Fluid mechanics fundamentals and applications – part 2

cen72367_appx1.qxd 11/17/04 4:34 PM Page 885 APPENDIX P R O P E R T Y TA B L E S A N D CHARTS (SI UNITS)* TABLE A–1 Molar Mass, Gas Constant, and Ideal-Gas Specfic Heats of Some Substances 886 TABLE A–2 Boiling and Freezing Point Properties 887 TABLE A–3 Properties of Saturated Water 888 TABLE A–4 Properties of Saturated Refrigerant-134a 889 TABLE A–5 Properties of Saturated Ammonia 890 TABLE A–6 Properties of Saturated Propane 891 TABLE A–7 Properties of Liquids 892 TABLE A–8 Properties of Liquid Metals 893 TABLE A–9 Properties of Air at atm Pressure 894 TABLE A–10 Properties of Gases at atm Pressure 895 TABLE A–11 Properties of the Atmosphere at High Altitude 897 FIGURE A–12 The Moody Chart for the Friction Factor for Fully Developed Flow in Circular Pipes 898 TABLE A–13 One-dimensional isentropic compressible flow functions for an ideal gas with k ϭ 1.4 899 TABLE A–14 One-dimensional normal shock functions for an ideal gas with k ϭ 1.4 900 TABLE A–15 Rayleigh flow functions for an ideal gas with k ϭ 1.4 901 TABLE A–16 Fanno flow functions for an ideal gas with k ϭ 1.4 902 * Most properties in the tables are obtained from the property database of EES, and the original sources are listed under the tables Properties are often listed to more significant digits than the claimed accuracy for the purpose of minimizing accumulated round-off error in hand calculations and ensuring a close match with the results obtained with EES 885 cen72367_appx1.qxd 11/17/04 4:34 PM Page 886 886 FLUID MECHANICS TA B L E A – Molar mass, gas constant, and ideal-gas specfic heats of some substances Substance Air Ammonia, NH3 Argon, Ar Bromine, Br2 Isobutane, C4H10 n-Butane, C4H10 Carbon dioxide, CO2 Carbon monoxide, CO Chlorine, Cl2 Chlorodifluoromethane (R-22), CHClF2 Ethane, C2H6 Ethylene, C2H4 Fluorine, F2 Helium, He n-Heptane, C7H16 n-Hexane, C6H14 Hydrogen, H2 Krypton, Kr Methane, CH4 Neon, Ne Nitrogen, N2 Nitric oxide, NO Nitrogen dioxide, NO2 Oxygen, O2 n-Pentane, C5H12 Propane, C3H8 Propylene, C3H6 Steam, H2O Sulfur dioxide, SO2 Tetrachloromethane, CCl4 Tetrafluoroethane (R-134a), C2H2F4 Trifluoroethane (R-143a), C2H3F3 Xenon, Xe Specific Heat Data at 25°C Molar Mass M, kg/kmol Gas Constant R, kJ/kg · K* cp, kJ/kg · K 28.97 17.03 39.95 159.81 58.12 58.12 44.01 28.01 70.905 86.47 30.070 28.054 38.00 4.003 100.20 86.18 2.016 83.80 16.04 20.183 28.01 30.006 46.006 32.00 72.15 44.097 42.08 18.015 64.06 153.82 102.03 84.04 131.30 0.2870 0.4882 0.2081 0.05202 0.1430 0.1430 0.1889 0.2968 0.1173 0.09615 0.2765 0.2964 0.2187 2.077 0.08297 0.09647 4.124 0.09921 0.5182 0.4119 0.2968 0.2771 0.1889 0.2598 0.1152 0.1885 0.1976 0.4615 0.1298 0.05405 0.08149 0.09893 0.06332 1.005 2.093 0.5203 0.2253 1.663 1.694 0.8439 1.039 0.4781 0.6496 1.744 1.527 0.8237 5.193 1.649 1.654 14.30 0.2480 2.226 1.030 1.040 0.9992 0.8060 0.9180 1.664 1.669 1.531 1.865 0.6228 0.5415 0.8334 0.9291 0.1583 cv, kJ/kg · K 0.7180 1.605 0.3122 0.1732 1.520 1.551 0.6550 0.7417 0.3608 0.5535 1.468 1.231 0.6050 3.116 1.566 1.558 10.18 0.1488 1.708 0.6180 0.7429 0.7221 0.6171 0.6582 1.549 1.480 1.333 1.403 0.4930 0.4875 0.7519 0.8302 0.09499 k ϭ cp /cv 1.400 1.304 1.667 1.300 1.094 1.092 1.288 1.400 1.325 1.174 1.188 1.241 1.362 1.667 1.053 1.062 1.405 1.667 1.303 1.667 1.400 1.384 1.306 1.395 1.074 1.127 1.148 1.329 1.263 1.111 1.108 1.119 1.667 * The unit kJ/kg · K is equivalent to kPa · m3/kg · K The gas constant is calculated from R ϭ Ru /M, where Ru ϭ 8.31447 kJ/kmol · K is the universal gas constant and M is the molar mass Source: Specific heat values are obtained primarily from the property routines prepared by The National Institute of Standards and Technology (NIST), Gaithersburg, MD cen72367_appx1.qxd 11/17/04 4:34 PM Page 887 887 APPENDIX TA B L E A – Boiling and freezing point properties Boiling Data at atm Substance Normal Boiling Point, °C Ammonia Ϫ33.3 Latent Heat of Vaporization hfg , kJ/kg 1357 Freezing Point, °C Ϫ77.7 Latent Heat of Fusion hif , kJ/kg 322.4 Argon Benzene Brine (20% sodium chloride by mass) n-Butane Carbon dioxide Ethanol Ethyl alcohol Ethylene glycol Glycerine Helium Hydrogen Isobutane Kerosene Mercury Methane 103.9 Ϫ0.5 Ϫ78.4* 78.2 78.6 198.1 179.9 Ϫ268.9 Ϫ252.8 Ϫ11.7 204–293 356.7 Ϫ161.5 — 385.2 230.5 (at 0°C) 838.3 855 800.1 974 22.8 445.7 367.1 251 294.7 510.4 Ϫ17.4 Ϫ138.5 Ϫ56.6 Ϫ114.2 Ϫ156 Ϫ10.8 18.9 — Ϫ259.2 Ϫ160 Ϫ24.9 Ϫ38.9 Ϫ182.2 109 108 181.1 200.6 — 59.5 105.7 — 11.4 58.4 Methanol Nitrogen 64.5 Ϫ195.8 1100 198.6 Ϫ97.7 Ϫ210 99.2 25.3 124.8 306.3 Ϫ57.5 180.7 Ϫ218.8 13.7 Ϫ187.7 80.0 Octane Oil (light) Oxygen Petroleum Propane Ϫ185.9 80.2 Ϫ183 — Ϫ42.1 Refrigerant-134a Ϫ26.1 Water 100 161.6 394 Freezing Data 212.7 230–384 427.8 216.8 2257 Ϫ189.3 5.5 Ϫ96.6 0.0 28 126 — 80.3 — 333.7 Liquid Properties Temperature, °C Density r, kg/m3 Specific Heat cp, kJ/kg · K Ϫ33.3 Ϫ20 25 Ϫ185.6 20 682 665 639 602 1394 879 4.43 4.52 4.60 4.80 1.14 1.72 20 Ϫ0.5 25 20 20 20 Ϫ268.9 Ϫ252.8 Ϫ11.7 20 25 Ϫ161.5 Ϫ100 25 Ϫ195.8 Ϫ160 20 25 Ϫ183 20 Ϫ42.1 50 Ϫ50 Ϫ26.1 25 25 50 75 100 1150 601 298 783 789 1109 1261 146.2 70.7 593.8 820 13,560 423 301 787 809 596 703 910 1141 640 581 529 449 1443 1374 1295 1207 1000 997 988 975 958 3.11 2.31 0.59 2.46 2.84 2.84 2.32 22.8 10.0 2.28 2.00 0.139 3.49 5.79 2.55 2.06 2.97 2.10 1.80 1.71 2.0 2.25 2.53 3.13 1.23 1.27 1.34 1.43 4.22 4.18 4.18 4.19 4.22 * Sublimation temperature (At pressures below the triple-point pressure of 518 kPa, carbon dioxide exists as a solid or gas Also, the freezing-point temperature of carbon dioxide is the triple-point temperature of Ϫ56.5°C.) cen72367_appx1.qxd 11/17/04 4:34 PM Page 888 888 FLUID MECHANICS TA B L E A – Properties of saturated water Temp T, °C Saturation Pressure Psat, kPa Density r, kg/m3 Enthalpy of Vaporization hfg, kJ/kg Specific Heat cp, J/kg · K Liquid Vapor Thermal Conductivity k, W/m · K Prandtl Number Pr Dynamic Viscosity m, kg/m · s Vapor Volume Expansion Coefficient b, 1/K Liquid Liquid Vapor Liquid Vapor Liquid Liquid Vapor 0.01 10 15 20 0.6113 0.8721 1.2276 1.7051 2.339 999.8 999.9 999.7 999.1 998.0 0.0048 0.0068 0.0094 0.0128 0.0173 2501 2490 2478 2466 2454 4217 4205 4194 4186 4182 1854 1857 1862 1863 1867 0.561 0.571 0.580 0.589 0.598 0.0171 0.0173 0.0176 0.0179 0.0182 1.792 ϫ 10Ϫ3 1.519 ϫ 10Ϫ3 1.307 ϫ 10Ϫ3 1.138 ϫ 10Ϫ3 1.002 ϫ 10Ϫ3 0.922 0.934 0.946 0.959 0.973 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 13.5 11.2 9.45 8.09 7.01 1.00 1.00 1.00 1.00 1.00 Ϫ0.068 0.015 0.733 0.138 0.195 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 25 30 35 40 45 3.169 4.246 5.628 7.384 9.593 997.0 996.0 994.0 992.1 990.1 0.0231 0.0304 0.0397 0.0512 0.0655 2442 2431 2419 2407 2395 4180 4178 4178 4179 4180 1870 1875 1880 1885 1892 0.607 0.615 0.623 0.631 0.637 0.0186 0.0189 0.0192 0.0196 0.0200 0.891 0.798 0.720 0.653 0.596 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 0.987 1.001 1.016 1.031 1.046 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 6.14 5.42 4.83 4.32 3.91 1.00 1.00 1.00 1.00 1.00 0.247 0.294 0.337 0.377 0.415 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 50 55 60 65 70 12.35 15.76 19.94 25.03 31.19 988.1 985.2 983.3 980.4 977.5 0.0831 0.1045 0.1304 0.1614 0.1983 2383 2371 2359 2346 2334 4181 4183 4185 4187 4190 1900 1908 1916 1926 1936 0.644 0.649 0.654 0.659 0.663 0.0204 0.0208 0.0212 0.0216 0.0221 0.547 0.504 0.467 0.433 0.404 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.062 1.077 1.093 1.110 1.126 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 3.55 3.25 2.99 2.75 2.55 1.00 1.00 1.00 1.00 1.00 0.451 0.484 0.517 0.548 0.578 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 75 80 85 90 95 38.58 47.39 57.83 70.14 84.55 974.7 971.8 968.1 965.3 961.5 0.2421 0.2935 0.3536 0.4235 0.5045 2321 2309 2296 2283 2270 4193 4197 4201 4206 4212 1948 1962 1977 1993 2010 0.667 0.670 0.673 0.675 0.677 0.0225 0.0230 0.0235 0.0240 0.0246 0.378 0.355 0.333 0.315 0.297 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.142 1.159 1.176 1.193 1.210 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.38 2.22 2.08 1.96 1.85 1.00 1.00 1.00 1.00 1.00 0.607 0.653 0.670 0.702 0.716 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 100 110 120 130 140 101.33 143.27 198.53 270.1 361.3 957.9 950.6 943.4 934.6 921.7 0.5978 0.8263 1.121 1.496 1.965 2257 2230 2203 2174 2145 4217 4229 4244 4263 4286 2029 2071 2120 2177 2244 0.679 0.682 0.683 0.684 0.683 0.0251 0.0262 0.0275 0.0288 0.0301 0.282 0.255 0.232 0.213 0.197 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.227 1.261 1.296 1.330 1.365 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.75 1.58 1.44 1.33 1.24 1.00 1.00 1.00 1.01 1.02 0.750 0.798 0.858 0.913 0.970 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 150 160 170 180 190 475.8 617.8 791.7 1,002.1 1,254.4 916.6 907.4 897.7 887.3 876.4 2.546 3.256 4.119 5.153 6.388 2114 2083 2050 2015 1979 4311 4340 4370 4410 4460 2314 2420 2490 2590 2710 0.682 0.680 0.677 0.673 0.669 0.0316 0.0331 0.0347 0.0364 0.0382 0.183 0.170 0.160 0.150 0.142 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.399 1.434 1.468 1.502 1.537 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.16 1.09 1.03 0.983 0.947 1.02 1.05 1.05 1.07 1.09 1.025 1.145 1.178 1.210 1.280 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 200 220 240 260 280 1,553.8 2,318 3,344 4,688 6,412 864.3 840.3 813.7 783.7 750.8 7.852 11.60 16.73 23.69 33.15 1941 1859 1767 1663 1544 4500 4610 4760 4970 5280 2840 3110 3520 4070 4835 0.663 0.650 0.632 0.609 0.581 0.0401 0.0442 0.0487 0.0540 0.0605 0.134 0.122 0.111 0.102 0.094 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.571 1.641 1.712 1.788 1.870 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.910 0.865 0.836 0.832 0.854 1.11 1.15 1.24 1.35 1.49 1.350 1.520 1.720 2.000 2.380 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1405 1239 1028 720 5750 6540 8240 14,690 — 5980 7900 11,870 25,800 — 0.548 0.509 0.469 0.427 — 0.0695 0.0836 0.110 0.178 — 0.086 0.078 0.070 0.060 0.043 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 1.965 2.084 2.255 2.571 4.313 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.902 1.00 1.23 2.06 1.69 1.97 2.43 3.73 2.950 ϫ 10Ϫ3 300 320 340 360 374.14 8,581 11,274 14,586 18,651 22,090 713.8 667.1 610.5 528.3 317.0 46.15 64.57 92.62 144.0 317.0 Note 1: Kinematic viscosity n and thermal diffusivity a can be calculated from their definitions, n ϭ m/r and a ϭ k/rcp ϭ n/Pr The temperatures 0.01°C, 100°C, and 374.14°C are the triple-, boiling-, and critical-point temperatures of water, respectively The properties listed above (except the vapor density) can be used at any pressure with negligible error except at temperatures near the critical-point value Note 2: The unit kJ/kg · °C for specific heat is equivalent to kJ/kg · K, and the unit W/m · °C for thermal conductivity is equivalent to W/m · K Source: Viscosity and thermal conductivity data are from J V Sengers and J T R Watson, Journal of Physical and Chemical Reference Data 15 (1986), pp 1291–1322 Other data are obtained from various sources or calculated cen72367_appx1.qxd 11/17/04 4:34 PM Page 889 889 APPENDIX TA B L E A – Properties of saturated refrigerant-134a Temp T, °C Saturation Pressure P, kPa Liquid Vapor Liquid Vapor Vapor Volume Expansion Coefficient b, 1/K Liquid Ϫ40 Ϫ35 Ϫ30 Ϫ25 Ϫ20 51.2 66.2 84.4 106.5 132.8 1418 1403 1389 1374 1359 2.773 3.524 4.429 5.509 6.787 225.9 222.7 219.5 216.3 213.0 1254 1264 1273 1283 1294 748.6 764.1 780.2 797.2 814.9 0.1101 0.1084 0.1066 0.1047 0.1028 0.00811 0.00862 0.00913 0.00963 0.01013 4.878 4.509 4.178 3.882 3.614 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 2.550 3.003 3.504 4.054 4.651 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 5.558 5.257 4.992 4.757 4.548 0.235 0.266 0.299 0.335 0.374 0.00205 0.00209 0.00215 0.00220 0.00227 0.01760 0.01682 0.01604 0.01527 0.01451 Ϫ15 Ϫ10 Ϫ5 164.0 200.7 243.5 293.0 349.9 1343 1327 1311 1295 1278 8.288 10.04 12.07 14.42 17.12 209.5 206.0 202.4 198.7 194.8 1306 1318 1330 1344 1358 833.5 853.1 873.8 895.6 918.7 0.1009 0.0989 0.0968 0.0947 0.0925 0.01063 0.01112 0.01161 0.01210 0.01259 3.371 3.150 2.947 2.761 2.589 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 5.295 5.982 6.709 7.471 8.264 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 4.363 4.198 4.051 3.919 3.802 0.415 0.459 0.505 0.553 0.603 0.00233 0.00241 0.00249 0.00258 0.00269 0.01376 0.01302 0.01229 0.01156 0.01084 10 15 20 25 30 414.9 488.7 572.1 665.8 770.6 1261 1244 1226 1207 1188 20.22 23.75 27.77 32.34 37.53 190.8 186.6 182.3 177.8 173.1 1374 1390 1408 1427 1448 943.2 969.4 997.6 1028 1061 0.0903 0.0880 0.0856 0.0833 0.0808 0.01308 0.01357 0.01406 0.01456 0.01507 2.430 2.281 2.142 2.012 1.888 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 9.081 9.915 1.075 1.160 1.244 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ5 10Ϫ5 10Ϫ5 3.697 3.604 3.521 3.448 3.383 0.655 0.708 0.763 0.819 0.877 0.00280 0.00293 0.00307 0.00324 0.00342 0.01014 0.00944 0.00876 0.00808 0.00742 35 40 45 50 55 887.5 1017.1 1160.5 1318.6 1492.3 1168 1147 1125 1102 1078 43.41 50.08 57.66 66.27 76.11 168.2 163.0 157.6 151.8 145.7 1471 1498 1529 1566 1608 1098 1138 1184 1237 1298 0.0783 0.0757 0.0731 0.0704 0.0676 0.01558 0.01610 0.01664 0.01720 0.01777 1.772 1.660 1.554 1.453 1.355 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 1.327 1.408 1.486 1.562 1.634 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 3.328 3.285 3.253 3.231 3.223 0.935 0.995 1.058 1.123 1.193 0.00364 0.00390 0.00420 0.00456 0.00500 0.00677 0.00613 0.00550 0.00489 0.00429 60 65 70 75 80 1682.8 1891.0 2118.2 2365.8 2635.2 1053 1026 996.2 964 928.2 87.38 100.4 115.6 133.6 155.3 139.1 132.1 124.4 115.9 106.4 1659 1722 1801 1907 2056 1372 1462 1577 1731 1948 0.0647 0.0618 0.0587 0.0555 0.0521 0.01838 0.01902 0.01972 0.02048 0.02133 1.260 1.167 1.077 9.891 9.011 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ5 10Ϫ5 1.704 1.771 1.839 1.908 1.982 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 3.229 3.255 3.307 3.400 3.558 1.272 1.362 1.471 1.612 1.810 0.00554 0.00624 0.00716 0.00843 0.01031 0.00372 0.00315 0.00261 0.00209 0.00160 85 90 95 100 2928.2 3246.9 3594.1 3975.1 887.1 837.7 772.5 651.7 95.4 82.2 64.9 33.9 2287 2701 3675 7959 2281 2865 4144 8785 0.0484 0.0444 0.0396 0.0322 0.02233 0.02357 0.02544 0.02989 8.124 7.203 6.190 4.765 ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.071 2.187 2.370 2.833 ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 3.837 4.385 5.746 11.77 2.116 2.658 3.862 8.326 0.01336 0.01911 0.03343 0.10047 0.00114 0.00071 0.00033 0.00004 Density r, kg/m3 Liquid Vapor 182.3 217.8 269.3 376.3 Enthalpy of Vaporization hfg, kJ/kg Specific Heat cp, J/kg · K Thermal Conductivity k, W/m · K Prandtl Number Pr Dynamic Viscosity m, kg/m · s Liquid Vapor Liquid Surface Tension, N/m Note 1: Kinematic viscosity n and thermal diffusivity a can be calculated from their definitions, n ϭ m/r and a ϭ k/rcp ϭ n/Pr The properties listed here (except the vapor density) can be used at any pressures with negligible error except at temperatures near the critical-point value Note 2: The unit kJ/kg · °C for specific heat is equivalent to kJ/kg · K, and the unit W/m · °C for thermal conductivity is equivalent to W/m · K Source: Data generated from the EES software developed by S A Klein and F L Alvarado Original sources: R Tillner-Roth and H D Baehr, “An International Standard Formulation for the Thermodynamic Properties of 1,1,1,2-Tetrafluoroethane (HFC-134a) for Temperatures from 170 K to 455 K and Pressures up to 70 MPa,” J Phys Chem, Ref Data, Vol 23, No 5, 1994; M J Assael, N K Dalaouti, A A Griva, and J H Dymond, “Viscosity and Thermal Conductivity of Halogenated Methane and Ethane Refrigerants,” IJR, Vol 22, pp 525–535, 1999; NIST REFPROP program (M O McLinden, S A Klein, E W Lemmon, and A P Peskin, Physical and Chemical Properties Division, National Institute of Standards and Technology, Boulder, CO 80303, 1995) cen72367_appx1.qxd 11/17/04 4:34 PM Page 890 890 FLUID MECHANICS TA B L E A – Properties of saturated ammonia Saturation Pressure P, kPa Liquid Liquid Vapor Liquid Vapor Volume Expansion Coefficient b, 1/K Liquid Ϫ40 Ϫ30 Ϫ25 Ϫ20 Ϫ15 71.66 119.4 151.5 190.1 236.2 690.2 677.8 671.5 665.1 658.6 0.6435 1.037 1.296 1.603 1.966 1389 1360 1345 1329 1313 4414 4465 4489 4514 4538 2242 2322 2369 2420 2476 — — 0.5968 0.5853 0.5737 0.01792 0.01898 0.01957 0.02015 0.02075 2.926 2.630 2.492 2.361 2.236 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 7.957 8.311 8.490 8.669 8.851 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 — — 1.875 1.821 1.769 0.9955 1.017 1.028 1.041 1.056 0.00176 0.00185 0.00190 0.00194 0.00199 0.03565 0.03341 0.03229 0.03118 0.03007 Ϫ10 Ϫ5 10 290.8 354.9 429.6 516 615.3 652.1 645.4 638.6 631.7 624.6 2.391 2.886 3.458 4.116 4.870 1297 1280 1262 1244 1226 4564 4589 4617 4645 4676 2536 2601 2672 2749 2831 0.5621 0.5505 0.5390 0.5274 0.5158 0.02138 0.02203 0.02270 0.02341 0.02415 2.117 2.003 1.896 1.794 1.697 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 9.034 9.218 9.405 9.593 9.784 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 1.718 1.670 1.624 1.580 1.539 1.072 1.089 1.107 1.126 1.147 0.00205 0.00210 0.00216 0.00223 0.00230 0.02896 0.02786 0.02676 0.02566 0.02457 15 20 25 30 35 728.8 857.8 1003 1167 1351 617.5 610.2 602.8 595.2 587.4 5.729 6.705 7.809 9.055 10.46 1206 1186 1166 1144 1122 4709 4745 4784 4828 4877 2920 3016 3120 3232 3354 0.5042 0.4927 0.4811 0.4695 0.4579 0.02492 0.02573 0.02658 0.02748 0.02843 1.606 1.519 1.438 1.361 1.288 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 9.978 1.017 1.037 1.057 1.078 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.500 1.463 1.430 1.399 1.372 1.169 1.193 1.218 1.244 1.272 0.00237 0.00245 0.00254 0.00264 0.00275 0.02348 0.02240 0.02132 0.02024 0.01917 40 45 50 55 60 1555 1782 2033 2310 2614 579.4 571.3 562.9 554.2 545.2 12.03 13.8 15.78 18.00 20.48 1099 1075 1051 1025 997.4 4932 4993 5063 5143 5234 3486 3631 3790 3967 4163 0.4464 0.4348 0.4232 0.4116 0.4001 0.02943 0.03049 0.03162 0.03283 0.03412 1.219 1.155 1.094 1.037 9.846 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ5 1.099 1.121 1.143 1.166 1.189 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.347 1.327 1.310 1.297 1.288 1.303 1.335 1.371 1.409 1.452 0.00287 0.00301 0.00316 0.00334 0.00354 0.01810 0.01704 0.01598 0.01493 0.01389 65 70 75 80 85 2948 3312 3709 4141 4609 536.0 526.3 516.2 505.7 494.5 23.26 26.39 29.90 33.87 38.36 968.9 939.0 907.5 874.1 838.6 5340 5463 5608 5780 5988 4384 4634 4923 5260 5659 0.3885 0.3769 0.3653 0.3538 0.3422 0.03550 0.03700 0.03862 0.04038 0.04232 9.347 8.879 8.440 8.030 7.645 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.213 1.238 1.264 1.292 1.322 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.285 1.287 1.296 1.312 1.338 1.499 1.551 1.612 1.683 1.768 0.00377 0.00404 0.00436 0.00474 0.00521 0.01285 0.01181 0.01079 0.00977 0.00876 90 95 100 5116 5665 6257 482.8 470.2 456.6 43.48 49.35 56.15 800.6 759.8 715.5 6242 6561 6972 6142 6740 7503 0.3306 0.3190 0.3075 0.04447 0.04687 0.04958 7.284 ϫ 10Ϫ5 6.946 ϫ 10Ϫ5 6.628 ϫ 10Ϫ5 1.354 ϫ 10Ϫ5 1.389 ϫ 10Ϫ5 1.429 ϫ 10Ϫ5 1.375 1.429 1.503 1.871 1.999 2.163 0.00579 0.00652 0.00749 0.00776 0.00677 0.00579 Temp T, °C Density r, kg/m3 Vapor Enthalpy of Vaporization hfg, kJ/kg Specific Heat cp, J/kg · K Liquid Vapor Thermal Conductivity k, W/m · K Prandtl Number Pr Dynamic Viscosity m, kg/m · s Liquid Vapor Surface Tension, N/m Note 1: Kinematic viscosity n and thermal diffusivity a can be calculated from their definitions, n ϭ m/r and a ϭ k/rcp ϭ n/Pr The properties listed here (except the vapor density) can be used at any pressures with negligible error except at temperatures near the critical-point value Note 2: The unit kJ/kg · °C for specific heat is equivalent to kJ/kg · K, and the unit W/m · °C for thermal conductivity is equivalent to W/m · K Source: Data generated from the EES software developed by S A Klein and F L Alvarado Original sources: Tillner-Roth, Harms-Watzenberg, and Baehr, “Eine neue Fundamentalgleichung fur Ammoniak,” DKV-Tagungsbericht 20:167–181, 1993; Liley and Desai, “Thermophysical Properties of Refrigerants,” ASHRAE, 1993, ISBN 1-1883413-10-9 cen72367_appx1.qxd 11/17/04 4:34 PM Page 891 891 APPENDIX TA B L E A – Properties of saturated propane Temp T, °C Ϫ120 Ϫ110 Ϫ100 Ϫ90 Ϫ80 Saturation Pressure P, kPa 0.4053 1.157 2.881 6.406 12.97 Liquid Vapor Liquid Vapor Vapor Volume Expansion Coefficient b, 1/K Liquid 664.7 654.5 644.2 633.8 623.2 0.01408 0.03776 0.08872 0.1870 0.3602 498.3 489.3 480.4 471.5 462.4 2003 2021 2044 2070 2100 1115 1148 1183 1221 1263 0.1802 0.1738 0.1672 0.1606 0.1539 0.00589 0.00645 0.00705 0.00769 0.00836 6.136 5.054 4.252 3.635 3.149 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 4.372 4.625 4.881 5.143 5.409 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 6.820 5.878 5.195 4.686 4.297 0.827 0.822 0.819 0.817 0.817 0.00153 0.00157 0.00161 0.00166 0.00171 0.02630 0.02486 0.02344 0.02202 0.02062 0.6439 1.081 1.724 2.629 3.864 453.1 443.5 433.6 423.1 412.1 2134 2173 2217 2258 2310 1308 1358 1412 1471 1535 0.1472 0.1407 0.1343 0.1281 0.1221 0.00908 0.00985 0.01067 0.01155 0.01250 2.755 2.430 2.158 1.926 1.726 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 5.680 5.956 6.239 6.529 6.827 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 3.994 3.755 3.563 3.395 3.266 0.818 0.821 0.825 0.831 0.839 0.00177 0.00184 0.00192 0.00201 0.00213 0.01923 0.01785 0.01649 0.01515 0.01382 Density r, kg/m3 Liquid Vapor Enthalpy of Vaporization hfg, kJ/kg Specific Heat cp, J/kg · K Thermal Conductivity k, W/m · K Prandtl Number Pr Dynamic Viscosity m, kg/m · s Liquid Vapor Liquid Surface Tension, N/m Ϫ70 Ϫ60 Ϫ50 Ϫ40 Ϫ30 24.26 42.46 70.24 110.7 167.3 612.5 601.5 590.3 578.8 567.0 Ϫ20 Ϫ10 10 243.8 344.4 473.3 549.8 635.1 554.7 542.0 528.7 521.8 514.7 5.503 7.635 10.36 11.99 13.81 400.3 387.8 374.2 367.0 359.5 2368 2433 2507 2547 2590 1605 1682 1768 1814 1864 0.1163 0.1107 0.1054 0.1028 0.1002 0.01351 0.01459 0.01576 0.01637 0.01701 1.551 ϫ 1.397 ϫ 1.259 ϫ 1.195 ϫ 1.135 ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 7.136 7.457 7.794 7.970 8.151 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 3.158 3.069 2.996 2.964 2.935 0.848 0.860 0.875 0.883 0.893 0.00226 0.00242 0.00262 0.00273 0.00286 0.01251 0.01122 0.00996 0.00934 0.00872 15 20 25 30 35 729.8 834.4 949.7 1076 1215 507.5 500.0 492.2 484.2 475.8 15.85 18.13 20.68 23.53 26.72 351.7 343.4 334.8 325.8 316.2 2637 2688 2742 2802 2869 1917 1974 2036 2104 2179 0.0977 0.0952 0.0928 0.0904 0.0881 0.01767 0.01836 0.01908 0.01982 0.02061 1.077 1.022 9.702 9.197 8.710 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ5 10Ϫ5 10Ϫ5 8.339 8.534 8.738 8.952 9.178 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 2.909 2.886 2.866 2.850 2.837 0.905 0.918 0.933 0.950 0.971 0.00301 0.00318 0.00337 0.00358 0.00384 0.00811 0.00751 0.00691 0.00633 0.00575 40 45 50 60 70 1366 1530 1708 2110 2580 467.1 458.0 448.5 427.5 403.2 30.29 34.29 38.79 49.66 64.02 306.1 295.3 283.9 258.4 228.0 2943 3026 3122 3283 3595 2264 2361 2473 2769 3241 0.0857 0.0834 0.0811 0.0765 0.0717 0.02142 0.02228 0.02319 0.02517 0.02746 8.240 7.785 7.343 6.487 5.649 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 9.417 9.674 9.950 1.058 1.138 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ5 10Ϫ5 2.828 2.824 2.826 2.784 2.834 0.995 1.025 1.061 1.164 1.343 0.00413 0.00448 0.00491 0.00609 0.00811 0.00518 0.00463 0.00408 0.00303 0.00204 80 90 3127 3769 373.0 329.1 84.28 118.6 189.7 133.2 4501 6977 4173 7239 0.0663 0.0595 0.03029 0.03441 4.790 ϫ 10Ϫ5 3.807 ϫ 10Ϫ5 1.249 ϫ 10Ϫ5 1.448 ϫ 10Ϫ5 3.251 4.465 1.722 3.047 0.01248 0.02847 0.00114 0.00037 Note 1: Kinematic viscosity n and thermal diffusivity a can be calculated from their definitions, n ϭ m/r and a ϭ k/rcp ϭ n/Pr The properties listed here (except the vapor density) can be used at any pressures with negligible error except at temperatures near the critical-point value Note 2: The unit kJ/kg · °C for specific heat is equivalent to kJ/kg · K, and the unit W/m · °C for thermal conductivity is equivalent to W/m · K Source: Data generated from the EES software developed by S A Klein and F L Alvarado Original sources: Reiner Tillner-Roth, “Fundamental Equations of State,” Shaker, Verlag, Aachan, 1998; B A Younglove and J F Ely, “Thermophysical Properties of Fluids II Methane, Ethane, Propane, Isobutane, and Normal Butane,” J Phys Chem Ref Data, Vol 16, No 4, 1987; G.R Somayajulu, “A Generalized Equation for Surface Tension from the Triple-Point to the CriticalPoint,” International Journal of Thermophysics, Vol 9, No 4, 1988 cen72367_appx1.qxd 11/17/04 4:34 PM Page 892 892 FLUID MECHANICS TA B L E A – Properties of liquids Temp T, °C Density r, kg/m3 Specific Heat cp, J/kg · K Thermal Conductivity k, W/m · K Thermal Diffusivity a, m2/s Dynamic Viscosity m, kg/m · s Kinematic Viscosity n, m2/s Prandtl Number Pr Volume Expansion Coeff b, 1/K Methane (CH4) Ϫ160 Ϫ150 Ϫ140 Ϫ130 Ϫ120 Ϫ110 Ϫ100 Ϫ90 420.2 405.0 388.8 371.1 351.4 328.8 301.0 261.7 3492 3580 3700 3875 4146 4611 5578 8902 0.1863 0.1703 0.1550 0.1402 0.1258 0.1115 0.0967 0.0797 1.270 1.174 1.077 9.749 8.634 7.356 5.761 3.423 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 1.133 9.169 7.551 6.288 5.257 4.377 3.577 2.761 10Ϫ4 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.699 2.264 1.942 1.694 1.496 1.331 1.188 1.055 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 2.126 1.927 1.803 1.738 1.732 1.810 2.063 3.082 0.00352 0.00391 0.00444 0.00520 0.00637 0.00841 0.01282 0.02922 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 7.429 6.531 5.795 5.185 4.677 4.250 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 7.414 6.622 5.980 5.453 5.018 4.655 0.00118 0.00120 0.00123 0.00127 0.00132 0.00137 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ5 10Ϫ5 1.360 8.531 5.942 4.420 3.432 2.743 2.233 1.836 1.509 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 12.65 8.167 6.079 4.963 4.304 3.880 3.582 3.363 3.256 0.00142 0.00150 0.00161 0.00177 0.00199 0.00232 0.00286 0.00385 0.00628 8.219 5.287 3.339 1.970 1.201 7.878 5.232 3.464 2.455 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 84,101 54,327 34,561 20,570 12,671 8,392 5,631 3,767 2,697 4.242 9.429 2.485 8.565 3.794 2.046 1.241 8.029 6.595 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ4 10Ϫ4 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ6 10Ϫ6 46,636 10,863 2,962 1,080 499.3 279.1 176.3 118.1 98.31 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ Methanol [CH3(OH)] 20 30 40 50 60 70 788.4 779.1 769.6 760.1 750.4 740.4 2515 2577 2644 2718 2798 2885 0.1987 0.1980 0.1972 0.1965 0.1957 0.1950 1.002 9.862 9.690 9.509 9.320 9.128 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 5.857 5.088 4.460 3.942 3.510 3.146 Isobutane (R600a) Ϫ100 Ϫ75 Ϫ50 Ϫ25 25 50 75 100 683.8 659.3 634.3 608.2 580.6 550.7 517.3 478.5 429.6 1881 1970 2069 2180 2306 2455 2640 2896 3361 0.1383 0.1357 0.1283 0.1181 0.1068 0.0956 0.0851 0.0757 0.0669 1.075 1.044 9.773 8.906 7.974 7.069 6.233 5.460 4.634 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ8 9.305 5.624 3.769 2.688 1.993 1.510 1.155 8.785 6.483 Glycerin 10 15 20 25 30 35 40 1276 1273 1270 1267 1264 1261 1258 1255 1252 2262 2288 2320 2354 2386 2416 2447 2478 2513 0.2820 0.2835 0.2846 0.2856 0.2860 0.2860 0.2860 0.2860 0.2863 9.773 9.732 9.662 9.576 9.484 9.388 9.291 9.195 9.101 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10.49 6.730 4.241 2.496 1.519 0.9934 0.6582 0.4347 0.3073 Engine Oil (unused) 20 40 60 80 100 120 140 150 899.0 888.1 876.0 863.9 852.0 840.0 828.9 816.8 810.3 1797 1881 1964 2048 2132 2220 2308 2395 2441 0.1469 0.1450 0.1444 0.1404 0.1380 0.1367 0.1347 0.1330 0.1327 9.097 8.680 8.391 7.934 7.599 7.330 7.042 6.798 6.708 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 3.814 0.8374 0.2177 0.07399 0.03232 0.01718 0.01029 0.006558 0.005344 Source: Data generated from the EES software developed by S A Klein and F L Alvarado Originally based on various sources 0.00070 0.00070 0.00070 0.00070 0.00070 0.00070 0.00070 0.00070 0.00070 cen72367_appx1.qxd 11/17/04 4:34 PM Page 893 893 APPENDIX TA B L E A – Properties of liquid metals Temp T, °C Density r, kg/m3 Specific Heat cp, J/kg · K Thermal Conductivity k, W/m · K Thermal Diffusivity a, m2/s Dynamic Viscosity m, kg/m · s Kinematic Viscosity n, m2/s Prandtl Number Pr Volume Expansion Coeff b, 1/K Mercury (Hg) Melting Point: Ϫ39°C 25 50 75 100 150 200 250 300 13595 13534 13473 13412 13351 13231 13112 12993 12873 140.4 139.4 138.6 137.8 137.1 136.1 135.5 135.3 135.3 8.18200 8.51533 8.83632 9.15632 9.46706 10.07780 10.65465 11.18150 11.68150 4.287 ϫ 4.514 ϫ 4.734 ϫ 4.956 ϫ 5.170 ϫ 5.595 ϫ 5.996 ϫ 6.363 ϫ 6.705 ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 1.687 1.534 1.423 1.316 1.245 1.126 1.043 9.820 9.336 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ4 10Ϫ4 1.241 1.133 1.056 9.819 9.326 8.514 7.959 7.558 7.252 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 10Ϫ8 0.0289 0.0251 0.0223 0.0198 0.0180 0.0152 0.0133 0.0119 0.0108 1.545 1.436 1.215 1.048 9.157 ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ8 0.01381 0.01310 0.01154 0.01022 0.00906 2.167 1.976 1.814 1.702 1.589 1.475 1.360 ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 0.02252 0.02048 0.01879 0.01771 0.01661 0.01549 0.01434 7.432 5.967 4.418 3.188 2.909 2.614 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 0.01106 0.008987 0.006751 0.004953 0.004593 0.004202 4.213 3.456 2.652 2.304 2.126 ϫ ϫ ϫ ϫ ϫ 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 0.006023 0.004906 0.00374 0.003309 0.003143 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 10Ϫ7 0.02102 0.01611 0.01161 0.00753 0.00665 0.00579 Bismuth (Bi) Melting Point: 271°C 350 400 500 600 700 9969 9908 9785 9663 9540 146.0 148.2 152.8 157.3 161.8 16.28 16.10 15.74 15.60 15.60 1.118 1.096 1.052 1.026 1.010 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.540 1.422 1.188 1.013 8.736 ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ4 Lead (Pb) Melting Point: 327°C 400 450 500 550 600 650 700 10506 10449 10390 10329 10267 10206 10145 158 156 155 155 155 155 155 15.97 15.74 15.54 15.39 15.23 15.07 14.91 9.623 9.649 9.651 9.610 9.568 9.526 9.483 ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ6 2.277 2.065 1.884 1.758 1.632 1.505 1.379 ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 10Ϫ3 Sodium (Na) Melting Point: 98°C 100 200 300 400 500 600 927.3 902.5 877.8 853.0 828.5 804.0 1378 1349 1320 1296 1284 1272 85.84 80.84 75.84 71.20 67.41 63.63 6.718 6.639 6.544 6.437 6.335 6.220 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 6.892 5.385 3.878 2.720 2.411 2.101 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 Potassium (K) Melting Point: 64°C 200 300 400 500 600 795.2 771.6 748.0 723.9 699.6 790.8 772.8 754.8 750.0 750.0 43.99 42.01 40.03 37.81 35.50 6.995 7.045 7.090 6.964 6.765 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 3.350 2.667 1.984 1.668 1.487 ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 Sodium–Potassium (%22Na-%78K) Melting Point: Ϫ11°C 100 200 300 400 500 600 847.3 823.2 799.1 775.0 751.5 728.0 944.4 922.5 900.6 879.0 880.1 881.2 25.64 26.27 26.89 27.50 27.89 28.28 3.205 3.459 3.736 4.037 4.217 4.408 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 5.707 4.587 3.467 2.357 2.108 1.859 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 6.736 5.572 4.339 3.041 2.805 2.553 ϫ ϫ ϫ ϫ ϫ ϫ Source: Data generated from the EES software developed by S A Klein and F L Alvarado Originally based on various sources 1.810 1.810 1.810 1.810 1.810 1.810 1.815 1.829 1.854 ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 cen72367_appx1.qxd 11/17/04 4:34 PM Page 894 894 FLUID MECHANICS TA B L E A – Properties of air at atm pressure Temp T, °C Density r, kg/m3 Specific Heat cp J/kg · K Thermal Conductivity k, W/m · K Thermal Diffusivity a, m2/s Dynamic Viscosity m, kg/m · s Kinematic Viscosity n, m2/s Prandtl Number Pr Ϫ150 Ϫ100 Ϫ50 Ϫ40 Ϫ30 2.866 2.038 1.582 1.514 1.451 983 966 999 1002 1004 0.01171 0.01582 0.01979 0.02057 0.02134 4.158 8.036 1.252 1.356 1.465 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ5 10Ϫ5 10Ϫ5 8.636 1.189 1.474 1.527 1.579 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ5 10Ϫ5 10Ϫ5 3.013 5.837 9.319 1.008 1.087 ϫ ϫ ϫ ϫ ϫ 10Ϫ6 10Ϫ6 10Ϫ6 10Ϫ5 10Ϫ5 0.7246 0.7263 0.7440 0.7436 0.7425 Ϫ20 Ϫ10 10 1.394 1.341 1.292 1.269 1.246 1005 1006 1006 1006 1006 0.02211 0.02288 0.02364 0.02401 0.02439 1.578 ϫ 1.696 ϫ 1.818 ϫ 1.880 ϫ 1.944 ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.630 1.680 1.729 1.754 1.778 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.169 1.252 1.338 1.382 1.426 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.7408 0.7387 0.7362 0.7350 0.7336 15 20 25 30 35 1.225 1.204 1.184 1.164 1.145 1007 1007 1007 1007 1007 0.02476 0.02514 0.02551 0.02588 0.02625 2.009 2.074 2.141 2.208 2.277 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.802 1.825 1.849 1.872 1.895 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.470 1.516 1.562 1.608 1.655 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.7323 0.7309 0.7296 0.7282 0.7268 40 45 50 60 70 1.127 1.109 1.092 1.059 1.028 1007 1007 1007 1007 1007 0.02662 0.02699 0.02735 0.02808 0.02881 2.346 2.416 2.487 2.632 2.780 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.918 1.941 1.963 2.008 2.052 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.702 1.750 1.798 1.896 1.995 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.7255 0.7241 0.7228 0.7202 0.7177 80 90 100 120 140 0.9994 0.9718 0.9458 0.8977 0.8542 1008 1008 1009 1011 1013 0.02953 0.03024 0.03095 0.03235 0.03374 2.931 3.086 3.243 3.565 3.898 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.096 2.139 2.181 2.264 2.345 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.097 2.201 2.306 2.522 2.745 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.7154 0.7132 0.7111 0.7073 0.7041 160 180 200 250 300 0.8148 0.7788 0.7459 0.6746 0.6158 1016 1019 1023 1033 1044 0.03511 0.03646 0.03779 0.04104 0.04418 4.241 4.593 4.954 5.890 6.871 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.420 2.504 2.577 2.760 2.934 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 2.975 3.212 3.455 4.091 4.765 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.7014 0.6992 0.6974 0.6946 0.6935 350 400 450 500 600 0.5664 0.5243 0.4880 0.4565 0.4042 1056 1069 1081 1093 1115 0.04721 0.05015 0.05298 0.05572 0.06093 7.892 8.951 1.004 1.117 1.352 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ4 10Ϫ4 10Ϫ4 3.101 3.261 3.415 3.563 3.846 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 5.475 6.219 6.997 7.806 9.515 ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 0.6937 0.6948 0.6965 0.6986 0.7037 700 800 900 1000 1500 2000 0.3627 0.3289 0.3008 0.2772 0.1990 0.1553 1135 1153 1169 1184 1234 1264 0.06581 0.07037 0.07465 0.07868 0.09599 0.11113 1.598 1.855 2.122 2.398 3.908 5.664 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 4.111 4.362 4.600 4.826 5.817 6.630 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 10Ϫ5 1.133 1.326 1.529 1.741 2.922 4.270 ϫ ϫ ϫ ϫ ϫ ϫ 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 10Ϫ4 0.7092 0.7149 0.7206 0.7260 0.7478 0.7539 Note: For ideal gases, the properties cp, k, m, and Pr are independent of pressure The properties r, n, and a at a pressure P (in atm) other than atm are determined by multiplying the values of r at the given temperature by P and by dividing n and a by P Source: Data generated from the EES software developed by S A Klein and F L Alvarado Original sources: Keenan, Chao, Keyes, Gas Tables, Wiley, 198; and Thermophysical Properties of Matter, Vol 3: Thermal Conductivity, Y S Touloukian, P E Liley, S C Saxena, Vol 11: Viscosity, Y S Touloukian, S C Saxena, and P Hestermans, IFI/Plenun, NY, 1970, ISBN 0-306067020-8 Chapter 15 Computational Fluid Dynamics 15-64 Solution We are to study the counter-rotating vortices in a turbulent pipe flow downstream of an elbow Analysis Velocity vectors at several cross sections are plotted in Fig There are no counter-rotating eddies upstream of the elbow However, they are formed as the fluid passes through the elbow, and are very strong just downstream of the elbow These vortices decay in strength down the pipe after the elbow, but they persist for a very long time, and may influence the accuracy of flow meters downstream of an elbow This is why many manufacturers of pipe flow meters recommend that their flow meter be installed at least 10 or 20 pipe diameters downstream of an elbow – to avoid influence of the counter-rotating eddies Downstream pipe section FIGURE Velocity vector plots at several cross sections of a pipe with an elbow at Re = × 104, as predicted by CFD (turbulent flow, using the standard k-ε turbulence model) Elbow Upstream pipe section Flow direction Discussion The counter-rotating eddies lead to additional irreversible head loss as they dissipate 15-65 Solution We are to calculate the minor loss coefficient through a pipe elbow in turbulent flow Analysis The value of KL given in Chap is 0.30 For the pipe with the elbow, the pressure drop calculated by the CFD code is 0.284 kPa For the straight pipe, the pressure drop is calculated to be 0.224 kPa Subtracting these and converting to minor loss coefficient, the value of KL predicted by our CFD calculation for the standard k-ε turbulence model is 0.295 – a difference of less than two percent, and well within the accuracy of the Colebrook equation, which is about 15%, and the accuracy of the tabulated minor loss coefficients, which is often much greater than 15% This agreement is better than expected, considering that this is a very complex 3-D flow, and turbulence models may not necessarily apply for such problems Discussion The agreement here is excellent – CFD does not always match so well with experiment, especially in turbulent flow when using turbulence models, since turbulence models are approximations that often lead to significant error 15-46 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-66 Solution We are to compare various turbulence models – how well they predict the minor loss in a pipe elbow Analysis The results for four turbulence models are listed in Table The standard k-ε turbulence model does the best job in predicting KL The standard k-ω turbulence model does the worst job The Spallart-Allmaras turbulence model calculations are not the worst, even though this is the simplest of the models One would hope TABLE Minor loss coefficient as a function of turbulence model for flow that the more complicated model (Reynolds stress through a 90o elbow in a pipe The error is in comparison to the model) would a better job than the simpler experimental value of 0.30 models, but this is not the case in the present problem (it does worse than the k-ε model, but Turbulence model KL Error (%) better than the other two All turbulence models Spallart-Allmaras (1 eq.) 0.204 -32% are approximations, with calibrated constants -1.7% kε (2 eq.) 0.295 While one model may a better job in a certain 34% k-ω (2 eq.) 0.401 flow, it may not such a good job in another Reynolds stress model (7 eq.) 0.338 13% flow This is the unfortunate state of affairs concerning turbulence models Discussion Newer versions of FlowLab may give slightly different results Although the RSM model does not seem to be too impressive based on this comparison, keep in mind that this is a very simple flow field There are flows (generally flows of very complex geometries and rotating flows) for which the RSM model does a much better job than any 1- or 2equation turbulence model, and is worth the required increase in computer resources 15-67 Solution We are to use CFD to calculate the lift and drag coefficients on an airfoil as a function of angle of attack Analysis The CFD analysis involves turbulent flow, using the standard k-ε turbulence model The results are tabulated and plotted The lift coefficient rises to 1.44 at α = 14o, beyond which the lift coefficient drops off So, the stall angle is about 14o Meanwhile, the drag coefficient increases slowly up to the stall location, and then rises significantly after stall 1.6 1.4 C L 1.2 & C D 0.8 0.6 0.4 0.2 CL α (degrees) -2 10 12 14 16 18 20 CL 0.138008 0.348498 0.560806 0.769169 0.867956 0.967494 1.14544 1.29188 1.39539 1.44135 1.41767 1.34726 1.29543 CD 0.0153666 0.0148594 0.0149519 0.0170382 0.0192945 0.0210042 0.0275433 0.0375832 0.0522318 0.0725146 0.100056 0.140424 0.274792 CD -5 10 Angle of attack α, degrees 15 20 Discussion We note that this airfoil is not symmetric, as can be verified by the fact that the lift coefficient is nonzero at zero angle of attack The lift coefficient does not drop as dramatically as is observed empirically Why? The flow becomes unsteady for angles of attack beyond the stall angle However, we are performing steady calculations For higher angles, the run does not even converge; the CFD calculation is stopped because it has exceeded the maximum number of allowable iterations, not because it has converged Thus the main reason for not capturing the sudden drop in CL after stall is because we are not accounting for the transient nature of the flow The airfoil used in these calculations is called a ClarkY airfoil 15-47 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-68 Solution We are to analyze the effect of Reynolds number on lift and drag coefficient Analysis The CFD analysis involves turbulent flow, using the standard k-ε turbulence model For this airfoil, which is different than the airfoil analyzed in the previous problem, and for the case in which Re = × 106, the lift coefficient rises to about 1.77 at about 18o, beyond which the lift coefficient drops off (see the first table) So, the stall angle is about 18o Meanwhile, the drag coefficient increases slowly up to the stall location, and then rises significantly after stall The data are also plotted below Re = × 106 α (degrees) CL 0.221797 0.65061 0.858744 1.05953 1.2501 1.42542 1.57862 1.69816 1.76686 1.75446 1.70497 10 12 14 16 18 20 22 Re = × 106 1.5 Re = × 106 CL CL & CD α (degrees) CD 10 15 Angle of attack α, degrees 20 25 Re = × 106 1.5 CL & CD CL 0.5 CD 0 10 15 Angle of attack α, degrees 20 25 CL 0.226013 0.659469 0.870578 1.07512 1.27094 1.4533 1.61589 1.74939 1.83901 1.85799 1.76048 10 12 14 16 18 20 22 0.5 CD 0.0118975 0.0166523 0.0212052 0.0273125 0.0351061 0.0447038 0.0562746 0.0702321 0.0875881 0.111326 0.178404 CD 0.0106894 0.015384 0.0198087 0.0256978 0.0331355 0.0422083 0.0530114 0.0657999 0.0812925 0.101563 0.138806 For the case in which Re = × 106, lift coefficient rises to about 1.86 at about 20o, beyond which the lift coefficient drops off (see the second table) So, the stall angle is about 20o Meanwhile, the drag coefficient increases slowly up to the stall location, and then rises significantly after stall These data are also plotted The maximum lift coefficient and the stall angle have both increased somewhat compared to those at Re = × 106 (half the Reynolds number) Apparently, the higher Reynolds number leads to a more vigorous turbulent boundary layer that is able to resist flow separation to a greater downstream distance than for the lower Reynolds number case For all angles of attack, the drag coefficient is slightly smaller for the higher Reynolds number case, reflecting the fact that the skin friction coefficient decreases with increasing Re along a wall, all else being equal [Airfoil drag (before stall) is due mostly to skin friction rather than pressure drag.] Finally, we plot the lift coefficient as a function of angle of attack for the two Reynolds numbers The airfoil clearly performs better at the higher Reynolds number 1.8 1.6 1.4 1.2 CL Re = × 106 Re = × 106 0.8 0.6 0.4 0.2 Discussion The behavior of the lift and drag coefficients beyond stall is not 10 20 30 α, degrees as dramatic as we might have expected Why? The flow becomes unsteady for angles of attack beyond the stall angle However, we are performing steady calculations For higher angles, the run does not even converge; the CFD calculation is stopped because it has exceeded the 15-48 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics maximum number of allowable iterations, not because it has converged Thus the main reason for not capturing the sudden drop in CL after stall is because we are not accounting for the transient nature of the flow We note that the airfoil used in this problem (a NACA2415 airfoil) is different than the one used in Problem 15-67 (a ClarkY airfoil) Comparing the two, the present one has better performance (higher maximum lift coefficient and higher stall angle, even though the Reynolds numbers here are lower than that of Problem 15-67 At higher Re, this airfoil may perform even better 15-69 Solution number We are to examine the effect of grid resolution on airfoil stall at a given angle of attack and Reynolds Analysis The CFD results are shown in Table for the case in which the airfoil is at a 15o angle of attack at a Reynolds number of × 107 The lift coefficient levels off to a value of 1.44 to three significant digits by a cell count of about 17,000 The drag coefficient levels off to a value of 0.849 to three significant digits by a cell count of about 22,000 Thus, we have shown how far the grid must be refined in order to achieve grid independence Thus, we have achieved grid independence for a cell count greater than about 20,000 As for the effect of grid resolution on stall angle, we see that with poor grid resolution, flow separation is not predicted accurately Indeed, when the grid resolution is poor (under 10,000 cells in this particular case), stall is not observed even though the angle of attack (15o) is above the stall angle (14o) for this airfoil at this Reynolds number (1 × 107) When the cell count is about 15,000, however, stall is observed Thus, yes, grid resolution does affect calculation of the stall angle – it is not predicted well unless the grid is sufficiently resolved Discussion TABLE Lift and drag coefficients as a function of cell count (higher cell count means finer grid resolution) for the case of flow over a 2-D airfoil at an angle of attack of 15o and Re = × 107 Cells 672 1344 2176 6264 12500 16800 21700 24320 27200 CL 1.09916 1.00129 1.06013 1.02186 1.42487 1.4356 1.43862 1.43719 1.43741 CD 0.242452 0.224211 0.196212 0.189023 0.0869799 0.0857791 0.0848678 0.0852304 0.0854769 Newer versions of FlowLab may give slightly different results 15-70 Solution We are to study the effect of computational domain extent on the calculation of drag in creeping flow R/L CD Analysis The drag coefficient is 334.067 listed as a function of R/L in the table The 10 289.152 data are also plotted From these data we 50 278.291 see that for R/L greater than about 50, the 100 277.754 drag coefficient has leveled off to a value 500 277.647 of about 278 to three significant digits 1000 277.776 This is rather surprising since the 2000 278.226 Reynolds number is so small, and the viscous effects are expected to influence the flow for tens of body lengths away from the body Discussion When analyzing creeping flow using CFD, it is important to extend the computational domain very far from the object of interest, since viscous effects influence the flow very far from the object This effect is not as great at high Reynolds numbers, where the inertial terms dominate the viscous terms 350 340 330 320 310 CD 300 290 280 270 260 250 1010 10 10 102 100 R/L 103 10000 104 1000 15-49 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-71 Solution We are to study the effect of Reynolds number on flow over an ellipsoid TABLE Drag coefficient as a function of Reynolds number for flow over a 2×1 ellipsoid 1000 Analysis The velocity profile 100 for creeping flow (Re < 1) shows a Re CD very slowly varying velocity from 0.1 552.89 CD zero at the wall to V eventually At 0.5 114.634 high Re, we expect a thin boundary 14.3342 layer and flow that accelerates 20 4.89852 10 around the body However, in 50 2.61481 creeping flow, there is negligible 100 1.691 inertia, and the flow does not accelerate around the body Instead, the body has significant impact on the flow to distances very far from the -1 10 10 10 103 0.1 10 100 body As Re increases, the drag coefficient drops sharply, as expected Re based on experimental data (see Chap 11) At the higher values of Re FIGURE (here, for Re = 50 and 100), inertial effects are becoming more Drag coefficient plotted as a function of significant than viscous effects, and the velocity flow disturbance caused Reynolds number for creeping flow over a by the body is confined more locally around the body compared to the 2×1 ellipsoid lower Reynolds number cases If Re were to be increased even more, very thin boundary layers would develop along the walls The data are also plotted in Fig The drop in drag coefficient with increasing Reynolds number is quite dramatic as Re ranges from 0.1 to 100 (CD decreases from more than 500 to nearly in that range) Thus, we use a log-log scale in Fig Discussion Newer versions of FlowLab may give slightly different results 15-50 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics General CFD Problems 15-72 Solution We are to generate three different coarse grids for the same geometry and node distribution, and then compare the cell count and grid quality Analysis The three meshes are shown in Fig The node distributions along the edges of the computational domain are identical in all three cases, and no smoothing of the mesh is performed The structured multi-block mesh is shown in Fig 1a We split the domain into four blocks for convenience, and to achieve cells with minimal skewing There are 1060 cells The unstructured triangular mesh is shown in Fig 1b There is only one block, and it contains 1996 cells The unstructured quad mesh is shown in Fig 1c It has 833 cells in its one block Comparing the three meshes, the triangular unstructured mesh has too many cells The unstructured quad mesh has the least number of cells, but the clustering of cells occurs in undesirable locations, such as at the outlets on the right The structured quad mesh seems to be the best choice for this geometry – it has only about 27% more cells than the unstructured quad mesh, but we have much more control on the clustering of the cells Skewness is not a problem with any of the meshes (a) (b) (c) FIGURE Comparison of three meshes: (a) structured multiblock, (b) unstructured triangular, and (c) unstructured quadrilateral Discussion results Depending on the grid generation software and the specified node distribution, students will get a variety of 15-51 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-73 Solution We are to run a laminar CFD calculation of flow through a wye, calculate the pressure drop and how the flow splits between the two branches Analysis We choose the structured grid for our CFD calculations The back pressure at both outlets is set to zero gage pressure, and the average pressure at the inlet is calculated to be -8.74 × 10-5 Pa The pressure drop through the wye is thus only 8.74 × 10-5 Pa (a negligible pressure drop) The streamlines are shown in Fig For this case, 57.8% of the flow goes out the upper branch, and 42.2% goes out the lower branch Discussion There appears to be some tendency for the flow to separate at the upper left corner of the branch, but there is no reverse flow at the outlet of either branch This case is compared to a turbulent flow case in the following problem Upper branch Lower branch FIGURE Streamlines for laminar flow through a wye 15-74 Solution We are to run a turbulent CFD calculation of flow through a wye, calculate the pressure drop and how the flow splits between the two branches Analysis We choose the structured grid for our CFD calculations The back pressure at both outlets is set to zero gage pressure, and the average pressure at the inlet is calculated to be -3.295 Pa The pressure drop through the wye is thus 3.295 Pa (a significantly higher pressure drop than that of the laminar flow, although we note that the inlet velocity for the laminar flow case was 0.002 times that of the turbulent flow case) The streamlines are shown in Fig For this case, 54.4% of the flow goes out the upper branch, and 45.6% goes out the lower branch Compared to the laminar case, a greater percentage of the flow goes out the lower branch for the turbulent case The streamlines at first look similar, but a closer look reveals that the spacing between streamlines in the turbulent case is more uniform, indicating that the velocity distribution is also more uniform (more “full”), as is expected for turbulent flow Upper branch Lower branch FIGURE Streamlines for turbulent flow through a wye The k-ε turbulence model is used Discussion There appears to be some tendency for the flow to separate at the upper left corner of the branch, but there is no reverse flow at the outlet of either branch 15-75 Solution We are to keep refining a grid until it becomes grid independent for the case of a laminar boundary layer Analysis Students will have varied results, depending on the grid generation code, CFD code, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-76 Solution We are to keep refining a grid until it becomes grid independent for the case of a turbulent boundary layer Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-52 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-77 Solution We are to study ventilation in a simple 2-D room using CFD, and using a structured rectangular grid Analysis Students will have varied results, depending on the grid generation code, CFD code, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-78 Solution We are to repeat the previous problem except use an unstructured grid, and we are to compare results Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-79 Solution We are to use CFD to analyze the effect of moving the supply and/or return vents in a room Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-80 Solution We are to use CFD to analyze a simple 2-D room with air conditioning and heat transfer Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-81 Solution We are to compare the CFD predictions for 2-D and 3-D ventilation Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-53 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-82 Solution We are to use CFD to study compressible flow through a converging nozzle with inviscid walls Specifically, we are to vary the pressure until we have choked flow conditions Analysis Students will have varied results, depending on the grid generation code, CFD code, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-83 Solution We are to repeat the previous problem, but allow friction at the wall, and also use a turbulence model We are then to compare the results to those of the previous problem to see the effect of wall friction and turbulence on the flow Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-84 Solution We are to generate a low-drag, streamlined, 2-D body, and try to get the smallest drag in laminar flow Analysis Students will have varied results, depending on the grid generation code, CFD code, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-85 Solution We are to generate a low-drag, streamlined, axisymmetric body, and try to get the smallest drag in laminar flow We are also to compare the axisymmetric case to the 2-D case of the previous problem Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-86 Solution We are to generate a low-drag, streamlined, axisymmetric body, and try to get the smallest drag in turbulent flow We are also to compare the turbulent drag coefficient to the laminar drag coefficient of the previous problem Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-54 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-87 Solution We are to use CFD to study Mach waves in supersonic flow We are also to compare the computed Mach angle with that predicted by theory Analysis Students will have varied results, depending on the grid generation code, CFD code, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-88 Solution theory We are to study the effect of Mach number on the Mach angle in supersonic flow, and we are to compare to Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results Review Problems 15-89C Solution (a) False: If the boundary conditions are not correct, if the computational domain is not large enough, etc., the solution can be erroneous and nonphysical no matter how fine the grid (b) True: Each component of the Navier-Stokes equation is a transport equation (c) True: The four-sided cells of a 2-D structured grid require less cells than the triangular cells of a 2-D unstructured grid (Note however, that some unstructured cells can be four-sided as well as three-sided.) (d) True: Turbulence models are approximations of the physics of a turbulent flow, and unfortunately are not universal in their application 15-90C Solution We are to discuss right-left symmetry as applied to a CFD simulation and to a potential flow solution Analysis In the time-averaged CFD simulation, we are not concerned about top-bottom fluctuations or periodicity Thus, top-bottom symmetry can be assumed However, fluid flows not have upstream-downstream symmetry in general, even if the geometry is perfectly symmetric fore and aft In the problem at hand for example, the flow in the channel develops downstream Also, the flow exiting the left channel enters the circular settling chamber like a jet, separating at the sharp corner At the opposite end, fluid leaves the settling chamber and enters the duct more like an inlet flow, without significant flow separation We certainly cannot expect fore-aft symmetry in a flow such as this On the other hand, potential flow of a symmetric geometry yields a symmetric flow, so it would be okay to cut our grid in half, invoking fore-aft symmetry Discussion If unsteady or oscillatory effects were important, we should not even specify top-bottom symmetry in this kind of flow field 15-55 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-91C Solution We are to discuss improvements to the given computational domain Analysis (a) Since Gerry is not interested in unsteady fluctuations (which may be unsymmetric), he could eliminate half of the domain In other words, he could assume that the axis is a plane of symmetry between the top and bottom of the channel Gerry’s grid would be cut in size by a factor of two, leading to approximately half the required CPU time, but yielding virtually identical results (b) The fundamental flaw is that the outflow boundary is not far enough downstream There will likely be flow separation at the corners of the sudden contraction With a duct that is only about three duct heights long, it is possible that there will be reverse flow at the outlet Even if there is no reverse flow, the duct is nowhere near long enough for the flow to achieve fully developed conditions Gerry should extend the outlet duct by many duct heights to allow the flow to develop downstream and to avoid possible reverse flow problems Discussion The inlet appears to be perhaps too short as well If Gerry specifies a fully developed channel flow velocity profile at the inlet, his results may be okay, but again it is better to extend the duct many duct heights beyond what Gerry has included in his computational domain 15-92C Solution desirable We are to discuss a feature of modern computer systems for which nearly equal size multiblock grids are Analysis The fastest computers are multi-processor computers In other words, the computer system contains more than one CPU – a parallel computer Modern parallel computers may combine 32, 64, 128, or more CPUs or nodes, all working together In such a situation it is natural to let each node operate on one block If all the nodes are identical (equal speed and equal RAM), the system is most efficient if the blocks are of similar size Discussion In such a situation there must be communication between the nodes At the interface between blocks, for example, information must pass during the CFD iteration process 15-93C Solution We are to discuss the difference between multigridding and multiblocking, and we are to discuss how they may be used to speed up a CFD calculation Then we are to discuss whether multigridding and multiblocking can be applied together Analysis Multigridding has to with the resolution of an established grid during CFD calculations With multigridding, solutions of the equations of motion are obtained on a coarse grid first, followed by successively finer grids This speeds up convergence because the gross features of the flow are quickly established on the coarse grid (which takes less CPU time), and then the iteration process on the finer grid requires less time Multiblocking is something totally different It refers to the creation of two or more separate blocks or zones, each with its own grid The grids from all the blocks collectively create the overall grid As discussed in the previous problem, multiblocking can have some speed advantages if using a parallel-processing computer In addition, some CFD calculations would require too much RAM if the entire computational domain were one large block In such cases, the grid can be split into multiple blocks, and the CFD code works on one block at a time This requires less RAM, although information from the dormant blocks must be stored on disk or solid state memory chips, and then swapped into and out of the computer’s RAM There is no reason why multigridding cannot be used on each block separately Thus, multigridding and multiblocking can be used together Discussion Although all the swapping in and out requires more CPU time and I/O time, for large grids multiblocking can sometimes mean the difference between being able to run and not being able to run at all 15-56 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-94C Solution We are to discuss why we should spend a lot of time developing a multiblock structured grid when we could just use an unstructured grid Analysis There are several reasons why a structured grid is “better” than an unstructured grid, even for a case in which the CFD code can handle unstructured grids First of all the structured grid can be made to have better resolution with fewer cells than the unstructured grid This is important if computer memory and CPU time are of concern Depending on the CFD code, the solution may converge more rapidly with a structured grid, and the results may be more accurate In addition, by creating multiple blocks, we can more easily cluster cells in certain blocks and locations where high resolution is necessary, since we have much more control over the final grid with a structured grid Discussion As mentioned in this chapter, time spent creating a good grid is usually time well spent 15-95 Solution We are to calculate flow through a single-stage heat exchanger Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-96 Solution exchanger We are to study the effect of heating element angle of attack on heat transfer through a single-stage heat Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-97 Solution We are to calculate flow through a single-stage heat exchanger Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-57 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-98 Solution exchanger We are to study the effect of heating element angle of attack on heat transfer through a two-stage heat Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-99 Solution We are to study the effect of spin on a cylinder using CFD, and in particular, analyze the lift force Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-100 Solution We are to study the effect of spin speed on a spinning cylinder using CFD, and in particular, analyze the lift force as a function of rotational speed in nondimensional variables Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-101 Solution We are to study flow into a slot along a wall using CFD Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results 15-102 Solution We are to calculate laminar flow into a 2-D slot, compare with irrotational flow theory, and with results of the previous problem, and discuss the vorticity field Assumptions The flow is steady and 2-D The flow is laminar Analysis The flow field does not change much from the previous problem, except that a thin boundary layer shows up along the floor The vorticity is confined to a region close to the floor – vorticity is negligibly small everywhere else, so the irrotational flow approximation is appropriate everywhere except close to the floor Discussion The irrotational flow approximation is very useful for suction-type flows, as in air pollution control applications (hoods, etc.) 15-58 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics 15-103 Solution We are to model the flow of air into a vacuum cleaner using CFD, and we are to compare the results to those obtained with the potential flow approximation V 2L Analysis We must include a second length scale in the problem, namely the width w of the vacuum nozzle For the CFD calculations, we set w = 2.0 mm and place the inlet plane of the vacuum nozzle at b = 2.0 cm above the floor (Fig 1) Only half of the flow is modeled since we can impose a symmetry boundary condition along the y-axis We use the same volumetric suction flow rate as in the example problem, i.e., V / L = 0.314 m2/s, but in the CFD analysis we specify only half of this value since we are modeling half of the flow field Results of the CFD calculations are shown in Fig Fig 2a shows a view of streamlines in the entire computational plane Clearly, the streamlines far from the inlet of the nozzle appear as rays into the origin; from “far away” the flow feels the effect of the vacuum nozzle in the same way as it would feel a line sink In Fig 2b is shown a close-up view of these same streamlines Qualitatively, the streamlines appear similar to those predicted by the irrotational flow approximation In Fig 2c we plot contours of the magnitude of vorticity Since irrotationality is defined by zero vorticity, these vorticity contours indicate where the irrotational flow approximation is valid – namely in regions where the magnitude of vorticity is negligibly small Far field Symmetry line P∞ y w Vacuum nozzle b x Floor FIGURE CFD model of air being sucked into a vertical vacuum nozzle; the y-axis is a line of symmetry (not to scale –the far field is actually much further away from the nozzle than is sketched here) V 2L Far field P∞ Symmetry line Vacuum nozzle y b Floor (a) y x Floor x (b) 0.005 CP V 2L FIGURE CFD calculations of flow into the nozzle of a vacuum cleaner; (a) streamlines in the entire flow domain, (b) close-up view of streamlines, (c) contours of constant magnitude of vorticity illustrating regions where the irrotational flow approximation is valid, and (d) comparison of pressure coefficient with that predicted by the irrotational flow approximation Irrotational flow approximation -0.005 -0.01 Irrotational flow region -0.015 Vacuum nozzle -0.02 -0.025 b y Rotational flow regions -0.03 x Floor CFD calculations -0.035 -0.04 x* (c) (d) We see from Fig 2c that vorticity is negligibly small everywhere in the flow field except close to the floor, along the vacuum nozzle wall, near the inlet of the nozzle, and inside the nozzle duct In these 15-59 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission Chapter 15 Computational Fluid Dynamics regions, net viscous forces are not small and fluid particles rotate as they move; the irrotational flow approximation is not valid in these regions Nevertheless, it appears that the irrotational flow approximation is valid throughout the majority of the flow field Finally, the pressure coefficient predicted by the irrotational flow approximation is compared to that calculated by CFD in Fig 2d Discussion For x* greater than about 2, the agreement is excellent However, the irrotational flow approximation is not very reliable close to the nozzle inlet Note that the irrotational flow prediction that the minimum pressure occurs at x* ≈ is verified by CFD 15-104 Solution We are to compare CFD calculations of flow into a vacuum cleaner for the case of laminar flow versus the inviscid flow approximation Analysis Students will have varied results, depending on the grid generation code, CFD code, turbulence model, and their choice of computational domain, etc Discussion Instructors can add more details to the problem statement, if desired, to ensure consistency among the students’ results KJ 15-60 PROPRIETARY MATERIAL © 2006 The McGraw-Hill Companies, Inc Limited distribution permitted only to teachers and educators for course preparation If you are a student using this Manual, you are using it without permission ... 5. 624 3.769 2. 688 1.993 1.510 1.155 8.785 6.483 Glycerin 10 15 20 25 30 35 40 127 6 127 3 127 0 126 7 126 4 126 1 125 8 125 5 125 2 22 62 228 8 23 20 23 54 23 86 24 16 24 47 24 78 25 13 0 .28 20 0 .28 35 0 .28 46 0 .28 56... 100 110 120 130 140 101.33 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