ISBN: 0-8247-0444-4 This book is printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above Copyright # 2001 by Marcel Dekker, Inc All Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit): 10 PRINTED IN THE UNITED STATES OF AMERICA Preface The objectives of this book are twofold: (1) for the student, to show how the fundamental principles underlying the behavior of fluids (with emphasis on one-dimensional macroscopic balances) can be applied in an organized and systematic manner to the solution of practical engineering problems, and (2) for the practicing engineer, to provide a ready reference of current information and basic methods for the analysis of a variety of problems encountered in practical engineering situations The scope of coverage includes internal flows of Newtonian and nonNewtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas–liquid, solid– liquid, and gas–solid) flows, external flows (e.g., drag), and flow in porous media Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g., iii iv Preface centrifugal, sedimentation, filtration), packed columns, fluidized beds, sedimentation, solids transport in slurry and pneumatic flow, and frozen and flashing two-phase gas–liquid flows The treatment is from the viewpoint of the process engineer, who is concerned with equipment operation, performance, sizing, and selection, as opposed to the details of mechanical design or the details of flow patterns in such situations For the student, this is a basic text for a first-level course in process engineering fluid mechanics, which emphasizes the systematic application of fundamental principles (e.g., macroscopic mass, energy, and momentum balances and economics) to the analysis of a variety of fluid problems of a practical nature Methods of analysis of many of these operations have been taken from the recent technical literature, and have not previously been available in textbooks This book includes numerous problems that illustrate these applications at the end of each chapter For the practicing engineer, this book serves as a useful reference for the working equations that govern many applications of practical interest, as well as a source for basic principles needed to analyze other fluid systems not covered explicitly in the book The objective here is not to provide a mindless set of recipes for rote application, however, but to demonstrate an organized approach to problem analysis beginning with basic principles and ending with results of very practical applicability Chemical Engineering Fluid Mechanics is based on notes that I have complied and continually revised while teaching the junior-level fluid mechanics course for chemical engineering students at Texas A&M University over the last 30 years It has been my experience that, when being introduced to a new subject, students learn best by starting with simple special cases that they can easily relate to physically, and then progressing to more generalized formulations and more complex problems That is the philosophy adopted in this book It will certainly be criticized by some, since it is contrary to the usual procedure followed by most textbooks, in which the basic principles are presented first in the most general and mathematical form (e.g., the divergence theorem, Reynolds transport theorem, Navier Stokes equations, etc.), and the special cases are then derived from these Esoterically, it is very appealing to progress from the general to the specific, rather than vice versa However, having taught from both perspectives, it is my observation that most beginning students not gain an appreciation or understanding from the very general, mathematically complex, theoretical vector expressions until they have gained a certain physical feel for how fluids behave, and the laws governing their behavior, in special situations to which they can easily relate They also understand and appreciate the principles much better if they see how they can be applied to the analysis of practical and useful situations, with results that actually work Preface v in practice That is why the multi-dimensional vector generalizations of the basic conservations laws have been eschewed in favor of the simpler component and one-dimensional form of these laws It is also important to maintain a balanced perspective between fundamental, or theoretical, and empirical information, for the practicing engineer must use both to be effective It has been said that all the tools of mathematics and physics in the world are not sufficient to calculate how much water will flow in a given time from a kitchen tap when it is opened However, by proper formulation and utilization of certain experimental observations, this is a routine problem for the engineer The engineer must be able to solve certain problems by direct application of theoretical principles only (e.g., laminar flow in uniform conduits), others by utilizing hypothetical models that account for a limited understanding of the basic flow phenomena by incorporation of empirical parameters (e.g., :turbulent flow in conduits and fittings), and still other problems in which important information is purely empirical (e.g., pump efficiencies, two-phase flow in packed columns) In many of these problems (of all types), application of dimensional analysis (or the principle of ‘‘conservation of dimensions’’) for generalizing the results of specific analysis, guiding experimental design, and scaling up both theoretical and experimental results can be a very powerful tool This second edition of the book includes a new chapter on two-phase flow, which deals with solid–liquid, solid–gas, and frozen and flashing liquid–gas systems, as well as revised, updated, and extended material throughout each chapter For example, the method for selecting the proper control valve trim to use with a given piping configuration is presented and illustrated by example in Chapter 10 The section on cyclone separators has been completely revised and updated, and new material has been incorporated in a revision of the material on particles in non-Newtonian fluids Changes have made throughout the book in an attempt to improve the clarity and utility of the presentation wherever possible For example, the equations for compressible flow in pipes have been reformulated in terms of variables that are easier to evaluate and represent in dimensionless form It is the aim of this book to provide a useful introduction to the simplified form of basic governing equations and an illustration of a consistent method of applying these to the analysis of a variety of practical flow problems Hopefully, the reader will use this as a starting point to delve more deeply into the limitless expanse of the world of fluid mechanics Ron Darby Contents Preface Unit Conversion Factors iii xvi BASIC CONCEPTS I 1 2 II III FUNDAMENTALS A Basic Laws B Experience OBJECTIVE PHENOMENOLOGICAL RATE OR TRANSPORT LAWS A Fourier’s Law of Heat Conduction B Fick’s Law of Diffusion C Ohm’s Law of Electrical Conductivity D Newton’s Law of Viscosity 5 vii viii Contents IV V THE ‘‘SYSTEM’’ TURBULENT MACROSCOPIC (CONVECTIVE) TRANSPORT MODELS PROBLEMS NOTATION DIMENSIONAL ANALYSIS AND SCALE-UP 15 INTRODUCTION UNITS AND DIMENSIONS A Dimensions B Units C Conversion Factors III CONSERVATION OF DIMENSIONS A Numerical Values B Consistent Units IV DIMENSIONAL ANALYSIS A Pipeline Analysis B Uniqueness C Dimensionless Variables D Problem Solution E Alternative Groups V SCALE-UP VI DIMENSIONLESS GROUPS IN FLUID MECHANICS VII ACCURACY AND PRECISION PROBLEMS NOTATION 10 11 13 I II 15 16 16 18 19 20 21 22 22 25 28 28 29 29 30 FLUID PROPERTIES IN PERSPECTIVE 55 I II III CLASSIFICATION OF MATERIALS AND FLUID PROPERTIES DETERMINATION OF FLUID VISCOUS (RHEOLOGICAL) PROPERTIES A Cup-and-Bob (Couette) Viscometer B Tube Flow (Poiseuille) Viscometer TYPES OF OBSERVED FLUID BEHAVIOR A Newtonian Fluid B Bingham Plastic Model C Power Law Model 35 35 40 52 55 59 60 63 64 65 65 66 Contents ix D Structural Viscosity Models TEMPERATURE DEPENDENCE OF VISCOSITY A Liquids B Gases V DENSITY PROBLEMS NOTATION REFERENCES 67 71 71 72 72 73 83 84 FLUID STATICS 85 IV I II STRESS AND PRESSURE THE BASIC EQUATION OF FLUID STATICS A Constant Density Fluids B Ideal Gas—Isothermal C Ideal Gas—Isentropic D The Standard Atmosphere III MOVING SYSTEMS A Vertical Acceleration B Horizontally Accelerating Free Surface C Rotating Fluid IV BUOYANCY V STATIC FORCES ON SOLID BOUNDARIES PROBLEMS NOTATION 85 86 88 89 90 90 91 91 92 93 94 94 96 104 CONSERVATION PRINCIPLES 105 I II 105 106 106 107 108 110 112 113 116 120 121 123 127 III IV V THE SYSTEM CONSERVATION OF MASS A Macroscopic Balance B Microscopic Balance CONSERVATION OF ENERGY A Internal Energy B Enthalpy IRREVERSIBLE EFFECTS A Kinetic Energy Correction CONSERVATION OF MOMENTUM A One-Dimensional Flow in a Tube B The Loss Coefficient C Conservation of Angular Momentum x Contents D Moving Boundary Systems and Relative Motion E Microscopic Momentum Balance PROBLEMS NOTATION 149 FLOW REGIMES GENERAL RELATIONS FOR PIPE FLOWS A Energy Balance B Momentum Balance C Continuity D Energy Dissipation III NEWTONIAN FLUIDS A Laminar Flow B Turbulent Flow C All Flow Regimes IV POWER LAW FLUIDS A Laminar Flow B Turbulent Flow C All Flow Regimes V BINGHAM PLASTICS A Laminar Flow B Turbulent Flow C All Reynolds Numbers VI PIPE FLOW PROBLEMS A Unknown Driving Force B Unknown Flow Rate C Unknown Diameter D Use of Tables VII TUBE FLOW (POISEUILLE) VISCOMETER VIII TURBULENT DRAG REDUCTION PROBLEMS NOTATION REFERENCES PIPE FLOW I II 128 130 134 146 149 151 151 152 153 153 154 154 155 164 164 165 166 166 167 168 169 169 169 170 172 174 177 177 178 184 192 193 INTERNAL FLOW APPLICATIONS 195 I 195 195 198 NONCIRCULAR CONDUITS A Laminar Flows B Turbulent Flows Two-Phase Flow 457 constant diameter pipe, as was done in Chapter for single-phase flow (see Fig 5-6) For steady uniform flow through area Ax , X Fx ẳ ẳ dFxp ỵ dFxg ỵ dFxw ẳ Ax dP ẵS "m ị ỵ G "m gAx dz ẵwS ỵ wG Wp dX ð15-31Þ where wS and wG are the effective wall stresses resulting from energy dissipation due to the particle–particle as well as particle–wall and gas–wall interaction, and Wp is the wetted perimeter Dividing by Ax , integrating, and solving for the pressure drop, ÀÁP ¼ P1 À P2 , ÀÁP ¼ ẵS "m ị ỵ G "m g z ỵ wS ỵ wG ị4L=Dh 15-32ị where Dh ẳ 4Ax =Wp is the hydraulic diameter The void fraction "m is the volume fraction of gas in the pipe, i.e., "m ¼ À _ x mS ¼ S VS A x ỵ S1 xịG =S 15-33ị The wall stresses are related to corresponding friction factors by wS ¼ ÁPfS fS S "m ịVS ẳ 4L=Dh 15-34aị wG ẳ PfG fG " V2 ẳ m G 4L=Dh ð15-34bÞ Here ÁPfG is the pressure drop due to ‘‘gas only’’ flow (i.e., the gas flowing alone in the full pipe cross section) Note that if the pressure drop is less than about 30% of P1 , the incompressible flow equations can be used to determine ÁPfG by using the average gas density Otherwise, the compressibility must be considered and the methods in Chapter used to determine ÁPfG The pressure drop is related to the pressure ratio P1 =P2 by   P2 ð15-35Þ P1 À P2 ¼ À P1 P1 The solids contribution to the pressure drop, ÁPfS , is a consequence of both the particle–wall and particle–particle interactions The latter is reflected in the dependence of the friction factor fS on the particle diameter, drag coefficient, density, and relative (slip) velocity by (Hinkel, 1953):      S D VG VS 15-36ị Cd fS ẳ G d VS 458 Chapter 15 A variety of other expressions for fS have been proposed by various authors (see, e.g., Klinzing et al., 1997), such as that of Yang (1983) for horizontal flow,  "  #À1:15 NRet VG =" 1À" p fS ẳ 0:117 "ị 15-37ị NRep gD "3 and for vertical flow,  1À" fS ¼ 0:0206 "3 " NRet ð1 À "Þ NRep #À0:869 ð15-38Þ where NRet ¼ dVt G ; G NRep ¼ dðVG =" À VS ÞG G ð15-39Þ and Vt is the particle terminal velocity Vertical Transport The principles governing vertical pneumatic transport are the same as those just given, and the method for determining the pressure drop is identical (with an appropriate expression for fP ) However, there is one major distinction in vertical transport, which occurs as the gas velocity is decreased As the velocity drops, the frictional pressure drop decreases but the slip increases, because the drag force exerted by the gas entraining the particles also decreases The result is an increase in the solids holdup, with a corresponding increase in the static head opposing the flow, which in turn causes an increase in the pressure drop A point will be reached at which the gas can no longer entrain all the solids and a slugging, fluidized bed results with large pressure fluctuations This condition is known as choking (not to be confused with the choking that occurs when the gas velocity reaches the speed of sound) and represents the lowest gas velocity at which vertical pneumatic transport can be attained at a specified solids mass flow rate The choking velocity, VC , and the corresponding void fraction, "C , are related by the two equations (Yang, 1983) VC VS ẳ1ỵ Vt Vt "C ị 15-40ị  2:2 2gD"4:7 1ị C o ẳ 6:81 10 S ðVC À Vt Þ2 ð15-41Þ and These two equations must be solved simultaneously for VC and "C Two-Phase Flow IV 459 GAS–LIQUID TWO-PHASE PIPE FLOW The two-phase flow of gases and liquids has been the subject of literally thousands of publications in the literature, and it is clear that we can provide only a brief introduction to the subject here Although the single phase flow of liquids and gases is relatively straightforward, the twophase combined flow is orders of magnitude more complex Two-phase gas–liquid flows are also more complex than fluid–solid flows because of the wider variety of possible flow regimes and the possibility that the liquid may be volatile and/or the gas a condensable vapor, with the result that the mass ratio of the two phases may change throughout the system A Flow Regimes The configuration or distribution of the two phases in a pipe depends on the phase ratio and the relative velocities of the phases These regimes can be described qualitatively as illustrated in Fig 15-5a for horizontal flow and in Fig 15-5b for vertical flow The patterns for horizontal flow are seen to be more complex than those for vertical flow because of the asymmetrical effect of gravity The boundaries or transitions between these regimes have been mapped by various investigators on the basis of observations in terms of various flow and property parameters A number of these maps have been compared by Rouhani and Sohal (1983) Typical flow regime maps for horizontal and vertical flow are shown in Figs 15-6a and 15-6b _ In Figures 15-5 and 15-6, GG ¼ mG =A is the mass flux of the gas, GL ¼ _ mL =A is the mass flux of the liquid, and  and È are fluid property correction factors:   1=2  L ẳ G 15-42ị A W "   #1=2 W L W ȼ L W L ð15-43Þ where  is the surface tension and the subscripts W and A refer to water and air, respectively, at 208C A quantitative model for predicting the flow regime map for horizontal flow in terms of five dimensionless variables was developed by Taitel and Duckler (1976) The momentum equation written for a differential length of pipe containing the two-phase mixture is similar to Eq (15-29), except that the rate of momentum changes along the tube due to the change in 460 FIGURE 15-5 Chapter 15 Flow regimes in (a) horizontal and (b) vertical gas–liquid flow Two-Phase Flow 461 FIGURE 15-6 Flow regime maps for (a) horizontal and (b) vertical gas–liquid flow (a, From Baker, 1954; b, from Hewitt and Roberts, 1969.) 462 Chapter 15 velocity as the gas or vapor expands For steady uniform flow through area Ax , X _ _ dFx ẳ dẵmG VG ỵ mL VL ẳ dFxP ỵ dFxG ỵ dFxW ẳ ẳ Ax dP ẵL "m ị ỵ G "m Ax dz ẵwL ỵ wG Wp dX 15-44ị where wP and wG are the stresses exerted by the particles and the gas on the wall and Wp is the wetted perimeter Dividing by Ax dx and solving for the pressure gradient, ÀdP=dX, gives   dP dz ẳ ẵL "m ị ỵ G "m g þ ðwL þ wG Þ À dX dX Dh þ d _ _ m V ỵ mL VL ị Ax dX G G 15-45ị where Dh ẳ 4Ax =Wp is the hydraulic diameter The total pressure gradient is seen to be composed of three terms resulting from the static head change (gravity), energy dissipation (friction loss), and acceleration (the change in kinetic energy):       dP dP dP dP ẳ 15-46ị dX dX g dX f dX acc This is comparable to Eq (9-14) for pure gas flow Homogeneous Gas–Liquid Models In principle, the energy dissipation (friction loss) associated with the gas– liquid, gas–wall, and liquid–wall interactions can be evaluated and summed separately However, even for distributed (nonhomogeneous) flows it is common practice to evaluate the friction loss as a single term, which, however, depends in a complex manner on the nature of the flow and fluid properties in both phases This is referred to as the ‘‘homogeneous’’ model: !   dP 4fm m Vm 2f G2 À ¼ 15-47ị ẳ m m dX f Dh  m Dh The homogeneous model also assumes that both phases are moving at the same velocity, i.e., no slip Because the total mass flux is constant, the acceleration (or kinetic energy change) term can be written   dP dV d 15-48ị ẳ m Vm m ẳ G2 m m dX acc dX dX Two-Phase Flow 463 where m ¼ 1=m is the average specific volume of the homogeneous twophase mixture: m ẳ x 1x ẳ ỵ ẳ G x ỵ xịL m G L 15-49ị and x is the quality (i.e., the mass fraction of gas) For ‘‘frozen’’ flows in which there is no phase change (e.g., air and cold water), the acceleration term is often negligible in steady pipe flow (although it can be appreciable in entrance flows and in nonuniform channels) However, if a phase change occurs (e.g., flashing of hot water or other volatile liquid), this term can be very significant Evaluating the derivative of m from Eq (15-49) gives dm d dx d dP dx ẳx G ỵ GL ẳ x G þ ðG À L Þ dX dX dX dX dP dx 15-50ị where GL ẳ G L The first term on the right describes the effect of the gas expansion on the acceleration for constant mass fraction, and the last term represents the additional acceleration resulting from a phase change from liquid to gas (e.g., a flashing liquid) Substituting the expressions for the acceleration and friction loss pressure gradients into Eq (15-45) and rearranging gives 2fm G2 dx dz m ỵ m g ỵ G2 GL m dP dX dX  D ¼ m À dG dX þ G2 x m dP ð15-51Þ Finding the pressure drop corresponding to a total mass flux Gm from this equation requires a stepwise procedure using physical property data from which the densities of both the gas phase and the mixture can be determined as a function of pressure For example, if the upstream pressure P1 and the mass flux Gm are known, the equation is used to evaluate the pressure gradient at point and hence the change in pressure ÁP over a finite length ÁL, and hence the pressure P1ỵi ẳ P1 P The densities are then determined at pressure P1ỵi And the process is repeated at successive increments until the end of the pipe is reached A number of special cases permit simplification of the equation For example, if the pressure is high and the pressure gradient moderate, the term in the denominator that represents the acceleration due to gas expansion can be neglected Likewise for ‘‘frozen’’ flow, for which there is no phase change (e.g., air and cold water), the quality x is constant and the second term in the numerator is zero For flashing flows, the change in quality with length (dx/dX) must be determined from a total energy balance from the pipe inlet (or stagnation) conditions, along with the appropriate vapor–liquid 464 Chapter 15 equilibrium data for the flashing liquid If the Clausius–Clapeyron equation is used, this becomes   2 Cp T @G GL ẳ @P T 2 GL 15-52ị where GL is the heat of vaporization and GL is the change in specific volume at vaporization For an ideal gas,   @G ; ¼À P @P T   P1=k @G ẳ @P s 1 kP1ỵkị=k 15-53ị It should be noted that the derivative is negative, so that at certain conditions the denominator of Eq (15-51) can be zero, resulting in an infinite pressure gradient This condition corresponds to the speed of sound, i.e., choked flow For a nonflashing liquid and an ideal gas mixture, the cor* responding maximum (choked) mass flux Gm follows directly from the definition of the speed of sound:    1=2 rffiffiffiffiffiffiffiffiffiffiffiffi @P m kP Gm ¼ cm m ¼ m k * ẳ @m T " 15-54ị The ratio of the sonic velocity in a homogeneous two-phase mixture to that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in a gas alone is cm =c ¼ G ="m ẳ L =L "1 "ị This ratio can be much smaller than unity, so choking can occur in a two-phase mixture at a significantly higher downstream pressure than for single phase gas flow (i.e., at a lower pressure drop and a correspondingly lower mass flux) Evaluation of each term in Eq (15-51) is straightforward, except for the friction factor One approach is to treat the two-phase mixture as a ‘‘pseudo-single phase’’ fluid, with appropriate properties The friction factor is then found from the usual Newtonian methods (Moody diagram, Churchill equation, etc.) using an appropriate Reynolds number: NRe;TP ¼ DGm m ð15-55Þ where m is an appropriate viscosity for the two-phase mixture A wide variety of methods have been proposed for estimating this viscosity, but one that seems logical is the local volume-weighted average (Duckler et al., 1964b): m ẳ "G ỵ "ịL 15-56ị Two-Phase Flow 465 The corresponding density  is the ‘‘no-slip’’ or equilibrium density of the mixture:  ẳ "G ỵ "ịL ẳ x=G ỵ xị=L 15-57ị Note that the frictional pressure gradient is inversely proportional to the fluid density:   @P 2f G2 15-58ị ẳ m m À @X f D The corresponding pressure gradient for purely liquid flow is   @P 2f G2 À ¼ L L @X fL L D ð15-59Þ Taking the reference liquid mass flux to be the same as that for the twophase flow (GL ¼ Gm ) and the friction factors to be the same ( fL ¼ fm ), then        @P  @P  @P ẳ L ẳ x Lỵ1x ð15-60Þ À @X fm  @X fL @X fL  A similar relationship could be written by taking the single phase gas flow as the reference instead of the liquid, i.e., GG ¼ Gm This is the basis for the two-phase multiplier method:     @P @P ¼ È2 À ð15-61Þ À R @X fm @X fR where R represents a reference single-phase flow, and È2 is the two-phase R multiplier There are four possible reference flows: R¼L The total mass flow is liquid (Gm ¼ GL ) R¼G The total mass flow is gas (Gm ¼ GG ) R ¼ LLm The total mass flow is that of the liquid only in the mixture ẵGLm ẳ xịGm R ẳ GGm The total mass flow is that of the gas only in the mixture ðGGm ¼ xGm ) The two-phase multiplier method is used primarily for separated flows, which will be discussed later Omega Method for Homogeneous Equilibrium Flow For homogeneous equilibrium (no-slip) flow in a uniform pipe, the governing equation is [equivalent to Eq (15-45)] 466 Chapter 15 dP d 2f  G2 ỵ G2 m ỵ m m m ỵ g z ẳ m dX dX 2D 15-62ị where m ¼ 1=m By integrating over the pipe length L, assuming the friction factor to be constant, this can be rearranged as follows: 4fm L m ỵ G2 dm =dPịdp m ẳ Kf ẳ D ẵG2 2 =2ị þ ðgD=4fm ÞðÁz=Lފ m m ð15-63Þ Leung (1996) used a linearized two-phase equation of state to evaluate m ẳ fnPị:   m P  ẳ! 01 ỵ1ẳ 15-64ị o P m where 0 is the two-phase density at the upstream (stagnation) pressure P0 The parameter ! represents the compressibility of the fluid and can be determined from property data for  ẳ fnPị at two pressures or estimated from the physical properties at the upstream (stagnation) state For flashing systems,     P0 GL0 CpL0 T0 P0 GL0 ỵ ! ẳ "0 GL0 0 GL0 ð15-65Þ and for nonflashing (frozen) flows, ! ẳ "0 =k 15-66ị Using Eq (15-64), Eq (15-63) can be written 4fm L ¼ Kf ¼ À D 2 1 ẵ1 !ị2 ỵ !1 G* !=2 ị d G* ẵ1 !ị ỵ !2 =2 ỵ 2 Nfi 15-67ị where  ẳ P=P0 ; G* ẳ Gm =P0 0 ị1=2 and NFi ẳ 0 g Áz P0 ð4fm L=DÞ ð15-68Þ is the ‘‘flow inclination number.’’ From the definition of the speed of sound, it follows that the exit pressure ratio at which choking occurs is given by pffiffiffiffi 2c ¼ Gm ! * ð15-69Þ Two-Phase Flow 467 For horizontal flow, Eq (15-67) can be evaluated analytically to give    4fm L 1 2 ! !ị2 ỵ ! ẳ ỵ ln D !ị1 ỵ ! G*2 À ! ð1 À !Þ2    ð1 !ị2 ỵ ! 1 ln !ị1 ỵ ! 2 15-70ị As ! ! [i.e., setting ! ¼ 1:001 in Eq (15-70)], this reduces to the solution for ideal isothermal gas flow [Eq (9-17)], and for ! ¼ it reduces to the incompressible flow solution For inclined pipes, Leung (1996) gives the solution of Eq (15-67) in graphical form for various values of NFi Numerical Solutions The Omega method is limited to systems for which the linearized two-phase equation of state [Eq (15-64)] is a good approximation to the two-phase density (i.e., single-component systems that are not too near the critical temperature or pressure and multicomponent mixtures of similar compounds) For other systems, the governing equations for homogeneous flow can be evaluated numerically using either experimental or thermodynamic data for the two-phase P À  relation or from a limited amount of data and a more complex nonlinear model for this relation As an example, the program for such a solution for both homogeneous pipe and nozzle flow is included on a CD that accompanies a CCPS Guidelines book on pressure relief and effluent handling (CCPS, systems 1998) This program is simple to use but does require input data for the density of the two-phase mixture at either two or three pressures Separated Flow Models The separated flow models consider that each phase occupies a specified fraction of the flow cross section and account for possible differences in the phase velocities (i.e., slip) There are a variety of such models in the literature, and many of these have been compared against data for various horizontal flow regimes by Duckler et al (1964a), and later by Ferguson and Spedding (1995) The ‘‘classic’’ Lockhart–Martinelli (1949) method is based on the twophase multiplier defined previously for either liquid-only (Lm ) or gas-only (Gm ) reference flows, i.e.,     @P @P ¼ È2 À À Lm @X fm @X fLm ð15-71Þ 468 Chapter 15 FIGURE 15-7 or Lockhart–Martinelli two-phase multiplier     @P @P ¼ È2 À À Gm @X fm @X fGm ð15-72Þ where the two-phase multiplier È is correlated as a function of the parameter  as shown in Fig 15-7 There are four curves for each multiplier, depending on the flow regime in each phase, i.e., both turbulent (tt), both laminar (vv), liquid turbulent and gas laminar (tv), or liquid laminar and gas turbulent (vt) The curves can also be represented by the equations È2 ¼ þ Lm C þ  2 ð15-73Þ and È2 ẳ ỵ C ỵ 2 Gm 15-74ị Two-Phase Flow 469 TABLE 15-2 Values of Constant C in Two-Phase Multiplier Equations Flow state tt vt tv vv Liquid Gas C Turbulent Laminar Turbulent Laminar Turbulent Turbulent Laminar Laminar 20 12 10 where the values of C for the various flow combinations are shown in Table 15-2 The Lockhart–Martinelli correlating parameter 2 is defined as     @P @P 2 ẳ 15-75ị @X fLm @X fGm where   @P 2f ð1 À xÞ2 G2 m ẳ Lm @X fLm L D and 15-76ị   @P 2f x2 G2 m À ¼ Gm @X fGm G D ð15-77Þ Here, fLm is the tube friction factor based on the ‘‘liquid-only’’ Reynolds number NReLm ¼ ð1 À xÞGm D=L and fGm is the friction factor based on the ‘‘gas-only’’ Reynolds number NReGm ¼ xGm D=G The curves cross at  ¼ 1, and it is best to use the ‘‘G’’ reference curves for  < and the ‘‘L’’ curves for  > Using similarity analysis, Duckler et al (1964b) deduced that     @P 2fL G2 L m À ð’Þ 15-78ị ẳ @X fm L D m or     @P 2fG G2 G m ẳ ị @X fm G D m ð15-79Þ which is equivalent to the Martinelli parameters   and ẩ2 ẳ G ị ... with basic principles and ending with results of very practical applicability Chemical Engineering Fluid Mechanics is based on notes that I have complied and continually revised while teaching... OF MATERIALS AND FLUID PROPERTIES DETERMINATION OF FLUID VISCOUS (RHEOLOGICAL) PROPERTIES A Cup -and- Bob (Couette) Viscometer B Tube Flow (Poiseuille) Viscometer TYPES OF OBSERVED FLUID BEHAVIOR... 72 72 73 83 84 FLUID STATICS 85 IV I II STRESS AND PRESSURE THE BASIC EQUATION OF FLUID STATICS A Constant Density Fluids B Ideal Gas—Isothermal C Ideal Gas—Isentropic D The Standard Atmosphere