- Preface
- Introduction to Transport Phenomena
- Thermodynamics of Multiphase Systems
- Generalized Governing Equations in Mutliphase Systems: Local Instance Formulations
- Generalized Governing Equations for Multiphase Systems : Averaging Formulations
- Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
- Melting and Solidification
- Sublimation and Vapor Deposition
- Condensation
- Evaporation
- Boiling
- Two-Phase Flow and Heat Filter
- Appendix A : Constants, Units and Conversion Factors
- Appendix B: Transport Properties
- Appendix C: Vectors and Tensors
- Index

Index
Δ operator, 1007–11 Ablation, 448-450 Adhesion, 19, 370 Adiabatic tube, sublimation, 538–43 Adiabatic wall evaporation, 685–701 horizontal films, 685–91 vertical falling film, 691–701 Adsorption, 344-348 Air properties, 957 Aluminum properties, 962 Annular condensation heat transfer, 643-647 Annular flow, 857–858, 924 condensation, 911–19 dispersed, 862 transition to stratified flow, 888 wispy, 858 See also Flow patterns Archimedes number, 483, 610 Area-averaged homogeneous model, 265–70 continuity equation, 266 energy equation, 266–67 momentum equation, 266 second law of thermodynamics, 267 species, 267–70 Area-averaged models, 264–73 Average kinetic energy, 13 Averages extrinsic, 240–41, 244, 245, 265 intrinsic, 240, 241, 244, 245, 265 volume, 241, 242, 278 Averaging Boltzmann statistical, 244, 303–17 Eulerian, 239–42 formulations, 238–317 Lagrangian, 243 methods, 239 molecular statistical, 243–44 overview, 239–44 time, 239–40 volumetric, 239–40 Avogadro’s number, 161, 164 Axially-grooved heat pipe (AGHP), 368 Axisymmetric film condensation, 918 Barrel reactors, 566–70 Beattie-Bridgeman equation, 135 Benedict-Webb-Rubin equation of state, 135–36 Bernard cellular flow, 338-339 Binary diffusion coefficients, 41, 993 Binary solution system solidification, 465–79 integral approximate method, 466–71 mixture model, 471–75 volume-averaging model, 475–79 Binary vapor mixtures, 600–605 Biot number, 434, 549–50, 574 Boilers, 76 Boiling, 56–57, 765–843 critical heat flux, 803–806 explosive, 97–98 film, 766, 811–33 flow, 72 forced convective, 765, 889–904, 920 in micro/miniature channels, 933–37 nucleate, 766, 770–802 pool, 72, 765, 767–69 in porous media, 833–43 saturated, 766, 897 subcooled, 765 subcooling effects, 817 transition, 766, 806–10 on wicked surface, 766 Boiling point line, 583 Boltzmann constant, 12, 13, 21, 561 Boltzmann equation, 18–19, 303–11 Boltzmann statistical averaging, 244, 303–17 LBM, 311–14 LBM for multiphase flows, 314–17
Index 1013
Boussinesq assumption, 193 Brinkman’s equation, 283–84 Bubble detachment, 786–92 See also Vapor bubbles Bubble dynamics, 775–86 See also Vapor bubbles Bubble growth, 772 equation of motion, 785 merger and, 792–96 pattern, 796 schematic, 783 stage, 775 in superheated liquid droplets, 782–86 in viscous fluid, 788 in volatile droplet, 782 Bubble lift-off, 895–99 Bubble point line, 140 Bubbly flow, 857–862 See also Flow patterns Buckingham’s theorem, 49, 51 Burnout point, 769 Caloric, 3–4 Capillary action, 88, 652-8 Capillary depression, 596 Capillary phenomenon, 88, 332–42, 345 Capillary pressure, 298, 332–34 Carbon dioxide, 958 Carbon steel properties, 965 Carnahan-Starling-van der Waals equation of state, 317 Centrifugal field via rotating disk, 649–52 Chapman-Enskog analysis, 317 Chemical equilibrium, 124–28 Chemical reactions combustion and, 123–24 mole number and, 125 process, 125 species source/sink due to, 253 system with, 123–28 thermodynamic equilibrium constant, 201 total mass of species, 188 Chemical reaction sublimation, 550–54 Chemical stability, 121–22 Chemical vapor deposition (CVD), 67–68, 202, 533 applications, 563–74 atmospheric reduced pressure (APCVD), 556 in barrel reactor, 567–70
in horizontal reactor, 563–67 introduction, 554–57 laser (LCVD), 533, 570–74 low pressure (LPCVD), 556 precursors, 557 processes, 555 reactors, 556 Chimney effect, 691 Churn flow, 857-58 Clapeyron equation, 142–45, 593 Clausius-Clapeyron equation, 144–45, 777, 793, 837 Cohesion, 19 Collisions, 311 elastic, 306 elementary invariants, 307 operator, 312–13 Combustion, 77, 223–30 analysis, 228 chemical reaction and, 123–24 energy equation for, 224 near liquid fuel droplet, 227 Combustors liquid fuel, 77 solid fuel, 77 turbofan, 77–79 Completely wetting, 344 Condensation, 55–56, 69–71, 581–667 annular film, 919 by capillary action in heat pipe, 652–58 complete, 918 counter-current, 217 direct contact, 70 dropwise, 70, 581, 587–99 filmwise, 70, 581, 582, 599–637 forced convective, 587, 883–89 gravity-dominated, 660–63 heat flow resistance in, 594 heterogeneous, 581, 582 homogeneous, 581, 583 immiscible fluids, 585 incomplete, 918 indirect contact, 70 interfacial resistance, 366–68 on liquid droplets, 583 from macroscopic viewpoint, 69 in miniature tubes, 92–93, 908 modes flowchart, 582 nongravitational condensate removal, 638– 58
1014 Transport Phenomena in Multiphase Systems
of pure vapor, 584 in tube with suction at porous wall, 638–43 turbulent film, 586, 619–24 Condensers, 74, 75 Conduction, 26–27 in multidimensional isotropic system, 27 in solid, 45 thermal penetration depth, 59 Cone evaporation. See Wedge/cone evaporation Contact angles, 342–43, 375, 586, 589 Contact melting, 479–84 Continuity equation, 183, 190, 226, 227 area-averaged homogeneous model, 266 differential, 190 integral, 183 volume-averaged homogeneous model, 258 volume-averaged multi-fluid model, 244–45 volume-averaging model, 176 Volume of Fluid method (VOF), 395 Continuum approach, 393–402 front tracking methods, 400–402 Lagrangian techniques, 393–94 phase interface fitted grid, 398–400 stationary grid, 394–98 Continuum flow limitations, 21–23 regimes, 22 Continuum regime, 22 Continuum surface force (CSF) model, 397, 398 Convection, 28 forced, 53–54 free, 54–55 natural, 423, 693 Convective heat transfer coefficient, 29, 49, 53 Convective mass transfer, 31, 41–44 Conversion factors, 951–53 Cool evaporation, 684 Cooling convective, freezing of slab under, 434 droplets, schematic, 254 electronics, 92–96 tower, 253 Copper properties, 962 Couette flow, 23–24 Counter-current condensation, 217 Creep flow, 192 Crispation number, 341 Critical droplet radius, 588–93 Critical heat flux (CHF), 94 boiling, 803–6
forced convective boiling, 903 phenomenon, 803 summary, 805 vapor back-flow near, 936 as warning point, 769 Critical Helmholtz velocity, 803, 804 Cryopreservation, 84 Cyclic dryout, 933 Cylindrical coordinate systems, 1008–9 melting, 450–54 solidification, 450–54 Damping functions, 695 Darcian velocity, 664 Darcy’s law, 276, 278, 282–85, 510, 750 Deformation tensor, 352 Deposition, 67 rate of monocrystalline silicon, 566 temperature distribution, 532 vapor, 533 Dew point, 583 Dew point line, 140, 583 Diffusion coefficients, 994–95 Diffusive mass fluxes, 35, 38, 39–40, 205, 299 binary mixture, 40 phase, 299, 301 Dimensional analysis, 48–58 Dimensionless numbers, 52 Direct contact evaporation, 681–82, 724–29 of liquid droplet in hot gas, 724–27 of liquid jet in pure vapor, 727–29 See also Evaporation Direct contact condensation, 70 Direct methanol fuel cells (DMFCs), 34 Disjoining pressure, 162–67, 347-49, 371, 381,740 Dispersed bubble flow, 861, 862 Dispersed phases, 64, 65 Dispersion forces, 16 Disturbance amplification, 392 Disturbance wavelength, 385 Dittus-Boelter/McAdams equation, 900 Dropwise condensation, 70, 587–99 Dufour effect, 196, 204, 566, 567 Dufour heat flux, 565 Dynamic viscosity, 604 EHD-induced flow, 894 EHD number, 893, 894 Elastic collision, of two particles, 306
Index 1015
Electro-discharge machining (EDM), 926 Electronics cooling, 92–96 condensation in miniature tubes, 92–93 heat sinks, 93–94 micro heat pipes/spreaders, 94–96 Electron temperature, 19 Electron thermal conductivity of, 514 Electrophoretic force, 893 Electrostatic forces, 15 Elongated bubbles, 934 Embryo droplets equilibrium radius, 591 mass, 590 Emissive power, 30 Energy advection, 301 bond, 20 internal, 15, 115, 116 kinetic, 13, 14, 218, 304 mechanical balance, 266 rotational, 14, 304 thermal, 11, 14 total, 14, 185, 194 vapor, 255 vibrational, 14, 304 Energy equations area-averaged homogeneous model, 266–67 for combustion, 224 CVD, 558 extrinsic phase-average of, 248 homogeneous two-phase flow, 865, 866 inclined microchannel evaporation, 745 macroscopic (integral) formulation, 185–86 microscopic (differential) formulation, 193– 98 in moving coordinate, 449 porous media, 285–87 separated flow model, 869–70 single component system, 206 solid phase, 457 stationary grid approach, 396 turbulent falling film, 714 vapor film, 841 volume-averaged homogeneous model, 259– 60 volume-averaged multi-fluid model, 248–51 volume-averaging model, 477 wedge/cone evaporation, 719 Energy flux, conversion factor, 951 Enthalpy, 19–21
conversion factor, 951 distribution, 458 during phase change, 21 intrinsic phase average, 250 liquid, 354 mass average, 259 method, 455–60 model for two-region melting/ solidification, 488–92 multicomponent system, 195 partial molar, 36 for saturated vapor, 218 sensible, 225 substantial derivative, 206 Entropy balance, 152 change, in closed system, 186 generation, 187, 211 inequality, 210 mass-averaged, 260 Equations of state, 131 Beattie-Bridgeman, 135 Benedict-Webb-Rubin, 135–36 ideal gas, 131 for pure substances, 131–36 Redlich-Kwong, 134 van der Waals, 132, 133 Equilibrium bubble radius, 158 chemical, 124–28 closed system, 146 compositions, 1002 constant temperature and pressure system, 113 constant-temperature and volume system, 112–13 constant-volume isolated system, 112 contact angles, 343 criteria, 111 criteria summary, 113–14 at interfaces, 150–52 interface shapes, 332, 334–37 liquid droplets, 160, 589 metastable, 147–50 multiphase multicomponent systems, 146– 47 of single-phase systems, 111–28 stable, 147 two-phase component systems, 141–42 unstable, 147
1016 Transport Phenomena in Multiphase Systems
Equilibrium molecular dynamics (EMD), 48 Equilibrium radius, 591-93 Equivalent heat capacity method, 460–64 Eulerian approach, 178 Eulerian averaging, 239–42, 265 Euler’s equation, 752 Evaporating film characteristics, 377 on fragment of the rough solid surface, 374 heat flux through, 379 Evaporation, 55, 678–755 from adiabatic wall, 685–701 classification, 681–84 coefficient determination, 403 cool, 684 criteria, 680, 681–84 defined, 71–72 direct contact, 681, 724–29 domains, 684 in energy systems, 76 falling film, on heated wall, 701–24 film, 678–79 heterogeneous, 681 from horizontal films, 685–91 hot, 683 illustrated, 71 in inclined microchannel, 744–47 into humid air, 697 from inverted meniscus, 747–55 latent heat, 381, 698 of liquid droplet in hot gas, 724–27 from liquid droplets (in vapor-gas mixture), 680–81 from liquid droplets (on heated wall), 679– 80 of liquid jet in pure vapor, 727–29 micro layer, 799 nucleate boiling and, 798 Nusselt, 702–6 in pores and slots/microchannels, 729–47 two-phase flow, 923–29 under heat flux, 738–44 from vertical falling film, 691–701 of working fluids, 367 Explosive boiling, 97–98 Extrinsic average velocity, 281 Extrinsic phase average, 240–41, 244–45, 248– 49, 251, 265 Falling film evaporation, 701–24
classical Nusselt, 702–13 on heated wall, 701–24 surface spray cooling, 715–18 turbulent falling film, 713–15 wedge/cone embedded in porous medium, 718–24 Feedwater heater, 75 Fick’s law, 36 Fictitious density, 614 Film boiling, 766, 811–33 analysis, 811–21 boundary value problem, 816–17 direct numerical simulation, 821–24 energy balance at liquid-vapor interface, 815 on heated horizontal surface, 823 Leidenfrost phenomena, 824–33 mass balance at liquid-vapor interface, 812, 813 mass flow rate, 815 near immersed body, 819 over horizontal large flat plate, 819–20 in porous media, 840–43 in porous media illustration, 841 subcooling effects, 817 transition, 807 on vertical surface, 811 See also Boiling Film condensation in porous media, 658–67 Film evaporators, 679, 691 Film thickness, 388, 389 Filmwise condensation, 70, 599–637 average heat transfer coefficient, 611 of binary vapor mixture, 600–605 configurations, 625–27 defined, 581 empirical correlations, 586 flooding limit, 633–37 flow regimes, 600 gravity-dominated, 660–63 heat flow resistance, 594 illustrated, 582 laminar flow regime, 605–9 laminar regime, 600 noncondensable gas effects, 627–33 in porous media, 658–67 regimes, 599–600 in stagnant pure vapor reservoir, 605–13 turbulent, 611–13, 619–24 wavy condensate regime, 610–11 See also Condensation
Index 1017
Fixed melting, 479 Flat miniature evaporator, 924–28 Flooding limit, 633–37 Kutateladze correlation, 635–37 Wallis correlation, 633–35 Flow boiling, 72 Flow condensation, 908–19 Flow evaporation, 923–29 Flow maps condensation in miniature/micro channels, 909 forced convective condensation, 884 regime, 859–60 Flow patterns, 854–64 annular-dispersed flow, 862 annular flow, 857, 858 based on low-pressure air-water flow data, 861 bubbly flow, 857, 858 churn flow, 857, 858 concepts and notations, 854–57 condensation inside tubes, 883 dispersed bubble flow, 861, 862 in horizontal tubes, 861–64 Kattan-Thome-Favrat, 892 plug flow, 857, 858, 861, 862 reconstructed, 895 slug flow, 857, 858, 862 stratified flow, 862 stratified wavy flow, 862 two-phase, 904–7 two-phase condensation, 908–10 in vertical tubes, 857–61 void fraction, 854, 855, 856 wispy annular flow, 858 See also Two-phase flow Flow regimes, 72 boundaries between, 860 convective condensation in tube, 885 map, 859, 861, 863 Fluid mechanics, 23–26 Fluids immiscible, 585, 586 interfacial region between, 151 stressor tensor components, 25 Fluxes, 42 Food freezing/thawing, 83–84 Forced convective boiling, 765, 889–904 bubble lift-off size, 895–99 configurations, 890
critical heat flux (CHF), 903 flow regime sequence, 890–91 heat transfer predictions, 899–904 regimes, 889–95 subcooled, 896, 899 in tubes, 889–904 See also Boiling Forced convective condensation, 587, 883–89 flow pattern map, 884 gravitational effects, 885 heat transfer predictions, 885–89 two-phase flow regime, 883–85 See also Condensation Forces attractive, 16 conversion factor, 951 dispersion, 16 electrostatic, 15 induction, 15 interactive, 246, 248 intermolecular, 15–19 interphase, 259 Forchheimer inertia coefficient, 489 Fourier’s law, 651 Free convection, 766 Free energy approach, 317 Free liquid surface curvature, 376 Free molecular flow regime, 22 Free surfaces, numerical simulation, 392–406 Freezing food, 83–84 of slab under convective cooling, 434 Frictional coefficient, 641 Frictional loss, 281, 282 Frictional pressure drop, 870–78 bounds on two-phase flow, 874–78 correlations based on homogeneous model, 870–71 correlations based on separated flow model, 871–74 prediction, 875 Frictional pressure gradients determining, 871, 872, 876, 877 of vapor/liquid phase flow, 876 See also Pressure gradients Front tracking methods, 400–402 Froude number, 52, 902, 908 Fuel burning rate, 228 Fuel cell, 79–82, 274, 295–99, 325 Fused silica properties, 963
1018 Transport Phenomena in Multiphase Systems
Gas diffusion layers (GDLs), 80 Gas dynamic model, 506 Gaussian error function, 367 Generalized Fick equation, 36 Generalized Maxwell-Stefan equations, 36 Gibbs-Duhem equation, 165, 350 Gibbs free energy, 122, 126, 127, 154, 588 liquid phase, 591 of two phases, 213 vapor phase, 591 Gibbs phase rule, 108, 111 Grashof numbers, 557, 564, 824 Gravity-dominated condensation, 660–63 Griffith’s correlation, 599 Hamaker constant, 164 Heat absorption, 108 caloric, 3 carriers in solids, 512 conduction, 26, 436–39 effective capacity, 286 for gas, 5 hidden, 6 Joule concept, 4 latent, 5–7 mechanical equivalent, 4 multiphase, 62 sensible, 3–7, 481 specific, 5 spreaders, 95–96 Heat pipes, 88–92 axially-grooved (AGHP), 368 capillary action in, 652–58 evaporating section, 368, 496 frozen startup, 495–99 gas-loaded, temperature profiles, 658 micro, 94–96 noncondensable configuration, 653 pulsating, 91–92, 380 wick, 88–89, 496 wick structure, 497 Heat sinks, 93–94, 923–29, 932 Heaviside function, 823 Helium properties, 958 Helmholtz free energy, 126 Helmholtz instability, 804 Henry’s constant, 359, 1000 Heterogeneous bubble growth, 780–82
Heterogeneous condensation, 581–82 Heterogeneous evaporation 682– 83 Heterogeneous nucleation, 581 Heterogeneous reaction, 202 Holdup void fraction, 266 Homogeneous bubble growth, 776–80 Homogeneous condensation, 581-86 Homogeneous flow model, 864–66 correlations based on, 870–71 energy equation, 865, 866 governing equations, 865 mixture density, 864–65 Homogeneous model, 242 area-averaged, 265–70 for channel with variable cross-section, 268 volume-averaged, 258–64 Homogeneous nucleation, 170, 581 Homogeneous reacting systems, 202 Horizontal film evaporation, 685–91 Horizontal reactors, 563–67 Horizontal two-phase flow, 863 Hot evaporation, 683–84 Hot gas evaporation, 728 Hot wall tubular LPCVD reactors, 556 Hydrogen properties, 959 Hydrophobic materials, 346 Hydrostatic pressure gradient, 614 Hyperbolic function, 220 Immiscible fluids, 585–86 Inclined microchannel evaporation, 744–47 Inconel properties, 963 Indirect contact condensation, 70 Induction forces, 15 Injection velocity, 537 Inorganic compounds, solubility, 1001 Interfaces, 331-406 boundary condition at, 213 continuity equation at, 214 dynamic behaviors of, 382–92 dynamics in, 315 energy balance at, 215 equilibrium at, 150–52 jump conditions at, 189, 206–18 liquid temperature, 215 locations, 177 mole number, 153 numerical simulation, 392–406 second law of thermodynamics at, 210–11 solid-liquid, 424–27
Index 1019
stability, 384 supplementary conditions at, 212–18 thermodynamics at, 150–71 thickness, 207 tracking, 238 transport effects at, 351–82 unit normal vectors at, 208 vapor temperature, 215 velocity, 401 Interface shapes at equilibrium, 332, 334–37 Interface tracking, 393–402 front tracking methods, 400–402 Lagrangian techniques, 393–94 method, 177 phase interface fitted grid, 398–400 stationary grid approach, 394–98 Interface velocity, 428 Interfacial displacement velocity, 213 Interfacial phenomena, 68–69, 331 Interfacial resistance, 366–68, 595–96 Interfacial temperature, 370, 633 Interfacial tension gradients, 337–42 Interfacial thickness, 403 Interfacial velocity, 213, 398, 401 Interline region, 373 Intermolecular forces, 15–19 Internal energy, 115 rate of gain, 195 of real gas, 15 representation, 116 reversible rate of, 195 surfaces, 153 thermal stability requirement, 119 Interphase forces, 259 Interphase phase change energy, 259 Intrinsic phase average, 240, 241, 244–46, 250, 265 Inverted meniscus evaporation, 747–55 Iron properties, 963 Isothermal compatibility, 120 Jakob number, 662, 687, 778, 843 Joule, heat concept, 4 Jump conditions, 189, 206–18 as boundary conditions, 207 conservation of mass, 207–8 energy balance, 209–10 momentum balance, 208–9 second law of thermodynamics, 210–11 species, 211–12
supplementary conditions, 212–18 Kapitza number, 712 Kattan-Thome-Favrat flow pattern, 892 Kelvin equation, 350–51 Kelvin-Helmholtz instability, 382–86 Kinetic energy, 218 average, 13, 304 average random, 304 of individual molecules, 243 modes, 14 particle, 309 rotational, 304 vibrational, 304 Kinetic theory, 10–15 Kucherov-Rikenglaz equation, 367 Kutateladze correlation, 635–37 Lagrangian approach, 178 Lagrangian averaging, 243 Lagrangian interface tracking technique, 393–94 Laminar condensate flow, 614–17 Laminar falling film, 706–13 Laminar film condensation, 599-638 assumption, 613 of binary vapor mixture, 600–605 classical analysis, 605 coordinate system, 601 governing equations, 602–3 Nusselt analysis, 639 physical conditions, 605 physical model, 601 Reynolds number, 608 on sphere, 625 Laplace operator, 1008 Laplace-Young equation, 157, 158, 926 Laser-assisted manufacturing, 84–88 Laser chemical vapor deposition (LCVD), 533, 555–56, 570–74 Laser drilling, 505–8 Laser machining, 86–87 Latent heat, 5–7 of evaporation, 381, 698 melting, 425 of sublimation, 531 thermal energy storage, 74, 492–95 of vapor-to-liquid conversion, 593 Lattice Boltzmann Equation (LBE), 312 Lattice Boltzmann method (LBM), 44, 311–17 Lattice gas automata (LGA), 311–12
1020 Transport Phenomena in Multiphase Systems
Leibniz’s rule, 189 Leidenfrost drops, 824– 32 Leidenfrost effect, 824 Leidenfrost phenomena, 766, 824–33 Lennard-Jones potential, 16–18, 163 Leverette function, 298, 299, 664 Lewis number, 543, 546 Liquid droplets condensation on, 583 contact angle, 586, 589 diameter, 256, 263 equilibrium, 160, 589 evaporation, in hot gas, 724–27 evaporation, in vapor-gas mixture, 680–81 evaporation, on heated wall, 679–80 heat flow constriction, 598 interactive force between, 248 minimum equilibrium size, 596 pressure inside, 159 size, 597 superheated, bubble growth within, 782–86 supersaturation and, 161 surface evaporation, 725 suspending vapor phase, 159 symmetric condition, 724 temperature, 591 unstable equilibrium, 593 Liquid fuel combustors, 77 Liquid fuel droplets, 226–29 Liquid jet, 727–29 Liquid slugs, 934 Liquid-vapor flow patterns, 854–64 Liquid-vapor interface boundary conditions, 604 curvature, 373 differential control volume at, 629 energy balance, 506, 630, 640, 815 evolution of, 824 interfacial boundary conditions, 363 mass balance, 351, 812, 813, 842 mechanical equilibrium, 752 meniscus, 734 momentum balance, 351 resistances, 581 shear stress, 620 surface tension, 588 temperature, 632, 793, 841, 917 turbulent transport, 624 variation of characteristics, 742, 743 velocity, 645
Liquid-vapor meniscus, 735 Local thermal non-equilibrium, 287 Lockhart-Martinelli correlations, 872, 900 Macroscopic (integral) formulation, 179–89 conservation of mass, 183 control volumes, 179, 181, 182 energy equation, 185–86 momentum equation, 183–85 second law of thermodynamics, 186–87 species, 187–89 Marangoni effect, 337–39 Bernard cellular flow resulting from, 339 influence, 341, 914 Marangoni number, 339, 340 Mass-averaged velocity, 33 Mass species equation, 472 Mass transfer, 31–44 blowing velocity, 538 convective, 41–44 diffusion, 31–41 in equivalent laminar film, 628 from gases to liquids/solids, 359–60 modes, 31 multiphase, 3 from solid/liquid to gas stream, 357–59 solids to liquids or liquids to solids, 360–61 in vapor boundary layer, 628 Maxwell-Boltzmann distribution, 310 Maxwell equations, 502 Maxwell relations, 114–15 Maxwell-Stefan diffusivity, 28, 35, 38 Mean-field potential energy, 163 Mean film thickness, 386 Mechanical energy balance equation, 194 Mechanical stability, 119–21 Melting, 57, 421–522 boundary conditions, 433 conduction-controlled, 423, 456 contact, 60, 479–84 convection-controlled, 423 in cylindrical coordinate systems, 450–54 fixed, 479 fronts, propagation, 487 latent heat, 425 of multicomponent substances, 67 Nusselt number for, 487 picosecond laser, 513 point, 67, 444, 456 in porous media, 484–92
Index 1021
process classification, 421 rapid, of metal, 96–97 of single-component substances, 66 in subcooled semi-infinite body, 443 of subcooled solid, 449 three-dimensional melting, 427 two-region, 422 ultrafast laser surface, 96 See also Solidification Membrane electrode assemblies (MEAs), 80 Metastable equilibrium, 147–50 See also Equilibrium Microchannel, 21–22, 729, 744–47, 904–37 Micro-electro-Mechanical Systems (MEMS), 908 Microfilm region, 373–76 Microscale heat transfer, 3, 19, 44, 512 Microscale phase change, 512–16 Microscale transport phenomena, 44–48 Microscopic (differential) formulation, 189–230 combustion, 223–30 conservation of mass, 190–91 energy equation, 193–98 jump conditions at interfaces, 206–18 momentum equation, 191–93 PDEs, 219–21 rarefied vapor self-diffusion model, 222–23 second law of thermodynamics, 198–99 species, 199–206 Microwaves, 502 Microwave thawing, 501–5 Miniature heat sinks, 929–33 Miniature/micro channels annular flow condensation, 911–19 boiling in, 933–37 condensation flow map, 909 geometric configurations, 920 heat transfer capacity, 910 nucleate boiling onset, 920–23 Mixture model, 471–75 energy equation, 472 mass species equation, 472 mixture continuity equation, 471 Moist air properties, 690 Molar-averaged velocity, 33 Molar fluxes, 33-35, 604 Molar fraction, 32, 39, 654 Molecular dynamic simulation (MDS), 311, 402–6 Molecular level presentation, 9–21
Molecular statistical averaging, 243–44 Molecular velocity, 11–12 Molecules collision diameter, 18 diameter, 12 distance between collisions, 11 intermolecular forces between, 19–20 motion, 10 number density, 12 size, 11 Molybdenum properties, 964 Momentum equations area-averaged homogeneous model, 266 CVD, 558 in gas region, 693 homogeneous bubble growth, 777 macroscopic (integral) formulation, 183–85 microscopic (differential) formulation, 191– 93 multi-fluid models, 301–2 separated flow model, 867–68 stationary grid approach, 395 turbulent falling film, 714 unsteady-state flow in liquid film, 391 vertical falling film evaporation, 698 volume-averaged homogeneous model, 258– 59 volume-averaging model, 476 Momentum production rate, 259 Morton number, 389 Moving boundary problem, 423 Multicomponent Fick diffusivities, 36 Multicomponent PCM, 422–23, 465 Multicomponent thermodiffusivity, 37 Multi-fluid models, 238, 242 continuity equation, 290 energy equation, 301 liquid/vapor interfacial regions, 290 main flow variables, 302 MMM comparison, 301–3 momentum equation, 301–2 obtaining, 238–39 porous media, 290–98 term solutions, 303 volume-averaged, 244–57 Multiphase flows, 62, 178 Multiphase mixture model (MMM), 289 development, 298 MFM comparison, 301–3 porous media, 298–301
1022 Transport Phenomena in Multiphase Systems
term solutions, 303 variables, 302 Multiphase mixtures bulk velocity, 258 continuity equation, 258 mass-averaged entropy, 260 mass average enthalpy, 259 Multiphase systems, 62–73 electronics cooling, 92–96 energy systems, 73–82 food/biological material processing, 83–84 generalized governing equations, 177–230, 238–320 heat pipes, 88–92 laser-assisted manufacturing, 84–88 microscale phase change heat transfer, 96– 98 with separated phases, 66 transport phenomena in, 73–98 Mush zone, 467, 469 dielectric properties, 503 temperature distribution, 469 thermal properties, 503 Nanoscale heat transfer, 19, 44 Nanoscale transport phenomena, 44–48 Naphthalene sublimation, 533 Navier-Stokes equation, 21–22, 221 Net surface reaction rate, 559 Neumann problem, 435 Neumann stability criterion, 459 Newton-Raphson/secant method, 23 Newton’s law of viscosity, 25, 308 Nickel properties, 964 Niobium properties, 964 Nitrogen properties, 959 Noncondensable gas effects on filmwise condensation, 627–33 falling film condensation of steam with, 631 heat flux across liquid film, 630 See also Gases Noncontinuum approach, 21-23, 402–6 Nonequilibrium molecular dynamics (NEMD), 48 Nonevaporating liquid film, 382 Nongravitational condensate removal, 638–58 annular condensation heat transfer, 643–48 capillary action in heat pipe, 652–58 centrifugal field via rotating disk, 649–52 condensation in tube with suction at porous
wall, 638–43 Nonwetting, 344 Nucleate boiling, 766, 770–802 bubble detachment, 786–90 bubble dynamics, 775–86 bubble growth and merger, 792–96 evaporation and, 798 heat flux, 800 heat transfer, 796–802 inception, 770–75 nucleate site density, 791–92 nucleation, 770–75 onset in microchannel, 920–23 site density, 791–92 superheat requirement, 772 suppression factor, 900 temperature distribution, 797 temperature profile, 797 transition, 807 transport phenomena, 792 in wicked surface, 833–37 See also Boiling Nucleate site density, 791–92 Nucleation, 770 characterization, 770 heterogeneous, 581 homogeneous, 170, 581 rate, temperature in, 171 site distribution, 799 Numerical simulation, 454–64 enthalpy method, 455–60 equivalent heat capacity method, 460–62 free surfaces, 293–406 temperature-transforming model, 462–64 of interface and free surface, 392 Nusselt evaporation, 702–6 Nusselt number, 50, 250, 487, 488, 538 average, 691 average, at outer edge of layer, 709 average, for evaporation of wavy falling film, 710 average, modified, 623 based on convective heat flux, 547 based on convective heat transfer, 538 based on convective heat transfer coefficient, 546 based on total heat by external fluid, 549 based on total heat flux, 546, 547 forced convective heat transfer, 888 laminar flow, 616
Index 1023
Lewis number and, 543 local, 662, 688, 746, 747 local, for evaporation of wavy falling film, 711 local, modified, 617 local, variations, 546 nondimensionalizing, 622–23 subcooling effects, 817 for sublimation inside adiabatic tube, 543 Ohm’s law, 502, 893 One-region problem, 421 convention-controlled melting, 484–88 exact solution, 430–35 integral approximate solution, 439–43 One-step model, 515, 516 Oxygen properties, 960 Partial differential equations (PDEs) classification of, 219–21 examination, 219 propagation of disturbance, 220 Partially wetting, 344 Particle number density, 243 Partition coefficient, 361 Peclet number, 574 Perkins tube, 88 Permeability, 280, 282 relative, 298 Phase-average stress tensor, 246 Phase Change Materials (PCMs), 58, 136, 421– 25 contact melting, 61 density, 424 horizontal projected length, 61 infinite liquid, 453 melting point, 451 properties, 421, 967 thermal conductivity, 456 transparent, 139 Phase diagrams copper-nickel isomorphous alloy, 138 immiscible pure liquid, 586 liquids with miscibility gap, 585 liquid-vapor phase change of binary system, 140 multicomponent systems, 136–41 p-T, 130, 136 p-v, 130 T-s, 130
Phase explosion, 97–98 Phase interface fitted grid, 398–400 Phases bond energy, 20 density, 315 dispersed, 64, 65 interaction potential between, 316 liquid, 248 mixed, 63–64, 65 pressure, 399 primary, 250 separated, 62–65 vapor, 157, 158, 248 velocity, 399 volume factions, 291 Physical boundaries, 46 Physical constants, 950 Physical vapor deposition (PVD), 533 Planck’s constant, 46 Plug flow, 857–62 Pool boiling, 72 bubble departure size, 895 curve, 767, 839 curve for saturated water, 767 defined, 765 regimes, 767–69 See also Boiling Pores evaporation, 729–47 cylindrical, 730–34 effective evaporative heat transfer coefficient, 736 energy equation, 735 fluid flow effect, 734–38 heat transfer, 729, 736 under high heat flux, 738–44 Porosity, 273–301, 509–612, 652–67, 718–33 Porous media, 273–303 conservation of mass, 275–76 conservation of momentum, 276–85 Darcy’s law, 276, 282 effective thermal conductivity, 489 energy equation, 285–87 evaporation from inverted meniscus, 747–55 film condensation in, 658–67 heat transfer, 492 melting in, 484–92 multiphase transport, 289–303 properties, 279 Reynolds number, 276 second law of thermodynamics, 287–88
1024 Transport Phenomena in Multiphase Systems
solidification in, 484–92 as solid matrix, 274, 286 species, 288 steady-state convection, 492 transport in, 273 volume fraction, 488 wedge/cone evaporation, 718–24 zone, modeling, 274 zone, properties, 280 Porous media boiling, 833–43 Power, conversion factor, 952 Power and refrigeration cycles, 74–76 Power law equation, 797 Prandtl number, 494, 651, 709, 710 Precursors, 557–59 Prefixes, 950 Preheating duration, 443–44 Pressure capillary, 298, 332–34 conversion factor, 952 disjoining, 371, 732 frictional, 870–78 saturation, 370, 372, 741 surface, 344 thermodynamic, 24 turbulent condensate flow, 617 vapor, 168, 335, 350–51, 771, 785 volume-average, 278 Pressure gradients frictional, 871, 872, 876, 877 mass flux versus, 875 normalized, 647 Proton exchange membrane fuel cell (PEMFC), 80 Puddle thickness, 827 Pulsating heat pipes (PHP), 91–92, 380 Pure substances equations of state, 131–36 phase diagrams, 129–31 P-v-T surface, 128, 129 Radiation, 30–31 Random velocity, of species, 308 Rarefied vapor self-diffusion model, 222–23 Rayleigh-Bénard natural convection, 563 Rayleigh number, 486 based on permeability, 665 on solid-liquid interface, 491 Rayleigh-Taylor instability, 382–86, 838 Redlich-Kwong equation, 134
Reference frame velocity, 207, 209 Reference velocity, 210, 212 Relative velocity, of two particles, 306 Rohsenow’s correlation, 801 Rotational energy, 14, 304 Runge-Kutta method, 377, 541, 664, 917 SALD vapor infiltration (SALDVI), 87–88 Saturated boiling, 766, 897 Saturation mass fraction, 363 phase, 294 properties, 584 temperature, 7–8, 583, 691, 741 vapor pressure, 731 Saturation pressure, 370, 372, 741 surface, 731 thermodynamic relation, 291 Scale analysis, 58–61 Scanning velocity, 573 Schmidt number, 632, 687 for dilute solution in water, 997 for vapors in dilute mixture, 996 Second law of thermodynamics, 109, 186, 110 area-averaged homogeneous model, 267 extrinsic phase average, 251 integral form, 187 macroscopic (integral) formulation, 186–87 microscopic (differential) formulation, 198– 99 porous media, 287–88 volume-averaged homogeneous model, 260 volume-averaged multi-fluid models, 251– 52 See also Thermodynamic laws Seebeck effect, 45 Selective area laser deposition (SALD), 87–88, 557, 570–71 Selective laser sintering (SLS), 84–86, 508–12 Self-diffusion coefficient, 222 Self-diffusion equation, 222 Sensible heat, 3–7, 481–82 Separated flow model, 270–73, 866–70 for channel with variable cross-section, 272 conservation of mass, 867 correlations based on, 871–74 energy equation, 869–70 features, 271 governing equations, 867–68 momentum equations, 867–68
Index 1025
See also Two-phase flow Sherwood number, 538, 546, 549 average, 691 Lewis number and, 543 local, 688 sublimation inside adiabatic tube, 543 Slip flow regime, 22 Slip ratio, 879 Slug flow, 857, 858 Smooth-surface model, 375, 733 Solid freeform fabrication (SFF), 84 Solid fuel combustor, 77 Solidification, 57, 421–522 around horizontal tube, 493 of binary solution system, 465–79 boundary conditions, 433 on cold isothermal surface, 469 contours, 475 in cylindrical coordinate systems, 450–54 of finite liquid PCM, 451 governing equations, 427–29 heat transfer, 9 of multicomponent PCM, 423 of multicomponent substances, 67 in porous media, 484–92 process classification, 421 of single-component substances, 66 of SNC-acetone solution, 478 two-region, 422 See also Melting Solid-liquid interface, 331–406 boundary conditions, 424–27, 440 differential equations, 453 energy balance, 486, 506 heat flux, 913 location, 448, 454, 458 Rayleigh number, 491 shape, 426 velocity, 514 wick/vapor boundary, 496 Solid-liquid phase change, 66–67 applications, 492–512 heat pipe startup, 495–99 laser drilling, 505–8 latent heat thermal energy storage, 492–95 microwave thawing of food, 501–5 PCM, 424 phase diagram, 532 in porous media, 488 in rectangular cavity, 490
SLS of metal powders, 508–12 thermal protection, 499–501 Solids mass transfer from gases to, 359–60 mass transfer to gas stream, 357–59 mass transit to liquids, 360–61 Solid-vapor phase change, 67–68, 532 Species area-averaged homogeneous model, 267–70 balance, 355–66 Boltzmann equation, 305 concentration, 357, 359, 360 conservation of mass, 187, 200 homogeneous equation, 263 macroscopic (integral) formulation, 187–89 mass fraction, 356 mass velocity, 188 mole, rate of increase, 201 porous media, 288 random velocity of, 308 source/sink due to chemical reaction, 253 total mass, chemical reaction, 188 velocities, 356 volume-averaged homogeneous model, 260– 64 volume-averaged multi-fluid model, 252–57 Species equation, 200, 205, 300, 362, 472, 478 Specific heats conversion factor, 952 gaseous mixture, 562 temperature dependence, 461 values, 5 Spherical coordinate system, 1009–11 Stability chemical, 121–22 concept, 107 criteria, 117–22 mechanical, 119–21 of single-phase systems, 111–28 thermal, 117–19 Stable equilibrium, 147 Stagnant vapor reservoir filmwise condensation, 605–13 Stainless steel properties, 965 Standard reference state, 224 Stationary grid approach, 394–98 Stefan-Boltzmann constant, 819 Stefan-Maxwell equation, 571 Stefan number, 428, 431, 434, 486, 500 Sticking coefficient, 559, 572
1026 Transport Phenomena in Multiphase Systems
Stokes drag force, 277 Stokes flow, 192 Stratified flow, 862 annular flow transition to, 888 cross-section, 886 See also Flow patterns Stratified wavy flow, 862 Stream functions, 535, 813 Stress-strain rate relationships, 26 Stress tensors phase-average, 246 reducing, 308 volume average, 278 Strong numerical solution, 455 Subcooled boiling, 765 forced convective, 896, 899 liquid temperature effect, 766 See also Boiling Subcooling boiling and, 817 effects on Nusselt number, 817 liquid film and, 607, 615 parameter, 450 Subeutectic concentration, 139 Sublimation, 57, 67, 531–54 Supereutectic concentration, 139 Superficial velocity, 855, 909 Superheat calculated value, 168 effective heat transfer coefficient versus, 738 temperature, 169 thermodynamic, 167–71 Superheated liquids, 159 equilibrium phase boundary, 167 initial bubbles, 167 vapor pressure, 168 Surface excess, 152 Surface porosity, 733 Surface pressure, 344 Surfaces arbitrarily-curved, 334 curvature, 349 free, numerical simulation, 392–406 internal energy, 153 rough, 373, 374 Surface spray cooling, 715–18 average path length of conduction, 717 scale analysis, 716 Surface tension, 152–55, 332–42
effects, 397 as function of temperature, 363 for liquids, 155 molecular perspective, 155 Surface waves, on liquid film flow, 386–92 Taitel-Dukler flow map, 875 Tantalum properties, 965 Taylor series, 305, 317 Temperature-property relations coefficients, 981–92 acetone, 982 ammonia, 982 cesium, 983 Dowtherm, 983 ethane, 984 ethanol, 984 Freon, 985–87 heptane, 987 lead, 988 lithium, 988 mercury, 989 methanol, 989 nitrogen, 990 potassium, 990 rubidium, 991 silver, 991 sodium, 992 water, 992 Temperature-transforming model, 455, 462–64 Tensors, 1011–12 deformation, 352 stress, 246, 278, 308 Thermal conductivity, 27 for anisotropic materials, 27 conversion factor, 952 electrical component, 47 of electrons, 514 of half-grid, 457 of isotropic materials, 27 mixtures, 472 of polyatomic gas, 561 of porous medium, 489 precursors, 559 in solids, 45 unsintered powder, 510 Thermal diffusivity, conversion factor, 951 Thermal expansion coefficient, 564 Thermal penetration depth, 437–38, 444 Thermal radiation, 30
Index 1027
Thermal resistances in condensation process, 594–97 due to capillary depression, 596 due to conduction through droplet, 597 to heat flow, 594 heat sinks, 929 interfacial, 595–96 of tube wall, 543 in vapor, 595 Thermal stability, 117–19 Thermodynamic equilibrium, 107 Thermodynamic laws, 108–9 first law, 108–9 second law, 109, 110, 186–89 Thermodynamic limit of superheat, 168–71 Thermodynamic pressure, 24 Thermodynamic relations, 109–10, 196, 291 Thermodynamic surfaces, 128–41 Thermophysical properties, 968–80 acetone, 968 ammonia, 969 cesium, 970 Dowtherm, 970 ethane, 971 ethanol, 971 Freon, 972–74 helium, 974 heptane, 975 lead, 975 lithium, 976 mercury, 976 methanol, 977 nitrogen, 977 potassium, 978 rubidium, 978 silver, 979 sodium, 980 water, 980 See also Transport properties Thermosyphon, 89 Thin liquid films disjoining pressure, 371, 732 evaporation from, 685 evaporation on vertical adiabatic surface, 692 in evaporator, 678 formation, 373 on fragment of rough solid surface, 374 heat transfer through, 368–73 on solid surface, 349
supercooled, 368 thickness, 371 ultra-thin, 347–49 Thomson effect, 45 Three-dimensional sintering, 509 Time averaging Eulerian, 239–40 Lagrangian, 243 Titanium properties, 966 Transfer driving force, 553 Transition boiling, 766, 806–10 Transition regime, 22 Transport effects (interface), 351–82 energy balance, 354–55 mass balance, 351 momentum balance, 351–54 species balance, 355–66 Transport properties, 954–1003 air, 957 aluminum, 962 aluminum alloy, 962 carbon dioxide, 958 carbon steel, 965 cartridge brass, 962 coefficients of temperature-property relations, 981–92 copper, 962 density and volume expansion coefficients, 961 fused silica, 963 helium, 958 hydrogen, 959 Inconel, 963 iron, 963 molybdenum, 964 nickel, 964 niobium, 964 nitrogen, 959 oxygen, 960 PCMs, 967 stainless steel, 965 tantalum, 965 thermophysical, 968–80 titanium, 966 tungsten, 966 volume expansion coefficients, 961 water vapor, 960 T-s-p surface, 129 Tube, external heating, sublimation, 543–50 Tungsten properties, 966
1028 Transport Phenomena in Multiphase Systems
Turbofan engine combustion, 77–79 Turbulence modeling, 221 Turbulent condensate flow, 617–19 Turbulent falling film, 713–15 Turbulent film condensation, 619–24 Turbulent film regime, 611–13 Two-component flows, 62 Two-fluid model, 242, 864–82 separated-flow, 270–73 Two-phase single-component systems, 141–42 Two-phase flow, 72–73, 853–937 bounds on, 874–78 characteristic flow behavior, 853 as complex phenomenon, 859 condensation, 908–19 defined, 72, 853 electric flow, 893 evaporation and boiling, 920–37 frictional pressure drop, 870–78 frictional pressure gradient, 871, 872 heat transfer, 858 homogeneous, 864–66 liquid-vapor flow patterns, 854–64 in micro-/minichannels, 904–37 models, 73, 864–82 phase combinations, 853 pressure drop, 858 regime map, 859–60 regimes, 883–85 separated, 866–70 steady state, 865, 869 total mass flux, 856 types of, 72 void fraction, 878–82 Two-phase flow patterns, 904–7 in horizontal microchannels, 907 in horizontal minichannels, 906 liquid ring flow, 907 Unsaturated flow theory (UFT), 303 Unstable equilibrium defined, 148 illustrated, 147 See also Equilibrium Van der Waals equation, 132, 134 Van der Waals limit, 169 Van der Waals polynomial, 133 Vapor bubbles, 770 demerger pattern, 796
departure radii, 789 departure velocity, 787 detachment, 786–92 diameter at departure, 786 dimensionless volume, 773 dynamics, 775–86 embryo, 772 force balance, 898 forces acting on, 788, 790 growth, 772, 792–96 growth, equation of motion, 785 growth, in viscous fluid, 788 growth in volatile droplet, 782 growth pattern, 796 growth schematic, 783 growth stage, 775 growth within superheated liquid droplets, 782–86 heterogeneous growth, 780–82 homogeneous growth, 776–80 homogeneous nucleation, 776 lateral merger, 795 liquid microlayer, 775 merger, 792–96 radii, 158, 771 radius variation, 774 release frequency, 787 size at departure, 786 suspending in liquid phase, 156 transient radius, 897 vapor pressure, 771 Vapor deposition, 531, 554–74 applications, 533 categories, 533 CVD, 533, 554–74 LCVD, 533, 570–74 PVD, 533 Vaporization, 678–747 heat requirement, 7 interfacial resistance, 366–68 temperature, 506 Vapor pressure, 335 bubble, 837 early time variation, 785 gradient, 256 over curved surface, 350–51 partial, 358, 363 saturated, 350–51, 731 superheated liquids, 168 temperature vs., 165
Index 1029
vapor bubble, 771 Vectors, 1006–7 Vertical falling film evaporation, 691–701 Vertical reactors, 556 Vibrational energy, 14, 304 Viscosity, 362 basis, 19 constant, 192 conversion factor, 953 divergence of, 25 dynamic, 604 estimation, 26 measurement, 23 in multicomponent system, 26 Newton’s law, 25, 308 precursors, 559 Viscous dissipation, 204 in cylindrical/spherical coordinate system, 198 determining, 197–98 Void fraction, 854, 867, 878–82 constants and exponents, 881 correlations, 878, 880 homogeneous flow, 856 model comparison, 881 prediction, 879 quality versus, 882 range, 855 slip ratio and, 879 stratification angle and, 887 upper bound, 882 Volume, conversion factor, 953 Volume average, 241, 242 control volume, 488 stress tensor, 278 Volume-averaged homogeneous model, 258–64 continuity equation, 258 energy equation, 259–60 momentum equation, 258–59 second law of thermodynamics, 260 species, 260–64 Volume-averaged multi-fluid models, 244–57 continuity equation, 244–45 energy equation, 248–51 momentum equation, 245–48 second law of thermodynamics, 251–52 species, 252–57 See also Multi-fluid models Volume-averaged velocity, 274, 293 Volume-average pressure, 278
Volume-averaging model, 475–79 continuity equation, 176 energy equation, 477 momentum equation, 476 species equation, 478 Volume flow rate, 953 Volume fractions, 291 Volume of Fluid method (VOF), 822 continuity equation, 395 defined, 395 drawback, 398 parasitic currents and, 398 Volumetric averaging, 240 Volumetric condensation rate, 297 Volumetric heat generation rate, 953 Wallis correlation, 633–35 Wall superheat, 923 Water droplets lifetime, 825–26 radius, 829 vapor film thickness, 829 Water vapor properties, 960 Wave equation, 220 Wave velocity, 388 Wavy condensate regime, 610–11 Wavy film analysis, 709 Wavy flows stratified, 862 of thin liquid films, 706 See also Flow patterns Weber number, 694, 925 Wedge/cone evaporation, 718–24 Wetting adsorption and, 344–47 completely, 344 minimum contact angle, 343 partially, 344 Wicked surfaces, 833–37 Wicks, 655 conduction-convection, 834 film boiling, 834 heat transfer modes, 833 nucleate boiling, 834 receding liquid, 834 submerged, heat transfer, 835 Wiedemann-Franz law, 47 Wispy annular flow, 858 Young-Laplace equation, 332–34, 384, 390
1030 Transport Phenomena in Multiphase Systems