temperature scale from the experimental pressurevolume properties of a real gas CELSIUS TEMPERATURE In practice temperatures are expressed on the Celsius scale in which the zero point is set at the ice point defined as the equilibrium temperature of ice and airsaturated water at a pressure of one atmosphere Experimentally the ice point is found to be 001 K below the triple point of water and so it is at 27315 K The Celsius temperature C is defined by TC T K 27315 On the Celsius scale the experimentally measured temperature of the steam point which is the boiling point of water at a pressure of one atmosphere is 10000C DECIMAL UNITS In the SI system a single unit is defined for each quantity but named decimal multiples and submultiples also are recognized They are listed in Appendix 1 Time may be expressed in the nondecimal units minutes min hours h or days d STANDARD GRAVITY For certain purposes the acceleration of free fall in the earths gravitational field is used From deductions based on Eq 12 this quantity denoted by g is nearly constant It varies slightly with latitude and height above sea level For precise calculations an arbitrary standard gn has been set defined by gn 980665 ms² The natural unit of pressure in the SI system is the newton per square meter This unit called the pascal Pa is inconveniently small and a multiple called the bar also is used It is defined by 1 bar 1 x 10⁵ Pa 1 x 10⁵ Nm² A more common empirical unit for pressure used with all systems of units is the standard atmosphere atm defined by 1 atm 101325 x 10⁵ Pa 101325 bars The older cgs system can be derived from the SI system by making certain arbitrary decisions The standard for mass is the gram g defined by 1 g 1 x 10³ kg The standard for length is the centimeter cm defined by 1 cm 1 x 10² m Standards for time temperature and the mole are unchanged As in the SI system constant k1 in Eq 11 is fixed at unity The unit of force is called the dyne dyn defined by 1 dyn 1 gcms² The unit for energy and work is the erg defined by 1 erg 1 dyncm 1 10⁷ J 119 Constant k₃ in Eq 13 is not unity A unit for heat called the calorie cal is used to convert the unit for heat to ergs Constant 1k₃ is replaced by J which denotes the quantity called the mechanical equivalent of heat and is measured in joules per calorie Equation 13 becomes Wₑ JQₑ 120 Two calories are defined The thermochemical calorie calₜ used in chemistry is defined by 1 cal 41840 10⁷ ergs 41840 J 121 The international steam table calorie calₜʳ used in engineering is defined by 1 calₜʳ 41868 10⁷ ergs 41868 J 122 The calorie is so defined that the specific heat of water is approximately 1 calgC The standard acceleration of free fall in cgs units is gₙ 980665 cms² 123 Gas Constant If mass is measured in kilograms or grams constant k₄ in Eq 14 differs from gas to gas But when the concept of the mole as a mass unit is used k₄ can be replaced by the universal gas constant R which by Avogadros law is the same for all gases The numerical value of R depends only on the units chosen for energy temperature and mass Then Eq 14 is written lim pVnT R 124 where n is the number of moles This equation applies also to mixtures of gases if n is the total number of moles of all the molecular species that make up the volume V The accepted experimental value of R is R 831447 JKmol 831447 10⁷ ergsKmol 125 Values of R in other units for energy temperature and mass are given in Appendix 2 Although the mole is defined as a mass in grams the concept of the mole is easily extended to other mass units Thus the kilogram mole kg mol is the usual molecular or atomic weight in kilograms and the pound mole lb mol is that in avoirdupois pounds When the mass unit is not specified the gram mole g mol is intended Molecular weight M is a pure number FPS Engineering Units In some countries a nondecimal gravitational unit system has long been used in commerce and engineering The system can be derived from the SI system by making the following decisions The standard for mass is the avoirdupois pound lb defined by 1 lb 045359237 kg 126 The standard for length is the inch in defined as 254 cm This is equivalent to defining the foot ft as 1 ft 254 12 10² m 03048 m 127 The standard for time remains the second s The thermodynamic temperature scale is called the Rankine scale in which temperatures are denoted by degrees Rankine and defined by 1R 118 K 128 The ice point on the Rankine scale is 27315 18 49167R The analog of the Celsius scale is the Fahrenheit scale in which readings are denoted by degrees Fahrenheit It is derived from the Rankine scale by setting its zero point exactly 32F below the ice point on the Rankine scale so that TF TR 49167 32 TR 45967 129 The relation between the Celsius and the Fahrenheit scales is given by the exact equation TF 32 18TC 130 From this equation temperature differences are related by ΔTC 18ΔTF ΔT K The steam point is 21200F POUND FORCE The fps system is characterized by a gravitational unit of force called the pound force lb The unit is so defined that a standard gravitational field exerts a force of one pound on a mass of one avoirdupois pound The standard acceleration of free fall in fps units is to five significant figures gₙ 980665 ms²03048 mft 32174 fts² 132 The pound force is defined by 1 lb 32174 lbfts² 133 Then Eq 11 gives F lb dmudt32174 lbfts² 134 Equation 11 can also be written with 1gₑ in place of k₁ F dmudtgₑ 135 Comparison of Eqs 134 and 135 shows that to preserve both numerical equality and consistency of units in these equations it is necessary to define gₑ called the Newtonslaw proportionality factor for the gravitational force unit by gₑ 32174 lbfts²lb 136 The unit for work and mechanical energy in the fps system is the footpound force ftlb Power is measured by an empirical unit the horsepower hp defined by 1 hp 550 ftlbs 137 The unit for heat is the British thermal unit Btu defined by the implicit relation 1 BtulbF 1 calgC 138 As in the cgs system constant k₃ in Eq 13 is replaced by 1J where J is the mechanical equivalent of heat equal to 77817 ftlbBtu The definition of the Btu requires that the numerical value of specific heat be the same in both systems and in each case the specific heat of water is approximately 10 maximum precision in the final calculation and to take advantage of possible cancellations of numbers during the calculation Example 11 Using only exact definitions and standards calculate factors for converting a newtons to pounds force b British thermal units to IT calories c atmospheres to pounds force per square inch and d horsepower to kilowatts Solution a From Eqs 16 126 and 127 1 N 1 kgms2 1 lbfts2 045359237 x 03048 From Eq 132 1 lbfts2 03048980665 lbf and so 03048 1 N lbf 980665 x 045359237 x 03048 lbf 0224809 lbf In Appendix 3 it is shown that to convert newtons to pound force one should multiply by 0224809 Clearly to convert from pounds force to newtons multiply by 980665 x 045359237 4448222 b From Eq 138 1 Btu 1 calir 1 lb 1F 1 g 1C 1 calir 1 lb 1 kg 1F 1 g 1C From Eqs 116 126 and 131 045359237 x 1000 1 Btu 1 calir 251996 calir 18 c From Eqs 16 114 and 115 1 atm 101325 x 105 kgms2m2 From Eqs 126 127 and 136 since 1 ft 12 in 1 lbs2 03048 1 atm 101325 x 105 x 045359237 ft 32174 x 045359237 x 122 146959 lbfin2 d From Eqs 133 and 137 1 hp 550 ftlbfs 550 x 32174 ft2lbs3 Using Eqs 126 and 127 gives 1 hp 550 x 32174 x 045359237 x 030482 74570 Js Substituting from Eq 18 and dividing by 1000 1 hp 074570 kW Although conversion factors may be calculated as needed it is more efficient to use tables of the common factors A table for the factors used in this book is given in Appendix 3 Units and Equations Although Eqs 11 to 14 are sufficient for the description of unit systems they are but a small fraction of the equations needed in this book Many such equations contain terms that represent properties of substances and these are introduced as needed All new quantities are measured in combinations of units already defined and all are expressible as functions of the five base units for mass length time temperature and mole PRECISION OF CALCULATIONS In the above discussion the values of experimental constants are given with the maximum number of significant digits consistent with present estimates of the precision with which they are known and all digits in the values of defined constants are retained In practice such extreme precision is seldom necessary and defined and experimental constants can be truncated to the number of digits appropriate to the problem at hand although the advent of the digital computer makes it possible to retain maximum precision at small cost The engineer should use judgment in setting a suitable level of precision for the particular problem to be solved GENERAL EQUATIONS Except for the appearance of the proportionality factors gc and J the equations for all three unit systems are alike In the SI system neither constant appears in the cgs system gc is omitted and J retained in the fps system both constants appear In this text to obtain equations in a general form for all systems gc and J are included for use with fps units then either gc or both gc and J may be equated to unity when the equations are used in the cgs or SI systems Important principles in testing the dimensional consistency of an equation are as follows 1 The sums of the exponents relating to any given dimension length for example must be the same on both sides of the equation 2 An exponent must itself be dimensionlessa pure number 3 All the factors in the equation must be collectible into a set of dimensionless groups The groups themselves may carry exponents of any magnitude not necessarily whole numbers It is these exponents and any numerical coefficients in the equation that are determined by experiment Equations of State of Gases A pure gas consisting of n mol and held at a temperature T and pressure p will fill a volume V If any of the three quantities are fixed the fourth also is determined and only three of these quantities are independent This can be expressed by the functional equation fp T V n 0 Specific forms of this relation are called equations of state Many such equations have been proposed and several are in common use The most satisfactory equations of state can be written in the form PV nRT 1 B Vn C Vn2 D Vn3 Summing the exponents on the righthand side of Eq 143 gives For H 067 033 i For I 08 067 047 1 For L 16 067 02 047 2 For T 067 033 1 For M 08 033 047 0 These sums equal the exponents on the lefthand side thus Eq 142 is dimensionally homogeneous Equation 142 is given in dimensionless form on p 341 Eq 1230 z p ρMRT 1 ρMB ρMC ρMD All partial pressures in a given mixture add to the total pressure This applies to mixtures of both ideal and nonideal gases Specific heat JgC or BtulbF PROBLEMS 11 Using defined constants and conversion factors for mass length time and temperature calculate conversion factors for a footpounds force to kilowatthours b gallons 1 gal 231 in3 to liters 103 cm3 c Btu per pound mole to joules per kilogram mole Answers See Appendix 3 12 The BeattieBridgman equation a famous equation of state for real gases may be written p RT1 cvT3v2 u B01 bv A0v21 av 151 where a A0 b B0 and c are experimental constants and u is the molar volume 1g mol a Show that this equation can be put into the form of Eq 146 and derive equations for the virial coefficients B C and D in terms of the constants in Eq 151 b For air the constants are a 001931 A0 13012 b 001101 B0 004611 and c x 104 6600 all in cgs units atmospheres liters gram moles kelvins with R 008206 Calculate values of the virial coefficients for air in SI units c Calculate z for air at a temperature of 300 K and a molar volume of 0200 m3kg mol 13 A mixture of 25 percent ammonia gas and 75 percent air dry basis is passed upward through a vertical scrubbing tower to the top of which water is pumped Scrubbed gas containing 05 percent ammonia leaves the top of the tower and an aqueous solution containing 10 percent ammonia by weight leaves the bottom Both entering and leaving gas streams are saturated with water vapor The gas enters the tower at 378C and leaves at 211C The pressure of both streams and throughout the tower is 102 atm gauge The airammonia mixture enters the tower at a rate of 2832 m3min measured as dry gas at 156C and 1 atm What percentage of the ammonia entering the tower is not absorbed by the water How many cubic meters of water per hour are pumped to the top of the tower Answers 15 271 m2h 14 Dry gas containing 75 percent air and 25 percent ammonia vapor enters the bottom of a cylindrical packed absorption tower that is 2 ft in diameter Nozzles in the top of the tower distribute water over the packing A solution of ammonia in water is drawn from the bottom of the column and scrubbed gas leaves the top The gas enters at 80F and 760 mm Hg pressure It leaves at 60F and 730 mm The leaving gas contains on the dry basis 10 percent ammonia a If the entering gas flows through the empty bottom of the column at an average velocity upward of 15 fts how many cubic feet of entering gas are treated per hour b How many pounds of ammonia are absorbed per hour Answers a 16965 ft3h b 177 lb 15 An evaporator is fed continuously with 25 t metric tonsh of a solution consisting of 10 percent NaOH 10 percent NaCl and 80 percent H2O During evaporation water is boiled off and salt precipitates as crystals which are settled and removed from the remaining liquor The concentrated liquor leaving the evaporator contains 50 percent NaOH 2 percent NaCl and 48 percent H2O Calculate a the kilograms of water evaporated per hour b the kilograms of salt precipitated per hour and c the kilograms of concentrated liquor produced per hour Answers a 17600 kgh b 2400 kgh c 5000 kgh 16 Air is flowing steadily through a horizontal heated tube The air enters at 40F and at a velocity of 50 fts It leaves the tube at 140F and 75 fts The average specific heat of air is 024 BtulbF How many Btu per pound of air are transferred through the wall of the tube Answer 241 Btulb REFERENCES 1 Austin G T Shreves Chemical Process Industries 5th ed McGrawHill New York 1984 2 Bridgman P W Dimensional Analysis AMS Press New York 1978 3 CRC Handbook of Chemistry and Physics 69th ed CRC Press Boca Raton Fla 1988 p F191 4 Halladay D and R Resnick Fundamentals of Physics 3rd ed Wiley New York 1988 p 4 5 Moldover M R et al J Res Natl Bur Stand 93285 1988 6 Natl Bur Stand Tech News Bull 553 March 1971 7 National Research Council Frontiers in Chemical Engineering National Academy Press Washington DC 1988 8 Prausnitz J M R N Lichtenhaler and E G de Azevedo Molecular Theory of FluidPhase Equilibria PrenticeHall Englewood Cliffs NJ 1986 The behavior of fluids is important to process engineering generally and constitutes one of the foundations for the study of the unit operations An understanding of fluids is essential not only for accurately treating problems on the movement of fluids through pipes pumps and all kinds of process equipment but also for the study of heat flow and the many separation operations that depend on diffusion and mass transfer The branch of engineering science that has to do with the behavior of fluidsand fluids are understood to include liquids gases and vaporsis called fluid mechanics Fluid mechanics in turn is part of a larger discipline called continuum mechanics which also includes the study of stressed solids Fluid mechanics has two branches important to the study of unit operations fluid statics which treats fluids in the equilibrium state of no shear stress and fluid dynamics which treats fluids when portions of the fluid are in motion relative to other parts The chapters of this section deal with those areas of fluid mechanics that are important to unit operations The choice of subject matter is but a sampling of the huge field of fluid mechanics generally Chapter 2 treats fluid statics and some of its important applications Chapter 3 discusses the important phenomena appearing in flowing fluids Chapter 4 deals with the basic quantitative laws and equations of fluid flow Chapter 5 treats flow of incompressible fluids through pipes and in thin layers Chap 6 is on compressible fluids in flow and Chap 7 describes flow past solids immersed in the flowing fluid Chapter 8 deals with the important engineering tasks of moving fluids through process equipment and of measuring and controlling fluids in flow Finally Chap 9 covers mixing agitation and dispersion operations which in essence are applied fluid mechanics NATURE OF FLUIDS A fluid is a substance that does not permanently resist distortion An attempt to change the shape of a mass of fluid results in layers of fluid sliding over one another until a new shape is attained During the change in shape shear stresses exist the magnitudes of which depend upon the viscosity of the fluid and the rate of sliding but when a final shape has been reached all shear stresses will have disappeared A fluid in equilibrium is free from shear stresses At a given temperature and pressure a fluid possesses a definite density which in engineering practice is usually measured in kilograms per cubic meter or pounds per cubic foot Although the density of all fluids depends on the temperature and pressure the variation in density with changes in these variables may be small or large If the density changes only slightly with moderate changes in temperature and pressure the fluid is said to be incompressible if the changes in density are significant the fluid is said to be compressible Liquids are generally considered to be incompressible and gases compressible The terms are relative however and the density of a liquid can change appreciably if pressure and temperature are changed over wide limits Also gases subjected to small percentage changes in pressure and temperature act as incompressible fluids and density changes under such conditions may be neglected without serious error PRESSURE CONCEPT The basic property of a static fluid is pressure Pressure is familiar as a surface force exerted by a fluid against the walls of its container Pressure also exists at every point within a volume of fluid A fundamental question is What kind of quantity is pressure Is pressure independent of direction or does it vary with direction For a static fluid as shown by the following analysis pressure turns out to be independent of the orientation of any internal surface on which the pressure is assumed to act Choose any point O in a mass of static fluid and as shown in Fig 21 construct a cartesian system of coordinate axes with O as the origin The x and y axes are in the horizontal plane and the z axis points vertically upward Construct a plane ABC cutting the x y and z axes at distances from the origin of Δx Δy and Δz respectively Planes ABC AOC COB and AOB form a tetrahedron Let θ be the angle between planes ABC and COB This angle is less than 90 but otherwise is chosen at random Imagine that the tetrahedron is isolated as a free body and consider all forces acting on it in the direction of the z axis either from outside the fluid or from the surrounding fluid Three forces are involved 1 the force of gravity acting downward 2 the pressure force on plane COB acting upward and 3 the vertical component of the pressure force on plane ABC acting downward Since the fluid is in equilibrium the resultant of these forces is zero Also since a fluid in equilibrium cannot support shear stresses all pressure forces are normal to the surface on which they act Otherwise there would be shearforce components parallel to the faces The volume of the column let the pressure be p and the density be ρ As in the preceding analysis the resultant of all forces on the small volume of fluid of height dZ and crosssectional area S must be zero Three vertical forces are acting on this volume 1 the force from pressure p acting in an upward direction which is pS 2 the force from pressure p dp acting in a downward direction which is p dpS 3 the force of gravity acting downward which is ggcρS dZ Then or between the two definite heights Za and Zb shown in Fig 22 The pressure drop over any ring of rotating liquid is calculated as follows Consider the ring of liquid shown in Fig 23 and the volume element of thickness dr at a radius r The manometer is an important device for measuring pressure differences Figure 24 shows the simplest form of manometer Assume that the shaded portion of the U tube is filled with liquid A having a density ρA and that the arms of the U tube above the liquid are filled with fluid B having a density ρB Fluid B is immiscible with liquid A and less dense than A it is often a gas such as air or nitrogen Example 21 A manometer of the type shown in Fig 24 is used to measure the pressure drop across an orifice see Fig 819 Liquid A is mercury density 13590 kgm³ and fluid B flowing through the orifice and filling the manometer leads is brine density 1260 kgm³ When the pressures at the taps are equal the level of the mercury in the manometer is 09 m below the orifice taps Under operating conditions the gauge pressure at the upstream tap is 014 bar the pressure at the downstream tap is 250 mm Hg below atmospheric What is the reading of the manometer in millimeters Solution Call atmospheric pressure zero and note that gc 1 then the numerical data for substitution in Eq 210 are Pa 014 x 10⁵ 14000 Pa From Eq 25 Pb Zgggc 2501000 9806651 13590 33318 Pa Substituting in Eq 210 gives 14000 33318 Rm 980665 13590 1260 Rm 0391 m or 391 mm For measuring small differences in pressure the inclined manometer shown in Fig 25 may be used In this type one leg of the manometer is inclined in such a manner that for a small magnitude of Rm the meniscus in the inclined tube must move a considerable distance along the tube This distance is Rm divided by the sine of α the angle of inclination By making α small the magnitude of Rm is multiplied into a long distance R1 and a large reading becomes equivalent to a small pressure difference so pa pb ggc R1ρA ρB sin α 211 In this type of pressure gauge it is necessary to provide an enlargement in the vertical leg so that the movement of the meniscus in the enlargement is negligible within the operating range of the instrument CONTINUOUS GRAVITY DECANTER A gravity decanter of the type shown in Fig 26 is used for the continuous separation of two immiscible liquids of differing densities The feed mixture enters at one end of the separator the two liquids flow slowly through the vessel separate into two layers and discharge through overflow lines at the other end of the separator Provided the overflow lines are so large that frictional resistance to the flow of the liquids is negligible and provided they discharge at the same pressure as that in the gas space above the liquid in the vessel the performance of the decanter can be analyzed by the principles of fluid statics For example in the decanter shown in Fig 26 let the density of the heavy liquid be ρA and that of the light liquid be ρB The depth of the layer of heavy liquid is ZA1 and that of the light liquid is ZB The total depth of liquid in the vessel ZT is fixed by the position of the overflow line for the light liquid Heavy liquid discharges through an overflow leg connected to the bottom of the vessel and rising to a height ZA2 above the vessel floor The overflow lines and the top of the vessel are all vented to the atmosphere Since there is negligible frictional resistance to flow in the discharge lines the column of heavy liquid in the heavyliquid overflow leg must balance the somewhat greater depth of the two liquids in the vessel A hydrostatic balance leads to the equation ZBρB ZA1ρA ZA2ρA 212 Solving Eq 212 for ZA1 gives ZA1 ZA2 ZB ρBρA ZA2 ZT ZA1 ρBρA where the total depth of liquid in the vessel is ZT ZB ZA1 From this ZA1 ZA2 ZTρBρA 1 ρBρA 214 Equation 214 shows that the position of the liquidliquid interface in the separator depends on the ratio of the densities of the two liquids and on the elevations of the overflow lines It is independent of the rates of flow of the liquids Equation 214 shows that as ρA approaches ρB the position of the interface becomes very sensitive to changes in ZA2 the height of the heavyliquid leg With liquids that differ widely in density this height is not critical but with liquids of nearly the same density it must be set with care Often the top of the leg is made movable so that in service it can be adjusted to give the best separation The size of a decanter is established by the time required for separation which in turn depends on the difference between the densities of the two liquids and on the viscosity of the continuous phase Provided the liquids are clean and do not form emulsions the separation time may be estimated from the empirical equation t 100μρA ρB 215 where t separation time h ρA ρB densities of liquids A and B kgm³ μ viscosity of the continuous phase cP Equation 215 is not dimensionless and the indicated units must be used Example 22 A horizontal cylindrical continuous decanter is to separate 1500 bbld day 993 m³h of a liquid petroleum fraction from an equal volume of wash acid The oil is the continuous phase and at the operating temperature has a viscosity of 11 cP and a density of 54 lbft³ 865 kgm³ The density of the acid is 72 lbft³ 1153 kgm³ Compute a the size of the vessel and b the height of the acid overflow above the vessel floor Solution a The vessel size is found from the separation time Substitution in Eq 215 gives 100 x 11 t 038 h 1153 865 or 23 min Since 1 bbl 42 gal the rate of flow of each stream is 1500 x 42 24 x 60 438 galmin The total liquid holdup is 2 x 438 x 23 2014 gal The vessel should be about 95 percent full so its volume is 2014095 or 2120 gal 803 m³ The length of the tank should be about 5 times its diameter A tank 4 ft 122 m in diameter and 22 ft 610 m long would be satisfactory with standard dished heads on the ends its total volume would be 2124 gal b The fraction of the tank volume occupied by the liquid will be 95 percent and for a horizontal cylinder this means that the liquid depth will be 90 percent of the tank diameter Thus Zₜ 090 x 4 36 ft If the interface is halfway between the vessel floor and the liquid surface Zₐ₁ 180 ft Solving Eq 214 for Zₐ₂ the height of the heavyliquid overflow gives Zₐ₂ 180 360 1805472 315 ft 096 m Successful operation of a decanter depends on both the sedimentation and the coalescence of the dispersed phase Equation 215 gives poor results if the liquids to be separated are not clean but contain particulates or polymeric films that reduce the rate of coalescence Such contaminants may also lead to the formation of a dirty layer of uncoalesced dropletscalled a ragat the liquidliquid interface Coalescing devices such as beds of porous solids membranes or highvoltage fields are often necessary for a satisfactory separation With cleaner liquids the size of a decanter can often be greatly reduced by putting in horizontal or slightly inclined pipes or flat plates so that the heavyphase droplets need to fall only a short distance before reaching a layer of the heavy liquid CENTRIFUGAL DECANTER When the difference between the densities of the two liquids is small the force of gravity may be too weak to separate the liquids in a reasonable time The separation may then be accomplished in a liquidliquid centrifuge shown diagrammatically in Fig 27 It consists of a cylindrical metal bowl usually mounted vertically that rotates about its axis at high speed In Fig 27a the bowl is at rest and contains a quantity of two immiscible liquids of differing densities The heavy liquid forms a layer on the floor of the bowl beneath a layer of light liquid If the bowl is now rotated as in Fig 27b the heavy liquid forms a layer denoted as zone A in the figure next to the inside wall of the bowl A layer of light liquid denoted as zone B forms inside the layer of heavy liquid A cylindrical interface of radius rᵢ separates the two layers Since the force of gravity can be neglected in comparison with the much greater centrifugal force this interface is vertical It is called the neutral zone In operation of the machine the feed is admitted continuously near the bottom of the bowl Light liquid discharges at point 2 through ports near the axis of the bowl heavy liquid passes under a ring inward toward the axis of rotation and discharges over a dam at point 1 If there is negligible frictional resistance to the flow of the liquids as they leave the bowl the position of the liquidliquid interface is established by a hydrostatic balance and the relative heights radial distances from the axis of the overflow ports at 1 and 2 Assume that the heavy liquid of density ρₐ overflows the dam at radius rₐ and the light liquid of density ρb leaves through ports at radius rb Then if both liquids rotate with the bowl and friction is negligible the pressure difference in the light liquid between rb and rᵢ must equal that in the heavy liquid between rₐ and rᵢ The principle is exactly the same as in a continuous gravity decanter Thus pᵢ pb pᵢ pₐ From Eq 29 pᵢ pb ω²ρbvb² vᵢ² 2gₐ and pᵢ pₐ ω²ρₐr₂² rₐ² 2gₐ Equating these pressure drops and simplifying leads to ρbrb² rᵢ² ρₐrᵢ² rₐ² Pressure Nm2 or lbft2 pA at surface of heavy liquid in centrifuge pB at surface of light liquid in centrifuge pa at location a pb at location b pi at liquidliquid interface px py pz in x y z directions p1 at free liquid surface p2 at wall of centrifuge bowl p average pressure A centrifuge bowl 250mm ID internal diameter is turning at 4000 rmin It contains a layer of chlorobenzene 50 mm thick If the density of the chlorobenzene is 1109 kgm3 and the pressure at the liquid surface is atmospheric what gauge pressure is exerted on the wall of the centrifuge bowl The behavior of a flowing fluid depends strongly on whether or not the fluid is under the influence of solid boundaries In the region where the influence of the wall is small the shear stress may be negligible and the fluid behavior may approach that of an ideal fluid one that is incompressible and has zero viscosity Within the current of an incompressible fluid under the influence of solid boundaries four important effects appear 1 the coupling of velocitygradient and shearstress fields 2 the onset of turbulence 3 the formation and growth of boundary layers and 4 the separation of boundary layers from contact with the solid boundary In the flow of compressible fluids past solid boundaries additional effects appear arising from the significant density changes characteristic of compressible fluids These are considered in Chap 6 on flow of compressible fluids THE VELOCITY FIELD When a stream of fluid is flowing in bulk past a solid wall the fluid adheres to the solid at the actual interface between solid and fluid The adhesion is a result of the force fields at the boundary which are also responsible for the interfacial tension between solid and fluid If therefore the wall is at rest in the reference frame chosen for the solidfluid system the velocity of the fluid at the interface is zero Since at distances away from the solid the velocity is finite there must be variations in velocity from point to point in the flowing stream Therefore the velocity at any point is a function of the space coordinates of that point and a velocity field exists in the space occupied by the fluid The velocity at a given location may also vary with time When the velocity at each location is constant the field is invariant with time and the flow is said to be steady Onedimensional flow Velocity is a vector and in general the velocity at a point has three components one for each space coordinate In many simple situations all velocity vectors in the field are parallel or practically so and only one velocity component which may be taken as a scalar is required This situation which obviously is much simpler than the general vector field is called onedimensional flow an example is steady flow through straight pipe The following discussion is based on the assumptions of steady onedimensional flow LAMINAR FLOW At low velocities fluids tend to flow without lateral mixing and adjacent layers slide past one another like playing cards There are neither crosscurrents nor eddies This regime is called laminar flow At higher velocities turbulence appears and eddies form which as discussed later lead to lateral mixing VELOCITY GRADIENT AND RATE OF SHEAR Consider the steady onedimensional laminar flow of an incompressible fluid along a solid plane surface Figure 31a shows the velocity profile for such a stream The abscissa u is the velocity and the ordinate y is the distance measured perpendicular from the wall and therefore at right angles to the direction of the velocity At y 0 u 0 and u increases with distance from the wall but at a decreasing rate Focus attention on the velocities on two nearby planes plane A and plane B a distance Δy apart Let the velocities along the planes be uA and uB respectively and assume that uB uA Call Δu uB uA Define the velocity gradient at yA dudy by dudy lim Δy0 ΔuΔy The velocity gradient is clearly the reciprocal of the slope of the velocity profile of Fig 31a The local velocity gradient is also called the shear rate or time rate of shear The velocity gradient is usually a function of position in the stream and therefore defines a field as illustrated in Fig 31b THE SHEARSTRESS FIELD Since an actual fluid resists shear a shear force must exist wherever there is a time rate of shear In onedimensional flow the shear force acts parallel to the plane of the shear For example at plane C at distance yC from the wall the shear force Fs shown in Fig 31a acts in the direction shown in the figure This force is exerted by the fluid outside of plane C on the fluid between plane C and the wall By Newtons third law an equal and opposite force Fs acts on the fluid outside of plane C from the fluid inside plane C It is convenient to use not total force Fs but the force per unit area of the shearing plane called the shear stress and denoted by τ or τ FsAs where As is the area of the plane Since τ varies with y the shear stress also constitutes a field Shear forces are generated in both laminar and turbulent flow The shear stress arising from viscous or laminar flow is denoted by τv The effect of turbulence is described later NEWTONIAN AND NONNEWTONIAN FLUIDS The relationships between the shear stress and shear rate in a real fluid are part of the science of rheology Figure 32 shows several examples of the rheological behavior of fluids The curves are plots of shear stress vs rate of shear and apply at constant temperature and pressure The simplest behavior is that shown by curve A which is a straight line passing through the origin Fluids following this simple linearity are called newtonian fluids Gases and most liquids are newtonian The other curves shown in Fig 32 represent the rheological behavior of liquids called nonnewtonian plastics Line C represents a pseudoplastic fluid The curve passes through the origin is concave downward at low shears and becomes nearly linear at high shears Rubber latex is an example of such a fluid Curve D represents a dilatant fluid The curve is concave upward at low shears and almost linear at high shears Quicksand and some sandfilled emulsions show this behavior Pseudoplastics are said to be shear rate thinning and dilatant fluids shear rate thickening Timedependent flow None of the curves in Fig 32 depends on the history of the fluid and a given sample of material shows the same behavior no matter how TABLE 31 Rheological characteristics of fluids Designation Effect of increasing shear rate Time dependent Pseudoplastic Thins No Thixotropic Thins Yes Newtonian None No Dilatant Thickens No Rheopectic Thickens Yes long the shearing stress has been applied Such is not the case for some nonnewtonian liquids whose stressvsrateofshear curves depend on how long the shear has been active Thixotropic liquids break down under continued shear and on mixing give lower shear stress for a given shear rate that is their apparent viscosity decreases with time Rheopectic substances behave in the reverse manner and the shear stress increases with time as does the apparent viscosity The original structures and apparent viscosities are usually recovered on standing The rheological characteristics of fluids are summarized in Table 31 VISCOSITY In a newtonian fluid the shear stress is proportional to the shear rate and the proportionality constant is called the viscosity τv μ du dy 33 In SI units τv is measured in newtons per square meter and μ in kilograms per metersecond or pascalsecond In the cgs system viscosity is expressed in grams per centimetersecond and this unit is called the poise P Viscosity data are generally reported in centipoises cP 001 P 1 mPas since most fluids have viscosities much less than 1 P In English units viscosity is defined using the Newtonslaw conversion factor gc and the units of μ are pounds per footsecond or pounds per foothour The defining equation is τv μ du gc dy 34 Conversion factors among the different systems are given in Table 32 VISCOSITY AND MOMENTUM FLUX Although Eq 33 serves to define the viscosity of a fluid it can also be interpreted in terms of momentum flux The moving fluid just above plane C in Fig 31 has slightly more momentum in the x direction than the fluid just below this plane By molecular collisions momentum is transferred from one layer to the other tending to speed up the slower moving layer and to slow down the faster moving one Thus momentum passes across plane C to the fluid in the layer below this layer transfers momentum to the next lower layer and so on Hence xdirection momentum is transferred in the y direction all the way to the wall bounding the fluid where u 0 and is delivered to the wall as wall shear Shear stress at the wall is denoted by τw The units of momentum flux τx are kgmsm2s or kgms2 the same as the units for τ since Nm2 equals kgms2 Equation 33 therefore states that the momentum flux normal to the direction of flow of the fluid is proportional to the velocity gradient with the viscosity as the proportionality factor Momentum transfer is analogous to conductive heat transfer resulting from a temperature gradient where the proportionality factor between the heat flux and temperature gradient is called the thermal conductivity In laminar flow momentum is transferred as a result of the velocity gradient and the viscosity may be regarded as the conductivity of momentum transferred by this mechanism VISCOSITIES OF GASES AND LIQUIDS The viscosity of a newtonian fluid depends primarily on temperature and to a lesser degree on pressure The viscosity of a gas increases with temperature approximately in accordance with an equation of the type μ μ0 T 273n where μ viscosity at absolute temperature T K μ0 viscosity at 0C 273 K n constant Viscocities of gases have been intensively studied in kinetic theory and accurate and elaborate tables of temperature coefficients are available Exponent n 065 for air it is approximately 09 for carbon dioxide and simple hydrocarbons and about 11 for sulfur dioxide and steam The viscosity of a gas is almost independent of pressure in the region of pressures where the gas laws apply In this region the viscosities of gases are generally between 001 and 01 cP see Appendix 8 for the viscosities of common gases at 1 atm At high pressures gas viscosity increases with pressure especially in the neighborhood of the critical point The viscosities of liquids are generally much greater than those of gases and cover several orders of magnitude Liquid viscosities decrease significantly when the temperature is raised For example the viscosity of water falls from 179 cP at 0C to 028 cP at 100C The viscosity of a liquid increases with pressure but the effect is generally insignificant at pressures less than 40 atm Data for common liquids over a range of temperatures are given in Appendix 9 The absolute viscosities of fluids vary over an enormous range of magnitudes from about 01 cP for liquids near their boiling point to as much as 106 P for polymer melts Most extremely viscous materials are nonnewtonian and possess no single viscosity independent of shear rate Kinematic viscosity The ratio of the absolute viscosity to the density of a fluid μ ρ is often useful This property is called the kinematic viscosity and designated by ν In the SI system the unit for ν is square meters per second In the cgs system the kinematic viscosity is called the stoke St defined as 1 cm2s The fps unit is square feet per second Conversion factors are 1 m2s 104 St 107639 ft2s For liquids kinematic viscosities vary with temperature over a somewhat narrower range than absolute viscosities For gases the kinematic viscosity increases more rapidly with temperature than does the absolute viscosity RATE OF SHEAR VERSUS SHEAR STRESS FOR NONNEWTONIAN FLUIDS Bingham plastics like that represented by curve B in Fig 32 follow a rheological equation of the type τv gc τ0 gc K du dy 36 where K is a constant Over some range of shear rates dilatant and pseudoplastic fluids often follow a power law also called the Ostwaldde Waele equation τv gc K du dyn 37 where K and n are constants called the flow consistency index and the flow behavior index respectively Such fluids are known as powerlaw fluids For pseudoplastics curve C n 1 and for dilatant fluids curve D n 1 Clearly n 1 for newtonian fluids TURBULENCE It has long been known that a fluid can flow through a pipe or conduit in two different ways At low flow rates the pressure drop in the fluid increases directly with the fluid velocity at high rates it increases much more rapidly roughly as the square of the velocity The distinction between the two types of flow was first demonstrated in a classic experiment by Osborne Reynolds reported in 1883 A horizontal glass tube was immersed in a glasswalled tank filled with water A controlled flow of water could be drawn through the tube by opening a valve The entrance to the tube was flared and provision was made to introduce a fine filament of colored water from the overhead flask into the stream at the tube entrance Reynolds found that at low flow rates the jet of colored water flowed intact along with the mainstream and no cross mixing occurred The behavior of the color band showed clearly that the water was flowing in parallel straight lines and that the flow was laminar When the flow rate was increased a velocity called the critical velocity was reached at which the thread of color became wavy and gradually disappeared as the dye spread uniformly throughout the entire cross section of the stream of water This behavior of the colored water showed that the water no longer flowed in laminar motion but moved erratically in the form of crosscurrents and eddies This type of motion is turbulent flow REYNOLDS NUMBER AND TRANSITION FROM LAMINAR TO TURBULENT FLOW Reynolds studied the conditions under which one type of flow changes into the other and found that the critical velocity at which laminar flow changes into turbulent flow depends on four quantities the diameter of the tube and the viscosity density and average linear velocity the liquid Furthermore he found that these four factors can be combined into one group and that the change in kind of flow occurs at a definite value of the group The grouping of variables so found was N Re D V ρ μ D V ν where D diameter of tube V average velocity of liquid Eq 44 μ viscosity of liquid ρ density of liquid ν kinematic viscosity of liquid The dimensionless group of variables defined by Eq 38 is called the Reynolds number N Re It is one of the named dimensionless groups listed in Appendix 4 Its magnitude is independent of the units used provided the units are consistent Additional observations have shown that the transition from laminar to turbulent flow actually may occur over a wide range of Reynolds numbers In a pipe flow is always laminar at Reynolds numbers below 2100 but laminar flow can persist up to Reynolds numbers of several thousand under special conditions of wellrounded tube entrance and very quiet liquid in the tank Under ordinary conditions the flow in a pipe or tube is turbulent at Reynolds numbers above about 4000 Between 2100 and 4000 a transition region is found where the flow may be either laminar or turbulent depending upon conditions at the entrance of the tube and on the distance from the entrance