BAPP01.qxd 2/21/11 6:06 PM Page 981 APPENDIX A Thermophysical Properties of Matter 1 Table Page A.1 Thermophysical Properties of Selected Metallic Solids A.2 A.3 Thermophysical Properties of Selected Nonmetallic Solids Thermophysical Properties of Common Materials Structural Building Materials Insulating Materials and Systems Industrial Insulation Other Materials Thermophysical Properties of Gases at Atmospheric Pressure Thermophysical Properties of Saturated Fluids Saturated Liquids Saturated Liquid–Vapor, 1 atm Thermophysical Properties of Saturated Water Thermophysical Properties of Liquid Metals Binary Diffusion Coefficients at One Atmosphere Henry’s Constant for Selected Gases in Water at Moderate Pressure The Solubility of Selected Gases and Solids A.4 A.5 A.6 A.7 A.8 A.9 A.10 1 The convention used to present numerical values of the properties is illustrated by this example: T (K) 300 ⫺7 where = 0.349 ⫻ 10 2 ⫺3 m /s and k ⫽ 521 ⫻ 10 䡠 107 (m2/s) k 䡠 103 (W/m 䡠 K) 0.349 521 ⫽ 0.521 W/m 䡠 K at 300 K. 983 987 989 989 990 991 993 995 1000 1000 1002 1003 1005 1006 1007 1007 BAPP01.qxd 2/21/11 982 6:06 PM Page 982 Appendix A A.11 A.12 䊏 Thermophysical Properties of Matter Total, Normal (n) or Hemispherical (h) Emissivity of Selected Surfaces Metallic Solids and Their Oxides Nonmetallic Substances Solar Radiative Properties for Selected Materials References 1008 1008 1009 1010 1011 BAPP01.qxd 2/21/11 TABLE A.1 Thermophysical Properties of Selected Metallic Solidsa Properties at Various Temperatures (K) 903 237 97.1 775 2770 875 177 73.0 2790 883 168 68.2 1550 1850 1825 200 545 9780 122 2573 2500 1107 27.0 Cadmium 594 8650 231 96.8 Chromium 2118 7160 449 93.7 Cobalt 1769 8862 421 99.2 Copper Pure 1358 8933 385 401 117 1293 8800 420 52 14 1104 8780 355 54 17 1188 8530 380 110 1493 8920 384 23 1211 5360 322 59.9 Alloy 2024-T6 (4.5% Cu, 1.5% Mg, 0.6% Mn) Alloy 195, Cast (4.5% Cu) Beryllium Bismuth Boron Commercial bronze (90% Cu, 10% Al) Phosphor gear bronze (89% Cu, 11% Sn) Cartridge brass (70% Cu, 30% Zn) Constantan (55% Cu, 45% Ni) Germanium 7.86 100 302 482 65 473 200 237 798 163 787 400 240 949 186 925 600 231 1033 186 1042 800 1000 1200 1500 2000 218 1146 174 185 — — 59.2 990 301 161 126 106 90.8 78.7 203 1114 2191 2604 2823 3018 3227 3519 6.59 16.5 9.69 7.04 112 120 127 9.76 190 55.5 16.8 10.6 9.60 9.85 128 600 1463 1892 2160 2338 48.4 203 99.3 94.7 198 222 242 29.1 159 111 90.9 80.7 71.3 65.4 61.9 57.2 49.4 192 384 484 542 581 616 682 779 937 26.6 167 122 85.4 67.4 58.2 52.1 49.3 42.5 236 379 450 503 550 628 733 674 33.9 6.71 34.7 482 252 75 17 237 232 190 413 356 42 785 41 — 95 360 19 362 96.8 290 2500 393 397 52 460 65 — 137 395 379 417 59 545 74 — 149 425 366 433 352 451 339 480 43.2 337 27.3 348 19.8 357 17.4 375 17.4 395 Page 983 2702 ␣ 䡠 106 (m2/s) Thermophysical Properties of Matter 933 Aluminum Pure k (W/m 䡠 K) 䊏 cp (J/ kg 䡠 K) Appendix A (kg/m3) Composition 6:06 PM k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K Melting Point (K) 983 129 317 Iridium 2720 22500 130 147 Iron Pure 1810 7870 447 80.2 7870 447 7854 Armco (99.75% pure) Carbon steels Plain carbon (Mn ⱕ 1%, Si ⱕ 0.1%) AISI 1010 Carbon–silicon (Mn ⱕ 1%, 0.1% ⬍ Si ⱕ 0.6%) Carbon–manganese– silicon (1% ⬍ Mn ⱕ 1.65%, 0.1% ⬍ Si ⱕ 0.6%) Chromium (low) steels Cr– Mo–Si (0.18% C, 0.65% Cr, 0.23% Mo, 0.6% Si) 1 Cr– Mo (0.16% C, 1% Cr, 0.54% Mo, 0.39% Si) 1 Cr–V (0.2% C, 1.02% Cr, 0.15% V) 127 200 400 600 800 1000 1200 1500 327 109 172 90 323 124 153 122 311 131 144 133 298 135 138 138 284 140 132 144 270 145 126 153 255 155 120 161 111 172 23.1 134 216 94.0 384 69.5 490 54.7 574 43.3 680 32.8 975 28.3 609 32.1 654 72.7 20.7 95.6 215 80.6 384 65.7 490 53.1 574 42.2 680 32.3 975 28.7 609 31.4 654 434 60.5 17.7 56.7 487 48.0 559 39.2 685 30.0 1169 7832 434 63.9 18.8 7817 446 51.9 14.9 58.7 487 49.8 501 48.8 559 44.0 582 39.2 685 37.4 699 31.3 1168 29.3 971 8131 434 41.0 11.6 42.2 487 39.7 559 35.0 685 27.6 1090 7822 444 37.7 10.9 38.2 492 36.7 575 33.3 688 26.9 969 7858 442 42.3 12.2 42.0 492 39.1 575 34.5 688 27.4 969 7836 443 48.9 14.1 46.8 492 42.1 575 36.3 688 28.2 969 50.3 2000 2500 Page 984 19300 100 6:06 PM 1336 ␣ 䡠 106 (m2/s) Thermophysical Properties of Matter Gold k (W/m 䡠 K) 䊏 cp (J/kg 䡠 K) Appendix A (kg/m3) Composition k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K Melting Point (K) 2/21/11 Properties at Various Temperatures (K) BAPP01.qxd Continued 984 TABLE A.1 BAPP01.qxd 3.91 7900 477 14.9 3.95 AISI 316 8238 468 13.4 3.48 AISI 347 7978 480 14.2 3.71 35.3 AISI 304 1670 36.7 125 159 934 143 224 164 232 107 383 129 Magnesium 923 1740 1024 156 87.6 Molybdenum 2894 10240 251 138 53.7 Nickel Pure 1728 8900 444 90.7 1672 8400 420 12 3.4 1665 8510 439 11.7 3.1 8.7 — 2741 8570 265 53.7 23.6 Palladium 1827 12020 244 71.8 24.5 Platinum Pure 2045 21450 133 71.6 25.1 Alloy 60Pt–40Rh (60% Pt, 40% Rh) Rhenium 1800 16630 162 47 17.4 3453 21100 136 47.9 16.7 Rhodium 2236 12450 243 150 49.6 Silicon 1685 2330 712 148 89.2 Silver 1235 10500 235 429 Tantalum 3269 16600 140 57.5 24.7 Thorium 2023 11700 118 54.0 39.1 505 7310 227 66.6 40.1 Tin 23.0 174 28.0 640 31.7 682 112 295 105 308 98 330 90 380 65.6 592 16 525 17.0 510 67.6 530 21 545 20.5 546 71.8 562 76.2 594 82.6 616 10.3 372 80.2 485 14 480 13.5 473 24.0 626 27.6 — 33.0 — 55.2 188 76.5 168 52.6 249 71.6 227 55.2 274 73.6 251 58.2 283 79.7 261 61.3 292 86.9 271 64.4 301 94.2 281 67.5 310 102 291 72.1 79.1 324 347 110 307 77.5 100 72.6 125 58.9 97 186 147 884 259 444 187 59.2 110 59.8 99 85.2 188 51.0 127 154 220 264 556 430 225 57.5 133 54.6 112 73.3 215 71.8 136 52 — 46.1 139 146 253 98.9 790 425 239 57.8 144 54.5 124 62.2 243 73.2 141 59 — 44.2 145 136 274 61.9 867 412 250 58.6 146 55.8 134 75.6 146 65 — 44.1 151 127 293 42.2 913 396 262 59.4 149 56.9 145 78.7 152 69 — 44.6 156 121 311 31.2 946 379 277 60.2 152 56.9 156 82.6 157 73 — 45.7 162 116 327 25.7 967 361 292 61.0 155 58.7 167 89.5 165 76 — 47.8 171 110 349 22.7 992 86 459 99.4 179 51.9 186 112 376 62.2 64.1 65.6 160 172 189 985 11340 25.4 606 25.4 611 24.2 602 24.7 606 Thermophysical Properties of Matter 39.7 118 169 649 179 141 601 20.0 22.8 559 585 19.8 22.6 557 582 18.3 21.3 550 576 18.9 21.9 559 585 31.4 142 149 146 1170 1267 126 118 275 285 䊏 12.6 402 Lead Nichrome (80% Ni, 20% Cr) Inconel X-750 (73% Ni, 15% Cr, 6.7% Fe) Niobium 24.1 9.2 272 17.3 512 16.6 515 15.2 504 15.8 513 34.0 132 153 1074 134 261 Page 985 15.1 6:06 PM 480 Appendix A 8055 2/21/11 Stainless steels AISI 302 k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K cp (J/kg 䡠 K) k (W/m 䡠 K) 522 Tungsten 3660 19300 132 Uranium 1406 19070 116 27.6 12.5 Vanadium 2192 6100 489 30.7 10.3 693 7140 389 2125 6570 278 Zirconium a Adapted from References 1–7. 174 116 22.7 9.32 68.3 41.8 12.4 400 600 800 1000 1200 1500 2000 30.5 300 208 87 21.7 94 35.8 258 117 297 33.2 205 24.5 465 186 122 25.1 108 31.3 430 118 367 25.2 264 20.4 551 159 137 29.6 125 31.3 515 111 402 21.6 300 19.4 591 137 142 34.0 146 33.3 540 103 436 20.7 322 19.7 633 125 145 38.8 176 35.7 563 20.7 675 118 148 43.9 180 38.2 597 22.0 620 113 152 49.0 161 40.8 645 24.5 686 107 100 157 167 44.6 50.9 714 867 21.6 342 23.7 362 26.0 344 28.8 33.0 344 344 2500 95 176 Page 986 4500 200 Thermophysical Properties of Matter 1953 100 䊏 Titanium Zinc 21.9 ␣ 䡠 106 (m2/s) 6:06 PM (kg/m3) Appendix A Composition Melting Point (K) 2/21/11 Properties at Various Temperatures (K) BAPP01.qxd Continued 986 TABLE A.1 BAPP01.qxd 2/21/11 TABLE A.2 Thermophysical Properties of Selected Nonmetallic Solidsa Properties at Various Temperatures (K) 765 46 15.1 2323 3970 765 36.0 11.9 2725 3000 1030 Boron 2573 2500 1105 590 2080 Boron fiber epoxy (30% vol) composite k, 储 to fibers k, ⬜ to fibers cp Carbon Amorphous Diamond, type IIa insulator Graphite, pyrolytic k, 储 to layers k, ⬜ to layers cp Graphite fiber epoxy (25% vol) composite k, heat flow 储 to fibers k, heat flow ⬜ to fibers cp Pyroceram, Corning 9606 1122 1500 1950 — — 3500 509 2273 2210 450 1400 709 27.6 2300 9.99 1623 2600 — 450 — 133 — 82 — 55 — 190 — 52.5 — 32.4 940 26.4 940 196 1350 18.7 1490 18.9 1110 15.8 1110 111 1690 11.3 1880 13.0 1180 10.4 1180 70 1865 8.1 2135 — 0.67 — 10,000 21 1950 5.70 4970 16.8 136 5.7 0.87 3.98 800 2.10 0.37 364 11.1 935 808 600 88.0 2.29 0.59 1.60 400 1.89 0.46 337 5.25 — 1000 1200 1500 2000 10.5 1225 7.85 6.55 5.66 6.00 1225 — — — 47 33 21.5 15 1975 2055 2145 2750 6.3 5.2 2350 2555 2.23 2.28 0.49 0.60 757 1431 1.18 — 4000 194 1.89 — 2.19 — 2.37 — 2.53 — 2.84 — 3.48 — 1540 853 2500 Thermophysical Properties of Matter 3970 200 䊏 2323 100 3230 1390 892 667 534 448 357 262 9.23 4.09 2.68 2.01 1.60 1.34 1.08 0.81 411 992 1406 1650 1793 1890 1974 2043 8.7 13.0 0.68 1.1 642 1216 4.78 3.64 3.28 3.08 2.96 2.87 2.79 — 908 1038 1122 1197 1264 1498 Page 987 ␣ 䡠 10 (m2/s) Aluminum oxide, sapphire Aluminum oxide, polycrystalline Beryllium oxide 272 6:06 PM k (W/m 䡠 K)/cp (J/kg 䡠 K) 6 Appendix A Composition Properties at 300 K Melting Point cp k (K) (kg/m3) (J/ kg 䡠 K) (W/m 䡠 K) 987 Silicon dioxide, crystalline (quartz) k, 储 to c axis k, ⬜ to c axis cp Silicon dioxide, polycrystalline (fused silica) Silicon nitride 1883 2650 675 2220 745 745 2173 2400 691 392 2070 708 Thorium dioxide 3573 9110 235 Titanium dioxide, polycrystalline 2133 4157 710 a 230 400 — 880 600 — 1050 800 — 1135 1000 87 1195 1200 58 1243 1500 2000 30 1310 2500 Adapted from References 1, 2, 3 and 6. 10.4 6.21 1.38 16.0 0.206 13 8.4 0.834 9.65 39 20.8 — 0.69 — 16.4 9.5 — 1.14 — 7.6 5.0 4.2 4.70 3.4 3.1 885 1075 1250 1.51 1.75 2.17 2.87 4.00 905 1040 1105 1155 1195 — — 13.9 — 578 778 0.141 0.165 0.185 403 606 6.1 10.2 255 2.8 7.01 805 11.3 937 6.6 274 5.02 880 9.88 8.76 8.00 7.16 6.20 1063 1155 1226 1306 1377 4.7 285 3.94 910 3.68 295 3.46 930 3.12 303 3.28 945 2.73 315 2.5 330 Thermophysical Properties of Matter 1883 Sulfur 490 200 Page 988 3160 100 6:06 PM 3100 ␣ 䡠 106 (m2/s) 䊏 Silicon carbide k (W/m 䡠 K)/cp (J/kg 䡠 K) Appendix A Composition Properties at 300 K Melting Point cp k (K) (kg/m3) (J/kg 䡠 K) (W/m 䡠 K) 2/21/11 Properties at Various Temperatures (K) BAPP01.qxd Continued 988 TABLE A.2 BAPP01.qxd 2/21/11 6:06 PM Page 989 Appendix A TABLE A.3 䊏 989 Thermophysical Properties of Matter Thermophysical Properties of Common Materialsa Structural Building Materials Typical Properties at 300 K Description/Composition Building Boards Asbestos–cement board Gypsum or plaster board Plywood Sheathing, regular density Acoustic tile Hardboard, siding Hardboard, high density Particle board, low density Particle board, high density Woods Hardwoods (oak, maple) Softwoods (fir, pine) Masonry Materials Cement mortar Brick, common Brick, face Clay tile, hollow 1 cell deep, 10 cm thick 3 cells deep, 30 cm thick Concrete block, 3 oval cores Sand/gravel, 20 cm thick Cinder aggregate, 20 cm thick Concrete block, rectangular core 2 cores, 20 cm thick, 16 kg Same with filled cores Plastering Materials Cement plaster, sand aggregate Gypsum plaster, sand aggregate Gypsum plaster, vermiculite aggregate Density, (kg/m3) Thermal Conductivity, k (W/m 䡠 K) Specific Heat, cp (J/kg 䡠 K) 1920 800 545 290 290 640 1010 590 1000 0.58 0.17 0.12 0.055 0.058 0.094 0.15 0.078 0.170 — — 1215 1300 1340 1170 1380 1300 1300 720 510 0.16 0.12 1255 1380 1860 1920 2083 0.72 0.72 1.3 780 835 — — — 0.52 0.69 — — — — 1.0 0.67 — — — — 1.1 0.60 — — 1860 1680 720 0.72 0.22 0.25 — 1085 — BAPP01.qxd 2/21/11 990 6:06 PM Page 990 Appendix A TABLE A.3 䊏 Thermophysical Properties of Matter Continued Insulating Materials and Systems Typical Properties at 300 K Description/Composition Blanket and Batt Glass fiber, paper faced Glass fiber, coated; duct liner Board and Slab Cellular glass Glass fiber, organic bonded Polystyrene, expanded Extruded (R-12) Molded beads Mineral fiberboard; roofing material Wood, shredded/cemented Cork Loose Fill Cork, granulated Diatomaceous silica, coarse Powder Diatomaceous silica, fine powder Glass fiber, poured or blown Vermiculite, flakes Formed/Foamed-in-Place Mineral wool granules with asbestos/inorganic binders, sprayed Polyvinyl acetate cork mastic; sprayed or troweled Urethane, two-part mixture; rigid foam Reflective Aluminum foil separating fluffy glass mats; 10–12 layers, evacuated; for cryogenic applications (150 K) Aluminum foil and glass paper laminate; 75–150 layers; evacuated; for cryogenic application (150 K) Typical silica powder, evacuated Density, (kg/m3) Thermal Conductivity, k (W/m 䡠 K) Specific Heat, cp (J/kg 䡠 K) 16 28 40 32 0.046 0.038 0.035 0.038 — — — 835 145 105 0.058 0.036 1000 795 55 16 265 0.027 0.040 0.049 1210 1210 — 350 120 0.087 0.039 1590 1800 160 350 400 200 275 16 80 160 0.045 0.069 0.091 0.052 0.061 0.043 0.068 0.063 — — — — — 835 835 1000 190 0.046 — — 0.100 — 70 0.026 1045 40 0.00016 — 120 0.000017 — 160 0.0017 — BAPP01.qxd 2/21/11 TABLE A.3 Continued Industrial Insulation 0.036 0.038 0.040 12 16 24 32 48 0.035 0.033 0.030 0.029 0.027 0.036 0.035 0.032 0.030 0.029 0.039 0.036 0.033 0.032 0.030 1530 480 730 48 64 96 128 50–125 50 920 120 420 420 420 590 920 190 255 300 185 190 0.023 215 0.025 230 0.026 240 0.027 255 0.029 270 0.035 0.030 285 300 310 365 420 530 0.043 0.048 0.038 0.035 0.052 0.046 0.045 0.076 0.056 0.058 0.078 0.088 0.042 0.039 0.036 0.033 0.032 0.046 0.042 0.039 0.036 0.033 0.049 0.046 0.040 0.038 0.035 0.069 0.062 0.053 0.048 0.045 0.036 0.032 0.038 0.033 0.078 0.071 0.068 0.039 0.035 0.082 0.074 0.071 0.051 0.055 0.051 0.051 0.098 0.085 0.082 0.055 0.059 645 750 0.071 0.059 0.052 0.049 0.105 0.087 0.076 0.068 0.150 0.125 0.100 0.091 0.051 0.065 0.087 0.061 0.063 0.075 0.089 0.063 0.079 0.104 Page 991 96–192 40–96 10 200 Thermophysical Properties of Matter Felt, semirigid; organic bonded Felt, laminated; no binder Blocks, Boards, and Pipe Insulations Asbestos paper, laminated and corrugated 4-ply 6-ply 8-ply Magnesia, 85% Calcium silicate 920 815 450 Typical Thermal Conductivity, k (W/m 䡠 K), at Various Temperatures (K) 䊏 Blanket, alumina– silica fiber Typical Density (kg/m3) Appendix A Blankets Blanket, mineral fiber, metal reinforced Blanket, mineral fiber, glass; fine fiber, organic bonded Maximum Service Temperature (K) 6:06 PM Description/ Composition 991 350 350 350 56 35 16 340 70 1255 430 922 560 — — 45 105 — 122 80 0.023 0.023 0.026 0.036 215 0.023 0.023 0.029 0.039 230 240 255 270 285 300 310 365 420 0.046 0.048 0.051 0.052 0.055 0.058 0.062 0.069 0.079 0.022 0.023 0.030 0.023 0.025 0.033 0.023 0.025 0.035 0.025 0.026 0.036 0.026 0.027 0.038 0.027 0.029 0.040 0.029 0.029 0.030 0.032 0.033 0.071 0.079 0.108 0.115 0.042 0.043 0.046 0.049 0.038 0.051 0.039 0.053 0.042 0.056 0.056 0.049 0.058 0.051 0.061 0.055 0.063 0.058 0.065 0.061 0.068 0.063 0.071 0.066 530 645 750 0.092 0.101 0.098 0.100 0.104 0.115 0.088 0.105 0.123 0.123 0.137 Page 992 145 345 385 200 Thermophysical Properties of Matter 700 1145 1310 Typical Thermal Conductivity, k (W/m 䡠 K), at Various Temperatures (K) 䊏 Typical Density (kg/m3) Appendix A Cellular glass Diatomaceous silica Polystyrene, rigid Extruded (R-12) Extruded (R-12) Molded beads Rubber, rigid foamed Insulating Cement Mineral fiber (rock, slag or glass) With clay binder With hydraulic setting binder Loose Fill Cellulose, wood or paper pulp Perlite, expanded Vermiculite, expanded Maximum Service Temperature (K) 6:06 PM Description/ Composition 2/21/11 Industrial Insulation (Continued) BAPP01.qxd Continued 992 TABLE A.3 BAPP01.qxd 2/21/11 6:06 PM Page 993 Appendix A 䊏 TABLE A.3 993 Thermophysical Properties of Matter Continued Other Materials Description/ Composition Asphalt Bakelite Brick, refractory Carborundum Chrome brick Diatomaceous silica, fired Fireclay, burnt 1600 K Fireclay, burnt 1725 K Fireclay brick Magnesite Clay Coal, anthracite Concrete (stone mix) Cotton Foodstuffs Banana (75.7% water content) Apple, red (75% water content) Cake, batter Cake, fully baked Chicken meat, white (74.4% water content) Glass Plate (soda lime) Pyrex Temperature (K) Density, (kg/m3) Thermal Conductivity, k (W/m 䡠 K) Specific Heat, cp (J/kg 䡠 K) 300 300 2115 1300 0.062 1.4 920 1465 872 1672 473 823 1173 478 1145 773 1073 1373 773 1073 1373 478 922 1478 478 922 1478 300 300 300 300 — — 3010 1460 1350 2300 80 300 980 0.481 3350 300 300 300 198 233 253 263 273 283 293 840 720 280 — — 0.513 0.223 0.121 1.60 1.49 1.35 1.20 0.476 0.480 0.489 3600 — — — 300 300 2500 2225 — — 2050 — — 2325 2645 — — 18.5 11.0 2.3 2.5 2.0 0.25 0.30 1.0 1.1 1.1 1.3 1.4 1.4 1.0 1.5 1.8 3.8 2.8 1.9 1.3 0.26 1.4 0.06 1.4 1.4 — — 835 — 960 960 960 1130 880 1260 880 1300 750 835 BAPP01.qxd 2/21/11 994 6:06 PM Page 994 Appendix A TABLE A.3 䊏 Thermophysical Properties of Matter Continued Other Materials (Continued) Description/ Composition Temperature (K) Density, (kg/m3) Thermal Conductivity, k (W/m 䡠 K) Specific Heat, cp (J/kg 䡠 K) 273 253 300 300 300 920 — 998 930 900 1.88 2.03 0.159 0.180 0.240 2040 1945 — 1340 2890 300 300 300 300 300 2630 2320 2680 2640 2150 2.79 2.15 2.80 5.38 2.90 775 810 830 1105 745 300 300 300 300 273 1100 1190 1515 2050 110 500 2200 0.13 0.16 0.27 0.52 0.049 0.190 0.35 0.45 2010 — 800 1840 — — — — Ice Leather (sole) Paper Paraffin Rock Granite, Barre Limestone, Salem Marble, Halston Quartzite, Sioux Sandstone, Berea Rubber, vulcanized Soft Hard Sand Soil Snow Teflon Tissue, human Skin Fat layer (adipose) Muscle Wood, cross grain Balsa Cypress Fir Oak Yellow pine White pine Wood, radial Oak Fir a Adapted from References 1 and 8–13. 300 400 300 300 300 — — — 0.37 0.2 0.5 — — — 300 300 300 300 300 300 140 465 415 545 640 435 0.055 0.097 0.11 0.17 0.15 0.11 — — 2720 2385 2805 — 300 300 545 420 0.19 0.14 2385 2720 BAPP01.qxd 2/21/11 6:06 PM Page 995 Appendix A 䊏 995 Thermophysical Properties of Matter TABLE A.4 Thermophysical Properties of Gases at Atmospheric Pressurea T (K) (kg /m3) cp (kJ/kg 䡠 K) 䡠 107 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 106 (m2/s) Pr Air, ᏹ ⴝ 28.97 kg/kmol 100 150 200 250 300 3.5562 2.3364 1.7458 1.3947 1.1614 1.032 1.012 1.007 1.006 1.007 71.1 103.4 132.5 159.6 184.6 2.00 4.426 7.590 11.44 15.89 9.34 13.8 18.1 22.3 26.3 2.54 5.84 10.3 15.9 22.5 0.786 0.758 0.737 0.720 0.707 350 400 450 500 550 0.9950 0.8711 0.7740 0.6964 0.6329 1.009 1.014 1.021 1.030 1.040 208.2 230.1 250.7 270.1 288.4 20.92 26.41 32.39 38.79 45.57 30.0 33.8 37.3 40.7 43.9 29.9 38.3 47.2 56.7 66.7 0.700 0.690 0.686 0.684 0.683 600 650 700 750 800 0.5804 0.5356 0.4975 0.4643 0.4354 1.051 1.063 1.075 1.087 1.099 305.8 322.5 338.8 354.6 369.8 52.69 60.21 68.10 76.37 84.93 46.9 49.7 52.4 54.9 57.3 76.9 87.3 98.0 109 120 0.685 0.690 0.695 0.702 0.709 850 900 950 1000 1100 0.4097 0.3868 0.3666 0.3482 0.3166 1.110 1.121 1.131 1.141 1.159 384.3 398.1 411.3 424.4 449.0 93.80 102.9 112.2 121.9 141.8 59.6 62.0 64.3 66.7 71.5 131 143 155 168 195 0.716 0.720 0.723 0.726 0.728 1200 1300 1400 1500 1600 0.2902 0.2679 0.2488 0.2322 0.2177 1.175 1.189 1.207 1.230 1.248 473.0 496.0 530 557 584 162.9 185.1 213 240 268 76.3 82 91 100 106 224 257 303 350 390 0.728 0.719 0.703 0.685 0.688 1700 1800 1900 2000 2100 0.2049 0.1935 0.1833 0.1741 0.1658 1.267 1.286 1.307 1.337 1.372 611 637 663 689 715 298 329 362 396 431 113 120 128 137 147 435 482 534 589 646 0.685 0.683 0.677 0.672 0.667 2200 2300 2400 2500 3000 0.1582 0.1513 0.1448 0.1389 0.1135 1.417 1.478 1.558 1.665 2.726 740 766 792 818 955 468 506 547 589 841 160 175 196 222 486 714 783 869 960 1570 0.655 0.647 0.630 0.613 0.536 Ammonia (NH3), ᏹ ⴝ 17.03 kg/kmol 300 320 340 360 380 0.6894 0.6448 0.6059 0.5716 0.5410 2.158 2.170 2.192 2.221 2.254 101.5 109 116.5 124 131 14.7 16.9 19.2 21.7 24.2 24.7 27.2 29.3 31.6 34.0 16.6 19.4 22.1 24.9 27.9 0.887 0.870 0.872 0.872 0.869 BAPP01.qxd 2/21/11 996 6:06 PM Page 996 Appendix A TABLE A.4 T (K) Thermophysical Properties of Matter 䊏 Continued (kg /m3) cp (kJ/kg 䡠 K) 䡠 107 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 106 (m2/s) Pr Ammonia (NH3) (continued) 400 420 440 460 480 0.5136 0.4888 0.4664 0.4460 0.4273 2.287 2.322 2.357 2.393 2.430 138 145 152.5 159 166.5 26.9 29.7 32.7 35.7 39.0 37.0 40.4 43.5 46.3 49.2 31.5 35.6 39.6 43.4 47.4 0.853 0.833 0.826 0.822 0.822 500 520 540 560 580 0.4101 0.3942 0.3795 0.3708 0.3533 2.467 2.504 2.540 2.577 2.613 173 180 186.5 193 199.5 42.2 45.7 49.1 52.0 56.5 52.5 54.5 57.5 60.6 63.8 51.9 55.2 59.7 63.4 69.1 0.813 0.827 0.824 0.827 0.817 Carbon Dioxide (CO2), ᏹ ⴝ 44.01 kg/kmol 280 300 320 340 360 1.9022 1.7730 1.6609 1.5618 1.4743 0.830 0.851 0.872 0.891 0.908 140 149 156 165 173 7.36 8.40 9.39 10.6 11.7 15.20 16.55 18.05 19.70 21.2 9.63 11.0 12.5 14.2 15.8 0.765 0.766 0.754 0.746 0.741 380 400 450 500 550 1.3961 1.3257 1.1782 1.0594 0.9625 0.926 0.942 0.981 1.02 1.05 181 190 210 231 251 13.0 14.3 17.8 21.8 26.1 22.75 24.3 28.3 32.5 36.6 17.6 19.5 24.5 30.1 36.2 0.737 0.737 0.728 0.725 0.721 600 650 700 750 800 0.8826 0.8143 0.7564 0.7057 0.6614 1.08 1.10 1.13 1.15 1.17 270 288 305 321 337 30.6 35.4 40.3 45.5 51.0 40.7 44.5 48.1 51.7 55.1 42.7 49.7 56.3 63.7 71.2 0.717 0.712 0.717 0.714 0.716 Carbon Monoxide (CO), ᏹ ⴝ 28.01 kg/kmol 200 220 240 260 280 1.6888 1.5341 1.4055 1.2967 1.2038 1.045 1.044 1.043 1.043 1.042 127 137 147 157 166 7.52 8.93 10.5 12.1 13.8 17.0 19.0 20.6 22.1 23.6 9.63 11.9 14.1 16.3 18.8 0.781 0.753 0.744 0.741 0.733 300 320 340 360 380 1.1233 1.0529 0.9909 0.9357 0.8864 1.043 1.043 1.044 1.045 1.047 175 184 193 202 210 15.6 17.5 19.5 21.6 23.7 25.0 26.3 27.8 29.1 30.5 21.3 23.9 26.9 29.8 32.9 0.730 0.730 0.725 0.725 0.729 400 450 500 550 600 0.8421 0.7483 0.67352 0.61226 0.56126 1.049 1.055 1.065 1.076 1.088 218 237 254 271 286 25.9 31.7 37.7 44.3 51.0 31.8 35.0 38.1 41.1 44.0 36.0 44.3 53.1 62.4 72.1 0.719 0.714 0.710 0.710 0.707 BAPP01.qxd 2/21/11 6:06 PM Page 997 Appendix A TABLE A.4 T (K) 997 Thermophysical Properties of Matter 䊏 Continued (kg /m3) cp (kJ/kg 䡠 K) 䡠 107 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 106 (m2/s) Pr 301 315 329 343 58.1 65.5 73.3 81.5 47.0 50.0 52.8 55.5 82.4 93.3 104 116 0.705 0.702 0.702 0.705 28.9 38.8 50.2 — 76.2 0.686 0.679 0.676 — 0.673 Carbon Monoxide (CO) (continued) 650 700 750 800 0.51806 0.48102 0.44899 0.42095 1.101 1.114 1.127 1.140 Helium (He), ᏹ ⴝ 4.003 kg/kmol 100 120 140 160 180 0.4871 0.4060 0.3481 — 0.2708 5.193 5.193 5.193 5.193 5.193 96.3 107 118 129 139 19.8 26.4 33.9 — 51.3 73.0 81.9 90.7 99.2 107.2 200 220 240 260 280 — 0.2216 — 0.1875 — 5.193 5.193 5.193 5.193 5.193 150 160 170 180 190 — 72.2 — 96.0 — 115.1 123.1 130 137 145 — 107 — 141 — — 0.675 — 0.682 — 300 350 400 450 500 0.1625 — 0.1219 — 0.09754 5.193 5.193 5.193 5.193 5.193 199 221 243 263 283 122 — 199 — 290 152 170 187 204 220 180 — 295 — 434 0.680 — 0.675 — 0.668 550 600 650 700 750 — — — 0.06969 — 5.193 5.193 5.193 5.193 5.193 — 320 332 350 364 — — — 502 — — 252 264 278 291 — — — 768 — — — — 0.654 — 800 900 1000 — — 0.04879 5.193 5.193 5.193 382 414 446 — — 914 304 330 354 — — 1400 — — 0.654 17.4 34.7 56.2 81.4 111 67.0 101 131 157 183 24.6 49.6 79.9 115 158 0.707 0.699 0.704 0.707 0.701 143 179 218 261 305 204 226 247 266 285 204 258 316 378 445 0.700 0.695 0.689 0.691 0.685 Hydrogen (H2 ), ᏹ ⴝ 2.016 kg/kmol 100 150 200 250 300 0.24255 0.16156 0.12115 0.09693 0.08078 11.23 12.60 13.54 14.06 14.31 42.1 56.0 68.1 78.9 89.6 350 400 450 500 550 0.06924 0.06059 0.05386 0.04848 0.04407 14.43 14.48 14.50 14.52 14.53 98.8 108.2 117.2 126.4 134.3 BAPP01.qxd 2/21/11 998 6:06 PM Page 998 Appendix A TABLE A.4 T (K) Thermophysical Properties of Matter 䊏 Continued (kg /m3) cp (kJ/kg 䡠 K) 䡠 107 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 106 (m2/s) Pr Hydrogen (H2 ) (continued) 600 700 800 900 1000 0.04040 0.03463 0.03030 0.02694 0.02424 14.55 14.61 14.70 14.83 14.99 142.4 157.8 172.4 186.5 201.3 352 456 569 692 830 305 342 378 412 448 519 676 849 1030 1230 0.678 0.675 0.670 0.671 0.673 1100 1200 1300 1400 1500 0.02204 0.02020 0.01865 0.01732 0.01616 15.17 15.37 15.59 15.81 16.02 213.0 226.2 238.5 250.7 262.7 966 1120 1279 1447 1626 488 528 568 610 655 1460 1700 1955 2230 2530 0.662 0.659 0.655 0.650 0.643 1600 1700 1800 1900 2000 0.0152 0.0143 0.0135 0.0128 0.0121 16.28 16.58 16.96 17.49 18.25 273.7 284.9 296.1 307.2 318.2 1801 1992 2193 2400 2630 697 742 786 835 878 2815 3130 3435 3730 3975 0.639 0.637 0.639 0.643 0.661 Nitrogen (N2 ), ᏹ ⴝ 28.01 kg/kmol 100 150 200 250 300 3.4388 2.2594 1.6883 1.3488 1.1233 1.070 1.050 1.043 1.042 1.041 68.8 100.6 129.2 154.9 178.2 2.00 4.45 7.65 11.48 15.86 9.58 13.9 18.3 22.2 25.9 2.60 5.86 10.4 15.8 22.1 0.768 0.759 0.736 0.727 0.716 350 400 450 500 550 0.9625 0.8425 0.7485 0.6739 0.6124 1.042 1.045 1.050 1.056 1.065 200.0 220.4 239.6 257.7 274.7 20.78 26.16 32.01 38.24 44.86 29.3 32.7 35.8 38.9 41.7 29.2 37.1 45.6 54.7 63.9 0.711 0.704 0.703 0.700 0.702 600 700 800 900 1000 0.5615 0.4812 0.4211 0.3743 0.3368 1.075 1.098 1.122 1.146 1.167 290.8 321.0 349.1 375.3 399.9 51.79 66.71 82.90 100.3 118.7 44.6 49.9 54.8 59.7 64.7 73.9 94.4 116 139 165 0.701 0.706 0.715 0.721 0.721 1100 1200 1300 0.3062 0.2807 0.2591 1.187 1.204 1.219 423.2 445.3 466.2 138.2 158.6 179.9 70.0 75.8 81.0 193 224 256 0.718 0.707 0.701 Oxygen (O2 ), ᏹ ⴝ 32.00 kg/kmol 100 150 200 250 300 3.945 2.585 1.930 1.542 1.284 0.962 0.921 0.915 0.915 0.920 76.4 114.8 147.5 178.6 207.2 1.94 4.44 7.64 11.58 16.14 9.25 13.8 18.3 22.6 26.8 2.44 5.80 10.4 16.0 22.7 0.796 0.766 0.737 0.723 0.711 BAPP01.qxd 2/21/11 6:06 PM Page 999 Appendix A TABLE A.4 T (K) 999 Thermophysical Properties of Matter 䊏 Continued (kg /m3) cp (kJ/kg 䡠 K) 䡠 107 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 106 (m2/s) Pr Oxygen (O2 ) (continued) 350 400 450 500 550 1.100 0.9620 0.8554 0.7698 0.6998 0.929 0.942 0.956 0.972 0.988 233.5 258.2 281.4 303.3 324.0 21.23 26.84 32.90 39.40 46.30 29.6 33.0 36.3 41.2 44.1 29.0 36.4 44.4 55.1 63.8 0.733 0.737 0.741 0.716 0.726 600 700 800 900 1000 0.6414 0.5498 0.4810 0.4275 0.3848 1.003 1.031 1.054 1.074 1.090 343.7 380.8 415.2 447.2 477.0 53.59 69.26 86.32 104.6 124.0 47.3 52.8 58.9 64.9 71.0 73.5 93.1 116 141 169 0.729 0.744 0.743 0.740 0.733 1100 1200 1300 0.3498 0.3206 0.2960 1.103 1.115 1.125 505.5 532.5 588.4 144.5 166.1 188.6 75.8 81.9 87.1 196 229 262 0.736 0.725 0.721 Water Vapor (Steam), ᏹ ⴝ 18.02 kg/kmol a 380 400 450 500 550 0.5863 0.5542 0.4902 0.4405 0.4005 2.060 2.014 1.980 1.985 1.997 127.1 134.4 152.5 170.4 188.4 21.68 24.25 31.11 38.68 47.04 24.6 26.1 29.9 33.9 37.9 20.4 23.4 30.8 38.8 47.4 1.06 1.04 1.01 0.998 0.993 600 650 700 750 800 850 0.3652 0.3380 0.3140 0.2931 0.2739 0.2579 2.026 2.056 2.085 2.119 2.152 2.186 206.7 224.7 242.6 260.4 278.6 296.9 56.60 66.48 77.26 88.84 101.7 115.1 42.2 46.4 50.5 54.9 59.2 63.7 57.0 66.8 77.1 88.4 100 113 0.993 0.996 1.00 1.00 1.01 1.02 Adapted from References 8, 14, and 15. BAPP01.qxd 2/21/11 6:06 PM 1000 Page 1000 Appendix A TABLE A.5 䊏 Thermophysical Properties of Matter Thermophysical Properties of Saturated Fluidsa Saturated Liquids T (K) (kg/m3) cp (kJ/kg 䡠 K) 䡠 102 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 107 (m2/s) Pr  䡠 103 (Kⴚ1) Engine Oil (Unused) 273 280 290 300 310 320 330 340 899.1 895.3 890.0 884.1 877.9 871.8 865.8 859.9 1.796 1.827 1.868 1.909 1.951 1.993 2.035 2.076 385 217 99.9 48.6 25.3 14.1 8.36 5.31 4280 2430 1120 550 288 161 96.6 61.7 147 144 145 145 145 143 141 139 0.910 0.880 0.872 0.859 0.847 0.823 0.800 0.779 47,000 27,500 12,900 6400 3400 1965 1205 793 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 350 360 370 380 390 853.9 847.8 841.8 836.0 830.6 2.118 2.161 2.206 2.250 2.294 3.56 2.52 1.86 1.41 1.10 41.7 29.7 22.0 16.9 13.3 138 138 137 136 135 0.763 0.753 0.738 0.723 0.709 546 395 300 233 187 0.70 0.70 0.70 0.70 0.70 400 410 420 430 825.1 818.9 812.1 806.5 2.337 2.381 2.427 2.471 0.874 0.698 0.564 0.470 10.6 8.52 6.94 5.83 134 133 133 132 0.695 0.682 0.675 0.662 152 125 103 88 0.70 0.70 0.70 0.70 Ethylene Glycol [C 2H4(OH)2] 273 280 290 1130.8 1125.8 1118.8 2.294 2.323 2.368 6.51 4.20 2.47 57.6 37.3 22.1 242 244 248 0.933 0.933 0.936 617 400 236 0.65 0.65 0.65 300 310 320 330 340 1114.4 1103.7 1096.2 1089.5 1083.8 2.415 2.460 2.505 2.549 2.592 1.57 1.07 0.757 0.561 0.431 14.1 9.65 6.91 5.15 3.98 252 255 258 260 261 0.939 0.939 0.940 0.936 0.929 151 103 73.5 55.0 42.8 0.65 0.65 0.65 0.65 0.65 350 360 370 373 1079.0 1074.0 1066.7 1058.5 2.637 2.682 2.728 2.742 0.342 0.278 0.228 0.215 3.17 2.59 2.14 2.03 261 261 262 263 0.917 0.906 0.900 0.906 34.6 28.6 23.7 22.4 0.65 0.65 0.65 0.65 282 284 286 286 286 287 0.977 0.972 0.955 0.935 0.916 0.897 Glycerin [C3H5(OH)3] 273 280 290 300 310 320 1276.0 1271.9 1265.8 1259.9 1253.9 1247.2 2.261 2.298 2.367 2.427 2.490 2.564 1060 534 185 79.9 35.2 21.0 8310 4200 1460 634 281 168 85,000 43,200 15,300 6780 3060 1870 0.47 0.47 0.48 0.48 0.49 0.50 BAPP01.qxd 2/21/11 6:06 PM Page 1001 Appendix A TABLE A.5 䊏 1001 Thermophysical Properties of Matter Continued Saturated Liquids (Continued) T (K) (kg/m3) cp (kJ/kg 䡠 K) 䡠 102 (N 䡠 s/m2) 䡠 106 (m2/s) k 䡠 103 (W/m 䡠 K) ␣ 䡠 107 (m2/s) Pr  䡠 103 (Kⴚ1) Refrigerant-134a (C2H2F4) 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 1426.8 1397.7 1367.9 1337.1 1305.1 1271.8 1236.8 1199.7 1159.9 1116.8 1069.1 1015.0 951.3 870.1 740.3 1.249 1.267 1.287 1.308 1.333 1.361 1.393 1.432 1.481 1.543 1.627 1.751 1.961 2.437 5.105 0.04912 0.04202 0.03633 0.03166 0.02775 0.02443 0.02156 0.01905 0.01680 0.01478 0.01292 0.01118 0.00951 0.00781 0.00580 0.3443 0.3006 0.2656 0.2368 0.2127 0.1921 0.1744 0.1588 0.1449 0.1323 0.1209 0.1102 0.1000 0.0898 0.0783 112.1 107.3 102.5 97.9 93.4 89.0 84.6 80.3 76.1 71.8 67.5 63.1 58.6 54.1 51.8 0.629 0.606 0.583 0.560 0.537 0.514 0.491 0.468 0.443 0.417 0.388 0.355 0.314 0.255 0.137 5.5 5.0 4.6 4.2 4.0 3.7 3.5 3.4 3.3 3.2 3.1 3.1 3.2 3.5 5.7 2.02 2.11 2.23 2.36 2.53 2.73 2.98 3.30 3.73 4.33 5.19 6.57 9.10 15.39 55.24 1.087 1.100 1.117 1.137 1.161 1.189 1.223 1.265 1.319 1.391 1.495 1.665 1.997 3.001 0.03558 0.03145 0.02796 0.02497 0.02235 0.02005 0.01798 0.01610 0.01438 0.01278 0.01127 0.00980 0.00831 0.00668 0.2513 0.2268 0.2062 0.1884 0.1730 0.1594 0.1472 0.1361 0.1259 0.1165 0.1075 0.0989 0.0904 0.0811 114.5 109.8 105.2 100.7 96.2 91.7 87.2 82.6 78.1 73.4 68.6 63.6 58.3 53.1 0.744 0.720 0.695 0.668 0.641 0.613 0.583 0.552 0.518 0.481 0.438 0.386 0.317 0.215 3.4 3.2 3.0 2.8 2.7 2.6 2.5 2.5 2.4 2.4 2.5 2.6 2.8 3.8 2.05 2.16 2.29 2.45 2.63 2.86 3.15 3.51 4.00 4.69 5.75 7.56 11.35 23.88 0.1404 0.1393 0.1377 0.1365 0.1357 0.1353 0.1352 0.1355 0.1688 0.1523 0.1309 0.1171 0.1075 0.1007 0.0953 0.0911 0.1240 0.1125 0.0976 0.0882 0.0816 0.0771 0.0737 0.0711 Refrigerant-22 (CHClF2) 230 240 250 260 270 280 290 300 310 320 330 340 350 360 1416.0 1386.6 1356.3 1324.9 1292.1 1257.9 1221.7 1183.4 1142.2 1097.4 1047.5 990.1 920.1 823.4 Mercury (Hg) 273 300 350 400 450 500 550 600 13,595 13,529 13,407 13,287 13,167 13,048 12,929 12,809 8180 8540 9180 9800 10,400 10,950 11,450 11,950 42.85 45.30 49.75 54.05 58.10 61.90 65.55 68.80 0.0290 0.0248 0.0196 0.0163 0.0140 0.0125 0.0112 0.0103 0.181 0.181 0.181 0.181 0.181 0.182 0.184 0.187 BAPP01.qxd 2/21/11 6:06 PM 1002 TABLE A.5 Page 1002 Appendix A 䊏 Thermophysical Properties of Matter Continued Saturated Liquid–Vapor, 1 atmb Fluid Tsat (K) hƒ g (kJ/kg) ƒ (kg/m3) g (kg/m3) 䡠 103 (N/m) Ethanol Ethylene glycol Glycerin Mercury Refrigerant R-134a Refrigerant R-22 351 470 563 630 247 232 846 812 974 301 217 234 757 1111c 1260c 12,740 1377 1409 1.44 — — 3.90 5.26 4.70 17.7 32.7 63.0c 417 15.4 18.1 a Adapted from References 15–19. Adapted from References 8, 20, and 21. c Property value corresponding to 300 K. b BAPP01.qxd Thermophysical Properties of Saturated Watera Pressure, p (bars)b Specific Volume (m3/kg) vƒ 䡠 10 3 vg Specific Heat (kJ/kg 䡠 K) Thermal Conductivity (W/m 䡠 K) 273.15 275 280 285 290 cp, g ƒ 䡠 106 g 䡠 106 kƒ 䡠 103 kg 䡠 103 Prƒ Prg 2502 2497 2485 2473 2461 4.217 4.211 4.198 4.189 4.184 1.854 1.855 1.858 1.861 1.864 1750 1652 1422 1225 1080 8.02 8.09 8.29 8.49 8.69 569 574 582 590 598 18.2 18.3 18.6 18.9 19.3 12.99 12.22 10.26 8.81 7.56 0.815 0.817 0.825 0.833 0.841 75.5 75.3 74.8 74.3 73.7 ⫺68.05 ⫺32.74 46.04 114.1 174.0 1.000 1.000 1.000 1.000 1.001 295 300 305 310 315 0.02617 0.03531 0.04712 0.06221 0.08132 1.002 1.003 1.005 1.007 1.009 51.94 39.13 29.74 22.93 17.82 2449 2438 2426 2414 2402 4.181 4.179 4.178 4.178 4.179 1.868 1.872 1.877 1.882 1.888 959 855 769 695 631 8.89 9.09 9.29 9.49 9.69 606 613 620 628 634 19.5 19.6 20.1 20.4 20.7 6.62 5.83 5.20 4.62 4.16 0.849 0.857 0.865 0.873 0.883 72.7 71.7 70.9 70.0 69.2 227.5 276.1 320.6 361.9 400.4 295 300 305 310 315 320 325 330 335 340 0.1053 0.1351 0.1719 0.2167 0.2713 1.011 1.013 1.016 1.018 1.021 13.98 11.06 8.82 7.09 5.74 2390 2378 2366 2354 2342 4.180 4.182 4.184 4.186 4.188 1.895 1.903 1.911 1.920 1.930 577 528 489 453 420 9.89 10.09 10.29 10.49 10.69 640 645 650 656 660 21.0 21.3 21.7 22.0 22.3 3.77 3.42 3.15 2.88 2.66 0.894 0.901 0.908 0.916 0.925 68.3 67.5 66.6 65.8 64.9 436.7 471.2 504.0 535.5 566.0 320 325 330 335 340 345 350 355 360 365 0.3372 0.4163 0.5100 0.6209 0.7514 1.024 1.027 1.030 1.034 1.038 4.683 3.846 3.180 2.645 2.212 2329 2317 2304 2291 2278 4.191 4.195 4.199 4.203 4.209 1.941 1.954 1.968 1.983 1.999 389 365 343 324 306 10.89 11.09 11.29 11.49 11.69 664 668 671 674 677 22.6 23.0 23.3 23.7 24.1 2.45 2.29 2.14 2.02 1.91 0.933 0.942 0.951 0.960 0.969 64.1 63.2 62.3 61.4 60.5 595.4 624.2 652.3 697.9 707.1 345 350 355 360 365 370 373.15 375 380 385 0.9040 1.0133 1.0815 1.2869 1.5233 1.041 1.044 1.045 1.049 1.053 1.861 1.679 1.574 1.337 1.142 2265 2257 2252 2239 2225 4.214 4.217 4.220 4.226 4.232 2.017 2.029 2.036 2.057 2.080 289 279 274 260 248 11.89 12.02 12.09 12.29 12.49 679 680 681 683 685 24.5 24.8 24.9 25.4 25.8 1.80 1.76 1.70 1.61 1.53 0.978 0.984 0.987 0.999 1.004 59.5 58.9 58.6 57.6 56.6 728.7 750.1 761 788 814 370 373.15 375 380 385 390 400 410 420 430 1.794 2.455 3.302 4.370 5.699 1.058 1.067 1.077 1.088 1.099 0.980 0.731 0.553 0.425 0.331 2212 2183 2153 2123 2091 4.239 4.256 4.278 4.302 4.331 2.104 2.158 2.221 2.291 2.369 237 217 200 185 173 12.69 13.05 13.42 13.79 14.14 686 688 688 688 685 26.3 27.2 28.2 29.8 30.4 1.47 1.34 1.24 1.16 1.09 1.013 1.033 1.054 1.075 1.10 55.6 53.6 51.5 49.4 47.2 841 896 952 1010 390 400 410 420 430 1003 0.00611 0.00697 0.00990 0.01387 0.01917 Thermophysical Properties of Matter 273.15 275 280 285 290 Page 1003 Temperature, T (K) cp,ƒ Prandtl Number 䊏 Expansion Coefficient, ƒ 䡠 106 (Kⴚ1) Surface Tension, ƒ 䡠 103 (N/m) Viscosity (N 䡠 s/m2) Appendix A 206.3 181.7 130.4 99.4 69.7 Heat of Vaporization, hƒ g (kJ/kg) 6:06 PM Temperature, T (K) 2/21/11 TABLE A.6 0.261 0.208 0.167 0.136 0.111 2059 2024 1989 1951 1912 4.36 4.40 4.44 4.48 4.53 2.46 2.56 2.68 2.79 2.94 162 152 143 136 129 14.50 14.85 15.19 15.54 15.88 682 678 673 667 660 490 500 510 520 530 21.83 26.40 31.66 37.70 44.58 1.184 1.203 1.222 1.244 1.268 0.0922 0.0766 0.0631 0.0525 0.0445 1870 1825 1779 1730 1679 4.59 4.66 4.74 4.84 4.95 3.10 3.27 3.47 3.70 3.96 124 118 113 108 104 16.23 16.59 16.95 17.33 17.72 540 550 560 570 580 52.38 61.19 71.08 82.16 94.51 1.294 1.323 1.355 1.392 1.433 0.0375 0.0317 0.0269 0.0228 0.0193 1622 1564 1499 1429 1353 5.08 5.24 5.43 5.68 6.00 4.27 4.64 5.09 5.67 6.40 101 97 94 91 88 31.7 33.1 34.6 36.3 38.1 1.04 0.99 0.95 0.92 0.89 1.12 1.14 1.17 1.20 1.23 45.1 42.9 40.7 38.5 36.2 651 642 631 621 608 40.1 42.3 44.7 47.5 50.6 0.87 0.86 0.85 0.84 0.85 1.25 1.28 1.31 1.35 1.39 33.9 31.6 29.3 26.9 24.5 — — — — — 490 500 510 520 530 18.1 18.6 19.1 19.7 20.4 594 580 563 548 528 54.0 58.3 63.7 76.7 76.7 0.86 0.87 0.90 0.94 0.99 1.43 1.47 1.52 1.59 1.68 22.1 19.7 17.3 15.0 12.8 — — — — — 540 550 560 570 580 1.05 1.14 1.30 1.52 1.65 1.84 2.15 2.60 3.46 4.20 10.5 8.4 6.3 4.5 3.5 — — — — — 590 600 610 620 625 2.0 4.8 2.7 6.0 4.2 9.6 12 26 앝 앝 2.6 1.5 0.8 0.1 0.0 — — — — — 630 635 640 645 647.3c g 䡠 106 kƒ 䡠 103 kg 䡠 103 590 600 610 620 625 108.3 123.5 137.3 159.1 169.1 1.482 1.541 1.612 1.705 1.778 0.0163 0.0137 0.0115 0.0094 0.0085 1274 1176 1068 941 858 6.41 7.00 7.85 9.35 10.6 7.35 8.75 11.1 15.4 18.3 84 81 77 72 70 21.5 22.7 24.1 25.9 27.0 513 497 467 444 430 84.1 92.9 103 114 121 630 635 640 645 647.3c 179.7 190.9 202.7 215.2 221.2 1.856 1.935 2.075 2.351 3.170 0.0075 0.0066 0.0057 0.0045 0.0032 781 683 560 361 0 12.6 16.4 26 90 앝 22.1 27.6 42 — 앝 67 64 59 54 45 28.0 30.0 32.0 37.0 45.0 412 392 367 331 238 130 141 155 178 238 a Adapted from Reference 22. 1 bar ⫽ 105 N/m2. c Critical temperature. b 440 450 460 470 480 Page 1004 1.110 1.123 1.137 1.152 1.167 ƒ 䡠 106 6:06 PM 7.333 9.319 11.71 14.55 17.90 Prg cp,g Temperature, T (K) 2/21/11 440 450 460 470 480 Expansion Coefficient, ƒ 䡠 106 (Kⴚ1) Prƒ cp,ƒ Prandtl Number Thermophysical Properties of Matter vg Thermal Conductivity (W/m 䡠 K) Surface Tension, ƒ 䡠 103 (N/m) Viscosity (N 䡠 s/m2) 䊏 vƒ 䡠 103 Temperature, T (K) Specific Heat (kJ/kg 䡠 K) Appendix A Pressure, p (bars)b Heat of Vaporization, hƒ g (kJ/kg) Specific Volume (m3/ kg) BAPP01.qxd Continued 1004 TABLE A.6 BAPP01.qxd 2/21/11 6:06 PM Page 1005 Appendix A TABLE A.7 Composition 1005 Thermophysical Properties of Matter Thermophysical Properties of Liquid Metalsa Melting Point (K) Bismuth 544 Lead 600 Potassium 337 Sodium 371 NaK, (45%/55%) 292 NaK, (22%/78%) 262 PbBi, (44.5%/55.5%) 398 Mercury 234 a 䊏 Adapted from Reference 23. T (K) (kg/m3) 589 811 1033 644 755 977 422 700 977 366 644 977 366 644 977 366 672 1033 422 644 922 10,011 9739 9467 10,540 10,412 10,140 807.3 741.7 674.4 929.1 860.2 778.5 887.4 821.7 740.1 849.0 775.3 690.4 10,524 10,236 9835 cp (kJ/kg 䡠 K) 0.1444 0.1545 0.1645 0.159 0.155 — 0.80 0.75 0.75 1.38 1.30 1.26 1.130 1.055 1.043 0.946 0.879 0.883 0.147 0.147 — See Table A.5 䡠 107 (m2/s) k (W/m 䡠 K) ␣ 䡠 105 (m2/s) Pr 1.617 1.133 0.8343 2.276 1.849 1.347 4.608 2.397 1.905 7.516 3.270 2.285 6.522 2.871 2.174 5.797 2.666 2.118 — 1.496 1.171 16.4 15.6 15.6 16.1 15.6 14.9 45.0 39.5 33.1 86.2 72.3 59.7 25.6 27.5 28.9 24.4 26.7 — 9.05 11.86 — 1.138 1.035 1.001 1.084 1.223 — 6.99 7.07 6.55 6.71 6.48 6.12 2.552 3.17 3.74 3.05 3.92 — 0.586 0.790 — 0.0142 0.0110 0.0083 0.024 0.017 — 0.0066 0.0034 0.0029 0.011 0.0051 0.0037 0.026 0.0091 0.0058 0.019 0.0068 — — 0.189 — BAPP01.qxd 2/21/11 1006 6:06 PM Page 1006 Appendix A 䊏 TABLE A.8 Substance A Thermophysical Properties of Matter Binary Diffusion Coefficients at One Atmospherea,b Substance B T (K) DAB (m2/s) Gases NH3 H2O CO2 H2 O2 Acetone Benzene Naphthalene Ar H2 H2 H2 CO2 CO2 O2 Air Air Air Air Air Air Air Air N2 O2 N2 CO2 N2 O2 N2 298 298 298 298 298 273 298 300 293 273 273 273 293 273 273 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 Dilute Solutions Caffeine Ethanol Glucose Glycerol Acetone CO2 O2 H2 N2 H2O H2O H2O H2O H2O H2O H2O H2O H2O 298 298 298 298 298 298 298 298 298 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 Solids O2 N2 CO2 He H2 Cd Al Rubber Rubber Rubber SiO2 Fe Cu Cu 298 298 298 293 293 293 293 0.21 ⫻ 10⫺9 0.15 ⫻ 10⫺9 0.11 ⫻ 10⫺9 0.4 ⫻ 10⫺13 0.26 ⫻ 10⫺12 0.27 ⫻ 10⫺18 0.13 ⫻ 10⫺33 a Adapted with permission from References 24, 25, and 26. Assuming ideal gas behavior, the pressure and temperature dependence of the diffusion coefficient for a binary mixture of gases may be estimated from the relation b DAB ⬀ p⫺1T 3/2 BAPP01.qxd 2/21/11 6:06 PM Page 1007 Appendix A TABLE A.9 䊏 1007 Thermophysical Properties of Matter Henry’s Constant for Selected Gases in Water at Moderate Pressurea H ⴝ pA,i /xA,i (bars) T (K) NH3 Cl2 H2S SO2 CO2 CH4 O2 H2 273 280 290 300 310 320 323 21 23 26 30 — — — 265 365 480 615 755 860 890 260 335 450 570 700 835 870 165 210 315 440 600 800 850 710 960 1300 1730 2175 2650 2870 22,880 27,800 35,200 42,800 50,000 56,300 58,000 25,500 30,500 37,600 45,700 52,500 56,800 58,000 58,000 61,500 66,500 71,600 76,000 78,600 79,000 a Adapted with permission from Reference 27. TABLE A.10 The Solubility of Selected Gases and Solidsa Gas Solid T (K) S ⴝ CA, i /pA, i (kmol/m3 䡠 bar) O2 N2 CO2 He H2 Rubber Rubber Rubber SiO2 Ni 298 298 298 293 358 3.12 ⫻ 10⫺3 1.56 ⫻ 10⫺3 40.15 ⫻ 10⫺3 0.45 ⫻ 10⫺3 9.01 ⫻ 10⫺3 a Adapted with permission from Reference 26. Emissivity, n or h, at Various Temperatures (K) 400 600 800 1000 (h) (h) (h) 0.02 0.06 0.03 0.06 0.04 0.07 0.82 0.05 0.06 (n) 0.05 0.07 0.10 0.12 0.14 0.03 0.03 0.04 0.50 0.04 0.58 0.04 0.80 0.03 0.07 0.03 0.04 0.05 0.06 (h) (h) (h) 0.06 0.25 0.80 0.08 0.28 0.82 (h) (h) 0.09 0.40 (h) (h) 0.01 0.06 0.02 0.07 1500 2000 2500 0.10 0.31 0.12 0.35 0.15 0.42 0.21 0.26 0.11 0.49 0.14 0.57 0.17 0.10 0.13 0.15 0.76 (h) 䊏 (h) (h) 1200 (h) 0.02 0.02 0.03 0.05 0.08 (n) (n) (n) (n) (n) 0.17 0.22 0.17 0.22 0.19 0.24 0.23 0.28 0.33 0.67 0.88 0.30 0.35 0.40 0.70 0.89 0.87 (h) (h) 0.10 0.18 0.76 0.90 0.11 0.17 0.23 0.28 0.13 0.18 0.25 0.29 Page 1008 300 Thermophysical Properties of Matter 200 Appendix A Aluminum Highly polished, film Foil, bright Anodized Chromium Polished or plated Copper Highly polished Stably oxidized Gold Highly polished or film Foil, bright Molybdenum Polished Shot-blasted, rough Stably oxidized Nickel Polished Stably oxidized Platinum Polished Silver Polished Stainless steels Typical, polished Typical, cleaned Typical, lightly oxidized Typical, highly oxidized AISI 347, stably oxidized Tantalum Polished Tungsten Polished 100 6:06 PM Description /Composition 2/21/11 Metallic Solids and Their Oxidesa BAPP01.qxd 1008 TABLE A.11 Total, Normal (n) or Hemispherical (h) Emissivity of Selected Surfaces BAPP01.qxd 2/21/11 6:06 PM Page 1009 Appendix A 䊏 1009 Thermophysical Properties of Matter TABLE A.11 Continued Nonmetallic Substancesb Description/Composition Temperature (K) Emissivity 0.69 0.55 0.41 0.85–0.93 Aluminum oxide (n) Asphalt pavement Building materials Asbestos sheet Brick, red Gypsum or plaster board Wood Cloth Concrete Glass, window Ice Paints Black (Parsons) White, acrylic White, zinc oxide Paper, white Pyrex (h) 600 1000 1500 300 (h) (h) (h) (h) (h) (h) (h) (h) 300 300 300 300 300 300 300 273 0.93–0.96 0.93–0.96 0.90–0.92 0.82–0.92 0.75–0.90 0.88–0.93 0.90–0.95 0.95–0.98 (h) (h) (h) (h) (n) Pyroceram (n) 300 300 300 300 300 600 1000 1200 300 600 1000 1500 0.98 0.90 0.92 0.92–0.97 0.82 0.80 0.71 0.62 0.85 0.78 0.69 0.57 Refractories (furnace liners) Alumina brick (n) 800 1000 1400 1600 800 1000 1400 1600 800 1200 1400 1600 300 600 1000 1500 300 273 0.40 0.33 0.28 0.33 0.45 0.36 0.31 0.40 0.70 0.57 0.47 0.53 0.90 0.87 0.87 0.85 0.95 0.82–0.90 Magnesia brick (n) Kaolin insulating brick (n) Sand Silicon carbide (h) (n) Skin Snow (h) (h) BAPP01.qxd 2/21/11 1010 6:06 PM Page 1010 Appendix A 䊏 Thermophysical Properties of Matter TABLE A.11 Continued Nonmetallic Substancesb Description/Composition Soil Rocks Teflon (h) (h) (h) Vegetation Water (h) (h) Temperature (K) Emissivity 300 300 300 400 500 300 300 0.93–0.96 0.88–0.95 0.85 0.87 0.92 0.92–0.96 0.96 a Adapted from Reference 1. Adapted from References 1, 9, 28, and 29. b TABLE A.12 Solar Radiative Properties for Selected Materialsa Description/Composition Aluminum Polished Anodized Quartz overcoated Foil Brick, red (Purdue) Concrete Galvanized sheet metal Clean, new Oxidized, weathered Glass, 3.2-mm thickness Float or tempered Low iron oxide type Metal, plated Black sulfide Black cobalt oxide Black nickel oxide Black chrome Mylar, 0.13-mm thickness Paints Black (Parsons) White, acrylic White, zinc oxide Plexiglas, 3.2-mm thickness Snow Fine particles, fresh Ice granules Tedlar, 0.10-mm thickness Teflon, 0.13-mm thickness a ␣S b ␣S/ 0.09 0.14 0.11 0.15 0.63 0.60 0.03 0.84 0.37 0.05 0.93 0.88 3.0 0.17 0.30 3.0 0.68 0.68 0.65 0.80 0.13 0.28 5.0 2.9 0.79 0.88 0.92 0.93 0.92 0.87 0.10 0.30 0.08 0.09 9.2 3.1 11 9.7 0.87 0.98 0.26 0.16 0.98 0.90 0.93 1.0 0.29 0.17 0.90 0.13 0.33 0.82 0.89 0.16 0.37 Adapted with permission from Reference 29. The emissivity values in this table correspond to a surface temperature of approximately 300 K. b S 0.92 0.92 BAPP01.qxd 2/21/11 6:06 PM Page 1011 Appendix A 䊏 Thermophysical Properties of Matter 1011 References 1. Touloukian, Y. S., and C. Y. Ho, Eds., Thermophysical Properties of Matter, Vol. 1, Thermal Conductivity of Metallic Solids; Vol. 2, Thermal Conductivity of Nonmetallic Solids; Vol. 4, Specific Heat of Metallic Solids; Vol. 5, Specific Heat of Nonmetallic Solids; Vol. 7, Thermal Radiative Properties of Metallic Solids; Vol. 8, Thermal Radiative Properties of Nonmetallic Solids; Vol. 9, Thermal Radiative Properties of Coatings, Plenum Press, New York, 1972. 2. Touloukian, Y. S., and C. Y. Ho, Eds., Thermophysical Properties of Selected Aerospace Materials, Part I: Thermal Radiative Properties; Part II: Thermophysical Properties of Seven Materials. Thermophysical and Electronic Properties Information Analysis Center, CINDAS, Purdue University, West Lafayette, IN, 1976. 3. Ho, C. Y., R. W. Powell, and P. E. Liley, J. Phys. Chem. Ref. Data, 3, Supplement 1, 1974. 4. Desai, P. D., T. K. Chu, R. H. Bogaard, M. W. Ackermann, and C. Y. Ho, Part I: Thermophysical Properties of Carbon Steels; Part II: Thermophysical Properties of Low Chromium Steels; Part III: Thermophysical Properties of Nickel Steels; Part IV: Thermophysical Properties of Stainless Steels. CINDAS Special Report, Purdue University, West Lafayette, IN, September 1976. 5. American Society for Metals, Metals Handbook, Vol. 1, Properties and Selection of Metals, 8th ed., ASM, Metals Park, OH, 1961. 6. Hultgren, R., P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelley, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements, American Society of Metals, Metals Park, OH, 1973. 7. Hultgren, R., P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, American Society of Metals, Metals Park, OH, 1973. 8. American Society of Heating, Refrigerating and Air Conditioning Engineers, ASHRAE Handbook of Fundamentals, ASHRAE, New York, 1981. 9. Mallory, J. F., Thermal Insulation, Van Nostrand Reinhold, New York, 1969. 10. Hanley, E. J., D. P. DeWitt, and R. E. Taylor, “The Thermal Transport Properties at Normal and Elevated Temperature of Eight Representative Rocks,” Proceedings of the Seventh Symposium on Thermophysical Properties, American Society of Mechanical Engineers, New York, 1977. 11. Sweat, V. E., “A Miniature Thermal Conductivity Probe for Foods,” American Society of Mechanical Engineers, Paper 76-HT-60, August 1976. 12. Kothandaraman, C. P., and S. Subramanyan, Heat and Mass Transfer Data Book, Halsted Press/Wiley, Hoboken, NJ, 1975. 13. Chapman, A. J., Heat Transfer, 4th ed., Macmillan, New York, 1984. 14. Vargaftik, N. B., Tables of Thermophysical Properties of Liquids and Gases, 2nd ed., Hemisphere Publishing, New York, 1975. 15. Eckert, E. R. G., and R. M. Drake, Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972. 16. Vukalovich, M. P., A. I. Ivanov, L. R. Fokin, and A. T. Yakovelev, Thermophysical Properties of Mercury, State Committee on Standards, State Service for Standards and Handbook Data, Monograph Series No. 9, Izd. Standartov, Moscow, 1971. 17. Tillner-Roth, R., and H. D. Baehr, J. Phys. Chem. Ref. Data, 23, 657, 1994. 18. Kamei, A., S. W. Beyerlein, and R. T. Jacobsen, Int. J. Thermophysics, 16, 1155, 1995. 19. Lemmon, E. W., M. O. McLinden, and M. L. Huber, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 7.0 National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2002. 20. Bolz, R. E., and G. L. Tuve, Eds., CRC Handbook of Tables for Applied Engineering Science, 2nd ed., CRC Press, Boca Raton, FL, 1979. 21. Liley, P. E., private communication, School of Mechanical Engineering, Purdue University, West Lafayette, IN, May 1984. 22. Liley, P. E., Steam Tables in SI Units, private communication, School of Mechanical Engineering, Purdue University, West Lafayette, IN, March 1984. 23. Liquid Materials Handbook, 23rd ed., The Atomic Energy Commission, Department of the Navy, Washington, DC, 1952. 24. Perry, J. H., Ed., Chemical Engineer’s Handbook, 4th ed., McGraw-Hill, New York, 1963. 25. Geankoplis, C. J., Mass Transport Phenomena, Holt, Rinehart & Winston, New York, 1972. 26. Barrer, R. M., Diffusion in and Through Solids, Macmillan, New York, 1941. 27. Spalding, D. B., Convective Mass Transfer, McGrawHill, New York, 1963. 28. Gubareff, G. G., J. E. Janssen, and R. H. Torborg, Thermal Radiation Properties Survey, Minneapolis-Honeywell Regulator Company, Minneapolis, MN, 1960. 29. Kreith, F., and J. F. Kreider, Principles of Solar Energy, Hemisphere Publishing, New York, 1978. FMPreface.qxd 2/21/11 6:11 PM Page x This page intentionally left blank BAPP02.qxd 2/21/11 6:07 PM Page 1013 APPENDIX B Mathematical Relations and Functions Section Page B.1 B.2 B.3 1014 1015 B.4 B.5 Hyperbolic Functions Gaussian Error Function The First Four Roots of the Transcendental Equation, n tan n ⫽ Bi, for Transient Conduction in a Plane Wall Bessel Functions of the First Kind Modified Bessel Functions of the First and Second Kinds 1016 1017 1018 BAPP02.qxd 2/21/11 1014 B.1 6:07 PM Page 1014 Appendix B 䊏 Mathematical Relations and Functions Hyperbolic Functions1 x sinh x cosh x tanh x x sinh x cosh x 0.00 0.10 0.20 0.30 0.40 0.0000 0.1002 0.2013 0.3045 0.4108 1.0000 1.0050 1.0201 1.0453 1.0811 0.00000 0.09967 0.19738 0.29131 0.37995 2.00 2.10 2.20 2.30 2.40 3.6269 4.0219 4.4571 4.9370 5.4662 3.7622 4.1443 4.5679 5.0372 5.5569 0.96403 0.97045 0.97574 0.98010 0.98367 0.50 0.60 0.70 0.80 0.90 0.5211 0.6367 0.7586 0.8881 1.0265 1.1276 1.1855 1.2552 1.3374 1.4331 0.46212 0.53705 0.60437 0.66404 0.71630 2.50 2.60 2.70 2.80 2.90 6.0502 6.6947 7.4063 8.1919 9.0596 6.1323 6.7690 7.4735 8.2527 9.1146 0.98661 0.98903 0.99101 0.99263 0.99396 1.00 1.10 1.20 1.30 1.40 1.1752 1.3356 1.5095 1.6984 1.9043 1.5431 1.6685 1.8107 1.9709 2.1509 0.76159 0.80050 0.83365 0.86172 0.88535 3.00 3.50 4.00 4.50 5.00 1.50 1.60 1.70 1.80 1.90 2.1293 2.3756 2.6456 2.9422 3.2682 2.3524 2.5775 2.8283 3.1075 3.4177 0.90515 0.92167 0.93541 0.94681 0.95624 6.00 7.00 8.00 9.00 10.000 10.018 16.543 27.290 45.003 74.203 tanh x 10.068 16.573 27.308 45.014 74.210 201.71 548.32 1490.5 4051.5 11013 201.72 548.32 1490.5 4051.5 11013 tanh x ⫽ e x ⫺ e⫺x sinh x ⫽ e x ⫹ e⫺x cosh x 1 The hyperbolic functions are defined as sinh x ⫽ 1 2 (e x ⫺ e⫺x ) cosh x ⫽ 1 2 (e x ⫹ e⫺x ) The derivatives of the hyperbolic functions of the variable u are given as d du (sinh u) ⫽ (cosh u) dx dx d du (cosh u) ⫽ (sinh u) dx dx 冢 冣 du 1 d (tanh u) ⫽ dx cosh2 u dx 0.99505 0.99818 0.99933 0.99975 0.99991 0.99999 1.0000 1.0000 1.0000 1.0000 BAPP02.qxd 2/21/11 6:07 PM Page 1015 Appendix B B.2 䊏 1015 Mathematical Relations and Functions Gaussian Error Function1 w erf w w erf w w erf w 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.00000 0.02256 0.04511 0.06762 0.09008 0.11246 0.13476 0.15695 0.17901 0.20094 0.22270 0.24430 0.26570 0.28690 0.30788 0.32863 0.34913 0.36936 0.36 0.38 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00 0.38933 0.40901 0.42839 0.46622 0.50275 0.53790 0.57162 0.60386 0.63459 0.66378 0.69143 0.71754 0.74210 0.76514 0.78669 0.80677 0.82542 0.84270 1.04 1.08 1.12 1.16 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 0.85865 0.87333 0.88679 0.89910 0.91031 0.93401 0.95228 0.96611 0.97635 0.98379 0.98909 0.99279 0.99532 0.99814 0.99931 0.99976 0.99992 0.99998 1 The Gaussian error function is defined as erf w ⫽ 2 兹 冕e w ⫺v2 dv 0 The complementary error function is defined as erfc w ⬅ 1 ⫺ erf w BAPP02.qxd 2/21/11 1016 6:07 PM Page 1016 Appendix B 䊏 Mathematical Relations and Functions B.3 The First Four Roots of the Transcendental Equation, n tan n ⴝ Bi, for Transient Conduction in a Plane Wall hL Bi ⴝ ᎏ k 1 2 3 4 0 0.001 0.002 0.004 0.006 0.008 0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 30.0 40.0 50.0 60.0 80.0 100.0 앝 0 0.0316 0.0447 0.0632 0.0774 0.0893 0.0998 0.1410 0.1987 0.2425 0.2791 0.3111 0.4328 0.5218 0.5932 0.6533 0.7051 0.7506 0.7910 0.8274 0.8603 0.9882 1.0769 1.1925 1.2646 1.3138 1.3496 1.3766 1.3978 1.4149 1.4289 1.4729 1.4961 1.5202 1.5325 1.5400 1.5451 1.5514 1.5552 1.5708 3.1416 3.1419 3.1422 3.1429 3.1435 3.1441 3.1448 3.1479 3.1543 3.1606 3.1668 3.1731 3.2039 3.2341 3.2636 3.2923 3.3204 3.3477 3.3744 3.4003 3.4256 3.5422 3.6436 3.8088 3.9352 4.0336 4.1116 4.1746 4.2264 4.2694 4.3058 4.4255 4.4915 4.5615 4.5979 4.6202 4.6353 4.6543 4.6658 4.7124 6.2832 6.2833 6.2835 6.2838 6.2841 6.2845 6.2848 6.2864 6.2895 6.2927 6.2959 6.2991 6.3148 6.3305 6.3461 6.3616 6.3770 6.3923 6.4074 6.4224 6.4373 6.5097 6.5783 6.7040 6.8140 6.9096 6.9924 7.0640 7.1263 7.1806 7.2281 7.3959 7.4954 7.6057 7.6647 7.7012 7.7259 7.7573 7.7764 7.8540 9.4248 9.4249 9.4250 9.4252 9.4254 9.4256 9.4258 9.4269 9.4290 9.4311 9.4333 9.4354 9.4459 9.4565 9.4670 9.4775 9.4879 9.4983 9.5087 9.5190 9.5293 9.5801 9.6296 9.7240 9.8119 9.8928 9.9667 10.0339 10.0949 10.1502 10.2003 10.3898 10.5117 10.6543 10.7334 10.7832 10.8172 10.8606 10.8871 10.9956 BAPP02.qxd 2/21/11 6:07 PM Page 1017 Appendix B B.4 䊏 Mathematical Relations and Functions Bessel Functions of the First Kind x J0(x) J1(x) 0.0 0.1 0.2 0.3 0.4 1.0000 0.9975 0.9900 0.9776 0.9604 0.0000 0.0499 0.0995 0.1483 0.1960 0.5 0.6 0.7 0.8 0.9 0.9385 0.9120 0.8812 0.8463 0.8075 0.2423 0.2867 0.3290 0.3688 0.4059 1.0 1.1 1.2 1.3 1.4 0.7652 0.7196 0.6711 0.6201 0.5669 0.4400 0.4709 0.4983 0.5220 0.5419 1.5 1.6 1.7 1.8 1.9 0.5118 0.4554 0.3980 0.3400 0.2818 0.5579 0.5699 0.5778 0.5815 0.5812 2.0 2.1 2.2 2.3 2.4 0.2239 0.1666 0.1104 0.0555 0.0025 0.5767 0.5683 0.5560 0.5399 0.5202 1017 BAPP02.qxd 2/21/11 1018 B.5 6:07 PM Page 1018 Appendix B 䊏 Mathematical Relations and Functions Modified Bessel Functions1 of the First and Second Kinds x eⴚxI0(x) eⴚxI1(x) exK0(x) exK1(x) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.4 6.8 7.2 7.6 8.0 8.4 8.8 9.2 9.6 10.0 1.0000 0.8269 0.6974 0.5993 0.5241 0.4657 0.4198 0.3831 0.3533 0.3289 0.3085 0.2913 0.2766 0.2639 0.2528 0.2430 0.2343 0.2264 0.2193 0.2129 0.2070 0.2016 0.1966 0.1919 0.1876 0.1835 0.1797 0.1762 0.1728 0.1696 0.1666 0.1611 0.1561 0.1515 0.1473 0.1434 0.1398 0.1365 0.1334 0.1305 0.1278 0.0000 0.0823 0.1368 0.1722 0.1945 0.2079 0.2152 0.2185 0.2190 0.2177 0.2153 0.2121 0.2085 0.2046 0.2007 0.1968 0.1930 0.1892 0.1856 0.1821 0.1787 0.1755 0.1724 0.1695 0.1667 0.1640 0.1614 0.1589 0.1565 0.1542 0.1520 0.1479 0.1441 0.1405 0.1372 0.1341 0.1312 0.1285 0.1260 0.1235 0.1213 ⬁ 2.1407 1.6627 1.4167 1.2582 1.1445 1.0575 0.9881 0.9309 0.8828 0.8416 0.8056 0.7740 0.7459 0.7206 0.6978 0.6770 0.6579 0.6404 0.6243 0.6093 0.5953 0.5823 0.5701 0.5586 0.5478 0.5376 0.5279 0.5188 0.5101 0.5019 0.4865 0.4724 0.4595 0.4476 0.4366 0.4264 0.4168 0.4079 0.3995 0.3916 ⬁ 5.8334 3.2587 2.3739 1.9179 1.6361 1.4429 1.3010 1.1919 1.1048 1.0335 0.9738 0.9229 0.8790 0.8405 0.8066 0.7763 0.7491 0.7245 0.7021 0.6816 0.6627 0.6453 0.6292 0.6142 0.6003 0.5872 0.5749 0.5633 0.5525 0.5422 0.5232 0.5060 0.4905 0.4762 0.4631 0.4511 0.4399 0.4295 0.4198 0.4108 1 In⫹1(x) ⫽ In⫺1(x) ⫺ (2n/x)In(x) BAPP03.qxd 2/24/11 1:15 PM Page 1019 APPENDIX C Thermal Conditions Associated with Uniform Energy Generation in One-Dimensional, Steady-State Systems BAPP03.qxd 2/24/11 1020 1:15 PM Page 1020 Appendix C 䊏 One-Dimensional, Steady-State Conduction with Generation In Section 3.5 the problem of conduction with thermal energy generation is considered for one-dimensional, steady-state conditions. The form of the heat equation differs, according to whether the system is a plane wall, a cylindrical shell, or a spherical shell (Figure C.1). In each case, there are several options for the boundary condition at each surface, and hence a greater number of possibilities for specific forms of the temperature distribution and heat rate (or heat flux). An alternative to solving the heat equation for each possible combination of boundary conditions involves obtaining a solution by prescribing boundary conditions of the rfi st kind, Equation 2.31, at both surfaces and then applying an energy balance to each surface at which the temperature is unknown. For the geometries of Figure C.1, with uniform temperatures Ts,1 and Ts,2 prescribed at each surface, solutions to appropriate forms of the heat equation are readily obtained and are summarized in Table C.1. The temperature distributions may be used with Fourier’s law to obtain corresponding distributions for the heat flux and heat rate. If Ts,1 and Ts,2 are both known for a particular problem, the expressions of Table C.1 provide all that is needed to completely determine related thermal conditions. If Ts,1 and/or Ts,2 are not known, the results may still be used with surface energy balances to determine the desired thermal conditions. Plane Wall x –L Ts,1 +L Ts,2 q• Cylindrical Wall r1 q• Ts,1 L Ts,2 r2 Spherical Wall Ts,1 r2 q• Ts,2 r1 FIGURE C.1 One-dimensional conduction systems with uniform thermal energy generation: a plane wall with asymmetric surface conditions, a cylindrical shell, and a spherical shell. BAPP03.qxd 2/24/11 1:15 PM Page 1021 Appendix C One-Dimensional, Steady-State Conduction with Generation 䊏 1021 TABLE C.1 One-Dimensional, Steady-State Solutions to the Heat Equation for Plane, Cylindrical, and Spherical Walls with Uniform Generation and Asymmetrical Surface Conditions Temperature Distribution Ts,2 ⫺ Ts,1 x Ts,1 ⫹ Ts,2 q˙L2 x2 ⫹ 1⫺ 2 ⫹ 2k 2 L 2 L 冢 冣 Plane Wall T(x) ⫽ Cylindrical Wall T(r) ⫽ Ts,2 ⫹ q˙r 22 q˙r 22 r2 1⫺ 2 ⫺ 4k 4k r2 Spherical Wall T(r) ⫽ Ts,2 ⫹ q˙r 22 (1/r) ⫺ (1/r2) q˙r 22 r 21 r2 1⫺ 2 ⫺ 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k 6k (1/r1) ⫺ (1/r2) r2 r2 冣 冤 冢 冢 1⫺ r 21 r 22 冣 冤 冢 冢 (C.1) 冣 ⫹ (T s,2 ln(r /r) 冥 ln(r /r ) (C.2) 冥 (C.3) ⫺ Ts,1) 冣 2 2 1 Heat Flux Plane Wall Cylindrical Wall Spherical Wall k (T ⫺ Ts,1) q⬙(x) ⫽ q˙x ⫺ 2L s,2 q⬙(r) ⫽ q˙r ⫺ 2 q˙r q⬙(r) ⫽ ⫺ 3 k 冤 冢 (C.4) 冥 q˙r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 4k r2 冣 (C.5) r ln(r2 /r1) k 冤q6kr 冢1 ⫺ rr 冣 ⫹ (T ˙ 22 2 1 2 2 s,2 冥 ⫺ Ts,1) (C.6) r 2[(1/r1) ⫺ (1/r2)] Heat Rate 冤 冥 Plane Wall k (T ⫺ Ts,1) Ax q(x) ⫽ q˙x ⫺ 2L s,2 Cylindrical Wall q(r) ⫽ q˙Lr 2 ⫺ Spherical Wall q(r) ⫽ q˙4r3 ⫺ 3 (C.7) 冤 冢 冥 q˙r 22 r 21 2Lk 䡠 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) ln(r2 /r1) 4k r2 冣 冤q6kr 冢1 ⫺ rr 冣 ⫹ (T 4k ˙ 22 2 1 2 2 s,2 (1/r1) ⫺ (1/r2) 冥 (C.8) ⫺ Ts,1) (C.9) Alternative surface conditions could involve specification of a uniform surface heat flux (boundary condition of the second kind, Equation 2.32 or 2.33) or a convection condition (boundary condition of the third kind, Equation 2.34). In each case, the surface temperature would not be known but could be determined by applying a surface energy balance. The forms that such balances may take are summarized in Table C.2. Note that, to accommodate situations for which a surface of interest may adjoin a composite wall in which there is no generation, the boundary condition of the third kind has been applied by using the overall heat transfer coefficient U in lieu of the convection coefficient h. BAPP03.qxd 2/24/11 1022 1:15 PM Page 1022 Appendix C 䊏 One-Dimensional, Steady-State Conduction with Generation TABLE C.2 Alternative Surface Conditions and Energy Balances for One-Dimensional, Steady-State Solutions to the Heat Equation for Plane, Cylindrical, and Spherical Walls with Uniform Generation Plane Wall Uniform Surface Heat Flux x ⫽ ⫺L: q⬙s,1 ⫽ ⫺q˙L ⫺ k (T ⫺ Ts,1) 2L s,2 (C.10) k (T ⫺ Ts,1) 2L s,2 Prescribed Transport Coefficient and Ambient Temperature k U1(T앝,1 ⫺ Ts,1) ⫽ ⫺q˙L ⫺ (T ⫺ Ts,1) x ⫽ ⫺L: 2L s,2 k x ⫽ ⫹L: U2(Ts,2 ⫺ T앝,2) ⫽ q˙L ⫺ (T ⫺ Ts,1) 2L s,2 x ⫽ ⫹L: q⬙s,2 ⫽ q˙L ⫺ (C.11) (C.12) (C.13) Cylindrical Wall Uniform Surface Heat Flux r ⫽ r1: r ⫽ r2: q⬙s,1 ⫽ q˙r1 ⫺ 2 q˙r2 ⫺ q⬙s,2 ⫽ 2 k q˙r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 4k r2 冤 冢 冥 冣 (C.14) r1 ln(r2/r1) k ˙ 22 冤 q4kr 冢1 ⫺ rr 冣 ⫹ (T 2 1 2 2 s,2 冥 ⫺ Ts,1) (C.15) r2 ln(r2/r1) Prescribed Transport Coefficient and Ambient Temperature r ⫽ r1: r ⫽ r2: U1(T앝,1 ⫺ Ts,1) ⫽ U2(Ts,2 ⫺ T앝,2) ⫽ q˙r1 ⫺ 2 q˙r2 ⫺ 2 k q˙r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 4k r2 冤 冢 冥 冣 r1 ln(r2/r1) k q˙r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 4k r2 冤 冢 冥 冣 r2 ln(r2 /r1) (C.16) (C.17) Spherical Wall Uniform Surface Heat Flux r ⫽ r1: r ⫽ r2: q⬙s,1 ⫽ q˙r1 ⫺ 3 q˙r2 ⫺ q⬙s,2 ⫽ 3 k ˙ 22 冤 q6kr 冢1 ⫺ rr 冣 ⫹ (T 2 1 2 2 s,2 冥 ⫺ Ts,1) r 21[(1/r1) ⫺ (1/r2)] k ˙ 22 冤 q6kr 冢1 ⫺ rr 冣 ⫹ (T 2 1 2 2 s,2 冥 (C.18) ⫺ Ts,1) r 22[(1/r1) ⫺ (1/r2)] (C.19) BAPP03.qxd 2/24/11 1:15 PM Page 1023 Appendix C 䊏 TABLE C.2 1023 One-Dimensional, Steady-State Conduction with Generation Continued Prescribed Transport Coefficient and Ambient Temperature U1(T앝,1 ⫺ Ts,1) ⫽ r ⫽ r1: q˙r1 ⫺ 3 q˙r2 ⫺ U2(Ts,2 ⫺ T앝,2) ⫽ 3 r ⫽ r2: k q˙r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k r2 冤 冢 k ˙ 22 冥 冣 r 21[(1/r1) ⫺ (1/r2)] 冤 q6kr 冢1 ⫺ rr 冣 ⫹ (T 2 1 2 2 s,2 冥 (C.20) ⫺ Ts,1) r 22[(1/r1) ⫺ (1/r2)] (C.21) As an example, consider a plane wall for which a uniform (known) surface temperature Ts,1 is prescribed at x ⫽ ⫺L and a uniform heat flux q⬙s,2 is prescribed at x ⫽ ⫹L. Equation C.11 may be used to evaluate Ts,2, and Equations C.1, C.4, and C.7 may then be used to determine the temperature, heat flux, and heat rate distributions, respectively. Special cases of the foregoing configurations involve a plane wall with one adiabatic surface, a solid cylinder (a circular rod), and a sphere (Figure C.2). Subject to the requirements that dT/dx冨x⫽0 ⫽ 0 and dT/dr冨r⫽0 ⫽ 0, the corresponding forms of the heat equation may be solved to obtain Equations C.22 through C.24 of Table C.3. The solutions are based on prescribing a Plane Wall x L q• Ts Solid cylinder ro Ts q• ro Solid sphere q • Ts FIGURE C.2 One-dimensional conduction systems with uniform thermal energy generation: a plane wall with one adiabatic surface, a cylindrical rod, and a sphere. BAPP03.qxd 2/24/11 1024 1:15 PM Page 1024 Appendix C 䊏 One-Dimensional, Steady-State Conduction with Generation TABLE C.3 One-Dimensional, Steady-State Solutions to the Heat Equation for Uniform Generation in a Plane Wall with One Adiabatic Surface, a Solid Cylinder, and a Solid Sphere Temperature Distribution Plane Wall T(x) ⫽ q˙L2 x2 1 ⫺ 2 ⫹ Ts 2k L (C.22) Circular Rod T(r) ⫽ q˙r 2o r2 1 ⫺ 2 ⫹ Ts 4k ro (C.23) Sphere T(r) ⫽ q˙r 2o r2 1 ⫺ 2 ⫹ Ts 6k ro (C.24) 冢 冢 冣 冣 冢 冣 Heat Flux Plane Wall q⬙(x) ⫽ q˙x (C.25) Circular Rod q⬙(r) ⫽ q˙r 2 (C.26) Sphere q⬙(r) ⫽ q˙r 3 (C.27) Plane Wall q(x) ⫽ q˙xAx Circular Rod q(r) ⫽ q˙Lr 2 Heat Rate Sphere q(r) ⫽ q˙4r 3 (C.28) (C.29) 3 (C.30) uniform temperature Ts at x ⫽ L and r ⫽ ro. Using Fourier’s law with the temperature distributions, the heat flux (Equations C.25 through C.27) and heat rate (Equations C.28 through C.30) distributions may also be obtained. If Ts is not known, it may be determined by applying a surface energy balance, appropriate forms of which are summarized in Table C.4. TABLE C.4 Alternative Surface Conditions and Energy Balances for One-Dimensional, SteadyState Solutions to the Heat Equation for Uniform Generation in a Plane Wall with One Adiabatic Surface, a Solid Cylinder, and a Solid Sphere Prescribed Transport Coefficient and Ambient Temperature Plane Wall x ⫽ L: q˙L ⫽ U(Ts ⫺ T앝) Circular Rod q˙ro r ⫽ ro: ⫽ U(Ts ⫺ T앝) 2 Sphere q˙ro ⫽ U(Ts ⫺ T앝) r ⫽ ro: 3 (C.31) (C.32) (C.33) BAPP04.qxd 2/21/11 6:08 PM Page 1025 APPENDIX D The Gauss–Seidel Method BAPP04.qxd 2/21/11 1026 6:08 PM Page 1026 Appendix D 䊏 The Gauss–Seidel Method The Gauss–Seidel method is an example of an iterative approach for solving systems of linear algebraic equations, such as that represented by Equation 4.47, reproduced below. a11T1 a12T2 a13T3 … a1NTN C1 a21T1 a22T2 a23T3 … a2NTN C2 ⯗ ⯗ ⯗ ⯗ ⯗ ⯗ … aN1T1 aN2T2 aN3T3 aNNTN CN (4.47) For small numbers of equations, Gauss–Seidel iteration can be performed by hand. Application of the Gauss–Seidel method to the system of equations represented by Equation 4.47 is facilitated by the following procedure. 1. To whatever extent possible, the equations should be reordered to provide diagonal elements whose magnitudes are larger than those of other elements in the same row. That is, it is desirable to sequence the equations such that 冨a11冨 冨a12冨, 冨a13冨, . . ., 冨a1N冨; 冨a22冨 冨a21冨, 冨a23冨, . . ., 冨a2N冨; and so on. 2. After reordering, each of the N equations should be written in explicit form for the temperature associated with its diagonal element. Each temperature in the solution vector would then be of the form N aij Ci i1 aij (k) (k1) T (k) T (D.1) i a j a a Tj ii j1 ii ji1 ii 兺 兺 where i 1, 2, . . ., N. The superscript k refers to the level of the iteration. 3. An initial (k 0) value is assumed for each temperature Ti. Subsequent computations may be reduced by selecting values based on rational estimates of the actual solution. 4. Setting k 1 in Equation D.1, values of Ti(1) are then calculated by substituting assumed (second summation, k 1 0) or new (first summation, k 1) values of Tj into the right-hand side. This step is the first (k 1) iteration. 5. Using Equation D.1, the iteration procedure is continued by calculating new values of Ti(k) from the Tj(k) values of the current iteration, where 1 j i 1, and the Tj(k1) values of the previous iteration, where i 1 j N. 6. The iteration is terminated when a prescribed convergence criterion is satisfied. The criterion may be expressed as 兩Ti(k) Ti(k1)兩 (D.2) where represents an error in the temperature that is considered to be acceptable. If step 1 can be accomplished for each equation, the resulting system is said to be diagonally dominant, and the rate of convergence is maximized (the number of required iterations is minimized). However, convergence may also be achieved in many situations for which diagonal dominance cannot be obtained, although the rate of convergence is slowed. The manner in which new values of Ti are computed (steps 4 and 5) should also be noted. Because the Ti for a particular iteration are calculated sequentially, each value can be computed by using the most recent estimates of the other Ti. This feature is implicit in Equation D.1, where the value of each unknown is updated as soon as possible, that is, for 1 j i 1. An example problem that utilizes the Gauss–Seidel method is included in Section 4S.2. BAPP05.qxd 2/21/11 6:09 PM Page 1027 APPENDIX E The Convection Transfer Equations BAPP05.qxd 2/21/11 1028 6:09 PM Page 1028 Appendix E 䊏 The Convection Transfer Equations In Chapter 2 we considered a stationary substance in which heat is transferred by conduction and developed means for determining the temperature distribution within the substance. We did so by applying conservation of energy to a differential control volume (Figure 2.11) and deriving a differential equation that was termed the heat equation. For a prescribed geometry and boundary conditions, the equation may be solved to determine the corresponding temperature distribution. If the substance is not stationary, conditions become more complex. For example, if conservation of energy is applied to a differential control volume in a moving fluid, the effects of fluid motion (advection) on energy transfer across the surfaces of the control volume must be considered, along with those of conduction. The resulting differential equation, which provides the basis for predicting the temperature distribution, now requires knowledge of the velocity equations derived by applying conservation of mass and Newton’s second law of motion to a differential control volume. In this appendix we consider conditions involving flow of a viscous ufl id in which there is concurrent heat and mass transfer. We restrict our attention to the steady, two-dimensional flow of an incompressible fluid with constant properties in the x- and y-directions of a Cartesian coordinate system, and present the differential equations that may be used to predict velocity, temperature, and species concentration fields within the fluid. These equations can be derived by applying Newton’s second law of motion and conservation of mass, energy, and species to a differential control volume in the fluid. E.1 Conservation of Mass One conservation law that is pertinent to the flow of a viscous fluid is that matter can be neither created nor destroyed. For steady flow, this law requires that the net rate at which mass enters a control volume (inflow outflow) must equal zero. Applying this law to a differential control volume in the flow yields ⭸u ⭸v 0 ⭸x ⭸y (E.1) where u and v are the x- and y-components of the mass average velocity. Equation E.1, the continuity equation, is a general expression of the overall mass conservation requirement, and it must be satisfied at every point in the fluid. The equation applies for a single species fluid, as well as for mixtures in which species diffusion and chemical reactions may be occurring, provided that the fluid can be approximated as incompressible, that is, constant density. E.2 Newton’s Second Law of Motion The second fundamental law that is pertinent to the flow of a viscous fluid is Newton’s second law of motion. For a differential control volume in the fluid, under steady conditions, this requirement states that the sum of all forces acting on the control volume must equal the net rate at which momentum leaves the control volume (outflow inflow). These equations are derived in Section 6S.1. BAPP05.qxd 2/21/11 6:09 PM Page 1029 Appendix E 䊏 1029 The Convection Transfer Equations Two kinds of forces may act on the fluid: body forces, which are proportional to the volume, and surface forces, which are proportional to area. Gravitational, centrifugal, magnetic, and/or electric fields may contribute to the total body force, and we designate the x- and y-components of this force per unit volume of fluid as X and Y, respectively. The surface forces are due to the fluid static pressure as well as to viscous stresses. Applying Newton’s second law of motion (in the x- and y-directions) to a differential control volume in the fluid, accounting for body and surface forces, yields u 冢 ⭸p ⭸u ⭸u ⭸2u ⭸2u X v ⭸x ⭸y ⭸x ⭸x 2 ⭸y2 冢 ⭸p ⭸v ⭸v ⭸2v ⭸2v Y v ⭸x ⭸y ⭸y ⭸x 2 ⭸y2 u 冣 冢 冣 (E.2) 冣 冢 冣 (E.3) where p is the pressure and is the fluid viscosity. We should not lose sight of the physics represented by Equations E.2 and E.3. The two terms on the left-hand side of each equation represent the net rate of momentum flow from the control volume. The terms on the right-hand side, taken in order, account for the net pressure force, the net viscous forces, and the body force. These equations must be satisfied at each point in the fluid, and with Equation E.1 they may be solved for the velocity field. E.3 Conservation of Energy As mentioned at the beginning of this Appendix, in Chapter 2 we considered a stationary substance in which heat is transferred by conduction and applied conservation of energy to a differential control volume (Figure 2.11) to derive the heat equation. When conservation of energy is applied to a differential control volume in a moving fluid under steady conditions, it expresses that the net rate at which energy enters the control volume, plus the rate at which heat is added, minus the rate at which work is done by the fluid in the control volume, is equal to zero. After much manipulation, the result can be rewritten as a thermal energy equation. For steady, two-dimensional flow of an incompressible fluid with constant properties, the resulting differential equation is 冢 cp u 冣 冢 冣 ⭸T ⭸T ⭸2T ⭸2T k q˙ v ⭸x ⭸y ⭸x 2 ⭸y 2 (E.4) where T is the temperature, cp is the specific heat at constant pressure, k is the thermal conductivity, q˙is the volumetric rate of thermal energy generation, and , the viscous dissipation, is defined as ⬅ 冦冢 ⭸u ⭸v ⭸y ⭸x 冣 冤冢 冣 冢 冣 冥冧 2 2 ⭸u ⭸x 2 ⭸v ⭸y 2 (E.5) The same form of the thermal energy equation, Equation E.4, also applies to an ideal gas with negligible pressure variation. In Equation E.4, the terms on the left-hand side account for the net rate at which thermal energy leaves the control volume due to bulk fluid motion (advection), while the terms on the right-hand side account for net inflow of energy due to conduction, viscous BAPP05.qxd 2/21/11 1030 6:09 PM Page 1030 Appendix E 䊏 The Convection Transfer Equations dissipation, and generation. Viscous dissipation represents the net rate at which mechanical work is irreversibly converted to thermal energy due to viscous effects in the fluid. The generation term characterizes conversion from other forms of energy (such as chemical, electrical, electromagnetic, or nuclear) to thermal energy. E.4 Conservation of Species If the viscous fluid consists of a binary mixture in which there are species concentration gradients, there will be relative transport of the species, and species conservation must be satisfied at each point in the fluid. For steady flow, this law requires that the net rate at which species A enters a control volume (inflow outflow) plus the rate at which species A is produced in the control volume (by chemical reactions) must equal zero. Applying this law to a differential control volume in the flow yields the following differential equation, which has been expressed on a molar basis: u 冢 冣 ⭸2CA ⭸2CA ⭸CA ⭸C N˙A v A DAB ⭸x ⭸y ⭸x 2 ⭸y2 (E.6) where CA is the molar concentration of species A, DAB is the binary diffusion coefficient, and N˙A is the molar rate of production of species A per unit volume. Again, this equation has been derived assuming steady, two-dimensional flow of an incompressible fluid with constant properties. Terms on the left-hand side account for net transport of species A due to bulk fluid motion (advection), while terms on the right-hand side account for net inflow due to diffusion and production due to chemical reactions. An example problem involving the solution of the convection transfer equations is included in Section 6S.1. BAPP06.qxd 2/21/11 6:06 PM Page 1031 APPENDIX F Boundary Layer Equations for Turbulent Flow BAPP06.qxd 2/21/11 1032 6:06 PM Page 1032 Appendix F 䊏 Boundary Layer Equations for Turbulent Flow It has been noted in Section 6.3 that turbulent flow is inherently unsteady. This behavior is shown in Figure F.1, where the variation of an arbitrary flow property P is plotted as a function of time at some location in a turbulent boundary layer. The property P could be a velocity component, the fluid temperature, or a species concentration, and at any instant it may be represented as the sum of a time-mean value P and a fluctuating component P⬘. The average is taken over a time that is large compared with the period of a typical fluctuation, and if P is independent of time, the time-mean flow is said to be steady. Since engineers are typically concerned with the time-mean properties, P , the difficulty of solving the time-dependent governing equations is often eliminated by averaging the equations over time. For steady (in the mean), incompressible, constant property, boundary layer flow with negligible viscous dissipation, using well-established time-averaging procedures [1], the following forms of the continuity, x-momentum, energy, and species conservation equations may be obtained: ⭸u ⭸v ⫹ ⫽0 ⭸x ⭸y u (F.1) 冢 冣 dp ⭸u ⭸u ⭸u ⭸ ⫹v ⫽ ⫺ 1 앝 ⫹ 1 ⫺ u⬘v⬘ ⭸x ⭸y ⭸y ⭸y dx 冢 ⭸C ⭸C ⭸ u ⫹v ⫽ 冢D ⭸x ⭸y ⭸y u 冣 ⫺ v⬘C ⬘ 冣 ⭸T ⭸T ⭸ ⭸T k ⫹v ⫽ 1 ⫺ cp v⬘T⬘ ⭸x ⭸y cp ⭸y ⭸y A A AB ⭸CA ⭸y A (F.2) (F.3) (F.4) The equations are like those for the laminar boundary layer, Equations 6.27 through 6.30 (after neglecting viscous dissipation), except for the presence of additional terms of the form a⬘b⬘. These terms account for the effect of the turbulent fluctuations on momentum, energy, and species transport. On the basis of the foregoing results, it is customary to speak of a total shear stress and total heat and species fluxes, which are defined as 冢 ⭸u⭸y ⫺ u⬘v⬘冣 ⭸T q⬙ ⫽ ⫺冢k ⫺ c v⬘T⬘冣 ⭸y ⭸C ⫺ v⬘C ⬘ 冣 N⬙ ⫽ ⫺冢D ⭸y tot ⫽ tot A, tot p AB A A (F.5) (F.6) (F.7) P P' P Time, t FIGURE F.1 Property variation with time at some point in a turbulent boundary layer. BAPP06.qxd 2/21/11 6:06 PM Page 1033 Appendix F 䊏 Boundary Layer Equations for Turbulent Flow 1033 and consist of contributions due to molecular diffusion and turbulent mixing. From the form of these equations we see how momentum, energy, and species transfer rates are enhanced by the existence of turbulence. The term ⫺u⬘v⬘ appearing in Equation F.5 represents the momentum flux due to the turbulent fluctuations, and it is often termed the Reynolds stress. The terms cpv⬘T⬘ and v⬘CA⬘ in Equations F.6 and F.7, respectively, represent the heat and species fluxes due to the turbulent fluctuations. Unfortunately, these new terms introduced by the time-averaging process are additional unknowns, so that the number of unknowns exceeds the number of equations. Resolving this problem is the subject of the field of turbulence modeling [2]. References 1. Kays, W. M., M. E. Crawford, and B. Weigand, Convective Heat and Mass Transfer, 4th ed., McGraw-Hill Higher Education, Boston, 2005. 2. Wilcox, D. C., Turbulence Modeling for CFD, 2nd ed., DCW Industries, La Cañada, 1998. FMPreface.qxd 2/21/11 6:11 PM Page x This page intentionally left blank BAPP07.qxd 2/21/11 6:09 PM Page 1035 APPENDIX G An Integral Laminar Boundary Layer Solution for Parallel Flow over a Flat Plate BAPP07.qxd 2/21/11 1036 6:09 PM Page 1036 Appendix G 䊏 An Integral Laminar Boundary Layer Solution An alternative approach to solving the boundary layer equations involves the use of an approximate integral method. The approach was originally proposed by von Kárman [1] in 1921 and first applied by Pohlhausen [2]. It is without the mathematical complications inherent in the exact (similarity) method of Section 7.2.1; yet it can be used to obtain reasonably accurate results for the key boundary layer parameters (␦, ␦t, ␦c, Cƒ, h, and hm). Although the method has been used with some success for a variety of flow conditions, we restrict our attention to parallel flow over a flat plate, subject to the same restrictions enumerated in Section 7.2.1, that is, incompressible laminar ofl w with constant ufl id properties and negligible viscous dissipation. To use the method, the boundary layer equations, Equations 7.4 through 7.7, must be cast in integral form. These forms are obtained by integrating the equations in the y-direction across the boundary layer. For example, integrating Equation 7.4, we obtain 冕 ⭸u⭸x dy ⫹ 冕 ⭸v⭸y dy ⫽ 0 ␦ ␦ 0 (G.1) 0 or, since ⫽ 0 at y ⫽ 0, 冕 ⭸u⭸x dy ␦ v(y ⫽ ␦) ⫽ ⫺ (G.2) 0 Similarly, from Equation 7.5, we obtain 冕 u ⭸u⭸x dy ⫹ 冕 v ⭸u⭸y dy ⫽ 冕 ⭸y⭸ 冢⭸u⭸y冣 dy ␦ ␦ 0 ␦ 0 0 or, integrating the second term on the left-hand side by parts, 冕 u ⭸u⭸x dy ⫹ uv 冏 ⫺ 冕 u ⭸v⭸y dy ⫽ ⭸u⭸y 冏 ␦ ␦ 0 0 ␦ ␦ 0 0 Substituting from Equations 7.4 and G.2, we obtain 冕 u ⭸u⭸x dy ⫺ u 冕 ⭸u⭸x dy ⫹ 冕 u ⭸u⭸x dy ⫽ ⫺ ⭸u⭸y 冏 ␦ ␦ 앝 0 or 0 0 y⫽0 冕 ⭸u⭸x dy ⫺ 冕 2u ⭸u⭸x dy ⫽ ⭸u⭸y 冏 ␦ u앝 Therefore ␦ ␦ 0 0 y⫽0 冕 ⭸x⭸ (u 䡠 u ⫺ u 䡠 u) dy ⫽ ⭸u⭸y 冏 ␦ 앝 0 y⫽0 Rearranging, we then obtain d dx 冤冕 (u ␦ 0 앝 冥 ⫺ u)u dy ⫽ ⭸u ⭸y 冏 (G.3) y⫽0 Equation G.3 is the integral form of the boundary layer momentum equation. In a similar fashion, the following integral forms of the boundary layer energy and species continuity equations may be obtained: d dx 冤冕 (T ␦t 0 앝 冥 ⫺ T )u dy ⫽ ␣ ⭸T ⭸y 冏 y⫽0 (G.4) BAPP07.qxd 2/21/11 6:09 PM Page 1037 Appendix G 䊏 1037 An Integral Laminar Boundary Layer Solution d dx 冤冕 ( ␦c 0 A,앝 ⫺ 冥 A)u dy ⫽ DAB ⭸A ⭸y 冏 (G.5) y⫽0 Equations G.3 through G.5 satisfy the x-momentum, the energy, and the species conservation requirements in an integral (or average) fashion over the entire boundary layer. In contrast, the original conservation equations, (7.5) through (7.7), satisfy the conservation requirements locally, that is, at each point in the boundary layer. The integral equations can be used to obtain approximate boundary layer solutions. The procedure involves first assuming reasonable functional forms for the unknowns u, T, and A in terms of the corresponding (unknown) boundary layer thicknesses. The assumed forms must satisfy appropriate boundary conditions. Substituting these forms into the integral equations, expressions for the boundary layer thicknesses may be determined and the assumed functional forms may then be completely specified. Although this method is approximate, it frequently leads to accurate results for the surface parameters. Consider the hydrodynamic boundary layer, for which appropriate boundary conditions are u(y ⫽ 0) ⫽ ⭸u ⭸y 冏 y⫽␦ ⫽0 u(y ⫽ ␦) ⫽ u앝 and From Equation 7.5 it also follows that, since u ⫽ v ⫽ 0 at y ⫽ 0, ⭸2u ⭸y2 冏 y⫽0 ⫽0 With the foregoing conditions, we could approximate the velocity profile as a third-degree polynomial of the form 冢冣 冢冣 y y u u앝 ⫽ a1 ⫹ a2 ␦ ⫹ a3 ␦ 2 ⫹ a4 冢冣 y ␦ 3 and apply the conditions to determine the coefficients a1 to a4. It is easily verified that 3 1 a1 ⫽ a3 ⫽ 0, a2 ⫽ 2 and a4 ⫽ ⫺2, in which case 冢冣 3y 1 y u u앝 ⫽ 2 ␦ ⫺ 2 ␦ 3 (G.6) The velocity profile is then specified in terms of the unknown boundary layer thickness ␦. This unknown may be determined by substituting Equation G.6 into G.3 and integrating over y to obtain 冢 冣 d 39 u2 ␦ ⫽ 3 u앝 dx 280 앝 2 ␦ Separating variables and integrating over x, we obtain ␦2 ⫽ 140 x ⫹ constant 2 13 u앝 However, since ␦ ⫽ 0 at the leading edge of the plate (x ⫽ 0), the integration constant must be zero and 冢 冣 x ␦ ⫽ 4.64 u 앝 1/2 ⫽ 4.64x Re1/2 x (G.7) BAPP07.qxd 2/21/11 6:09 PM 1038 Page 1038 Appendix G 䊏 An Integral Laminar Boundary Layer Solution Substituting Equation G.7 into Equation G.6 and evaluating s ⫽ (⭸u/⭸y)s, we also obtain Cf,x ⫽ s ⫽ 0.646 u2앝/2 Re1/2 x (G.8) Despite the approximate nature of the foregoing procedure, Equations G.7 and G.8 compare quite well with results obtained from the exact solution, Equations 7.19 and 7.20. In a similar fashion one could assume a temperature profile of the form T* ⫽ 冢 冣 冢 冣 ⫹ b 冢␦y 冣 T ⫺ Ts y y ⫽ b1 ⫹ b2 ⫹ b3 T앝 ⫺ Ts ␦t ␦t and determine the coefficients from the conditions T *(y ⫽ 0) ⫽ ⭸T * ⭸y 冏 y⫽␦t 2 3 4 t ⫽0 T * ( y ⫽ ␦t) ⫽ 1 as well as ⭸2T * ⭸y2 冏 y⫽0 ⫽0 which is inferred from the energy equation (7.6). We then obtain 冢冣 y y T* ⫽ 3 ⫺ 1 2 ␦t 2 ␦t 3 (G.9) Substituting Equations G.6 and G.9 into Equation G.4, we obtain, after some manipulation and assuming Pr ⲏ 1, ␦t Pr⫺1/3 ⫽ ␦ 1.026 (G.10) This result is in good agreement with that obtained from the exact solution, Equation 7.24. Moreover, the heat transfer coefficient may then be computed from h⫽ ⫺k ⭸T/⭸y兩y⫽0 Ts ⫺ T앝 ⫽3 k 2 ␦t Substituting from Equations G.7 and G.10, we obtain 1/3 Nux ⫽ hx ⫽ 0.332 Re1/2 x Pr k (G.11) This result agrees precisely with that obtained from the exact solution, Equation 7.23. Using the same procedures, analogous results may be obtained for the concentration boundary layer. References 1. von Kárman, T., Z. Angew. Math. Mech., 1, 232, 1921. 2. Pohlhausen, K., Z. Angew. Math. Mech., 1, 252, 1921. bindex.qxd 2/25/11 6:21 AM Page 1039 Index NOTE: Page references preceded by a “W” refer to pages that are located on the Web site www.wiley.com/college/incropera. Page numbers followed by “n” refer to footnotes on the page. A Absolute species flux, 939–942 Absolute temperature, 9 Absorption: gaseous, 897–901 volumetric, 896–897 Absorptivity, 9, 802–803 Accommodation coefficient: momentum, 378n, 558–559 thermal, 189, 380n, 558 Adiabatic surfaces, 91, 230, 246 Adiabats, 230 plotting, W1–W2 Advection, 13, W25, 378, 381, 396, 398, 940, 943 definition of, 6 American Society of Mechanical Engineers (ASME), on SI units, 36 Analogies: Chilton-Colburn, 417 heat and mass transfer, 410–416, 934, 947, 966 heat diffusion and electrical charge, 114–115 Reynolds analogy, 416–417 Angle: azimuthal, 774 plane, 773 solid, 773 zenith, 774 Annular fins, 155–156, 167, 685 Azimuthal angle, 774 B Band emission, 785–792 Beer’s law, 897 Bessel equations, modified, 167–168 Bessel functions: of the first kind (table), 1017 modified, of the first and second kinds (table), 1018 Binary diffusion coefficients, 381, 937 at one atmosphere (table), 1006 Bioheat equation, 178–182 Biot number, 283–284, 408, 966 Blackbodies: concept of, 782–783 definition of, 9 Blackbody radiation, 9, 782–792 and band emission, 785–792 and Kirchhoff’s law, 810–811 Planck distribution and, 783–784 radiation exchange, 872–876 and the Stefan-Boltzmann law, 784–785 and Wien’s displacement law, 784 Body forces, W26, 594, 1029 Boiling, 7, 8, 15, 653–673 convection coefficients, typical (table), 8 dimensionless parameters in, 654–655, 672 forced convection, 655, 669–673 two-phase flow in, 670–673 modes of, 655 pool boiling, see Pool boiling saturated and subcooled, 655 Boiling crisis, 660 Boiling curve, in pool boiling, 656–657 Bond number, 408, 655 Boundary conditions, 90–91 adiabatic, 91 catalytic surface, 960–962 Dirichlet, 90 discontinuous, 954–956 of the first kind, 90, 1020–1021 mass diffusion and, 954–962 Neumann, 91 of the second kind, 91, 150, 1021–1023 of the third kind, 91, 150, 1021–1023 Boundary layer(s), 378–418, 434–468 approximations, 395–396 concentration boundary layer, 380–382 dimensionless parameters in, 398–402, 407–409, 598–599 equations, 394–397, 398–406, 597–598, 1031–1033, 1035–1038 evaporative cooling, 413–416 heat and mass transfer analogy, 410–416 hydrodynamic, 6 laminar and turbulent flow, 389–393 mixed conditions in external flow, 444–445 normalized equations, 398–402 functional forms, 400–406 similarity parameters, 398–400 bindex.qxd 2/25/11 1040 6:21 AM 䊏 Page 1040 Index Boundary layer(s) (continued) Reynolds analogy, 416–417 separation, 455–457, 465 significance of, 382 thermal boundary layer, 6, 379–380 velocity boundary layer, 378–379 Boussinesq approximation, 598 Bulk fluid motion, W29 Bulk temperature, 524–525 Buoyancy forces, 6–7, 594, 654 Buoyant jets, 595–596 Burnout point, 660 C Carnot efficiency, 32–36 Catalytic surface reactions, 960–962 Celsius temperature scale, 37 Characteristic length, 238, 284–285, 398 Chemical component, of internal energy, 15 Chemical reactions, 960–965 Chilton-Colburn analogies, 417 Circular tubes, see Tubes Coefficient of friction, see Friction coefficient Coiled tubes, 555–558 Colburn j factors, 409, 417 Cold plates, 93 Columns, evaporation in, 942–947 Compact heat exchangers, W44–W49, 708, 739 Complementary error function, 314, 1015n Composite wall systems: heat transfer in, 115–119 porous media as, 119–121 thermal contact resistance in, 117–119, 120 Compressible flow, 397 Concentration boundary layer, 380–382 and laminar or turbulent flow, 391–393 Concentration entry length, 563 Concentration penetration depth, 970 Concentric tube annulus, 553–555 Concentric tube heat exchangers, 706 Condensation, 7, 8, 15, 673–691 convection coefficients, typical (table), 8 dimensionless parameters in, 654–655, 689 dropwise, 690 film laminar, 675–679 on radial systems, 684–688 turbulent, 679–683 in horizontal tubes, 689–690 mechanisms of, 673–675 types of, 674 Conduction, 2–5, 46 analysis methods, 112–114, 132–135 and boundary/initial conditions, 90–93 definition of, 2 and Fourier’s law, 4, 68–70, 86–87 and heat diffusion equation, 82–90 micro- and nanoscale effects, 72–75, 77–78, 90, 189–190 one-dimensional steady-state, see One-dimensional steady-state conduction rate equation, 4, 46 shape factors, W3–W5, 235–240 in surface energy balance, 27–30 with thermal energy generation, see Thermal energy generation, conduction with and thermophysical properties of matter, 70–79 transient, see Transient conduction two-dimensional steady-state, see Two-dimensional steady-state conduction Conduction rate equation (Fourier’s law), 4, 68–70, 86–87 Conduction shape factor(s), W3–W5, 235–240 for selected systems (table), 236–237 Configuration factor(s), view factor, 862–872 Confinement number, 663, 672, 673 Conservation of energy, 12–31, W29–W31, 83–87 application methodology, 31 for control volumes, 13–31, W29–W31, 394–397, 1029–1030 equations, 14, 16, 17 surface energy balance, 27–30 Conservation of mass, W25–W26, 1028 Conservation of species, W32–W36 and boundary layer equations, 394–397 for nonstationary media, 1030 for stationary media, 947–954 Constriction resistance, 690 Contact resistance, 117–119, 120 Continuity equation, W26 Control surface, 13 Control volume(s): definition of, 13, 31 differential, 31, 83–85, 394, 948–949 Convection, 377–418. See also Boiling; Condensation; External flow; Free convection; Internal flow boundary conditions (table), 91 boundary layers concentration boundary layer, 380–382 dimensionless parameters, 398–402, 407–409 equations for, W25–W36, 394–406, 1027–1030, 1031–1033 evaporative cooling, 413–416 heat and mass transfer analogy, 410–416 laminar and turbulent, 389–393 normalized equations, 398–406 Reynolds analogy, 416–417 significance of, 382 thermal boundary layer, 379–380 velocity boundary layer, 378–379 coefficients, 8, 289, 380–385, 400–406 definition of, 2 dimensionless parameter significance, 407–409 forced, 6–7, 398. See also Boiling, forced convection; External flow; Internal flow free (natural), see Free convection laminar flow and boundary layers, 389–393 mass and heat transfer analogy, 378 micro- and nanoscale effects, 558–562 mixed, 7, 628 problem of, 385 rate equation, 8, 46 in surface energy balance, 27–28 transfer equations, W25–W36, 1027–1030 turbulent flow and boundary layers, 389–393 Convection heat transfer coefficient, 8, 289, 380, 382–383, 385, 400–401 local and average, 382–383 Convection mass transfer coefficient, 381–382, 383–385, 401–402 local and average, 383–385 Cooling, evaporative, 413–416 Counterflow heat exchangers, 706–707, 714–715, 722–727 Creeping flow, 465 Critical film thickness for microscale conduction, 73–74 Critical heat flux, 658, 659, 662–663, 670, 673 bindex.qxd 2/25/11 6:21 AM Page 1041 䊏 1041 Index Cross-flow heat exchangers, 706–707, 715,724–727 Cylinder(s): in cross flow, 455–465 flow considerations, 455–456 heat and mass transfer (convection), 457–465 free convection with concentric cylinders, 624–625 long horizontal cylinder, 613–616, 618 one-dimensional steady-state conduction in, 136–141, 1019–1024 shape factors for, 236–237 transient conduction in, 300–301, 303–307, 318–320 graphical representation of, W12, W14–W15 summary (table), 321–322 D Dalton’s law of partial pressures, 936 Darcy friction factor, for internal flow, 522–523 Density, 78 gradients, 594, 654 mass, 935 Differential control volumes, 31, 83–85, 394, 948–949 Diffuse emitters, 776, 782, 794 Diffuse radiation, 823 Diffusion: energy transfer by, 3, 6, W30 mass, see Mass diffusion Diffusion-limited processes, 962 Diffusive reflectors, 782 Diffusive species flux, 939–942 Diffusivity mass, 937 momentum, 407 thermal, 78 Dilute gas or liquid, 947 Dimensionless conduction heat rate, 235–240, 317–322 Dimensionless parameters: boiling and condensation, 654–655 boundary layers, 379, 390, 398–402, 407–409 conduction, 284–319 free convection, 598–599 of heat and mass transfer (table), 408–409 Dimensions, 36–38 Direct radiation, 823 Dirichlet conditions, 90 Discontinuous boundary conditions, 954–956 Discretization of the heat equation: explicit method of, 330–337 implicit method of, 337–345 Dittus-Boelter equation, 544–545 Drag coefficient, 456 Dropwise condensation, 674–675, 690 Dynamic viscosity, 80, 379 E Eckert number, 408 Effective thermal conductivity, 119–121 Effectiveness fin, 164 heat exchanger, 722–723 Effectiveness-NTU analysis method, 722–730, 739–746 definitions in, 722–723 Efficiency: Carnot, 32–36 fin, 165–172 of heat engines, 31–36 Eigenvalues, 300 Electrical energy, and thermoelectric power, 182–188 Electromagnetic spectrum, 769–770 Electromagnetic waves, 769–770 Emission, 768–770 band, 785–792 gaseous, 897–901 and intensity, 774–779 Emissive power, 9, 771, 775–776, 784–785 of a blackbody, 9, 784–785 Emissivity, 9–10 definition of, 792 of real surfaces, 792–796 representative values (table), 796 of selected surfaces (table), 1008–1010 Empirical method, 435–436 Enclosed fluids, free convection with, 621–627 Energy balance: atmospheric radiation, 821–823 for internal flow, 529–536 method for discretization, 243–249 surface, 27–30 Energy carriers, 71 Energy generation, 14–16, 84, 142–154, 182–188 Energy sources, 42–43, 84, 183–184 Energy storage, 14, 84 Energy use and sustainability, 41–43, 182–188 Enhancement, heat transfer boiling, 665 condensation, 685 fins, 155, 165 internal flow, 555–558 Enhancement surface(s), 665 Enthalpy, and steady-flow energy equation, 16–17 Entry length(s): concentration, 563 hydrodynamic, 519 thermal, 524 Entry region(s): hydrodynamic, 518–519 and internal flow, 542–544 thermal and combined, 524, 542–544 Environmental radiation, 818–826 atmospheric irradiation, 824 atmospheric radiation balance, 821–823 extraterrestrial solar, 819 scattering, 821 solar, 818–821 solar constant, 819 spectral distributions, 820 terrestrial solar irradiation, 823–824 Error function, 313, 1015 Evaporation, 15. See also Boiling in column, 942–947 cooling and, 413–416 mass transfer and, 563–565, 955 Evaporators, 654 Excess temperature, 158, 655 Extended surfaces, heat transfer from, 112, 154–178 conduction analysis, 156–158 fin characteristics and parameters, 154–156 fin effectiveness, 164 fin efficiency, 165–172 overall surface efficiency, 170–178, 709–710 nonuniform cross-sectional area fins, 167–170 uniform cross-sectional area fins, 158–164 bindex.qxd 2/25/11 1042 6:21 AM 䊏 Page 1042 Index External flow, 433–486 across banks of tubes, 468–476 cylinder in cross flow, 455–465 flow considerations, 455–456 heat and mass transfer (convection) in, 457–465 empirical method for, 434, 435–436 flat plate in parallel flow, 436–447 with constant heat flux conditions, 446 laminar flow, 437–443 with mixed boundary layer conditions, 444–445 turbulent flow, 443 with unheated starting length, 445–446 forced convection boiling, 669–670 free convection horizontal cylinder, 613–616 inclined and horizontal plates, 608–613 over vertical plate, 605–608 spheres, 617–618 friction coefficients of, 379 heat transfer correlations (table), 484–485 impinging jet(s) considerations, 477–478 heat and mass transfer (convection) in, 477–482 methodology for convection calculation, 447 over sphere, 465–468 packed bed(s), 482–483 similarity method for, 434–435, 437–443 F Fanning friction factor, 522 Fick’s law, 381–382, 936–937 Film boiling, 658–660, 663–665 Film condensation, 674–690 definition of, 674 laminar, 675–679 in tubes, 689–690 on tubes, 684–686 turbulent, 679–681 wavy, 680 Film temperature, 414, 436 Film(s), thermal conductivity of, 73–75, 77, 190 Finite control volumes, energy conservation of, 31 Finite-difference method: transient conduction explicit method of discretization of the heat equation, 330–337 implicit method of discretization of the heat equation, 337–345 two-dimensional steady-state conduction, 241–256 energy balance method in, 243–249 Gauss-Seidel iteration method, W5–W9, 250, 1025–1026 heat equation form, 242–243 nodal network selection, 241–242 solving, 250–256 Fins, 154–178 annular, 155–156, 167, 685 conduction analysis, 156–158 effectiveness, 164 efficiency, 165–172 film condensation on, 684–686, 690 free convection with, 618 of nonuniform cross-sectional area, 167–170 overall surface efficiency, 170–178, 709–710 performance measures, 164–167 pin, 155–156 straight, 155–156, 166 of uniform cross-sectional area, 158–164 First law of thermodynamics, 12–14 First-order chemical reactions, 961–964 Flat plate: boundary layers and, 378–382 parallel flow over, 436–447 with constant heat flux conditions, 446 integral boundary layer solution for, 1035–1038 laminar flow, 389–393, 437–443 with mixed boundary layer conditions, 444–445 turbulent flow over, 389–393, 443 with unheated starting length, 445–446 Flow. See also External flow; Internal flow compressible, 397 creeping, 465 steady, two-dimensional, W25–W36, 394, 1027–1030 Flow work, 16 Fluidized beds, 482 Fluids: convection and, 378 free convection with enclosed, 621–627 incompressible, 394, 1028 nanofluid, 77, 80–82 Newtonian, W28, 379 and problem of convection, 385 thermal conductivity of, 75–77 thermophysical properties of (table), 1000–1005 viscous, W25–W36, 1027–1030 Flux-plotting method, W1–W5, 231 Forced convection, 6–7, 398, 669–673 combined free and forced, 627–628 and external flow, see External flow and internal flow, see Internal flow Forced convection boiling, 655, 669–673 external, 669–670 two-phase flow, 670–673 flow regimes, 671 Form drag, 456 Fouling: in condensation, 675 in heat exchangers, 709–711 Fouling factor, 709 Fourier number, 285, 408 Fourier’s law, 4–5, 68–70, 86–87 Free boundary flows, 595–596 Free convection, 6–8, 593–631 applications of, 594 buoyancy and, 594–596 combined free and forced, 627–628 dimensionless parameters for, 598–599 empirical correlations (table), 617–618 with enclosed fluids, 621–627 concentric cylinders, 624–625 concentric spheres, 625–627 rectangular cavities, 621–624 external flows, 604–618 horizontal cylinder, 613–616 inclined and horizontal plates, 608–613 spheres, 617–618 vertical plate, 605–608 free convection boiling, 657–658 governing equations, 597–598 laminar free convection on a vertical surface, 599–602 and mass transfer, 628–629 mixed convection, 627–628 physical considerations of, 594–596 bindex.qxd 2/25/11 6:21 AM Page 1043 䊏 1043 Index turbulence effects, 602–604 within parallel plate channels, 618–621 inclined channels, 621 vertical channels, 619–621 Free convection boiling, 657–658 Free stream, 379 Freezing, 15 Friction coefficient, 379, 382, 400, 408, 440, 442, 443, 444, 522 Friction drag, 456 Friction factor, 408 for external flow, 472–473 for internal flow, 522–523, 553, 557 Froude number, 672 G Gas(es): conduction in, 3 convection coefficients, typical (table), 8 emission from, 768–769 ideal, thermal energy equations for, 16–20 mass diffusion in, 934–935 micro- and nanoscale conduction effects, 189–190 micro- and nanoscale convection effects, 558–559 radiation exchange with, 896–901 solubility of, 955–960, 1007 thermal conductivity of, 75–78 thermal radiation and, 10 thermophysical properties of (table), 995–999 Gauss-Seidel iteration method, 250, 1025–1026 example, W5–W9 Gaussian error function, 313, 1015 Generation, see Thermal energy generation Graphical methods: for two-dimensional steady-state conduction, 231 conduction shape factors, W3–W5 flux-plot construction, W1–W2 heat transfer rate determination, W2–W3 Grashof number, 408–409, 599, 628 Gravitational field, and pool boiling, 664 Gray surfaces: radiation behavior, 812–814 radiation exchange, 876–893 net radiation exchange, 877–878 radiation shields, 886 reradiating surfaces, 888–893 surface radiation exchanges, 878–880 thermal radiation and, 10 H Heat diffusion equation (heat equation), 82–91 boundary conditions, 90–91 finite-difference form, 242–243, 330–345 microscale effects, 90 Heat engines, efficiency of, 31–36 Heat equation, see Heat diffusion equation Heat exchangers, 705–748 compact, W44–W49, 708, 739 design problems, 730 effectiveness (table), 724 effectiveness-NTU analysis method, 722–730, 739–746 definitions in, 722–723 relations, 723–727 log mean temperature difference (LMTD) analysis, 711–721 analysis with, 711–712, 739–746 for counterflow heat exchangers, 714–715 for multipass and cross-flow heat exchangers, W40–W44 for parallel-flow heat exchangers, 712–714 NTU (table), 725 overall heat transfer coefficient for, 708–711 performance calculation problems, 730 types of, 706–708 Heat flow lines, 230 plotting, W1–W2 Heat flux, 4–5, 8, 9–12, 85 critical, 658, 659, 662–663, 673 radiation fluxes, 771–772 Heat rate, 4–5, 10, 33 Heat sinks, 44 Heat transfer: in convection, 382–383 definition of, 2 dimensionless groups in, 407–409 efficiency and, 32–36 enhancement research, 739 from extended surfaces, 112, 154–178 conduction analysis, 156–158 fin characteristics and parameters, 154–156 fin effectiveness, 164 fin efficiency, 165–172 overall surface efficiency, 170–178, 709–710 nonuniform cross-sectional area fins, 167–170 uniform cross-sectional area fins, 158–164 in insulation systems, 77–78 methodology for problem-solving, 38–41, 114 multimode, 893–895 physical mechanisms of, 3–12 rate determination (two-dimensional steady-state conduction), W2–W3 relevance of, 41–45 summary of modes (table), 46 thermodynamics vs., 12–13 Henry’s constant, 956 for selected gases in water (table), 1007 Henry’s law, 956 Heterogeneous chemical reactions, 960–962 Homogeneous chemical reactions, 949, 960n, 962–965 Hydraulic diameter, 552 Hydrodynamic boundary layers, 6. See also Velocity boundary layer Hydrodynamic considerations: with impinging jet(s), 477–478 with internal flow, 518–523 Hydrodynamic entry length, 519 Hyperbolic functions (table), 1014 I Ideal gases, 16–17 Impingement zones, 477–478 Impinging jet(s): considerations, 477–478 heat and mass transfer (convection) through, 478–482 nozzle considerations, 480–482 Incident radiation, 779 Incompressible liquids, 16–17, W26, W29, 394, 1027–1030 Initial conditions, 90–91 Insulation: micro- and nanoscale effects, 77–78 systems and types, 77 thermophysical properties of (table), 990–992 typical thermal conductivities, 71 Intensity, radiation, 773–782 Internal energy, 13–15, W31 bindex.qxd 2/25/11 1044 6:21 AM 䊏 Page 1044 Index Internal flow, 517–568 in circular tubes convection correlations (table), 567 laminar flow, 537–544 turbulent flow, 544–552 in coiled tubes, 556–558 convection mass transfer, 563–565 energy balance in, 529–536 with constant surface heat flux, 530–533 with constant surface temperature, 533–536 general considerations, 529–530 heat transfer enhancement in, 555–558 hydrodynamic considerations, 518–523 flow conditions, 518–519 friction factor, 522–523 mean velocity, 519–520 velocity profile, 520–522 micro- and nanoscale effects, 558–562 in noncircular tubes, 552–555 thermal considerations, 523–529 with fully developed conditions, 525–527 mean temperature, 524–525 Newton’s law of cooling in, 525 Irradiation, 9–12, 771, 779–781, 801 Isothermal surfaces, 69 Isotherms, 69–70, 230, 235 Isotropic media, 70 effective thermal conductivities in, 121 J Jakob number, 409, 655 Jet(s): in boiling, 658–659 buoyant, 595–596 impinging, see Impinging jet(s) Joule heating, see Ohmic heating K Kelvin, 37 Kelvin-Planck statement, 31 Kinematic viscosity, 407 Kirchhoff’s law, 810–811 L Laminar boundary layer, 389–393 Laminar film condensation, 675–679 Laminar flow: boundary layers and equations, 389–397, 597–598 in circular tubes, 537–544 in noncircular tubes, 552–555 over flat plate, 437–443 Latent component, of internal energy, 15 Latent energy, in convection, 7 Latent heat, in boiling/condensation, 654 Latent heat of fusion, 26–27 Lattice waves, conduction and, 4, 71–72 Leidenfrost point, 660 Length, units for, 36–37 Lewis number, 407–409 Liquid metals: convection coefficients for, 442–443, 546 thermophysical properties of (table), 1005 Liquid(s): conduction in, 3–4 convection coefficients, typical (table), 8 gas solubility in, 955–960 mass diffusion in, 935 microscale convection in, 559–560 radiation from, 768–769 thermal conductivity of, 75–77 thermal energy equations for, 16–17 thermal radiation and, 8, 10 Log mean temperature difference method (LMTD), 711–721, 739–746 for counterflow heat exchangers, 714–715 for multipass and cross-flow heat exchangers, W40–W44 for parallel-flow heat exchangers, 712–714 Longitudinal pitch, 468–469 Lumped capacitance method, 280–297 calculations for, 281–283 conditions for, 280–281 general lumped capacitance analysis, 287–297 validity of, 283–286 Lumped thermal capacitance, 282 M Mach number, 409 Martinelli parameter, 689 Mass: conservation of, see Conservation of mass units for, 36–37 Mass diffusion, 933–972 boundary conditions and discontinuous interface concentrations, 954–962 catalytic surface reactions, 960–962 evaporation and sublimation, 955 solubility of gases in liquids and solids, 955–960 with homogeneous chemical reactions, 962–965 mass diffusion equation, 948–950 in nonstationary media, 939–947 absolute and diffusive species fluxes, 939–942 evaporation in column, 942–947 physical process of, 934–935 Fick’s law and, 936–937 mass diffusivity, 937–939 mixture composition, 935–936 in stationary media, 947–954 conservation of species for control volumes, 948 mass diffusion equation, 948–950 with specified surface concentrations, 950–954 stationary medium approximation, 947 transient diffusion, 965–971 Mass diffusion equation, 948–950 Mass diffusivity, 937–939 Mass flow rate, 16, 17 Mass transfer by convection, 383–385 dimensionless groups in, 407–409 external flow, 434, 441–444, 447 cylinder in cross flow, 457–465 impinging jet(s), 477–482 packed bed(s), 482–483 in free convection, 628–629 heat transfer analogy, 410–416 internal flow, 563–565 Matrix equation method, 250 Mean beam length, 900 Mean free path, 71, 73–75 Mean temperature, of internal flow, 524–525 Mean velocity, of internal flow, 519–520 Melting, 15 Metabolic heat generation, 178–182 bindex.qxd 2/25/11 6:21 AM Page 1045 䊏 1045 Index Metals and metallic solids: emissivity of (table), 1008 thermal conductivity of, 71–72, 77 thermophysical properties of, 983–986, 1005, 1008 Microchannels in boiling, 673 in condensation, 690 effects, 378 in internal flow, 558–560 Microfluidic devices, 558 Microscale effects: in conduction, 72–75, 77–78, 90, 189–190 in convection, 380n, 558–562 Mie scattering, 821 Mixed convection, 7, 628 Mixtures, characteristics of, 935–936 Modes of heat transfer, definition of, 2 Modified Bessel equations, 167–168 Molar concentration, 935 Momentum accommodation coefficients, 378n, 558–559 Momentum diffusivity, 407 Moody diagram, 523 Moody friction factor, for internal flow, 522–523 Multimode heat transfer, 893–895 Multipass heat exchangers, W40–W44, 708, 715 N Nanofluid, 77, 80–82 Nanoscale effects: in conduction, 72–75, 77–78, 189–190 in convection, 380n, 560 in radiation, 769 Nanostructured materials, 74, 77–78, 186 Natural convection, see Free convection Net radiation exchange, 877–878 Net radiative flux, 771–772, 782 Neumann conditions, 90–91 Newtonian fluids, W28, 379 Newton’s law of cooling, 8, 115, 380, 525, 655 Newton’s second law of motion, W26–W29, 1028–1029 Nodal network, 241–242, 879–880 Nodal points, 241–242, 879–880 Noncircular tubes, see Tubes Nonmetallic materials: emissivity of solids (table), 1009–1010 thermal conductivity of, 71–72, 76–77 thermophysical properties of solids, 987–988 Nonparticipating media, 862 Nonstationary media: absolute and diffusive species fluxes, 939–942 evaporation in column, 942–947 Nuclear component, of internal energy, 15 Nucleate boiling, 658–659, 660–664 Number of transfer units (NTU), 723–725 Nusselt number, 401, 409 O Ohmic heating, 143 One-dimensional steady-state conduction, 111–193 alternative analysis approach, 132–135, 141–142 bioheat equation, 178–182 extended surfaces and, see Extended surface(s), heat transfer from micro- and nanoscale effects, 189–190 in plane wall systems composite walls, 115–117 contact resistance in, 117–119, 120 temperature distribution, 112–114 with thermal energy generation, 143–149 thermal resistance in, 114–115, 708–709 within porous media, 119–125 in radial systems, 136–142 cylinders, 136–141 spheres, 141–142 with thermal energy generation, 149–150 summary solutions (table), 143 temperature distribution in, 4–5, 85 with thermal energy generation, 142–154 in plane wall systems, 143–149 in radial systems, 149–154 thermal conditions with uniform generation, 1019–1024 and thermoelectric power generation, 182–188 uniform generation thermal conditions, 1019–1024 Opaque media, 772, 781–782, 805–806 Open systems, 13–17 Ordinary diffusion, 937 Orthogonal functions, 233–234 Overall heat transfer coefficient, 116, 137–138 and heat exchangers, 708–711 Overall surface efficiency, 170–178, 709–710 P Packed bed(s): definition of, 119 heat and mass transfer (convection) through, 482–483 Parallel-flow heat exchangers, 706–707, 712–714, 723–727 Parallel plates, free convection with, 618–621 Parameter sensitivity study, 38 Participating media, 862 radiation exchange with, 896–901 Peclet number, 409 Peltier effect, 183–184 Penetration depth: concentration, 970 thermal, 314 Pennes equation, 178–182 Perfusion, and bioheat equation, 178–182 Phase change, 7, 15 convection coefficients, typical (table), 8 Phonons, 71–75 Photons, 769 Pin fins, 155–156 Pitch (tubes), 468–469 Planck constant, 783 Planck distribution, 783–784 Planck’s law, 783–784, 827 Plane angle, 773 Plane wall systems: one-dimensional steady-state conduction in, 112–132 composite walls, 115–117 contact resistance in, 117–119, 120 temperature distribution, 112–114 with thermal energy generation, 143–149, 1019–1024 thermal resistance in, 114–115, 708–709 within porous media, 119–121 shape factors for, W3–W4, 236 transient conduction in, 283–286, 298–303, 318–323 approximate solution, 300–301, 318–323 with convection, 299–303 exact solution, 300 graphical representation of, W12–W13 roots of transcendental equation for, 1016 summary (table), 321–322 bindex.qxd 2/25/11 1046 6:21 AM 䊏 Page 1046 Index Plumes, 595–596 Pool boiling, 655, 656–669 boiling curve and, 656–657 critical heat flux, 658, 659, 662–663 film boiling, 658, 660, 663–664 free convection boiling, 657–658 Leidenfrost point, 660 minimum heat flux, 658, 660, 663 nucleate boiling, 658–659, 660–663 parametric effects on, 664–665 transition boiling, 658, 659–660 Porosity, 483 Porous media, conduction in, 119–121 Power-controlled heating, 656–657 Prandtl number, 398–399, 407–409 Problems, methodology for analysis, 38 Q Quality of fluid, 671 Quanta, 769 Quasi-steady approximation, 616 Quenching, 283 Quiescent fluid(s), 596, 596n R Radial systems: film condensation in, 684–688 one-dimensional steady-state conduction in, 136–142 cylinders, 136–141 spheres, 141–142 with thermal energy generation, 149–154 transient conduction in, 303–310, 318–322 Radiation. See also Radiation exchange and absorptivity, 802–803 blackbody, see Blackbody radiation emission from real surfaces, 792–800 environmental, see Environmental radiation gaseous, 896–901 gray surface, see Gray surfaces heat fluxes, 771–772 intensity, 773–782 definitions in, 773–774 and emission, 774–779 and irradiation, 779–781 and net radiative flux, 782 and radiosity, 781–782 and Kirchhoff’s law, 810–811 nature and properties of, 768–770 rate equation, 10, 46 and reflectivity, 803–804 surface characteristics considerations, 805–806 in surface energy balance, 27–30 terminology glossary (table), 827–828 thermal, see Thermal radiation and transmissivity, 805 Radiation balance (atmospheric), 821–823 Radiation exchange, 861–902 between diffuse gray surfaces (enclosed), 876–893 net radiation exchange, 877–878 radiation shields, 886 reradiating surfaces, 888–893 surface radiation exchanges, 878–880 two-surface enclosures, 884–885 blackbody radiation, 872–876 gaseous, 896–901 emission and absorption, 897–901 volumetric absorption, 896–897 and multimode heat transfer, 893–895 view factors in, 862–872 definition, 862 for two-dimensional geometries (table), 865–867 view factor integral, 862–863 view factor relations, 863–870 Radiation heat transfer coefficient, 10 Radiation intensity, see Radiation, intensity Radiative resistance, 877–879 Radiosity, 771–772, 781–782 Raoult’s law, 955 Rate equations: for conduction, 4–5 for convection, 8 for radiation heat transfer, 10 summary (table), 46 Rayleigh number, 603 Rayleigh scattering, 821 Reaction-limited processes, 962 Reciprocity relation, 863 Rectangular cavities, free convection in, 621–624 Reflection, 558–559, 801–802 and reflectivity, 772 Reflectivity, 803–804 Reradiating surfaces, 888–893 Resistance: constriction, 690 contact, 117–119, 120 fin, 165 radiative, 877–879 thermal, 12, 114–115, 137, 142 Resistance heating, see Ohmic heating Reynolds analogy, 416–417 Reynolds number, 390, 398–399, 407–409 Reynolds stress, 1033 S Saturated boiling, 655, 656, 671 Saturated porous media, 119–120 Schmidt number, 398–399, 407–409 Second law of thermodynamics, 31–36 Seebeck effect and coefficient, 182–188 Semi-infinite solid(s): transient conduction in, 310–318, 319 solutions summarized, 313–314 Semitransparent media, 771, 805–806 Sensible energy, 7, 15, 84 Separation of variables, method, 231–235, 299 Separation point(s), 455 Shape factor(s): conduction, W3–W5, 235–240 view factor, 862–872 Shear stresses, 379 Shell-and-tube heat exchangers, W40–W44, 707, 723–727 Sherwood number, 402, 409 Shields, radiation, 886–888 SI (Système International d’Unités) system, 36–38 Similarity solution(s), 438, 600 Similarity variable(s), 311, 438 Simplified steady-flow thermal energy equation, 17 Sinks (energy), 16, 84, 183–184 bindex.qxd 2/25/11 6:21 AM Page 1047 䊏 1047 Index Solar radiation, 818–824 properties for selected materials (table), 1010 representative values for surfaces (table), 824 Solid angle, 773 Solidification, 15 Solid(s): conduction in, 3–5, 118–119, 190 gas solubility in, 955–960 mass diffusion in, 935 radiation from, 9–12, 768–769 semi-infinite, see Semi-infinite solid(s) solubility of (table), 1007 thermal conductivity of, 71–75 micro- and nanoscale effects, 72–75, 190 Solubility: of gases in liquids and solids, 955–960 of selected gases and solids (table), 1007 Species: characteristics of, 934–936 concentration in mass transfer, 563–565 conservation of, see Conservation of species Species fluxes, 939–942 Specific heat, 78 Spectral absorptivity, 802 Spectral emission, 775 Spectral emissivity, 793 Spectral intensity, 774–775 Spectral irradiation, 779, 801 Spectral radiosity, 781–782 Spectral reflectivity, 804 Sphere(s): dimensionless conduction heat rate for, 238 film condensation on, 684 free convection with, 617–618 concentric spheres, 625–626 heat and mass transfer (convection) from, 465–468 one-dimensional steady-state conduction in, 141–142, 1019–1024 shape factors for, 236–238 transient conduction in, 300–301, 303–305, 308–310, 318–320 graphical representation of, W12, W15–W16 summary (table), 321–322 Stagnation point(s), 455 Stagnation zone(s), 477–478 Stanton number, 409, 416–417 Stationary media: diffusion approximation for, 947 mass diffusion in, 947–954 with specified surface concentrations, 950–954 Steady-state conditions, 4, 14, 16, 112 Stefan-Boltzmann constant, 9 Stefan-Boltzmann law, 9, 784–785 Stokes’ law, 465 Straight fins, 155–156, 166 Stratification parameter, 672 Streaks, 389 Stresses: shear, 379 viscous, W26–W29, 1029 Structural building materials, thermophysical properties of (table), 989 Subcooled boiling, 655, 664–665, 670–671 Sublimation, mass transfer and, 563–565, 955 Summation rule, 864 Surface energy balance, 27–30 Surface forces, W26–W29, 1029 Surface friction, and boundary layers, 382 Surface phenomena, 16 radiation as, 769, 801–802 Surface roughness, 665 Surface tension, 654, 655 Surface(s): radiation exchange between gray, 876–893 surface energy balance, 27–30 Surroundings, 9–10 T Temperature: conduction and, 2–5 and efficiency, 32–33 excess, 158, 655 film, 414, 436 mean, of internal flow, 524–525 scales, 37 units for, 36–37 Temperature distribution, 82 during thermal treatment, 45 one-dimensional steady-state conduction, 4–5, 112–114 two-dimensional steady-state conduction, 230–231, 231–232, 242–243 Thermal accommodation coefficient, 189–190, 380n Thermal boundary layer, 6, 379–380, 382 and laminar or turbulent flow, 391–393 Thermal circuits, 112–117, 171–172 Thermal conductivity, 70–78 bulk solid, 72 conduction and, 4–5 effective, 119–121 of fluids, 75–78 and Fourier’s law, 68–70 of insulation systems, 77–78 of porous media, 119–121 of solids, 71–75 Thermal contact resistance, 117–119, 120, 171–172 Thermal diffusivity, 78–80, 85 Thermal energy, components of, 15 Thermal energy equation, W31 Thermal energy generation: conduction with, 142–154, 1019–1024 bioheat, 178–182 in plane wall systems, 143–149 in radial systems, 149–154 Thermal entry length, 542–544 Thermal penetration depth, 314 Thermal radiation, 8–12 and boiling, 663–664 definition of, 2, 769–770 emission of, 768–770 resistance for, 115 Thermal resistance, 12, 114–115, 137–142 fouling factor, 709 in plane wall systems, 114–117, 708–709 thermal contact resistance, 117–119, 120 Thermal time constant, 282 Thermodynamic properties, 78–82 Thermodynamics, heat transfer vs., 12–13 Thermoelectric power generation, 182–188 Thermophysical properties, 78–82, 981–1010 of common materials (table), 989–994 industrial insulation, 991–992 bindex.qxd 2/25/11 1048 6:21 AM 䊏 Page 1048 Index Thermophysical properties (continued) insulating materials/systems, 990 structural building materials, 989 of gases at atmospheric pressure (table), 995–999 of liquid metals (table), 1005 of saturated fluids (table), 1000–1002 of saturated water (table), 1003–1004 of selected metallic solids (table), 983–986 of selected nonmetallic solids (table), 987–988 of thermoelectric modules, 183–186 Thermoregulation, 28–30, 44–45, 121–125 Time, units for, 36–37 Transient conduction, 279–346 coefficients for one-dimensional conduction (table), 301 finite-difference methods for explicit method of discretization of the heat equation, 330–337 implicit method of discretization of the heat equation, 337–345 graphical representation of, W12–W22 lumped capacitance method, 280–297 multidimensional effects with, W16–W22 roots of transcendental equation (plane wall), 1016 objects with constant surface heat flux, 319–320, 322 objects with constant surface temperature, 317–319, 321 periodic heating, 327–330 plane wall with convection, 299–303 solutions for, W12–W13, 300–301 radial systems with convection, 303–310 solutions for, W14–W16, 303–304 in semi-infinite solids, 310–317 solutions summarized, 313–314 spatial effects, 298–299 Transient diffusion, 965–971 Transition boiling, 658, 659–660 Transition to turbulence, 389–391 Transmissivity, 771–772, 805 Transport properties, 70, 78–79 Transverse pitch, 468–469 Triangular fins, 168–170 Tubes. See also Heat exchangers arrangements of, 468–470 banks, 468–477 boiling in, 672–673 boiling on, 664 circular convection correlations (table), 567 laminar flow in, 537–544 turbulent flow in, 544–552 concentric tube annulus, 553–555 condensation in, 689–690 condensation on, 684–688 in cross flow, 468–476 configurations, 468–469 flow conditions, 468–470 noncircular, 552–555 rough vs. smooth, 545–546 Turbulent boundary layer, 389–391, 602 Turbulent film condensation, 679–683 Turbulent flow: and boundary layers, 389–393, 602–604, 1031–1033 in circular tubes, 544–552 across cylinders, 455–459 over flat plate, 443 Two-dimensional steady flow, heat and mass transfer in, W25–W36, 1027–1030 Two-dimensional steady-state conduction, 229–257 alternative approaches to, 230–231 conduction shape factors in, W3–W5, 235–240 dimensionless conduction heat rate in, 235–240 finite-difference method for, 241–256 solving, 250–256 graphical method for conduction shape factors, W3–W5 flux-plot construction, W1–W2 heat transfer rate determination, W2–W3 separation of variables method with, 231–235 Two-phase flow, forced convection boiling, 670–673 U Unheated starting length, 445 Unit mass, in flow work, 16 Units: derived, 37 English system, 36 SI system, 36–38 Unsaturated porous media, 119 V Vapor blanket, 660, 663 Vaporization, 15 Velocity boundary layer, 378–379, 382 and laminar or turbulent flow, 389–391 Velocity profile, boundary layer, 379 Velocity profile(s), for internal flow, 519–522 View factor(s), 862–872 definition of, 862 integral, 862–863 for two-dimensional geometries (table), 865–867 view factor relations, 863–870 Viscosity: dynamic, 80, 379 kinematic, 78 Viscous dissipation, 17, W31, 396, 1029 Viscous fluids, heat and mass transfer in, W25–W36, 1027–1030 Viscous stresses, W26–W29, 1029 Void fraction, 483 Volumetric flow rate, 17 Volumetric heat capacity, 78 Volumetric phenomena, 15–16 radiation, 768–769, 801, 896–901 Volumetric thermal expansion coefficient, 597 W Wall jet(s), 477–478 Water, thermophysical properties of (saturated), 1003–1004 Weber number, 409, 670 Wien’s displacement law, 784 Z Zenith angle, 774, 819 Zero-order chemical reactions, 962–963 BMConversionFactors.qxd 2/21/11 6:07 PM Page 2 Conversion Factors Acceleration Area 1 m/s2 1 m2 Density Energy Force Heat transfer rate Heat flux Heat generation rate Heat transfer coefficient Kinematic viscosity and diffusivities Latent heat Length 1 kg/m3 1 J (0.2388 cal) 1N 1W 1 W/m2 1 W/m3 1 W/m2 • K ⫽ 4.2520 ⫻ 107 ft/h2 ⫽ 1550.0 in.2 ⫽ 10.764 ft2 ⫽ 0.06243 lbm/ft3 ⫽ 9.4782 ⫻ 10⫺4 Btu ⫽ 0.22481 lbf ⫽ 3.4121 Btu/h ⫽ 0.3170 Btu/h • ft2 ⫽ 0.09662 Btu/h • ft3 ⫽ 0.17611 Btu/h • ft2 • ⬚F 1 m2/s ⫽ 3.875 ⫻ 104 ft2/h 1 J/kg 1m ⫽ 4.2992 ⫻ 10⫺4 Btu/lbm ⫽ 39.370 in. ⫽ 3.2808 ft ⫽ 0.62137 mile ⫽ 2.2046 lbm ⫽ 0.06243 lbm/ft3 ⫽ 7936.6 lbm/h ⫽ 1.1811 ⫻ 104 ft/h Mass Mass density Mass flow rate Mass transfer coefficient Power 1 km 1 kg 1 kg/m3 1 kg/s 1 m/s 1 kW Pressure and stress1 1 N/m2 (1 Pa) Specific heat Temperature 1.0133 ⫻ 105 N/m2 1 ⫻ 105 N/m2 1 kJ/kg • K K Temperature difference 1K Thermal conductivity Thermal resistance Viscosity (dynamic)2 1 W/m • K 1 K/W 1 N • s/m2 Volume 1 m3 Volume flow rate 1 m3/s 1 2 ⫽ 3412.1 Btu/h ⫽ 1.341 hp ⫽ 0.020885 lbf /ft2 ⫽ 1.4504 ⫻ 10⫺4 lbf /in.2 ⫽ 4.015 ⫻ 10⫺3 in. water ⫽ 2.953 ⫻ 10⫺4 in. Hg ⫽ 1 standard atmosphere ⫽ 1 bar ⫽ 0.2388 Btu/lbm • ⬚F ⫽ (5/9)⬚R ⫽ (5/9)(⬚F ⫹ 459.67) ⫽ ⬚C ⫹ 273.15 ⫽ 1⬚C ⫽ (9/5)⬚R ⫽ (9/5)°F ⫽ 0.57779 Btu/h • ft •⬚F ⫽ 0.52753 ⬚F/h • Btu ⫽ 2419.1 lbm/ft • h ⫽ 5.8015 ⫻ 10⫺6 lbf • h/ft2 ⫽ 6.1023 ⫻ 104 in.3 ⫽ 35.315 ft3 ⫽ 264.17 gal (U.S.) ⫽ 1.2713 ⫻ 105 ft3/h ⫽ 2.1189 ⫻ 103 ft3/min ⫽ 1.5850 ⫻ 104 gal/min The SI name for the quantity pressure is pascal (Pa) having units N/m2 or kg/m • s2. Also expressed in equivalent units of kg/s • m. BMPhysicalConstants.qxd 2/21/11 6:08 PM Page 3 Physical Constants Universal Gas Constant: ⫽ 8.205 ⫻ 10⫺2 m3 • atm/kmol • K ⫽ 8.314 ⫻ 10⫺2 m3• bar/kmol • K ⫽ 8.315 kJ/kmol • K ⫽ 1545 ft• lbf /lbmole • °R ⫽ 1.986 Btu/lbmole • °R Avogadro’s Number: ᏺ ⫽ 6.022 ⫻ 1023 molecules/mol Planck’s Constant: h ⫽ 6.626 ⫻ 10⫺34 J • s Boltzmann’s Constant: kB ⫽ 1.381 ⫻ 10⫺23 J/K Speed of Light in Vacuum: co ⫽ 2.998 ⫻ 108 m/s Stefan-Boltzmann Constant: ⫽ 5.670 ⫻ 10⫺8 W/m2 • K4 Blackbody Radiation Constants: C1 ⫽ 3.742 ⫻ 108 W • m4/m2 C2 ⫽ 1.439 ⫻ 104 m • K C3 ⫽ 2898 m • K Solar Constant: Sc ⫽ 1368 W/m2 Gravitational Acceleration (Sea Level): g ⫽ 9.807 m/s2 ⫽ 32.174 ft/s2 Standard Atmospheric Pressure: p ⫽ 101,325 N/m2 ⫽ 101.3 kPa Heat of Fusion of Water at Atmospheric Pressure: hsf ⫽ 333.7 kJ/kg Heat of Vaporization of Water at Atmospheric Pressure: hfg ⫽ 2257 kJ/kg
