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/ A PPENDIX A Properties o f S aturated Water '" Properties of saturated water T e C) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 120 140 160 180 200 220 240 260 280 300 320 340 360 p ( kglm3 ) 999:8 1000.0 999.8 999.2 998.3 997.1 995.7 994.1 992.3 990.2 998.0 985.7 983.1 980.5 977.7 974.7 971.6 968.4 965.1 961.7 958.1 942.8 925.9 907.3 886.9 864.7 840.4 813.6 783.9 750.5 712.2 666.9 610.2 526.2 p. x 1()3 (kglm-s) 1.791 1.520 1.308 1.139 1.003 0.8908 0.7978 0 .7196 0.6531 0.5962 0.5471 0.5043 0.4668 0.4338 0.4044 0.3783 0.35~0 v xl()6 cp kJlkg-OC 4.218 4.203 4.193 4.187 4.182 4.180 4.180 4.179 4.179 4.182 4.182 4.184 4.186 4.187 4.191 4.191 4.195 4.201 4.203 4.210 4.215 4.246 4.282 4.339 4.411 4.498 4.608 4.770 4.991 5.294 5.758 6.566 8.234 16.138 m 2/s 1.792 1.520 1.308 1.140 1.004 0.8933 0.8012 0.7238 0.6582 0.6021 0.5537 0.5116 0.4748 0.4424 0.4137 0.3881 0.3653 0.3448 0.3284 0.3097 0.2945 0.2461 0.2118 0.1869 0.1684 0.1545 0.1439 0.1358 0.1295 0.1245 0.1205 0.1174 0.1151 0.1139 k W/m-OC 0.5619 0.5723 0.5820 0.5911 0.5996 0.6076 0.6150 0.6221 0.6286 0.6347 0.6405 0.6458 0.6507 0.6553 0.6594 0.6633 0.6668 0.6699 0.6727 0.6753 0.6775 0.6833 0.6845 0.6815 0.6745 0.6634 0.6483 0.6292 0.6059 0.5780 0.5450 0.5063 0.4611 0.4115, P x 1()3 11K -0.0853 0.0052 0.0821 0.148 0.207 0.259 0.306 0.349 0.389 0.427 0.462 0.496 0.529 0.560 0.590 0.619 0.647 0.675 0.702 0.728 0.755 0.859 0.966 1.084 1.216 1.372 1.563 1.806 2.130 2 .589 3.293 4.511 7.170 21.28 Pr 13.45 11.16 9.42 8.07 6.99 6.13 5.42 4.83 4.34 3.93 3.57 3.27 3.00 2.77 2.57 2.39 2.23 2.09 1.97 1.86 1.76 1.44 1.23 1.08 0.98 0.91 0.86 0.84 0.84 0.86 0.91 1.02 1.25 2.35 _/ 0.1339 0.3150 0.2978 0.2822 0.2321 0.1961 0.1695 0.1494 0.1336 0.1210 0.1105 0.1015 0.0934 0.0858 0.0783 0.0702 0.0600 / 606 A PPENDIX B Properties of Dry Air at Standard Atmospheric Pressure Properties o f d ry air a t s tandard atmospheric pressure T eC) - 50 - 40 - 30 - 20 p ( kglm3) p, X 106 (kglm-s) 14.63 1'5.17 15.59 16.20 16.71 17.20 17.69 18.17 18.65 19.11 19.57, 20.03 \ 20.47 20.92 21.35 21.78 22.20 22.62 23.03 23.44 23,84 24.24 24.63 25.03 25.41 25.79 26.17 26.54 26.91 27.27 27.64 27.99 VX 106 (m2/s) 9.25 10.02 10.81 11.62 12.46 13.31 14.19 15.09 16.01 1 6.96 17.92 18.90 19.90 20.92 21.96 . 23.02 24.10 25.19, 26.31 27.44 28.58 29.75 30.93 32.13 33.34 34.57 35.82 37.08 38.36 , 39.65 \ 40.96 42.28 cp (kJlkg_°C) k x lC)3 (W/m-°C) Pr 0.735 0.731 0.728 0.724 0.721 0.718 0.716 0.713 0.712 0.710 0.709 0.708 0.707 0.706 0.705 0.704 0.704 0.703 0.702 0.702 0.701 0.701 0.700 0.700 0.699 0.699 0.699 0.699 0.698 0.698 0.698 0.698 - 10 0 10 20 30 40 50 60 70 80 90 100 110 120 qo /' 140 150 160 170 180 190 200 210 220 230 240 250 260 1.5819 1.5141 1.4518 1.3944 1.3414 1.2923 1.2467 1.2042 1.1644 1.1273 1.0924 1.0596 1.0287 0.9996 0.9721 0.9460 0.9213 0.8979 0.8756 0.8544 0.8342 . 0.8150 0.7966 0.7790 0.7622 0.7461 0.7306 0.7158 0.7016 0.6879 0.6748 0.6621 1.0064 1.0060 1.0058 1.0057 1.0056 1.0057 1.0058 1.0061 1.0064 1.0068 1.0074 1.0080 1.0087 1.0095 1.0103 1.0113 1.0123 1.0134 1.0146 1.0159 1.0172 1.0186 1.0201 1.0217 1.0233 1.0250 1.0268 1.0286 1.0305 1.0324 1.0344 1.0365 20.04 20.86 21.68 22.49 23.29 24.08 24.87 25.64 26.38 27.10 27.81 28.52 29.22 29.91 30.59 31.27 31.94 32.61 33.28 33.94 34.59 35.25 35.89 36.54 37.18 37.81 38.45 39.08 39.71 40.33 40.95 41.57 607 608 Introduction to Convective Heat Transfer Analysis Properties o f dry a ir a t s tandard at~ospheric pressure (continued) T p (kglm3) 0;6499 0 .6382 0.6268 0.6159 0.6053 0.5951 0.5853 0.5757 0.5665 0.5575 0.5489 0.5405 0.5323 0 .5244 0.5167 0.5093 0 .5020 0 .4950 0 .4882 0.4815 0 .4750 0 .4687 0 .4626 0 .4566 0.4451 0.4395 0.4341 6:4288 0.4237 0.4187 0.4138 0 .4090 0.4043 0 .3997/ 0 .3952 0.3908 0 .3866 0 .3824 p. x 1 0' v xl()6 (m2/s) cp (kJlkg_°C) 1.0386 1.0407 1.0429 1.0452 1.0475 1.0499 1.0523 1.0544 1.0568 1.0591 1.0615 1.0639 1.0662, 1.0686 1.0710 1.0734 1.0758 1.0782 1.0806 1.0830 1.0854 1.0878 1.0902 1.0926 1.0973 1.0996 1.1020 1.1043 1.1066 1.1088 1.1111 1.1133 1.1155 1.1177 1.1198 1.1219 1.1240 1.1260 k x 1()3 (W1m_0C) eC) 2 70 2 80 2 90 3 00 3 10 3 20 3 30 3 40 3 50 3 60 3 70 3 80 3 90 4 00 4 10 4 20 4 30 4 40 4 50 4 60 4 70 4 80 4 90 5 00 5 20 5 30 5 40 5 50 5 60 5 70 5 80 5 90 6 00 6 10 6 20 6 30 6 40 6 50 (kglm-s) 28.35 28.70 29.05 29.39 29.73 30.07 30.41 30.74 31.07 31.40 31.72 32.04 32.36 32.68 32.99 33.30 33.61 33.92 34.22 34.52 34.82 35.12 35.42 35.71 36.29 36.58 36.87 37.15 37.43 37.71 37.99 3 8.27 38.54 38.81 39.09 39.36 39.62 39.89 Pr 0.698 0.698 0.698 0.698 0.698 0.698 0.69,8 0.699 0.700 0.701 0.701 0.702 0.702 0 .702 0 .70J 0.703 0.703 0.703 0.703 0.703 0.703 0.704 0.704 0.704 0.704 0.704 0.704 0.704 0.705 0.705 0.705 0.705 0.705 0.706 0.706 0.706 0.707 0.707 43.62 44.97 4 6.34 47.72 49.12 50.53 51.95 53.39 54.85 56.31 57.79 59.29 60.79 62.:U 63.85 65.39 66.95 68.52 70.11 71.70 73.31 74.93 76.57 78.22 81.54 83.23 84.92 86.63 88.35 90.07 91.82 93.57 95.33 97.11 98.89 100.69 102.50 104.32 4 2.18 ' 42.79 4 3.40 44.01 44.61 ' 45.21 4 5.84 4 6.38 4 6.92 4 7.47 48.02 4 8.58 4 9.15 4 9.72 50.29 50.86 5 1.44 5 2.02 5 2.59 5 3.16 5 3.73 34.31 5 4.87 5 5.44 5 6.57 5 7.13 5 7.68 5 8.24 5 8.79 5 9.33 5 9.87 60.41 6 0.94 61.47 6 2.00 62.52 63.03 6 3.55 " Index Adiabatic wall temperature, 142-144 A iding flow, 429 A ir properties, 27, 607-608 Analogy Solutions, 244-245 external flow, 254-271 internal flow, 304-322 Reynolds, 2 54-262,304-312 Taylor-Prandtl, 262-267 , three-layer, 267-271,.312-322 Apparent conductivity, 497 Apparent heat flux, 56-57, 229 Apparent shear stress, 5~55, 229 Apparent thermal diffusivity, 497, 546 A rea averaged velocity, 488-489 A spect ratio, 391, 537-538 . Assisting flow, 429, 431 Axial heat conduction, 160-161, 180, 190 \ Boundary layer, laminar forced arbitrary wall temperature, 98-105 flat plate, 83-105 flat plate heat transfer rate, 9 2-94 flat plate temperature profile, 91 flat plate velocity profile, 88 . integral equation solutions, 114-123 numerical solution, 123-140 specified heat flux, 138-140 unheated starting length, 121-123 uniform heat flux, 100 uniform wall temperature, 83-94 viscous dissipation effects, 140-150 Boundary layer, laminar, combined, 431-444 Boundary layer, laminar, natural, 349-366 Boundary layer, po~ous media, 498-521 cylinder, 512, 519-521 flat plate, forced convection, 500-503, .' B enard cell, 406 B enard convection, 4 06 in porous media, 544-545 Blasiu~ solution, 8 3-94 B lock,heated o n wall, 336 . B ody fitted coordinates, 66 Body force, 4, 12-15 Boundary conditions, 354 Boundary layer, 7, 6 0-61 Boundary layer assumptions, 60-61 Boundary layer, film condensation, 516-517 integral equation solutions, 514-519 natural convection, 526-531 numerical solution, 507-513 similarity solutions, 500-507, 526-531 stagnation point, 505-507, 519 ,Boundary layer, turbulent forced, 255-281, 296-297 \ integral equation solution, 272-281 \ numerical solution, f 81-299. 586-596 609 . 610 Introduction to Convective Heat Transfer Analysis Boundary layer, turbulent f orced-Cont. Nusselt number, 257, 260, 2 65-266, 270-271, 280 Reynolds analogy, 2 55-262 Taylor-Prandtl analogy, 262-267 three layer analogy, 267-271 unheated leading edge. section, 276-281 viscous dissipation effects, 296-297 Boundary layer, turbulent natural, 4 07-414 Nusselt number, 412-413 Boundary layer equations, 6 1-69 boundary conditions, 6 7-69 energy, 65-66, 70 flow over a curved surface, 6 6 integral equations, 7 1-80, 1 14-123,408 laminar, 6 1-69 laminar, numerical solution, 1 23-140 momentum, 63-65, 7 0 natural convection, 349-354 order of magnitude analysis, 6 2-66 porous media, 498-499 pressure change across, 6 5,70 turbulent, 69-71, 228-231 turbulent kinetic energy, 71, 241-243 two-dimensional flow equations, 6 6 velocity and temperature profiles, 62 Boundary layerjheory, 6 1-80 Boundary layer thickness displacement, 272, 331-332 laminar flow, flat plate, 89 momentum, 272, 331-333 natural convection, 3 59-360,408-412 porous media ". cylinder, 521 ' flat plate, 517 stagnation point, 519 temperature, 61-62 thermal, 6 1-62 turbulent flow, 273, 278 velocity, 61-62, 278 Bousinesq approximation, 342-344 Brinkman extension o f Darcy model, 546 Brinkman number, 23 Buffer region (layer) 247, 268, 317-318 Bulk temperature, 7, 18 Buoyancy force, 4, 12-15 Buoyancy force parameter, 431 Centrifugal forces, 4, 342-343 condensation, 5 97-599 Channel flow laminar, 158-219 natural, 366-385 porous, 524 turbulent, 3 04-337 Channel flow, natural convection, 366-385 flow rate, 379 heat transfer, 3 80 pressure variation, 368 Closure problem, 5 7 Coefficient of bulk expansion, 14-15 Coefficient of cubical expansion, 14-15 Combined convection, 4, 426-477 aiding flow, 429 assisting flow, 4 29 . buoyancy force parameter, 431 correlation o f heat transfer results, 449-452 . dimensional analysis, 428-429 flowdirection,effecto~427-429 flow separation, effect o f buoyancy f orces,428-429,449 forced convection limit, 451 free convection limit, 451-452 governing equations, 430-431 governing parameters, 426-429 horizontal p late,446 indices i n correlation equation, 451 internal flow, 4 64-476 forced convection limit, 465-466 free convection limit, 465-466 full equations, solutions, 446-449 fully-developed flow, 466-471 horizontal pipe, 464, 466, 474-476 vertical duct, 466-471 vertical\ pipe, 465 laminar boundary layer flow, vertical plate, 4 31-444 forced convection limit, 435-436, 444, 463 free convection limit, 440-44,1, 444, 463 numerical solution, 442-444 Nusselt number, 434-435, 440-441, 4 42-443 series solution, 4 33-442 wall shear stress, 434-435 opposing flo'IV, 4 28-429, 431 Index Combined convection-Cont. turbulent flow, 4 55-463 forced and free convection limits, 463 Combined modes o f heat transfer, 3 Compressibility effects, 20 Condensation, 5 55-600 centrifugal force, 597-599 dropwise, 555-558 film (filmwise), 555-558 film laminar boundary layer type analysis, 586-596 effect o f subcooling, 567-570 film thickness, 562, 598 heat transfer rate, 562, 577 horizontal cylinders, 574-579 horizontal rotating plate, 597-599 improved analysis, 586-596 inclined plate, 5 66-567 mass flow rate, 561, 576 non-gravitational,597-599 Nusselt analysis, 558-563 assumptions, 558 Nusselt number, 577, 563, 594-596, 598-599 similarity solution, 586-596 surface shear stress, 579-582 temperature distribution, 558 tubes, outside, 565, 574-579 vapor drag, 5 79-582 velocity distribution, 560, 575 vertical cylinder, 565 vertical plate, 558"-'563 forced convection, 579-582 heat transfer rate, 557 inertia, effect of, 596 non-condensable gases~~ 5 85-586 non-gravitational, 597-599 Nusselt analysis, 558-563 assUmptions, 558 promoters, 557 subcooling, 5 67-570 surface shear stress, 579-582 transition, 570-572 tube banks, 577-578 turbulent film: vertical plate, 5 70-572 wavy film, 57{}-;572 wettability, 557 Condensation, f ilm I transition to turbulence, 5 70-572 I 611 Conduction, heat transfer by, 1, 3 Conservation principles, 31 energy, 31 mass, 31 momentum, 31 Continuity equation, 3 1-35 cartesian coordinates, 33 cylindrical coordinates, 34 numerical solution, 130-131 porous media, 492 turbulent flow,'52 Control volume, 32 Convection, 1 combined, 4, 426-427 forced,4 free, 4 '. mixed, 4, 426 natural, 4, 342-416 ", Convection, heat transfer by, 1 Convective h eat transfer, 1 forced combined with natural, 4 interaction with radiant heat transfer, 2 . porous media, 487-545 Convective heat transfer coefficient, 5, 6 mean, 9 Correlation o f combined convection re~ults,. 4 49-452· indices in correlation equation, 451 Core region, 330 Critical Rayleigh number, 4 04-406 Cylinder boundary flow over, 68 combined convection, 428-429, 451 film condensation, 574-579, 565 horizontal, 574-.-578 square, 151-15~, 4 47-449, 452 vertical, 565 . Cylindrical coorQinates, 3 4-35, 232 Damping factor, 288 Darcy model (or law) 4 94,545 deviations, 5 45-546 Darcy-modified Rayleigh number, 528 . c ritical,545 Density,12 a ir,27,607-608 water, 28, 606 , Density changes, 4 , 1 3 - 1 5 ' . D EVDUCT,200 / 612 Introduction to Convective Heat Transfer Analysis DEVPIPE, 195, 197 Diffusivity, thermal, 4 97,576 Dimensional analysis, 11-23 Dimensionless numbers, 16, 23 Brinkman, 23 Eckert, 23 Froude, 20, 23 Graetz, 190 Grashof, 18, 23 Mach, 1 5,20,23, 1 44-145 Nusselt, 18, 23 P eelet,23 Prandtl, 18, 23 turbulent, 231 Rayleigh, 23 Darcy-modified, 528 Reynolds, 18, 23 Stanton, 23 Weber, 23 Dimensionless numbers, from similarity analysis, 4 4-46 Dimensionless numbers, physical interpretation, 2 3-26 Dimensionless numbers, table listing, 23 Dimensions, 15 Dissipation function, 41 Dissipation rate, turbulent flow, 2 40-243 equation for, 243 turbulence model, 2 40-243 Dropwise condensation, 555-558 promoters, 557 Duct, turn-around, 336 Duct flow, laminar, 1 57-220 developing flow, 68 fully developed flow, 59--60, 1 58-188 Graetz problem, 189:"'197 natural convectio~, 3 66-385 pipe flow, developing, 1 89-197,201-212 pipe flow, fully-developed, 158-167 pipe flow, thermally developing, 189-197 plane duct, developing, 212-219 plane duct, fully-developed, 169-179 plane duct, thermally developing, 197-201 rectangular duct, fully-developed, 179-188 slug flow, 1 64-165 transition to turbulence, 250 Duct flow, porous media, 521-525 Duct flow, turbulent, 3 09-327 DUCTSYM, 217 Dynamic pressure, 63Dynamic similarity, 18 Eckert number, 23, 44-45, 66 Eddies, turbulent, 3, 4 9 Eddy conductivity, 2 30 diffusivity, 230, 237 viscosity, 230, 2 36 Effective leading edge, 2 60 Enclosures, natural convection in approximate solution, 4 01-402 boundarylayerregnne, 401-402 heating from below, 4 03-407 horizontal, 4 03-406 inclined, 5 39-540 isotherms, 399 porous media filled, 5 31-540 streamlines, 398 tilted (inclined) 386, 5 39-540 ENCLREC, 3 98,404 ENCLPOR, 537 Energy equation, 31, 35-41 boundary layer, laminar, 6 5-66 boundary layer, turbulent, 229-23'1 cartesian coordinates, 41 cylindrical coordinates, 41 integral, 278 porous media, 4 95-497 Enthalpy, 36 specific, 36 total,36 Entrance region laminar pipe, 1 89-197,201-212 plane duct, 1 97-201,212-219 t hermal,189-201 turbulent, 3 29-336 Error function, 502, 507 Expansion coefficient, thermal, 13-14 Explicit solution, 1 24-125,204-210, 3 73-380 instability of, 2 09-210,218-219,379 External flow, 5 boundary layer flow, 60--61 combined convection, 4 31-464 laminar forced convection, 83-152 natural convection, 354-366, 526-531 porous media, 498-521 Index External f low-Cont. porous media, natural convection, 526-531 solution to full equations, 150-152, 299 turbulent forced convection, 254-299 EXTSQCYL, 151 Falkner-Skan flows, 106-11 0 F ilm coefficient, 6 F ilm conductance, 6 Film (filmwise) condensation, 555-558 boundary layer type analysis, 586-596 horizontal cylinder, 574-579 heat transfer rate, 577 mass flow rate, 576 Nusselt number, 577 velocity distribution, 575 improved analysis, 586-596 inclined plate, 5 66-567 non-condensible gases, effect of, 585-586 non-gravitational, 597-599 Nusselt analysis, 558-563 surface shear stress, 579-582 transition, 5 70-572 vertical cylinder, 565 vertical flat plate, 558-563 film thickness, 562 heat transfer rate, 562 mass flow rate, 561 Nusselt number, 563 surface shear stress, 579-582 turbulent flow, 570-572 wavy film, 5 70-572 Finite differences solutiopS· enclosure, 3 91-398 laminar boundary layer, forced, 123-140 laminar duct flow, forced, 179 lamirlar duct flow, free, 366-380 laminar pipe flow, forced, 167,201-212 turbulent boundary layer, forced, 2 81-296 turbulent pipe flow, forced, 322. . J 27 . / . F irst law of thermodynamics, 31 F lat plate combined convection, 431-446, 451 high speed flow, 1 40-150,296-297 l aminar boundary layer flow, 83-106 laminar natural convection, 354-365 turbulent boundary layer flow, 254-271 613 Flow reversal, mixed convection in duct, 471 Flow variables, 31 Fluid Dynamics International, 336 Fluid properties, 26, 31, 606-608 temperature for, 19, 149-150 variation with pressure, 26 variation with temperature, 19 Fluid temperature, 5 -1 internal flow, 7 -8 mean, 10, 19 Forced convection limit, mixed convection, 4 35-436,444,451,465-466 Forced velocity, definition, 12 Forchheimer extension o f Darcy model, 546 Form factor, 272-273, 277, 331 equation for, 273, 331 Fourier's law, 3 Free convection, 4, 342, 366-385; see also .. Natural convection Free convection limit, mixed convection, 4 40-441,444,451-452,465-466 F ree surface flows, 20-21 Freestream turbulence, 248-249 Friction factor, 308 equations, 309-311 Moody chart, 310 relation to wall shear stress, 309 roughness values, 31 Friction velocity, 246, 267, 288-290, 316 Froude number, 20, 23 F ull equations, solutions to laminar flow, combined, 446-449 laminar flow, external, i 50-152 laminar flow, internal, 219-220 turbulent flow, external, 299, 336-337 Fully developed duct flow, 5 9-60 turbulent, 304-321 uniform h eat flux, laminar, 169-174 uniform temperature, laminar, 178-179 ° °. Governing equations, 31....:80, 344-345, . 349-354 Graetz number, 190 GraetZ problem, 189-197 Grashof number, 18, 23, 25, 347, 470 film condensation, 565 Gravitational forces; 4, 342 / /. 614 Introduction to Convective Heat Transfer Analysis o Odlines laminar boundary layer solution, 124, 127, 132-133 Grid modification, 132-133 Heat conduction, 1 ,3 H eat flux, 3 Heat convection, 1 Heat transfer combined convective, 4 b y combined modes, 1 condensation,557,562 b y conduction, 1 b y convection, 1 entrance region, 2 01-219 forced convective, 4 free convective, 4, 380 mixed convective, 4 modes of, 1 natural convective, 4, 3 80 porous media, 487-488 b y radiation, 1 thermal entrance region, 193, 196-197, 200 viscous dissipation effects, 1 40-150 Heat transfer coefficient, 6 horizontal cylinder, film condensation, 574-579 heat transferr,ate, 577 mass flow rate, 576 Nusselt number, 577 velocity distribution, 575 mean, 10 typical values, 10 units of, 6 variables influencing, 11-12 Heat transfer rate, 3 condensation, 5 57,562 Horizontal pipe, mixed convection, 4 64-466,474-477 Horizontal porous layers, 5 40-545 critical Darcy-modified Rayleigh number, 545 experimental results, 545 instability, 544 small disturbance analysis, 542-545 Hydraulic diameter, 177-178, 1 88,250 film condensation, 570 Hydrostatic pressure, 1 3-14 , Implicit solution, 124-125 ,Inclination, effect on natural convection: . porous media filled enclosures, 5 39-540 Incompressible flow, 14 Inertia force, 23-25, 6 4 Inlet conditions, 2 02-204, 2 13-214, 366-367 , Inner region (layer) 268, 288, 317 Instability Benard flow, porous media, 545 , explicit numerical solution, 2 09-2io, 2 18-219,379 Insulatirig material, 4 87-488 Integral equation solutions developing turbulent duct flow, 3 29-335 laminar boundary layer, forced, 114-123 turbulent boundary layer, 272-281 turbulent natural convection, 4 07-414 Integral equations for boundary layer flow, 7 1-80 derivation from boundary layer equations, 7 8-80 energy integral equation, 7 5-77 laminar flow, 1 14-123 momentum integral equation, 7 3-75 natural convection, 4 08 turbulent flow, 272, 278 Internal flow, 5 combined convection, 4 64-476 developing, 189-219, 329-335 forced convection, 3 04-337 fully-developed,304-322 laminar, 1 57-220 developing, 189-219 forced convection, 157-220 fully-developed, 5 9-60 natural convection, 3 66-385 porous media, 521-525 turbulent, 3 04-337 Inviscid flow, 61, 68, 4 92 Irrotational flow, 492 Isotherms, 538 Isotropic porous media, 492 Jakob number, 565, 569 Index K-E turbulence model, 242-244 Kinematic viscosity, 18, 26 Kinetic energy, turbulent, 59, 239 equation for, 5 7-59, 239-242 turbulence model, 2 39-244 Kinetic energy changes, 15 615 / L AMBOUN,135 LAMBOUQ,140 LAMBMIX, 442 LAMBNAT, 365 Laminar boundary layer boundary layer thickness, 61-62, 89, 117-120 Falkner-Skan equation, 106-110 flat plate adiabatic wall temperature, 140-145 boundary layer thickness, thermal, 92, 119-121 flow perpendicular to, 11 0 integral method, 114-123 numerical solution, 1 23-140 Nusselt number, 9 2-94, 100 profiles, 88, 91 recovery factor, 144-145 shooting method, 88 similarity solution, 8 3-104 temperature distribution, 91 unheated starting length, 121-123 velocity distribution, 88 viscous dissipation effects, 1 40-150 Mach number, 15, 23 integral method, 114-123 / Mixed convection; see Combined numerical solutions, 123:. .:140 convection Nusselt number, 9 2-94, 98, 100, 104, Mixing-cup temperature, 8 113, 121, 123, 134, 148-149 Mixing length turbulence model, 234-239, recovery factor, 144-145 287-289 Runge-Kutta procedure, 91 effect o f buoyancy forces, 4 55-461 similarity solution, 8 3-110 M IXSQCYL,447 unheated starting length, 121-123 .. Models viscous dissipation, 1 40-150 o f porous media, 4 90-492, 497, 545-547 / wedge, 106-110 o f turbulence, 2 34-244,455-461 'Laminar duct flow Modes o f heat transfer axial conduction, 1 60-161 combined modes, 1 non-circular, 179-188, 188 conduction, 1 Nusselt number, 164, 167, 177-179, convection, 1 1 88,195-197,200,208-210, radiation, 1 2 17,219 pipe developing, 2 01-212, fully developed flow, 1 58-167 slug flow, 1 64-165 thermally-developing, 189-197 wall heat flux specified, 167 wall temperature specified, 164 plane duct developing, 212-219 fully developed flow, 1 69-179 natural convection, 366-385 thermally-developing, 197-201 wall heat flux specified, 179 wall temperature specified, 174 rectangular, 179-188 simultaneous v~locity and temperature development, 2 01-219 Laminar flow external, solutions to full equations, 1 50-152,299 internal, solutions to full equations, 2 19-220 natural, enclosures, 3 85-407 Laminar flow, condensation, 5 58-570, 5 74-599 vertical plate, 558-563 Latent heat, 562, 569 modified, 569 Leading edge conditions, 134-135 Length scales, 19 Liquid metals, 6 2 Local shearing stress coefficient, 257 Logarithmic velocity profile, 247 616 Introduction to Convective Heat Transfer Analysis I Momentum equation, 3 1-35 Momentum equation, laminar boundary layer, 63-65 derivation, 63-65 similarity solution, 83-113, 354-361, 4 98-507 Moody chart, 3 10 i N ATCHAN,379-380 Natural convection, laminar Benard cells, 406 boundary layer equations, 349-354 centrifugal force, 4 , 3 42-343 channel flow, 366-385 combined natural and forced, 4, 426-427 enclosures, 3 85-407 boundary conditions, 387, 3 89-390, 3 93-395 governing equations, 388-391 heat transfer rate, 3 97-399,401-402 horizontal, 4 03-407 numerical solution procedure, 391-398 flat plate, 354-365 boundary layer thickness, 3 59-360 combined convection, 4, 426-427 integral method, 4 08-414 Nusseltnumber, 3 59-360,412-413 similarity solution, 354-365 temperaturedistribution, 360 velocity distribution, 359': vertical, 354-361 governing equations, 344-345 heat transfer rate, 3 80 horizontal e nclosure,403-407 Benard cells, 406 Benard convection, 4 06 critical Darcy-modified Rayleigh number, 545 experimental results, 545 instability, 4 04-406, 5 44 Nusselt number variation, 405 porous media, 5 40-545 small disturbance analysis, 542-545 stability of flow, 4 04-406, 544 inlet conditions, 366-367 integral equation solution, 4 08-414 laminar-to-turbulence transition, 407 limiting solutions, 3 80-384 mean velocity, 379 Natural convection, l aminar-Cont. numerical solution, 365 Nusselt number, 1 8,23, 24 numerical, 133-134 outlet pressure, 368 porous medium, 5 26-540 pressure variation, 368 similarity analysis, 345-348 temperature distribution, 360, 407 transition to turbulence, 407 velocity distribution, 359, 4 09 Natural convection, porous media boundary layer, 526-531 enclosures, 5 31-540 horizontal layers, 5 40-545 critical Darcy-modified Rayleigh number, 545 experimental results, 545 instability, 544 small disturbance analysis, 542-545 Natural convection, turbulent, 4 07-414 ~ Navier-Stokesequations, 3 1-35 cartesian coordinates, 3 4 cylindrical coordinates, 35 enclosures, natural convection, 386 numerical solutions, 150--152,219-220, 2 99,446-449 turbulent flow, 5 2-54 Near-wall region, 2 44-247 Newtonian fluid, stress-strain rate relation, 33 Nodal points, 124-127, 1 30,391-392 Non-gravitational condensation, 597-599 Non-circular ducts, laminar flow, 1 79-188 Non-condensible gases, effect on condensation,585-586 Non-Newtonian fluid, 31 Number Brinkman, Q3 Eckert, 23 Froude, 20, 23 Graetz, 190 Grashof, 18, 23 Nusselt, 1 8,23 P eclet,23 P randtl,18,23 turbulent, 231 Rayleigh,23 Darcy-modified, 528 Reynolds, 18, 23 ( / Index 617 N umber-Cont. Stanton, 23 Weber, 23 Numerical solutions combined convection, boundary layer, 442-444 full equations combined convection, laminar, 446-449 laminar external flow, 1 50-152 laminar internal flow, 2 19-220 turbulent external flow, 299 turbulent internal flow, 3 36-337 laminar de.'.'eloping pipe flow, 193-197, 2 01-212 laminar d,uct flow, 179, 182-188, 1 97-201,212-219 laminar pipe flow, 167, 193-197, 2 01-212 natural convection boundary layer, 365-366 natural convection enclosure, 3 91-406 natural convection plane duct, 373-379 porous media boundary layer, 5 08-513 enclosure, 5 32-538 turbulent boundary layer, 2 81-296 turbulent developing pipe flow, 3 22-329 turbulent duct flow, 3 22-335 turbulent pipe flow, 3 22-329 Nusselt analysis, film condensation, 558-563 assumptions, 5 58 Nusselt number cOfl1.bined convection correlation equations, 449-452 square cylinder, 4 48 turbulent flow, 461-462 vertical plate, 434-435, 440-443 d uct flow, 164, 167, 177-179 film condensation boundary layer type analysis, 594-596 modified, 5 72 non-gravitational, 5 98-599 vertical plate, 563, 572 Nusselt n umber-Cont. laminar forced convection boundary layer flow, 9 2-94, 98, 104, 112-113 developing duct flow, 2 16-217, 219 developing pipe flow, 210 integral equation solution, 121, 123 pipe flow, 164, 167 plane duct flow, 174, 177-179,200, 219 rectangular ducts, 187-188 thermally developing duct flow, 197-201 thermally developing pipe flow, 192, 196-197 viscous dissipation effects, 148-149 '. natural convection, 348 channel flow, 3 82-383 enclosures, 3 97-399,401-402,405 flat plate, 359-361 turbulent boundary layer, 412-413 porous media cylinder, forced convection, 512-513, 521 enclosure, 537-538 flat plate, forced convection, 503, 517 flat plate, natural convection, 530-531 fully-developed duct flow, 524 horizontal layer, 537-538 stagnation point, forced convection, 5 07,519 turbulent forced convection boundary layer flow, 257, 259, 2 65-267,270-271,280,292 developing duct flow, 334 developing flow, 327, 334 developing pipe flow, 327 integral' equation solution, 280 pipe flow, 309, 311,320, 327 Opposing flow, 4 28-429, 431 Outerregion, 269, 289, 3 18-319 Parabolic equations, 3 69 Parabolized equations, 369 Peclet number, 23, 501 P ermeability,491-494 model for, 494 typical values, 493 . , 618 Introduction to Convective Heat Transfer Analysis PiPe, laminar flow developing, 2 01-212 fully-developed, 158-167 thermally-developing, 189-197 Pipe, turbulent flow equations, 231 Pipe flow, 158-167 PIPEFLOW, 209 PIPETEM, 167 Plane duct, laminar flow developing, 2 01-212 thermally-developing, 1 97-201 PORCYL,512 Porosity, 493 typical values, 493 Porous media, 487-488 apparent conductivity, 4 97 apparent thermal diffusivity, 497 area averaged velocity, 4 88-489 boundarylaye~498-521 \Porous media-Cont. , porosity, 493 typical values, 493 velocity a t surface, 492 Potential flow, 6 0-61,68,492 Power law profiles, 320, 409 Prandtl number, 1 8,23,24-25,264-267• . 2 70-271,347 turbulent, 231, 237, 244, 269-271 Properties o f fluids; see Fluid properties Radiant heat transfer, 1, 2 interaction with other modes, 2 Radiation, thermal, 1, 2 Rayleigh number, 23, 390, 3 99,402,465, 466 ' Benard flow, 4 06 critical, 4 04-406 Rayleigh number, Darcy-modified, 528, 533 critical, 545 RECDUCT, 187 Rectangular ducts, laminar flow, 179-188 Rectangular enclosure, 3 85-407 Recovery factor laminar flow, 1 44-145 turbulent flow, 296 Relaxation factor, 395, 5 35-536 Reynolds analogy assumptions, 2 56-257,306 . turbulent boundary layer flow, 2 55-260 turbulent internal flow, 3 05-312 Reynolds number, 1 8,23,24 condensed film, 5 70-572 Reynolds stresses, 5 4-55 Roughness values, 310 Runge-Kutta method, 91 Separation, floW, 68, 111 Separation o f variables, 191-193, 197, 198 Shearing stress coefficient, 257, 273 Shearing stresses, 33 equation for, 3 31,410 surface, effect on film condensation, 579-582 Similar flows, 41, 46 forced convection, 4 1-46 natural convection, 345-348 Similarity, 4 1-46 ,/ cylinder, 512-513, 519-521 flat plate, forced convection, 5 00-503, 5 16-517 integral equation solutions, 514-519 natural convection, 526-531 similarity solutions, 5 00-507, 5 26-531 stagnation point, 5 05-507, 519 Brinkman extension of Darcy model, 546 Darcy model, 490-494, 545 duct flow, fully-developed, 5 21-525 examples, 487-488 enclosure, 5 31-540 energy equation, 4 95-497 forced convection, 498-525 Forchheimer extension o f Darcy model, 546 horizontal layers, 5 40-545 critical Darcy-modified Rayleigh number, 545 experimental results, 545 small disturbance analysis, 542-545 instability, 544 isotropic, 492 natural convection, 526-545 non-Darcy effects, 545-547 numerical solution, 5 07-513 penneability, 491-494 model for, 494 typical values, 493 pore Reynolds number, 545 Index Similarity solutions Falkner-Skan flow, 105-113 flat plate, 8 3-105 forced convection boundary layer, 83-114, 141-143 natural convection, 354-361 porous media, 498-507, 526-531 stagnation point, 112-113 SIMPLATE, 91 S IMPLCOM,435 Simplifying assumptions used in analysis o f convection, 5 9-61 SIMPLNAT, 359 SIMVART,99 Skin-friction coefficient; see Shearing stress coefficient Slug flow, 1 64-165 Soil,488 Specific heat, 12 Sphere, combined convection, 451 Stability Benard flow porous media, 545 explicit numerical solution, 209-210, 2 18-219,379 Stagnation point heat transfer, 112-113 Stanton number, 23 Stream function, 48, 388, 493, 527, 530, 5 32,588 S treamlines,538 . Stress, turbulent or Reynolds, 55,. 229-231 Surface temperature; s ee Wall temperature Surface tension, 20 . . ./ .. 619 Taylor-Prandtl analogy ~sumptions,262-264 tUrbulent boundary layer flow, 2 62-267 Temperature adiabatic wall, 142-144 bulk, 7 -8 f luid,5-10 for fluid properties, 19, 149-150, 297 mean fluid, 1 0 mean wall, 1 0 mixing cup, 8 wall, 5 ,9 Temperature profile laminar, integral equation analysis, 117 natural convection, 360, 409 porous media, 507, 516, 523-524, 530 power law, 276 similarity solution, 91 Thermal conductivity, 11 apparent, 497 Thermal diffusivity o f porous media, 497, 546 Thermal dispersion, 547 Thermally developing flow, 189-201 Three-layer analogy, 267-271 Time-averaged equations, 49-59, 227-232 Time-averaging, 4 9-50 Transition, laminar to turbulent flow, 2 47-250,258-259,407 boundary layer flow, 2 48-249 condensed film, 5 70-572 duct flow, 250 effect o f freestream turbulence, 248-249 / natural convection, 407 pipe flow, 249 Transition region, 249, 259, 407 Triangular cross-section duct, 188 Tridiagonal matrix, 128-130, 139,286, 2 95,325,510 Tube banks, condensation, 577-578 Tunnelling, 547 TURBINRK, 274 TURBOUND,291 TURBOUNQ, 327 Turbulence kinetic energy, 59 kinetic energy equation, 5 7-59 modeling, 57 Prandtl number, turbulent, 231 time-averaged equations, 49-59, 227-232 transition, laminar-to-turbulent, 247-250, 2 58-259,407 Turbulence models, 57, 234-244 dissipation o f kinetic energy, 2 40-244 kinetic energy based, 239-244 mixing-length, 234-239, 246-247, 455-461 Van Driest's relation, 288 Turbulent boundary layer adiabatic wall temperature, 296 boundary layer thickness, 273, 278 . 620 Introduction to Convective Heat Transfer Analysis Turbulent boundary layer-Conte displacement thickness, 272 flat plate, 254-271 integral equation solution, 272-281 momentum thickness, 272, 331-332 numerical solution, 281-299 Nusselt number, 257, 259, 265-267, 2 70-271,280,292 Prandtl number, turbulent, 231 recovery factor, 296 Reynolds analogy, 2 55-260 transition, 2 59-260 unheated starting length, 276-281 Turbulent duct flow, 3 04-327 entrance region, 329-335 Nusselt number, 309, 3 11,320,327,334 relation between velocity and temperature profiles, 3 06-307 Reynolds analogy, 3 05-312 Turbulent flow, 49; see Turbulence analogy solutions, 2 44-245 closure problem, 57 combined convection, 455-463 condensed film, 5 70-572 continuity equation, 52 energy equation, 5 5-57 e quations,49-59,227-232 fluctuating value o f variables, 4 9-50, 2 27-228 friction velocity, 246 laws o f averaging, 51 mean value o f variables, 4 9-50· Navier-Stokes equation, 52-54, 299 near-wall region, 245-247 steady turbulent flow, 50 time-averaging, 5 0-51 . . Turbulent heat transfer terms, 5 6,229-231 Turbulent Prandtl number, 231, 237, 244 Turbulent shearing stress., 5 5,229-231 Turbulent stresses, 54 TURDUCD, 327 TURINDEV, 335 Underrelaxation, 395-396, 535-536 Wnheated starting length '; laminar flow, 1 21-123 turbulent flow, 2 76-281 Unit thermal convective conductance, 6 Van Driest model, 288 damping factor, 288 Velocity, area averaged, 488-489 Velocity, turbulent flow logarithmic, 247 power law, 277 Velocity profile laminar, integral equation analysis, 115 mixed convection, 470-471 natural convection, 3 59,409 power law, 277 Viscosity, 11 a rr,607-608 water, 606 Viscosity, kinematic, 18 air, 27 water, 28 Viscous dissipation, 38-41 effect on heat transfer, 148-149 laminar boundary layer, 1 40-150 turbulent boundary layer, 296-297 Vorticity, 4 6-49,388 Vorticity-streamfunction formulation, 388-389 Wall temperature, 5, 10 Wall layer, 268, 288 Wall region, 268, 288 Water properties, 6 07-608 Wavy condensed film, 5 70-572 Weber number, 23 Wettability, 557 Wetted perimeter, 17}-178, 188, 250, 570 W ork,38 9 w ww.mhhe.com