Modeling Transport Phenomena in Porous Media with Applications

250 Pages·2018·7.155 MB·English

by  Malay K. Das,Partha P. Mukherjee,K. Muralidhar (auth.)

Mechanical Engineering Series
Malay K. Das
Partha P. Mukherjee
K. Muralidhar
Modeling
Transport
Phenomena in
Porous Media with
Applications
Mechanical Engineering Series
Series editor
Francis A. Kulacki, University of Minnesota
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More information about this series at http://www.springer.com/series/1161
Malay K. Das Partha P. Mukherjee
(cid:129)
K. Muralidhar
Modeling Transport
Phenomena in Porous Media
with Applications
123
Malay K.Das K.Muralidhar
Department ofMechanical Engineering Department ofMechanical Engineering
Indian Institute of Technology Kanpur Indian Institute of Technology Kanpur
Kanpur,Uttar Pradesh Kanpur,Uttar Pradesh
India India
ParthaP. Mukherjee
Schoolof MechanicalEngineering
PurdueUniversity
West Lafayette, IN
USA
ISSN 0941-5122 ISSN 2192-063X (electronic)
MechanicalEngineering Series
ISBN978-3-319-69864-9 ISBN978-3-319-69866-3 (eBook)
https://doi.org/10.1007/978-3-319-69866-3
LibraryofCongressControlNumber:2017956310
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Preface
In recent years, transport phenomena in porous media have attracted significant
attention from multiple branches of science and engineering. The reasons for such
renewed interest are twofold. First, the past two decades have experienced the
emergence of a variety of porous media applications, primarily involving energy,
environment, and biological systems. Second, the rapid increase in computing
powerhas facilitated theoreticaladvancement in porous media transport modeling.
In this context, the present monograph appends classical theories of porous media
transport with recent advancements in the domain. The monograph aims at pro-
vidingastrongsupporttothegraduateandseniorundergraduatestudentsinterested
in porous media research.
Thisbookisprimarilydividedintotwoparts:theoryandapplications.Typesof
porous media and the importance of mathematical modeling of transport through
such media are discussed inChap. 1. While Chaps. 2 and 3discuss thetheories of
transport phenomena in porous media, Chaps. 4–7 deal with engineering applica-
tions and their modeling considerations. Finally, Chap. 8 summarizes the present
understandingaswellastheimminentchallengesofmodelingtransportphenomena
through porous media.
Modeling of transport through porous media may be undertaken at various
scales.Chapter2examinesthefundamentaldistinctionofcontinuum-topore-scale
models. The chapter then follows the historical development from the Darcian
transportmodeltothecontemporaryphenomenologicalformalisms.Chapter2also
includes the treatment of multiphase flow and heat transfer through porous media.
Finally, charge transport and chemical reactions are also included. In continuation
with the continuum models developed in Chaps. 2 and 3 outlines the mesoscale
models for transport in porous media. In particular, Chap. 3 delineates the appli-
cationofLatticeBoltzmannmethod(LBM)inunderstandingtransportphenomena
at the pore scale. The chapter compares relative advantages of LBM in complex
geometries, particularly in porous media with the continuum formulation.
Thetheoreticaldevelopment,discussedinChaps.2and3,providestheessential
backgroundfortheapplicationscontainedinChaps.4–7.Chapters4and5describe
models of transport phenomena in electrochemical and biological systems. While
v
vi Preface
Chap. 4 includes the modeling offuel cells and batteries, Chap. 5 describes blood
flowthroughaneurysmsinhumanarteries. Chapter6includesanovelapproachof
modeling oscillatory flow in heat exchangers within a porous media framework.
Finally, Chap. 7 encompasses the reservoir simulation for CO sequestration in
2
hydrate form. Overall, Chaps. 5–7 cover broad areas of contemporary applications
inporousmedia,encouraginginterestedresearcherstodelveintotheirchosenareas.
While this book provides theoretical and modeling support with examples to
porous media researchers, it does not intend to substitute well-established mono-
graphs on transport phenomena. Fundamental knowledge offluid mechanics, heat
andmasstransfer,andelectrochemicalsystemsisnecessarytoefficientlyutilizethis
book.Finally,thereferencesincludedinthisbookwillprovideadditionalresources
of study for early researchers in the subject.
TheauthorsgratefullyacknowledgetheCurriculum DevelopmentforTechnical
Education (CDTE) cell of Indian Institute of Technology Kanpur for financial
support.TheauthorsaregratefultotheinstituteauthoritiesatIITKanpurandTexas
A&M University for providing a conducive research environment.
Kanpur, India Malay K. Das
West Lafayette, USA Partha P. Mukherjee
Kanpur, India K. Muralidhar
March 2017
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
K. Muralidhar
1.1 Physical Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Representative Elementary Volume . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Mathematical Modeling of Fluid Flow. . . . . . . . . . . . . . . . . . . . 4
1.4 Darcy’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Microscale Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Equations Governing Flow and Transport in Porous Media . . . . . . 15
K. Muralidhar
2.1 Darcy’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Cartesian and Cylindrical Coordinate Systems . . . . . . . . 18
2.1.2 Inhomogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 Compressible Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5 Effect of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Brinkman-Corrected Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Forschheimer-Extended Darcy’s Law. . . . . . . . . . . . . . . . . . . . . 25
2.4 Non-darcy Model of Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Special Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Compressible Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.4 Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Thermal Non-equilibrium Model. . . . . . . . . . . . . . . . . . 39
2.5.2 Compressible Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
vii
viii Contents
2.6 Unsaturated Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.1 Oil–Water Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.2 Multiphase Multicomponent Flow. . . . . . . . . . . . . . . . . 49
2.7 Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.8 Combined Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . 53
2.9 Flow, Heat, and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.10 Nanoscale Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.11 Multiscale Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 Mesoscale Interactions of Transport Phenomena in Polymer
Electrolyte Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Partha P. Mukherjee, Ankit Verma, and Aashutosh Mistry
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Description of Charge Transport in Porous Media . . . . . . . . . . . 68
3.2.1 Special Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Mesoscale Models in Porous Media. . . . . . . . . . . . . . . . . . . . . . 72
3.4 Microstructure Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Lattice Boltzmann Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.2 Representative Highlights . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Electrochemistry-Coupled Direct Numerical Simulation . . . . . . . 81
3.6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6.2 Representative Results . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Porous Media Applications: Electrochemical Systems. . . . . . . . . . . . 93
Partha P. Mukherjee, Aashutosh Mistry, and Ankit Verma
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Thermodynamic, Kinetic, and Transport Behavior
of Li-Ion Battery Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.1 Open Circuit Potential . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.2 Entropic Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.3 Cell Capacity and C-Rate . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.4 Intercalation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.5 Electrolyte Transport Properties. . . . . . . . . . . . . . . . . . . 100
4.3 Modeling Isothermal Operation of a Li-Ion Cell. . . . . . . . . . . . . 102
4.3.1 Macrohomogeneous Description . . . . . . . . . . . . . . . . . . 104
4.3.2 Comments on Mathematical Nature of Governing
Equations and Solution. . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . 107
Contents ix
4.4 Importance of Thermal Effects and Its Influence on
Electrochemical Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4.1 Energy Equation to Define Temperature Changes. . . . . . 113
4.4.2 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.5.2 Microstructural Effects: Properties for Composite
Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5 Porous Media Applications: Biological Systems . . . . . . . . . . . . . . . . 123
Malay K. Das and Chandan Paul
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.1 Aneurysm and the Treatment Options . . . . . . . . . . . . . . 123
5.1.2 Blood Flow in Coil-Embolized Aneurysm:
Role of Modeling and Simulation . . . . . . . . . . . . . . . . . 124
5.2 Analytical Solutions of Flow in a Channel and a Tube . . . . . . . . 125
5.2.1 Pulsatile Flow Through a Tube with Clear Media . . . . . 125
5.2.2 Steady Flow Through a Channel Filled with Porous
Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.3 Steady Flow Through a Tube Filled
with Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.4 Pulsatile Flow Through a Channel Filled
with Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.5 Pulsatile Flow Through a Tube Filled with Porous
Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Pulsatile Flow in a Porous Bulge. . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3.2 Flow Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.3 Pulsatile Flow in a Porous Bulge:
Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3.4 Validation of the Finite Volume Solver . . . . . . . . . . . . . 134
5.3.5 Pulsatile Flow in a Bulge . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.6 Pulsatile Flow in a Patient-Specific Geometry . . . . . . . . 144
5.4 Rheology of Biological Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.1 Role of RBC in Blood Rheology . . . . . . . . . . . . . . . . . 148
5.4.2 Realistic Blood Model . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151