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

1.

25

Zin

c, p

ure

420

7,14

4 0.

3843

11

2.2

4.10

6 11

4 11

2 10

9 10

6 10

0 93

~

'-'

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