Back to Advanced Heat and Mass Transfer Home
Table of Contents
LIST OF APPENDICES Appendix A Constants, Units and Conversion Factors 876 Appendix B Transport Properties of Solids 880 Appendix C Transport Properties of Gases and Liquids at Atmospheric Pressure 888 Appendix D Transport Properties for Phase Change 895 Appendix E Mass Transfer Properties 899 Appendix F Configuration Factors and Surface Properties for Radiation 911 Appendix G Mathematical Relations 916 Advanced Heat and Mass Transfer 875 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX A CONSTANTS, UNITS AND CONVERSION FACTORS Table A.1 Physical constants Universal gas constant Boltzmann constant Stefan-Boltzmann constant Atmospheric pressure Gravitational acceleration Ru = 8314.34 J/(kmol-K) kb = 1.38054 × 10−23 J/K σ SB = 5.67 × 10−8 W/(m 2 -K 4 ) patm = 1.013 × 105 Pa g = 9.807 m/s 2 Table A.2 Prefixes Factor 1018 1015 1012 109 106 103 102 10 Prefix exa peta tera giga mega kilo hecto deka Symbol E P T G M k h da Factor 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 Prefix deci centi milli micro nano pico femto atto Symbol d c m μ n p f a 876 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table A.3 Conversion factors Physical quantity Acceleration Area ft/s2 m/s2 ft2 cm2 m2 lbm/in3 lbm/ft3 kg/m3 g/m3 Btu Btu Btu ft-lbf kW-h kcal kJ HP-h Btu/h-ft2 kcal/h-m2 cal/s-cm2 W/cm2 W/m2 W/m2 Btu/lbm J/kg lbf lbf kgf kgf N Btu/h-ft2-°F kcal/h-m2-°C W/m2-K W/m2-K ft2/h ft2/s m2/s m2/s m2/s in ft in mm m Conversion factor 0.30480 3.2808 0.092903 0.15500 10.764 1728.0 16.018 0.062428 62.428 1.0551 0.0002930 0.25200 0.0012851 3412.8 3.9683 0.94782 2544.4 3.1546 0.36867 13272.0 3170.0 0.8598 0.31700 2324.4 0.00043021 32.1740 4.448 2.2046 9.80665 0.22481 5.6782 0.20482 0.17611 0.8598 2.5807×10-5 0.092905 10000 38750 10.764 25.4 0.3048 0.08333 0.039370 3.2808 m/s2 ft/s2 m2 in2 ft2 lbm/ft3 kg/m3 lbm/ft3 lbm/ft3 kJ kW-h kcal Btu Btu Btu Btu Btu W/m2 Btu/h-ft2 Btu/h-ft2 Btu/h-ft2 kcal/h-m2 Btu/h-ft2 J/kg Btu/lbm lbm-ft/s2 N lbf N lbf W/m2-K Btu/h-ft2-°F Btu/h-ft2-°F kcal/h-m2-°C m2/s m2/s cm2/s(stokes) ft2/h ft2/s mm m ft in ft Density Energy Energy flux Enthalpy Force Heat transfer coefficient Kinematic viscosity (ν) Thermal diffusivity (α) Mass diffusivity (D) Length Appendix A Constants, Units, and Conversion Factors 877 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table A.3 Conversion factors (cont’d) Physical quantity Mass Mass flow rate Conversion factor 0.45359 2.2046 0.45359 3600 7936.6 2.2046 1.055 0.293 3.412 9.48×10-4 0.746 0.707 6895 1.013×105 14.696 105 1.000 133.32 27.68 0.4335 4.1868 1.000 0.23885 T(°C)=5/9[T(°F)-32] T(°F)=9/5 T(°C)+32 T(K)=T(°C)+273.15 T(°R)=T(°F)+459.67 1.7307 418.68 0.5778 0.30480 0.27778 1.609 1 0.1 1.4882 4.1338×10-4 0.67195 2419.08 1 0.001 0.02832 16.39 0.7646 35.313 Power Pressure Specific heat, specific entropy Temperature Thermal conductivity Velocity Viscosity Volume lbm kg lbm/h lbm/s kg/s kg/h Btu/s Btu/h W W HP HP psi atm atm bar torr torr psi ft-H2O Btu/lbm-°F kcal/kg-°C kJ/kg-K °F °C °C °F Btu/h-ft-°F cal/cm-s-°C W/m-K ft/s km/h mile/h kg/s-m posi lbm/s-ft lbm/h-ft N-s/m2 N-s/m2 L L ft 3 in3 y d3 m3 kg lbm kg/h lbm/h lbm/h lbm/h kW W Btu/h Btu/s kW Btu/s Pa Pa psi Pa mmHg Pa in H2O psi kJ/kg-K Btu/lbm-°F Btu/lbm-°F °C °F K °R W/m-K W/m-K Btu/h-ft-°F m/s m/s km/h N-s/m2 N-s/m2 N-s/m2 N-s/m2 lbm/s-ft lbm/h-ft dm3 m3 m3 cm3 m3 ft3 878 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table A.3 Conversion factors (cont’d) Physical quantity Volume Conversion factor 3.785 4.546 0.4732 0.5683 4.7196×10-4 0.0028318 2118.8 10.35 0.0966 Volume flow rate Volumetric heat generation rate Gal (U.S.) Gal (IMP) Pint (U.S.) Pint (IMP) ft3/min ft3/s m3/s Btu/h-ft3 W/m3 L L L L m3/s m3/s ft3/min W/m3 Btu/h-ft3 Appendix A Constants, Units, and Conversion Factors 879 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX B TRANSPORT PROPERTIES OF SOLIDS List of Properties Tables Table B.1 Aluminum and Aluminum Alloy ...................................................... 881 Table B.2 Beryllium........................................................................................... 881 Table B.3 Chromium ......................................................................................... 881 Table B.4 Copper and Alloy .............................................................................. 882 Table B.5 Gold ................................................................................................... 882 Table B.6 Iron and Steel .................................................................................... 883 Table B.7 Molybdenum ..................................................................................... 883 Table B.8 Nickel and alloy ................................................................................ 884 Table B.9 Niobium............................................................................................. 884 Table B.10 Tantalum ......................................................................................... 885 Table B.11 Titanium .......................................................................................... 885 Table B.12 Tungsten .......................................................................................... 885 Table B.13 Nonmetallic solids ........................................................................... 886 Table B.14 Insulation Materials ......................................................................... 887 880 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press B.1 Metallic Solids Table B.1 Aluminum and Aluminum Alloy Aluminum, Al, Tm = 933 K (Rohsenow et al., 1985) T (K) 100 150 200 250 300 400 600 800 100 200 300 400 600 ρ (kg/m³) 2732 2726 2719 2710 2701 2681 2639 2591 c (kJ/kg-K) 0.481 0.683 0.797 0.859 0.902 0.949 1.042 1.134 0.473 0.787 0.875 0.925 1.042 k (W/m-K) 300 250 237 235 237 240 231 218 65 163 177 186 186 Aluminum Alloy, 2024-T6 (4.5% Cu, 1.5% Mg, 0.6% Mn), Tm = 775 K (Incropera et al., 2007) 2770 Table B.2 Beryllium Beryllium, Tm = 1550 K (Incropera et al., 2007) T (K) 100 200 300 400 600 800 1000 1200 ρ (kg/m³) c (kJ/kg-K) 0.203 1.114 1.825 2.191 2.604 2.823 3.018 3.227 k (W/m-K) 990 301 200 161 126 106 90.8 78.7 1850 Table B.3 Chromium Chromium, Tm = 2118 K (Incropera et al., 2007) T (K) 100 200 300 400 600 800 1000 1200 1500 2000 ρ (kg/m³) c (kJ/kg-K) 0.192 0.384 0.449 0.484 0.542 0.581 0.616 0.682 0.779 0.937 k (W/m-K) 159 111 93.7 90.9 80.7 71.3 65.4 61.9 57.2 49.4 7160 Appendix B. Transport Properties of Solids 881 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table B.4 Copper and Alloy Copper, Cu, Tm = 1358 K (Rohsenow et al., 1985) T (K) 100 150 200 250 300 400 600 800 1000 1200 200 300 400 600 100 200 300 400 600 ρ (kg/m³) 9009 8992 8973 8951 8930 8884 8787 8642 8568 8458 c (kJ/kg-K) 0.254 0.323 0.357 0.377 0.386 0.396 0.431 0.448 0.446 0.480 0.785 0.420 0.460 0.545 k (W/m-K) 480 429 413 406 401 393 379 366 352 339 42 52 52 59 75 95 110 137 149 Commercial bronze (90% Cu, 10% Al), Tm = 1293 K (Incropera et al., 2007) 8800 Cartridge brass (70% Cu, 30% Zn), Tm = 1188 K (Incropera et al., 2007) 8530 0.360 0.380 0.395 0.425 Table B.5 Gold Gold, Au, Tm = 1336 K (Incropera et al., 2007) T (K) 100 200 300 400 600 800 1000 1200 ρ (kg/m³) c (kJ/kg-K) 0.109 0.124 0.129 0.131 0.135 0.140 0.145 0.155 k (W/m-K) 327 323 317 311 298 284 270 255 19300 882 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table B.6 Iron and Steel Iron, Fe, Tm = 1810 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) 100 7900 150 7890 200 7880 250 7870 300 7860 400 7830 600 7760 800 7690 1000 7650 1200 7620 1400 7520 1600 7420 1800 7420 Plain Carbon Steel, Tm=1480 °C (Incropera et al., 2007) 300 7854 400 600 800 1000 Stainless Steel 304, Tm = 1670 K (Incropera et al., 2007) 100 200 300 7900 400 600 800 1000 1200 1500 c (kJ/kg-K) 0.216 0.324 0.384 0.422 0.450 0.491 0.555 0.692 1.034 k (W/m-K) 134 104 94 87 80 70 55 43 32 28 31 0.434 0.487 0.559 0.685 1.169 0.272 0.402 0.477 0.515 0.557 0.582 0.611 0.640 0.682 60.5 56.7 48.0 39.2 30.0 9.2 12.6 14.9 16.6 19.8 22.6 25.4 28.0 31.7 Table B.7 Molybdenum Molybdenum, Mo, Tm = 2892 K (Rohsenow et al., 1985) T (K) 100 150 200 250 300 400 500 600 800 1000 1200 1400 ρ (kg/m³) 10260 10250 10250 10250 10240 10220 10210 10190 10160 10120 10080 10040 c (kJ/kg-K) 0.140 0.196 0.223 0.241 0.248 0.261 0.268 0.274 0.280 0.292 k (W/m-K) 180 149 143 140 138 134 130 126 118 112 105 100 Appendix B. Transport Properties of Solids 883 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table B.8 Nickel and alloy Nickel, Ni, Tm = 1728 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) c (kJ/kg-K) 100 8960 0.323 150 8940 0.329 200 8930 0.383 250 8910 0.416 300 8900 0.444 400 8860 0.490 600 8780 0.590 800 8690 0.530 1000 8610 0.556 1200 8510 0.582 1400 8410 1600 8320 Inconel X-750 (73% Ni, 15% Cr, 6.7% Fe), Tm = 1665 K (Incropera et al., 2007) 100 200 0.372 300 8510 0.439 400 0.473 600 0.510 800 0.546 1000 0.626 1200 1500 k (W/m-K) 165 120 105 98 91 80 66 68 72 76 80 8.7 10.3 11.7 13.5 17.0 20.5 24.0 27.6 33.0 Table B.9 Niobium Niobium, Nb, Tm = 2740 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) 100 8600 150 8590 200 8580 250 8570 300 8570 400 8550 500 8530 600 8510 800 8470 1000 8430 1200 8380 1400 8340 c (kJ/kg-K) 0.202 0.238 0.254 0.263 0.268 0.272 0.277 0.281 0.290 0.298 0.307 k (W/m-K) 55 53 53 53 54 55 57 58 61 64 68 71 884 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table B.10 Tantalum Tantalum, Ta, Tm = 3252 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) 100 16490 150 16480 200 16460 250 16450 300 16440 400 16410 500 16370 600 16340 800 16270 1000 16200 1200 16130 1400 16060 c (kJ/kg-K) 0.108 0.125 0.132 0.137 0.141 0.145 0.148 0.149 0.152 0.160 k (W/m-K) 59 58 58 57 58 58 59 59 59 60 61 62 Table B.11 Titanium Titanium, Ti, Tm = 1953 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) 100 4510 150 4515 200 4520 250 4515 300 4510 400 4490 600 4470 800 4440 1000 4410 1200 4380 1400 4350 1600 4320 c (kJ/kg-K) 0.295 0.406 0.464 0.501 0.525 0.555 0.597 0.627 0.652 k (W/m-K) 31 27 25 23 21 20 19 19 21 22 24 Table B.12 Tungsten Tungsten, W, Tm = 3660 K (Rohsenow et al., 1985) T (K) ρ (kg/m³) 100 19310 150 19300 200 19290 250 19280 300 19270 400 19240 500 19220 600 19190 800 19130 1000 19080 1200 19020 1400 18950 c (kJ/kg-K) 0.089 0.113 0.125 0.131 0.135 0.137 0.139 0.140 0.144 0.148 k (W/m-K) 208 192 185 180 174 159 146 137 125 118 112 108 Appendix B. Transport Properties of Solids 885 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press B.2 Nonmetallic Solids Table B.13 Nonmetals (Ozisik, 1993) Materials Asphalt Brick Building brick, common face Carborundum brick Chrome brick Diatomaceous earth, molded and fired Fireclay brick, burned 1330°C Clay Cement, portland Coal. anthracite Concrete, cinder Stone 1-2-4 mix Cotton Glass, window Pyrex Paper Paraffin Plaster, gypsum Rubber, vulcanized Soft Hard Sand Stone Granite Limestone Marble Sandstone Teflon Tissue, humana Skin Fat layer (adipose) Muscle Wood (across grain) Balsa Cypress Fir Maple or oak Yellow pine White pine a T(°C) 20-55 20 600 1400 200 900 200 870 500 800 30 23 30 23 20 20 20 30 30 30 20 30 30 30 k (W/m-K) 0.74-0.76 0.69 1.32 18.5 11.1 2.32 1.99 0.24 0.31 1.04 1.07 1.3 0.29 0.26 0.76 1.37 0.06 0.78 (avg) 1.4 0.011 0.020 0.48 0.012 0.013 0.027 1.73-3.98 1.26-1.33 2.07-2.94 1.83 0.35 0.37 0.2 0.5 0.055 0.097 0.11 0.166 0.147 0.112 ρ (kg/m3) c (J/kg-K) α (10-7m2/s) 1600 2000 0.84 5.2 3000 0.84 9.2 7.9 2000 1460 1500 1200-1500 1900-2300 80 2700 2225 930 900 1440 1100 1190 1515 2640 2500 2500-2700 2160-2300 2200 0.96 0.88 1.26 0.88 1.30 0.84 0.835 1.340 2.890 0.84 2.010 0.800 0.82 0.90 0.80 0.71 5.4 8.2-6.8 3.4 4.0 100-300 40 30 27 27 27 30 30 23 30 23 30 8-18 5.6-5.9 10-13.6 11.2-11.9 140 460 420 540 640 430 2.72 2.4 2.8 0.96 1.28 0.82 Incropera et al., (2007) 886 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table B.14 Insulating Materials (Ozisik, 1993) Materials Asbestos Loosely packed Asbestos-cement boards Sheets Felt, 40 laminations in Balsam wool Board and slab Cellular glass Glass fiber, organic bonded Polystyrene, expanded extruded (R-12) Mineral fiberboard; roofing material Cardboard, corrugated Celotex Corkboard Diatomaceous earth (Sil-o-cel) Felt, hair Wool Fiber, insulating board Glass wool Loose fill Cork, granulated Glass fiber, poured or blown Vermiculite, flakes Magnesia, 85% T(°C) 0 100 20 51 38 32 30 30 30 k (W/m-K) 0.154 0.161 0.74 0.166 0.057 0.04 0.058 0.036 0.027 ρ (kg/m3) 470-570 c (J/kg-K) 0.816 α (10-7m2/s) 3.3-4 35 145 105 55 1.000 0.795 1.210 30 32 30 0 30 30 20 23 30 30 30 38 150 204 32 150 260 23 32 23 0.049 0.064 0.048 0.043 0.061 0.036 0.052 0.048 0.038 0.045 0.043 0.068 0.067 0.074 0.080 0.040 0.067 0.087 0.059 0.024 0.059 265 160 320 130-200 330 240 24 160 16 80 270 0.7 22.6 0.835 0.835 Rock wool, 10 lb/ft3 Loosely packed Sawdust Silica aerogel Wood shavings 160 64 140 References Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York, NY. Ott, L., 1984, An Introduction to Statistical Methods and Data Analysis, Duxbury Press, Boston, MA. Ozisik, M.N., 1993, Heat Conduction, 2nd ed., Wiley-Interscience, New York. Rohsenow, W.N., Hartnett, J.P., and Ganic, E.N. eds., 1985, Handbook of Heat Transfer Fundamentals, McGraw-Hill, New York, NY. Appendix B. Transport Properties of Solids 887 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX C TRANSPORT PROPERTIES OF GASES AND LIQUIDS AT ATMOSPHERIC PRESSURE List of Properties Tables Table C.1 Air at 1 atm ....................................................................................... 889 Table C.2 Carbon dioxide (CO2) at 1 atm ......................................................... 890 Table C.3 Helium (He) at 1 atm ........................................................................ 890 Table C.4 Hydrogen (H2) at 1 atm .................................................................... 891 Table C.5 Nitrogen (N2) at 1 atm ...................................................................... 891 Table C.6 Oxygen (O2) at 1 atm ....................................................................... 892 Table C.7 Water (H2O) vapor at 1 atm ............................................................. 892 Table C.8 Liquid Water (H2O) at 1 atm ............................................................ 893 Table C.9 Liquid Gasoline at 1 atm .................................................................. 893 Table C.10 Engine Oil, Unused ........................................................................ 893 Table C.11 Volume expansion coefficients for liquids ..................................... 894 888 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press C.1 Gases Table C.1 Air at 1 atm (Incropera et al., 2007) T Temp. (K) 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 3000 ρ density (kg/m3) 3.5562 2.3364 1.7458 1.3947 1.1614 0.995 0.8711 0.7740 0.6964 0.6329 0.5804 0.5356 0.4975 0.4643 0.4354 0.4097 0.3868 0.3666 0.3482 0.3166 0.2902 0.2679 0.2488 0.2322 0.2177 0.2049 0.1935 0.1833 0.1741 0.1658 0.1582 0.1513 0.1448 0.1389 0.1135 cp specific heat (kJ/Kg-K) 1.032 1.012 1.007 1.006 1.007 1.009 1.014 1.021 1.030 1.040 1.051 1.063 1.075 1.087 1.099 1.110 1.121 1.131 1.141 1.159 1.175 1.189 1.207 1.23 1.248 1.267 1.286 1.307 1.337 1.372 1.417 1.478 1.558 1.665 2.726 μ viscosity (10-7N-s/m²) 71.1 103.4 132.5 159.6 184.6 208.2 230.1 250.7 270.1 288.4 305.8 322.5 338.8 354.6 369.8 384.3 398.1 411.3 424.4 449 473 496 530 557 584 611 637 663 689 715 740 766 792 818 955 ν kinematic viscosity (10-6m2/s) 2.00 4.426 7.59 11.44 15.89 20.92 26.41 32.39 38.79 45.57 52.69 60.21 68.10 76.37 84.93 93.80 102.9 112.2 121.9 141.8 162.9 185.1 213 240 268 298 329 362 396 431 468 506 547 589 841 k thermal conductivity (10-3W/m-K) 9.34 13.8 18.1 22.3 26.3 30.0 33.8 37.3 40.7 43.9 46.9 49.7 52.4 54.9 57.3 59.6 62.0 64.3 66.7 71.5 76.3 82 91 100 106 113 120 128 137 147 160 175 196 222 486 α thermal diffusivity (10-6m2/s) 2.54 5.84 10.3 15.9 22.5 29.9 38.3 47.2 56.7 66.7 76.9 87.3 98 109 120 131 143 155 168 195 224 238 303 350 390 435 482 534 589 646 714 783 869 960 1570 Pr Prandtl number 0.786 0.758 0.737 0.72 0.707 0.700 0.690 0.686 0.684 0.683 0.685 0.690 0.695 0.702 0.709 0.716 0.720 0.723 0.726 0.728 0.728 0.719 0.703 0.685 0.688 0.685 0.683 0.677 0.672 0.667 0.655 0.647 0.63 0.613 0.536 Appendix C. Transport Properties of Gases and Liquids at atmospheric Pressure 889 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table C.2 Carbon dioxide (CO2) at 1 atm (Incropera et al., 2007) T Temp. (K) 280 300 320 340 360 380 400 450 500 550 600 650 700 750 800 ρ density (kg/m3) 1.9022 1.7730 1.6609 1.5618 1.4743 1.3961 1.3257 1.1782 1.0594 0.9625 0.8826 0.8143 0.7564 0.7057 0.6614 cp specific heat (kJ/Kg-K) 0.830 0.851 0.872 0.891 0.908 0.926 0.942 0.981 1.020 1.050 1.080 1.100 1.130 1.150 1.170 μ viscosity (10-7N-s/m²) 140 149 156 165 173 181 190 210 231 251 270 288 305 321 337 ν kinematic viscosity (10-6m2/s) 7.36 8.40 9.39 10.60 11.70 13.00 14.30 17.80 21.80 26.10 30.60 35.40 40.30 45.50 51.00 k thermal conductivity (10-3W/m-K) 15.20 16.55 18.05 19.70 21.20 22.75 24.30 28.30 32.50 36.60 40.70 44.50 48.10 51.70 55.10 α thermal diffusivity (10-6m2/s) 9.63 11.00 12.50 14.20 15.80 17.60 19.50 24.50 30.10 36.20 42.70 49.70 56.30 63.70 71.20 Pr Prandtl number 0.765 0.766 0.754 0.746 0.741 0.737 0.737 0.728 0.725 0.721 0.717 0.712 0.717 0.714 0.716 Table C.3 Helium (He) at 1 atm (Bejan, 2004) T Temp. (K) 4.22 7 10 20 30 60 100 200 300 600 1000 ρ density (kg/m3) 16.900 7.530 5.020 2.440 1.620 0.811 0.487 0.244 0.162 0.0818 0.0487 cp specific heat (kJ/Kg-K) 9.78 5.71 5.41 5.25 5.22 5.20 5.20 5.19 5.19 5.19 5.19 μ viscosity (10-6N-s/m²) 1.25 1.76 2.26 3.58 4.63 7.12 9.78 15.1 19.9 32.2 46.3 ν kinematic viscosity (10-6m2/s) 0.0739 0.234 0.449 1.470 2.860 8.800 20.10 62.20 122.0 396.0 946.0 k thermal conductivity (10-3W/m-K) 0.011 0.014 0.018 0.027 0.034 0.053 0.074 0.118 0.155 0.251 0.360 α thermal diffusivity (10-6m2/s) 0.00064 0.00321 0.00642 0.0209 0.0403 0.125 0.291 0.932 1.830 5.940 14.20 Pr Prandtl number 1.15 0.73 0.70 0.70 0.71 0.70 0.69 0.67 0.67 0.67 0.67 890 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table C.4 Hydrogen (H2) at 1 atm (Incropera et al., 2007) T Temp. (K) 100 150 200 250 300 350 400 450 500 550 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 ρ density (kg/m3) 0.24255 0.16156 0.12115 0.09693 0.08078 0.06924 0.06059 0.05386 0.04848 0.04407 0.04040 0.03463 0.03030 0.02694 0.02424 0.02204 0.02020 0.01865 0.01732 0.01616 0.01520 0.01430 0.01350 0.01280 0.01210 cp specific heat (kJ/Kg-K) 11.230 12.600 13.540 14.060 14.310 14.430 14.480 14.500 14.520 14.530 14.550 14.610 14.700 14.830 14.990 15.170 15.370 15.590 15.810 16.020 16.280 16.580 16.960 17.490 18.250 μ viscosity (10-7N-s/m²) 42.1 56.0 68.1 78.9 89.6 98.8 108.2 117.2 126.4 134.3 142.4 157.8 172.4 186.5 201.3 213.0 226.2 238.5 250.7 262.7 273.7 284.9 296.1 307.2 318.2 ν kinematic viscosity (10-6m2/s) 17.4 34.7 56.2 81.4 111 143 179 218 261 305 352 456 569 692 830 966 1120 1279 1447 1626 1801 1992 2193 2400 2630 k thermal conductivity (10-3W/m-K) 67 101 131 157 183 204 226 247 266 285 305 342 378 412 448 488 528 568 610 655 697 742 786 835 878 α thermal diffusivity (10-6m2/s) 24.6 49.6 79.9 115 158 204 258 316 378 445 519 676 849 1030 1230 1460 1700 1955 2230 2530 2815 3130 3435 3730 3975 Pr Prandtl number 0.707 0.699 0.704 0.707 0.701 0.700 0.695 0.689 0.691 0.685 0.678 0.675 0.670 0.671 0.673 0.662 0.659 0.655 0.650 0.643 0.639 0.637 0.639 0.643 0.661 Table C.5 Nitrogen (N2) at 1 atm (Incropera et al., 2007) T Temp. (K) 100 150 200 250 300 350 400 450 500 550 600 700 800 900 1000 1100 1200 1300 ρ density (kg/m3) 3.4388 2.2594 1.6883 1.3488 1.1233 0.9625 0.8425 0.7485 0.6739 0.6124 0.5615 0.4812 0.4211 0.3743 0.3368 0.3062 0.2807 0.2591 cp specific heat (kJ/Kg-K) 1.070 1.050 1.043 1.042 1.041 1.042 1.045 1.050 1.056 1.065 1.075 1.098 1.220 1.146 1.167 1.187 1.204 1.219 μ viscosity (10-7N-s/m²) 68.8 100.6 129.2 154.9 178.2 200.0 220.4 239.6 257.7 274.7 290.8 321.0 349.1 375.3 399.9 423.2 445.3 466.2 ν kinematic viscosity (10-6m2/s) 2.00 4.45 7.65 11.48 15.86 20.78 26.16 32.01 38.24 44.86 51.79 66.71 82.9 100.3 118.7 138.2 158.6 179.9 k thermal conductivity (10-3W/m-K) 9.58 13.9 18.3 22.2 25.9 29.3 32.7 35.8 38.9 41.7 44.6 49.9 54.8 59.7 64.7 70.0 75.8 81.0 α thermal diffusivity (10-6m2/s) 2.6 5.86 10.4 15.8 22.1 29.2 37.1 45.6 54.7 63.9 73.9 94.4 116 139 165 193 224 256 Pr Prandtl number 0.768 0.759 0.736 0.727 0.716 0.711 0.704 0.703 0.700 0.702 0.701 0.706 0.715 0.721 0.721 0.718 0.707 0.701 Appendix C. Transport Properties of Gases and Liquids at atmospheric Pressure 891 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table C.6 Oxygen (O2) at 1 atm (Incropera et al., 2007) T Temp. (K) 100 150 200 250 300 350 400 450 500 550 600 700 800 900 1000 1100 1200 1300 ρ density (kg/m3) 3.9450 2.5850 1.9300 1.5420 1.2840 1.1000 0.9620 0.8554 0.7698 0.6998 0.6414 0.5498 0.4810 0.4275 0.3848 0.3498 0.3206 0.2960 cp specific heat (kJ/Kg-K) 0.9620 0.9210 0.9150 0.9150 0.9200 0.9290 0.9420 0.9560 0.9720 0.9880 1.0030 1.0310 1.0540 1.0740 1.0900 1.1030 1.1150 1.1250 μ viscosity (10-7N-s/m²) 76.4 114.8 147.5 178.6 207.2 233.5 258.2 281.4 303.3 324.0 343.7 380.8 415.2 447.2 477.0 505.5 532.5 588.4 ν kinematic viscosity (10-6m2/s) 1.94 4.44 7.64 11.58 16.14 21.23 26.84 32.90 39.40 46.30 53.59 69.26 86.32 104.6 124.0 144.5 166.1 188.6 k thermal conductivity (10-3W/m-K) 9.25 13.8 18.3 22.6 26.8 29.6 33.0 36.3 41.2 44.1 47.3 52.8 58.9 64.9 71.0 75.8 81.9 87.1 α thermal diffusivity (10-6m2/s) 2.44 5.80 10.4 16.0 22.7 29.0 36.4 44.4 55.1 63.8 73.5 93.1 116 141 169 196 229 262 Pr Prandtl number 0.796 0.766 0.737 0.723 0.711 0.733 0.737 0.741 0.716 0.726 0.729 0.744 0.743 0.740 0.733 0.736 0.725 0.721 Table C.7 Water (H2O) vapor at 1 atm (Incropera et al., 2007) T Temp. (K) 380 400 450 500 550 600 650 700 750 800 850 ρ density (kg/m3) 0.5863 0.5542 0.4902 0.4405 0.4005 0.3652 0.3380 0.3140 0.2931 0.2739 0.2579 cp specific heat (kJ/Kg-K) 2.060 2.014 1.980 1.985 1.997 2.026 2.056 2.085 2.119 2.152 2.186 μ viscosity (10-7N-s/m²) 127.1 134.4 152.5 170.4 188.4 206.7 224.7 242.6 260.4 278.6 296.9 ν kinematic viscosity (10-6m2/s) 21.68 24.25 31.11 38.68 47.04 56.60 66.48 77.26 88.84 101.7 115.1 k thermal conductivity (10-3W/m-K) 24.6 26.1 29.9 33.9 37.9 42.2 46.4 50.5 54.9 59.2 63.7 α thermal diffusivity (10-6m2/s) 20.4 23.4 30.8 38.8 47.4 57.0 66.8 77.1 88.4 100.0 113.0 Pr Prandtl number 1.060 1.040 1.010 0.998 0.993 0.993 0.996 1.000 1.000 1.010 1.020 892 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press C.2 Liquid Table C.8 T Temp. (°C) 0 5 10 15 20 25 30 35 40 50 60 70 80 90 100 Liquid Water (H2O) at 1 atm (Bejan, 2004) cp specific heat (kJ/Kg-K) 4.217 4.202 4.192 4.186 4.182 4.179 4.178 4.178 4.178 4.180 4.184 4.189 4.196 4.205 4.216 μ viscosity (10-3N-s/m²) 1.787 1.514 1.304 1.137 1.002 0.891 0.798 0.720 0.654 0.548 0.467 0.405 0.355 0.316 0.283 ν kinematic viscosity (10-6m2/s) 1.787 1.514 1.304 1.138 1.004 0.894 0.802 0.725 0.659 0.554 0.475 0.414 0.366 0.327 0.295 k thermal conductivity (W/m-K) 0.56 0.57 0.58 0.59 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.67 0.68 α thermal diffusivity (10-6m2/s) 0.133 0.136 0.138 0.140 0.142 0.144 0.146 0.149 0.152 0.155 0.158 0.161 0.164 0.165 0.166 Pr Prandtl number 13.44 11.13 9.45 8.13 7.07 6.21 5.49 4.87 4.34 3.57 3.01 2.57 2.23 1.98 1.78 β volume expansion coefficient (10-4/K) -0.6 +0.1 0.9 1.5 2.1 2.6 3.0 3.4 3.8 4.5 5.1 5.7 6.2 6.7 7.1 ρ density (kg/m3) 999.9 1000 999.7 999.1 998.2 997.1 995.7 994.1 992.3 988.1 983.2 977.8 971.8 965.3 958.4 Table C.9 Liquid Gasoline at 1 atm (Bejan, 2004) T Temp. (°C) 20 50 100 150 200 ρ density (kg/m3) 751 721 681 628 570 cp specific heat (kJ/Kg-K) 2.06 2.20 2.46 2.74 3.04 μ viscosity (10-4N-s/m²) 5.29 3.70 2.25 1.56 1.11 ν kinematic viscosity (10-7m2/s) 7.04 5.13 3.30 2.48 1.95 k thermal conductivity (W/m-K) 0.1164 0.1105 0.1005 0.0919 0.0800 α thermal diffusivity (10-7m2/s) 0.752 0.697 0.600 0.534 0.462 Pr Prandtl number 9.4 7.4 5.5 4.6 4.2 Table C.10 Engine Oil, Unused (Bejan, 2004) T Temp. (K) ρ density (kg/m3) cp specific heat (kJ/Kg-K) μ viscosity (N-s/m²) ν kinematic viscosity (10-4m2/s) k thermal conductivity (W/m-K) α thermal diffusivity (10-8m2/s) Pr Prandtl number β volume expansion coefficient (10-4/K) 7 7 7 7 7 7 7 7 260 280 300 320 340 360 380 400 908 896 884 872 860 848 836 824 1.76 1.83 1.91 1.99 2.08 2.16 2.25 2.34 12.23 2.17 0.486 0.141 0.053 0.025 0.014 0.009 135 24.2 5.50 1.62 0.62 0.30 0.17 0.11 0.149 0.146 0.144 0.141 0.139 0.137 0.136 0.134 9.32 8.90 8.53 8.13 7.77 7.48 7.23 6.95 144500 27200 6450 1990 795 395 230 155 Appendix C. Transport Properties of Gases and Liquids at atmospheric Pressure 893 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table C.11 Volume expansion coefficients for liquids (Mills, 1999) Liquid Ammonia Engine oil (SAE 50) Ethylene glycol C2H4(OH)2 Refrigerant-22 T (K) 293 273 430 273 373 250 260 270 280 290 300 310 320 330 340 350 230 240 250 260 270 280 290 300 310 320 330 340 350 280 300 320 β × 103 (1/K) 2.45 0.70 0.70 0.65 0.65 2.27 2.41 2.58 2.78 3.03 3.35 3.75 4.30 5.09 6.34 8.64 2.00 2.09 2.20 2.32 2.47 2.65 2.86 3.13 3.48 3.95 4.61 5.60 7.32 0.47 0.48 0.50 Liquid Hydrogen Mercury Nitrogen T (K) 20.3 273 550 70 77.4 80 90 100 110 120 89 366 230 250 300 350 400 450 500 550 β × 103 (1/K) 15.1 0.18 0.18 4.9 5.7 5.9 7.2 9.0 12 24 2.0 0.27 0.79 0.75 0.70 0.70 0.76 0.84 0.96 1.1 Oxygen Sodium Therminol® 60 Refrigerant-134a Glycerin C3H5(OH)3 References Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, New York, NY. Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York, NY. Mills, A.F., 1999, Basic Heat and Mass Transfer, 2nd Ed., Prentice Hall, Upper Saddle River, NJ. 894 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX D TRANSPORT PROPERTIES FOR PHASE CHANGE List of Properties Tables Table D.1 Phase Change Materials (PCMs) ...................................................... 896 Table D.2 Thermophysical properties at saturation for Freon®-134a ............... 896 Table D.3 Thermophysical properties at saturation for water............................ 897 References .......................................................................................................... 898 Appendix D. Transport Properties for Phase Change 895 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table D.1 Phase Change Materials (PCMs) hs PCMs chemical formula latent heat (kJ/kg) Tm melting point (°C) ρs solid density (kg/m³) ρℓ liquid density (kg/m³) μℓ liquid viscosity (10-3 N-s/m²) cp,s solid specific heat (kJ/kg-K) cp,ℓ liquid specific heat (kJ/kg-K) ks solid thermal conductivity (W/mK) kℓ liquid thermal conductivity (W/mK) β liquid thermal expansion coefficient (10-41/K) 3.9 1.81 1.3 1.75 1.31 0.514 0.476 33.5 238 1.88 896 Advanced Heat and Mass Transfer 5.5 18.2 27.5 36.40 29.78 660.4 0 16.7 0.358 1.80 2.15 1.92 0.340 c 1.076 2.04 2.040 1.690 36 280 1520 1446 226 228.9 244 247.3 80.16 395 333.7 g 187 825 833 814 a 815 6095 2702 920 1214 771 774 774 a 780 6093 2380 e 1000 1050 0.15 0.1505 0.152 0.150 32.0 94.03 0.569 0.l8 2.31 2.18 2.46 0.382 1.08 4.23 1.960 1.940 8.5 8.5 1.2 1.2 -0.6805 Amir Faghri, Yuwen Zhang, and John Howell 4.35 Freon-134a, CF3CH2F, Molecular Mass: 102.0, (Tsat = -26.4°C; Tm = -101 °C; ASHRAE, 2001) μv kℓ μℓ kv σ ρv ρℓ liquid vapor liquid vapor liquid liquid vapor viscosity viscosity thermal thermal surface density density (10-3 (10-7 conductivity conductivity tension 3 (10 kg/m³) (kg/m³) N-s/m²) N-s/m²) (10-3N/m) (W/m-K) (W/m-K) 1.474 0.9268 0.663 83.0 0.121 0.00656 20.80 1.418 2.769 0.472 91.2 0.111 0.00817 17.60 1.358 6.785 0.353 99.2 0.101 0.00982 14.51 1.295 14.428 0.271 107.3 0.0920 0.01151 11.56 1.225 27.778 0.211 115.81 0.0833 0.01333 8.76 1.147 50.075 0.163 125.5 0.0747 0.01544 6.13 1.053 81.413 0.124 137.9 0.0661 0.01831 3.72 cp,ℓ liquid specific heat (kJ/kg-K) 1.223 1.255 1.293 1.341 1.405 1.498 1.660 C14H30 C16H34 C18H38 C20H42 Ga Al H2O CH3COOH a Hale et al. (1971); b Humphries and Griggs (1977); c Bennon and Incropera (1988); d Brent et al. (1988); e Iida and Guthrie (1988); f Incropera et al. (2007); g Cengel and Boles (2002) n-Tetradecane a n-Hexadecane b n-Octadecane c n-Eicosane b Gallium d Aluminum c Water f Acetic Acid a Sodium Hydrogen Phosphate Dodecahydrate a Na2HPO4· 12H2O Table D.2 Thermophysical properties at saturation for Freon®-134a T Temp. ˚C pv saturation pressure (105 Pa) hv latent heat (kJ/kg) Copyright © 2010 Global Digital Press -60 -40 -20 0 20 40 60 0.1591 0.5121 1.3273 2.9280 5.7171 10.166 16.818 237.95 225.86 212.91 198.60 182.28 163.02 139.13 cp,v vapor specific heat (kJ/kg-K) 0.692 0.749 0.816 0.897 1.001 1.145 1.387 Table D.3 Thermophysical properties at saturation for water Amir Faghri, Yuwen Zhang, and John Howell Water, H2O, Molecular Mass: 18.0, (Tsat = 100 °C; Tm = 0.0 °C; Vargaftik, 1975) μv kℓ kv μℓ ρℓ ρv liquid vapor liquid vapor liquid vapor viscosity viscosity thermal thermal density density (10-7 (10-7 conductivity conductivitya (kg/m³) (kg/m³) N-s/m²) N-s/m²) (W/m-K) (W/m-K) 999.0 0.01729 10015 88.5 0.602 0.0188 993.05 0.05110 6513 96.6 0.630 0.0201 983.28 0.13020 4630 105.0 0.653 0.0216 971.82 0.29320 3510 113.0 0.669 0.0231 958.77 0.59740 2790 121.0 0.680 0.0248 943.39 1.12100 2300 128.0 0.685 0.0267 925.93 1.96560 1950 135.0 0.687 0.0288 907.44 3.25890 1690 142.0 0.684 0.0313 887.31 5.15970 1493 149.0 0.676 0.0341 865.05 7.86530 1338 156.0 0.664 0.0375 σ liquid surface tension (10-3/m) 72.88 69.48 66.07 62.69 58.91 54.96 50.79 46.51 42.19 37.77 cp,ℓ liquid specific heat a (kJ/kg-K) 4.182 4.179 4.185 4.197 4.216 4.245 4.285 4.339 4.408 4.497 T Temp. °C pv saturation pressure (105 Pa) hv latent heat (kJ/kg) 20 40 60 80 100 120 140 160 180 200 0.023368 0.073749 0.199190 0.473590 1.013250 1.985400 3.613600 6.180400 10.02700 15.55100 2453.8 2406.5 2358.4 2308.9 2251.2 2202.9 2144.9 2082.2 2014.0 1939.0 cp,v vapor specific heat a (kJ/kg-K) 1.874 1.894 1.924 1.969 2.034 2.124 2.245 2.406 2.615 2.883 Appendix D. Transport Properties for Phase Change 897 Copyright © 2010 Global Digital Press References American Society of Heating, Refrigerating and Air Conditioning Engineers, 2001, ASHRAE Handbook of Fundamentals, ASHRAE, New York, NY. Bennon, W.D., and Incropera, F.P., 1988, “Developing Laminar Mixed Convection with Solidification in a Vertical Channel,” ASME Journal of Heat Transfer, Vol. 110, pp. 410-415. Brent, A.D., Voller, V.R., Reid, K.J., 1988, “Enthalpy-Porosity Technique for Modeling Convection-Diffusion Phase Change: Application to the Melting of Pure Metal,” Numerical Heat Transfer, Part B, Vol. 13, pp. 297-318. Cengel, Y.A., and Boles, M.A., 2002, Thermodynamics – An Engineering Approach, 4th ed., McGraw-Hill, New York, NY. Hale, D.V., Hoovers, M.J., and O’Nell, M.J., 1971, Phase Change Materials Handbook, NASA-CR-61363. Humphries, W.R., and Griggs, E.I., 1977, A Design Handbook for Phase Change Thermal Control and Energy Storage Devices, NASA-TP-1074. Iida, T., and Guthrie, R.I.L., 1988, The Physical Properties of Liquid Metals, Oxford University Press, Oxford, UK. Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York, NY. Vargaftik, N.B., 1975, Handbook of Physical Properties of Liquids and Gases, Hemisphere, New York, NY. 898 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX E MASS TRANSFER PROPERTIES List of Properties Tables Table E.1 Binary diffusion coefficients at 1 atm ............................................... 900 Table E.2 Diffusion coefficients in air at 1 atm ................................................. 901 Table E.3 Diffusion coefficients in solids.......................................................... 902 Table E.4 Schmidt number for vapors in dilute mixture in air at normal temperature, enthalpy of vaporization, and boiling point at 1 atm .............. 903 Table E.5 Schmidt numbers for dilute solution in water at 300 K ..................... 904 Table E.6 Solubility and permeability of gases in solids ................................... 905 Table E.7 Henry’s constant for selected gases in water at moderate pressure ... 906 Table E.8 The solubility of selected gases and solids ........................................ 906 Table E.9 Solubility of inorganic compounds in water ..................................... 907 Table E.10 Equilibrium compositions for the NH3-water system...................... 908 Table E.11 Equilibrium compositions for the SO2-water system ...................... 908 Table E.12 Thermodynamic properties of water vapor-air mixtures at 1 atm .. 909 Appendix E. Mass Transfer Properties 899 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.1 Binary diffusion coefficients at 1 atma (Incropera et al., 2007) Substance A Gases NH3 H2O CO2 H2 O2 Acetone Benzene Naphthalene Ar H2 H2 H2 CO2 CO2 O2 Dilute solutions Caffeine Ethanol Glucose Glycerol Acetone CO2 O2 H2 N2 Solids O2 N2 CO2 He H2 Cd Al a Substance B Air Air Air Air Air Air Air Air N2 O2 N2 CO2 N2 O2 N2 H2O H2O H2 O H2O H2O H2O H2O H2O H2O Rubber Rubber Rubber SiO2 Fe Cu Cu T(K) 298 298 298 298 298 273 298 300 293 273 273 273 293 273 273 298 298 298 298 298 298 298 298 298 298 298 298 293 293 293 293 DAB (m2/s) 0.28 × 10-4 0.26 × 10-4 0.16 × 10-4 0.41 × 10-4 0.21 × 10-4 0.11 × 10-4 0.88 × 10-5 0.62 × 10-5 0.19 × 10-4 0.70 × 10-4 0.68 × 10-4 0.55 × 10-4 0.16 × 10-4 0.14 × 10-4 0.18 × 10-4 0.63 × 10-9 0.12 × 10-8 0.69 × 10-9 0.94 × 10-9 0.13 × 10-8 0.20 × 10-8 0.24 × 10-8 0.63 × 10-8 0.26 × 10-8 0.21 × 10-9 0.15 × 10-9 0.11 × 10-9 0.4 × 10-13 0.26 × 10-12 0.27 x 10-18 0.13 x 10-33 Assuming ideal gas behavior, the pressure and temperature dependence of the diffusion coefficient for a binary mixture of gases may be estimated form the relation DAB ∝ p-1 T 3/2. 900 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.2 Diffusion coefficients in air at 1 atm (1.013 × 105 Pa)a (Mills, 1999) T [K] 200 300 400 500 600 700 800 900 1000 1200 1400 1600 1800 2000 O2 0.095 0.188 0.325 0.475 0.646 0.838 1.05 1.26 1.52 2.06 2.66 3.32 4.03 4.80 CO2 0.074 0.157 0.263 0.385 0.537 0.684 0.857 1.05 1.24 1.69 2.17 2.75 3.28 3.94 Binary Diffusion Coefficient (m2/s × 104) CO C7H16 H2 NO 0.098 0.036 0.375 0.088 0.202 0.075 0.777 0.180 0.332 0.128 1.25 0.303 0.485 0.194 1.71 0.443 0.659 0.270 2.44 0.603 0.854 0.354 3.17 0.782 1.06 0.442 3.93 0.978 1.28 0.538 4.77 1.18 1.54 0.641 5.69 1.41 2.09 0.881 7.77 1.92 2.70 1.13 9.90 2.45 3.37 1.41 12.5 3.04 4.10 1.72 15.2 3.70 4.87 2.06 18.0 4.48 SO2 0.058 0.126 0.214 0.326 0.440 0.576 0.724 0.887 1.06 1.44 1.87 2.34 2.85 3.36 He 0.363 0.713 1.14 1.66 2.26 2.91 3.64 4.42 5.26 7.12 9.20 11.5 13.9 16.6 a Owing to the practical importance of water vapor-air mixtures, engineers have used convenient empirical formulas for DH 2 O air. A formula that has been widely used is  p  T  m2/s; DH 2 O, air = 1.97 × 10−5  0    273K < T < 373K  p  T0  where p0 = 1 atm; T0 = 256 K. The following formula has also found increasing use (Marrero and Mason,1972); 1.685 DH 2 O air = 1.87 × 10−10 T = 2.75 × 10−9 2.072 p ; 280K < T < 450K 450K < T < 1070K T 1.632 ; p for p in atmospheres and T in Kelvins. Over the temperature range 290-330 K, the discrepancy between the two formulas is less than 2.5%. For small concentrations of water vapor in air, the older formula gives a constant value of ScH2 O air = 0.61 over the temperature range 273-373 K. On the other hand, the Marrero and Mason (1972) formula gives values of Sc H 2O air that vary from 0.63 at 280 K to 0.57 at 373 K. Appendix E. Mass Transfer Properties 901 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.3 Diffusion coefficients in solids, D = D0 exp ( − Ea / RuT ) (Mills, 1999) D0 System Oxygen-Pyrex glass Oxygen-fused silica glass Oxygen-titanium Oxygen-titanium alloy (Ti-6Al-4V) Oxygen-zirconium Hydrogen-iron Hydrogen-α-titanium Hydrogen-β-titanium Hydrogen-zirconium Hydrogen-Zircaloy-4 Deuterium-Pyrex glass Deuterium-fused silica glass Helium-Pyrex glass Helium-fused silica glass Helium-borosilicate Neon-borosilicate Carbon-FCC iron Carbon-BCC iron a Eaa kJ/kmol m /s 2 6.19 × 10−8 2.61 × 10−9 5.0 × 10−3 5.82 × 10−2 4.68 × 10−5 7.60 × 10−8 1.80 × 10−6 1.95 × 10−7 1.09 × 10−7 1.27 × 10−5 6.19 × 10−8 2.61 × 10−9 4.76 × 10−8 5.29 × 10−8 1.94 × 10−8 1.02 × 10−10 2.3 × 10−5 1.1 × 10−6 4.69 × 104 3.77 × 104 2.13 × 105 2.59 × 105 7.06 × 105 5.60 × 103 5.18 × 104 2.78 × 104 4.81 × 104 6.05 × 105 4.69 × 104 3.77 × 104 2.72 × 104 2.55 × 104 2.34 × 104 3.77 × 104 1.378 × 105 8.75 × 104 Activation energy 902 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.4 Schmidt number for vapors in dilute mixture in air at normal temperature, enthalpy of vaporization, and boiling point at 1 atma (Mills, 1999) Chemical Vapor Acetone Ammonia Benzene Carbon dioxide Carbon monoxide Chlorine Ethanol Helium Heptane Hydrogen Hydrogen sulfide Methanol Napthalenec Nitric oxide Octane Oxygen Pentane Sulfur dioxide Water vapor a hv Scb 1.42 0.61 1.79 1.00 0.77 1.42 1.32 0.22 2.0 0.20 0.94 0.98 2.35 0.87 2.66 0.83 1.49 1.24 0.61 0.340 0.454 0.548 1.100 0.567 0.465 0.303 0.214 0.357 0.398 2.257 J/kg × 10-6 0.527 1.370 0.395 0.398 0.217 0.288 0.854 Boiling point temperature K 329 240 354 194 81 238 352 4.3 372 20.3 213 338 491 121 399 90.6 309 263 373 Formula CH3COCH3 NH3 C6H6 CO2 CO Cl2 CH3CH2OH He C7H16 H2 H2S CH3OH C10H8 NO C8H18 O2 C5H12 SO2 H2O b With the Clausius-Clapeyron relation, one may estimate vapor pressure as  Mh  1 1    psat  exp  − v  −   atm, for T  TBP  Ru  T TBP     The Schmidt number is defined as Sc = μ / ρ D = v / D . Since the vapors are in small concentrations, values for μ, ρ and v can be taken as pure air values. c Cho et al. (1992); hv = 0.567 × 106 J/K is at 300 K. Appendix E. Mass Transfer Properties 903 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.5 Schmidt numbers for dilute solution in water at 300 Ka (Mills, 1999) Solute Helium Hydrogen Nitrogen Water Nitric Oxide Carbon monoxide Oxygen Ammonia Carbon dioxide Hydrogen sulfide Ethylene Methane Nitrous oxide Sulfur dioxide Sodium chloride Sodium hydroxide Acetic acid Acetone Methanol Ethanol Chlorine Benzene Ethylene glycol n-Propanol i-Propanol Propane Aniline Benzoic acid Glycerol Sucrose a Schmidt number, Sc 120 190 280 340 350 360 400 410 420 430 450 490 490 520 540 490 620 630 640 640 670 720 720 730 730 750 800 830 1040 1670 Molecular mass, M (kg/kmol) 4.003 2.016 28.02 18.016 30.01 28.01 32.00 17.03 44.01 34.08 28.05 16.04 44.02 64.06 58.45 40.00 60.05 58.08 32.04 46.07 70.90 78.11 62.07 60.09 60.09 44.09 93.13 122.12 92.09 342.3 For other temperatures use Sc / Sc300 K  ( μ 2 / ρT ) /( μ 2 / ρT )300 K , where μ and ρ are for water, and T is absolute temperature. For chemically similar solutes of different molecular weights use Sc 2 / Sc1  ( M 2 / M1 )0.4 . A table of ( μ 2 / ρT ) /( μ 2 / ρT )300 K for water follows. T[K] 290 300 310 320 330 ( μ 2 / ρT ) /( μ 2 / ρT )300 K 1.66 1.00 0.623 0.429 0.296 T [K] 340 350 360 370 ( μ 2 / ρT ) /( μ 2 / ρT )300 K 0.221 0.167 0.123 0.097 Spalding (1963). 904 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.6 Solubility and permeability of gases in solids (Mills, 1999) Gas H2 Solid Vulcanized rubber Vulcanized neoprene Silicone rubber Natural rubber Polyethylene Polycarbonate Fused silica Nickel T(K) 300 290 300 300 300 300 400 800 360 440 He Silicone rubber Natural rubber Polycarbonate Nylon 66 Teflon Fused silica Pyrex glass 7740 glass (94% SiO2+B2O3+P2O5 5% Na2O+Li2+K2O 1% other oxides) 7056 glass (90% SiO2+B2O3+P2O5 8% Na2O+Li2+K2O 1% PbO, 0.5% other oxides) Vulcanized rubber Silicone rubber Natural rubber Polyethylene Polycarbonate Silicone-polycarbonate copolymer (57% silicone) Ethyl cellulose 300 300 300 300 300 300 800 300 800 470 580 720 390 680 300 300 300 300 300 300 300 S ′ [m3 (STP)/m3 atm] or S’a Permeabilityb m3(STP)/m2s (atm/m) 0.34 × 10-10 0.053 × 10-10 4.2 × 10-10 0.37 × 10-10 0.065 × 10-10 0.091 × 10-10 S ′ = 0.040 S ′ = 0.051 S ′′ ≅ 0.035 S ′′ ≅ 0.030 S ′′ ≅ 0.202 S ′′ ≅ 0.192 2.3 × 10-10 0.24 × 10-10 0.11 × 10-10 0.0076 × 10-10 0.047 × 10-10 S ′′ ≅ 0.018 S ′′ ≅ 0.026 S ′′ ≅ 0.006 S ′′ ≅ 0.024 S ′ = 0.0084 S ′ = 0.0038 S ′ = 0.0046 S ′ = 0.0039 S ′ = 0.0059 S ′ = 0.070 4.6 × 10-13 1.6 × 10-12 6.4 × 10-12 1.2 × 10-14 1.0 × 10-12 0.15 × 10-10 3.8 × 10-10 0.18 × 10-10 4.2 × 10-12 0.011 × 10-10 1.2 × 10-10 0.09 × 10-10 O2 Appendix E. Mass Transfer Properties 905 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.6 Solubility and permeability of gases in solids (Mills, 1999) (cont’d) Gas N2 Solid Vulcanized rubber Silicone rubber Natural rubber Silicone-polycarbonate copolymer (57% silicone) Teflon® Vulcanized rubber Silicone rubber Natural rubber Silicone-polycarbonate copolymer (57% silicone) Nylon 66 H2O Ne Ar a T(K) 300 300 300 300 300 300 300 300 300 300 310 300-1200 900-1200 S ′ [m3 (STP)/m3 atm] or S’a Permeabilityb m3(STP)/m2s (atm/m) 0.054 × 10-10 1.9 × 10-10 0.062 × 10-10 0.53 × 10-10 0.019 × 10-10 1.0 × 10-10 21 × 10-10 1.0 × 10-10 7.4 × 10-10 0.0013 × 10-10 0.91-1.8 × 10-10 S ′ = 0.035 CO2 S ′ = 0.90 Cellophane Fused silica Fused silica S ′′ ≅ 0.002 S ′′ ≅ 0.01 Solubility S ′ = Volume of solute gas (0 °C, 1 atm) dissolved in unit volume of solid when the gas is at 1 atm partial pressure. Solubility coefficient S ′′ = c1, g / c2 . Permeability K = DAB S ′ . b From various sources, including Geankoplis (1993), Doremus (1973), and Altemose (1961). Table E.7 Henry’s constant for selected gases in water at moderate pressurea H = p A,i / x A,i (bars) T (K) 273 280 290 300 310 320 323 a NH3 21 23 26 30 ---- C l2 265 365 480 615 755 860 890 H2S 260 335 450 570 700 835 870 SO2 165 210 315 440 600 800 850 CO2 710 960 1300 1730 2175 2650 2870 CH4 22,880 27,800 35,200 42,800 50,000 56,300 58,000 O2 25,500 30,500 37,600 45,700 52,500 56,800 58,000 H2 58,000 61,500 66,500 71,600 76,000 78,600 79,000 Incropera and DeWitt (2001) and Spalding (1963). Table E.8 The solubility of selected gases and solids (Incropera eta al., 2007) Gas O2 N2 CO2 He H2 Solid Rubber Rubber Rubber SiO2 Ni T (K) 298 298 298 293 358 S = c A, s / p A, g (kmol/m3-bar) 3.12 x 10-3 1.56 x 10-3 40.15 x 10-3 0.45 x 10-3 9.01 x 10-3 906 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.9 Solubility of inorganic compounds in watera (Mills, 1999) Solute Aluminum sulfate Calcium bicarbonate Calcium chloride Calcium hydroxide Potassium chloride Potassium nitrate Potassium sulfate Sodium bicarbonate Sodium carbonate Sodium chloride Sodium nitrate Sodium sulfate Formula Al2(SO4)3 Ca(HCO3)2 CaCl2 CaCl2 Ca(OH)2 KCl KNO3 K2SO4 NaHCO3 Na2O3 Na2CO3 NaCl NaNO2 Na2SO4 Na2SO4 Na2SO4 a Solid Phase 18H2O 6H2O 2H2O 10H2O 1H2O 10H2O 7H2O - 273.15 280 31.2 32.8 290 35.5 300 39.1 310 44.3 320 50.3 T (K) 330 57.0 340 63.9 350 70.8 360 78.3 370 84.6 373.15 89.0 18.40 159.0 0.077 56.7 246.0 24.1 45.5 39.8 180 42.5 16.15 16.30 16.53 16.75 16.98 17.20 17.43 17.65 17.88 59.5 63.3 71.5 93.3 137.2 - 18.10 18.33 - 134.6 140.2 145.3 150.9 157.0 0.088 0.080 53.1 55.8 0.185 0.179 0.168 0.157 0.145 0.132 0.120 0.109 0.098 27.6 13.3 7.35 6.9 7 35.7 73 5.0 19.5 29.9 18.5 33.1 28.2 36.1 41.3 39.1 58.2 41.8 44.6 47.4 50.2 78.7 102.3 129.2 159.2 191.6 232.1 22.36 23.69 45.7 38.8 159 43.3 45.55 39.5 175 42.7 8.63 10.51 12.38 14.20 15.95 17.64 19.25 20.88 7.76 10.8 35.8 78 7.7 26.7 9.14 10.63 12.20 13.90 15.79 18.7 35.9 85 16.1 39.6 33.4 36.2 93 34.1 49.1 36.5 101 49.6 47.8 36.9 111 47.4 46.7 37.2 121 45.7 46.2 37.6 132 43.7 45.9 38.2 144 44.0 Solubility expressed in kilograms of anhydrous substance that is soluble in 100 kg water. Appendix E. Mass Transfer Properties 907 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.10 Equilibrium compositions for the NH3-water system (Mills, 1999) p A, g (atm) 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0 x A, 290 K 0.030 0.056 0.078 0.096 0.11 0.18 0.26 0.31 0.35 300 K 0.019 0.036 0.052 0.064 0.079 0.14 0.21 0.26 0.29 0.32 310 K 0.012 0.024 0.035 0.046 0.056 0.099 0.16 0.20 0.23 0.27 320 K 0.008 0.016 0.024 0.032 0.040 0.057 0.12 0.16 0.19 0.22 330 K 0.006 0.012 0.017 0.023 0.029 0.052 0.092 0.13 0.15 0.17 Table E.11 Equilibrium compositions for the SO2-water systema (Mills, 1999) p A, g (atm) 0.001 0.003 0.01 0.03 0.1 0.3 1.0 a x A, × 103 290 K 0.12 0.25 0.62 1.4 4.1 11.0 33.0 300 K 0.084 0.18 0.42 1.1 2.9 7.9 24.0 310 K 0.059 0.13 0.31 0.73 2.0 5.6 18.0 320 K 0.042 0.093 0.22 0.51 1.4 3.9 12.0 Notice that Henry’s law is invalid for the SO2-water system, even at very dilute concentrations. 908 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table E.12 Thermodynamic properties of water vapor-air mixtures at 1 atm (Mills, 1999) Specific Volume (m3/kg) Temp. (°C) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 a b Enthalpya, b (KJ/kg) Liquid Water 42.13 46.32 50.52 54.71 58.90 63.08 67.27 71.45 75.64 79.82 83.99 88.17 92.35 96.53 100.71 104.89 109.07 113.25 117.43 121.61 125.79 129.97 134.15 138.32 142.50 146.68 150.86 155.04 159.22 163.40 167.58 171.76 175.94 180.12 184.29 188.47 192.65 196.83 201.01 205.19 Dry Air 10.059 11.065 12.071 13.077 14.083 15.089 16.095 17.101 18.107 19.113 20.120 21.128 22.134 23.140 24.147 25.153 26.159 27.166 28.172 29.178 30.185 31.191 32.198 33.204 34.211 35.218 36.224 37.231 38.238 39.245 40.252 41.259 42.266 43.273 44.280 45.287 46.294 47.301 48.308 49.316 Saturated Air 29.145 31.481 33.898 36.401 38.995 41.684 44.473 47.367 50.372 53.493 56.736 60.107 63.612 67.259 71.054 75.004 79.116 83.400 87.862 92.511 97.357 102.408 107.674 113.166 118.893 124.868 131.100 137.604 144.389 151.471 158.862 166.577 174.630 183.037 191.815 200.980 210.550 220.543 230.980 241.881 Saturation Mass Fraction 0.007608 0.008136 0.008696 0.009289 0.009918 0.01058 0.01129 0.01204 0.01283 0.01366 0.01455 0.01548 0.01647 0.01751 0.01861 0.01978 0.02100 0.02229 0.02366 0.02509 0.02660 0.02820 0.02987 0.03164 0.03350 0.03545 0.03751 0.03967 0.04194 0.04432 0.04683 0.04946 0.05222 0.05512 0.05817 0.06137 0.06472 0.06842 0.07193 0.07580 Dry Air 0.8018 0.8046 0.8075 0.8103 0.8131 0.8160 0.8188 0.8217 0.8245 0.8273 0.8302 0.8330 0.8359 0.8387 0.8415 0.8444 0.8472 0.8500 0.8529 0.8557 0.8586 0.8614 0.8642 0.8671 0.8699 0.8728 0.8756 0.8784 0.8813 0.8841 0.8870 0.8898 0.8926 0.8955 0.8983 0.9012 0.9040 0.9068 0.9097 0.9125 Saturated Air 0.8054 0.8086 0.8117 0.8148 0.8180 0.8212 0.8244 0.8276 0.8309 0.8341 0.8374 0.8408 0.8441 0.8475 0.8510 0.8544 0.8579 0.8615 0.8650 0.8686 0.8723 0.8760 0.8798 0.8836 0.8874 0.8914 0.8953 0.8994 0.9035 0.9077 0.9119 0.9162 0.9206 0.9251 0.9297 0.9343 0.9391 0.9439 0.9489 0.9539 The enthalpies of dry air and liquid water are set equal to zero at a datum temperature of 0 °C The enthalpy of an unsaturated water vapor-air mixture can be calculated as h = hdry air + (m1 / m1, sat )(hsat − hdry air ) . Appendix E. Mass Transfer Properties 909 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press References Altemose, V.O., 1961, “Helium Diffusion through Glass,” Journal of Applied Physics, Vol. 32, pp. 1309-1316. Cho, C., Irvine, T.F., Jr., and Karni, J., 1992, “Measurement of the Diffusion Coefficient of Naphthalene into Air,” International Journal of Heat and Mass Transfer, Vol. 35, pp. 957-966. Doremus, R.H., 1973, Glass Science, Wiley, New York. Geankoplis, C. J., 1993, Transport Processes and Unit Operations, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ. Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York, NY. Marrero, T.R., and Mason, E.A., 1972, “Gaseous Diffusion Coefficients,” Journal of Physical and Chemical Reference Data, Vol. pp. 1, 3-118. Mills, A.F., 1999, Basic Heat and Mass Transfer, 2nd Ed., Prentice Hall, Upper Saddle River, NJ. Spalding, D.B., 1963, Convective Mass Transfer, McGraw-Hill, New York, NY. 910 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX F CONFIGURATION FACTORS AND SURFACE PROPERTIES FOR RADIATION Table F.1 Selected configuration factor relations Factor 1) Differential strip dA1 of any length to parallel cylindrical surface A2 of infinite length. Geometry Relation Fd 1− 2 = 1 2 ( sin θ 2 − sin θ1 ) 2) Plane element dA1 to plane parallel rectangle A2: normal to element passes through corner of rectangle X=a/c Y=b/c Fd 1− 2 = + 1 2π Y 1+ Y 2 ( X 1+ X tan −1 2 tan Y −1 X 1+ Y 2 1+ X 2 ) 3) Infinitely long directly opposed parallel plates of equal width H = h/W F1− 2 = 1 − H − H 2 Appendix F. Configuration Factors and Surface Properties for Radiation 911 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table F.1 Selected configuration factor relations (cont’d) Factor Geometry F1− 2 = Relation X=a/c Y = b/c 2 4) Identical parallel directly opposed rectangles π XY {ln[ 2 (1 + X )(1 + Y ) 1+ X + Y −1 2 2 2 2 ] 1/ 2 +X 1 + Y tan 2 X 1+ Y 2 +Y 1 + X tan −1 −1 Y 1+ X −1 2 - X tan X − Y tan Y 5) Infinitely long plates having a common edge with angle of 90o 6) Infinitely long enclosure formed of three plane areas H = h/w F1− 2 = 1 2 (1 + H − 1 + H 2 F1− 2 = A1 + A2 − A3 2 A1 7) Infinitely long plates of equal width with a common edge and included angle α F1− 2 = 1 − sin (α / 2 ) 8) Parallel circular disks of unequal radius with common axis R1 = r1 / h X = 1+ R1 2 R2 = r2 / h 2 1 + R2 R  F1− 2 =  X − X − 4  2  2  R1   2 1 2     912 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table F.1 Selected configuration factor relations (cont’d) Factor Geometry Relation 9) Infinitely long plate of finite width to parallel infinitely long cylinder F1− 2 =  tan −1 b − tan −1 a    b−a c c r 10) Infinitely long parallel cylinders of same diameter X = 1 + s / 2r F1− 2 =  X 2 − 1 + sin −1 1 − X    π X  1 11) Concentric cylinders of infinite length F1− 2 = 1 F2 −1 = ( r1 / r2 ) F2 − 2 = 1 − ( r1 / r2 ) 12) Sphere to disk; normal to disk center passes through sphere center F1− 2 = R2 = r2/h 1  1 1 −  2 2 1 + R2    13) Concentric spheres F1− 2 = 1 r  F2 −1 =  1   r2  2 r  F2 − 2 = 1 −  1   r2  2 Appendix F. Configuration Factors and Surface Properties for Radiation 913 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press TABLE F.2 Solar Normal Total Absorptivity and Total Hemispherical Emissivity of Metals and Deposited Coatings on Metals MATERIAL Aluminum, polished a Chromium, polished CrOx on aluminum foil b Copper, polished a CuO on aluminum sheet b Gold, polished Iron, polished Nickel, polished c NiS+ZnS on steel b Nickel dendrites on stainless steel b Platinum, polished a Silver, polished Stainless steel, 301 Titanium a Tungsten, polished c Zinc oxide on stainless steel b Solar Normal Total Absorptivity Total Hemispherical Emissivity at Moderate Temperature αn, solar 0.10 0.415 a 0.964 0.29 0.90 0.19 a 0.445 a 0.15 0.88 0.99 0.307 0.07c 0.37c 0.63 0.37 0.95 ε 0.018 * 0.08 c 0.023 0.019 * 0.15 0.018 c * 0.05 c * 0.04 * 0.10 0.26 0.050 * 0.017 a * 0.05 a * 0.134 * 0.03 * 0.08 a Touloukian and DeWitt (1970); b Agnihotri and Gupta (1981); c Siegel and Howell (2002) * Normal value, usually up to 10 percent smaller than hemispherical value for metals  α n,solar  ε  5.6 5.2 42 15 6.0 11 8.9 3.8 8.8 3.8 6.1 7 7.4 4.7 12 11.6     914 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press TABLE F.3 Solar Normal Total Absorptivity and Total Hemispherical Emissivity of Dielectrics and Other Common Materials MATERIAL Solar Normal Total Absorptivity Total Hemispherical Emissivity at Moderate Temperature, αn, solar Aluminum Oxide, Al2O3 Bricks, red c Bricks, white refractory d Carbon black, lampblack c Magnesium Carbonate, MgCO3 c Magnesium Oxide, MgO d Paints d Magnesium oxide white Titanium oxide white Zinc oxide white Carbon black paint Oil, light green Oil, light gray Snow, fine, fresh c Snow, ice granules c Soil, black loam Soil, plowed 0.06-0.23 d 0.55 0.29 0.98 0.025-0.04 0.15 0.09 0.20 0.15 0.96 0.50 0.75 0.13 0.33 0.9 c 0.75 d 0.92 0.69 0.95 0.79 0.6* ε  α n,solar  ε      0.60 0.42 1.0 0.03-0.05 0.22 a Touloukian and DeWitt (1970); b Agnihotri and Gupta (1981); c Kreider and Kreith (1981); d Siegel and Howell (2002) * Normal value, usually up to 10 percent smaller than hemispherical value for metals 0.88 c 0.92-0.96 * 0.92-0.96 * 0.82 0.89 0.66 d* 0.90 c 1.1 0.53 0.80 0.16 0.37 1.50 0.83 References Agnihotri, O.P. and Gupta, B.K., 1981, Solar Selective Surfaces, John Wiley & Sons, New York. Kreider, J.F. and Kreith, F., 1981, Solar Energy Handbook, McGraw-Hill, New York, 1981. Siegel, R. and Howell, J.R., 2002, Thermal Radiation Heat Transfer, Taylor and Francis, New York. Touloukian, Y.S. and DeWitt, D.P., (eds.), 1970-[79], Thermophysical Properties of Matter, Vol. 7, Thermal radiative properties: metallic elements and alloys, and Vol. 8. Thermal radiative properties: non-metallic solids, IFI/Plenum, New York. Appendix F. Configuration Factors and Surface Properties for Radiation 915 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press APPENDIX G MATHEMATICAL RELATIONS G.1 Vectors The term scalar refers to a single real number used to describe the magnitude of a quantity. Pressure, temperature, and internal energy are all scalar. A vector is defined as an entity that possesses both magnitude and direction or as a directed line segment subject to the parallelogram law of addition. In three-dimensional space, a vector may be specified by three vector components. A unit vector is a vector whose magnitude is unity. A vector in a three-dimensional Cartesian coordinate system can be expressed as A = iAx + jAy + kAz (G.1) where i, j, and k are unit vectors in the x-, y-, and z-directions. The vector components in the x-, y-, and z-directions are Ax, Ay, and Az, respectively. A vector in a Cartesian coordinate system is shown in Fig. G.1. It can be seen that the magnitude of the vector is 2 2 2 A = Ax + Ay + Az (G.2) Figure G.1 Vector and its components. 916 Advances of Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press and that its three components are the projections of the vector on the x, y, and zaxes. A vector can also be represented in matrix form:  Ax   A =  Ay  A   z (G.3) The dot product (also referred to as the scalar or inner product) of two vectors A and B is a scalar; it is obtained by summation of the product of each of their corresponding components, i.e., A ⋅ B = Ax Bx + Ay By + Az Bz (G.4) The cross product (or vector product) of two vectors A and B is a vector, i.e., i A × B = Ax Bx j Ay By k Az Bz (G.5) = i ( Ay Bz − Az By ) + j( Az Bx − Ax Bz ) + k ( Ax By − Ay Bx ) G.2 Operations with the ∇ Operator G.2.1 Cartesian Coordinate System An important vector for fluid mechanics and heat transfer is the ∇ operator, which is defined as ∇=i ∂ ∂ ∂ + j +k ∂x ∂y ∂z (G.6) in a three-dimensional Cartesian coordinate system. It can be applied to a scalar function, φ , to obtain its gradient, ∂φ ∂φ ∂φ grad φ = ∇φ = i (G.7) + j +k ∂x ∂y ∂z which is a vector. The ∇ operator can also be applied to a vector function such as velocity, V = iu + jv + kw (G.8) to get its divergence: div V = ∇ ⋅ V = ∂u ∂v ∂w + + ∂x ∂y ∂z (G.9) or its curl:  ∂w ∂v   ∂u ∂w   ∂v ∂u  − + j − + k −  curl V = ∇ × V = i  ∂y ∂z   ∂z ∂x      ∂x ∂y  (G.10) Appendix G Mathematical Relations 917 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Another related operator that is very useful in fluid mechanics and heat transfer is the Laplacian operator, defined as ∇ 2 = ∇ ⋅∇ = ∂2 ∂2 ∂2 + 2+ 2 ∂x 2 ∂y ∂z (G.11) Application of the Laplace operator to a scalar function φ results in ∇ 2φ = ∂ 2φ ∂ 2φ ∂ 2φ + + ∂x 2 ∂y 2 ∂z 2 (G.12) Forming the dot product of the velocity and the gradient of a scalar φ results in a scalar: ∂φ ∂φ ∂φ V ⋅∇φ = u (G.13) +v +w ∂x ∂y ∂z which is used to describe the advection of the property φ . The operation ∂  ∂φ  ∂  ∂φ  ∂  ∂φ  ∇ ⋅ α∇ φ =  α  + α  + α  ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  (G.14) results in a scalar that describes the diffusion of the property φ . Application of the Laplace operator to a velocity vector, V, results in a vector: ∇ 2 V = i∇ 2 u + j∇ 2 v + k∇ 2 w (G.15) G.2.2 Cylindrical Coordinate System The cylindrical coordinate system (r, ϕ , z) shown in Fig. G.2 is related to Cartesian coordinates (x, y, z) by x = r cos ϕ y = r sin ϕ z = z (G.16) The velocity vector, V, in a cylindrical coordinate system has three components, i.e., V = k r Vr + k ϕ Vϕ + k z Vz (G.17) where k r , k ϕ and k z are the unit vectors in r-, ϕ -, and z-directions, respectively. The operations involving the ∇ operator and either the general scalar function φ or the vector V in a cylindrical coordinate system are summarized below: ∂φ 1 ∂φ ∂φ grad φ = ∇φ = k r + kϕ + kz (G.18) ∂r ∂z r ∂ϕ div V = ∇ ⋅ V = 1 ∂(rVr ) 1 ∂Vϕ ∂Vz + + ∂z r ∂r r ∂ϕ (G.19) 918 Advances of Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure G.2 Relationship between cylindrical and Cartesian coordinates. ∇ 2φ = 1 ∂  ∂φ  1 ∂ 2φ ∂ 2φ r + + r ∂r  ∂r  r 2 ∂ϕ 2 ∂z 2   ∂φ Vϕ ∂φ ∂φ + + Vz ∂r r ∂ϕ ∂z (G.20) (G.21) (G.22) V ⋅∇φ = Vr ∇ ⋅ α∇ φ = 1 ∂  ∂φ  1 ∂  α ∂φ  ∂  ∂φ  rα + α  + r ∂r  ∂r  r ∂ϕ  r ∂ϕ  ∂z  ∂z       ∂ 1 ∂ ∂ 2V  1 ∂ 2V 2 ∂V ∇2 V = k r   ( rVr ) + 2 2r − 2 ϕ + 2r   r ∂ϕ r ∂ϕ ∂z    ∂r  r ∂r 2 2  ∂ 1 ∂   1 ∂ Vϕ 2 ∂Vr ∂ Vϕ   rVϕ )  + 2 +k ϕ   +2 + (  ∂r  r ∂r r ∂ϕ 2 r ∂ϕ ∂z 2       1 ∂  ∂Vz +k z  r  r ∂r  ∂r 2 2  1 ∂ Vz ∂ Vz  +2 +   2 ∂z 2   r ∂ϕ (G.23) G.2.3 Spherical Coordinate System The spherical coordinate system (r, θ , ϕ ) shown in Fig. G.3 is related to the Cartesian coordinate system by x = r sinθ cosϕ y = r sinθ sin ϕ z = r cosθ (G.24) A velocity vector, V, has three components, i.e., Appendix G Mathematical Relations 919 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure G.3 Relationship between spherical and Cartesian coordinates. (G.25) where k r , k θ , and k ϕ are the unit vectors in r, θ , and ϕ directions, respectively. The operations involving the ∇ operator and either the general scalar function φ or the vector V in a spherical coordinate system are summarized below: ∂φ 1 ∂φ 1 ∂φ (G.26) grad φ = ∇φ = k r + kθ + kϕ r ∂θ r sinθ ∂ϕ ∂r 1 ∂(r 2Vr ) 1 ∂(Vθ sinθ ) 1 ∂Vϕ div V = ∇ ⋅ V = 2 + + (G.27) ∂r r sinθ ∂θ r sinθ ∂ϕ r 1 ∂  ∂φ  1 ∂ ∂φ  1 ∂ 2φ (G.28) ∇ 2φ = 2  r 2 + 2  sinθ ∂θ  + 2 2 2 r ∂r  ∂r  r sinθ ∂θ   r sin θ ∂ϕ V = k r Vr + k θ Vθ + k ϕ Vϕ V ⋅∇φ = Vr Vϕ ∂φ ∂φ Vθ ∂φ + + ∂r r ∂θ r sin θ ∂ϕ (G.29) ∇ ⋅ α ∇φ = + 1 ∂  2 ∂φ  1 ∂ ∂φ  ( sinθ )α   r α ∂r  + 2 ∂θ  r 2 ∂r  r sinθ ∂θ    ∂  ∂φ  1 α  r 2 sin 2 θ ∂ϕ  ∂ϕ  (G.30) 920 Advances of Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press  2V cotθ 2V 2 ∂V 2 ∂Vϕ  ∇ 2 V = k r ∇ 2Vr − 2r − 2 θ − θ 2 −2  r r ∂θ r r sinθ ∂ϕ    V 2 ∂Vr 2 cosθ ∂Vϕ  +k θ ∇ 2Vθ + 2 −2θ −2  r ∂θ r sin θ r sin 2 θ ∂ϕ   Vϕ  2 ∂Vr 2 cosθ ∂Vθ  +k z ∇ 2Vφ − 2 +2 +2  2 r sin θ r sinθ ∂ϕ r sin 2 θ ∂ϕ   (G.31) G.3 Tensors A tensor of rank n in the Cartesian coordinate system has 3n components. A scalar can be considered as a tensor of rank 0 because it has only one component. A vector is a tensor of rank 1 since it has 3 components. As was demonstrated in Section 1.3.1, the normal and shear stresses in a fluid can be described by a tensor of rank 2. The defining characteristics of a tensor is the manner in which its components transform under a rotation of the coordinate system where the components are defined. The transformation law for the components of tensor of a rank two is given as τ i′j = α ikα jlτ kl (G.32) where the prime denotes the tensor components in the rotated coordinate, and α ik and α jl represent, respectively, the cosines of the angles between the ith rotated axis and the kth original axis, and between the jth rotated axis and the lth original axis. It should be noted that it is not the tensor itself that transforms under this change in the reference coordinate system but, rather, the coordinates that describe the tensor. The application of the ∇ operator to each component of the velocity vector V = iu + jv + kw also yields a tensor of rank two:  ∂u  ∂x   ∂u ∇V =  ∂y   ∂u  ∂z  ∂v ∂x ∂v ∂y ∂v ∂z ∂w  ∂x   ∂w  ∂y   ∂w  ∂z   (G.33) which can be used to determine the strain rate tensor. The dot product of a vector and a tensor of rank 2 is a vector. For example, the dot product of the ∇ operator and a stress tensor is Appendix G Mathematical Relations 921 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press  ∂τ xx ∂τ xy ∂τ xz  ∂ + +    ∂x  ∂x ∂y ∂z  τ xx τ xy τ xz      ∂τ yx ∂τ yy ∂τ yz   ∂   ∇ ⋅ τ =   τ yx τ yy τ yz  =  + + ∂y ∂y ∂z    ∂x   τ τ zy τ zz    ∂τ  ∂   zx  ∂τ zx + zy + ∂τ zz   ∂z    ∂x ∂y ∂z    (G.34) The contraction of two tensors of rank two a and b is obtained by summing the products of the corresponding components from both tensors: a : b = axx bxx + axy bxy + axz bxz + ayx byx + ayy byy + ayz byz + azx bzx + azy bzy + azz bzz which can also be written as a : b = aij bij (G.35) (G.36) using the summation convention of tensors. According to the summation convention, the repetition of an index in a term denotes a summation with respect to that index over its range (i, j=x, y, z). The definitions and operations of the vectors and tensors reviewed here provide foundations for the governing equations for multiphase systems. Additional information about tensors and their associated operations can be found in a continuum mechanics textbook, such as Fung (1994). G.4 Bessel Functions Bessel functions are very useful for heat conduction in cylindrical coordinate system (see Chapter 3). They are the solutions of the following differential equation: x 2 y′′ + xy′ + ( x 2 −ν 2 )y = 0 (G.37) which is referred to as Bessel’s differential of order ν . The solution of eq. (G.37) can be obtained by using the method of Frobenius (Myers, 1998). Assuming the solution of eq. (G.37) has the following form: y( x ) = x c  an x n n =0 ∞ (G.38) where c and an are unspecified constant. Substituting eq. (G.38) into eq. (G.37), the following condition for the constant c must be satisfied c2 =ν 2 i.e., c = ±ν (G.39) If we choose c = ν the following solution can be obtained y( x ) = Jν ( x ) =  ∞ m =0 2 ν + 2m ( −1)m xν + 2m m !Γ(ν + m + 1) (G.40) 922 Advances of Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press which is referred to as Bessel function of the first kind of order ν . Since eq. (G.37) is a second-order ordinary differential equation, there must be two linearly independent solutions. It follows from eq. (G.40) that if c = −ν is chosen, another solution can be obtained as ( −1)m x −ν + 2 m (G.41) m !Γ( −ν + m + 1) m =0 2 It should be pointed out the for ν = n where n is an integer, J n ( x ) and J − n ( x ) y( x ) = J −ν ( x ) =  ∞ −ν + 2 m are not linearly independent and are related to each other by (G.42) If ν is not an integer, Jν ( x ) and J −ν ( x ) are linearly independent so that they can be used to construct general solution of eq. (G.37): ′ y( x ) = C1′Jν ( x ) + C 2 J −ν ( x ) (G.43) which can also be rewritten as: y( x ) = C1 Jν ( x ) + C 2Yν ( x ) (G.44) where cos(νπ ) J v ( x ) − J − v ( x ) Yν ( x ) = (G.45) sin(νπ ) is referred to as Bessel function of the second kind of order ν . When ν is an integer n, the Bessel function of the order n is: Yn ( x ) = lim Yν ( x ) (G.46) ν →n J − n ( x ) = (−1)n J n ( x ) which is linearly independent from Jn(x). Therefore, eq. (G.44) is the general solution of Bessel differential equation (G.37) whether ν is an integer or not. Modified Bessel functions are solutions of the following differential equation: x 2 y′′ + xy′ − ( x 2 + ν 2 )y = 0 (G.47) which is referred to as Bessel’s modified differential of order ν . The general solution of eq. (G.47) can be expressed as: y( x ) = C1 Iν ( x ) + C 2 Kν ( x ) (G.48) where Iν ( x ) = i −ν Jν (ix ) (G.49) is the modified Bessel function of the first kind of order ν and π  I ( x ) − I −ν ( x )  Kν ( x ) =  ν (G.50) 2  sin(νπ )   is the modified Bessel function of the second kind of order ν . The following first and second order derivatives of Bessel functions are useful for heat conduction in cylindrical coordinate system (Ozisik, 1993; Myers, 1998): ν Jν′ ( x ) = Jν −1 ( x ) − Jν ( x ) (G.51) x Yν′( x ) = Yν −1 ( x ) − Yν ( x ) x ν (G.52) Appendix G Mathematical Relations 923 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Iν′ ( x ) = Iν −1 ( x ) − ν x Iν ( x ) (G.53) (G.54) (G.55) (G.56) (G.57) (G.58) Kν′ ( x ) = Kν −1 ( x ) − ν x Kν ( x ) 1  ν (ν + 1)  Jν′′( x ) = − Jν −1 ( x ) − 1 − Jν ( x ) x x2    1  ν (ν + 1)  Yν′′( x ) = − Yν −1 ( x ) − 1 − Yν ( x ) x x2    1  ν (ν + 1)  Iν′′( x ) = − Iν −1 ( x ) + 1 − Iν ( x ) x x2    1  ν (ν + 1)  Kν′′( x ) = − Kν −1 ( x ) + 1 − Kν ( x ) x x2    References Fung, Y.C., 1994, First Course in Continuum Mechanics, 3rd edition, Prentice Hall. Myers, G.E., 1998, Analytical Methods in Conduction Heat Transfer, 2nd ed., AMCHT Publications, Madison, WI. Ozisik, M.N., 1993, Heat Conduction, 2nd ed., Wiley-Interscience, New York. 924 Advances of Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press