Transport phenomena in multiphase flows

458 Pages·2015·3.477 MB·English

by  Mauri, Roberto

Fluid Mechanics and Its Applications
Roberto Mauri
Transport
Phenomena
in Multiphase
Flows
Fluid Mechanics and Its Applications
Volume 112
Series editor
André Thess, German Aerospace Center, Institute of Engineering
Thermodynamics, Stuttgart, Germany
Founding Editor
René Moreau, Ecole Nationale Supérieure d’Hydraulique de Grenoble,
Saint Martin d’Hères Cedex, France
Aims and Scope of the Series
The purpose of this series is to focus on subjects in which fluid mechanics plays a
fundamental role.
As well as the more traditional applications of aeronautics, hydraulics, heat and
mass transfer etc., books will be published dealing with topics which are currently
in a state of rapid development, such as turbulence, suspensions and multiphase
fluids, super and hypersonic flows and numerical modeling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive
intense scientific attention, bringing them to the forefront of technological
advancement. Fluids have the ability to transport matter and its properties as well
astotransmitforce,thereforefluidmechanicsisasubjectthatisparticularlyopento
crossfertilizationwithothersciencesanddisciplinesofengineering.Thesubjectof
fluidmechanicswillbehighlyrelevantindomainssuchaschemical,metallurgical,
biological and ecological engineering. This series is particularly open to such new
multidisciplinary domains.
Themedianlevelofpresentationisthefirstyeargraduatestudent.Sometextsare
monographs defining the current state of a field; others are accessible to final year
undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/5980
Roberto Mauri
Transport Phenomena
in Multiphase Flows
123
RobertoMauri
DICI
UniversitàdegliStudidi Pisa
Pisa
Italy
ISSN 0926-5112 ISSN 2215-0056 (electronic)
FluidMechanics andIts Applications
ISBN 978-3-319-15792-4 ISBN 978-3-319-15793-1 (eBook)
DOI 10.1007/978-3-319-15793-1
LibraryofCongressControlNumber:2015935217
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2015
Translation from the Italian language edition: Elementi di fenomeni di trasporto by Roberto Mauri,
© Pisa University Press 2005. All rights reserved
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or
part of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by
similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
from the relevant protective laws and regulations and therefore free for general use.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made.
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
Preface
This textbook is an introduction to the transport of the three quantities that are
conservedinnature,namelymass,momentum,andenergy(thetransportofelectric
chargecanbeseenasaparticularcaseofthetransportofmass).Itistheoutgrowth
of 30 years of teaching this material to students of chemical, mechanical, nuclear,
and biomedical engineering, both in the U.S. and in Italy.
No transport phenomena textbook could be written without referring to the
first modern book on this subject, namely Transport Phenomena, published by
R.B. Bird, W.E. Stewart, and E.N. Lightfoot in 1960, which has constituted the
gold standard for all textbooks that have been written after it. In that book, the
authors intended to answer the “current demand in engineering education, to put
more emphasisonunderstandingbasicphysicalprinciplesthan ontheblinduseof
empiricism.” To understand the meaning of this statement, one has only to look at
the typical heat and mass transfer book of the 1940s: no mathematics is required
beyond elementary algebra and no partial differential equations can be found in
thosetexts.Infact,Bird,Stewart,andLightfootwenttheoppositeway,astheyfirst
applied the mathematical framework of continuum mechanics to derive the fun-
damental governing equations at the microscopic level and only afterward, by
integrating them, they obtain the macroscopic equations of mass, momentum, and
energy balance. Therefore, microscopic balances precede any coarse-grained
analysis: for example, the Navier-Stokes equation is derived in Chap. 3, while for
the Bernoulli equation, one has to wait until Chap. 7.
This very rigorous approach is coupled to an extremely powerful idea: that of
unifying all types of transport phenomena, describing them within a common
framework, which is in terms of cause and effect, respectively, represented by the
driving force and the flux of the transported quantity.
Inthistextbookweretainthisbasicidea.However,Idecidedtoreversetheway
this material is presented to students. Fifty-five years have passed since the
appearanceofBird,Stewart,andLightfoot’stextbook,andengineeringstudentsare
nowmuchmoreproficientinadvancedmathematics.Ontheotherhand,Ifeel,they
have not been exposed enough to “common sense” physics. In fact, often I find
myself presenting this subject to students who had no previous exposure to fluid
v
vi Preface
dynamics or heat and mass transfer and have no idea of what a pressure drop or a
heat flux are. Therefore, I try to strike a balance between a rigorous explanation
of the fundamental laws that govern these subjects and an intuitive approach that
stresseswheretheselawscomefrom.PresentingtheNavier-Stokesequationbefore
any basic macroscopic balance would be like explaining electromagnetism by
deriving Maxwell’s equation before showing the basic experimental results by
FaradayandAmpere.ThisiswhyIthinkthat,fromadidacticalpointofviewandin
light of the type of students we are dealing with today, it is better to describe
phenomena first from a macroscopic point of view, even at the expense of math-
ematical rigor,andonlyafterwards,moreadvancedtreatmentscanbecarriedover.
This textbook has been written as a teaching tool for a two-semester course (or
two one-semester courses) on transport phenomena. It is a modular teaching
tool, though. So, for example, if transport phenomena are taught in a single
one-semestercourse,oneshouldfollowonlythemacroscopictreatments,described
in Chaps. 1–5, 9–11, 14–16, postponing the study of the material covered in the
otherchapterstoasubsequent,moreadvancedcourse.Inanycase,thebookoffers
anabundantresourceinthesensethatthematerialcoveredismuchmorethanwhat
one can hope to cover in two semesters. So, it is up to the instructor to choose
which subjects should be favored, also based on the students’ needs: mechanical
engineering students will be more interested in turbulence, while biomedical
engineering students will tend to prefer surface phenomena.
Specialmentionshouldbemadeoftheproblemsthatareproposedattheendof
each chapter. First of all, their importance cannot be underestimated: a student
cannot claim to understand a subject until she/he can solve problems. However,
while in Anglo-Saxon universities, problem solving is part of the students’
homework assignments, elsewhere, they are solved in class, and therefore, they
becomeanintegralpartofthecourse.Thisiswhyherethesolutionoftheproblems
are provided in the Appendix.
Finally, I would like to answer a common complaint that I heard from my
students, namely that in this textbook, they cannot find all the physical properties
that are required to solve some of the proposed problems. This deficiency is
deliberate, being motivated by the fact that in this way the students are forced to
look outside and find the missing data.
Pisa, November 2014 Roberto Mauri
Acknowledgments
I would like to thank the many generations of students whose questions and
complaintshavecontributedtoshapethistextbook.Iwouldalsoliketothankafew
colleagues who have enlightened me on how transport phenomena should be
taught.Inparticular,IwouldliketomentionJohnF.Brady,whoshowedmehowto
find the correct scaling of different problems in very intuitive ways, and Andrea
Lamorgese, for his line-by-line comments of the text.
Pisa, November 2014 Roberto Mauri
vii
Contents
1 Thermodynamics and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Statics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Local Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Introduction to Continuum Mechanics. . . . . . . . . . . . . . . . . . 6
1.3.1 Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Convection and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Convective Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Diffusive Fluxes and Constitutive Relations. . . . . . . . 12
1.5 Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Molecular Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Molecular Diffusion as an Example of Random Walk. . . . . . . 18
1.9 Examples of Diffusive Processes. . . . . . . . . . . . . . . . . . . . . . 20
1.10 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Statics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Hydrostatic Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Manometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The Young-Laplace Equation. . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Thermodynamic Approach. . . . . . . . . . . . . . . . . . . . 30
2.4.2 Mechanical Approach . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Contact Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
ix
x Contents
3 General Features of Fluid Mechanics. . . . . . . . . . . . . . . . . . . . . . 39
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 The Reynolds Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Boundary Layer and Viscous Resistance . . . . . . . . . . . . . . . . 42
3.4 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Macroscopic Balances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Mass Balance and Continuity Equation . . . . . . . . . . . . . . . . . 49
4.2 Mechanical Energy Balance and Bernoulli Equation . . . . . . . . 51
4.2.1 Example: The Pitot Tube. . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Generalization of the Bernoulli Equation . . . . . . . . . . 53
4.3 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Recapitulation of the Bernoulli Equation . . . . . . . . . . . . . . . . 57
4.4.1 Effect of the Non-Uniformity of the Velocity Field. . . 58
4.4.2 Effect of the Friction Forces. . . . . . . . . . . . . . . . . . . 59
4.4.3 Effect of Pumps and Turbines . . . . . . . . . . . . . . . . . 59
4.5 Pressure Drops in Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1 Fanning vs. Darcy Friction Factor. . . . . . . . . . . . . . . 61
4.6 Localized Pressure Drops. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1 Example: Flow Through a Sudden Enlargement . . . . . 65
4.7 Flow Around a Submerged Object . . . . . . . . . . . . . . . . . . . . 66
4.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Laminar Flow Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Fully Developed Flow of a Newtonian Fluid in a Pipe . . . . . . 75
5.1.1 Thermodynamic and Modified Pressure. . . . . . . . . . . 78
5.1.2 Couette Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Fluid Rheology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1 Time-Dependent Rheology. . . . . . . . . . . . . . . . . . . . 80
5.3 Flow of Non-Newtonian Fluids in Circular Pipes . . . . . . . . . . 81
5.4 Flow in Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Packed Beds and Fluidized Beds . . . . . . . . . . . . . . . 87
5.4.2 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Quasi Steady Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.6 Capillary Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 The Governing Equations of a Simple Fluid. . . . . . . . . . . . . . . . . 97
6.1 General Microscopic Balance Equation . . . . . . . . . . . . . . . . . 97
6.2 Mass Balance: The Continuity Equation . . . . . . . . . . . . . . . . 99
6.3 Momentum Balance: Cauchy’s Equation . . . . . . . . . . . . . . . . 102
6.4 Angular Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . 105