fluid flow for chemical engineers (2nd edition)

Fluid Flow for Chemical Engineers Second edition Professor F. A. Holland Overseas Educational Development Off ice University of Salford Dr R. Bragg Department of Chemical Engineering University of Manchester Institute of Science and Technology A member of the Hodder Headline Group LONDON First published in Great Britain 1973 Published in Great Britain 1995 by Edward Arnold, a division of Hodder Headline PLC, 338 Euston Road, London NW1 3BH 0 1995 F. A. Holland and R. Bragg All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronically or mechanically, including photocopying, recording or any information storage or retrieval system, without either prior permission in writing from the publisher or a licence permitting restricted copying. In the United Kingdom such licences are issued by the Copyright Licensing Agency: 90 Tottenham Court Road, London W1P 9HE. Whilst the advice and information in this book is believed to be true and accurate at the date of going to press, neither the authors nor the publisher can accept any legal responsibility or liability for any errors or omissions that may be made. British Libraty Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 340 61058 I 2 3 4 5 6 789 10 Typeset in 10/13pt Plantin by Wearset, Boldon, Tyne and Wear Printed and bound in Replika Press Pvt Ltd. 100% EOU, Delhi-110 040. India Preface to the second edition In preparing the second edition of this book, the authors have been concerned to maintain or expand those aspects of the subject that are specific to chemical and process engineering. Thus, the chapter on gas-liquid two-phase flow has been greatly extended to cover flow in the bubble regime as well as to provide an introduction to the homogeneous model and separated flow model for the other flow regimes. The chapter on non-Newtonian flow has also been extended to provide a greater emphasis on the Rabinowitsch-Mooney equation and its modification to deal with cases of apparent wall slip often encountered in the flow of suspensions. An elementary discussion of viscoelasticity has also been given. A second aim has been to make the book more nearly self-contained and to this end a substantial introductory chapter has been written. In addition to the material provided in the first edition, the principles of continuity, momentum of a flowing fluid, and stresses in fluids are discussed. There is also an elementary treatment of turbulence. Throughout the book there is more explanation than in the first edition. One result of this is a lengthening of the text and it has been necessary to omit the examples of applications of the Navier-Stokes equations that were given in the first edition. However, derivation of the Navier-Stokes equations and related material has been provided in an appendix. The authors wish to acknowledge the help given by Miss S. A. Petherick in undertaking much of the word processing of the manuscript for this edition. It is hoped that this book will continue to serve as a useful undergradu- ate text for students of chemical engineering and related disciplines. F. A. Holland R. Bragg May 1994 xi Contents List of examples Preface to the second edition Nomenclature 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Fluids in motion Units and dimensions Description of fluids and fluid flow Types of flow Conservation of mass Energy relationships and the Bernoulli equation Momentum of a flowing fluid Stress in fluids Sign conventions for stress Stress components Volumetric flow rate and average velocity in a pipe Momentum transfer in laminar flow Non-Newtonian behaviour Turbulence and boundary layers Flow of incompressible Newtonian fluids in pipes and channels Reynolds number and flow patterns in pipes and tubes Shear stress in a pipe Friction factor and pressure drop Pressure drop in fittings and curved pipes Equivalent diameter for non-circular pipes Velocity profile for laminar Newtonian flow in a pipe Kinetic energy in laminar flow Velocity distribution for turbulent flow in a pipe ix xi 1 1 1 4 7 9 17 27 36 43 45 46 48 55 70 70 71 71 80 84 85 86 86 V vi CONTENTS 2.9 2.10 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6 6.1 6.2 Universal velocity distribution for turbulent flow in a pipe Flow in open channels Flow of incompressible non-Newtonian fluids in pipes Elementary viscometry Rabinowitsch-Mooney equation Calculation of flow rate-pressure drop relationship for laminar flow using 7-j data Wall shear stress-flow characteristic curves and scale-up for laminar flow Generalized Reynolds number for flow in pipes Turbulent flow of inelastic non-Newtonian fluids in pipes Power law fluids Pressure drop for Bingham plastics in laminar flow Laminar flow of concentrated suspensions and apparent slip at the pipe wall Viscoelasticity Pumping of liquids Pumps and pumping System heads Centrifugal pumps Centrifugal pump relations Centrifugal pumps in series and in parallel Positive displacement pumps Pumping efficiencies Factors in pump selection Mixing of liquids in tanks Mixers and mixing Small blade high speed agitators Large blade low speed agitators Dimensionless groups for mixing Power curves Scale-up of liquid mixing systems The purging of stirred tank systems Flow of compressible fluids in conduits Energy relationships Equations of state 89 94 96 96 102 108 110 114 115 118 123 125 131 140 140 140 143 152 156 159 160 162 164 164 165 170 173 174 181 185 189 189 193 CONTENTS vii 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8 8.1 8.2 8.3 8.4 8.5 9 9.1 9.2 9.3 9.4 9.5 9.6 10 10.1 10.2 Isothermal flow of an ideal gas in a horizontal pipe Non-isothermal flow of an ideal gas in a horizontal pipe Adiabatic flow of an ideal gas in a horizontal pipe Speed of sound in a fluid Maximum flow rate in a pipe of constant cross-sectional area Adiabatic stagnation temperature for an ideal gas Gas compression and compressors Compressible flow through nozzles and constrictions Gas-liquid two-phase flow Flow patterns and flow regime maps Momentum equation for two-phase flow Flow in bubble columns Slug flow in vertical tubes The homogeneous model for two-phase flow Two-phase multiplier Separated flow models Flow measurement Flowmeters and flow measurement Head flowmeters in closed conduits Head flowmeters in open conduits Mechanical and electromagnetic flowmeters Scale errors in flow measurement Fluid motion in the presence of solid particles Relative motion between a fluid and a single particle Relative motion between a fluid and a concentration of particles Fluid flow through packed beds Fluidization Slurry transport Filtration Introduction to unsteady flow Quasi-steady flow Incremental calculation: time to discharge an ideal gas from a tank 195 199 200 202 203 205 206 209 219 219 224 227 235 239 249 25 1 268 268 270 278 282 284 288 288 292 294 298 300 303 305 305 308 viii CONTENTS 10.3 Time for a solid spherical particle to reach 99 per cent of its terminal velocity when falling from rest in the Stokes regime Suddenly accelerated plate in a Newtonian fluid 10.4 10.5 Pressure surge in pipelines Appendix 1 The Navier-Stokes equations Appendix I1 Further problems Answers to problems Conversion factors Friction factor charts 311 312 317 322 332 345 348 349 Index 35 1 Nomenclature a a A b C C C C Cd CP c7J d de D De e L E E EO f F F Fr g G h H H He blade width, m propagation speed of pressure wave in equation 10.39, m/s area, m' width, m speed of sound, m/s couple, N m Chezy coefficient (2g/j-)1'2, m1'2/s constant, usually dimensionless solute concentration, kg/m3 or kmol/m3 drag coefficient or discharge coefficient, dimensionless specific heat capacity at constant pressure, J/(kg K) specific heat capacity at constant volume, J/(kg K) diameter, m equivalent diameter of annulus, D, - do, m diameter, m Deborah number, dimensionless roughness of pipe wall, m 1 PA /v efficiency function (- ) (+ ) , m3/ J ~~ total energy per unit mass, J/kg or m2/s2 Eotvos number, dimensionless Fanning friction factor, dimensionless energy per unit mass required to overcome friction, J/kg force, N Froude number, dimensionless gravitational acceleration, 9.81 m/s2 mass flux, kg/(s m2) head, m height, m specific enthalpy, J/kg Hedstrom number, dimensionless xii N 0 MEN CLATU RE xiii IT i J/ J k k K K K Kc K' KE L e In log m M Ma n n' N NPSH P P PA PB PE Po 4 4 r r r/ R R R' Re RMM Q tank turnovers per unit time in equation 5.8, s-l volumetric flux, m/s basic friction factorjf = j72, dimensionless molar diffusional flux in equation 1.70, kmoY(mzs) index of polytropic change, dimensionless proportionality constant in equation 5.1 , dimensionless consistency coefficient, Pa S" number of velocity heads in equation 2.23 proportionality constant in equation 2.64, dimensionless parameter in Carman-Kozeny equation, dimensionless consistency coefficient for pipe flow, Pa s" kinetic energy flow rate, W length of pipe or tube, m mixing length, m log,, dimensionless log , dimensionless mass of fluid, kg mass flow rate of fluid, kg/s Mach number, dimensionless power law index, dimensionless flow behaviour index in equation 3.26, dimensionless rotational speed, reds or rev/min net positive suction head, m pitch, m pressure, Pa agitator power, W brake power, W power, W power number, dimensionless heat energy per unit mass, J/kg heat flux in equation 1.69, W/m2 volumetric flow rate, m3/s blade length, m radius, m recovery factor in equation 6.85 universal gas constant, 8314.3 J/(kmol K) radius of viscometer element specific gas constant, J/(kg K) Reynolds number, dimensionless relative molecular mass conversion factor, kg/kmol xiv N 0 MEN C LATU R E S S S S SO T TO t U UG,L ut UT U V V W We X Y Y Z V W X z a CY P Y Y & E 77 A P v P (T T 4 distance, m scale reading in equation 8.39, dimensionless slope, sine, dimensionless cross-sectional flow area, m2 surface area per unit volume, m-' time, s temperature, K stagnation temperature in equation 6.85, K volumetric average velocity, m/s characteristic velocity in equation 7.29, ds terminal velocity, m/s tip speed, m/s internal energy per unit mass, J/kg or m2/s2 point velocity, m/s volume, m3 specific volume, m3/kg weight fraction, dimensionless work per unit mass, J/kg or m2/s2 Weber number, dimensionless distance, m Martinelli parameter in equation 7.84, dimensionless distance, m yield number for Bingham plastic, dimensionless distance, m compressibility factor, dimensionless velocity distribution factor in equation 1.14, dimensionless void fraction, dimensionless coefficient of rigidity of Bingham plastic in equation 1.73, Pa s ratio of heat capacities C,/C,, dimensionless shear rate, s-' eddy kinematic viscosity, m2/s void fraction of continuous phase, dimensionless efficiency, dimensionless relaxation time, s dynamic viscosity, Pa s kinematic viscosity, m2/s density, kg/m3 surface tension, N/m shear stress, Pa power function in equation 5.18, dimensionless [...]... of a flowing fluid Although Newton's second law of motion 1 8 FLUID FLOW FOR CHEMICAL ENGINEERS net force = rate of change of momentum applies to an element of fluid, it is difficult to follow the motion of such an element as it flows It is more convenient to formulate a version of Newton’s law that can be applied to a succession of fluid elements flowing through a particular region, for example flowing... the flow rate when the flow is laminar However, as shown in Figure 1.2, when the flow is turbulent the pressure drop increases more rapidly, almost as the square of the flow rate Turbulent flow has the advantage of Figure 1.1 Now regimes in a pipe shown by dye injection (a) Laminar flow @) TurMent flow 6 FLUID FLOW FOR CHEMICAL ENGINEERS Flow rate Figure 1.2 Therelationship between pressure drop and flow. .. kinematic viscosity for turbulent flow 62 Chapter 2 2.1 Calculation of pressure drop for turbulent flow in a pipe 2.2 Calculation of flow rate for given pressure drop 75 78 Chapter 3 3.1 Use of the Rabinowitsch-Mooney equation to calculate the flow curve for a non-Newtonian liquid flowing in a pipe 3.2 Calculation of flow rate from viscometric data 3.3 Calculation of flow rate using flow characteristic... be written as mass flow rate in = mass flow rate out rate of accumulation within section + that is or 8 FLUID FLOW FOR CHEMICAL ENGINEERS Figure 1.3 Flow through a pipe of changing diameter where Vis the constant volume of the section between planes 1 and 2, and pavis the density of the fluid averaged over the volume V This equation represents the conservation of mass of the flowing fluid: it is frequently... the form in which the work terms are zero, it states that the total mechanical energy remains constant along a streamline Fluids flowing along different streamlines have different total energies For example, for laminar flow in a horizontal pipe, the pressure energy and potential energy for an element of fluid flowing in the centre of the pipe will be virtually identical to those for an element flowing... entrance length, it exhibits a higher pressure gradient Developing flow is more difficult to analyse than fully developed flow owing to the variation along the flow direction 1.2.4 Paths, streaklines and streamlines The pictorial representation of fluid flow is very helpful, whether this be 4 FLUID FLOW FOR CHEMICAL ENGINEERS done by experimental flow visualization or by calculating the velocity field The... Newton’s law a positive force must act on the fluid in the section, ie a force in the positive x-direction If the flow were reversed, the force would be reversed The above example shows the effect of a change in pipe diameter, and therefore flow area, on the momentum flow rate It is clear that for steady, fully developed, incompressible flow in a pipe of constant diameter, the fluid s momentum must remain... one or more of the terms on the right hand side of equation 1.16 will be zero, or may be negative For downward flow the hydrostatic pressure imeuses in the direction of flow and for decelerating flow the loss of kinetic energy produces an increase in pressure (pressure recovery) 14 FLUID FLOW FOR CHEMICAL ENGINEERS Denoting the total pressure drop ( P I- Pz) by AP,it can be written as lv = hp,+AP,+APf... is v2/2, where v is the velocity of the fluid relative to some fixed body Total energy Summing these components, the total energy E per unit mass of fluid is given by the equation P v2 E = U+zg+-+P 2 1 0 FLUID FLOW FOR CHEMICAL ENGINEERS where each term has the dimensions of force times distance per unit mass, ie ( M L I T ~ I L I ML ~ I T ~ or Consider fluid flowing from point 1 to point 2 as shown... Mu2 M u , (1.22) 20 FLUID FLOW FOR CHEMICAL ENGINEERS Thus a force equal to M(u2- ul) must be applied to the fluid This force is measured as positive in the positive x-direction These equations are valid when there is no accumulation of momentum within the section When accumulation of momentum occurs within the section, the momentum equation must be written as net force acting on the fluid = rate of change . two-phase flow Flow in bubble columns Slug flow in vertical tubes The homogeneous model for two-phase flow Two-phase multiplier Separated flow models Flow measurement Flowmeters and flow measurement. by dye injection (a) Laminar flow @) TurMent flow 6 FLUID FLOW FOR CHEMICAL ENGINEERS Flow rate Figure 1.2 The relationship between pressure drop and flow rate in a pipe promoting. streamlines The pictorial representation of fluid flow is very helpful, whether this be 4 FLUID FLOW FOR CHEMICAL ENGINEERS done by experimental flow visualization or by calculating the velocity