Modern Fluid Dynamics FLUID MECHANICS AND ITS APPLICATIONS Volume 87 Series Editor: R MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Bte Postale 95 38402 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 as to transmit force, therefore fluid mechanics is a subject that is particularly open to cross fertilization with other sciences and disciplines of engineering The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering This series is particularly open to such new multidisciplinary domains The median level of presentation is the first year graduate student Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity For other titles published in this series, go to www.springer.com/series/5980 Clement Kleinstreuer Modern Fluid Dynamics Basic Theory and Selected Applications in Macro- and Micro-Fluidics Clement Kleinstreuer Department of Mechanical and Aerospace Engineering North Carolina State University Raleigh, NC 27695-7910 USA ck@eos.ncsu.edu ISBN 978-1-4020-8669-4 e-ISBN 978-1-4020-8670-0 DOI 10.1007/978-1-4020-8670-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009934512 © Springer Science + Business Media B.V 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose ofbeing entered and executed on a computer system, for exclusive use by the purchaser of the work Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my family, Christin, Nicole, and Joshua Contents Preface xiii Part A: Fluid Dynamics Essentials Review of Basic Engineering Concepts 1.1 1.2 1.3 1.4 1.5 Approaches, Definitions and Concepts The Continuum Mechanics Assumption 13 Fluid Flow Description 14 Thermodynamic Properties and Constitutive Equations 24 Homework Assignments 36 1.5.1 Concepts, Derivations and Insight 36 1.5.2 Problems 39 Fundamental Equations and Solutions 41 2.1 Introduction 41 2.2 The Reynolds Transport Theorem 47 2.3 Fluid-Mass Conservation 51 2.3.1 Mass Conservation in Integral Form 51 2.3.2 Mass Conservation in Differential Form 56 2.3.3 Continuity Derived from a Mass Balance 57 2.4 Momentum Conservation 61 2.4.1 Momentum Conservation in Integral Form 61 2.4.2 Momentum Conservation in Differential Form 67 2.4.3 Special Cases of the Equation of Motion 75 2.5 Conservation Laws of Energy and Species Mass 82 2.5.1 Global Energy Balance 83 2.5.2 Energy Conservation in Integral Form 85 2.5.3 Energy and Species Mass Conservation in Differential Form 86 2.6 Homework Assignments 93 2.6.1 Text-Embedded Insight and Problems 93 2.6.2 Additional Problems 97 Introductory Fluid Dynamics Cases 99 3.1 Inviscid Flow Along a Streamline 99 3.2 Quasi-unidirectional Viscous Flows 105 3.2.1 Steady 1-D Laminar Incompressible Flows 105 3.2.2 Nearly Parallel Flows 122 3.3 Transient One-Dimensional Flows 123 3.3.1 Stokes’ First Problem: Thin Shear-Layer Development 123 vii Contents viii 3.3.2 Transient Pipe Flow 126 3.4 Simple Porous Media Flow 129 3.5 One-Dimensional Compressible Flow 139 3.5.1 First and Second Law of Thermodynamics for Steady Open Systems 140 3.5.2 Sound Waves and Shock Waves 143 3.5.3 Normal Shock Waves in Tubes 150 3.5.4 Isentropic Nozzle Flow 153 3.6 Forced Convection Heat Transfer 159 3.6.1 Convection Heat Transfer Coefficient ………………………161 3.6.2 Turbulent Pipe Flow Correlations 171 3.7 Entropy Generation Analysis 173 3.7.1 Background Information 173 3.7.2 Entropy Generation Derivation 174 3.8 Homework Assignments 182 3.8.1 Physical Insight 182 3.8.2 Problems 185 References (Part A) 191 Part B: Conventional Applications Internal Flow 195 4.1 Introduction 195 4.2 Laminar and Turbulent Pipe Flows 198 4.2.1 Analytical Solutions to Laminar Thermal Flows 198 4.2.2 Turbulent Pipe Flow 206 4.3 Basic Lubrication Systems 221 4.3.1 Lubrication Approximations 223 4.3.2 The Reynolds Lubrication Equation 232 4.4 Compartmental Modeling 238 4.4.1 Compartments in Parallel 241 4.4.2 Compartments in Series 241 4.5 Homework Assignments 247 4.5.1 Text-Embedded Insight Questions and Problems 248 4.5.2 Problems 249 External Flow 253 5.1 Introduction 253 5.2 Laminar and Turbulent Boundary-Layer Flows 255 5.2.1 Solution Methods for Flat-Plate Boundary-Layer Flows 255 5.2.2 Turbulent Flat-Plate Boundary-Layer Flow 261 5.3 Drag and Lift Computations 267 5.4 Film Drawing and Surface Coating 274 Contents ix 5.4.1 Drawing and Coating Processes 274 5.4.2 Fluid-Interface Mechanics 276 5.5 Homework Assignments 297 5.5.1 Text-Embedded Insight Questions and Problems 297 5.5.2 Problems 298 References (Part B) 303 Part C: Modern Fluid Dynamics Topics Dilute Particle Suspensions 307 6.1 Introduction 307 6.2 Modeling Approaches 309 6.2.1 Definitions 309 6.2.2 Homogeneous Flow Equations 317 6.3 Non-Newtonian Fluid Flow 320 6.3.1 Generalized Newtonian Liquids 322 6.4 Particle Transport 332 6.4.1 Particle Trajectory Models 332 6.4.2 Nanoparticle Transport 337 6.5 Homework Assignments and Course Projects 341 6.5.1 Guideline for Project Report Writing 341 6.5.2 Text-Embedded Insight Questions and Problems 342 6.5.3 Problems 344 6.5.4 Projects 346 Microsystems and Microfluidics 349 7.1 Introduction 349 7.2 Microfluidics Modeling Aspects 354 7.2.1 Molecular Movement and Impaction 354 7.2.2 Movement and Impaction of Spherical Micron Particles 363 7.2.3 Pumps Based on Microscale Surface Effects 369 7.2.4 Microchannel Flow Effects 377 7.2.5 Wall Boundary Conditions 379 7.3 Electro-hydrodynamics in Microchannels 395 7.3.1 Electro-osmosis 397 7.3.2 Electrophoresis 407 7.4 Entropy Generation in Microfluidic Systems 409 7.4.1 Entropy Minimization 411 7.5 Nanotechnology and Nanofluid Flow in Microchannels 416 7.5.1 Microscale Heat-Sinks with Nano-coolants 417 7.5.2 Nanofluid Flow in Bio-MEMS 423 7.6 Homework Assignments and Course Projects 428 7.6.1 Guideline for Project Report Writing 429 Contents x 7.6.2 Homework Problems and Mini-Projects 430 7.6.3 Course Projects 432 Fluid–Structure Interaction 435 8.1 Introduction 435 8.2 Solid Mechanics Review 437 8.2.1 Stresses in Solid Structures 437 8.2.2 Equilibrium Conditions 443 8.2.3 Stress–Strain Relationships 445 8.3 Slender-Body Dynamics 453 8.4 Flow-Induced Vibration 460 8.4.1 Harmonic Response to Free Vibration 465 8.4.2 Harmonic Response to Forced Vibration 473 8.5 Homework Assignments and Course Projects 477 8.5.1 Guideline for Project Report Writing 477 8.5.2 Text-embedded Insight Questions and Problems 478 8.5.3 Projects 479 Biofluid Flow and Heat Transfer 481 9.1 9.2 9.3 9.4 9.5 9.6 Introduction 481 Modeling Aspects 484 Arterial Hemodynamics 490 Lung-Aerosol Dynamics 505 Bioheat Equation 514 Group Assignments and Course Projects 518 9.6.1 Guideline for Project Report Writing 519 9.6.2 Text-Embedded Insight Questions and Problems 520 9.6.3 Projects 521 10 Computational Fluid Dynamics and System Design 523 10.1 Introduction 523 10.2 Modeling Objectives and Numerical Tools 524 10.2.1 Problem Recognition and System Identification 525 10.2.2 Mathematical Modeling and Data Needs 526 10.2.3 Computational Fluid Dynamics 526 10.2.4 Result Interpretation 531 10.2.5 Computational Design Aspects 533 10.3 Model Validation Examples 534 10.3.1 Microsphere Deposition in a Double Bifurcation 534 10.3.2 Microsphere Transport Through an Asymmetric Bifurcation 536 10.4 Example of Internal Flow 537 10.4.1 Introduction 537 10.4.2 Methodology 537 Contents xi 10.4.3 Results and Discussion 542 10.4.4 Conclusions 548 10.5 Example of External Flow 550 10.5.1 Background Information 550 10.5.2 Theory 551 10.5.3 One-Way FSI Simulation of 2D-Flow over a Tall Building 554 10.6 Group Assignments and Project Suggestions 567 10.6.1 Group Assignments 567 10.6.2 Project Suggestions 569 References (Part C) 571 Appendices 577 A Review of Tensor Calculus, Differential Operations, Integral Transformations, and ODE Solutions Plus Removable B Equation Sheets 579 Fluid Properties, CD-Correlations, MOODY Chart and Turbulent Velocity Profiles 605 Index 615 Modern Fluid Dynamics • 605 Steady turbulent flow u = u + u′ ; Internal Flow Extended Bernoulli where, 2 ∫ T u = (1/T ) udt , and u′ = 0 p1 ρ + 0.5α1V1 + gz1 = p ρ + 0.5α V2 + gz + gh τ total = τ lam + τ turb = μ lam (du dy ) + μ turb (du dy ) hlT is total losses due to friction (major) τ turb = −ρu ′v ′ ; r μ turb = fct (k , ε, ∇v, geo., etc.) and form changes (minor) as expressed via K-values v2 ⎛ L ⎞ = ⎜f ( ) + ∑ K⎟ 2g ⎝ D ⎠ h lT = h f + ∑ h m v avg D 64 ; Re D = Re D ν Overlap log-law Turbulent: use the Moody chart: f = fct(ReD, e/D) Hydraulic diameter D H = 4A / P ( P = peˆ rimeter) Laminar (Poiseuille) pipe flow u (r ) = u max − (r R ) ; [ ] ( ) u max = v avg = − R 4μ (dp / dx ) Q = v avg A = ∫ R τ rx = μ du dr = u ( r )dA = πΔpD /( 128μL) r ∂p ∂x velocity>; y + ≡ yu τ /(μ / ρ) + + Laminar wall layer: u = y (0≤ y+ ≤ 5) Laminar pipe flow f = u + ≡ u / u τ ; u τ ≡ τ w / ρ < friction layer u ⎛ yu τ ⎞ ⎟+B = ln⎜ u τ κ ⎜ (μ / ρ) ⎟ ⎝ ⎠ κ ≈ 0.41; B ≈ 5.0 Empirical smooth pipe flow approximation u u max = ( y R )1 n = (1 − r R )1 n , n ≈ for Re D ≈ 10 when v av / u max = 0.817 v av ≈ (1 + 1.3 f ) −1 ; f=f(ReD, e/D) u max Appendix B B.1 Conversion Factors Dimension Metric Acceleration m/s2 = 100 cm/s2 Metric/English –6 Area m = 10 cm = 10 mm = 10 km Density g/cm3 = kg/L = 1,000 kg/m3 Energy, heat, work, internal energy, enthalpy kJ = 1,000 J = 1,000 N · m = kPa · m3 kJ/kg = 1,000 m2/s2 kWh = 3,600 kJ calb = 4.184 J IT calb = 4.1868 J calb = 4.1868 kJ Force N = kg · m/s2 = 105 dyn kgf = 9.80665 N W/cm2 = 104 W/m2 W/m2 · °C = W/m2 · K m = 100 cm = 1,000 mm = 106µm km = 1,000 m Heat flux Heat transfer coefficient Length Mass Power, heat transfer rate W = J/s kW = 1,000 W = 1.341 hp hpc = 745.7 W Pressure Pa = N/m2 kPa = 103 Pa = 10–3 MPa atm = 101.325 kPa = 1.01325 bar = 760 mm Hg at 0°C = 1.03323 kgf/cm2 mm Hg = 0.1333 kPa Specific heat a kg = 1,000 g metric ton = 1,000 kg kJ/kg · °C = kJ/kg · K = J/g · °C m/s2 = 3.2808 ft/s2 ft/s2 = 0.3048a m/s2 m2 = 1550 in.2 = 10.764 ft2 ft2 = 144 in.2 = 0.09290304a m2 g/cm3 = 62.428 lbm/ft3 = 0.036127 lbm/in.3 lbm/in.3 = 1728 lbm/ft3 kg/m3 = 0.062428 lbm/ft3 kJ = 0.94782 Btu Btu = 1.055056 kJ = 5.40395 psia · ft3 = 778.169 lbf · ft Btu/lbm = 25,037 ft2/s2 = 2.326a kJ/kg kJ/kg = 0.430 Btu/lbm kWh = 3412.14 Btu therm = 105 Btu = 1.055 × 105 kJ (natural gas) N = 0.22481 lbf lbf = 32.174 lbm · ft/s2 = 4.44822 N W/m2 = 0.3171 Btu/h · ft2 W/m2 · °C = 0.17612 Btu/h · ft2 · °F m = 39.370 in = 3.2808 ft = 1.0926 yd ft = 12 in = 0.3048a m mi = 5280 ft = 1.6093 km in = 2.54a cm kg = 2.2046226 lbm lbm = 0.45359237a kg oz = 28.3495 g slug = 32.174 lbm = 14.5939 kg short ton = 2000 lbm = 907.1847 kg kW = 3412.14 Btu/h = 737.56 lbf · ft/s hp = 550 lbf · ft/s = 0.7068 Btu/s = 42.41 Btu/min = 2544.5 Btu/h = 0.74570 kW boiler hp = 33,475 Btu/h Btu/h = 1.055056 kJ/h ton of refrigeration = 200 Btu/min Pa = 1.4504 × 10–4 psia = 0.020886 lbf/ft2 psi = 144 lbf/ft2 = 6.894757 kPa atm = 14.696 psia = 29.92 in Hg at 30°F in Hg = 3.387 kPa Btu/lbm · °F = 4.1868 kJ/kg · °C Btu/lbmol · R = 4.1868 kJ/kmol · K kJ/kg · °C = 0.23885 Btu/lbm · °F = 0.23885 Btu/lbm · R Exact conversion factor between metric and English units b Calorie is originally defined as the amount of heat needed to raise the temperature of g of water by 1°C, but it varies with temperature The international steam table (IT) calorie (generally preferred by engineers) is exactly 4.1868 J by definition and corresponds to the specific heat of water at 15°C The thermochemical calorie (generally preferred by physicists) is exactly 4.184 J by definition and corresponds to the specific heat of water at room temperature The difference between the two is about 0.06%, which is negligible The capitalized calorie used by nutritionists is actually a kilocalorie (1,000 IT cal) 607 Appendix B 608 Dimension Metric Metric/English Specific volume m /kg = 1,000 L/kg = 1,000 cm /g m3/kg = 16.02 ft3/lbm ft3/lbm = 0.062428 m3/kg Temperature T(K) = T(°C) + 273.15 ΔT(K) = Δ T(°C) T(R) = T(°F) + 459.67 = 1.8T(K) T(°F) = 1.8 T(°C) + 32 ΔT(°F) = ΔT(R) = 1.8 ΔT(K) Thermal conductivity W/m · °C = 0.57782 Btu/h · ft · °F m/s = 3.60 km/h m/s = 3.2808 ft/s = 2.237 mi/h mi/h = 1.4667 ft/s mi/h = 1.6093 km/h Volume m3 = 1,000 L = 106 cm3 (cc) m3 = 6.1024 × 104 in.3 = 35.315 ft3 = 264.17 gal (U.S.) U.S gal = 231 in.3 = 3.7854 L fl oz = 29.5735 cm3 = 0.0295735 L U.S gal = 128 fl oz Volume flow rate c W/m · °C = W/m · K Velocity m3/s = 60,000 L/min = 106 cm3/s m3/s = 15,850 gal/min (gpm) = 35.315 ft3/s = 2118.9 ft3/min (cfm) Mechanical horsepower The electrical horsepower is taken to be exactly 746 W Some Physical Constants Universal gas constant Standard acceleration of gravity Standard atmospheric pressure Stefan–Boltzmann constant Boltzmann’s constant Speed of light in vacuum Speed of sound in dry air at 0°C and atm Heat of fusion of water at atm Enthalpy of vaporization of water at atm Ru = 8.31447 kJ/kmol · K = 8.31447 kPa · m3/kmol · K = 0.0831447 bar · m3/kmol · K = 82.05 L · atm/kmol · K = 1.9858 Btu/lbmol · R = 1545.37 ft · lbf/lbmol · R = 10.73 psia · ft3/lbmol · R g = 9.80665 m/s2 = 32.174 ft/s2 atm = 101.325 kPa = 1.01325 bar = 14.696 psia = 760 mm Hg (0°C) = 29.9213 in Hg (32°F) = 10.3323 m H2O (4°C) σ = 5.6704 × 10–8 W/m2 · K4 = 0.1714 × 10–8 Btu/h · ft2 · R4 k = 1.380650 × 10–23 J/K co = 2.9979 × 108 m/s = 9.836 × 108 ft/s c = 331.36 m/s = 1089 ft/s hif = 333.7 kJ/kg = 143.5 Btu/lbm hft = 2256.5 kJ/kg = 970.12 Btu/lbm Modern Fluid Dynamics 609 B.2 Properties Table B.2-1 Molar mass, gas constant, and critical-point properties Substance Air Ammonia Argon Benzene Bromine n-Butane Carbon dioxide Carbon monoxide Carbon tetrachloride Chlorine Chloroform Dichlorodifluoromethane (R-12) Dichlorofluoromethane (R-21) Ethane Ethyl alcohol Ethylene Helium n-Hexane Hydrogen (normal) Krypton Methane Methyl alcohol Methyl chloride Neon Nitrogen Nitrous oxide Oxygen Propane Propylene Sulfur dioxide Tetrafluoroethane (R-134a) Trichlorofluoromethane (R-11) Water Xenon Formula – NH3 Ar C6H6 Br2 C4H10 CO2 CO CCl4 Cl2 CHCl3 CCl2F2 CHCl2F C2H6 C2H5OH C2H4 He C6H14 H2 Kr CH4 CH3OH CH3Cl Ne N2 N2O O2 C3H8 C3H6 SO2 CF3CH2F CCl3F H2O Xe Molar mass, M kg/kmol 28.97 17.03 39.948 78.115 159.808 58.124 44.01 28.011 153.82 70.906 119.38 120.91 102.92 30.070 46.07 28.054 4.003 86.179 2.016 83.80 16.043 32.042 50.488 20.183 28.013 44.013 31.999 44.097 42.081 64.063 102.03 137.37 18.015 131.30 Gas constant, R kJ/kg · Ka 0.2870 0.4882 0.2081 0.1064 0.0520 0.1430 0.1889 0.2968 0.05405 0.1173 0.06964 0.06876 0.08078 0.2765 0.1805 0.2964 2.0769 0.09647 4.1240 0.09921 0.5182 0.2595 0.1647 0.4119 0.2968 0.1889 0.2598 0.1885 0.1976 0.1298 0.08149 0.06052 0.4615 0.06332 Critical-point properties Temperature, K 132.5 405.5 151 562 584 425.2 304.2 133 556.4 417 536.6 384.7 451.7 305.5 516 282.4 5.3 507.9 33.3 209.4 191.1 513.2 416.3 44.5 126.2 309.7 154.8 370 365 430.7 374.2 471.2 647.1 289.8 Pressure, MPa 3.77 11.28 4.86 4.92 10.34 3.80 7.39 3.50 4.56 7.71 5.47 4.01 5.17 4.48 6.38 5.12 0.23 3.03 1.30 5.50 4.64 7.95 6.68 2.73 3.39 7.27 5.08 4.26 4.62 7.88 4.059 4.38 22.06 5.88 Volume, m3/kmol 0.0883 0.0724 0.0749 0.2603 0.1355 0.2547 0.0943 0.0930 0.2759 0.1242 0.2403 0.2179 0.1973 0.1480 0.1673 0.1242 0.0578 0.3677 0.0649 0.0924 0.0993 0.1180 0.1430 0.0417 0.0899 0.0961 0.0780 0.1998 0.1810 0.1217 0.1993 0.2478 0.0560 0.1186 a The unit kJ/kg · K is equivalent to kPa · m3/kg · K The gas constant is calculated from R = Ru/M, where Ru = 8.31447 kJ/kmol · K and M is the molar mass Source: K A Kobe and R E Lynn, Jr., Chemical Review 52 (1953), pp 117–236; and ASHRAE, Handbook of Fundamentals (Atlanta, GA: American Society of Heating, Refrigerating and AirConditioning Engineers, Inc., 1993), pp 16.4 and 36.1 Table B.2-2 Ideal-gas specific heats of various common gases (a) At 300 K Gas Air Argon Butane Carbon dioxide Carbon monoxide Ethane Ethylene Helium Hydrogen Methane Neon Nitrogen Octane Oxygen Propane Steam Formula – Ar C4H10 CO2 CO C2H6 C2H4 He H2 CH4 Ne N2 C8H28 O2 C3H8 H2O Gas constant, R kJ/kg · K 0.2870 0.2081 0.1433 0.1889 0.2968 0.2765 0.2964 2.0769 4.1240 0.5182 0.4119 0.2968 0.0729 0.2598 0.1885 0.4615 Cp kJ/kg · K 1.005 0.5203 1.7164 0.846 1.040 1.7662 1.5482 5.1926 14.307 2.2537 1.0299 1.039 1.7113 0.918 1.6794 1.8723 Cv kJ/kg · K 0.718 0.3122 1.5734 0.657 0.744 1.4897 1.2518 3.1156 10.183 1.7354 0.6179 0.743 1.6385 0.658 1.4909 1.4108 k 1.400 1.667 1.091 1.289 1.400 1.186 1.237 1.667 1.405 1.299 1.667 1.400 1.044 1.395 1.126 1.327 Note: The unit kJ/kg · K is equivalent to kJ/kg · °C Source: Chemical and Process Thermodynamics 3/E by Kyte, B G., © 2000 (Adapted by permission of Pearson Education, Inc., Upper Saddle River, NJ) Appendix B 610 (b) At various temperatures Temperature, K 250 300 350 400 450 500 550 600 650 700 750 800 900 1,000 cp kJ/kg · K cv kJ/kg · K 1.003 1.005 1.008 1.013 1.020 1.029 1.040 1.051 1.063 1.075 1.087 1.099 1.121 1.142 Air 0.716 0.718 0.721 0.726 0.733 0.742 0.753 0.764 0.776 0.788 0.800 0.812 0.834 0.855 250 300 350 400 450 500 550 600 650 700 750 800 900 1,000 14.051 14.307 14.427 14.476 14.501 14.513 14.530 14.546 14.571 14.604 14.645 14.695 14.822 14.983 k cp kJ/kg · K 1.401 1.400 1.398 1.395 1.391 1.387 1.381 1.376 1.370 1.364 1.359 1.354 1.344 1.336 0.791 0.846 0.895 0.939 0.978 1.014 1.046 1.075 1.102 1.126 1.148 1.169 1.204 1.234 1.416 1.405 1.400 1.398 1.398 1.397 1.396 1.396 1.395 1.394 1.392 1.390 1.385 1.380 1.039 1.039 1.041 1.044 1.049 1.056 1.065 1.075 1.086 1.098 1.110 1.121 1.145 1.167 Hydrogen, H2 9.927 10.183 10.302 10.352 10.377 10.389 10.405 10.422 10.447 10.480 10.521 10.570 10.698 10.859 cv kJ/kg · K k Carbon dioxide, CO2 0.602 0.657 0.706 0.750 0.790 0.825 0.857 0.886 0.913 0.937 0.959 0.980 1.015 1.045 cp kJ/kg · K 1.314 1.288 1.268 1.252 1.239 1.229 1.220 1.213 1.207 1.202 1.197 1.193 1.186 1.181 1.039 1.040 1.043 1.047 1.054 1.063 1.075 1.087 1.100 1.113 1.126 1.139 1.163 1.185 1.400 1.400 1.399 1.397 1.395 1.391 1.387 1.382 1.376 1.371 1.365 1.360 1.349 1.341 0.913 0.918 0.928 0.941 0.956 0.972 0.988 1.003 1.017 1.031 1.043 1.054 1.074 1.090 Nitrogen, N2 0.742 0.743 0.744 0.747 0.752 0.759 0.768 0.778 0.789 0.801 0.813 0.825 0.849 0.870 cv kJ/kg · K k Carbon monoxide, CO 0.743 0.744 0.746 0.751 0.757 0.767 0.778 0.790 0.803 0.816 0.829 0.842 0.866 0.888 1.400 1.399 1.398 1.395 1.392 1.387 1.382 1.376 1.370 1.364 1.358 1.353 1.343 1.335 Oxygen, O2 0.653 0.658 0.668 0.681 0.696 0.712 0.728 0.743 0.758 0.771 0.783 0.794 0.814 0.830 1.398 1.395 1.389 1.382 1.373 1.365 1.358 1.350 1.343 1.337 1.332 1.327 1.319 1.313 Source: Kenneth Wark, Thermodynamics, 4th ed (New York: McGraw-Hill, 1983), p 783, Table A4M Originally published in Tables of Thermal Properties of Gases, NBS Circular 564, 1955 Table B.2.3 Properties of common liquids, solids, and foods (a) Liquids Boiling data at atm Substance Normal boiling point, °C Latent heat of vaporization hfg, kJ/kg Freezing data Freezing point, °C Latent heat of fusion hif, kJ/kg Liquid properties Temperature, °C Density, ρ , kg/m Specific heat cp, kJ/kg · K –33.3 –20 25 –185.6 20 20 682 665 639 602 1394 879 1150 4.43 4.52 4.60 4.80 1.14 1.72 3.11 Ammonia –33.3 1357 –77.7 322.4 Argon Benzene Brine (20% sodium chloride by mass) n-Butane Carbon dioxide Ethanol Ethyl alcohol Ethylene glycol Glycerine Helium Hydrogen Isobutane Kerosene Mercury Methane –185.9 80.2 103.9 161.6 394 – –189.3 5.5 –17.4 28 126 – –0.5 –78.4a 385.2 230.5 (at 0°C) –138.5 –56.6 80.3 –0.5 601 298 2.31 0.59 78.2 78.6 838.3 855 –114.2 –156 109 108 25 20 783 789 2.46 2.84 198.1 800.1 –10.8 181.1 20 1109 2.84 179.9 –268.9 –252.8 –11.7 204–293 356.7 –161.5 974 22.8 445.7 367.1 251 294.7 510.4 18.9 – –259.2 –160 –24.9 –38.9 –182.2 200.6 – 59.5 105.7 – 11.4 58.4 Methanol Nitrogen 64.5 –195.8 1100 198.6 –97.7 –210 99.2 25.3 Octane Oil (light) Oxygen Petroleum Propane 124.8 306.3 –57.5 180.7 –183 – –42.1 212.7 230–384 427.8 –218.8 13.7 –187.7 80.0 Refrigerant134a –26.1 217.0 –96.6 – 20 –268.9 –252.8 –11.7 20 25 –161.5 –100 25 –195.8 –160 20 25 –183 20 –42.1 50 –50 –26.1 25 1261 146.2 70.7 593.8 820 13,560 423 301 787 809 596 703 910 1141 640 581 529 449 1443 1374 1295 1207 2.32 22.8 10.0 2.28 2.00 0.139 3.49 5.79 2.55 2.06 2.97 2.10 1.80 1.71 2.0 2.25 2.53 3.13 1.23 1.27 1.34 1.43 Modern Fluid Dynamics Water 100 611 2257 0.0 333.7 25 50 75 100 1000 997 988 975 958 4.22 4.18 4.18 4.19 4.22 a Sublimation temperature, (At pressures below the triple-point pressure of 518 kPa, carbon dioxide exists as a solid or gas Also, the freezing-point temperature of carbon dioxide is the triple-point temperature of –56.5°C.) Table B.2-3 Properties of common liquids, solids, and foods (concluded) (b) Solids (values are for room temperature unless indicated otherwise) Metals Aluminum 200 K 250 K 300 K Density, ρ kg/m3 Specific heat, cp kJ/kg K 2110 1922 2300 2300 0.920 0.79 0.960 0.653 1000 2420 2700 2230 2500 2700 800 0.920 0.616 0.800 0.840 0.711 1.017 1.09 0.254 0.342 0.367 0.381 0.386 0.393 0.403 Clay Diamond Glass, window Glass, pyrex Graphite Granite Gypsum or plaster board Ice 200 K 220 K 240 K 260 K 273 K Limestone Marble Plywood (Douglas Fir) 921 1650 2600 545 1.56 1.71 1.86 2.01 2.11 0.909 0.880 1.21 1.840 Specific heat, cp kJ/kg · K Density, ρ kg/m3 Substance 2,700 0.929 0.949 0.973 0.997 0.400 8,310 Nonmetals Asphalt Brick, common Brick, fireclay (500°C) Concrete 0.797 0.859 0.902 8,280 350 K 400 K 450 K 500 K Bronze (76% Cu, 2% Zn, 2% Al) Brass, yellow (65% Cu, 35% Zn) Copper –173°C –100°C –50°C 0°C 27°C 100°C 200°C Substance 0.400 8,900 Iron 7,840 0.45 Rubber (soft) 1100 Lead 11,310 0.128 Rubber (hard) 1150 2.009 Magnesium 1,730 1.000 Sand 1520 0.800 Nickel 8,890 0.440 Stone 1500 0.800 Silver 10,470 0.235 Woods, hard (maple, oak, etc.) Woods, soft (fir, pine, etc.) 721 1.26 513 1.38 Steel, mild 7,830 0.500 Tungsten 19,400 0.130 (c) Foods Food Apples Water content, % (mass) 84 Specific heat, kJ/kg K Freezing point, °C –1.1 Above freezing 3.65 Below freezing 1.90 Latent heat of fusion, kJ/kg 281 Specific heat, kJ/kg K Food Lettuce Water content, Freezing % (mass) point, °C 95 –0.2 Above freezing 4.02 Below freezing 2.04 Latent heat of fusion, kJ/kg 317 Bananas 75 –0.8 3.35 1.78 251 Milk, whole 88 –0.6 3.79 1.95 294 Beet round 67 – 3.08 1.68 224 Oranges 87 –0.8 3.75 1.94 291 Broccoli 90 –0.6 3.86 1.97 301 Potatoes 78 –0.6 3.45 1.82 261 Butter 16 – – 1.04 53 Salmon fish 64 –2.2 2.98 1.65 214 Cheese, swiss 39 –10.0 2.15 1.33 130 Shrimp 83 –2.2 3.62 1.89 277 Cherries 80 –1.8 3.52 1.85 267 Spinach 93 –0.3 3.96 2.01 311 Chicken 74 –2.8 3.32 1.77 247 Strawberries 90 –0.8 3.86 1.97 301 Corn, sweet 74 –0.6 3.32 1.77 247 94 –0.5 3.99 2.02 314 Eggs, whole 74 –0.6 3.32 1.77 247 Tomatoes, ripe Turkey 64 – 2.98 1.65 214 Ice cream 63 –5.6 2.95 1.63 210 Watermelon 93 –0.4 3.96 2.01 311 612 Appendix B Source: Values are obtained from various handbooks and other sources or are calculated Water content and freezing-point data of foods are from ASHRAE, Handbook of Fundamentals, SI version (Atlanta, GA: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1993), Chapter 30, Table Freezing point is the temperature at which freezing starts for fruits and vegetables, and the average freezing temperature for other foods B.3 Drag Coefficient: (A) smooth sphere and (B) an infinite cylinder as a function of Reynolds number Modern Fluid Dynamics B.4 Moody Chart 613 Appendix B 614 B.5 Turbulent Velocity Profiles in Pipes B.5.1 Composite (Log-Law) Profile • Law of the wall: u + = y + , u + ≡ yu u , y+ ≡ τ ν uτ where y = R − r is the wall coordinate, u τ ≡ τ wall / ρ is the friction velocity, and ν = μ / ρ is the kinematic viscosity Note: The linear profile u + = y + holds within the viscous sublayer ≤ y + ≤ , i.e., y = δsublayer = 5ν / u τ • Logarithmic law: u + = 2.5 ln y + + 5.0 ; y + > 5.0 Note: The log-law matches experimental data outside the viscous sublayer ≤ y + ≤ B.5.2 One-Seventh Power-Law ( n ≈ ): u u max y r = ( )1/ n = (1 − )1/ n R R Note: The power-law fails to: (i) generate a zero slope at the pipe centerline and (ii) calculate the wall shear stress However, it is easy to use, and when selecting n = n (Re D ) it provides reasonable profiles • Power-law exponent n (Re D ) Re D = u avg D ν n u average u max × 103 2.3 × 104 1.1 × 105 1.1 × 106 3.2 × 105 6.0 6.6 7.0 8.8 10.0 0.791 0.807 0.817 0.850 0.865 Index A turbulent, 264, 272, 298 Aerosol acceleration, 512 turbulent flat-plate, 261 Apparent viscosity, 543 Brownian motion, 357 Approach C differential, 7, 46 Capillary number, 371 integral, 6, 46 Capillary pump, 372 molecular dynamics, 6, 46 Casson model, 488, 489 phenomenological, 7, 46 CAUCHY equation, 598 Closed system, 21 Approaches Commercial software, 530 derivation, 6, 46 Arterial hemodynamics, 490 Compartmental modeling, 238 Axial force, 445 Compartments B in parallel, 241 Balance principles, 13 in series, 241 Bending, 445 Compressibility factor, 147 BERNOULLI equation, 78, 143, 604 Compressible flow one-dimensional, 139 Bifurcation, 536 Bingham fluid, 323 Computational fluid dynamics, 526 Bioheat equation, 514 Concentrated-mass oscillations, 457 Bio-MEMS, 351, 352 Conditions of compatibility, 445 Blasius flow, 258 Constitutive equations, 25 Blood properties, 485 Contact angle, 384 Blood rheology, 520 Continuity equation, 59, 597 Bond number, 371 Continuous phase, 309 Boundary conditions, 531 Continuum mechanics, 13 assumption, 373 Boundary layer, 255 Couette flow, 69, 167, 381, 391, 478 Boundary-layer flow, 261 with a Bingham plastic, 328 flat-plate, 261 Creeping flow, 135 laminar, 258, 263 615 Index 616 Cunningham slip correction factor, 338, in differential form, 67 509 in integral form, 85 D Entrance effects, 377 Damped vibration, 465 Entropy generation, 181 Damping, 479 derivation, 184 coefficient, 493, 494 due to heat transfer, 410 ratio, 492 in microfluidic systems, 409 Darcy’s experiment, 132 thermal pipe flow, 179 Debye length, 403 Entropy minimization, 411 Deformation analysis, 445 Equation discretization, 530 Diffusion coefficient, 33, 340, 494 Equilibrium conditions, 443 Diffusivity, 357 Error function, 590 Dimensionless groups, 10, 16, 371 EULER equation, 77 Dispersed phase, 309 Euler number, 10 Dissipation function, 602 Eulerian frame, 24 Divergence theorem, 588 External Flow, 8, 253 Drag F coefficient, 271, 518 Film coating, 108 form, 273 Film drawing, 274, 286 frictional, 272 Flow assumption, 18 computation, 267 Flow dynamics, 485 Driving forces, Flow-induced vibration, 460 Dynamics, 14, 481 Fluid, E Fluid dynamics, Electric double layer (EDL), 397 Fluid flow Effective thermal conductivity, 418 non-Newtonian, 320 Elastic chamber, 502 Fluid phase, 309 Electro-hydrodynamics, 395 Fluid rheology, 485 Electrokinetics, 407 Fluid-interface mechanics, 281 Electrolyte liquid, 395 Fluid-particle dynamics, 485, 492 Electro-osmosis, 397 Fluid-structure interaction (FSI), 500, 531, Electrophoresis, 407 Energy balance, 79 Energy conservation, 85 568 simulation, 569 Flux vector, 33 Modern Fluid Dynamics 617 Force balance derivation, 72 K Forces Kelvin model, 449 normal/tangential, Kinetics, 13 Fourier’s law, 25, 34 Knudsen number, 377 Friction factor, 171 L turbulent, 219 Lagrangian description, 19 Fundamental assumptions, 45 Lamé constants, 449 G Laminar flow fully-developed, 117 Gas flow, 391 steady 1-D incompressible, 105 characteristics, 355 in a microchannel, 413 Laplacian operator, 583 Geometric configurations, 490 Law of thermodynamics, 140 H Leibniz rule, 590 Harmonic response Lift, 271 of an underdamped system, 474 coefficient, 271 to forced vibration, 473 computations, 267 to free vibration, 465 Liquid flow, 357, 384 Heat exchanger basic shell and tube, 219 in a microchannel, 413 Lubrication approximations, 223 systems, 221 single tube, 217 Heat transfer Lung-aerosol dynamics, 505 convection coefficient, 190 M forced convection, 159 Mach number, 356 Homogeneous flow equations, 317 Marangoni effect, 373 Hooke’s law, 25, 450 Mass conservation, 51 Hydraulic diameter, 196 in differential form, 56 Hydrodynamic in integral form, 51 entrance length, 206 lubrication, 232 Hyperbolic functions, 588 Maxwell’s first-order slip-velocity model, 386 Microchannel I cross section, 378 Internal flow, 8, 195 entrance flow, 377 Inviscid flow, 99 flow effects, 377 Isothermal flow, 206 surface roughness, 378 Index 618 Microfluidics, 351 Nozzle flow, 153 modeling aspects, 354 devices, 358 converging-diverging, 157 Numerical validation, 532 Micro-mixers, 427 Nusselt number, 10, 162 Microparticle dynamics, 507 O Microscale heat-sinks, 417 Open system, 19, 84 Microscopic models, 14 Ordinary differential equation (ODE), 594 Microsphere P deposition, 536 Parallel flow, 109, 122, 124 transport, 553 Parallel-plate flow, 382, 385 Microsystem modeling assumption, 352 Particle phase, 308, 309 Particle trajectory model, 332 Mixed-phase flow, 307 Particle transport, 332 Mixture flow Phonons, 358 Poiseuille-type, 314 Pipe flow Mixture viscosity, 314 bubbly, 318 Model validation, 478, 520, 534 non-isothermal turbulent, 214 Modeling aspects, 484 Pipe Flow Moments, 443 Laminar, 198 Momentum conservation, 61 turbulent, 214 in differential form, 67 Plane stress, 452 in integral Form, 61 Poiseuille blood flow, 502 Momentum equation, 599 Poiseuille flow, 81, 118, 403, 414 N Porous media flow, 129 Nanofluid flow Power-law fluid flow in a slightly-tapered tube, in bio-MEMS, 423 323 with heat transfer, 420 modeling, 337 Nanofluids, 433 Nanoparticle PRANDTL boundary-layer equations, 76 mixing, 424 Q transport, 354 Quemada model, 487, 540 Nanotechnology, 416 R Navier’s slip-length model, 385 Radial flow, 137 NAVIER-STOKES equations, 75, 600 Result interpretation, 531 Modern Fluid Dynamics 619 Stress Reynolds lubrication equation, 232 in solid structures, 437 transport theorem, 47, 48, 49 tensor, 31 vector, 31, 360 Reynolds number, 10, 377, 508 Reynolds-Colburn analogy, 163, 165 Stress-strain relations, 448 S Strouhal number, 10, 11 Scale analysis, 10, 91 Sudden impact force, 365 Schmidt number, 10 Surface Separated flow, 320 coating, 282, 291 Shear effects, 384 force, 461 roughness effects, 377 modulus, 450 tension, 384 stress, 26, 69 T Sherwood number, 10 Thermal conduction, 358 Shock wave Thermal diffusivity, 33 induced velocity, 152 Thermal pipe flow, 87 in tube, 150 Thermocapillary effect, 374 Simulation accuracy, 528 Thermodynamic properties, 24 Slender-body dynamics, 453 Thin-film flow, 382 Slender-body oscillations, 455, 459 Torsion, 441 Slip flows, 385 Transient flow Slot flow, 225 one-dimensional, 123 Sound waves, 143 pipe, 129 Spin coating, 291 Transport and deposition Squeeze film force, 365 micronparticle, 498 Stern layer, 398 nanomaterial, 506 STOKES equation, 97 Transport phenomena, Stokes number, 10, 511, 533 Tribology, 221 Stokes’ Turbulent first problem, 123 eddy viscosity, 262 law, 364 pipe flow, 171 STOKES-EINSTEIN equation, 357 Streamlines, thermal pipe flow, 172 Two-fluid flow, 382 Two-phase flow, 317 Index 620 U W Undamped vibration, 465 Wall boundary conditions, 381 V for macro-scale flows, 380 Viscosity, 26 for micro-conduit flows, 383 Viscous dissipation, 163, 379 Weber number, 10, 372 in microchannels, 395 Windkessel model, 521 Viscous flows, 105 Womersley number, 10 Volume fraction, 311 Y Von Karman integral method, 591 Young’s modulus, 450 ... readability and clarity For other titles published in this series, go to www.springer.com/series/5980 Clement Kleinstreuer Modern Fluid Dynamics Basic Theory and Selected Applications in Macro- and Micro-Fluidics. .. traditional and modern fluid dynamics, i.e., the fundamentals of and basic applications in fluid mechanics and convection heat transfer with brief excursions into fluid- particle dynamics and solid... phenomena in terms of differential or integral equations The (analytical or numerical) solutions to the describing C Kleinstreuer, Modern Fluid Dynamics: Basic Theory and Selected Applications in Macro-