简介
Biotransport: Principles and Applications is written primarily for biomedical engineering and bioengineering students at the introductory level, but should prove useful for anyone interested in quantitative analysis of transport in living systems.聽 It is important that bioengineering students be exposed to the principles and subtleties of transport phenomena within the context of problems that arise in living systems.聽 These tend to have constitutive properties, compositions, and geometries that are quite distinct from those of typical inanimate systems.聽 The book derives its genesis from a novel Engineering Research Center (ERC) in Bioengineering Educational Technologies sponsored by the National Science Foundation.聽 This ERC was a multi-institutional consortium among Vanderbilt, Northwestern, Texas and Harvard/MIT Universities (VaNTH) based on collaboration among bioengineers, learning scientists and learning technologists. An objective was to develop state-of-the-art learning materials for students in bioengineering.聽 This text is an outgrowth of the VaNTH ERC and was designed with dual objectives: to provide a coherent and concise pedagogical exposition of biotransport that includes the domains of fluid, heat and mass flows, and to present a guide for teaching and studying in the "How People Learn" (HPL) framework, with appropriate supporting materials for students and teachers.聽 There is no other text that meets the latter objective.聽 The text is designed for use in either a traditional didactic course or in an active learning environment in which a course is organized around a series of open ended challenge problems.聽 The main portion of the text presents enduring concepts and analogies that form the foundations of biotransport.聽 Sections on biofluid, bioheat and biomass transport are further subdivided into chapters that progressively cover principles and applications of biotransport fundamentals, macroscopic biotransport, 1-D steady and unsteady state transport, and general multidimensional microscopic transport. Biotransport: Principles and Applications should serve as a clear and effective resource for students to learn the basic components of biotransport, so that class time can be freed to allow student-faculty interactions which focus on development of skills in adaptive thinking and solving open ended problems.聽 The text provides numerous example problems with detailed numerical solutions to help students learn effectively during self study.聽 Intermediate steps in derivations are included to make it easier for students to follow.聽 The text includes extensive examples of various learning challenges that have been written by the authors for use in their own biotransport courses.聽 Chapter summaries, review questions and over 230 problems are included at the end of chapters.
目录
Foreword 5
Contents 11
Part I: Fundamentals of How People Learn (HPL) 21
Chapter 1: Introduction to HPL Methodology 22
1.1 Introduction 22
1.2 Adaptive Expertise 23
1.3 Learning for Adaptive Expertise 24
1.4 Principles of Effective Learning 24
1.5 Challenge-Based Instruction 25
1.6 STAR.Legacy (SL) Cycle for Inquiry Learning 27
1.7 Developing Innovation 28
1.7.1 How to Use the Generate Ideas Model 28
1.7.2 How to Use This Textbook to Develop Innovation 45
1.8 Learning to Gain Understanding 46
References 48
Part II: Fundamental Concepts in Biotransport 50
Chapter 2: Fundamental Concepts in Biotransport 51
2.1 Introduction 51
2.2 The System and Its Environment 52
2.3 Transport Scales in Time and Space 53
2.3.1 Continuum Concepts 55
2.4 Conservation Principles 57
2.5 Transport Mechanisms 58
2.5.1 Molecular Transport Mechanisms 59
2.5.1.1 Basic Constitutive Equations for Molecular Transport 61
2.5.1.2 Molecular Transport Properties 63
2.5.1.3 1D Molecular Transport Analogies 64
2.5.2 Convective Transport Mechanisms 66
2.6 Macroscopic Transport Coefficients 67
2.7 Interphase Transport 70
2.8 Transport in Biological Systems: Some Unique Aspects 74
2.9 Summary of Key Concepts 76
2.10 Questions 78
2.11 Problems 79
2.11.1 Transport Examples 79
2.11.2 Convective and Diffusive Fluxes 80
2.11.3 Relation Between Macroscopic and Microscopic Transport Coefficients, Heat Transfer 80
2.11.4 Relation Between Macroscopic and Microscopic Transport Coefficients, Mass Transfer 80
2.11.5 Conservation Principles 81
2.11.6 Conservation and Chemical Reaction 81
2.11.7 Measurement of Lung Gas Volumes 81
2.11.8 Measurement of FRC or RV 82
2.11.9 Continuum Concepts 82
2.12 Challenges 82
2.12.1 Preliminary Research, Design of a Heart-Lung Machine 82
2.12.2 Alternate Applications for a Device 83
2.12.3 Design of a Heart-Lung Machine 83
2.12.4 Heat Loss by Conduction 83
2.12.5 Respiratory Heat Loss 84
References 84
Chapter 3: Modeling and Solving Biotransport Problems 85
3.1 Introduction 85
3.2 Theoretical Approach 86
3.2.1 Geometric Considerations 87
3.2.2 Governing Equations 88
3.2.3 Solution Procedures 88
3.2.4 Presentation of Results 90
3.2.5 Scaling: Identification of Important Dimensionless Parameters 90
3.2.6 Examples of the Theoretical Approach 93
3.3 Empirical Approach 102
3.3.1 The Buckingham Pi Theorem: Dimensional Analysis 102
3.4 Summary of Key Concepts 112
3.5 Questions 113
3.6 Problems 114
3.6.1 Setting Up a Transport Problem 114
3.6.2 Flow Between Reservoirs 115
3.6.3 Hematocrit in Reservoir 2 115
3.6.4 Temperature in Reservoir 2 116
3.6.5 Transport Principles 116
3.6.6 Tapered Tube 116
3.6.7 External Flow 116
3.6.8 Dimensional Analysis, Lung Alveoli 117
3.6.9 Dimensional Analysis, Parallel Plates 117
3.6.10 Dimensional Analysis, Cylindrical Tube 118
3.6.11 Dimensional Analysis, Cylindrical Cell 118
3.6.12 Flow Measurement 119
3.6.13 Dimensional Analysis: Flow Through an Elliptical Blood Vessel 119
3.6.14 Forced Convection from a Circular Cylinder 119
3.6.15 Natural Convection from a Circular Cylinder 120
3.7 Challenges 121
3.7.1 Heat Loss to the Environment 121
3.7.2 Application of Buckingham Pi Theorem to Heart-Lung Machine 121
References 121
Part III: Biofluid Transport 122
Chapter 4: Rheology of Biological Fluids 123
4.1 Introduction 123
4.2 Solids and Fluids 123
4.3 Flow Regimes: Laminar and Turbulent Flow 126
4.4 Boundary Conditions 126
4.5 Viscous Properties of Fluids 128
4.6 Viscous Momentum Flux and Shear Stress 128
4.7 Viscometers 131
4.8 Newtonian and Non-Newtonian Fluid Models 135
4.8.1 Newtonian Fluid Model 136
4.8.2 Non-Newtonian Fluid Models 140
4.8.2.1 Power Law Model 141
4.8.2.2 Bingham Fluid Model 145
4.8.2.3 Casson Fluid Model 146
4.8.2.4 Herschel-Bulkley Fluid Model 148
4.8.3 Identification of Constitutive Model Equations 149
4.9 Rheology of Biological Fluids 154
4.9.1 Rheological Properties of Extravascular Body Fluids 155
4.9.2 Blood Rheology 158
4.9.2.1 Blood Plasma 158
4.9.2.2 Erythrocytes 159
4.9.2.3 Leukocytes 160
4.9.2.4 Whole Blood 161
4.9.3 Biorheology and Disease 170
4.9.3.1 Polycythemia 171
4.9.3.2 Cancer 172
4.9.3.3 Sickle Cell Anemia 172
4.9.3.4 Cystic Fibrosis 173
4.10 Summary of Key Concepts 174
4.11 Questions 175
4.12 Problems 177
4.12.1 Cytoplasm Viscosity 177
4.12.2 Herschel-Bulkley Fluid 177
4.12.3 Yield Stress 177
4.12.4 Constitutive Relationships 177
4.12.5 Apparent Viscosity 178
4.12.6 Blood Yield Stress 178
4.12.7 Casson Fluid Parameters 178
4.12.8 Fahraeus-Lindqvist Effect 179
4.12.9 Apparent Viscosity of Blood 179
4.12.10 Apparent Viscosity of Blood 179
4.12.11 Tube Hematocrit 179
4.12.12 Casson Fluid 179
4.12.13 Determination of Fluid Properties 180
4.12.14 Fahraeus Lindqvist Effect 180
4.12.15 Blood Rheology in a Small Tube 181
4.12.16 Reynolds Numbers for Blood Vessels 181
4.13 Challenges 181
4.13.1 Respiratory Clearance 181
4.13.2 Blood Rheology 182
4.13.3 Blood Doping 182
References 182
Chapter 5: Macroscopic Approach for Biofluid Transport 185
5.1 Introduction 185
5.2 Conservation of Mass 185
5.3 Conservation of Momentum 196
5.4 Conservation of Energy 204
5.5 Engineering Bernoulli Equation 210
5.6 Friction Loss in Conduits 215
5.7 Friction Loss Factors, Flow Through Fittings 229
5.8 Laminar Flow and Flow Resistance in Noncircular Conduits 239
5.9 Flow in Packed Beds 244
5.10 External Flow: Drag and Lift 246
5.11 Blood Flow in Microvessels 251
5.12 Steady Flow Through a Network of Rigid Conduits 253
5.13 Compliance and Resistance of Flexible Conduits 259
5.14 Flow in Collapsible Tubes 268
5.15 Fluid Inertia 277
5.16 Blood Flow in Organs 286
5.17 Osmotic Pressure and Flow 291
5.18 Summary of Key Concepts 305
5.19 Questions 307
5.20 Problems 310
5.20.1 Conservation of Mass 310
5.20.2 Organ Reperfusion 310
5.20.3 Flow Out of a Tank 311
5.20.4 Flow Between Reservoirs 311
5.20.5 Mass and Momentum Balance 312
5.20.6 Isolated Perfused Organ 313
5.20.7 Flow Through a Curved, Tapered Vessel 313
5.20.8 Laminar Flow Through an Aortic Aneurysm 314
5.20.9 Steady Flow in a Blood Oxygenator 314
5.20.10 Pressure and Flow in a Vein 315
5.20.11 Blood Flow in a Microvessel 315
5.20.12 Pressure Drop Across a Blood Oxygenator 315
5.20.13 Friction in a Vascular Network 316
5.20.14 Friction Losses in Branching Networks 316
5.20.15 Laminar Flow of Blood Through a Vein 317
5.20.16 Flow Resistance in Laminar and Turbulent Flow 317
5.20.17 Friction Factor in a Rough Tube 318
5.20.18 Pressure Drop, Circular Tube 318
5.20.19 Flow Rate, Circular Tube 318
5.20.20 Pressure and Flow 318
5.20.21 Blood Flow in the Portal Vein 319
5.20.22 Flow Through a Semicircular Duct 319
5.20.23 Flow Through an Annulus 319
5.20.24 Pressure Drop in a Fluid System 320
5.20.25 Flow in a Channel 320
5.20.26 Design of a Device to Study Chemical Reactions 321
5.20.27 Syringe Selection 321
5.20.28 Pressure Drop in Fittings 321
5.20.29 Pressure Drop Across a Blood Oxygenator 322
5.20.30 Pressure Drop in an Extracorporeal Device 322
5.20.31 Blood Flow Through an In Vitro Network 322
5.20.32 Vascular Compliance 323
5.20.33 Vascular Compliance 323
5.20.34 Left Atrial Filling 324
5.20.35 Simple Ventilation Model 324
5.20.36 Air Flow to Compliant Lungs with Different Airway Resistances 325
5.20.37 Response of a Fluid-Filled Compliant Pressure Transducer 325
5.20.38 Aortic Pressure and Flow in a Compliant System 326
5.20.39 Venous Occlusion Experiment 326
5.20.40 Flow in a Collapsed Venule 327
5.20.41 Lung Zones 327
5.20.42 Lung Zones 328
5.20.43 Osmotic Pressure 328
5.20.44 Osmotic Pressure 328
5.20.45 Osmotic Flow 328
5.20.46 Osmotic flow 329
5.20.47 Cell Fluid Loss 329
5.20.48 Concentrating Proteins 330
5.20.49 Renal Vascular Resistance and GFR 330
5.20.50 Osmotic Water Flow 331
5.21 Challenges 332
5.21.1 Rapid Reaction Experiment 332
5.21.2 Vascular Resistance and GFR 332
5.21.3 Osmotic Shock 333
References 333
Chapter 6: Shell Balance Approach for One-Dimensional Biofluid Transport 334
6.1 Introduction 334
6.2 General Approach 335
6.2.1 Selecting an Appropriate Shell 336
6.2.2 Fluid Mass Balance 337
6.2.3 Fluid Momentum Balance 338
6.2.4 Application of the Fluid Constitutive Relation to Find Fluid Velocity 343
6.2.5 Examining and Applying Solutions for Shear Stress and Velocity 344
6.2.6 Additional Shell Balances in Rectangular Coordinates 347
6.3 One-Dimensional Shell Balances in Cylindrical Coordinates 361
6.3.1 Flow of a Newtonian Fluid Through a Circular Cylinder 361
6.3.2 Flow of a Newtonian Fluid in an Annulus with Inner Wall Moving 371
6.3.3 Flow Through an Inclined Tube or Annulus 374
6.3.4 Flow of a Casson Fluid Through a Circular Cylinder 377
6.3.5 Osmotic Pressure and Flow in a Cylindrical Pore 381
6.3.5.1 Accounting for Multiple Solutes 385
6.4 Unsteady-State 1-D Shell Balances 388
6.5 Summary of Key Concepts 392
6.6 Questions 394
6.7 Problems 395
6.7.1 Marginal Zone Theory 395
6.7.2 Power Law Fluid 395
6.7.3 Respiratory Mucus Transport 396
6.7.4 Boundary Conditions at Air-Mucus Interface 396
6.7.5 Mucus Plug 397
6.7.6 Power Law Fluid 397
6.7.7 Thin Film Blood Oxygenator 397
6.7.8 Flow Between Inclined Parallel Plates 397
6.7.9 Flow in the Lung Microcirculation 398
6.7.10 Flow of a Casson Fluid Between Two Plates 398
6.7.11 Laminar Flow in a Tube 399
6.7.12 Newtonian Film Oxygenator 399
6.7.13 Non-Newtonian Film Oxygenator 399
6.7.14 Alternate Shell Approach 399
6.7.15 Marginal Zone Theory 400
6.7.16 Marginal Zone Theory 400
6.7.17 Yield Stress 400
6.7.18 Power Law Fluid in a Cylindrical Tube 401
6.7.19 Bingham Fluid in a Cylindrical Tube 401
6.7.20 Bingham Fluid in an Annulus 401
6.7.21 Flow of Air and Mucus Through a Small Bronchiole 401
6.7.22 Respiratory Flow of Non-Newtonian Fluid 402
6.7.23 Restricted Fluid Flow Through a Narrow Slit 402
6.7.24 Starling麓s Law for Flow Through a Narrow Slit 402
6.8 Challenges 402
6.8.1 Blood Flow in Vessels 402
6.8.2 Modeling Respiratory Clearance 403
References 403
Chapter 7: General Microscopic Approach for Biofluid Transport 404
7.1 Introduction 404
7.2 Conservation of Mass 404
7.3 Conservation of Linear Momentum 406
7.4 Moment Equations 409
7.5 General Constitutive Relationship for a Newtonian Fluid 410
7.6 Substantial Derivative 413
7.7 Modified Pressure, 415
7.8 Equations of Motion for Newtonian Fluids 415
7.9 The Stream Function and Streamlines for Two-Dimensional Incompressible Flow 417
7.10 Use of Navier-Stokes Equations in Rectangular Coordinates 419
7.10.1 Hydrostatics 419
7.10.2 Reduction of the Equations of Motion 421
7.11 Navier-Stokes Equations in Cylindrical and Spherical Coordinate Systems 430
7.12 Use of Navier-Stokes Equations in Cylindrical and Spherical Coordinates 435
7.13 Scaling the Navier-Stokes Equation 451
7.14 General Momentum Equations for Use with Non-Newtonian Fluids 470
7.15 Constitutive Relationships for Non-Newtonian Fluids 472
7.15.1 Power Law Fluid 474
7.15.2 Bingham Fluid 476
7.15.3 Casson Fluid 477
7.15.4 Herschel-Bulkley Fluid 478
7.16 Setting Up and Solving Non-Newtonian Problems 479
7.17 Summary of Key Concepts 489
7.18 Questions 491
7.19 Problems 492
7.19.1 Falling Film 492
7.19.2 Immiscible Fluids 492
7.19.3 Cylindrical Tube 492
7.19.4 Annulus Flow 492
7.19.5 Start-Up Flow in a Circular Tube 492
7.19.6 Pressure Drop Across a Red Blood Cell 493
7.19.7 Addition of Pulsations to a Bioreactor 493
7.19.8 Blood Flow in a Slightly Convergent Channel 494
7.19.9 Blood Flow in Alveolar Wall of the Lung 495
7.19.10 Scaling 495
7.19.11 Scaling and Blood Flow in the Lung Microcirculation 496
7.19.12 Flow of a Viscous Fluid Through a Compliant Tube 496
7.19.13 Casson Fluid 497
7.19.14 Herschel-Bulkley Fluid 497
7.19.15 Flow of Bingham Fluid Between Parallel Plates 497
7.19.16 Power Law Fluid in Couette Viscometer 497
7.19.17 Blood as a Casson Fluid 497
7.19.18 Bingham Fluid in a Couette Viscometer 498
7.19.19 Flow Past a Cylinder 499
7.20 Challenges 499
7.20.1 Laminar Flow in Noncircular Tubes 499
References 500
Part IV: Bioheat Transport 501
Chapter 8: Heat Transfer Fundamentals 502
8.1 Introduction 502
8.2 Conduction 502
8.2.1 Thermal Resistance in Conduction 505
8.3 Convection 506
8.3.1 Four Principle Characteristics of Convective Processes 507
8.3.2 Fundamentals of Convective Processes 508
8.3.2.1 The Constitutive Equation for Convection: Newton麓s Law of Cooling 508
8.3.2.2 Temperature and Velocity Boundary Layers 510
8.3.2.3 Dimensionless Parameters of Convection 513
8.3.3 Forced Convection Analysis 516
8.3.3.1 Internal Flow Geometries 516
Introductory Concepts and Background 516
Correlation Equations for Internal Convection Flow 520
Convection in Fully Developed Laminar Flow 520
Convection in the Laminar Flow Entrance Length 521
Convection in Turbulent Internal Flow 522
8.3.3.2 External Flow Geometries 523
Laminar External Flow over a Flat Plate 524
Turbulent External Flow over a Flat Plate 524
External Flow over a Perpendicular Cylinder 524
External Flow over a Sphere 525
Impingent Flow from a Round Jet onto a Planar Perpendicular Surface 526
8.3.4 Free Convection Processes 527
8.3.4.1 Free Convection over a Vertical Plate 527
8.3.4.2 Free Convection over a Horizontal Plate 528
8.3.4.3 Free Convection over a Horizontal Cylinder 528
8.3.4.4 Free Convection over a Sphere 529
8.3.4.5 Free Convection Inside Closed Cavities 529
Horizontal Concentric Cylinders 529
Concentric Spheres 531
Enclosed Straight Sided Spaces 531
8.3.5 Thermal Resistance in Convection 535
8.3.6 Biot Number 536
8.4 Thermal Radiation 537
8.4.1 Three Governing Characteristics of Thermal Radiation Processes 537
8.4.2 The Role of Surface Temperature in Thermal Radiation 537
8.4.3 The Role of Surface Properties in Thermal Radiation 542
8.4.4 The Role of Geometric Sizes, Shapes, Separation, and Orientation in Thermal Radiation 545
8.4.5 Electrical Resistance Model for Radiation 552
8.5 Common Heat Transfer Boundary Conditions 560
8.6 Summary of Key Concepts 562
8.7 Questions 564
8.8 Problems 565
8.8.1 Radiation During Barbequing 565
8.8.2 Heat Transfer Through an Insulated Window 566
8.8.3 Steady-State Temperature Distribution in the Arm 566
8.8.4 Blood Bank Refrigerator 566
8.8.5 Cooling Coffee When It Is Poured into a Cup 567
8.9 Challenges 567
8.9.1 Debunking Brain Freeze Myth from Reality 567
8.9.2 Jet Impingement Cooling of Skin During Laser Surgery 568
References 569
Chapter 9: Macroscopic Approach to Bioheat Transport 571
9.1 Introduction 571
9.2 General Macroscopic Energy Relation 571
9.3 Steady-State Applications of the Macroscopic Energy Balance 573
9.3.1 Thermal Resistances 573
9.3.2 Heat Transfer Coefficients 580
9.3.3 Convective Heat Transport 586
9.3.4 Biomedical Applications of Thermal Radiation 588
9.3.4.1 Steady-State Radiation Exchange 588
9.3.4.2 Environmental Radiation Load on the Human Body 592
9.3.4.3 Radiation Insulation 597
9.3.5 Heat Transfer with Phase Change 599
9.4 Unsteady-State Macroscopic Heat Transfer Applications 600
9.4.1 Lumped Parameter Analysis of Transient Diffusion with Convection 601
9.4.2 Thermal Compartmental Analysis 607
9.5 Multiple System Interactions 610
9.5.1 Convection: Multiple Well-Mixed Compartments 610
9.5.2 Combined Conduction and Convection 613
9.5.3 Radiation: Flame Burn Injury 614
9.5.4 Human Thermoregulation 626
9.5.4.1 Conservation of Energy 628
9.5.4.2 Interactive Garments: Space Suits and Protective Clothing 629
9.6 Summary of Key Concepts 630
9.7 Questions 631
9.8 Problems 632
9.8.1 Conservation of Energy 632
9.8.2 Scalding Water in a Bath Tub 632
9.8.3 Insulating Properties of Clothing 633
9.8.4 Heat Exchange Between Man and Environment 633
9.8.5 Heat Transfer in the Left Ventricle of a Mouse 634
9.8.6 Blood Warmer 634
9.8.7 Steady Conduction Through Multilayered Skin 635
9.8.8 Thermal Mixing of Blood 635
9.8.9 Postmortem Interval 636
9.9 Challenges 636
9.9.1 Postmortem Interval Analysis 636
9.9.2 Heart-Lung Machine Perfusion of Patient During Heart Surgery 637
9.9.3 Design of a Combined Heat and Mass Exchanger 638
References 638
Chapter 10: Shell Balance Approach for One-Dimensional Bioheat Transport 640
10.1 Introduction 640
10.2 General Approach 640
10.3 Steady-State Conduction with Heat Generation 641
10.3.1 Steady-State Conduction with Heat Generation in a Slab 641
10.3.2 Steady-State Conduction with Heat Generation in a Cylinder 644
10.3.3 Steady-State Conduction with Heat Generation in a Sphere 650
10.4 Steady-State One-Dimensional Problems Involving Convection 651
10.4.1 Internal Flow Convection with a Constant Temperature Boundary Condition 653
10.4.2 Internal Flow Convection with a Constant Heat Flux Boundary Condition 657
10.4.3 Heat Exchangers 659
10.4.3.1 Cocurrent and Counter-Current Heat Exchangers 661
10.4.3.2 Effectiveness-NTU Analysis Method 667
10.5 One-Dimensional Steady-State Heat Conduction 680
10.5.1 Heat Conduction with Convection or Radiation at Extended Surfaces 680
10.5.2 Heat Exchange in Tissue: Transient and Steady-State Pennes Equation 689
10.6 Transient Diffusion Processes with Internal Thermal Gradients 691
10.6.1 Symmetric Geometries: Exact and Approximate Solutions for Negligible Heat Generation 693
10.6.2 Semi-Infinite Geometry 703
10.6.3 Graphical Methods 709
10.6.3.1 Schmidt Plot 710
10.6.3.2 Heisler Charts 713
10.7 Summary of Key Concepts 720
10.8 Questions 722
10.9 Problems 723
10.9.1 Cryotherapy Safety 723
10.9.2 Warming a Three-Dimensional Stick 723
10.9.3 Heating in a Muscle Cell 724
10.9.4 Heat Conduction from the Body 724
10.9.5 Thanksgiving Festivities 724
10.9.6 Hot Surface Burn 725
10.9.7 Heat Loss by Walking on the Floor 725
10.9.8 Transient Temperature Caused by Quenching 725
10.9.9 Cooling the Skin 726
10.9.10 Contact Burn 726
10.9.11 Cooling an Isolated Muscle 726
10.9.12 Brewing a Coffee Bean 727
10.9.13 Protecting an Orange from a Freeze 727
10.9.14 Postmortem Interval Case Revisited 727
10.9.15 Hibernation Temperature 728
10.9.16 Frostbite to a Mountain Climber 729
10.9.17 Heat Exchanger 729
10.9.18 Steady-State Temperature Distribution in the Boundary Layer 730
10.10 Challenges 730
10.10.1 Heat Transfer with Blood as it Flows Through Vessels 730
References 731
Chapter 11: General Microscopic Approach for Bioheat Transport 733
11.1 General Microscopic Formulation of Conservation of Energy 733
11.1.1 Derivation of Conservation of Energy for Combined Conduction and Convection 733
11.1.2 Simplifying the General Microscopic Energy Equation 736
11.1.2.1 One-Dimensional Steady-State Conduction in Tissue with Heat Generation 737
11.1.2.2 Cooling of a Cylinder 737
11.1.2.3 Steady Flow Through a Tube with Constant Heat Flux at the Boundary 738
11.2 Numerical Methods for Transient Conduction: Finite Difference Analysis 739
11.2.1 Forward Finite Difference Method 743
11.2.2 Backward Finite Difference Method 757
11.3 Thermal Injury Mechanisms and Analysis 761
11.3.1 Burn Injury 761
11.3.2 Therapeutic Applications of Hyperthermia 771
11.4 Laser Irradiation of Tissue 774
11.4.1 Distributed Energy Absorption 774
11.4.2 Time Constant Analysis of the Transient Temperature Field 776
11.4.3 Surface Cooling During Irradiation 779
11.5 Summary of Key Concepts 786
11.6 Questions 789
11.7 Problems 789
11.7.1 Vulcanization Process 789
11.7.2 Metabolic Heat Generation 790
11.7.3 Internal Temperature Gradient 790
11.7.4 Temperature Gradient in Stagnant Air 790
11.7.5 Heat Transfer in a Hollow Fiber 791
11.7.6 Hyperthermia Therapy for Tumors 791
11.7.7 Heat Exchanger to Coagulate Blood 792
11.8 Challenges 793
11.8.1 Kangaroo Care for Enhancing Neonatal Thermoregulatory Function 793
References 794
Part V: Biological Mass Transport 796
Chapter 12: Mass Transfer Fundamentals 797
12.1 Average and Local Mass and Molar Concentrations 797
12.2 Phase Equilibrium 803
12.2.1 Liquid-Gas Equilibrium 803
12.2.2 Liquid-Liquid, Gas-Solid, Liquid-Solid, Solid-Solid Equilibrium 811
12.3 Species Transport Between Phases 814
12.4 Species Transport Within a Single Phase 816
12.4.1 Species Fluxes and Velocities 817
12.4.2 Diffusion Fluxes and Velocities 818
12.4.3 Convective and Diffusive Transport 819
12.4.4 Total Mass and Molar Fluxes 820
12.4.5 Molecular Diffusion and Fick麓s Law of Diffusion 825
12.4.5.1 Stokes-Einstein Relation 835
12.4.6 Mass Transfer Coefficients 837
12.4.7 Experimental Approach to Determining Mass Transfer Coefficients 838
12.4.7.1 External Mass Transfer Coefficients 839
Forced Convection, Sphere 839
Free or Natural Mass Transfer from a Sphere 844
Forced Convective Mass Transfer from a Cylinder 844
Natural Convection from a Cylinder 845
Forced Mass Transfer from a Flat Plate 845
12.4.7.2 Internal Mass Transfer Coefficients 846
Forced Convective Mass Transfer in Conduits 847
Forced Convective Mass Transfer in a Packed Column 847
12.5 Relation Between Individual and Overall Mass Transfer Coefficients 848
12.6 Permeability of Nonporous Materials 850
12.6.1 Membrane Permeability 850
12.6.2 Vessel or Hollow Fiber Permeability 853
12.6.3 Comparison of Internal and External Resistances to Mass Transfer 857
12.7 Transport of Electrically Charged Species 859
12.8 Chemical Reactions 863
12.8.1 Hemoglobin and Blood Oxygen Transport 866
12.8.2 Blood CO2 Transport and pH 872
12.8.3 Enzyme Kinetics 874
12.8.4 Ligand-Receptor Binding Kinetics 880
12.9 Cellular Transport Mechanisms 884
12.9.1 Carrier-Mediated Transport 885
12.9.2 Active Transport 888
12.10 Mass Transfer Boundary Conditions 889
12.10.1 Mass or Molar Concentration Specified at a Boundary 889
12.10.2 Mass or Molar Flux Specified at a Boundary 890
12.10.3 No-Flux Boundary Condition 891
12.10.4 Concentration and Flux at an Interface 891
12.10.5 Heterogeneous Reaction at a Surface 891
12.11 Summary of Key Concepts 892
12.12 Questions 895
12.13 Problems 897
12.13.1 Local and Average Concentration 897
12.13.2 Concentrations in an Ideal Gas 897
12.13.3 Water-Gas Equilibrium 897
12.13.4 Liquid Equilibrium 897
12.13.5 Partition Coefficient, Solubility, Boundary Conditions 898
12.13.6 Membrane Permeability 898
12.13.7 Partition Coefficient 899
12.13.8 Maximum Depth of a Gopher Tunnel 899
12.13.9 Binary Diffusion Coefficient 899
12.13.10 Heterogeneous Reaction 899
12.13.11 Membrane Design 900
12.13.12 Empirical Relationships for Mass Transfer 900
12.13.13 Mass Transfer from a Sphere 901
12.13.14 Internal Mass Transfer Coefficients 901
12.13.15 Membrane Transport of Charged Species 901
12.13.16 Nernst Equation and Donnan Equilibrium 901
12.13.17 Goldman Equation 902
12.13.18 Goldman-Hodgkin-Katz Equation 902
12.13.19 Bound and Dissolved Oxygen in Blood 902
12.13.20 Blood Hemoglobin 902
12.13.21 Blood-Doping Ethics 903
12.14 Challenges 903
12.14.1 Enzyme Kinetics 903
12.14.2 Blood Doping 903
References 904
Chapter 13: Macroscopic Approach to Biomass Transport 905
13.1 Introduction 905
13.2 Species Conservation 905
13.3 Compartmental Analysis 909
13.3.1 Single Compartment 909
13.3.1.1 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow, Constant Rate of Infusion 909
13.3.1.2 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow, Bolus Injection 910
13.3.1.3 Single Compartment, Variable Volume, Single Inlet and Outlet, Constant Inlet Flow 910
13.3.1.4 Single Compartment, Constant Volume, Two Inlets and Single Outlet, Constant Inlet Flow 914
13.3.1.5 Single Compartment, Constant Volume, Single Inlet and Outlet, Oscillating Inlet Flow, Constant Inlet Concentration 915
13.3.1.6 Single Compartment, Constant Volume, Mass Flow Through a Permeable Wall 916
13.3.1.7 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow and Inlet Concentration, Permeable Wall 917
13.3.2 Two Compartments 918
13.3.2.1 Two Compartments in Series 918
13.3.2.2 Two Compartments in Parallel 921
13.3.2.3 Exchange Between Two Well Mixed Compartments Across a Membrane 923
13.3.2.4 Exchange Between Blood and Tissue Compartments 925
13.3.3 Multiple Compartments 930
13.4 Indicator Dilution Methods 931
13.4.1 Stewart-Hamilton Relation for Measuring Flow Through a System 932
13.4.2 Volume Measurements 934
13.4.3 Permeability-Surface Area Measurements 935
13.5 Chemical Reactions and Bioreactors 942
13.5.1 Homogeneous Chemical Reactions 942
13.5.1.1 Zeroth Order Homogeneous Reaction 942
13.5.1.2 First-Order Irreversible Homogeneous Reaction 943
13.5.1.3 Second-Order Reversible Homogeneous Reaction 943
13.5.1.4 Second-Order Reversible Homogeneous Reaction with Convection 946
13.5.1.5 Oxygen-Hemoglobin Reactions 948
Oxygenation and Deoxygenation of Red Cells 948
Pulmonary Shunt Fraction 952
Cardiac Output Using the Fick Principle 954
13.5.1.6 Enzyme Kinetics 955
13.5.2 Heterogeneous Reactions 959
13.5.2.1 Heterogeneous Reactions at an Endothelial Surface 959
13.6 Pharmacokinetics 960
13.6.1 Renal Excretion 961
13.6.2 Drug Delivery to Tissue, Two Compartment Model 965
13.6.2.1 Bolus Injection 965
13.6.2.2 Constant Infusion 968
13.6.2.3 Loading Dose Followed by Constant Infusion 969
13.6.2.4 Oral Administration 971
13.6.3 More Complex Pharmacokinetics Models 975
13.7 Mass Transfer Coefficient Applications 976
13.8 Solute Flow Through Pores in Capillary Walls 979
13.8.1 Small Solute Transport 980
13.8.2 Large Solute Transport Through Pores 981
13.9 Summary of Key Concepts 988
13.10 Questions 989
13.11 Problems 991
13.11.1 Unsteady-State Mass Transfer 991
13.11.2 Formaldehyde and Eye Irritation 991
13.11.3 Hematocrit Value 992
13.11.4 Hematocrit Value 992
13.11.5 Hematocrit Value 992
13.11.6 Hematocrit Value 993
13.11.7 Macroscopic Mass Transfer 993
13.11.8 Steady-State Removal of a Toxin 993
13.11.9 Toxic Waste 994
13.11.10 Toxic Waste 994
13.11.11 Unsteady-State Mass Transfer from a Cell 994
13.11.12 Blood Flow and Tissue Volumes of White and Gray Matter in the Brain 995
13.11.13 Pulmonary Circulation 996
13.11.14 Compartmental Analysis 996
13.11.15 Cocaine Exchange 997
13.11.16 Exchange in a Well-Mixed Hemodialyzer 997
13.11.17 Compartmental Modeling 997
13.11.18 Flow Measurement 998
13.11.19 Flow Measurement 999
13.11.20 Oxygen Transport to a Bioartificial Organ 999
13.11.21 Physiological Shunt, Neglecting Dissolved Oxygen 1000
13.11.22 Physiological Shunt, Including Dissolved Oxygen 1000
13.11.23 Oxygen Delivery to an Isolated Perfused Organ 1001
13.11.24 Heart Muscle O2 Consumption 1001
13.11.25 Oxygen Exchange and Organ Resistance for an Isolated Perfused Organ 1001
13.11.26 Pharmacokinetics 1001
13.11.27 Absorption of Aspirin from the Gut 1002
13.11.28 Pharmacokinetics of Ampicillin 1002
13.11.29 Oral Drug Administration 1003
13.11.30 Pharmacokinetics 1003
13.11.31 Pharmacokinetics 1004
13.11.32 Pharmacokinetics 1004
13.11.33 Pharmacokinetics 1005
13.11.34 Pharmacokinetics 1006
13.11.35 Pharmacokinetics/Compartmental Analysis 1006
13.11.36 Complex Chemotherapy Model 1007
13.11.37 Dissolution of a Sucrose Rod 1008
13.11.38 Two Pore System 1008
13.11.39 Steric Partition Coefficient 1008
13.11.40 Solute Flow Through a Narrow Slit 1008
13.11.41 Dual Tracer Study 1009
13.12 Challenges 1009
13.12.1 Maternal-Fetal Exchange Across the Placenta 1009
13.12.2 Pharmacokinetics of Aspirin 1010
13.12.3 Chemotherapy 1010
References 1010
Chapter 14: Shell Balance Approach for One-Dimensional Biomass Transport 1012
14.1 Introduction 1012
14.2 Microscopic Species Conservation 1012
14.3 One-Dimensional Steady-State Diffusion Through a Membrane 1013
14.4 1D Diffusion with Homogeneous Chemical Reaction 1021
14.4.1 Zeroth Order Reaction 1021
14.4.1.1 Rectangular Shaped Cell 1021
14.4.1.2 Cylindrical Cell 1027
14.4.1.3 Spherical Cell 1031
14.4.2 First-Order Reaction 1032
14.4.3 Michaelis-Menten Kinetics 1038
14.4.4 Diffusion and Reaction in a Porous Particle Containing Immobilized Enzymes 1039
14.4.4.1 Simplification for High and Low Values of beta=Csp(Rp)/Km 1045
14.4.4.2 Effectiveness Factor, eta 1046
14.5 Convection and Diffusion 1048
14.5.1 Conduits with Constant Wall Concentration 1049
14.5.2 Hollow Fiber Devices 1052
14.5.2.1 Solute Exchange with a Well-Mixed External Compartment 1052
14.5.2.2 Cocurrent Mass Exchanger 1057
14.5.2.3 Counter-Current Mass Exchanger 1061
14.5.2.4 Effect of Axial Diffusion on the Rate of Solute Exchange 1063
14.5.3 Capillary Exchange of Non-Reacting Solutes 1066
14.5.3.1 Small Solute and Inert Gas Exchange in Lung Capillaries 1066
14.5.3.2 Solute Removal by Tissue Capillaries 1067
14.6 Convection, Diffusion, and Chemical Reaction 1069
14.6.1 Transcapillary Exchange of O2 and CO2 1069
14.6.1.1 Oxygen Exchange in Lung Capillaries 1070
14.6.1.2 Oxygen Exchange in Tissue Capillaries 1073
Internal Resistance to Oxygen Exchange in Capillaries 1074
14.6.1.3 Carbon Dioxide Exchange in Lung Capillaries 1076
14.6.1.4 Carbon Dioxide Exchange in Tissue Capillaries 1078
14.6.2 Tissue Solute Exchange, Krogh Cylinder 1079
14.6.2.1 Oxygen Exchange in a Krogh Cylinder 1080
14.6.3 Bioreactors 1086
14.6.3.1 Analysis of an Imbedded Enzyme Bioreactor 1087
14.6.3.2 Bioreactor: Analysis of the Mobile Phase 1088
14.6.3.3 Zeroth-Order Reaction in the Stationary Phase (beta1) 1091
14.6.3.4 First-Order Reaction in the Stationary Phase (beta1) 1092
14.6.3.5 Michaelis-Menten Kinetics in the Stationary Phase 1094
14.7 One-Dimensional Unsteady-State Shell Balance Applications 1102
14.7.1 Diffusion to Tissue 1102
14.7.1.1 Diffusion in a Semi-Infinite Slab 1103
14.7.1.2 Diffusion Between Two Semi-Infinite Slabs 1106
14.7.1.3 Diffusion to a Semi-Infinite Slab with Finite External Resistance to Mass Transfer 1108
14.7.1.4 Unsteady-State Diffusion to a Slab with Finite Thickness and with Non-zero Surface Resistance 1110
14.7.1.5 Unsteady-State Diffusion in a Long Cylinder 1118
14.7.1.6 Unsteady-State Diffusion in a Sphere 1121
14.7.2 Unsteady-State 1D Convection and Diffusion 1123
14.7.2.1 Indicator Dilution Applications 1123
Intravascular Tracer, cR 1125
Non-Returning Diffusible Tracer, cD 1125
Flow-Limited Diffusible Tracer, cD 1130
14.7.2.2 Chromatography 1131
14.8 Summary of Key Concepts 1135
14.9 Questions 1136
14.10 Problems 1138
14.10.1 Steady-State Diffusion of an Inert Gas Through the Wall of a Tube 1138
14.10.2 Bioreactor 1139
14.10.3 Steady-State Removal of a Toxin 1139
14.10.4 CO2 Diffusion in Cell Culture Media 1140
14.10.5 Anesthetic Gas Exchange in the Lung 1140
14.10.6 Inert Gas Exchange in Lung Capillaries 1140
14.10.7 Exchange of Inert Gas in Lungs 1140
14.10.8 O2 Consumption by Cells 1141
14.10.9 Steady-State Capillary Filtration (1D) 1141
14.10.10 Distributed Consumption Rate 1142
14.10.11 Diffusion of Drug in a Tumor 1143
14.10.12 Radial Variation in Consumption Rate 1144
14.10.13 Steady-State Removal of a Waste Product from Tissue 1144
14.10.14 Disk-Shaped Particles with Immobilized Enzymes 1145
14.10.15 Counter-Current Mass Exchanger 1145
14.10.16 Dialysis Fluid Concentration in a Counter-Current Mass Exchanger 1145
14.10.17 Kidney Dialysis 1145
14.10.18 Oxygen Exchange in an Organ 1146
14.10.19 Hollow Fiber Reactor 1146
14.10.20 Blood Doping 1146
14.10.21 CO2 Exchange in Tissue Capillaries 1147
14.10.22 Carbon Dioxide Exchange in the Lung 1147
14.10.23 Facilitated Diffusion with Consumption 1147
14.10.24 Facilitated Transport of Oxygen in a Hemoglobin Solution 1147
14.10.25 Urea Production: Krogh Cylinder 1148
14.10.26 Production of Species in a Bioreactor 1148
14.10.27 Batch Reactor 1149
14.10.28 Immobilized Enzyme Bioreactor 1149
14.10.29 Mobile Phase of Reactor 1150
14.10.30 Krogh Cylinder 1150
14.10.31 Oxygen Exchange from a HbSS Solution 1150
14.10.32 Carbon Dioxide Transport in a Bioreactor 1151
14.10.33 Mass Transfer from a Muscle Fiber 1152
14.10.34 Mass Transfer from a Finite Slab 1152
14.10.35 Transient Inert Gas Exchange from Blood to Gas in the Lung 1152
14.10.36 Indicator Dilution and Chromatography 1153
14.10.37 Multiple Indicator Dilution Experiment 1153
14.11 Challenges 1154
14.11.1 Cell Size 1154
14.11.2 Membrane Resistance to Oxygen Transport 1154
14.11.3 Modeling Blood Doping 1154
References 1155
Chapter 15: General Microscopic Approach for Biomass Transport 1156
15.1 Introduction 1156
15.2 3-D, Unsteady-State Species Conservation 1156
15.2.1 Comparison of the General Species Continuity Equation and the One-Dimensional Shell Balance Approach 1162
15.3 Diffusion 1165
15.3.1 Steady-State, Multidimensional Diffusion 1165
15.3.2 Steady-State Diffusion and Superposition 1169
15.3.3 Unsteady-State, Multidimensional Diffusion 1171
15.4 Diffusion and Chemical Reaction 1177
15.5 Convection and Diffusion 1182
15.5.1 Steady-State, Multidimensional Convection and Diffusion 1186
15.5.1.1 Mass Transfer with Flow Past a Flat Surface 1186
15.5.1.2 Analogies Between Momentum Transport and Convective Heat and Mass Transport 1190
15.5.1.3 Constant Solute Flux to Fluid Flowing in a Tube 1192
15.5.1.4 Flow Between Parallel Plates with Constant Wall Concentration 1198
15.6 Convection, Diffusion, and Chemical Reaction 1205
15.6.1 Blood Oxygenation in a Hollow Fiber 1205
15.7 Summary of Key Concepts 1213
15.8 Questions 1214
15.9 Problems 1215
15.9.1 Tissue CO2 Exchange 1215
15.9.2 Diffusion of Carbon Dioxide in a Tapered Lung Capillary Channel 1216
15.9.3 Tracer Diffusion Through a Vessel Wall 1216
15.9.4 Mass Transfer from a Finite Cylinder: Product Solution 1217
15.9.5 Mass Transfer from a Finite Cylinder 1217
15.9.6 Constant Mass Flux to Fluid Flowing Between Parallel Plates 1217
15.9.7 Carbon Dioxide Transport for Blood Flowing in a Hollow Fiber 1218
15.9.8 Mass Transfer from a Finite Cylinder: Product Solution 1218
15.9.9 Design of a Membrane Dialysis Device 1218
15.9.10 Diffusion of CO2 from Lung Capillaries to Alveolar Gas 1219
15.9.11 Hollow Fiber Design for O2 Transport 1220
15.9.12 Hollow Fiber Design for CO2 Transport 1220
15.10 Challenges 1220
15.10.1 Kidney Dialysis Device 1220
15.10.2 Extracorporeal Membrane Oxygenation (ECMO) 1221
References 1221
Appendix 1223
Appendix A Nomenclature 1223
Appendix B.1 Physical Constants 1242
Appendix B.2 Prefixes and Multipliers for SI Units 1242
Appendix B.3 Conversion Factors 1243
Appendix C Transport Properties 1246
Fluid Properties 1246
Flow Properties of Selected Fluids 1246
Normal Blood Perfusion Rates in Human Tissue 1247
Thermal Properties 1247
Thermal Properties of Selected Materials 1247
Mass Transfer Properties 1250
Diffusion Coefficients in Gases at Atmospheric Pressure 1250
Diffusion Coefficients and Bunsen Solubility Coefficients for Dissolved Gases in Various Media at Atmospheric Pressure 1251
Diffusion Coefficients and Solubility Coefficients for Non-Gaseous Solutes in Various Media at Atmospheric Pressure 1253
Partition Coefficients for Solute A in Material B Relative to Material C at 37C (PhiABC=(cAB)eq/(cAC)eq=1/PhiACB) 1254
References 1255
Appendix D Charts for Unsteady Conduction and Diffusion 1257
D.1 Introduction 1257
D.1.1 Finding the Concentration or Temperature at the Center of the Material at a Given Time 1258
D.1.2 Finding the Surface Concentration or Temperature of the Solid at a Particular Time 1259
D.1.3. Finding the Temperature or Concentration at a Position in the Material Other than at the Center or the Surface 1259
D.1.4 Finding the Flux of Heat or Mass Across the Solid-Fluid Interface at Any Time 1260
D.1.5 Finding the Amount of Heat or Mass Which Has Accumulated in the Solid After a Given Time 1260
D.2 Charts for a Slab 1261
D.3 Charts for a Cylinder 1264
D.4 Charts for a Sphere 1266
Index 1269
Contents 11
Part I: Fundamentals of How People Learn (HPL) 21
Chapter 1: Introduction to HPL Methodology 22
1.1 Introduction 22
1.2 Adaptive Expertise 23
1.3 Learning for Adaptive Expertise 24
1.4 Principles of Effective Learning 24
1.5 Challenge-Based Instruction 25
1.6 STAR.Legacy (SL) Cycle for Inquiry Learning 27
1.7 Developing Innovation 28
1.7.1 How to Use the Generate Ideas Model 28
1.7.2 How to Use This Textbook to Develop Innovation 45
1.8 Learning to Gain Understanding 46
References 48
Part II: Fundamental Concepts in Biotransport 50
Chapter 2: Fundamental Concepts in Biotransport 51
2.1 Introduction 51
2.2 The System and Its Environment 52
2.3 Transport Scales in Time and Space 53
2.3.1 Continuum Concepts 55
2.4 Conservation Principles 57
2.5 Transport Mechanisms 58
2.5.1 Molecular Transport Mechanisms 59
2.5.1.1 Basic Constitutive Equations for Molecular Transport 61
2.5.1.2 Molecular Transport Properties 63
2.5.1.3 1D Molecular Transport Analogies 64
2.5.2 Convective Transport Mechanisms 66
2.6 Macroscopic Transport Coefficients 67
2.7 Interphase Transport 70
2.8 Transport in Biological Systems: Some Unique Aspects 74
2.9 Summary of Key Concepts 76
2.10 Questions 78
2.11 Problems 79
2.11.1 Transport Examples 79
2.11.2 Convective and Diffusive Fluxes 80
2.11.3 Relation Between Macroscopic and Microscopic Transport Coefficients, Heat Transfer 80
2.11.4 Relation Between Macroscopic and Microscopic Transport Coefficients, Mass Transfer 80
2.11.5 Conservation Principles 81
2.11.6 Conservation and Chemical Reaction 81
2.11.7 Measurement of Lung Gas Volumes 81
2.11.8 Measurement of FRC or RV 82
2.11.9 Continuum Concepts 82
2.12 Challenges 82
2.12.1 Preliminary Research, Design of a Heart-Lung Machine 82
2.12.2 Alternate Applications for a Device 83
2.12.3 Design of a Heart-Lung Machine 83
2.12.4 Heat Loss by Conduction 83
2.12.5 Respiratory Heat Loss 84
References 84
Chapter 3: Modeling and Solving Biotransport Problems 85
3.1 Introduction 85
3.2 Theoretical Approach 86
3.2.1 Geometric Considerations 87
3.2.2 Governing Equations 88
3.2.3 Solution Procedures 88
3.2.4 Presentation of Results 90
3.2.5 Scaling: Identification of Important Dimensionless Parameters 90
3.2.6 Examples of the Theoretical Approach 93
3.3 Empirical Approach 102
3.3.1 The Buckingham Pi Theorem: Dimensional Analysis 102
3.4 Summary of Key Concepts 112
3.5 Questions 113
3.6 Problems 114
3.6.1 Setting Up a Transport Problem 114
3.6.2 Flow Between Reservoirs 115
3.6.3 Hematocrit in Reservoir 2 115
3.6.4 Temperature in Reservoir 2 116
3.6.5 Transport Principles 116
3.6.6 Tapered Tube 116
3.6.7 External Flow 116
3.6.8 Dimensional Analysis, Lung Alveoli 117
3.6.9 Dimensional Analysis, Parallel Plates 117
3.6.10 Dimensional Analysis, Cylindrical Tube 118
3.6.11 Dimensional Analysis, Cylindrical Cell 118
3.6.12 Flow Measurement 119
3.6.13 Dimensional Analysis: Flow Through an Elliptical Blood Vessel 119
3.6.14 Forced Convection from a Circular Cylinder 119
3.6.15 Natural Convection from a Circular Cylinder 120
3.7 Challenges 121
3.7.1 Heat Loss to the Environment 121
3.7.2 Application of Buckingham Pi Theorem to Heart-Lung Machine 121
References 121
Part III: Biofluid Transport 122
Chapter 4: Rheology of Biological Fluids 123
4.1 Introduction 123
4.2 Solids and Fluids 123
4.3 Flow Regimes: Laminar and Turbulent Flow 126
4.4 Boundary Conditions 126
4.5 Viscous Properties of Fluids 128
4.6 Viscous Momentum Flux and Shear Stress 128
4.7 Viscometers 131
4.8 Newtonian and Non-Newtonian Fluid Models 135
4.8.1 Newtonian Fluid Model 136
4.8.2 Non-Newtonian Fluid Models 140
4.8.2.1 Power Law Model 141
4.8.2.2 Bingham Fluid Model 145
4.8.2.3 Casson Fluid Model 146
4.8.2.4 Herschel-Bulkley Fluid Model 148
4.8.3 Identification of Constitutive Model Equations 149
4.9 Rheology of Biological Fluids 154
4.9.1 Rheological Properties of Extravascular Body Fluids 155
4.9.2 Blood Rheology 158
4.9.2.1 Blood Plasma 158
4.9.2.2 Erythrocytes 159
4.9.2.3 Leukocytes 160
4.9.2.4 Whole Blood 161
4.9.3 Biorheology and Disease 170
4.9.3.1 Polycythemia 171
4.9.3.2 Cancer 172
4.9.3.3 Sickle Cell Anemia 172
4.9.3.4 Cystic Fibrosis 173
4.10 Summary of Key Concepts 174
4.11 Questions 175
4.12 Problems 177
4.12.1 Cytoplasm Viscosity 177
4.12.2 Herschel-Bulkley Fluid 177
4.12.3 Yield Stress 177
4.12.4 Constitutive Relationships 177
4.12.5 Apparent Viscosity 178
4.12.6 Blood Yield Stress 178
4.12.7 Casson Fluid Parameters 178
4.12.8 Fahraeus-Lindqvist Effect 179
4.12.9 Apparent Viscosity of Blood 179
4.12.10 Apparent Viscosity of Blood 179
4.12.11 Tube Hematocrit 179
4.12.12 Casson Fluid 179
4.12.13 Determination of Fluid Properties 180
4.12.14 Fahraeus Lindqvist Effect 180
4.12.15 Blood Rheology in a Small Tube 181
4.12.16 Reynolds Numbers for Blood Vessels 181
4.13 Challenges 181
4.13.1 Respiratory Clearance 181
4.13.2 Blood Rheology 182
4.13.3 Blood Doping 182
References 182
Chapter 5: Macroscopic Approach for Biofluid Transport 185
5.1 Introduction 185
5.2 Conservation of Mass 185
5.3 Conservation of Momentum 196
5.4 Conservation of Energy 204
5.5 Engineering Bernoulli Equation 210
5.6 Friction Loss in Conduits 215
5.7 Friction Loss Factors, Flow Through Fittings 229
5.8 Laminar Flow and Flow Resistance in Noncircular Conduits 239
5.9 Flow in Packed Beds 244
5.10 External Flow: Drag and Lift 246
5.11 Blood Flow in Microvessels 251
5.12 Steady Flow Through a Network of Rigid Conduits 253
5.13 Compliance and Resistance of Flexible Conduits 259
5.14 Flow in Collapsible Tubes 268
5.15 Fluid Inertia 277
5.16 Blood Flow in Organs 286
5.17 Osmotic Pressure and Flow 291
5.18 Summary of Key Concepts 305
5.19 Questions 307
5.20 Problems 310
5.20.1 Conservation of Mass 310
5.20.2 Organ Reperfusion 310
5.20.3 Flow Out of a Tank 311
5.20.4 Flow Between Reservoirs 311
5.20.5 Mass and Momentum Balance 312
5.20.6 Isolated Perfused Organ 313
5.20.7 Flow Through a Curved, Tapered Vessel 313
5.20.8 Laminar Flow Through an Aortic Aneurysm 314
5.20.9 Steady Flow in a Blood Oxygenator 314
5.20.10 Pressure and Flow in a Vein 315
5.20.11 Blood Flow in a Microvessel 315
5.20.12 Pressure Drop Across a Blood Oxygenator 315
5.20.13 Friction in a Vascular Network 316
5.20.14 Friction Losses in Branching Networks 316
5.20.15 Laminar Flow of Blood Through a Vein 317
5.20.16 Flow Resistance in Laminar and Turbulent Flow 317
5.20.17 Friction Factor in a Rough Tube 318
5.20.18 Pressure Drop, Circular Tube 318
5.20.19 Flow Rate, Circular Tube 318
5.20.20 Pressure and Flow 318
5.20.21 Blood Flow in the Portal Vein 319
5.20.22 Flow Through a Semicircular Duct 319
5.20.23 Flow Through an Annulus 319
5.20.24 Pressure Drop in a Fluid System 320
5.20.25 Flow in a Channel 320
5.20.26 Design of a Device to Study Chemical Reactions 321
5.20.27 Syringe Selection 321
5.20.28 Pressure Drop in Fittings 321
5.20.29 Pressure Drop Across a Blood Oxygenator 322
5.20.30 Pressure Drop in an Extracorporeal Device 322
5.20.31 Blood Flow Through an In Vitro Network 322
5.20.32 Vascular Compliance 323
5.20.33 Vascular Compliance 323
5.20.34 Left Atrial Filling 324
5.20.35 Simple Ventilation Model 324
5.20.36 Air Flow to Compliant Lungs with Different Airway Resistances 325
5.20.37 Response of a Fluid-Filled Compliant Pressure Transducer 325
5.20.38 Aortic Pressure and Flow in a Compliant System 326
5.20.39 Venous Occlusion Experiment 326
5.20.40 Flow in a Collapsed Venule 327
5.20.41 Lung Zones 327
5.20.42 Lung Zones 328
5.20.43 Osmotic Pressure 328
5.20.44 Osmotic Pressure 328
5.20.45 Osmotic Flow 328
5.20.46 Osmotic flow 329
5.20.47 Cell Fluid Loss 329
5.20.48 Concentrating Proteins 330
5.20.49 Renal Vascular Resistance and GFR 330
5.20.50 Osmotic Water Flow 331
5.21 Challenges 332
5.21.1 Rapid Reaction Experiment 332
5.21.2 Vascular Resistance and GFR 332
5.21.3 Osmotic Shock 333
References 333
Chapter 6: Shell Balance Approach for One-Dimensional Biofluid Transport 334
6.1 Introduction 334
6.2 General Approach 335
6.2.1 Selecting an Appropriate Shell 336
6.2.2 Fluid Mass Balance 337
6.2.3 Fluid Momentum Balance 338
6.2.4 Application of the Fluid Constitutive Relation to Find Fluid Velocity 343
6.2.5 Examining and Applying Solutions for Shear Stress and Velocity 344
6.2.6 Additional Shell Balances in Rectangular Coordinates 347
6.3 One-Dimensional Shell Balances in Cylindrical Coordinates 361
6.3.1 Flow of a Newtonian Fluid Through a Circular Cylinder 361
6.3.2 Flow of a Newtonian Fluid in an Annulus with Inner Wall Moving 371
6.3.3 Flow Through an Inclined Tube or Annulus 374
6.3.4 Flow of a Casson Fluid Through a Circular Cylinder 377
6.3.5 Osmotic Pressure and Flow in a Cylindrical Pore 381
6.3.5.1 Accounting for Multiple Solutes 385
6.4 Unsteady-State 1-D Shell Balances 388
6.5 Summary of Key Concepts 392
6.6 Questions 394
6.7 Problems 395
6.7.1 Marginal Zone Theory 395
6.7.2 Power Law Fluid 395
6.7.3 Respiratory Mucus Transport 396
6.7.4 Boundary Conditions at Air-Mucus Interface 396
6.7.5 Mucus Plug 397
6.7.6 Power Law Fluid 397
6.7.7 Thin Film Blood Oxygenator 397
6.7.8 Flow Between Inclined Parallel Plates 397
6.7.9 Flow in the Lung Microcirculation 398
6.7.10 Flow of a Casson Fluid Between Two Plates 398
6.7.11 Laminar Flow in a Tube 399
6.7.12 Newtonian Film Oxygenator 399
6.7.13 Non-Newtonian Film Oxygenator 399
6.7.14 Alternate Shell Approach 399
6.7.15 Marginal Zone Theory 400
6.7.16 Marginal Zone Theory 400
6.7.17 Yield Stress 400
6.7.18 Power Law Fluid in a Cylindrical Tube 401
6.7.19 Bingham Fluid in a Cylindrical Tube 401
6.7.20 Bingham Fluid in an Annulus 401
6.7.21 Flow of Air and Mucus Through a Small Bronchiole 401
6.7.22 Respiratory Flow of Non-Newtonian Fluid 402
6.7.23 Restricted Fluid Flow Through a Narrow Slit 402
6.7.24 Starling麓s Law for Flow Through a Narrow Slit 402
6.8 Challenges 402
6.8.1 Blood Flow in Vessels 402
6.8.2 Modeling Respiratory Clearance 403
References 403
Chapter 7: General Microscopic Approach for Biofluid Transport 404
7.1 Introduction 404
7.2 Conservation of Mass 404
7.3 Conservation of Linear Momentum 406
7.4 Moment Equations 409
7.5 General Constitutive Relationship for a Newtonian Fluid 410
7.6 Substantial Derivative 413
7.7 Modified Pressure, 415
7.8 Equations of Motion for Newtonian Fluids 415
7.9 The Stream Function and Streamlines for Two-Dimensional Incompressible Flow 417
7.10 Use of Navier-Stokes Equations in Rectangular Coordinates 419
7.10.1 Hydrostatics 419
7.10.2 Reduction of the Equations of Motion 421
7.11 Navier-Stokes Equations in Cylindrical and Spherical Coordinate Systems 430
7.12 Use of Navier-Stokes Equations in Cylindrical and Spherical Coordinates 435
7.13 Scaling the Navier-Stokes Equation 451
7.14 General Momentum Equations for Use with Non-Newtonian Fluids 470
7.15 Constitutive Relationships for Non-Newtonian Fluids 472
7.15.1 Power Law Fluid 474
7.15.2 Bingham Fluid 476
7.15.3 Casson Fluid 477
7.15.4 Herschel-Bulkley Fluid 478
7.16 Setting Up and Solving Non-Newtonian Problems 479
7.17 Summary of Key Concepts 489
7.18 Questions 491
7.19 Problems 492
7.19.1 Falling Film 492
7.19.2 Immiscible Fluids 492
7.19.3 Cylindrical Tube 492
7.19.4 Annulus Flow 492
7.19.5 Start-Up Flow in a Circular Tube 492
7.19.6 Pressure Drop Across a Red Blood Cell 493
7.19.7 Addition of Pulsations to a Bioreactor 493
7.19.8 Blood Flow in a Slightly Convergent Channel 494
7.19.9 Blood Flow in Alveolar Wall of the Lung 495
7.19.10 Scaling 495
7.19.11 Scaling and Blood Flow in the Lung Microcirculation 496
7.19.12 Flow of a Viscous Fluid Through a Compliant Tube 496
7.19.13 Casson Fluid 497
7.19.14 Herschel-Bulkley Fluid 497
7.19.15 Flow of Bingham Fluid Between Parallel Plates 497
7.19.16 Power Law Fluid in Couette Viscometer 497
7.19.17 Blood as a Casson Fluid 497
7.19.18 Bingham Fluid in a Couette Viscometer 498
7.19.19 Flow Past a Cylinder 499
7.20 Challenges 499
7.20.1 Laminar Flow in Noncircular Tubes 499
References 500
Part IV: Bioheat Transport 501
Chapter 8: Heat Transfer Fundamentals 502
8.1 Introduction 502
8.2 Conduction 502
8.2.1 Thermal Resistance in Conduction 505
8.3 Convection 506
8.3.1 Four Principle Characteristics of Convective Processes 507
8.3.2 Fundamentals of Convective Processes 508
8.3.2.1 The Constitutive Equation for Convection: Newton麓s Law of Cooling 508
8.3.2.2 Temperature and Velocity Boundary Layers 510
8.3.2.3 Dimensionless Parameters of Convection 513
8.3.3 Forced Convection Analysis 516
8.3.3.1 Internal Flow Geometries 516
Introductory Concepts and Background 516
Correlation Equations for Internal Convection Flow 520
Convection in Fully Developed Laminar Flow 520
Convection in the Laminar Flow Entrance Length 521
Convection in Turbulent Internal Flow 522
8.3.3.2 External Flow Geometries 523
Laminar External Flow over a Flat Plate 524
Turbulent External Flow over a Flat Plate 524
External Flow over a Perpendicular Cylinder 524
External Flow over a Sphere 525
Impingent Flow from a Round Jet onto a Planar Perpendicular Surface 526
8.3.4 Free Convection Processes 527
8.3.4.1 Free Convection over a Vertical Plate 527
8.3.4.2 Free Convection over a Horizontal Plate 528
8.3.4.3 Free Convection over a Horizontal Cylinder 528
8.3.4.4 Free Convection over a Sphere 529
8.3.4.5 Free Convection Inside Closed Cavities 529
Horizontal Concentric Cylinders 529
Concentric Spheres 531
Enclosed Straight Sided Spaces 531
8.3.5 Thermal Resistance in Convection 535
8.3.6 Biot Number 536
8.4 Thermal Radiation 537
8.4.1 Three Governing Characteristics of Thermal Radiation Processes 537
8.4.2 The Role of Surface Temperature in Thermal Radiation 537
8.4.3 The Role of Surface Properties in Thermal Radiation 542
8.4.4 The Role of Geometric Sizes, Shapes, Separation, and Orientation in Thermal Radiation 545
8.4.5 Electrical Resistance Model for Radiation 552
8.5 Common Heat Transfer Boundary Conditions 560
8.6 Summary of Key Concepts 562
8.7 Questions 564
8.8 Problems 565
8.8.1 Radiation During Barbequing 565
8.8.2 Heat Transfer Through an Insulated Window 566
8.8.3 Steady-State Temperature Distribution in the Arm 566
8.8.4 Blood Bank Refrigerator 566
8.8.5 Cooling Coffee When It Is Poured into a Cup 567
8.9 Challenges 567
8.9.1 Debunking Brain Freeze Myth from Reality 567
8.9.2 Jet Impingement Cooling of Skin During Laser Surgery 568
References 569
Chapter 9: Macroscopic Approach to Bioheat Transport 571
9.1 Introduction 571
9.2 General Macroscopic Energy Relation 571
9.3 Steady-State Applications of the Macroscopic Energy Balance 573
9.3.1 Thermal Resistances 573
9.3.2 Heat Transfer Coefficients 580
9.3.3 Convective Heat Transport 586
9.3.4 Biomedical Applications of Thermal Radiation 588
9.3.4.1 Steady-State Radiation Exchange 588
9.3.4.2 Environmental Radiation Load on the Human Body 592
9.3.4.3 Radiation Insulation 597
9.3.5 Heat Transfer with Phase Change 599
9.4 Unsteady-State Macroscopic Heat Transfer Applications 600
9.4.1 Lumped Parameter Analysis of Transient Diffusion with Convection 601
9.4.2 Thermal Compartmental Analysis 607
9.5 Multiple System Interactions 610
9.5.1 Convection: Multiple Well-Mixed Compartments 610
9.5.2 Combined Conduction and Convection 613
9.5.3 Radiation: Flame Burn Injury 614
9.5.4 Human Thermoregulation 626
9.5.4.1 Conservation of Energy 628
9.5.4.2 Interactive Garments: Space Suits and Protective Clothing 629
9.6 Summary of Key Concepts 630
9.7 Questions 631
9.8 Problems 632
9.8.1 Conservation of Energy 632
9.8.2 Scalding Water in a Bath Tub 632
9.8.3 Insulating Properties of Clothing 633
9.8.4 Heat Exchange Between Man and Environment 633
9.8.5 Heat Transfer in the Left Ventricle of a Mouse 634
9.8.6 Blood Warmer 634
9.8.7 Steady Conduction Through Multilayered Skin 635
9.8.8 Thermal Mixing of Blood 635
9.8.9 Postmortem Interval 636
9.9 Challenges 636
9.9.1 Postmortem Interval Analysis 636
9.9.2 Heart-Lung Machine Perfusion of Patient During Heart Surgery 637
9.9.3 Design of a Combined Heat and Mass Exchanger 638
References 638
Chapter 10: Shell Balance Approach for One-Dimensional Bioheat Transport 640
10.1 Introduction 640
10.2 General Approach 640
10.3 Steady-State Conduction with Heat Generation 641
10.3.1 Steady-State Conduction with Heat Generation in a Slab 641
10.3.2 Steady-State Conduction with Heat Generation in a Cylinder 644
10.3.3 Steady-State Conduction with Heat Generation in a Sphere 650
10.4 Steady-State One-Dimensional Problems Involving Convection 651
10.4.1 Internal Flow Convection with a Constant Temperature Boundary Condition 653
10.4.2 Internal Flow Convection with a Constant Heat Flux Boundary Condition 657
10.4.3 Heat Exchangers 659
10.4.3.1 Cocurrent and Counter-Current Heat Exchangers 661
10.4.3.2 Effectiveness-NTU Analysis Method 667
10.5 One-Dimensional Steady-State Heat Conduction 680
10.5.1 Heat Conduction with Convection or Radiation at Extended Surfaces 680
10.5.2 Heat Exchange in Tissue: Transient and Steady-State Pennes Equation 689
10.6 Transient Diffusion Processes with Internal Thermal Gradients 691
10.6.1 Symmetric Geometries: Exact and Approximate Solutions for Negligible Heat Generation 693
10.6.2 Semi-Infinite Geometry 703
10.6.3 Graphical Methods 709
10.6.3.1 Schmidt Plot 710
10.6.3.2 Heisler Charts 713
10.7 Summary of Key Concepts 720
10.8 Questions 722
10.9 Problems 723
10.9.1 Cryotherapy Safety 723
10.9.2 Warming a Three-Dimensional Stick 723
10.9.3 Heating in a Muscle Cell 724
10.9.4 Heat Conduction from the Body 724
10.9.5 Thanksgiving Festivities 724
10.9.6 Hot Surface Burn 725
10.9.7 Heat Loss by Walking on the Floor 725
10.9.8 Transient Temperature Caused by Quenching 725
10.9.9 Cooling the Skin 726
10.9.10 Contact Burn 726
10.9.11 Cooling an Isolated Muscle 726
10.9.12 Brewing a Coffee Bean 727
10.9.13 Protecting an Orange from a Freeze 727
10.9.14 Postmortem Interval Case Revisited 727
10.9.15 Hibernation Temperature 728
10.9.16 Frostbite to a Mountain Climber 729
10.9.17 Heat Exchanger 729
10.9.18 Steady-State Temperature Distribution in the Boundary Layer 730
10.10 Challenges 730
10.10.1 Heat Transfer with Blood as it Flows Through Vessels 730
References 731
Chapter 11: General Microscopic Approach for Bioheat Transport 733
11.1 General Microscopic Formulation of Conservation of Energy 733
11.1.1 Derivation of Conservation of Energy for Combined Conduction and Convection 733
11.1.2 Simplifying the General Microscopic Energy Equation 736
11.1.2.1 One-Dimensional Steady-State Conduction in Tissue with Heat Generation 737
11.1.2.2 Cooling of a Cylinder 737
11.1.2.3 Steady Flow Through a Tube with Constant Heat Flux at the Boundary 738
11.2 Numerical Methods for Transient Conduction: Finite Difference Analysis 739
11.2.1 Forward Finite Difference Method 743
11.2.2 Backward Finite Difference Method 757
11.3 Thermal Injury Mechanisms and Analysis 761
11.3.1 Burn Injury 761
11.3.2 Therapeutic Applications of Hyperthermia 771
11.4 Laser Irradiation of Tissue 774
11.4.1 Distributed Energy Absorption 774
11.4.2 Time Constant Analysis of the Transient Temperature Field 776
11.4.3 Surface Cooling During Irradiation 779
11.5 Summary of Key Concepts 786
11.6 Questions 789
11.7 Problems 789
11.7.1 Vulcanization Process 789
11.7.2 Metabolic Heat Generation 790
11.7.3 Internal Temperature Gradient 790
11.7.4 Temperature Gradient in Stagnant Air 790
11.7.5 Heat Transfer in a Hollow Fiber 791
11.7.6 Hyperthermia Therapy for Tumors 791
11.7.7 Heat Exchanger to Coagulate Blood 792
11.8 Challenges 793
11.8.1 Kangaroo Care for Enhancing Neonatal Thermoregulatory Function 793
References 794
Part V: Biological Mass Transport 796
Chapter 12: Mass Transfer Fundamentals 797
12.1 Average and Local Mass and Molar Concentrations 797
12.2 Phase Equilibrium 803
12.2.1 Liquid-Gas Equilibrium 803
12.2.2 Liquid-Liquid, Gas-Solid, Liquid-Solid, Solid-Solid Equilibrium 811
12.3 Species Transport Between Phases 814
12.4 Species Transport Within a Single Phase 816
12.4.1 Species Fluxes and Velocities 817
12.4.2 Diffusion Fluxes and Velocities 818
12.4.3 Convective and Diffusive Transport 819
12.4.4 Total Mass and Molar Fluxes 820
12.4.5 Molecular Diffusion and Fick麓s Law of Diffusion 825
12.4.5.1 Stokes-Einstein Relation 835
12.4.6 Mass Transfer Coefficients 837
12.4.7 Experimental Approach to Determining Mass Transfer Coefficients 838
12.4.7.1 External Mass Transfer Coefficients 839
Forced Convection, Sphere 839
Free or Natural Mass Transfer from a Sphere 844
Forced Convective Mass Transfer from a Cylinder 844
Natural Convection from a Cylinder 845
Forced Mass Transfer from a Flat Plate 845
12.4.7.2 Internal Mass Transfer Coefficients 846
Forced Convective Mass Transfer in Conduits 847
Forced Convective Mass Transfer in a Packed Column 847
12.5 Relation Between Individual and Overall Mass Transfer Coefficients 848
12.6 Permeability of Nonporous Materials 850
12.6.1 Membrane Permeability 850
12.6.2 Vessel or Hollow Fiber Permeability 853
12.6.3 Comparison of Internal and External Resistances to Mass Transfer 857
12.7 Transport of Electrically Charged Species 859
12.8 Chemical Reactions 863
12.8.1 Hemoglobin and Blood Oxygen Transport 866
12.8.2 Blood CO2 Transport and pH 872
12.8.3 Enzyme Kinetics 874
12.8.4 Ligand-Receptor Binding Kinetics 880
12.9 Cellular Transport Mechanisms 884
12.9.1 Carrier-Mediated Transport 885
12.9.2 Active Transport 888
12.10 Mass Transfer Boundary Conditions 889
12.10.1 Mass or Molar Concentration Specified at a Boundary 889
12.10.2 Mass or Molar Flux Specified at a Boundary 890
12.10.3 No-Flux Boundary Condition 891
12.10.4 Concentration and Flux at an Interface 891
12.10.5 Heterogeneous Reaction at a Surface 891
12.11 Summary of Key Concepts 892
12.12 Questions 895
12.13 Problems 897
12.13.1 Local and Average Concentration 897
12.13.2 Concentrations in an Ideal Gas 897
12.13.3 Water-Gas Equilibrium 897
12.13.4 Liquid Equilibrium 897
12.13.5 Partition Coefficient, Solubility, Boundary Conditions 898
12.13.6 Membrane Permeability 898
12.13.7 Partition Coefficient 899
12.13.8 Maximum Depth of a Gopher Tunnel 899
12.13.9 Binary Diffusion Coefficient 899
12.13.10 Heterogeneous Reaction 899
12.13.11 Membrane Design 900
12.13.12 Empirical Relationships for Mass Transfer 900
12.13.13 Mass Transfer from a Sphere 901
12.13.14 Internal Mass Transfer Coefficients 901
12.13.15 Membrane Transport of Charged Species 901
12.13.16 Nernst Equation and Donnan Equilibrium 901
12.13.17 Goldman Equation 902
12.13.18 Goldman-Hodgkin-Katz Equation 902
12.13.19 Bound and Dissolved Oxygen in Blood 902
12.13.20 Blood Hemoglobin 902
12.13.21 Blood-Doping Ethics 903
12.14 Challenges 903
12.14.1 Enzyme Kinetics 903
12.14.2 Blood Doping 903
References 904
Chapter 13: Macroscopic Approach to Biomass Transport 905
13.1 Introduction 905
13.2 Species Conservation 905
13.3 Compartmental Analysis 909
13.3.1 Single Compartment 909
13.3.1.1 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow, Constant Rate of Infusion 909
13.3.1.2 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow, Bolus Injection 910
13.3.1.3 Single Compartment, Variable Volume, Single Inlet and Outlet, Constant Inlet Flow 910
13.3.1.4 Single Compartment, Constant Volume, Two Inlets and Single Outlet, Constant Inlet Flow 914
13.3.1.5 Single Compartment, Constant Volume, Single Inlet and Outlet, Oscillating Inlet Flow, Constant Inlet Concentration 915
13.3.1.6 Single Compartment, Constant Volume, Mass Flow Through a Permeable Wall 916
13.3.1.7 Single Compartment, Constant Volume, Single Inlet and Outlet, Constant Flow and Inlet Concentration, Permeable Wall 917
13.3.2 Two Compartments 918
13.3.2.1 Two Compartments in Series 918
13.3.2.2 Two Compartments in Parallel 921
13.3.2.3 Exchange Between Two Well Mixed Compartments Across a Membrane 923
13.3.2.4 Exchange Between Blood and Tissue Compartments 925
13.3.3 Multiple Compartments 930
13.4 Indicator Dilution Methods 931
13.4.1 Stewart-Hamilton Relation for Measuring Flow Through a System 932
13.4.2 Volume Measurements 934
13.4.3 Permeability-Surface Area Measurements 935
13.5 Chemical Reactions and Bioreactors 942
13.5.1 Homogeneous Chemical Reactions 942
13.5.1.1 Zeroth Order Homogeneous Reaction 942
13.5.1.2 First-Order Irreversible Homogeneous Reaction 943
13.5.1.3 Second-Order Reversible Homogeneous Reaction 943
13.5.1.4 Second-Order Reversible Homogeneous Reaction with Convection 946
13.5.1.5 Oxygen-Hemoglobin Reactions 948
Oxygenation and Deoxygenation of Red Cells 948
Pulmonary Shunt Fraction 952
Cardiac Output Using the Fick Principle 954
13.5.1.6 Enzyme Kinetics 955
13.5.2 Heterogeneous Reactions 959
13.5.2.1 Heterogeneous Reactions at an Endothelial Surface 959
13.6 Pharmacokinetics 960
13.6.1 Renal Excretion 961
13.6.2 Drug Delivery to Tissue, Two Compartment Model 965
13.6.2.1 Bolus Injection 965
13.6.2.2 Constant Infusion 968
13.6.2.3 Loading Dose Followed by Constant Infusion 969
13.6.2.4 Oral Administration 971
13.6.3 More Complex Pharmacokinetics Models 975
13.7 Mass Transfer Coefficient Applications 976
13.8 Solute Flow Through Pores in Capillary Walls 979
13.8.1 Small Solute Transport 980
13.8.2 Large Solute Transport Through Pores 981
13.9 Summary of Key Concepts 988
13.10 Questions 989
13.11 Problems 991
13.11.1 Unsteady-State Mass Transfer 991
13.11.2 Formaldehyde and Eye Irritation 991
13.11.3 Hematocrit Value 992
13.11.4 Hematocrit Value 992
13.11.5 Hematocrit Value 992
13.11.6 Hematocrit Value 993
13.11.7 Macroscopic Mass Transfer 993
13.11.8 Steady-State Removal of a Toxin 993
13.11.9 Toxic Waste 994
13.11.10 Toxic Waste 994
13.11.11 Unsteady-State Mass Transfer from a Cell 994
13.11.12 Blood Flow and Tissue Volumes of White and Gray Matter in the Brain 995
13.11.13 Pulmonary Circulation 996
13.11.14 Compartmental Analysis 996
13.11.15 Cocaine Exchange 997
13.11.16 Exchange in a Well-Mixed Hemodialyzer 997
13.11.17 Compartmental Modeling 997
13.11.18 Flow Measurement 998
13.11.19 Flow Measurement 999
13.11.20 Oxygen Transport to a Bioartificial Organ 999
13.11.21 Physiological Shunt, Neglecting Dissolved Oxygen 1000
13.11.22 Physiological Shunt, Including Dissolved Oxygen 1000
13.11.23 Oxygen Delivery to an Isolated Perfused Organ 1001
13.11.24 Heart Muscle O2 Consumption 1001
13.11.25 Oxygen Exchange and Organ Resistance for an Isolated Perfused Organ 1001
13.11.26 Pharmacokinetics 1001
13.11.27 Absorption of Aspirin from the Gut 1002
13.11.28 Pharmacokinetics of Ampicillin 1002
13.11.29 Oral Drug Administration 1003
13.11.30 Pharmacokinetics 1003
13.11.31 Pharmacokinetics 1004
13.11.32 Pharmacokinetics 1004
13.11.33 Pharmacokinetics 1005
13.11.34 Pharmacokinetics 1006
13.11.35 Pharmacokinetics/Compartmental Analysis 1006
13.11.36 Complex Chemotherapy Model 1007
13.11.37 Dissolution of a Sucrose Rod 1008
13.11.38 Two Pore System 1008
13.11.39 Steric Partition Coefficient 1008
13.11.40 Solute Flow Through a Narrow Slit 1008
13.11.41 Dual Tracer Study 1009
13.12 Challenges 1009
13.12.1 Maternal-Fetal Exchange Across the Placenta 1009
13.12.2 Pharmacokinetics of Aspirin 1010
13.12.3 Chemotherapy 1010
References 1010
Chapter 14: Shell Balance Approach for One-Dimensional Biomass Transport 1012
14.1 Introduction 1012
14.2 Microscopic Species Conservation 1012
14.3 One-Dimensional Steady-State Diffusion Through a Membrane 1013
14.4 1D Diffusion with Homogeneous Chemical Reaction 1021
14.4.1 Zeroth Order Reaction 1021
14.4.1.1 Rectangular Shaped Cell 1021
14.4.1.2 Cylindrical Cell 1027
14.4.1.3 Spherical Cell 1031
14.4.2 First-Order Reaction 1032
14.4.3 Michaelis-Menten Kinetics 1038
14.4.4 Diffusion and Reaction in a Porous Particle Containing Immobilized Enzymes 1039
14.4.4.1 Simplification for High and Low Values of beta=Csp(Rp)/Km 1045
14.4.4.2 Effectiveness Factor, eta 1046
14.5 Convection and Diffusion 1048
14.5.1 Conduits with Constant Wall Concentration 1049
14.5.2 Hollow Fiber Devices 1052
14.5.2.1 Solute Exchange with a Well-Mixed External Compartment 1052
14.5.2.2 Cocurrent Mass Exchanger 1057
14.5.2.3 Counter-Current Mass Exchanger 1061
14.5.2.4 Effect of Axial Diffusion on the Rate of Solute Exchange 1063
14.5.3 Capillary Exchange of Non-Reacting Solutes 1066
14.5.3.1 Small Solute and Inert Gas Exchange in Lung Capillaries 1066
14.5.3.2 Solute Removal by Tissue Capillaries 1067
14.6 Convection, Diffusion, and Chemical Reaction 1069
14.6.1 Transcapillary Exchange of O2 and CO2 1069
14.6.1.1 Oxygen Exchange in Lung Capillaries 1070
14.6.1.2 Oxygen Exchange in Tissue Capillaries 1073
Internal Resistance to Oxygen Exchange in Capillaries 1074
14.6.1.3 Carbon Dioxide Exchange in Lung Capillaries 1076
14.6.1.4 Carbon Dioxide Exchange in Tissue Capillaries 1078
14.6.2 Tissue Solute Exchange, Krogh Cylinder 1079
14.6.2.1 Oxygen Exchange in a Krogh Cylinder 1080
14.6.3 Bioreactors 1086
14.6.3.1 Analysis of an Imbedded Enzyme Bioreactor 1087
14.6.3.2 Bioreactor: Analysis of the Mobile Phase 1088
14.6.3.3 Zeroth-Order Reaction in the Stationary Phase (beta1) 1091
14.6.3.4 First-Order Reaction in the Stationary Phase (beta1) 1092
14.6.3.5 Michaelis-Menten Kinetics in the Stationary Phase 1094
14.7 One-Dimensional Unsteady-State Shell Balance Applications 1102
14.7.1 Diffusion to Tissue 1102
14.7.1.1 Diffusion in a Semi-Infinite Slab 1103
14.7.1.2 Diffusion Between Two Semi-Infinite Slabs 1106
14.7.1.3 Diffusion to a Semi-Infinite Slab with Finite External Resistance to Mass Transfer 1108
14.7.1.4 Unsteady-State Diffusion to a Slab with Finite Thickness and with Non-zero Surface Resistance 1110
14.7.1.5 Unsteady-State Diffusion in a Long Cylinder 1118
14.7.1.6 Unsteady-State Diffusion in a Sphere 1121
14.7.2 Unsteady-State 1D Convection and Diffusion 1123
14.7.2.1 Indicator Dilution Applications 1123
Intravascular Tracer, cR 1125
Non-Returning Diffusible Tracer, cD 1125
Flow-Limited Diffusible Tracer, cD 1130
14.7.2.2 Chromatography 1131
14.8 Summary of Key Concepts 1135
14.9 Questions 1136
14.10 Problems 1138
14.10.1 Steady-State Diffusion of an Inert Gas Through the Wall of a Tube 1138
14.10.2 Bioreactor 1139
14.10.3 Steady-State Removal of a Toxin 1139
14.10.4 CO2 Diffusion in Cell Culture Media 1140
14.10.5 Anesthetic Gas Exchange in the Lung 1140
14.10.6 Inert Gas Exchange in Lung Capillaries 1140
14.10.7 Exchange of Inert Gas in Lungs 1140
14.10.8 O2 Consumption by Cells 1141
14.10.9 Steady-State Capillary Filtration (1D) 1141
14.10.10 Distributed Consumption Rate 1142
14.10.11 Diffusion of Drug in a Tumor 1143
14.10.12 Radial Variation in Consumption Rate 1144
14.10.13 Steady-State Removal of a Waste Product from Tissue 1144
14.10.14 Disk-Shaped Particles with Immobilized Enzymes 1145
14.10.15 Counter-Current Mass Exchanger 1145
14.10.16 Dialysis Fluid Concentration in a Counter-Current Mass Exchanger 1145
14.10.17 Kidney Dialysis 1145
14.10.18 Oxygen Exchange in an Organ 1146
14.10.19 Hollow Fiber Reactor 1146
14.10.20 Blood Doping 1146
14.10.21 CO2 Exchange in Tissue Capillaries 1147
14.10.22 Carbon Dioxide Exchange in the Lung 1147
14.10.23 Facilitated Diffusion with Consumption 1147
14.10.24 Facilitated Transport of Oxygen in a Hemoglobin Solution 1147
14.10.25 Urea Production: Krogh Cylinder 1148
14.10.26 Production of Species in a Bioreactor 1148
14.10.27 Batch Reactor 1149
14.10.28 Immobilized Enzyme Bioreactor 1149
14.10.29 Mobile Phase of Reactor 1150
14.10.30 Krogh Cylinder 1150
14.10.31 Oxygen Exchange from a HbSS Solution 1150
14.10.32 Carbon Dioxide Transport in a Bioreactor 1151
14.10.33 Mass Transfer from a Muscle Fiber 1152
14.10.34 Mass Transfer from a Finite Slab 1152
14.10.35 Transient Inert Gas Exchange from Blood to Gas in the Lung 1152
14.10.36 Indicator Dilution and Chromatography 1153
14.10.37 Multiple Indicator Dilution Experiment 1153
14.11 Challenges 1154
14.11.1 Cell Size 1154
14.11.2 Membrane Resistance to Oxygen Transport 1154
14.11.3 Modeling Blood Doping 1154
References 1155
Chapter 15: General Microscopic Approach for Biomass Transport 1156
15.1 Introduction 1156
15.2 3-D, Unsteady-State Species Conservation 1156
15.2.1 Comparison of the General Species Continuity Equation and the One-Dimensional Shell Balance Approach 1162
15.3 Diffusion 1165
15.3.1 Steady-State, Multidimensional Diffusion 1165
15.3.2 Steady-State Diffusion and Superposition 1169
15.3.3 Unsteady-State, Multidimensional Diffusion 1171
15.4 Diffusion and Chemical Reaction 1177
15.5 Convection and Diffusion 1182
15.5.1 Steady-State, Multidimensional Convection and Diffusion 1186
15.5.1.1 Mass Transfer with Flow Past a Flat Surface 1186
15.5.1.2 Analogies Between Momentum Transport and Convective Heat and Mass Transport 1190
15.5.1.3 Constant Solute Flux to Fluid Flowing in a Tube 1192
15.5.1.4 Flow Between Parallel Plates with Constant Wall Concentration 1198
15.6 Convection, Diffusion, and Chemical Reaction 1205
15.6.1 Blood Oxygenation in a Hollow Fiber 1205
15.7 Summary of Key Concepts 1213
15.8 Questions 1214
15.9 Problems 1215
15.9.1 Tissue CO2 Exchange 1215
15.9.2 Diffusion of Carbon Dioxide in a Tapered Lung Capillary Channel 1216
15.9.3 Tracer Diffusion Through a Vessel Wall 1216
15.9.4 Mass Transfer from a Finite Cylinder: Product Solution 1217
15.9.5 Mass Transfer from a Finite Cylinder 1217
15.9.6 Constant Mass Flux to Fluid Flowing Between Parallel Plates 1217
15.9.7 Carbon Dioxide Transport for Blood Flowing in a Hollow Fiber 1218
15.9.8 Mass Transfer from a Finite Cylinder: Product Solution 1218
15.9.9 Design of a Membrane Dialysis Device 1218
15.9.10 Diffusion of CO2 from Lung Capillaries to Alveolar Gas 1219
15.9.11 Hollow Fiber Design for O2 Transport 1220
15.9.12 Hollow Fiber Design for CO2 Transport 1220
15.10 Challenges 1220
15.10.1 Kidney Dialysis Device 1220
15.10.2 Extracorporeal Membrane Oxygenation (ECMO) 1221
References 1221
Appendix 1223
Appendix A Nomenclature 1223
Appendix B.1 Physical Constants 1242
Appendix B.2 Prefixes and Multipliers for SI Units 1242
Appendix B.3 Conversion Factors 1243
Appendix C Transport Properties 1246
Fluid Properties 1246
Flow Properties of Selected Fluids 1246
Normal Blood Perfusion Rates in Human Tissue 1247
Thermal Properties 1247
Thermal Properties of Selected Materials 1247
Mass Transfer Properties 1250
Diffusion Coefficients in Gases at Atmospheric Pressure 1250
Diffusion Coefficients and Bunsen Solubility Coefficients for Dissolved Gases in Various Media at Atmospheric Pressure 1251
Diffusion Coefficients and Solubility Coefficients for Non-Gaseous Solutes in Various Media at Atmospheric Pressure 1253
Partition Coefficients for Solute A in Material B Relative to Material C at 37C (PhiABC=(cAB)eq/(cAC)eq=1/PhiACB) 1254
References 1255
Appendix D Charts for Unsteady Conduction and Diffusion 1257
D.1 Introduction 1257
D.1.1 Finding the Concentration or Temperature at the Center of the Material at a Given Time 1258
D.1.2 Finding the Surface Concentration or Temperature of the Solid at a Particular Time 1259
D.1.3. Finding the Temperature or Concentration at a Position in the Material Other than at the Center or the Surface 1259
D.1.4 Finding the Flux of Heat or Mass Across the Solid-Fluid Interface at Any Time 1260
D.1.5 Finding the Amount of Heat or Mass Which Has Accumulated in the Solid After a Given Time 1260
D.2 Charts for a Slab 1261
D.3 Charts for a Cylinder 1264
D.4 Charts for a Sphere 1266
Index 1269
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