APPENDIX A Thermophysical Properties of Matter 1 Table A.1 A.2 A.3 A.4 A.5 A.6 A.7 1 Page Thermophysical Properties of Selected Metallic Solids 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 The convention used to present numerical values of the properties is illustrated by this example: T (K) 300 ⫺7 where = 0.349 ⫻ 10 ⫺3 m /s and k ⫽ 521 ⫻ 10 2 䡠107 (m2/s) k 䡠103 (W/m 䡠 K) 0.349 521 ⫽ 0.521 W/m 䡠 K at 300 K. 899 903 905 905 906 907 909 911 916 916 918 919 921 898 Appendix A A.8 A.9 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 922 922 923 924 925 TABLE A.1 Thermophysical Properties of Selected Metallic Solidsa Properties at Various Temperatures (K) k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K 933 2702 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 Aluminum Pure 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 k (W/m 䡠 K) 7.86 ␣ 䡠 106 (m2/s) 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 Thermophysical Properties of Matter cp (J/kg 䡠 K) 䊏 (kg/m3) Composition Appendix A Melting Point (K) 899 Continued 900 TABLE A.1 Properties at Various Temperatures (K) 1336 19300 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) ␣ 䡠 10 (m2/s) 127 100 200 400 600 800 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 Thermophysical Properties of Matter Gold k (W/m 䡠 K) 䊏 cp (J/kg 䡠 K) 6 Appendix A (kg/m3) Composition k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K Melting Point (K) Stainless steels AISI 302 480 15.1 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 11340 129 24.1 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 36.7 125 159 934 143 224 164 232 107 383 28.0 640 31.7 682 112 295 105 308 98 330 Nichrome (80% Ni, 20% Cr) Inconel X-750 (73% Ni, 15% Cr, 6.7% Fe) Niobium 1665 8510 439 11.7 3.1 8.7 — 10.3 372 2741 8570 265 53.7 23.6 Palladium 1827 12020 244 71.8 24.5 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 Platinum Pure 2045 21450 133 71.6 25.1 77.5 100 72.6 125 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 1235 10500 235 429 Tantalum 3269 16600 140 57.5 24.7 Thorium 2023 11700 118 54.0 39.1 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 58.7 167 505 7310 227 66.6 40.1 51.0 127 154 220 264 556 430 225 57.5 133 54.6 112 73.3 215 73.2 141 59 — 44.2 145 136 274 61.9 867 412 250 58.6 146 55.8 134 89.5 165 76 — 47.8 171 110 349 22.7 992 Silver 58.9 97 186 147 884 259 444 187 59.2 110 59.8 99 85.2 188 71.8 136 52 — 46.1 139 146 253 98.9 790 425 239 57.8 144 54.5 124 62.2 243 174 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 24.0 626 27.6 — 33.0 — 90 380 80.2 485 14 480 13.5 473 86 459 99.4 179 51.9 186 112 376 62.2 64.1 65.6 160 172 189 901 Tin 23.0 39.7 118 169 649 179 141 25.4 606 25.4 611 24.2 602 24.7 606 Thermophysical Properties of Matter 601 12.6 402 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 Lead 9.2 272 17.3 512 16.6 515 15.2 504 15.8 513 34.0 132 153 1074 134 261 Appendix A 8055 Continued 902 TABLE A.1 Properties at Various Temperatures (K) k (W/m 䡠 K)/cp (J/kg 䡠 K) Properties at 300 K (kg/m3) cp (J/kg 䡠 K) k (W/m 䡠 K) 4500 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 12.4 200 400 600 800 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 21.6 342 23.7 362 26.0 344 28.8 33.0 344 344 44.6 50.9 714 867 2500 95 176 Thermophysical Properties of Matter 1953 100 䊏 Titanium Zinc 21.9 ␣ 䡠 10 (m2/s) 6 Appendix A Composition Melting Point (K) TABLE A.2 Thermophysical Properties of Selected Nonmetallic Solidsa Properties at Various Temperatures (K) k (W/m 䡠 K)/cp (J/kg 䡠 K) ␣ 䡠 106 (m2/s) 3970 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 272 27.6 9.99 — — 3500 509 2273 2210 1.60 2300 — — 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 10,000 21 4970 16.8 136 709 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 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 1400 11.1 5.7 0.87 1623 450 — 133 — 0.67 — 1950 5.70 450 800 2.10 0.37 364 1122 1950 600 88.0 2.29 0.59 1500 400 Thermophysical Properties of Matter 2323 200 䊏 Aluminum oxide, sapphire Aluminum oxide, polycrystalline Beryllium oxide 100 Appendix A Composition Properties at 300 K Melting Point cp k (K) (kg/m3) (J/kg 䡠 K) (W/m 䡠 K) 2600 935 808 3.98 1.89 0.46 337 5.25 — 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 903 Continued 904 TABLE A.2 Properties at Various Temperatures (K) 3100 3160 Silicon dioxide, crystalline (quartz) k, 储 to c axis k, ⬜ to c axis cp Silicon dioxide, polycrystalline (fused silica) Silicon nitride 1883 2650 675 200 400 — 880 600 — 1050 800 — 1135 1000 87 1195 1200 58 1243 1500 2000 30 1310 2500 745 745 2173 2400 691 392 2070 708 Thorium dioxide 3573 9110 235 Titanium dioxide, polycrystalline 2133 4157 710 Adapted from References 1, 2, 3 and 6. 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 2220 a 100 230 10.4 6.21 1883 Sulfur 490 ␣ 䡠 10 (m2/s) 䊏 Silicon carbide 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) Appendix A TABLE A.3 䊏 905 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 — 906 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 — TABLE A.3 Continued Industrial Insulation Description/ Composition 96 –192 40–96 10 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 0.030 215 230 240 255 270 285 300 310 365 420 530 645 750 0.078 0.088 0.048 0.046 0.045 0.076 0.056 0.058 0.043 0.038 0.035 0.052 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 1530 480 730 48 64 96 128 50 –125 50 0.071 0.059 0.052 0.049 0.105 0.087 0.076 0.068 0.150 0.125 0.100 0.091 920 120 420 420 420 590 920 190 255 300 185 190 0.023 0.025 0.026 0.027 0.029 0.035 0.030 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 0.051 0.065 0.087 0.098 0.085 0.082 0.055 0.059 0.061 0.063 0.075 0.089 0.063 0.079 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 920 815 450 200 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 Typical Thermal Conductivity, k (W/m 䡠 K), at Various Temperatures (K) Maximum Service Temperature (K) 0.104 907 Continued 908 TABLE A.3 Industrial Insulation (Continued) Description/ Composition Typical Density (kg/m3) 700 1145 1310 145 345 385 350 350 350 56 35 16 340 70 1255 430 0.071 0.079 922 560 0.108 0.115 — — 45 105 — 122 80 200 0.039 240 255 270 285 300 310 365 420 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 0.042 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 Thermophysical Properties of Matter 0.036 0.023 0.023 0.029 230 䊏 0.023 0.023 0.026 215 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 Typical Thermal Conductivity, k (W/m 䡠 K), at Various Temperatures (K) Maximum Service Temperature (K) Appendix A 䊏 TABLE A.3 909 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 — — 2050 — — 2325 2645 — — 1460 1350 2300 80 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 — — 835 — 960 960 960 1130 880 1260 880 1300 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 1.4 1.4 750 835 910 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 Appendix A 䊏 911 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 0.869 912 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 Appendix A TABLE A.4 T (K) 913 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 98.8 108.2 117.2 126.4 134.3 914 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 Appendix A TABLE A.4 T (K) 915 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. 916 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 Appendix A TABLE A.5 䊏 917 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 0.187 918 TABLE A.5 Appendix A 䊏 Thermophysical Properties of Matter Continued Saturated Liquid–Vapor, 1 atmb Fluid Tsat (K) hƒg (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 TABLE A.6 Temperature, T (K) Thermophysical Properties of Saturated Watera Specific Volume (m3/kg) Pressure, p (bars)b vƒ 䡠 10 3 vg cp, g ƒ 䡠 106 g 䡠 106 kƒ 䡠 103 kg 䡠 103 Prƒ 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 Viscosity (N 䡠 s/m2) Thermal Conductivity (W/m 䡠 K) Prg Surface Tension, ƒ 䡠 103 (N/m) Expansion Coefficient, ƒ 䡠 106 (Kⴚ1) Temperature, T (K) 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 46.04 114.1 174.0 273.15 275 280 285 290 Prandtl Number 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 390 400 410 420 430 919 0.00611 0.00697 0.00990 0.01387 0.01917 Thermophysical Properties of Matter 273.15 275 280 285 290 䊏 cp,ƒ Specific Heat (kJ/kg 䡠 K) Appendix A 206.3 181.7 130.4 99.4 69.7 Heat of Vaporization, hƒg (kJ/kg) Continued 920 TABLE A.6 Pressure, p (bars)b vƒ 䡠 103 vg Heat of Vaporization, hƒg (kJ/kg) 440 450 460 470 480 7.333 9.319 11.71 14.55 17.90 1.110 1.123 1.137 1.152 1.167 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 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 36.2 490 500 510 520 530 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 17.72 651 642 631 621 608 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 — — — — — 490 500 510 520 530 540 550 560 570 580 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 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 Temperature, T (K) Specific Volume (m3/kg) Specific Heat (kJ/kg 䡠 K) Thermal Conductivity (W/m 䡠 K) cp,ƒ cp,g ƒ 䡠 106 g 䡠 106 kƒ 䡠 103 kg 䡠 103 Prƒ Prg Surface Tension, ƒ 䡠 103 (N/m) Viscosity (N 䡠 s/m2) 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 Thermophysical Properties of Matter 1.482 1.541 1.612 1.705 1.778 Temperature, T (K) 䊏 108.3 123.5 137.3 159.1 169.1 Expansion Coefficient, ƒ 䡠 106 (Kⴚ1) Appendix A 590 600 610 620 625 Prandtl Number Appendix A TABLE A.7 Composition 921 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 — Total, Normal (n) or Hemispherical (h) Emissivity of Selected Surfaces 922 TABLE A.8 Metallic Solids and Their Oxidesa Emissivity, n or h, at Various Temperatures (K) Description /Composition 300 400 600 800 1000 (h) (h) (h) 0.02 0.06 0.03 0.06 0.04 0.07 0.82 0.05 0.06 0.76 (n) 0.05 0.10 0.12 0.14 0.03 0.03 0.03 0.07 0.03 1200 1500 2000 2500 0.04 0.50 0.04 0.58 0.04 0.80 0.04 0.05 0.06 (h) (h) (h) 0.06 0.25 0.80 0.08 0.28 0.82 0.10 0.31 0.12 0.35 0.15 0.42 0.21 0.26 (h) (h) 0.09 0.40 0.11 0.49 0.14 0.57 0.17 0.10 0.13 0.15 䊏 200 Appendix A 0.07 (h) (h) (h) (h) 0.01 0.06 0.02 0.07 (h) (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 Thermophysical Properties of Matter 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 Appendix A 䊏 TABLE A.8 923 Thermophysical Properties of Matter 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) 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 0.57 800 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 Pyroceram (n) Refractories (furnace liners) Alumina brick (n) Magnesia brick (n) Kaolin insulating brick (n) Sand Silicon carbide (h) (n) Skin Snow (h) (h) 924 Appendix A TABLE A.8 䊏 Thermophysical Properties of Matter 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, 24, and 25. b TABLE A.9 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 25. The emissivity values in this table correspond to a surface temperature of approximately 300 K. b S 0.92 0.92 Appendix A 䊏 Thermophysical Properties of Matter 925 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. 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. 8. American Society of Heating, Refrigerating and Air Conditioning Engineers, ASHRAE Handbook of Fundamentals, ASHRAE, New York, 1981. 22. Liley, P. E., Steam Tables in SI Units, private communication, School of Mechanical Engineering, Purdue University, West Lafayette, IN, March 1984. 9. Mallory, J. F., Thermal Insulation, Van Nostrand Reinhold, New York, 1969. 23. Liquid Materials Handbook, 23rd ed., The Atomic Energy Commission, Department of the Navy, Washington, DC, 1952. 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. 24. Gubareff, G. G., J. E. Janssen, and R. H. Torborg, Thermal Radiation Properties Survey, Minneapolis-Honeywell Regulator Company, Minneapolis, MN, 1960. 25. Kreith, F., and J. F. Kreider, Principles of Solar Energy, Hemisphere Publishing, New York, 1978. This page intentionally left blank APPENDIX B Mathematical Relations and Functions Section B.1 B.2 B.3 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 Page 928 929 930 931 932 928 B.1 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 Appendix B B.2 䊏 929 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 930 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 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 931 932 B.5 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 In⫹1(x) ⫽ In⫺1(x) ⫺ (2n/x)In(x) 1 APPENDIX C Thermal Conditions Associated with Uniform Energy Generation in One-Dimensional, Steady-State Systems 934 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 first 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 q• Ts,2 Cylindrical Wall r1 q• Ts,1 L Ts,2 r2 Spherical Wall Ts,1 q• r2 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. Appendix C One-Dimensional, Steady-State Conduction with Generation 䊏 935 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 q̇L2 x2 1⫺ 2 ⫹ ⫹ 2k 2 L 2 L 冢 Plane Wall T(x) ⫽ Cylindrical Wall T(r) ⫽ Ts,2 ⫹ Spherical Wall 冣 (C.1) 冢 冣 冤 冢 1 ⫺ rr 冣 ⫹ (T 冢 冣 冤 冢 q̇r 22 q̇r 22 r2 1⫺ 2 ⫺ 4k 4k r2 2 1 2 2 s,2 ln(r /r) 冥 ln(r /r ) (C.2) 冥 (C.3) ⫺ Ts,1) 2 2 冣 1 q̇r 22 (1/r) ⫺ (1/r2) q̇r 22 r 21 r2 1⫺ 2 ⫺ 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k 6k (1/r r2 r2 1) ⫺ (1/r2) T(r) ⫽ Ts,2 ⫹ Heat Flux Plane Wall Cylindrical Wall Spherical Wall q⬙(x) ⫽ q̇x ⫺ q̇r q⬙(r) ⫽ ⫺ 2 q⬙(r) ⫽ q̇r ⫺ 3 k (T ⫺ Ts,1) 2L s,2 (C.4) 冤q̇r4k 冢1 ⫺ rr 冣 ⫹ (T 2 2 k 2 1 2 2 s,2 冥 ⫺ Ts,1) (C.5) r ln(r2 /r1) k 冤 冢 冥 冣 q̇r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k r2 (C.6) r [(1/r1) ⫺ (1/r2)] 2 Heat Rate 冤 q(x) ⫽ q̇x ⫺ Cylindrical Wall q(r) ⫽ q̇Lr 2 ⫺ 3 Spherical Wall 冥 k (T ⫺ Ts,1) Ax 2L s,2 Plane Wall q̇4r q(r) ⫽ ⫺ 3 (C.7) 冤 冢 冥 冣 q̇r 22 r 21 2Lk 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 䡠 ln(r2 /r1) 4k r2 4k 冤q̇r6k 冢1 ⫺ rr 冣 ⫹ (T 2 2 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. 936 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 ⫽ ⫺q̇L ⫺ 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) ⫽ ⫺q̇L ⫺ (T ⫺ Ts,1) x ⫽ ⫺L: 2L s,2 k U2(Ts,2 ⫺ T앝,2) ⫽ q̇L ⫺ (T ⫺ Ts,1) x ⫽ ⫹L: 2L s,2 x ⫽ ⫹L: q⬙s,2 ⫽ q̇L ⫺ (C.11) (C.12) (C.13) Cylindrical Wall Uniform Surface Heat Flux r ⫽ r 1: r ⫽ r2: q̇r1 q⬙s,1 ⫽ ⫺ 2 q⬙s,2 ⫽ q̇r2 ⫺ 2 冤 q̇r4k 冢1 ⫺ rr 冣 ⫹ (T 2 2 k 2 1 2 2 s,2 冥 ⫺ Ts,1) (C.14) r1 ln(r2/r1) 冤 冢 k 冥 冣 q̇r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 4k r2 (C.15) r2 ln(r2/r1) Prescribed Transport Coefficient and Ambient Temperature r ⫽ r 1: r ⫽ r 2: U1(T앝,1 ⫺ Ts,1) ⫽ q̇r1 ⫺ 2 q̇r2 U2(Ts,2 ⫺ T앝,2) ⫽ ⫺ 2 k 冤 q̇r4k 冢1 ⫺ rr 冣 ⫹ (T 2 2 2 1 2 2 s,2 冥 ⫺ Ts,1) r1 ln(r2/r1) k 冤 q̇r4k 冢1 ⫺ rr 冣 ⫹ (T 2 2 2 1 2 2 s,2 r2 ln(r2 /r1) 冥 (C.16) ⫺ Ts,1) (C.17) Spherical Wall Uniform Surface Heat Flux r ⫽ r 1: r ⫽ r2: q⬙s,1 ⫽ q⬙s,2 ⫽ q̇r1 ⫺ 3 q̇r2 ⫺ 3 冤 冢 冥 冣 q̇r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k r2 k r 21[(1/r1) 冤 冢 k ⫺ (1/r2)] 冣 冥 q̇r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k r2 r 22[(1/r1) ⫺ (1/r2)] (C.18) (C.19) Appendix C 䊏 TABLE C.2 937 One-Dimensional, Steady-State Conduction with Generation Continued Prescribed Transport Coefficient and Ambient Temperature q̇r1 U1(T앝,1 ⫺ Ts,1) ⫽ ⫺ 3 r ⫽ r1: r ⫽ r 2: U2(Ts,2 ⫺ T앝,2) ⫽ q̇r2 ⫺ 3 k 冤 q̇r6k 冢1 ⫺ rr 冣 ⫹ (T 2 2 2 1 2 2 s,2 冥 ⫺ Ts,1) r 21[(1/r1) ⫺ (1/r2)] 冤 冢 冥 冣 q̇r 22 r 21 1 ⫺ 2 ⫹ (Ts,2 ⫺ Ts,1) 6k r2 k r 22[(1/r1) ⫺ (1/r2)] (C.20) (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. 938 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 冢 冣 (C.22) 冢 冣 (C.23) 冢 冣 (C.24) Plane Wall T(x) ⫽ q̇L2 x2 1 ⫺ 2 ⫹ Ts 2k L Circular Rod T(r) ⫽ q̇r 2o r2 1 ⫺ 2 ⫹ Ts 4k ro Sphere T(r) ⫽ q̇r 2o r2 1 ⫺ 2 ⫹ Ts 6k ro Heat Flux Plane Wall q⬙(x) ⫽ q̇x (C.25) Circular Rod q⬙(r) ⫽ q̇r 2 (C.26) Sphere q⬙(r) ⫽ q̇r 3 (C.27) Heat Rate Plane Wall q(x) ⫽ q̇xAx Circular Rod q(r) ⫽ q̇Lr Sphere q(r) ⫽ (C.28) 2 (C.29) q̇4r 3 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: q̇L ⫽ U(Ts ⫺ T앝) (C.31) Circular Rod r ⫽ ro: q̇ro ⫽ U(Ts ⫺ T앝) 2 (C.32) q̇ro ⫽ U(Ts ⫺ T앝) 3 (C.33) Sphere r ⫽ ro: APPENDIX D The Gauss–Seidel Method 940 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 aii j1 aii j a Tj ji1 ii 兺 3. 4. 5. 6. 兺 where i 1, 2, . . ., N. The superscript k refers to the level of the iteration. 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. 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. 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. 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. APPENDIX E The Convection Transfer Equations 942 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 fluid in which there is concurrent heat 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 and temperature fields within the fluid. These equations can be derived by applying Newton’s second law of motion and conservation of mass and energy 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, 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). 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 These equations are derived in Section 6S.1. Appendix E 䊏 943 The Convection Transfer Equations 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 ⭸2v ⭸2v ⭸v 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 ⭸2T ⭸2T ⭸T k q̇ 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, q̇ is the volumetric rate of thermal energy generation, and , the viscous dissipation, is defined as ⬅ 冦冢⭸u⭸y ⭸v⭸x冣 2冤冢⭸u⭸x冣 冢⭸v⭸y冣 冥冧 2 2 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 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. This page intentionally left blank APPENDIX F Boundary Layer Equations for Turbulent Flow 946 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 or the fluid temperature, 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, and energy 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 u 冢 冣 ⭸T ⭸T 1 ⭸ k ⭸T ⫺ c v⬘T⬘ ⫹v ⫽ c p ⭸x ⭸y ⭸y p ⭸y (F.2) (F.3) The equations are like those for the laminar boundary layer, Equations 6.15 through 6.17 (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 and energy transport. On the basis of the foregoing results, it is customary to speak of a total shear stress and total heat flux, which are defined as 冢 ⭸u⭸y ⫺ u⬘v⬘冣 ⭸T q⬙ ⫽ ⫺冢k ⫺ c v⬘T⬘冣 ⭸y tot ⫽ tot p (F.4) (F.5) and consist of contributions due to molecular diffusion and turbulent mixing. From the form of these equations we see how momentum and energy transfer rates are enhanced by the existence of turbulence. The term ⫺u⬘v⬘ appearing in Equation F.4 represents the momentum P P' P Time, t FIGURE F.1 Property variation with time at some point in a turbulent boundary layer. Appendix F 䊏 Boundary Layer Equations for Turbulent Flow 947 flux due to the turbulent fluctuations, and it is often termed the Reynolds stress. The term cpv⬘T⬘ in Equation F.5 represents the heat flux 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. This page intentionally left blank APPENDIX G An Integral Laminar Boundary Layer Solution for Parallel Flow over a Flat Plate 950 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, Cƒ, and h). 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 flow with constant fluid properties and negligible viscous dissipation. To use the method, the boundary layer equations, Equations 7.3 through 7.5, 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 7.3, 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 7.4, 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 7.3 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 冕 ⭸u⭸x dy ⫺ 冕 2u ⭸u⭸x dy ⫽ ⭸u⭸y 冏 ␦ u앝 Therefore y⫽0 ␦ 0 0 冕 ⭸x⭸ (u 䡠 u ⫺ u 䡠 u) dy ⫽ ⭸u⭸y 冏 y⫽0 ␦ 앝 0 Rearranging, we then obtain d dx 冤冕 (u ␦ 0 앝 冥 ⫺ u)u dy ⫽ ⭸u ⭸y 冏 y⫽0 (G.3) y⫽0 Equation G.3 is the integral form of the boundary layer momentum equation. In a similar fashion, the following integral form of the boundary layer energy equation may be obtained: d dx 冤冕 (T ␦t 0 앝 冥 ⫺ T )u dy ⫽ ␣ ⭸T ⭸y 冏 (G.4) y⫽0 Appendix G 䊏 951 An Integral Laminar Boundary Layer Solution Equations G.3 through G.4 satisfy the x-momentum and energy requirements in an integral (or average) fashion over the entire boundary layer. In contrast, the original conservation equations, (7.4) and (7.5), 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 and T 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 ⫽0 u(y ⫽ ␦) ⫽ u앝 and y⫽␦ From Equation 7.4 it also follows that, since u ⫽ v ⫽ 0 at y ⫽ 0, ⭸2u ⭸y2 冏 ⫽0 y⫽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.5) The velocity profile is then specified in terms of the unknown boundary layer thickness ␦. This unknown may be determined by substituting Equation G.5 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.6) 952 Appendix G 䊏 An Integral Laminar Boundary Layer Solution Substituting Equation G.6 into Equation G.5 and evaluating s ⫽ (⭸u/⭸y)s, we also obtain Cf,x ⫽ s ⫽ 0.646 2 u앝/2 Re1/2 x (G.7) Despite the approximate nature of the foregoing procedure, Equations G.6 and G.7 compare quite well with results obtained from the exact solution, Equations 7.17 and 7.18. In a similar fashion one could assume a temperature profile of the form T* ⫽ 冢冣 冢冣 T ⫺ Ts y y ⫽ b1 ⫹ b2 ⫹ b3 T앝 ⫺ T s ␦t ␦t and determine the coefficients from the conditions T *(y ⫽ 0) ⫽ ⭸T * ⭸y 冏 2 ⫹ b4 冢冣 y ␦t 3 ⫽0 y⫽␦t T * ( y ⫽ ␦t) ⫽ 1 as well as ⭸2T * ⭸y2 冏 ⫽0 y⫽0 which is inferred from the energy equation (7.5). We then obtain 冢冣 y y T* ⫽ 3 ⫺ 1 2 ␦t 2 ␦t 3 (G.8) Substituting Equations G.5 and G.8 into Equation G.4, we obtain, after some manipulation and assuming Pr ⲏ 1, ␦t Pr⫺1/3 ⫽ ␦ 1.026 (G.9) This result is in good agreement with that obtained from the exact solution, Equation 7.22. 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.6 and G.9, we obtain 1/3 Nux ⫽ hx ⫽ 0.332 Re1/2 x Pr k (G.10) This result agrees precisely with that obtained from the exact solution, Equation 7.21. References 1. von Kárman, T., Z. Angew. Math. Mech., 1, 232, 1921. 2. Pohlhausen, K., Z. Angew. Math. Mech., 1, 252, 1921. 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 temperature, 9 Absorption: gaseous, 863–867 volumetric, 862–863 Absorptivity, 9–12, 768–769 Accommodation coefficients: momentum, 530 thermal, 189–190 Adiabatic surfaces, 91, 230, 246 Adiabats, 230 plotting, W1–W2 Advection, 13, W25 definition of, 6 American Society of Mechanical Engineers (ASME), on SI units, 36 Analogies: Chilton-Colburn, 404 heat diffusion and electrical charge, 114–115 Reynolds analogy, 402–404 Angle: azimuthal, 739–740 plane, 739 solid, 739–740 zenith, 739–740, 785 Annular fins, 155–156, 167, 651 Azimuthal angle, 739–740 B Band emission, 751–758 Beer’s law, 863 Bessel equations, modified, 167–168 Bessel functions: of the first kind (table), 931 modified, of the first and second kinds (table), 932 Bioheat equation, 178–182 Biot number, 283–284 Blackbodies, 776–777 concepts of, 748–749 definition of, 9 Blackbody radiation, 748–758 and band emission, 751–758 and Kirchhoff’s law, 776–777 Planck distribution and, 749–750 radiation exchange, 838–842 and the Stefan-Boltzmann law, 750–751 and Wien’s displacement law, 750 Body forces, W26 Boiling, 7, 8, 15 dimensionless parameters in, 620–621, 638, 654–655 forced convection, 621, 635–639 two-phase flow in, 636–639 modes of, 621 pool boiling, see Pool boiling saturated and subcooled, 621 Boiling crisis, 626 Boiling curve, in pool boiling, 622–623 Bond number, 402, 621 Boundary conditions: adiabatic, 90–91 Dirichlet, 90–91 of the first kind, 90–91, 933–938 Neumann, 91 of the second kind, 91, 150, 935–938 of the third kind, 91, 150, 935–938 Boundary layer(s), 378–405, 415–465 approximations, 394–397 dimensionless parameters in, 379, 385, 400–402, 566–567 equations, 388–396 for laminar flow, 949–952 for turbulent flow, 945–947 and external flow, 416. See also External flow hydrodynamic, 6, 8 laminar and turbulent flow, 383–386, 565–566, 570–572 mixed conditions in external flow, 425 normalized equations, 392–396 functional forms, 393–396 similarity parameters, 392–393 Reynolds analogy, 402–404 separation, 458–461 significance of, 380–381 thermal, 6, 379–380, 385–386, 496 velocity, 378–379, 383–385 Boussinesq approximation, 566 Bulk fluid motion, W30 Bulk temperature, 496 Buoyancy forces, 7, 562, 620 Buoyant jets, 563–564 Burnout point, 626 C Carnot efficiency, 32–36 Celsius temperature scale, 37 Characteristic length, 238, 284–285, 392, 567 Chemical component, of internal energy, 15 Chilton-Colburn analogy, 404 Circular tubes, see Tubes Coefficient of friction, see Friction coefficient Coiled tubes, 527–530 Colburn j factor, 402 Cold plate, 93 954 䊏 Index Compact heat exchangers, W42–W47, 674, 705 Complementary error function, 314 Composite wall systems: heat transfer in, 115–119 porous media in, 119–121 thermal contact resistance in, 117–119, 120 Compressible flow, 391–392 Compressive stresses, W27 Concentric tube annulus, 525–526 Concentric tube heat exchangers, 672 Condensation, 7, 8, 15, 639–657 convection coefficients, typical (table), 8 dimensionless parameters in, 620–621, 655 dropwise, 656 film laminar, 641–645 on radial systems, 650–654 turbulent, 645–649 in horizontal tubes, 655–656 mechanisms of, 639–641 types of, 640 Conduction, 2–5, 46, 67–95 analysis methods, 112–114, 132–135, 136–142 and boundary/initial conditions, 90–91 definition of, 2 Fourier’s Law and, 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), 68–70, 86–87 Conduction shape factor(s), W3–W5, 235–240 for selected systems (table), 236–237 Configuration factor(s), view factor, 828–838 Confinement number, 639 Conservation of energy, 12–31, W29–W31, 83–87 application methodology, 31 for control volumes, 13–31, W29–W31, 941–943 equations, 14, 17 surface energy balance, 27–30 Conservation of mass, W25–W26, 942 Constriction resistance, 656 Contact resistance, 117–119, 120 Continuity equation, W26 Control surface, 13 Control volume(s): definition of, 13, 31 differential, 31, 83–85 Convection, 377–405. See also Boiling; Condensation; External flow; Free convection boundary conditions (table), 91 boundary layers dimensionless parameters, 400–402, 566–567 equations for, W25–W35, 388–396 laminar and turbulent, 383–386, 570–572 normalized equations, 392–396 Reynolds analogy, 402–404 significance of, 380–381 thermal, 379–380 thermal boundary layer, 379–380 velocity, 378–379 velocity boundary layer, 378–379 coefficients, 8, 289 definition of, 2 dimensionless parameter significance, 400–402 forced, 6–7, 635–639. See also Boiling, forced convection; External flow; Internal flow free (natural), see Free convection laminar flow and boundary layers, 389–392 micro- and nanoscale effects, 530–534 mixed, 7, 8, 595–596 problem of, 382–383 rate equation, 8, 46 in surface energy balance, 27–28, 91 transfer equations, W25–W35, 941–943 turbulent flow and boundary layers, 389–392 Convection heat transfer coefficient, 8, 154–155, 289, 403–404 local and average, 381–383 Counterflow heat exchangers, 672–673, 672–674, 680–681, 690–692 Creeping flow, 444 Critical film thickness for microscale conduction, 73–74 Critical heat flux, 624, 625, 628–629, 639 Cross-flow heat exchangers, W38–W42, 672–673, 690–693 Cylinder(s): in cross flow, 433–443 flow considerations, 433–435 heat transfer (convection), 436–443 free convection with, 581–584, 592–593 concentric cylinders, 592–593 long horizontal cylinder, 581–584 one-dimensional steady-state conduction in, 136–141, 933–938 shape factors for, W3–W5, 236–237 transient conduction in, 300–301, 303–307, 318–320 graphical representation of, W12, W14–W15 summary (table), 321–322 D Darcy friction factor, for internal flow, 494–495 Density, 78 gradients, 562, 620 Differential control volumes, energy conservation in, 31 Diffuse emitters, 742, 760 Diffuse reflectors, 748 Diffusion, energy transfer by, 6 Diffusivity: momentum, 401 thermal, 78 Dimensionless conduction heat rate, 235–240, 317–320 Dimensionless parameters: boiling and condensation, 620–621 boundary layers, 379, 385, 392–396, 400–402, 566–567 conduction, 284–319 for free convection, 566–567 friction factor, 494–495 of heat transfer (table), 402 number of transfer units (NTU), 663 physical interpretations of, 400–402 time, 285 Dimensions, 36–38 Dirichlet conditions, 90–91 Discretization of the heat equation: explicit method of, 330–337 implicit method of, 337–345 Dittus-Boelter equation, 516 Drag coefficient, 435 Dropwise condensation, 640, 656 Dynamic viscosity, 379 E Eckert number, 402 Effective thermal conductivity, 119–121, 593 Effectiveness: definition of, 662–663 fin, 164 heat exchanger, 688–689 Effectiveness-NTU analysis method, 662–670, 679–686 definitions in, 662–663 䊏 955 Index Efficiency: Carnot, 32–36 fin, 165–170 heat exchanger, 675–676 Eigenvalues, 300 Electrical energy, and thermoelectric power, 182–188 Electromagnetic spectrum, 735–736 Electromagnetic waves, 735–736 Emission, 734–736 band, 751–758 gaseous, 863–867 and intensity, 740–745 Emissive power, 9, 737, 741–742, 750–751 of a blackbody, 9, 759 Emissivity, 9–12 definition of, 758 from real surfaces, 758–762 representative values (table), 762 Empirical method, 416–418 Enclosed fluids, energy balance with, 501–508 Energy, conservation of, see Conservation of energy Energy balance: atmospheric radiation, 787–789 for internal flow, 501–508 surface, 27–30, 91 Energy balance method, 243–249 Energy carriers, 71 Energy generation, 14–16, 84, 143, 182–188 Energy sources, 84, 183–184 Energy storage, 84 Energy use and sustainability, 41–45, 182–188 Enhancement, heat transfer: boiling, 631 condensation, 651 fins, 155, 165 internal flow, 527–530 Enhancement surface(s), 631 Enthalpy, and steady-flow energy equation, 16–17 Entry length(s): hydrodynamic, 491 thermal, 496 Entry region(s): hydrodynamic, 490–492 and internal flow, 514–516 thermal and combined, 514–516 Environmental radiation, 784–792 atmospheric radiation balance, 787–789 solar, 784–787 terrestrial solar irradiation, 789–790 Error function, 313 Evaporation, 15. See also Boiling Evaporators, 620 Excess temperature, 158, 621 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 fin overall surface efficiency, 170–178, 649–650 nonuniform cross-sectional area fins, 167–170 uniform cross-sectional area fins, 158–164 External flow, 415–465 across banks of tubes, 447–455 cylinder in cross flow, 433–443 flow considerations, 433–435 heat transfer (convection), 436–443 empirical method for, 416–418 flat plate in parallel flow, 418–427 with constant heat flux conditions, 427 laminar flow, 418–424, 949–952 with mixed boundary layer conditions, 425 with unheated starting length, 426 forced convection boiling, 635–636 free convection, 572–586 friction coefficients of, 379 heat transfer correlations (table), 463–464 impinging jets, 455–461 considerations, 456–458 heat transfer (convection) through, 458–461 methodology for convection calculation, 428–433 over sphere, 443–446 similarity method for, 418–424 through packed bed(s), 461–462 F Fanning friction factor, 522 Film boiling, 624, 626, 629–631 Film condensation: definition of, 640 in horizontal tubes, 655–656 laminar, 641–645 on radial systems, 650–654 turbulent, 645–649 wavy, 645 Film temperature, 418 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 heat equation form, 242–243 nodal network selection, 241–242 solving, 250–256 Fins, 154–156 annular, 155–156, 167, 651 conduction analysis, 156–158 effectiveness, 164 efficiency, 165–172 film condensation on, 650–651, 656 free convection with, 586–589 of nonuniform cross-sectional area, 167–170 overall surface efficiency, 170–178, 649–650 performance measures, 164–167 pin, 155–156 straight, 155–156, 166 of uniform cross-sectional area, 158–164 First law of thermodynamics, 12–14 Flat plate: boundary layers and, 378–381 heat transfer correlations (table), 463 with mixed boundary layer conditions, 425 parallel flow over, 418–427 boundary layer solution for, 949–952 with constant heat flux conditions, 427 laminar flow, 383–386, 418–424 turbulent flow, 383–386, 424 with unheated starting length, 426 Flow. See also External flow; Internal flow compressible, 392 creeping, 444 steady, two-dimensional, W25–W35, 942–943 Flow work, 16 Fluidized bed(s), 461 Fluids: free convection with enclosed, 589–595 incompressible, 16–17, W29, 389 microfluidic devices, 530 Newtonian, W28, 379 and problem of convection, 382–383 thermal conductivity of, 75–78 thermophysical properties of (saturated), 916–920 viscous, W25–W36 956 䊏 Index Flux-plotting method, W1–W2, 231 Forced convection, 6–7, 635–639 combined free and forced, 595–596 and external flow, see External flow and internal flow, see Internal flow Forced convection boiling, 621, 635–639 external, 635–636 two-phase flow, 636–639 flow regimes, 637 Form drag, 434 Fouling: in condensation, 675 in heat exchangers, 675–677 Fouling factor, 649 Fourier number, 285, 402 Fourier’s law, 4–5, 68–70, 86–87 Free boundary flows, 563 Free convection, 6–7, 561–597 buoyancy forces in, 562–564 combined free and forced, 595–596 dimensionless parameters for, 566–567 empirical correlations (table), 585–586 with enclosed fluids, 589–595 concentric cylinders, 592–593 concentric spheres, 593–595 for rectangular cavities, 589–592 equations governing, 565–566 external flows, 572–586 for inclined/horizontal plates, 576–581 for long horizontal cylinders, 581–584 for spheres, 585–586 for vertical plates, 573–576 free convection boiling, 623–624 laminar free convection on a vertical surface, 567–570 physical considerations of, 562–564 turbulence effects, 570–572 vertical plate, 567–570 within parallel plate channels, 586–589 Free convection boiling, 623–624 Free stream, 378 Freezing, 15 Friction coefficient, 379, 393, 402, 494 Friction drag, 456 Friction factor, 402 for external flow, 451–452 for internal flow, 494–495, 525 Froude number, 638 Fully developed flow regions, 490–491, 497–501, 509–513 G Gas(es): emission from, 734–735 micro- and nanoscale conduction effects, 189–190 radiation exchange with, 862–867 thermal conductivity of, 75–78 thermal energy equations for, 16–20 thermal radiation and, 10–12 thermophysical properties of (table), 911–915 Gauss-Seidel iteration method, 250, 939–940 example, W5–W9 Gaussian error function, 313, 929 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, 402, 567 Gravitational field, and pool boiling, 630 Gray surfaces: radiation behavior, 778–780, 842–859 radiation exchange, net radiation exchange, 843–844 radiation exchange radiation shields, 852 reradiating surfaces, 854–859 surface radiation exchanges, 844–846 thermal radiation and, 10 H Heat diffusion equation (heat equation), 82–91 boundary conditions, 90–91 finite-difference method, 242–243 microscale effects, 90 Heat engines, efficiency of, 31–36 Heat equation, see Heat diffusion equation (heat equation) Heat exchangers, 671–714 compact, W42–W47, 674, 705 design problems, 696 effectiveness (table), 690 effectiveness-NTU analysis method, 688–696 definitions in, 688–689 relations, 689–693 log mean temperature difference (LMTD) analysis, 677–681 analysis with, 677–678 for counterflow heat exchangers, 680–681 for multipass and cross-flow heat exchangers, W38–W42 for parallel-flow heat exchangers, 678–680 overall heat transfer coefficient for, 674–677 performance calculation problems, 696–704 research and development in, 705 types of, 672–674 Heat flow lines, 230 plotting, W1–W2 Heat flux, 4–5, 8, 9–12, 85 critical, 624, 625, 628–629, 639 radiation fluxes, 737–738 Heat rate, by conduction, 4–5 Heat sinks, 44, 183–184 Heat transfer: in convection, 381–383 definition of, 2 efficiency and, 32–36 enhancement in, 527–530, 679 enhancement research, 679 from extended surfaces, 154–178 conduction analysis, 156–158 fin characteristics, 154–156 fin overall surface efficiency, 649–650 fin performance measures, 164–167 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, 859–861 rate determination (two-dimensional steady-state conduction), W2–W3 relevance of, 41–45 summary of modes (table), 46 thermodynamics vs., 12–13 Hydraulic diameter, 524 Hydrodynamic boundary layer, 6, 8. See also Velocity boundary layer Hydrodynamic considerations: with impinging jet(s), 456–458 with internal flow, 490–495 Hydrodynamic entry length, 491 Hyperbolic functions (table), 928 I Ideal gases, 16–17 Ideal radiator, see Blackbodies Impingement zone(s), 456 Impinging jet(s): considerations, 456–458 heat transfer (convection) through, 458–461 heat transfer correlations (table), 464 Incident radiation, 745 Incompressible liquids, 16–17, W29, 389 䊏 957 Index Influence coefficients, 527 Initial conditions, 90–91 Insulation: micro-and nanoscale effects, 77–78 systems and types, 77 thermal conductivity of, 77–78 thermophysical properties of (table), 906 Intensity, radiation, 739–748 Internal energy, 13–15, W29 Internal flow, 489–537 in circular tubes convection correlations (table), 536 laminar flow, 509–516 turbulent flow, 516–524 in coiled tubes, 527–530 energy balance in, 501–508 with constant surface heat flux, 502–505 with constant surface temperature, 505–508 general considerations, 501–502 heat transfer enhancement in, 527–530 hydrodynamic considerations, 490–495 flow conditions, 490–491 friction factor, 494–495 mean velocity, 491–492 velocity profile, 492–494 micro- and nanoscale effects, 530–534 in noncircular tubes, 524–527 thermal considerations, 495–501 with fully developed conditions, 497–499 mean temperature, 496–497 Newton’s law of cooling in, 497 Irradiation, 9–12, 737, 745–747, 767 Isothermal surfaces, 69 Isotherms, 69–70, 230, 235 Isotropic media, 70 effective thermal conductivities in, 121 thermal radiation and, 8, 10 Log mean temperature difference method (LMTD), W38–W47, 651–652, 651–661, 679–686 for counterflow heat exchangers, 680–681 for multipass and cross-flow heat exchangers, W38–W42 for parallel-flow heat exchangers, 652–654 Longitudinal pitch, 448 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 Kelvin temperature scale, 37 Kelvin-Planck statement, 31 Kinematic viscosity, 78, 946 Kirchhoff’s law, 776–777 Mach number, 402 Martinelli parameter, 655 Mass: conservation of, see Conservation of mass units for, 36–37 Mass flow rate, 16, 17 Matrix equation method, 250 Mean beam length, 866 Mean free path, 71, 73–75 Mean temperature, in internal flow, 496–497, 506 Mean velocity, of internal flow, 491–492 Melting, 15 Metabolic heat generation, 178–182 Metals and metallic solids: emissivity of (table), 922 thermal conductivity of, 71–72, 77 thermophysical properties of, 899–904, 921, 922 Microchannels: in boiling, 639 in condensation, 656 in internal flow, 531–532 Microfluidic devices, 530 Microscale effects: in conduction, 72–75, 77–78, 90, 189–190 in convection, 530–532 Mie scattering, 787 Mixed convection, 7, 8, 595–596 Modes of heat transfer, definition of, 2 Modified Bessel equations, 167–168 Momentum accommodation coefficients, 530 Moody diagrams, 494–495 Moody friction factor, for internal flow, 494–495 Multimode heat transfer, 859–861 Multipass heat exchangers, W38–W42, 672 L N Laminar boundary layer, 378–381 Laminar film condensation, 641–645 Laminar flow: boundary layers and equations, 383–386, 389–391, 565–566 in circular tubes, 509–516 in noncircular tubes, 524–527 over flat plate, 418–424, 949–952 Latent component, of internal energy, 15 Latent energy, in convection, 7 Latent heat, in boiling/condensation, 620 Latent heat of fusion, 26–27 Lattice waves, conduction and, 4, 71–72 Leidenfrost point, 626 Length, units for, 36–37 Liquid metals, 423–424 thermophysical properties of (table), 921 Liquid(s): conduction in, 3–5 convection coefficients, typical (table), 8 microscale convection in, 531–532 radiation from, 734–735 thermal conductivity of, 75–78 thermal energy equations for, 16–17 Nanofluids, 77–78 Nanoscale effects: in conduction, 72–75, 77–78, 90, 189–190 in convection, 532–534 in radiation, 735 Nanoshells, 323 Nanostructured materials, 74, 77–78, 186 Natural convection, see Free convection Net radiation exchange, 843–844 Net radiative flux, 737–738, 748 Neumann conditions, 90–91 Newtonian fluids, W28, 379 Newton’s law of cooling, 8, 115, 497, 621 Newton’s second law of motion, W26–W29, 389–390, 942–943 Nodal network, 241–242, 845–846 Nodal points, 241–242, 845–846 Noncircular tubes, see Tubes Nonmetallic materials: emissivity of solids (table), 923–924 thermal conductivity of, 71–72, 76–77 thermophysical properties of solids, 903–904 Nonparticipating media, 828 Nuclear component, of internal energy, 15 J Jakob number, 402, 621 Jet(s): in boiling, 624–625 impinging, see Impinging jet(s) Joule heating, see Ohmic heating K 958 䊏 Index Nucleate boiling, 624–625, 626–630 Number of transfer units (NTU), 663–665 Nusselt number, 395–396, 402, 525, 526 O Ohmic heating, 143 One-dimensional steady-state conduction, 111–228 alternative analysis approach, 132–135, 141–142 bioheat equation, 178–182 extended surfaces and, see Extended surfaces, 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, 648–649 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 and thermoelectric power generation, 182–188 uniform generation thermal conditions, 933–938 Opaque media, 738, 747–748, 771–772 Open systems, 13–17 Orthogonal functions, 233–234 Overall heat transfer coefficient, 116, 137–138 and heat exchangers, 674–677 Overall surface efficiency, 170–178, 649–650 P Packed bed(s): definition of, 119 heat transfer (convection) through, 461–462 Parallel plates, free convection with, 562–564 Parallel-flow heat exchangers, 672–673, 678–680, 689–692 Parameter sensitivity study, 38 Participating media, 828 radiation exchange with, 862–867 Peclet number, 402 Peltier effect, 183–184 Penetration depth, thermal, 314 Pennes equation, 178–182 Perfusion, and bioheat equation, 178–182 Phase changes, 15 convection coefficients, typical (table), 8 Phonons, 71–75 Photons, 735 Pin fins, 155–156 Pitch (tubes), 448 Planck constant, 749 Planck distribution, 749–750 Plane angle, 739 Plane wall systems: one-dimensional steady-state conduction composite walls, 115–117 contact resistance in, 117–119, 120 temperature distribution, 112–114 with thermal energy generation, 143–149, 933–938 thermal resistance in, 114–115 within porous media, 119–125 one-dimensional steady-state conduction in, thermal resistance in, 648–649 shape factors for, W3–W5, 237 transient conduction in, 283–286, 318–320, 321–322 approximate solution, 300–301 with convection, 299–303 exact solution, 300 graphical representation of, W12–W13 roots of transcendental equation for, 930 summary (table), 321–322 Plumes, 563–564 Pool boiling, 621, 622–635 boiling curve and, 622–623 film boiling, 624, 626, 629–631 free convection boiling, 623–624 Leidenfrost point, 626 nucleate boiling, 624–625, 626–630 parametric effects on, 630–631 transition boiling, 624, 625–626 Porosity, 461 Porous media, conduction in, 119–125 Power-controlled heating, 622–623 Prandtl number, 393, 394, 402 Problems, analysis methodology, 38 Q Quality of fluid, 671n Quanta, 735 Quasi-steady approximation, 584 Quenching, 283 Quiescent fluid(s), 564 R Radial systems: film condensation in, 650–654 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 Radiation. See also Radiation exchange and absorptivity, 768–769 blackbody, see Blackbody radiation emission from real surfaces, 758–766 environmental, 784–792 atmospheric radiation balance, 787–789 solar, 784–787 terrestrial solar, 789–790 gaseous, 862–867 gray surface, 778–780 heat fluxes, 737–738 intensity, 739–748 definitions in, 739–740 and emission, 740–745 and irradiation, 745–747 and net radiative flux, 748 and radiosity, 747–748 and Kirchhoff’s law, 776–777 nature and properties of, 734–736 rate equation, 10, 46 and reflectivity, 769–770 surface characteristics considerations, 771–772 in surface energy balance, 27–30 terminology glossary (table), 793–794 thermal, see Thermal radiation and transmissivity, 771 Radiation balance (atmospheric), 787–789 Radiation exchange, 827–868 between diffuse gray surfaces (enclosed), 842–859 net radiation exchange, 843–844 radiation shields, 852 reradiating surfaces, 854–859 surface radiation exchanges, 844–846 two-surface enclosures, 850–851 blackbody radiation, 838–842 gaseous, 862–867 emission and absorption, 863–867 volumetric absorption, 862–863 and multimode heat transfer, 859–861 view factors in, 828–838 䊏 959 Index definition, 828 factor integral, 828–829 factor relations, 829–836 for two-dimensional geometries (table), 831–833 Radiation heat transfer coefficient, 10 Radiation intensity, see Radiation, intensity Radiative resistance, 844 Radiosity, 737–738, 747–748 Rate equations: for conduction, 4–5 for convection, 8 for radiation heat transfer, 10 summary (table), 46 Rayleigh number, 571 Rayleigh scattering, 787 Reciprocity relation, 829 Rectangular cavities, 589–592 Reflection, 530–531, 767–768 and reflectivity, 738 Reflectivity, 769–770 Reradiating surfaces, 854–859 Resistance: constriction, 656 contact, 117–119, 120 fin, 165 radiative, 844 thermal, 12, 15–16, 32–36, 114–115 Resistance heating, 143 Reynolds analogy, 402–404 Reynolds number, 385, 393, 394, 402, 491, 567 Reynolds stress, 947 S Saturated boiling, 621, 622, 637 Saturated porous media, 119–120 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 use in practical problems, 311 Semitransparent media, 737, 771–772 Sensible energy, 7, 15, 84 Separation point(s), 434 Separation of variables, method, 231–235, 299 Shape factor(s): conduction, W3–W5, 235–240 view factor, 828–838 Shear stresses, W26, 378–379, 946–947 Shell-and-tube heat exchangers, 673, 689–692 Shields, radiation, 852–854 SI (Système International d’Unités) system, 36–38 Similarity solution(s), 419 Similarity variable(s), 311, 419 Simplified steady-flow thermal energy equation, 17 Sinks (energy), 16, 44, 84, 183–184 Solar radiation, 784–790 properties for selected materials (table), 924 representative values for surfaces (table), 790 Solid angle, 739–740 Solidification, 15 Solid(s): conduction in, 3–5, 118–119, 190 radiation from, 734–735 semi-infinite, see Semi-infinite solid(s) thermal conductivity of, 71–75 micro- and nanoscale effects, 72–75, 190 thermal radiation and, 9–12 Specific heat, 78 Spectral absorptivity, 768 Spectral emissivity, 759 Spectral intensity, 740–741 Spectral irradiation, 745, 767 Spectral radiosity, 747–748 Spectral reflectivity, 738 Sphere(s): dimensionless conduction heat rate for, 238 film condensation on, 650 free convection with, 585–586, 593–595 heat transfer correlations (table), 463 one-dimensional steady-state conduction in, 141–142, 933–938 shape factors for, W3–W5, 236–237 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), 456 Stanton number, 402 Steady-state conditions, 4, 14, 16, 112 conduction and, 933–938 Stefan-Boltzmann constant, 9 Stefan-Boltzmann law, 9, 750–751 Stokes’ law, 465 Straight fins, 155–156, 166 Stratification parameter, 638 Streaks, 384 Stresses: Reynolds, 947 shear, 378–379, 946–947 viscous, W26–W29 Structural building materials, thermophysical properties of (table), 905 Subcooled boiling, 621, 630–631, 636–637 Summation rule, 830 Surface energy balance, 27–30, 91 Surface forces, W26 Surface friction, and boundary layers, 380–381 Surface phenomena, 16 radiation as, 735, 767–768 Surface roughness, 631 Surface tension, 620, 621 Surface(s): radiation exchange between gray, 842–859 surface energy balance, 27–30, 91 T Temperature: absolute, 9 conduction and, 2–5 and efficiency, 32–33 excess, 621 film, 418 mean, 496–497 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 Temperature effectiveness, 649–650 Thermal accommodation coefficient, 189–190 Thermal boundary layer, 6, 379–380 and laminar or turbulent flow, 385–386, 496 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, 15 960 䊏 Index Thermal energy equation, W–31 Thermal energy generation: conduction with, 142–154 bioheat, 178–182 in plane wall systems, 143–149 in radial systems, 149–154 resistance for, 115 Thermal entry length, 496 Thermal penetration depth, 314 Thermal radiation, 8–12 and boiling, 629–630 definition of, 2, 735–736 emission of, 734, 736 resistance for, 115 Thermal resistance, 12, 15–16, 32–36 fouling factor, 649 in plane wall systems, 114–117, 648–649 thermal contact resistance, 117–119, 120 Thermal time constant, 282 Thermodynamic properties, 78–82, 565 Thermodynamics, heat transfer vs., 12–13 Thermoelectric power generation, 182–188 Thermophysical properties, 78–82, 897–924 of common materials (table) industrial insulation, 907–908 insulating materials/systems (table), 906 structural building materials, 905 emissivity of selected substances, 922–924 of gases at atmospheric pressure (table), 911–915 of liquid metals (table), 921 of saturated fluids (table), 916–920 of saturated water (table), 919–920 of selected metallic solids (table), 899–902 of selected nonmetallic solids (table), 903–904 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 plane wall system, 287–290 multidimensional effects with, W16–W22 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 Transition boiling, 624, 625–626 Transition to turbulence, 389 Transmissivity, 737–738, 771 Transport properties, 70, 78–79 Transverse pitch, 448 Triangular fins, 168–170 Tubes. See also Heat exchangers arrangements of, 447 banks, 447–455 circular convection correlations (table), 536 laminar flow in, 509–516 turbulent flow in, 516–524 concentric tube annulus, 525–527 condensation in, 640–641 condensation on, 650–652 in cross flow configurations, 447–448 flow conditions, 448–452 film condensation in, 651–656 heat transfer correlations (table), 464 noncircular, 524–527 rough vs. smooth, 517–518 Turbulent boundary layer, 383–386, 570–572 Turbulent film condensation, 645–649 Turbulent flow: across cylinders, 433–438 and boundary layers, 383–386, 570–572, 945–947 in circular tubes, 516–524 in noncircular tubes, 524–527 over flat plate, 424 over vertical plate, 570–572 Two-dimensional steady flow, heat transfer in, W25–W35, 942–943 Two-dimensional steady-state conduction, 229–277 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 energy balance method, 243–249 solving, 250–256 graphical method for conduction shape factors, W3–W5 energy balance method in, 243–249 flux-plot construction, W1–W2 heat transfer rate determination, W2–W3 separation of variables method with, 231–235 Two-phase flow, forced convection boiling, 636–639 U Unheated starting length, 445–446 Unit mass, in flow work, 16 Units: derived, 37 English system, 36 SI system, 36–38 Unsaturated porous media, 119 V Vapor blanket, 626, 629 Vaporization, 15 Velocity boundary layer, 378–379 and laminar or turbulent flow, 383–385 Velocity profile(s), for internal flow, 492–494 View factor(s), 828–838 definition of, 828 integral, 828–829 for two-dimensional geometries (table), 831–833 view factor relations, 829–836 Viscosity: dynamic, 80, 379 kinematic, 78, 946 Viscous dissipation, 17 Viscous fluids, heat transfer in, W25–W36, 942–943 Viscous stresses, W26, 943 Void fraction, 461 Volumetric flow rate, 17 Volumetric heat capacity, 78 Volumetric phenomena, 15–16 radiation, 742–744, 862–867 Volumetric thermal expansion coefficient, 565 W Wall jet(s), 456–458 Water, thermophysical properties of (saturated), 919–920 Weber number, 402, 636 Wien’s displacement law, 750 Z Zenith angle, 739–740, 785 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 Mass Mass density Mass flow rate Power 1 km 1 kg 1 kg/m3 1 kg/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 ⫽ 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 ⫽ 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 1 2 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. 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
