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APPENDIX A
Conversion Factors
1. Acceleration 1 ft S-2 = 0.3048 m S-2 1 m S-2 = 3.2808 ft S-2
2. Area 1 in2 = 6.4516 cm2
1 ft2 = 0.0929 m2
1 m2 =10.764 ft 2
3. Density lib in- 3 = 27.680 g cm- 3
lIb ft- 3 =16.019 kg m- 3
1 kg m- 3 = 0.06243 lb ft- 3
1 slug ft- 3 = 515.38 kg m- 3
4. Diffusivity (heat, mass, momentum) 1 ft 2 S-1 = 0.0929 m2 S-1 1 ft 2 h - 1 = 0.2581 X 10 - 4 m2 s - 1 1 m2 S-1 =10.7639 ft 2 S-1 1 cm2 S-1 = 3.8745 ft 2 h- 1
5. Energy, heat, power 1J=lWs=lNm 1 J =107 erg
456
1 Btu = 1055.04 J 1 Btu = 1055.04 W s 1 Btu = 252 cal 1 Btu = 778.161 ft lbf 1 Btu h- 1 = 0.2931 W 1 Btu h- 1 = 3.93xlO- 4 hp 1 cal = 4.1868 J (or W s or N m) 1 cal=3.968X10- 3 Btu 1 hp = 550 ft lbf S-1 1 hp = 745.7 W = 745.7 N m S-1 1 Wh = 3.413 Btu
6. Heat capacity, heat per unit mass, specific heat 1 Btu h- 1 °F- 1 = 0.5274 W °C- 1 1 W °C- 1 =1.8961 Btuh- 1 °F- 1
1 Btu Ib- 1 = 2325.9 J kg- 1
1 Btulb- 1 °F- 1= 4.18669 kJ kg-l °C- 1 (or J g-1 °C- 1)
1 Btu Ib- 1 °F- 1 =1 cal g-1 °C- 1 =1 kcal kg- 1 °C- 1
7. Heat flux 1 Btu h- 1 ft- 2 = 3.1537X10- 3 kW m- 2
1 W m- 2 = 0.31709 Btu h- 1 ft- 2
8. Heat-generation rate 1 Btu h- 1 ft- 3 =10.35 W m- 3
1 W m- 3 = 0.0966 Btu h-1 ft- 3
9. Heat-transfer coefficient 1 Btu h- 1 ft- 2 °F-1 = 5.677 W m- 2 °C- 1
1 W m- 2 °C-1 = 0.1761 Btuh- 1 ft- 2 °F- 1
1 Btuh-1 ft- 2 °F-1= 4.882 kcal h- 1 m- 2 °C- 1
10. Length 1 in = 2.54 cm 1 in = 2.54 X 10- 2 m 1 ft = 0.3048 m 1 m = 3.2808 ft 1 mile = 1609.34 m 1 mile = 5280 ft
11. Mass 1 oz = 28.35 g 11b=16oz lIb = 453.6 g 1 lb = 0.4536 kg 1 kg = 2.2046 lb 1 g = 15.432 grains 1 slug = 32.1739 lb
12. Mass flux lIb ft- 2 h- 1 = 1.3563 X 10-3 kg m- 2 S-l lIb ft- 2 S-l = 4.882 kg m- 2 S-l 1 kg m- 2 S-l = 737.3lb ft- 2 h- 1 1 kg m- 2 S-l = 0.20481b ft- 2 S-l
13. Pressure, force 1N=lkgms-2
1 N = 0.2248 I lbf 1 N = 7.2333 poundals 1 N=10 5 dyn 1 N m- 2 =1 Pa 1lbf = 32.174 ft lb S-2 1 lbf = 4.4482 N 1 lbf = 32.1739 poundals 1 lbf in - 2 == (1 psi) = 6894.76 Pa 1 lbf fe 2 = 47.880 Pa 1 bar =10 5 Pa
Conversion Factors 457
1 atm=14.696Ibf in- 2
1 atm = 2116.2 lbf ft- 2
1 atm = 1.0132 X 105 Pa
14. Specific heat 1 Btu Ib-1 °F-1 =1 kcal kg- 1 °C-1 =1 cal g-l °C-1 1 Btu Ib- 1 °F- 1 = 4.18669 J g-l °K-1 (or W s g-l °C-1) 1 J g-l °C-1 = 0.23885 Btu Ib- 1 °F- 1
(cal g-l °C-1 or kcal kg- 10C- I )
15. Speed 1 ft s-1=0.3048 m S-l 1 m s-1=3.2808 ft S-l 1 mile h -1 = 1.4667 ft S-l 1 mile h -[ = 0.44704 m S-l
16. Temperature 1 ° K =1.8°R T(OF) = 1.WK - 273) + 32
T(OK) = 1~8 (OF-32)+273
T(0C) = _1 (OR - 492) 1.8
17. Thermal conductivity 1 Btu h- 1 ft- 1°F-1 =1.7303 W m- 1 °C- 1
1 Btu h- 1 ft- 1°F-1 = 0.4132 cal S-l m- 1 °C- 1
1 W m - 1oC -1 = 0.5779 Btu h -1 rei °F-1
18. Thermal resistance 1 h- 1 °F- 1 Btu- I =1.896 °C w-1 1 °C w- 1 = 0.528 h OF Btu-1
19. Viscosity 1 poise =1 g cm-1 S-l 1 poise = 102 centipoise 1 poise = 241.91b rei h- 1 lIb ft- 1 S-l =1.4882 kg m- 1 S-l lIb ft- 1 S-I =14.882 poises lIb ft- 1 h-1 = 0.4134 X 10- 3 kg m- I S-I lIb ft- I h- I = 0.4134X10- 2 poise
20. Volume 1 in3 = 16.387 cm3
1 cm3 = 0.06102 in3
1 oz (U.S. fluid) = 29.573 c~
458 Appendix A
1 ft3 = 0.0283168 m3
1 ft3 = 28.3168 liters 1 ft 3 = 7.4805 gal (U.S.) 1 m3 = 35.315 ft 3
1 gal (U.S.) = 3.7854 liters 1 gal (U.S.) = 3.7854XI0- 3 m3
1 gal (U.S.) = 0.13368 ft 3
Constants
g, = gravitational acceleration conversion factor
= 32.1739 ft lb lbj - I S-2
= 4.1697 X 10 8 ft lb lbj - I h- 2
= 1 g cm dyn - 1 S - 2
=1 kg m N- 1 S-2
= 1 lb ft poundal- 1 s - 2
=1 slug ft lbj - I S-2
J = mechanical equivalent of heat = 778.16 ft lbj Btu- 1
9f = gas constant = 1544 ft lbj lb -I mol-loR -I = 8.314 N m g-l mol- 1 °K- 1
=1.987 cal g-I mol- I °K- 1
APPENDIX B
Physical Properties of Gases, Liquids, Liquid Metals, and Metals
Table B-1 Physical properties of gases at atmospheric pressure
v, K,
P Cp ' p., m2 s- 1 k, m2 s- 1
T, oK kgm- 3 kJ kg- l °K- l kg m-ls- l X10 6 W m- l °K- l XI0 4
Air
100 3.6010 1.0266 0.6924 X 10 - 5 1.923 0.009246 0.02501 150 2.3675 1.0099 1.0283 4.343 0.013735 0.05745 200 1.7684 1.0061 1.3289 7.490 0.01809 0.10165 250 1.4128 1.0053 1.488 9.49 0.02227 0.13161 300 1.1774 1.0057 1.983 15.68 0.02624 0.22160 350 0.9980 1.0090 2.075 20.76 0.03003 0.2983 400 0.8826 1.0140 2.286 25.90 0.03365 0.3760 450 0.7833 1.0207 2.484 28.86 0.03707 0.4222 500 0.7048 1.0295 2.671 37.90 0.04038 0.5564 550 0.6423 1.0392 2.848 44.34 0.04360 0.6532 600 0.5870 1.0551 3.018 51.34 0.04659 0.7512 650 0.5430 1.0635 3.177 58.51 0.04953 0.8578 700 0.5030 1.0752 3.332 66.25 0.05230 0.9672 750 0.4709 1.0856 3.481 73.91 0.05509 1.0774 800 0.4405 1.0978 3.625 82.29 0.05779 1.1951
459
Pr
0.770 0.753 0.739 0.722 0.708 0.697 0.689 0.683 0.680 0.680 0.680 0.682 0.684 0.686 0.689
460 Appendix B
Table B-1 (continued)
P, H:,
P cP ' }L, m2 s- 1 k, m2 S-1
T, oK kgm- 3 kJ kg- 1 °K- 1 kg m- 1s- 1 XlO 6 W m- 1 °K- 1 XlO 4 Pr
Air
850 0.4149 1.1095 3.765 90.75 0.06028 1.3097 0.692 900 0.3925 1.1212 3.899 99.3 0.06279 1.4271 0.696 950 0.3716 1.1321 4.023 lO8.2 0.06525 1.55lO 0.699
1000 0.3524 1.1417 4.152 117.8 0.06752 l.6779 0.702 1100 0.3204 1.160 4.44 138.6 0.0732 l.969 0.704 1200 0.2947 1.179 4.69 159.1 0.0782 2.251 0.707 l300 0.2707 1.197 4.93 182.1 0.0837 2.583 0.705 1400 0.2515 1.214 5.17 205.5 0.0891 2.920 0.705 1500 0.2355 1.230 5.40 229.1 0.0946 3.262 0.705 1600 0.2211 1.248 5.63 254.5 0.100 3.609 0.705 1700 0.2082 1.267 5.85 280.5 0.105 3.977 0.705 1800 0.1970 1.287 6.07 308.1 0.111 4.379 0.704 1900 0.1858 1.309 6.29 338.5 0.117 4.811 0.704 2000 0.l762 1.338 6.50 369.0 0.124 5.260 0.702 2100 0.1682 1.372 6.72 399.6 0.l31 5.715 0.700 2200 0.1602 1.419 6.93 432.6 0.l39 6.120 0.707 2300 0.1538 1.482 7.14 464.0 0.149 6.540 0.710 2400 0.l458 1.574 7.35 504.0 0.161 7.020 0.718 2500 0.1394 1.688 7.57 543.5 0.175 7.441 0.730
Helium
3 5.200 8.42X10- 7 0.0106 33 l.4657 5.200 50.2 3.42 0.0353 0.04625 0.74
144 3.3799 5.200 125.5 37.11 0.0928 0.5275 0.70 200 0.2435 5.200 156.6 64.38 O.ll77 0.9288 0.694 255 0.1906 5.200 181.7 95.50 0.l357 1.3675 0.70 366 0.13280 5.200 230.5 173.6 0.1691 2.449 0.71 477 0.10204 5.200 275.0 269.3 0.197 3.716 0.72 589 0.08282 5.200 311.3 375.8 0.225 5.215 0.72 700 0.07032 5.200 347.5 494.2 0.251 6.661 0.72 800 0.06023 5.200 381.7 634.1 0.275 8.774 0.72 900 0.05286 5.200 413.6 781.3 0.298 10.834 0.72
Carbon dioxide
220 2.4733 0.783 l1.105xlO- 6 4.490 0.010805 0.05920 0.818 250 2.1657 0.804 12.590 5.813 0.012884 0.07401 0.793 300 1.7973 0.871 14.958 8.321 0.016572 0.10588 0.770 350 1.5362 0.900 17.205 ll.l9 0.02047 0.14808 0.755 400 1.3424 0.942 19.32 14.39 0.02461 0.19463 0.738 450 1.1918 0.980 21.34 17.90 0.02897 0.24813 0.721 500 1.0732 1.0l3 23.26 21.67 0.03352 0.3084 0.702 550 0.9739 1.047 25.08 25.74 0.03821 0.3750 0.685 600 0.8938 1.076 26.83 30.02 0.043ll 0.4483 0.668
Physical Properties of Gases, Liquids, Liquid Metals, and Metals 461
Table B-1 (continued)
v, 1(,
P cP ' /L, m2 s- I k, m2 s- I
T, oK kgm- 3 kJ kg- I °K- I kg m-Is- I X 106 W m- I °K- I x10 4 Pr
Carbon monoxide
220 1.55363 1.0429 13.832XlO- 6 8.903 0.01906 0.11760 0.758 250 0.8410 1.0425 15.40 11.28 0.02144 0.15063 0.750 300 1.13876 1.0421 17.843 15.67 0.02525 0.21280 0.737 350 0.97425 1.0434 20.09 20.62 0.02883 0.2836 0.728 400 0.85363 1.0484 22.19 25.99 0.03226 0.3605 0.722 450 0.75848 1.0551 24.18 31.88 0.0436 0.4439 0.718 500 0.68223 1.0635 26.06 38.19 0.03863 0.5324 0.718 550 0.62024 1.0756 27.89 44.97 0.04162 0.6240 0.721 600 0.56850 1.0877 29.60 52.06 0.04446 0.7190 0.724
Ammonia, NH3
220 0.3828 2.198 7.255XlO- 6 19.0 0.0171 0.2054 0.93 273 0.7929 2.177 9.353 11.8 0.0220 0.1308 0.90 323 0.6487 2.177 11.035 17.0 0.0270 0.1920 0.88 373 0.5590 2.236 12.886 23.0 0.0327 0.2619 0.87 423 0.4934 2.315 14.672 29.7 0.0391 0.3432 0.87 473 0.4405 2.395 16.49 37.4 0.0467 0.4421 0.84
Steam (H 20 vapor)
380 0.5863 2.060 12.71 XlO- 6 21.6 0.0246 0.2036 1.060 400 0.5542 2.014 13.44 24.2 0.0261 0.2338 1.040 450 0.4902 1.980 15.25 31.1 0.0299 0.307 1.010 500 0.4405 1.985 17.04 38.6 0.0339 0.387 0.996 550 0.4005 1.997 18.84 47.0 0.0379 0.475 0.991 600 0.3652 2.026 20.67 56.6 0.0422 0.573 0.986 650 0.3380 2.056 22.47 64.4 0.0464 0.666 0.995 700 0.3140 2.085 24.26 77.2 0.0505 0.772 1.000 750 0.2931 2.119 26.04 88.8 0.0549 0.883 1.005 800 0.2739 2.152 27.86 102.0 0.0592 1.001 1.010 850 0.2579 2.186 29.69 115.2 0.0637 1.130 1.019
Hydrogen
30 0.84722 10.840 1.606 X 10- 6 1.895 0.0228 0.02493 0.759 50 0.50955 10.501 2.516 4.880 0.0362 0.0676 0.721
100 0.24572 11.229 4.212 17.14 0.0665 0.2408 0.712 150 0.16371 12.602 5.595 34.18 0.0981 0.475 0.718 200 0.12270 13.540 6.8l3 55.53 0.1282 0.772 0.719 250 0.09819 14.059 7.919 80.64 0.1561 1.130 0.713 300 0.08185 14.314 8.963 109.5 0.182 1.554 0.706 350 0.07016 14.436 9.954 141.9 0.206 2.031 0.697 400 0.06l35 14.491 10.864 177.1 0.228 2.568 0.690 450 0.05462 14.499 11.779 215.6 0.251 3.164 0.682
462 Appendix B
Table B-1 (continued)
v, IC,
P cp ' 1-', m2s-1 k, m2s- 1
T, oK kgm- 3 kJ kg- 1 °K-1 kgm-1s- 1 X10 6 W m- 1 °K- 1 X10 4 Pr
Hydrogen
500 0.04918 14.507 12.636 257.0 0.272 3.817 0.675 550 0.04469 14.532 13.475 301.6 0.292 4.516 0.668 600 0.04085 14.537 14.285 349.7 0.315 5.306 0.664 700 0.03492 14.574 15.89 455.1 0.351 6.903 0.659 800 0.03060 14.675 17.40 569 0.384 8.563 0.664 900 0.02723 14.821 18.78 690 0.412 10.217 0.676
1000 0.02451 14.968 20.16 822 0.440 11.997 0.686 1100 0.02227 15.165 21.46 965 0.464 13.726 0.703 1200 0.02050 15.366 22.75 1107 0.488 15.484 0.715 1300 0.01890 15.575 24.08 1273 0.512 17.394 0.733 1333 0.01842 15.638 24.44 1328 0.519 18.013 0.736
Oxygen
100 3.9918 0.9479 7.768XlO- 6 1.946 0.00903 0.023876 0.815 150 2.6190 0.9178 11.490 4.387 0.01367 0.05688 0.773 200 1.9559 0.9131 14.850 7.593 0.01824 0.10214 0.745 250 1.5618 0.9157 17.87 11.45 0.02259 0.15794 0.725 300 1.3007 0.9203 20.63 15.86 0.02676 0.22353 0.709 350 1.1133 0.9291 23.16 20.80 0.03070 0.2968 0.702 400 0.9755 0.9420 25.54 26.18 0.03461 0.3768 0.695 450 0.8682 0.9567 27.77 31.99 0.03828 0.4609 0.694 500 0.7801 0.9722 29.91 38.34 0.04173 0.5502 0.697 550 0.7096 0.9881 31.97 45.05 0.04517 0.6441 0.700 600 0.6504 1.0044 33.92 52.15 0.04832 0.7399 0.704
Nitrogen
100 3.4808 1.0722 6.862XlO- 6 1.971 0.009450 0.025319 0.786 200 1.7108 1.0429 12.947 7.568 0.01824 0.10224 0.747 300 1.1421 1.0408 17.84 15.63 0.02620 0.22044 0.713 400 0.8538 1.0459 21.98 25.74 0.03335 0.3734 0.691 500 0.6824 1.0555 25.70 37.66 0.03984 0.5530 0.684 600 0.5687 1.0756 29.11 51.19 0.04580 0.7486 0.686 700 0.4934 1.0969 32.13 65.13 0.05123 0.9466 0.691 800 0.4277 1.1225 34.84 81.46 0.05609 1.1685 0.700 900 0.3796 1.1464 37.49 91.06 0.06070 1.3946 0.711
1000 0.3412 1.1677 40.00 117.2 0.06475 1.6250 0.724 1100 0.3108 1.1857 42.28 136.0 0.06850 1.8591 0.736 1200 0.2851 1.2037 44.50 156.1 0.07184 2.0932 0.748
Tab
le B
-2 P
hysi
cal p
rope
rtie
s o
f sa
tura
ted
liqu
ids
c p,
k,
K,
m2
S-1
t,OC
p
, kg
m-3
kJ
kg
-1°K
-1
1',
m2
S-1
W
m-1
°K-1
x
10
7 P
r /3
,oK
-1
Am
mon
ia, N
H 3
-50
70
3.69
4.
463
0.4
35
XlO
-6
0.54
7 1.
742
2.60
-4
0
691.
68
4.46
7 0.
406
0.54
7 1.
775
2.28
-3
0
679.
34
4.47
6 0.
387
0.54
9 1.
801
2.15
-2
0
666.
69
4.50
9 0.
381
0.54
7 1.
819
2.09
-1
0
653.
55
4.56
4 0.
378
0.54
3 1.
825
2.07
0 64
0.10
4.
635
0.37
3 0.
540
1.81
9 2.
05
10
626.
16
4.71
4 0.
368
0.53
1 1.
801
2.04
20
61
1.75
4.
798
0.35
9 0.
521
1.77
5 2.
02
2.4
5X
lO-3
30
596.
37
4.89
0 0.
349
0.50
7 1.
742
2.01
40
58
0.99
4.
999
0.34
0 0.
493
1.70
1 2.
00
50
564.
33
5.11
6 0.
330
0.47
6 1.
654
1.99
Car
bon
diox
ide,
CO
2
-50
1,
156.
34
1.84
0.
119
X 1
0-6
0.
0855
0.
4021
2.
96
-40
1,
117.
77
1.88
0.
118
0.10
11
0.48
10
2.46
-3
0
1,07
6.76
1.
97
0.11
7 0.
1116
0.
5272
2.
22
-20
1,
032.
39
2.05
0.
115
0.11
51
0.54
45
2.12
-1
0
983.
38
2.18
0.
113
0.10
99
0.51
33
2.20
0 92
6.99
2.
47
0.10
8 0.
1045
0.
4578
2.
38
10
860.
03
3.14
0.
101
0.09
71
0.36
08
2.80
20
77
2.57
5.
0 0.
091
0.08
72
0.22
19
4.10
14
.00
X 1
0-3
~
30
597.
81
36.4
0.
080
0.07
03
0.02
79
28.7
0
-w
Tab
le D
-2 (
Con
tinu
ed)
~
c p,
k,
K, m
2 S
-l
0\
t,O
C
p,
kg m
-3
kJ k
g-l
°K
-1
p,m
2s-1
W
m-l
°K
-1
X 1
07
p,o
K-1
~
Pr
Dic
hlor
odif
tuor
omet
hane
(F
reon
), C
CI1
F 1
-50
1,
546.
75
0.87
50
0.3
10
XlO
-6
0.06
7 0.
501
6.2
2.6
3X
10
-3
-40
1,
518.
71
0.88
47
0.27
9 0.
069
0.51
4 5.
4 -3
0
1,48
9.56
0.
8956
0.
253
0.06
9 0.
526
4.8
-20
1,
460,
57
0.90
73
0.23
5 0.
071
0.53
9 4.
4 -1
0
1,42
9.49
0.
9203
0.
221
0.07
3 0.
550
4.0
0 1,
397.
45
0.93
45
0.21
4 0.
073
0.55
7 3.
8 10
1,
364.
30
0.94
96
0.20
3 0.
073
0.56
0 3.
6 20
1,
330.
18
0.96
59
0.19
8 0.
073
0.56
0 3.
5 30
1,
295.
10
0.98
35
0.19
4 0.
071
0.56
0 3.
5 40
1,
257.
13
1.00
19
0.19
1 0.
069
0.55
5 3.
5 50
1,
215.
96
1.02
16
0.19
0 0.
067
0.54
5 3.
5
Eng
ine
oil (
unus
ed)
0 89
9.12
1.
796
0.00
428
0.14
7 0.
911
47,1
00
20
888.
23
1.88
0 0.
0009
0 0.
145
0.87
2 10
,400
0.
70 X
10
-3
40
876.
05
1.96
4 0.
0002
4 0.
144
0.83
4 2,
870
60
864.
04
2.04
7 0
.83
9X
10
-4
0.14
0 0.
800
1,05
0 80
85
2.02
2.
131
0.37
5 0.
138
0.76
9 49
0
100
840.
01
2.21
9 0.
203
0.13
7 0.
738
276
120
828.
96
2.30
7 0.
124
0.13
5 0.
710
175
140
816.
94
2.39
5 0.
080
0.13
3 0.
686
116
160
805.
89
2.48
3 0.
056
0.13
2 0.
663
84
Eth
ylen
e gl
ycol
, C1H
4(O
H1)
0 1,
130.
75
2.29
4 5
7.5
3x
lO-6
0.
242
0.93
4 61
5 20
1,
116.
65
2.38
2 19
.18
0.24
9 0.
939
204
0.6
5X
lO-3
40
1,10
1.43
20
474
8.69
0.
256
0.93
9 93
60
1,
087.
66
2,56
2 4.
75
0.26
0 0.
932
51
80
1,07
7.56
2.
650
2.98
0.
261
0.92
1 32
04
100
1,05
8.50
2.
742
2.03
0.
263
0.90
8 22
04
Eut
ecti
c ca
lciu
m c
hlor
ide
solu
tion
, 29.
9% C
aCI 2
-50
1,
319.
76
2.60
8 36
.35
X 1
0-6
00
402
1.16
6 31
2 -4
0
1,31
4.96
2.
6356
24
.97
0041
5 1.
200
208
-30
1,
310.
15
2.66
11
17.1
8 00
429
1.23
4 13
9 -2
0
1,30
5.51
2.
688
11.0
4 0.
445
1.26
7 87
.1
-10
1,
300.
70
2.71
3 6.
96
0045
9 1.
300
53.6
0 1,
296.
06
2.73
8 4.
39
0047
2 1.
332
33.0
10
1,
291.
41
2.76
3 3.
35
0048
5 1.
363
24.6
20
1,
286.
61
2.78
8 2.
72
0049
8 1.
394
19.6
30
1,
281.
96
2.81
4 2.
27
0.51
1 1.
419
16.0
40
1,
277.
16
2.83
9 1.
92
0.52
3 1.
445
13.3
50
1,
272.
51
2.86
8 1.
65
0.53
5 1.
468
11.3
Gly
ceri
n, C
3H
6(O
Hh
0 1,
276.
03
2.26
1 0.
0083
1 0.
282
0.98
3 84
.7X
I03
10
1,27
0.11
2.
319
0.00
300
0.28
4 0.
965
31.0
20
1,
264.
02
2.38
6 0.
0011
8 0.
286
0.94
7 12
.5
0.5
0x
1O
-3
30
1,25
8.09
2.
445
0.00
050
0.28
6 0.
929
5.38
40
1,
252.
01
2.51
2 0.
0002
2 0.
286
0.91
4 20
45
50
1,24
4.96
2.
583
0.00
015
0.28
7 0.
893
1.63
Mer
cury
, H
g
0 13
,628
.22
0.14
03
0.12
4X1O
-6
8.20
42
.99
0.02
88
20
13,5
79.0
4 0.
1394
0.
114
8.69
46
.06
0.02
49
1.82
X1O
-4
"'" 50
13
,505
.84
0.13
86
0.10
4 90
40
50.2
2 0.
0207
0
-10
0 13
,384
.58
0.13
73
0.09
28
10.5
1 57
.16
0.01
62
Vl
150
13,2
64.2
8 0.
1365
0.
0853
11
049
63.5
4 0.
0134
200
13,1
44.9
4 0.
1570
0.
0802
12
.34
69.0
8 0.
0116
Tab
le D
-2 (
Con
tinu
ed)
~
c p,
k,
K, m
2 S
-I
a- a-t,O
C
p,
kg m
-3
kJ k
g-I
°K
-I
JI, m
2 S
-I
W m
-I
°K-I
X
10
7 P
r {3
,oK
-I
250
13,0
25.6
0 0.
1357
0.
0765
13
.07
74.0
6 0.
0103
31
5.5
12,8
47
0.13
4 0.
0673
14
.02
81.5
0.
0083
Met
hyl c
hlor
ide,
CH
30
-50
1,
052.
58
1.47
59
0.32
0 X
10
-6
0.21
5 1.
388
2.31
-4
0
1,03
3.35
1.
4826
0.
318
0.20
9 1.
368
2.32
-3
0
1,01
6.53
1.
4922
0.
314
0.20
2 1.
337
2.35
-2
0
999.
39
1.50
43
0.30
9 0.
196
1.30
1 2.
38
-10
98
1.45
1.
5194
0.
306
0.18
7 1.
257
2.43
0 96
2.39
1.
5378
0.
302
0.17
8 1.
213
2.49
10
94
2.36
1.
5600
0.
297
0.17
1 1.
166
2.55
20
92
3.31
1.
5860
0.
293
0.16
3 1.
112
2.63
30
90
3.12
1.
6161
0.
288
0.15
4 1.
058
2.72
40
88
3.10
1.
6504
0.
281
0.14
4 0.
996
2.83
50
86
1.15
1.
6890
0.
274
0.13
3 0.
921
2.97
Sulfu
r D
ioxi
de, S
02
-50
1,
560.
84
1.35
95
0.48
4 X
10
-6
0.24
2 1.
141
4.24
-4
0
1,53
6.81
1.
3607
0.
424
0.23
5 1.
130
3.74
-3
0
1,52
0.64
1.
3616
0.
371
0.23
0 1.
117
3.31
-2
0
1,48
8.60
1.
3624
0.
324
0.22
5 1.
107
2.93
-1
0
1,46
3.61
1.
3628
0.
288
0.21
8 1.
097
2.62
0 1,
438.
46
1.36
36
0.25
7 0.
211
1.08
1 2.
38
10
1,41
2.51
1.
3645
0.
232
0.20
4 1.
066
2.18
20
1,
386.
40
1.36
53
0.21
0 0.
199
1.05
0 2.
00
1.9
4X
lO-3
30
1,35
9.33
1.
3662
0.
190
0.19
2 1.
035
1.83
40
1,
329.
22
1.36
74
0.17
3 0.
185
1.01
9 1.
70
50
1,29
9.10
1.
3683
0.
162
0.17
7 0.
999
1.61
Wat
er, H
20
0 1,
002.
28
4.21
78
1.7
88
XlO
-6
0.55
2 1.
308
13.6
20
1,
000.
52
4.18
18
1.00
6 0.
597
1.43
0 7.
02
0.1
8X
lO-3
40
994.
59
4.17
84
0.65
8 0.
628
1.51
2 4.
34
60
985.
46
4.18
43
0.47
8 0.
651
1.55
4 3.
02
80
974.
08
4.19
64
0.36
4 0.
668
1.63
6 2.
22
100
960.
63
4.21
61
0.29
4 0.
680
1.68
0 1.
74
120
945.
25
4.25
0 0.
247
0.68
5 1.
708
1.44
6 14
0 92
8.27
4.
283
0.21
4 0.
684
1.72
4 1.
241
160
909.
69
4.34
2 0.
190
0.68
0 1.
729
1.09
9 18
0 88
9.03
4.
417
0.17
3 0.
675
1.72
4 1.
004
200
866.
76
4.50
5 0.
160
0.66
5 1.
706
0.93
7 22
0 84
2.41
4.
610
0.15
0 0.
652
1.68
0 0.
891
240
815.
66
4.75
6 0.
143
0.63
5 1.
639
0.87
1 26
0 78
5.87
4.
949
0.13
7 0.
611
1.57
7 0.
874
280.
6 75
2.55
5.
208
0.13
5 0.
580
1.48
1 0.
910
300
714.
26
5.72
8 0.
135
0.54
0 1.
324
1.01
9
~
-..J
.".
Tab
le B
-3 P
hysi
cal p
rope
rtie
s of
liqu
id m
etal
s 0
-0
0
Mel
ting
B
oilin
g JL
X 1
04
V X
l06
k K
Xl0
6 P
oint
P
oint
T
p
c p
Met
al
°C
°C
°C
kg
m-3
kJ
kg
-1 °
C-1
kg
m-1
S-1
m
2 S
-1
W m
-1 °
C-1
m
2s-
1 P
r
Bis
mut
h 27
1 14
77
315
10,0
11
0.14
4 16
.2
0.16
0 16
.4
11.2
5 0.
0142
53
8 97
39
0.15
5 11
.0
0.11
3 15
.6
10.3
4 0.
0110
76
0 94
67
0.16
5 7.
9 0.
083
15.6
9.
98
0.00
83
Lea
d 32
7 17
37
371
10,5
40
0.15
9 2.
40
0.02
3 16
.1
9.61
0.
024
704
10,1
40
0.15
5 1.
37
0.01
4 14
.9
9.48
0.
0143
Lit
hium
17
9 13
17
204.
4 50
9.2
4.36
5 5.
416
1.10
98
46.3
7 20
.96
0.05
1 31
5.6
498.
8 4.
270
4.46
5 0.
8982
43
.08
20.3
2 0.
0443
42
6.7
489.
1 4.
211
3.92
7 0.
8053
38
.24
18.6
5 0.
0432
53
7.8
476.
3 4.
171
3.47
3 0.
7304
30
.45
15.4
0 0.
0476
Mer
cury
-3
8.9
35
7 -1
7.8
13
,707
.1
0.14
15
18.3
34
0.13
42
9.76
5.
038
0.02
66
93.3
13
,409
.4
0.13
65
12.2
24
0.09
03
10.3
8 5.
619
0.01
61
204.
4 13
,168
.1
0.13
56
10.0
46
0.07
48
12.6
3 7.
087
0.01
08
Sod
ium
97
.8
883
93.3
93
1.6
1.38
4 7.
131
0.76
89
84.9
6 56
.29
0.01
16
204.
4 90
7.5
1.33
9 4.
521
0.50
10
80.8
1 66
.80
0.00
75
315.
6 87
8.5
1.30
4 3.
294
0.37
66
75.7
8 66
.47
0.00
567
426.
7 85
2.8
1.27
7 2.
522
0.29
68
69.3
9 64
.05
0.00
464
537.
8 82
3.8
1.26
4 2.
315
0.28
21
64.3
7 62
.09
0.00
455
648.
9 79
0.0
1.26
1 1.
964
0.24
96
60.5
6 61
.10
0.00
408
760.
0 76
7.5
1.27
0 1.
716
0.22
45
56.5
8 58
.34
0.00
385
Pot
assi
um
63.9
76
0 42
6.7
741.
7 0.
766
2.10
8 0.
2839
39
.45
69.7
4 0.
0041
53
7.8
714.
4 0.
762
1.71
1 0.
2400
36
.51
67.3
9 0.
0036
64
8.9
690.
3 0.
766
1.46
3 0.
2116
33
.74
64.1
0 0.
0033
76
0.0
667.
7 0.
783
1.33
1 0.
1987
31
.15
59.8
6 0.
0033
NaK
-1
1.1
78
4 93
.3
889.
8 1.
130
5.62
2 0.
6347
25
.78
25.7
6 0.
0246
(5
6% N
a,
204.
4 86
5.6
1.08
9 3.
803
0.44
14
26.4
7 28
.23
0.01
55
44%
K)
315.
6 83
8.3
1.06
8 2.
935
0.35
15
27.1
7 30
.50
0.01
15
426.
7 81
4.2
1.05
1 2.
150
0.26
52
27.6
8 32
.52
0.00
81
537.
8 78
8.4
1.04
7 2.
026
0.25
81
27.6
8 33
.71
0.00
76
648.
9 75
9.5
1.05
1 1.
695
0.22
40
27.6
8 34
.86
0.00
64
Tab
le B
-4 P
hysi
cal p
rope
rtie
s of
met
als
Pro
pert
ies
at 2
0°C
T
herm
al c
ondu
ctiv
ity
k, W
m-
1 °C
-1
Mel
ting
K
,
Poi
nt
p,
c p,
k,
m2s-
1
Met
al
°C
kg
m-3
k
Jkg
-1
°C-
1 W
m-
1 °C
-1
XlO
s -1
00
°C O
°C
100°
C 2
00°C
300
°C 4
00°C
600
°C 8
00°C
l0
00°C
Alu
min
um:
Pur
e 66
0 2,
707
0.89
6 20
4 8.
418
215
202
206
215
228
249
AI-
Cu
(Dur
alum
in),
94
-96%
AI,
3-5%
C
u, t
race
Mg
2,78
7 0.
883
164
6.67
6 12
6 15
9 18
2 19
4 A
I-Si
(S
ilum
in,
copp
er-b
eari
ng),
86
.5%
AI,
1% C
u 2,
659
0.86
7 13
7 5.
933
119
137
144
152
161
AI-
Si (
Alu
sil)
, 78
-80
% A
I, 20
-22%
Si
2,
627
0.85
4 16
1 7.
172
144
157
168
175
178
AI-
Mg-
Si, 9
7% A
I, 1%
Mg
,l%
Si,
1% M
n 2,
707
0.89
2 17
7 7.
311
175
189
204
Ber
ylli
um
1277
1,
850
1.82
5 20
0 5.
92
Bis
mut
h 27
2 9.
780
0.12
2 7.
86
0.66
C
adm
ium
32
1 8,
650
0.23
1 96
.8
4.84
C
oppe
r:
Pur
e 10
85
8.95
4 0.
3831
38
6 11
.234
40
7 38
6 37
9 37
4 36
9 36
3 35
3 A
lum
inum
bro
nze
95%
Cu,
5%
AI
8,66
6 0.
410
83
2.33
0 B
ronz
e 75
% C
u,
25%
Sn
8,66
6 0.
343
26
0.85
9 R
ed b
rass
85%
Cu,
9%
Sn,
6% Z
n
8,71
4 0.
385
61
1.80
4 59
71
""" B
rass
70%
Cu,
0
1
30%
Zn
8,
522
0.38
5 II
I 3.
412
88
128
144
147
147
'CI
Tab
le 8
04 (
cont
inue
d)
.j::.
P
rope
rtie
s at
20°
C
The
rmal
con
duct
ivit
y k,
W m
-1
°C-
1 -.
J 0
Mel
ting
cp
' IC
, P
oint
p,
k,
m
2s-
1
Met
al
°C
kg
m-3
k
Jkg
-1
°C-1
W
m-
1 °C
-1
XI0
5 -1
00
°C O
°C 1
00°C
200
°C 3
00°C
4O
Q°C
600
°C 8
00°C
lO
00°C
Ger
man
sil
ver
62%
C
u, 1
5% N
i,
22%
Zn
8,
618
0.39
4 24
.9
0.73
3 19
.2
31
40
45
48
Con
stan
tan
60%
C
u,40
% N
i 8,
922
0.41
0 22
.7
0.6l
2 21
22
.2
26
Iron
: P
ure
1537
7,
897
0.45
2 73
2.
034
87
73
67
62
55
48
40
36
35
Wro
ught
iron
, 0.5
% C
7,
849
0.46
59
1.
626
59
57
52
48
45
36
33
33
Ste
el
(C m
ax =
1.5
%):
C
arbo
n st
eel
C=
0.5
%
7,83
3 0.
465
54
1.47
4 55
52
48
45
42
35
31
29
1.
0%
7,80
1 0.
473
43
1.17
2 43
43
42
40
36
33
29
28
1.
5%
7,75
3 0.
486
36
0.97
0 36
36
36
35
33
31
28
28
N
icke
l ste
el
Ni=
0%
7,
897
0.45
2 73
2.
026
20%
7,
933
0.46
19
0.
526
40%
8,
169
0.46
10
0.
279
80%
8,
618
0.46
35
0.
872
Inva
r36%
Ni
8,13
7 0.
46
10.7
0.
286
Chr
ome
stee
l C
r=
0%
7,89
7 0.
452
73
2.02
6 87
73
67
62
55
48
40
36
35
1%
7,
865
0.46
61
1.
665
62
55
52
47
42
36
33
33
5%
7,83
3 0.
46
40
1.11
0 40
38
36
36
33
29
29
29
20
%
7,68
9 0.
46
22
0.63
5 22
22
22
22
24
24
26
29
C
r-N
i (ch
rom
e-ni
ckel
): 1
5% C
r,
10%
Ni
7,86
5 0.
46
19
0.52
7 18
% C
r, 8
% N
i (V
2A)
7,81
7 0.
46
16.3
0.
444
16.3
17
17
19
19
22
27
31
20
% C
r, 1
5% N
i 7,
833
0.46
15
.1
0.41
5 25
% C
r, 2
0% N
i 7,
865
0.46
12
.8
0.36
1
Tun
gste
n st
eel
W=
0%
7,
897
0.45
2 73
2.
026
1%
7,91
3 0.
448
66
1.85
8 5%
8,
073
0.43
5 54
1.
525
10%
8,
314
0.41
9 48
1.
391
Lea
d 32
8 11
,373
0.
130
35
2.34
3 36
.9
35.1
33
.4
31.5
29
.8
Mag
nesi
um:
Pur
e 65
0 1,
746
1.01
3 17
1 9.
708
178
171
168
163
157
Mg-
AI
(ele
ctro
ly-
tic)
6-8
% A
I, 1
-2%
Zn
1,
810
1.00
66
3.
605
52
62
74
83
Mol
ybde
num
2,
621
10,2
20
0.25
1 12
3 4.
790
138
125
118
114
111
109
106
102
99
Nic
kel:
P
ure
(99.
9%)
1,45
5 8,
906
0.44
59
90
2.26
6 10
4 93
83
73
64
59
N
i-C
r 90
% N
i,
10%
Cr
8,66
6 0.
444
17
0.44
4 17
.1
18.9
20
.9
22.8
24
.6
80%
Ni,
20%
Cr
8,31
4 0.
444
12.6
0.
343
12.3
13
.8
15.6
17
.1
18.0
22
.5
Silv
er:
Pur
est
962
10,5
24
0.23
40
419
17.0
04
419
417
415
412
Pur
e (9
9.9%
) 10
,525
0.
2340
40
7 16
.563
41
9 41
0 41
5 37
4 36
2 36
0 T
in,
pure
23
2 7,
304
0.22
65
64
3.88
4 74
65
.9
59
57
Tun
gste
n 3,
387
19,3
50
0.13
44
163
6.27
1 16
6 15
1 14
2 13
3 12
6 11
2 76
U
rani
um
1,13
3 19
,070
0.
116
27.6
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Zin
c, p
ure
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7,14
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4.10
6 11
4 11
2 10
9 10
6 10
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~
'-'
APPENDIX C
Gamma, Beta and Incomplete Beta Functions
Gamma function definition
Recursion formula:
a f(a)
1.00 1.0000 1.05 0.9735 1.10 0.9514 1.15 0.9330 1.20 0.9182 1.25 0.9064 1.30 0.8975
Beta function definition:
472
f(o:) = l°Ota-le-tdt o
f ( 0: + 1) = o:f ( 0:) a f(a)
1.35 0.8912 1.40 0.8873 1.45 0.8857 1.50 0.8862 1.55 0.8889 1.60 0.8935 1.65 0.9001
a f(a)
1.70 0.9086 1.75 0.9191 1.80 0.9314 1.85 0.9456 1.90 0.9618 1.95 0.9799 2.00 1.0000
Appendix C 473
Incomplete Beta function definition:
Bx(o.,f3) = lXta-l(l-t)fJ-ldt 0
Recursion formula:
BAa, f3) = Bl (a, f3) - BI-A a, f3)
The following table [1] gives the functional ratios Ix(o., f3) =
BX< a, f3)/ Bl (a, f3) for typical combinations of a and f3:
Incomplete beta function ratios Ix( a, P)
0.=1/3 0.=1/3 0.=1/3 0.=2/3 0.=1/9 0.=1/9 0.=1/9 0.=8/9 x P=2/3 P=4/3 P=8/3 P=4/3 P=8/9 P =10/9 p= 20/9 P=1O/9
0 0 0 0 0 0 0 0 0 0.02 0.2249 0.3068 0.4007 0.0912 0.6346 0.6588 0.7281 0.0342 0.04 0.2838 0.3859 0.5007 0.1443 0.6856 0.7113 0.7845 0.0628 0.06 0.3254 0.4410 0.5684 0.1886 0.7173 0.7439 0.8186 0.0917 0.08 0.3588 0.4845 0.6204 0.2278 0.7407 0.7679 0.8431 0.1174 0.10 0.3872 0.5210 0.6627 0.2636 0.7595 0.7870 0.8622 0.1416 0.20 0.4924 0.6506 0.8008 0.4124 0.8213 0.8490 0.9199 0.2607 0.30 0.5694 0.7377 0.8793 0.5321 0.8603 0.8870 0.9506 0.3715 0.40 0.6337 0.8038 0.9284 0.6339 0.8895 0.9146 0.9696 0.4765 0.50 0.6911 0.8566 0.9599 0.7225 0.9133 0.9362 0.9820 0.5767 0.60 0.7448 0.8998 0.9796 0.7999 0.9335 0.9538 0.9901 0.6725 0.70 0.7970 0.9352 0.9912 0.8671 0.9515 0.9686 0.9952 0.7640 0.80 0.8501 0.9640 0.9972 0.9244 0.9679 0.9812 0.9982 0.8507 0.90 0.9084 0.9863 0.9996 0.9706 0.9835 0.9917 0.9996 0.9313 1.00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Bl (a, P) 3.6275 2.6499 2.0153 1.2092 9.1853 8.8439 7.9839 1.0206
References
[1] Baxter, D. C, and Reynolds, W. c.: Fundamental solutions for heat transfer from nonisothermal flat plates. J. Aero. Sci. 25:403 (1958).
APPENDIX D
Fortran Program for Head's Method
In this Appendix we present a computer program for predicting the turbulent boundary-layer development on two-dimensional bodies by Head's method. The code uses FORTRAN IV and is based on the integration of Eqs. (3.68) and (6.117) by a fourth-order Runge-Kutta method [1], which requires the specification of the following arguments to solve the equations:
X The dependent variable x. The initial value of x must be input. B The number of dependent variables to be computed. For
example,
[Ue:HJ = [ ;gn Initial values of (J, (ue(JH1) must be input.
C Denotes the right-hand sides of the differential equations. For example
[ cf (J dUe 1 [ 1 --(H+2)-- _ C(l) 2 Ue dx -
ueF C(2)
DX The increment in x. DX must be input. N The number of simultaneous equations to be integrated.
474
Fortran Program for Head's Method 475
F The array used by the subroutine (RKM) to store values of the array B. It is dimensioned N.
G An array that contains intermediate values computed by the subroutine. Four entries of G are used to compute one entry of G. G is dimensioned 4N.
IS A code variable that must be set to zero to initialize the subroutine. It is automatically stepped through the values 1, 2, 3 and 4 and is reset to zero by the subroutine after the variables for X and the array B are computed.
Since the solution of Eq. (3.68) requires the specification of the external velocity distribution, we input ue(x)/uoo ' UE(I), as a function of surface distance x / L, X(I), with Uoo denoting the reference freestream velocity and L a reference length. The initial conditions consist of a dimensionless momentum thickness, () / L, T(l), and shape factor H, H(l), at the first station. In addition, we specify a reference Reynolds number R L = Uoo L /'" RL, and the total number of x-stations, NXT. In the code, the derivative of external velocity du e / dx is computed by using a three-point Lagrange-interpolation formula.
COMMON/SHARE/ NXT,RL,X(41),UE(41),DUEDX(41), T(41),S(41),H(41), 1 RTH(41),CF(41) ·C-----------------------------------
C
RGNG (Xl, X2, X3, Yl, Y2, Y3, Z) = YU (2. D*Z-X2-X3) / (Xl-X2) / (Xl-X3) + 1 Y2*(2.0*Z-Xl-X3)/(X2-Xl)/(X2-X3)+Y3*(2.0*Z-Xl-X2)/(X3-Xl) 1 /(X3-X2)
1 READ (5, 2, END=20) NXT, RL, T (1), H (1) READ(5,3) (X(I),I=l,NXT) READ(5,3) (UE(I),I=l,NXT) X(NXT+l) =2.*X(NXT)-X(NXT-l)
C CALCULATE THE VELOCITY GRADIENT BY THREE POINT LAGRANGIAN FORMULA NXTMl = NXT-l
C
C
DUEDX (1) = RGNG (X (1) ,X (2), X (3), UE (1), UE (2), UE (3), X (1)) DUEDX (NXT) = RGNG (X (NXT-2), X (NXT-l), x (NXT) ,UE (NXT-2), UE (NXT-l),
1 UE (NXT) ,X (NXT)) DO 10 I=2,NXTMl
10 DUEDX (I) = RGNG (X (1-1), X (I), X (1+1), UE (1-1), UE (I), UE (1+1), X (I))
S (1) = UE (1) *T (1) *HOFHl (-H (1)) CALL STNDRD
WRITE (6, 4) DO 15 I=l,NXT Hl = S (I) /UE (I) IT (I) DELST = T(I)*H(I) DELTA = T(I)*Hl+DELST WRITE(6,5) I,X(I),UE(I),DUEDX(I), T(I),H(I),DELST,DELTA,CF(I),
1 RTH(I) 15 CONTINUE
GDTO 1 20 STOP
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2 FORMAT(I5,3Fl0.D) 3 FORMAT(6F1D.D) 4 FORMAT(lHl,lX,2HNX,6X,lHX,12X,2HUE,lDX,5HDUEDX,9X,5HTHETA,llX,
1 lHH,11X,5HDELST,9X,5HDELTA,11X,2HCF,lDX,6HRTHETA,/) 5 FORMAT(lH ,I3,9E14.6)
END
SUBROUTINE STNDRD COMMON/SHARE/ NXT, RL, X (41) ,UE (41), DUEDX (41), T (41), S (41) ,H (41),
1 RTH (41) ,CF (41) DIMENSION C (2), B (2), Z (2) ,G (8)
476 Appendix D
C-- ---------------------------N 2 IS 0 XX x (1) B(l) T(l) B(2) S(l) UI UE (1) UP DUEDX (1) OX x (2) -x (1) NXTPl NXT'l DO 100 I=2,NXTPl DO 110 LL=l, 4 GO TO (18, lS, 18, 17), LL
lS UI = (UI'UE(I))/2.0.DX*(UP-DUEDX(I))/8.0 UP = (UP.DUEDX(I))/2.0 GO TO 18
17 UI = UE(I) UP = DUEDX (I)
18 CONTINUE Hl B(2)/B(1)/UI IF' ( Hl .LE. 3.0 ) GO TO 120 HB HOF'Hi(Hl) RTHE UI*B(l)*RL CF'02 0.123/10.0.*(0.S78*HB)/RTHE •• 0.2S8 C(l) -(HB'2.0)*B(1)/UI.UP.CF'02 C(2) UI*0.0306/(Hl-3.0)*.0.6169 IF(LL .GT. 1) GO TO 19 H(I-i)= HB RTH(I-l) = RTHE CF'(I-l) 2.0*CF'02 IF'(I .GT. NXT) GO TO 100
19 CONTINUE 110 CALL RKM(XX,B,C,OX,N,Z,G,IS)
T(I) = B(l) S (I) = B (2) ox = X(I'l)-X(I)
100 CONTINUE RETURN
120 WRITE(S,10)I 10 F'ORMAT(lHO, ' •• WARNING: PROGRAM F'AILS AT NX=' ,13,
1 ' INDICATING POSSIBILITY OF' F'LOW SEPARATION •• ',/) NXT = 1-1 RETURN END F'UNCTION HOF'Hl(A)
C H = F'UNCT(Hl=A), INVERSE IF' A NEGATIVE, C Hl= F'UNCT(H=-A)
REAL Cl/1.SS01/,C2/0.S778/,C3/-3.0S4/,C4/3.3/,CS/0.8234/, 1 CS/l.l/,C71-1.2871
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - -HOF'Hl = 0.0 II"(A .LT. -C6) GO TO 2 IF'(A .LE. C4) RETURN IF'(A .LT. S.3) GO TO 3 HOFHl = ((A-C4)/CS).*(1.0/C7).CS RETURN
3 HOFHl = ((A-C4)/C1).*(1.0/C3).C2 RETURN
2 IF'(A .LT. -l.S) GO TO 4 HOF'Hl = CS.(-A-CS) •• C7.C4 RETURN
4 HOF'Hl = Cl*(-A-C2)**C3.C4 RETURN END
SUBROUTINE RKM(A,B,C,DX,N,F',G,IS) DIMENSION B (1), C(1), F' (1" G (1)
C ---------------------------------IS = 15'1 GO TO (10,30,60,80), IS
C FIRST ENTRY 10 E = A
00 20 I=l,N 1"(1) = B (!) G (4*1-3) = C (I) *DX
20 B(I) = F(I)'G(4*I-3)/2.0 GO TO SO
C SECOND ENTRY 3D DO 40 I=l,N
G(4*I-2) = C(I)*OX 406(1) = F(I)-G(UI-2)/2.0 50 A = E-OX/2.0
GO TO 100 C THIRD ENTRY
60 DO 70 I=l,N G(4*I-l) = C(I)*OX
70 6(1) = F(I)-G(4*I-l) A = E-OX GO TO 100
C FOURTH ENTRY 80 DO 90 I=l,N
G (4*1) = C (I) *OX 6 (I) = G (4*1-3) -2.0* (G (4*1-2) -G (4)1-1))
90 6 (I) (6 (I) -G (4*1)) /6. O-F (I) IS 0
100 RETURN END
References
Appendix D 477
[1] Hildebrand, F. B.: Advanced Calculus for Application. Prentice-Hall, NJ, 1962.
Index
Acceleration 4, 25 Accuracy
experimental 151 numerical 78, 88, 386-391,406
Adiabatic (wall) flow 51, 303, 307, 345
Adiabatic (wall) temperature 106 "Aided" duct flow with buoyancy 289,
292 Analogy between
eddies and molecular motion 4-5 heat and momentum transfer
(Reynolds' analogy) 9-12,50, 85,151,168,177,222,254,349
mass and momentum transfer 11 Average, time 4-6, 30-32 Axisymmetric body 47,99-105 Axisymmetric duct see Pipe Axisymmetric flow, equations for
47-48,61-62,99-101 Axisymmetric jet 243-244,259-260
Bernoulli's equation 44, 46 Beta function 181,472-473 Blasius flow (constant pressure laminar
boundary layer) 80, 90, 99, 281 Block-elimination method 385,
394-395, 397-398 Blowing, through surface 81,107 Blunt-nosed body 102
Body force 15,22; see also Buoyancy Body of revolution 47,99-105 Boundary conditions 1,11,12,15,
37,41,43,62-66,74,386 for buoyant flows 268, 280, 285 for coupled flow 66, 302 for duct flow 65, 126, 128, 129 for external flows 72-73, 89 for free shear layers 63, 240, 241,
246,247 for internal flows 65, 126 for turbulent flows 64-65, 66 for wall jet 108 two-point 82
Boundary layer 1-2,11,15,41 and passim
approximation 41-46 control 105; see also Transpiration,
Wall jet equations 15,41-67 laminar 71-113 thermal 1-2,42 thickness 1, 42, 63 turbulent 3,150-167,333-371,
415-427 response to perturbations
154-156 Boussinesq approximation 13, 265 "Box" numerical scheme 391-395,
406-415
479
480 Index
Bulk-average properties 71, 125, 376 Bulk-average velocity 125, 126, 219 Bulk viscosity 23, 53 Buoyancy 2,15,256,263-300,
430-439
Cebeci-Smith eddy viscosity model 187-189,273,422-424
Cells, convection 264, 279 Channel see Internal flow Chapman-Rubesin viscosity law
303-305, 324 Circular cylinder 95-97 Circular duct see Pipe Combustion 8, 263-264 Compressible (variable-density) flow
21,256-259; see also Coupled flows Compressibility, effect on turbulence
256-259 Compression-work 55 Computer programs for
buoyant duct flows 430-439 wall jets 439-444
coupled boundary layers 415-427 duct flows 430-439 free shear layers 444-455
energy equation in duct flow 398-406
Head's method for turbulent boundary layers 474-476
pipe-flow Nusselt number (Problem 5.5) 144
Smith-Spalding Stanton number formula (Problem 4.28) 121
Conduction longitudinal 45, 98 thickness 94
Conductive sublayer 36, 161-165, 169, 186-189, 335-339
Conductivity eddy 16, 38, 133 thermal 1, 8, 9, 26 thermometric 9
Conical nose 102 Conservation (or transport) see
Energy, Enthalpy, Internal energy, Kinetic energy, Mass, Momentum
Constant-density flow 20; see also Low speed flow
Constant-pressure boundary layer 1-2, 10,11,64,80,90,99,172-184,281
Constant-property approximation 12, 13-14,25
Continuity (mass conservation) 19-21, 33,43,72-73,136,137
global 136 Control volume analysis 19-30 Convection
forced, in buoyant flow 263-264, 280-294
free or natural 16,263, 267-280, 430-439
Convective heat transfer see Heat transfer
Conversion factors, SIjImperial units 456-458
Coordinates, transformation of see Transformation
Correlation, data 5, 13, 17, 151, 153, 198
Couette flow 143 Coupled flow 13-14,15,17,254-259,
301-332,333-371; see also Buoyancy, Compressible flow
equations for 32, 53-62 Crank-Nicolson numerical scheme
389-391 Crocco integral 305-306 Curvature, transverse 48, 68 Cylinder, circular 95
Damping length (Van Driest damping factor) 162, 164, 169, 187, 217-218
Defect law 168-170,340-341 for temperature 169-170
Density fluctuations 17, 33, 37, 48-53, 255
effect on turbulence 17,48-53, 255, 333
Developing flow see Entry length Differences, finite 386-387 Differential equations
ordinary 16,41,58-62,72-78,88, 474
partial 15, 19, 41 and passim Differential methods 88, 97-99,
184-189 Diffusivity of mass (diffusion coefficient)
8,11,31 Diffusivity, thermal 9
Dimensional analysis 156-159, 165-166, 168,208
Dimensions 9 Direct stress (" normal stress") 4 Discontinuity of surface temperature
see Step Dissipation of kinetic energy
mean 16,29-30,36,54-55,159, 301,337
turbulent 35-36, 56, 337 Displacement-interaction region 230 Displacement thickness 59-61, 80 Domain of computation (integration)
62 Duct (non-circular) 3, 129-132,
171-172 compressible flow in 372-384 laminar flow in 124-149,376-381 turbulent flow in 216-237,381-384 two-dimensional 124 vertical, heated 288-294, 430-439
Dye, Schmidt number for diffusion of 11
Eddy conductivity 16, 38, 133 diffusivity 16; see also Turbulent
Prandtl number viscosity 16, 37-38, 151-153, 185,
202, 216, 250, 252, 273 empirical formulas for 187-189,
218-220,230,273,395,422-423 Effective origin see Virtual origin Effectiveness of film cooling 106, 201 Eigenvalue method 141, 290 Ellipse, flow over 189 Elliptic PDEs 44
and upstream influence 320 Empirical data for turbulence 5 "Energy"
internal 7 kinetic 7
turbulent 36 thermal internal 7 (enthalpy) thickness 60-61, 90
"Energy" (enthalpy) equation 12, 30-31,35-37,45,48,302
finite difference solution of 386-395 integral 61, 90 (similarity) transformed 77,137
Index 481
Enthalpy 7 equation for 30-31,35-37,45,46,
48 thickness 60-61, 90 total 31
equation for 39, 55-57 "integral" equation for 60-62,
90 Entrainment 196,211 Entropy increase in shock wave
315-316 Entry flow 136-142 Entry length 136-142,230-233
thermal 132-142,227-233; see also Unheated starting length
Equations of motion (summary) 37-38 Error function 84, 256 Error, numerical 78,386-391,406 Expansion fan in supersonic flow 316 Explicit finite-difference methods
388-389 External flow 1-2, 72; see also
Boundary layer, Jet, etc. Extra diffusivity see Eddy diffusivity
Falkner-Skan transformation 75-76, 99, 108, 133, 301
Favre averaging 32 Film cooling 106-113,201-207 Film temperature 71 Finite-difference methods 385-428 Finite-difference errors 88 First law of thermodynamics 7 First order system 392 FLARE approximation 327-328 Flat plate, buoyant flow over 266-280 "Flat plate" (constant pressure flow)
1-2,10,11,64,172-184 in laminar flow 83, 86-88, 89-92;
see also Blasius flow in turbulent flow 168-184
Flow entry 136-142 external 1,72; see also Boundary
layer, Jet, etc. index 47, 99, 129, 216 internal 124-142
Fluctuations definition of 32 density 17, 33, 37, 48-53, 255
482 Index
Mach number 257-258 pressure 48-49 temperature 8,48-53 total temperature 49-50 velocity 32-33 viscosity 33
Fluid element 23 Forced convection in buoyant flow
263-264, 280-294 Free convection 16,263,267-280,
430-439 Free interaction 320-328 Free shear flows 2,238-262; see also
Jet, Mixing layer, Wake Free stream 11; see also External flow Friction factor in duct flow 126,
130-131,217-220,224-225; see also Skin friction
Friction, Mach number 335 temperature 159 velocity 156
Fully-developed duct or pipe flow 124-132,216-227, 398-406
Fully-developed temperature field 124 Fully-rough surface 167
Gamma function 181, 472-473 Gas law 31, 48
properties 459-462 Gases, kinetic theory of 4, 9 Global continuity relation 136, 138 Gortler parameter 253, 256-257 Gradient
of pressure see Pressure gradient of stress 4, 24, 42 transport 4, 8, 23
Grashof number 267, 270, 272 Gravity 22; see also Buoyancy Green's "lag-entrainment" method
197-198 Grid, finite-difference 387-389,392 Group theory 73,238
Hagen-Poiseuille equation 142 "Half jet" see Mixing layer Head's method for turbulent boundary
layers 196-197,213,474-476 Heat 7
conduction law 8, 30
conductivity see Thermal conductivity
flux rate 7-8, 36 equations for 8, 36
specific 9 transfer 1-3,7-13,26 and passim
coefficient 91, 128 formulas
for "flat plates" 177-184 for pipes 222-224
Hiemenz (plane stagnation) flow 80, 95
High Prandtl number, effect on heat transfer of 138, 142
High speed flow 14, 16; see also Compressible flow, Coupled flow
equivalence to low-speed flow with heat addition 301
Howarth's flow 119,312-315,357-360 Hydraulic diameter 131-132 "Hydraulically smooth" surface 167 Hydrodynamic (as distinct from" ther-
mal") 1 Hydrostatic pressure 265
Illingworth-Stewartson transformation 302-312, 329
Impinging jet 2-3 Implicit finite difference methods
388-395 Inclined plate, natural-convection flow
over 276-277 Incompressible flow see Constant-den
sity flow Initial conditions 97-99, 421
for Thwaites' method 93, 103 for wall jet 108-110
Inner law for temperature 158-159 for velocity 156-158
Inner layer 152, 154, 156-168, 335-340
"Integral" equations 16,41,58-62, 196, 199
"Integral" methods 58,88-97,172, 196-201,474-476
"Integral" thicknesses 59-62, 80, 174, 176, 307-309
Interaction free 320-328 viscous-inviscid 230, 320, 327
Intermediate temperature formulas 125,343
Intermittency of transitional or turbulent flow 187-188,232
Internal energy (thermal plus kinetic) 7
equation for 25-29 thermal
equation for 29-30 Internal flow 124-142 "Inviscid" flow 1,63
Jet 2-3, 11-12,41,64, 73 impinging 2-3 interaction with shock wave 316,
318 laminar 73, 239-246 turbulent 249-251, 444-453 wall 72,105-113,201-207
Karman constant in logarithmic law 159
Karman-Schoenherr skin friction formula 174-175
Kinetic energy 7 equation for 29, 39 turbulent 36
Kinetic heating 14, 16, 159, 161, 186, 208
Kinetic theory of gases 4, 9
"Lag-entrainment" method 197-198 Laminar flow 6, 11, 71; see also
Boundary layer, Duct, etc. Law
defect 168-170,340-341 gas 31,48 inner 156-159 of the wall 156, 171
for temperature 158-159,186 Leveque solution 119, 235 Liquid metals 84, 468
low Prandtl number of 10 Logarithmic law
for rough surfaces 165-168 for temperature profile 159-161,
186 for velocity profile 159-161, 170,
186, 203, 338
Index 483
Low Prandtl number, effect on heat transfer of 142, 145, 163
Low Reynolds number, effect on turbulent flow of 169,171,188,
341, 367 Low speed flow (low Mach number)
17,31 with large temperature or density
differences 50-51,254-256 Ludwieg-Tillmann skin friction law
197
Mach number 14, 53, 54-55, 256 fluctuation 257-258 friction 335
Mangler transformation 48, 68, 99-102,137
"Marching" methods for PDEs 43-44, 62,66,385-395
Mass conservation ("continuity") 19-21, 33, 136, 139
Mass diffusivity 8, 11, 31 Mass fraction 8 Mass transfer 8,11,31,81-82; see
also Transpiration analogy with heat transfer 8, 11
Mass-weighted averages 32 Matrix-elimination method (dimen
sional analysis) 157 Mean (time average) values 4-6,31 Mixed-mean temperature 125,128;
see also Bulk-average properties Mixing layer 191,246-249,319-320,
453-455 effect of density differences on
255-259 effect of Mach number on 256-259
Mixinglength 187,194,202,217 Momentum conservation 3-4 Momentum equation 21-25,33-35
integral 58-60, 88, 89, 172, 196 Momentum thickness 59-61, 80 Momentum transfer 1-6 and passim
analogy with heat transfer 9-12
Natural convection 263-280 Navier-Stokes equations 21,43-44 Net, finite-difference 387-389,392 Newton's method for nonlinear alge-
braic equations 409
484 Index
Newton's second law 3-5, 21 Newtonian fluid 23 Nonsimilar flow 72, 75, 88-99 Normal stress 4, 23-24
turbulent 152 Nose
blunt 102 conical 102
Numerical errors 88 Numerical methods 386-395 Nusselt number 84, 128
Oil 85,91 high Prandtl number of 10, 91
"Opposed" duct flow with buoyancy 289-292
Ordinary differential equations (ODEs) 16,41,58-62,72-78,88,474
Origin, "effective" or "virtual" 174, 245-246
Outer layer 154,156,168-170, 340-342
Parabolic PDEs 44 Partial differential equations (PDEs)
15, 19, 37 and passim Peclet number 136, 159-160, 172 Physical properties of fluids 5, 6, 9,
376, 459-471 Pipe (circular duct) 3,124-129
calculated Nusselt number in 128, 135,140,221-229,232-233,406
Plate flat see Flat plate inclined or horizontal heated
276-280 vertical heated 267-276
Plume 16,263-264 Poiseuille equation 142 Pohlhausen's method 89-92 Poiseuille flow see Duct flow Pollutant transfer 2 Polynomial profile "family" 89,90 Potential energy 7 Power law velocity profile family 89,
176-177 Power law temperature profile family
178 Prandtl number 9-11,42,56,160
effect on buoyant flow 282-284, 292
effect on laminar heat transfer 82-88,91,94-97,140-142,244,
248-249 effect on turbulent heat transfer 165,
168,177-181,223 solutions for large 85-86, 91, 138 solutions for small 84-85, 86, 92 turbulent 11,151-153,169,185,
189,216,250,252,254,338,348, 395
Pressure 11,13,22 fluctuation 11, 17,48-49,255,258 gradient 11, 24, 172
calculation of, in duct flow 141 effect on boundary layer 80,
184-194, 306-315, 355-365 normal 43,45,54,63 parameter 76, 306
work-term 28, 55 Primitive variables 134, 138 Production of turbulent energy 56 Profile see Temperature, Velocity Properties, thermodynamic, of common
fluids 5, 6, 9, 376, 459-471
Ramp-induced separation 365-367 Reattachment 12, 320, 325 Recovery factor 51, 340 Recovery temperature 51 Reference temperature 71 Reversed flow 292, 320, 321 Reynolds analogy 12, 50, 86, 151, 170,
199 factor 12,85,95,177-178,349
Reynolds equations 33 Reynolds number 13, 42, 44, 172; see
also Transition for turbulent inner layer 159 effects of low 169, 171, 188, 341,
367 roughness 166,168
Reynolds stress 34,152,238 Richardson number 263,267,285-287,
297 Roughness, sand grain 167,195
surface 154,165-168,194-195, 224-227
effect on heat transfer 168, 194-195,224-227
inner layer formulas for 165-168 heat transfer formula for 168 skin friction formula for 182-184,
350 Runge-Kutta integration method 82,
474
Salt, Schmidt number for diffusion of 11
Sand-grain roughness 167, 195 Schmidt number 11 Schoenherr see Karman-Schoenherr Secondary flow 129 Separation 12, 71, 97, 191, 281, 292,
312-315, 327 due to opposing buoyancy force
282,292 effect of heat transfer on 313-314,
358-360 non-zero heat transfer at 88, 96 shock-induced 316-319
Sharp-nosed body 102 Shape factor (shape parameter) 59,
93, 196, 199, 309 Shear layer 4, 15
thin 4,41-43,67; see also Boundary layer
Shear stress 4 and passim Shock wave 255, 301
interaction with boundary layer 315-328, 365-369
normal 316-319 oblique 316-319
Similar profiles 11, 16, 64 Similarity 64, 71-88, 101-102, 238,
240-244, 278-279 in axisymmetric flow 101-102 in compressible (coupled) flow 306 inner layer 156-168, 335-340 slow approach to 154-155,238 variables 73, 101, 133, 153, 240,
247, 268, 278, 280, 302, 306 Singularity at separation 328 Skin friction (surface shear stress) 1 Skin friction coefficient 12, 59, 80,
93, 151, 176-177 formulas for 174-177,343-355
Index 485
on rough surface 184,350-355 Slot (jet nozzle) 107-110,202-203 Smith-Spalding method 94, 103-105 Smoke, Schmidt number for diffusion
of 11 Sound, speed of 304 Spalding-Chi skin friction formula
343-344 Specific heat 7, 30 Specified surface (wall) heat flux 77,
127, 135, 137, 181-183 Specified surface (wall) temperature
76,135,137,177-180 Sphere, laminar flow over 103-105 Stability
hydrodynamic see Transition numerical 389
Stagnation flow axisymmetric 102 plane 93,95
Stagnation point 12, 93, 95 Stanton number 12, 61, 85
effect of pressure gradient on 189-194, 359-363
empirical formuhts for 95, 103, 177 -183; see also Reynolds analogy factor
Starting length, unheated 97-99, 178-181
Static enthalpy thickness 61 Step, backward-facing 319-320
in wall temperature 1-2,45, 98-99, 180, 183
Strain, rate of 4, 23, 30, 36 fluctuating 36
Stream function 73, 137, 240, 246 Stress
turbulent (Reynolds) 34,152,238 viscous 4, 23
Stress gradient 42 Sublayer
conductive 36, 161-165, 169, 186-189, 335-339
~scous 64,154, 161-165, 169, 186-189, 335-339
Substantial derivative 24, 153 Suction see Transpiration Superlayer, viscous 172 Superposition solution of linear equa-
486 Index
tions 82, 180 Surface see also Wall
heat transfer rate (" heat flux"), specified 77, l34, 181
shear stress 88 Sutherland viscosity law 4
Taylor's method for dimensional analysis 157-158
Temperature 1 fluctuation 8, 48-53 profile formula 158-165, 169-172 total 31
Thermal boundary layer 1-2,42,78 Thermal conductivity 1, 8, 9, 26
temperature dependence of 9 Thermal diffusivity 9
turbulent see Turbulent Prandtl number
Thermal internal energy 7 equation for 29-30
Thermodynamics, first law of 25 Thickness, of shear layer 1, 78, 80-81,
240, 242, 248, 250; see also Momentum, Displacement, etc.
Thin shear layer approximation 88; see also Boundary layer approxima
tion Three-dimensional flow 19 Thwaites method 92-94, 103-105,
198 Time averaging 4-6,31-32 Time-dependent flow see Unsteady
flow Total enthalpy 7, 31, 55
equation 31, 39, 55-56, 302 "integral" equation 60-62 thickness 61-62
Total temperature 31 Transfer
heat 7-9 mass 8
Transformation Coles 342-343 compressibility 16,302-312,329,
340-343 Falkner-Skan 72-78,99-103, l37,
184-187,240-246,280,301 compressible 301-303,356,
373-374
Illingworth-Stewartson 302-312, 329
Mangler 48,68,99-102, l37, 373 Van Driest 338-342 Von Mises 114
Transformed variables 73, 101, l33, l37
Transition 71, 172-l75, 187-188, 191-192, 274, 279
effect of heat transfer on 175, 274, 381-382
empirical formula for 189 "Transitional" roughness 167 Transpiration, surface 81-82, 107,
161, 187; see also Film cooling Transport equation 25; see also Con
servation Transport operator (" substantial deriva
tive") 24, 153 Transposition theorem 322 Transverse curvature parameter 48,
68, 100, l37 Tridiagonal matrix 391, 394 Triple deck 325, 367 Tube see Pipe Turbulence 3,4-5,31-38
analogy with molecular motion 4-5 effect of viscosity on 64; see also
Viscous sublayer, Low Reynolds number
in buoyant flow 273-276 models 5-6,153,207,258,267; see
also Eddy viscosity, Turbulent Prandtl number
Turbulent flow, equations for 31-37, 46
Turbulent heat-flux rates 8, 36, 46, 151, 152-153, 185, 238
Turbulent kinetic energy 36 Turbulent Prandtl number 11,
151-153,169,185,189,216,250, 252, 254, 338, 348, 395
Turbulent stresses (Reynolds stresses) 4-6, 34
equations for 5, 8 Two-dimensional flow 19, 21, 37 Two-point boundary conditions 82
Uncoupled flow l3, l7, 71 equations for 25,31,34-35,37,
42-48
Unheated starting length 1-2, 89, 97-99,132-136,405-406
Units 14 Unsteady flow 21, 268 Upstream influence 43-44, 62,
320-328
Van Driest skin friction formulas 344-349
Van Driest sublayer formula 162-165 Van Driest transformation 338-342 Velocity profile formula 170-17l;
see also Inner layer, Outer layer Virtual origin of jet 245-246
of turbulent boundary layer 174
Viscosity 1, 4-6 Chapman-Rubesin law for 303-305 bulk 23,53 data for 5, 6, 376, 459-468 eddy 16 kinematic 4 Sutherland law for 4 temperature dependence of 4-6,
303-304, 335 Viscous effects on turbulent flow 159,
161-165
Viscous stress 1, 22-24 law 4,23-24 gradient 24, 44
Index 487
Viscous sublayer 34,64, 154, 161-165, 169, 186-189, 335-339
Viscous superlayer 172 Von Karman constant 159
for temperature profile 160 Von Mises transformation 11
VVake 41,73,256,258-259,318-319 function 170-1 7l law 170-171,340,368 parameter 170-17l, 188, 208, 218,
340-342 Wall jet 72,105-113,201-207
buoyant 284-288,439-444 Wall law 156, 171 Wall plus wake velocity profile formula
170-172,174-175 Wall shear stress 1-2,54; see also
Surface Water, properties of 5,6,9,469 Wedge flow 79-80 Work 7,11,28-30,55
Recommended