Fluid Mechanics F.M. White Apendices a-D

8/19/2019 Fluid Mechanics F.M. White Apendices a-D

1/26

Fig. A.1 Absolute viscosity of com-mon fluids at 1 atm.

Appendix A

Physical Propertiesof Fluids

A b s o l u t e v i s c o s i t y µ ,

N ⋅ s / m 2

0.50.4

0.3

0.2

0.1

0.06

0.040.03

0.02

0.01

6

43

2

1 × 10 – 3

1 × 10 – 4

1 × 10 – 5

5– 20 0 20 40 60 80 100 120

Temperature, °C

Aniline

Mercury

Kerosine

Air

6

43

2

643

2

Hydrogen

Carbon dioxide

Helium

Benzene

Gasoline (SG 0.68)

Water

Ethyl alcohol

C a r b o n t e t r a c h l o r i d e

Crude oil (SG 0.86)

Castor oil

SAE 30 oil

Glycerin

SAE 10 oil

769


2/26

Fig. A.2 Kinematic viscosity of

common fluids at 1 atm.

770 Appendix A

K i n e m a t i c v i s c o s i t y ν

, m 2 / s

1 × 10 – 38

6

4

3

2

1 × 10 – 4

1 × 10 – 5

1 × 10 – 6

1 × 10 – 7

–20 0 20 40 60 80 100 120

Temperature, °C

8

6

4

3

2

8

6

4

3

2

8

6

4

3

2

Mercury

Gasoline (SG 0.68) Carbon

tetrachloride

Water

Ethyl alcohol

Kerosine

Benzene

Crude oil (SG 0.86)

Carbon dioxide

SAE 30 oil

Air and oxygen

Hydrogen

SAE 10 oil

Glycerin

Helium


3/26

Table A.1 Viscosity and Density of

Water at 1 atm

Suggested curve fits for water in the range 0 T 100°C:

(kg/m3) 1000 0.0178T °C 4°C1.7 0.2%

ln

0

1.704 5.306 z 7.003 z2

z2

T

73

K

K 0 1.788 E-3 kg/(m s)

Physical Properties of Fluids 771

T , °C , kg/m3 , N s/m2 , m2 /s T , °F , slug/ft3 , lb s/ft2 , ft2 /s

0 1000 1.788 E-3 1.788 E-6 32 1.940 3.73 E-5 1.925 E-5

10 1000 1.307 E-3 1.307 E-6 50 1.940 2.73 E-5 1.407 E-5

20 998 1.003 E-3 1.005 E-6 68 1.937 2.09 E-5 1.082 E-530 996 0.799 E-3 0.802 E-6 86 1.932 1.67 E-5 0.864 E-5

40 992 0.657 E-3 0.662 E-6 104 1.925 1.37 E-5 0.713 E-5

50 988 0.548 E-3 0.555 E-6 122 1.917 1.14 E-5 0.597 E-5

60 983 0.467 E-3 0.475 E-6 140 1.908 0.975 E-5 0.511 E-5

70 978 0.405 E-3 0.414 E-6 158 1.897 0.846 E-5 0.446 E-5

80 972 0.355 E-3 0.365 E-6 176 1.886 0.741 E-5 0.393 E-5

90 965 0.316 E-3 0.327 E-6 194 1.873 0.660 E-5 0.352 E-5

100 958 0.283 E-3 0.295 E-6 212 1.859 0.591 E-5 0.318 E-5

Table A.2 Viscosity and Density of

Air at 1 atm T , °C , kg/m3 , N s/m2 , m2 /s T , °F , slug/ft3 , lb s/ft2 , ft2 /s

40 1.520 1.51 E-5 0.99 E-5 40 2.94 E-3 3.16 E-7 1.07 E-4

0 1.290 1.71 E-5 1.33 E-5 32 2.51 E-3 3.58 E-7 1.43 E-4

20 1.200 1.80 E-5 1.50 E-5 68 2.34 E-3 3.76 E-7 1.61 E-4

50 1.090 1.95 E-5 1.79 E-5 122 2.12 E-3 4.08 E-7 1.93 E-4

100 0.946 2.17 E-5 2.30 E-5 212 1.84 E-3 4.54 E-7 2.47 E-4

150 0.835 2.38 E-5 2.85 E-5 302 1.62 E-3 4.97 E-7 3.07 E-4

200 0.746 2.57 E-5 3.45 E-5 392 1.45 E-3 5.37 E-7 3.71 E-4

250 0.675 2.75 E-5 4.08 E-5 482 1.31 E-3 5.75 E-7 4.39 E-4

300 0.616 2.93 E-5 4.75 E-5 572 1.20 E-3 6.11 E-7 5.12 E-4

400 0.525 3.25 E-5 6.20 E-5 752 1.02 E-3 6.79 E-7 6.67 E-4

500 0.457 3.55 E-5 7.77 E-5 932 0.89 E-3 7.41 E-7 8.37 E-4

Suggested curve fits for air:

R pT Rair 287 J/(kg K)

Power law:

0

T T

0

0.7

Sutherland law:

0

T T

0

3/2

T T 0

S

S S air 110.4 K

with T 0 273 K, 0 1.71 E-5 kg/(m s), and T in kelvins.


4/26

Table A.3 Properties of Common

Liquids at 1 atm and 20°C (68°F)

Molecular Specific-heat Power-lawGas weight R, m2 /(s2 K) g, N/m3 , N s/m2 ratio exponent n†

H2 2.016 4124 00.822 9.05 E-6 1.41 0.68

He 4.003 2077 01.630 1.97 E-5 1.66 0.67

H2O 18.020 0461 07.350 1.02 E-5 1.33 1.15

Ar 39.944 0208 16.300 2.24 E-5 1.67 0.72

Dry air 28.960 0287 11.800 1.80 E-5 1.40 0.67

CO2 44.010 0189 17.900 1.48 E-5 1.30 0.79

CO 28.010 0297 11.400 1.82 E-5 1.40 0.71

N2 28.020 0297 11.400 1.76 E-5 1.40 0.67

O2 32.000 0260 13.100 2.00 E-5 1.40 0.69

NO 30.010 0277 12.100 1.90 E-5 1.40 0.78

N2O 44.020 0189 17.900 1.45 E-5 1.31 0.89

Cl2 70.910 0117 28.900 1.03 E-5 1.34 1.00

CH4 16.040 0518 06.540 1.34 E-5 1.32 0.87

†The power-law curve fit, Eq. (1.27), / 293K (T /293)

n, fits these gases to within 4 percent in the range 250 T

1000 K. The temperature must be in kelvins.

772 Appendix A

Bulk modulus, ViscosityLiquid , kg/m3 , kg/(m s) , N/m* p

, N/m2 N/m2 parameter C †

Ammonia 13,608 2.20 E-4 2.13 E-2 9.10 E5 — 1.05

Benzene 13,881 6.51 E-4 2.88 E-2 1.01 E4 1.43 E9 4.34Carbon tetrachloride 31,590 9.67 E-4 2.70 E-2 1.20 E4 9.65 E8 4.45

Ethanol 13,789 1.20 E-3 2.28 E-2 5.73 E3 9.03 E8 5.72

Ethylene glycol 31,117 2.14 E-2 4.84 E-2 1.23 E1 — 11.7Freon 12 31,327 2.62 E-4 — — — 1.76Gasoline 13,680 2.92 E-4 2.16 E-2 5.51 E4 9.58 E80 3.68

Glycerin 31,260 1.49 6.33 E-2 1.43 E-2 4.34 E90 28.0

Kerosine 13,804 1.92 E-3 2.83 E-2 3.11 E3 1.63 E90 5.56

Mercury 13,550 1.56 E-3 4.84 E-1 1.13 E-3 2.55 E10 1.07

Methanol 13,791 5.98 E-4 2.25 E-2 1.34 E4 8.33 E80 4.63

SAE 10W oil 13,870 1.04 E-1

‡

3.63 E-2 — 1.31 E

90 15.7SAE 10W30 oil 13,876 1.7 E-1‡ — — — 14.0SAE 30W oil 13,891 2.9 E-1‡ 3.53 E-2 — 1.38 E90 18.3SAE 50W oil 13,902 8.6 E-1‡ — — — 20.2Water 13,998 1.00 E-3 7.28 E-2 2.34 E3 2.19 E90 Table A.1

Seawater (30%) 31,025 1.07 E-3 7.28 E-2 2.34 E3 2.33 E90 7.28

*In contact with air.†The viscosity-temperature variation of these liquids may be fitted to the empirical expression

2

0°C

exp C 2T 93

K

K 1

with accuracy of 6 percent in the range 0 T 100°C.‡Representative values. The SAE oil classifications allow a viscosity variation of up to 50 percent, especially at lower

temperatures.

Table A.4 Properties of Common

Gases at 1 atm and 20°C (68°F)


5/26

Table A.5 Surface

Tension, Vapor

Pressure, and Sound

Speed of Water

z, m T , K p, Pa , kg/m3 a, m/s

0500 291.41 107,508 1.2854 342.2

0,0000 288.16 101,350 1.2255 340.3

0,0500 284.91 095,480 1.1677 338.401,000 281.66 089,889 1.1120 336.5

01,500 278.41 084,565 1.0583 334.5

02,000 275.16 079,500 1.0067 332.6

02,500 271.91 074,684 0.9570 330.6

03,000 268.66 070,107 0.9092 328.6

03,500 265.41 065,759 0.8633 326.6

04,000 262.16 061,633 0.8191 324.6

04,500 258.91 057,718 0.7768 322.6

05,000 255.66 054,008 0.7361 320.6

05,500 252.41 050,493 0.6970 318.506,000 249.16 047,166 0.6596 316.5

06,500 245.91 044,018 0.6237 314.4

07,000 242.66 041,043 0.5893 312.3

07,500 239.41 038,233 0.5564 310.2

08,000 236.16 035,581 0.5250 308.1

08,500 232.91 033,080 0.4949 306.0

09,000 229.66 030,723 0.4661 303.8

09,500 226.41 028,504 0.4387 301.7

10,000 223.16 026,416 0.4125 299.5

10,500 219.91 024,455 0.3875 297.311,000 216.66 022,612 0.3637 295.1

11,500 216.66 020,897 0.3361 295.1

12,000 216.66 019,312 0.3106 295.1

12,500 216.66 017,847 0.2870 295.1

13,000 216.66 016,494 0.2652 295.1

13,500 216.66 015,243 0.2451 295.1

14,000 216.66 014,087 0.2265 295.1

14,500 216.66 013,018 0.2094 295.1

15,000 216.66 012,031 0.1935 295.1

15,500 216.66 011,118 0.1788 295.1

16,000 216.66 010,275 0.1652 295.1

16,500 216.66 009,496 0.1527 295.1

17,000 216.66 008,775 0.1411 295.1

17,500 216.66 008,110 0.1304 295.1

18,000 216.66 007,495 0.1205 295.1

18,500 216.66 006,926 0.1114 295.1

19,000 216.66 006,401 0.1029 295.1

19,500 216.66 005,915 0.0951 295.1

20,000 216.66 005,467 0.0879 295.1

22,000 218.60 004,048 0.0645 296.4

24,000 220.60 002,972 0.0469 297.8

26,000 222.50 002,189 0.0343 299.1

28,000 224.50 001,616 0.0251 300.4

30,000 226.50 001,197 0.0184 301.7

40,000 250.40 000,287 0.0040 317.2

50,000 270.70 000,080 0.0010 329.9

60,000 255.70 000,022 0.0003 320.6

70,000 219.70 000,006 0.0001 297.2

Physical Properties of Fluids 773

T, °C , N/m p , kPa a, m/s

0 0.0756 0.611 1402

10 0.0742 1.227 1447

20 0.0728 2.337 148230 0.0712 4.242 1509

40 0.0696 7.375 1529

50 0.0679 12.34 1542

60 0.0662 19.92 1551

70 0.0644 31.16 1553

80 0.0626 47.35 1554

90 0.0608 70.11 1550

100 0.0589 101.3 1543

120 0.0550 198.5 1518140 0.0509 361.3 1483

160 0.0466 617.8 1440

180 0.0422 1,002 1389

200 0.0377 1,554 1334

220 0.0331 2,318 1268

240 0.0284 3,344 1192

260 0.0237 4,688 1110

280 0.0190 6,412 1022

300 0.0144 8,581 920

320 0.0099 11,274 800340 0.0056 14,586 630

360 0.0019 18,651 370

374* 0.0*19 22,090* 0*

*Critical point.

Table A.6 Prop-

erties of the

Standard

Atmosphere


6/26

Table B.1Isentropic Flow

of a Perfect

Gas, k 1.4

Appendix B

Compressible-Flow Tables

Ma p / p0 / 0 T / T 0 A / A*

0.0 1.0 1.0 1.0

0.02 0.9997 0.9998 0.9999 28.9421

0.04 0.9989 0.9992 0.9997 14.4815

0.06 0.9975 0.9982 0.9993 9.6659

0.08 0.9955 0.9968 0.9987 7.2616

0.1 0.9930 0.9950 0.9980 5.8218

0.12 0.9900 0.9928 0.9971 4.8643

0.14 0.9864 0.9903 0.9961 4.1824

0.16 0.9823 0.9873 0.9949 3.67270.18 0.9776 0.9840 0.9936 3.2779

0.2 0.9725 0.9803 0.9921 2.9635

0.22 0.9668 0.9762 0.9904 2.7076

0.24 0.9607 0.9718 0.9886 2.4956

0.26 0.9541 0.9670 0.9867 2.3173

0.28 0.9470 0.9619 0.9846 2.1656

0.3 0.9395 0.9564 0.9823 2.0351

0.32 0.9315 0.9506 0.9799 1.9219

0.34 0.9231 0.9445 0.9774 1.8229

0.36 0.9143 0.9380 0.9747 1.73580.38 0.9052 0.9313 0.9719 1.6587

0.4 0.8956 0.9243 0.9690 1.5901

0.42 0.8857 0.9170 0.9659 1.5289

0.44 0.8755 0.9094 0.9627 1.4740

0.46 0.8650 0.9016 0.9594 1.4246

0.48 0.8541 0.8935 0.9559 1.3801

0.5 0.8430 0.8852 0.9524 1.3398

0.52 0.8317 0.8766 0.9487 1.3034

0.54 0.8201 0.8679 0.9449 1.2703

0.56 0.8082 0.8589 0.9410 1.24030.58 0.7962 0.8498 0.9370 1.2130

0.6 0.7840 0.8405 0.9328 1.1882

0.62 0.7716 0.8310 0.9286 1.1656

0.64 0.7591 0.8213 0.9243 1.1451

0.66 0.7465 0.8115 0.9199 1.1265

0.68 0.7338 0.8016 0.9153 1.1097

0.7 0.7209 0.7916 0.9107 1.0944

0.72 0.7080 0.7814 0.9061 1.0806

Ma p / p0 / 0 T / T 0 A / A*

0.74 0.6951 0.7712 0.9013 1.0681

0.76 0.6821 0.7609 0.8964 1.0570

0.78 0.6690 0.7505 0.8915 1.0471

0.8 0.6560 0.7400 0.8865 1.0382

0.82 0.6430 0.7295 0.8815 1.0305

0.84 0.6300 0.7189 0.8763 1.0237

0.86 0.6170 0.7083 0.8711 1.0179

0.88 0.6041 0.6977 0.8659 1.0129

0.9 0.5913 0.6870 0.8606 1.00890.92 0.5785 0.6764 0.8552 1.0056

0.94 0.5658 0.6658 0.8498 1.0031

0.96 0.5532 0.6551 0.8444 1.0014

0.98 0.5407 0.6445 0.8389 1.0003

1.0 0.5283 0.6339 0.8333 1.0000

1.02 0.5160 0.6234 0.8278 1.0003

1.04 0.5039 0.6129 0.8222 1.0013

1.06 0.4919 0.6024 0.8165 1.0029

1.08 0.4800 0.5920 0.8108 1.0051

1.1 0.4684 0.5817 0.8052 1.00791.12 0.4568 0.5714 0.7994 1.0113

1.14 0.4455 0.5612 0.7937 1.0153

1.16 0.4343 0.5511 0.7879 1.0198

1.18 0.4232 0.5411 0.7822 1.0248

1.2 0.4124 0.5311 0.7764 1.0304

1.22 0.4017 0.5213 0.7706 1.0366

1.24 0.3912 0.5115 0.7648 1.0432

1.26 0.3809 0.5019 0.7590 1.0504

1.28 0.3708 0.4923 0.7532 1.0581

1.3 0.3609 0.4829 0.7474 1.06631.32 0.3512 0.4736 0.7416 1.0750

1.34 0.3417 0.4644 0.7358 1.0842

1.36 0.3323 0.4553 0.7300 1.0940

1.38 0.3232 0.4463 0.7242 1.1042

1.4 0.3142 0.4374 0.7184 1.1149

1.42 0.3055 0.4287 0.7126 1.1262

1.44 0.2969 0.4201 0.7069 1.1379

1.46 0.2886 0.4116 0.7011 1.1501

774


7/26

Ma p / p0 / 0 T / T 0 A / A*

2.56 0.0533 0.1232 0.4328 2.7891

2.58 0.0517 0.1205 0.4289 2.8420

2.6 0.0501 0.1179 0.4252 2.89602.62 0.0486 0.1153 0.4214 2.9511

2.64 0.0471 0.1128 0.4177 3.0073

2.66 0.0457 0.1103 0.4141 3.0647

2.68 0.0443 0.1079 0.4104 3.1233

2.7 0.0430 0.1056 0.4068 3.1830

2.72 0.0417 0.1033 0.4033 3.2440

2.74 0.0404 0.1010 0.3998 3.3061

2.76 0.0392 0.0989 0.3963 3.3695

2.78 0.0380 0.0967 0.3928 3.4342

2.8 0.0368 0.0946 0.3894 3.50012.82 0.0357 0.0926 0.3860 3.5674

2.84 0.0347 0.0906 0.3827 3.6359

2.86 0.0336 0.0886 0.3794 3.7058

2.88 0.0326 0.0867 0.3761 3.7771

2.9 0.0317 0.0849 0.3729 3.8498

2.92 0.0307 0.0831 0.3696 3.9238

2.94 0.0298 0.0813 0.3665 3.9993

2.96 0.0289 0.0796 0.3633 4.0763

2.98 0.0281 0.0779 0.3602 4.1547

3.0 0.0272 0.0762 0.3571 4.23463.02 0.0264 0.0746 0.3541 4.3160

3.04 0.0256 0.0730 0.3511 4.3990

3.06 0.0249 0.0715 0.3481 4.4835

3.08 0.0242 0.0700 0.3452 4.5696

3.1 0.0234 0.0685 0.3422 4.6573

3.12 0.0228 0.0671 0.3393 4.7467

3.14 0.0221 0.0657 0.3365 4.8377

3.16 0.0215 0.0643 0.3337 4.9304

3.18 0.0208 0.0630 0.3309 5.0248

3.2 0.0202 0.0617 0.3281 5.1210

3.22 0.0196 0.0604 0.3253 5.2189

3.24 0.0191 0.0591 0.3226 5.3186

3.26 0.0185 0.0579 0.3199 5.4201

3.28 0.0180 0.0567 0.3173 5.5234

3.3 0.0175 0.0555 0.3147 5.6286

3.32 0.0170 0.0544 0.3121 5.7358

3.34 0.0165 0.0533 0.3095 5.8448

3.36 0.0160 0.0522 0.3069 5.9558

3.38 0.0156 0.0511 0.3044 6.0687

3.4 0.0151 0.0501 0.3019 6.1837

3.42 0.0147 0.0491 0.2995 6.3007

3.44 0.0143 0.0481 0.2970 6.4198

3.46 0.0139 0.0471 0.2946 6.5409

3.48 0.0135 0.0462 0.2922 6.6642

3.5 0.0131 0.0452 0.2899 6.7896

3.52 0.0127 0.0443 0.2875 6.9172

3.54 0.0124 0.0434 0.2852 7.0471

3.56 0.0120 0.0426 0.2829 7.1791

3.58 0.0117 0.0417 0.2806 7.3135

3.6 0.0114 0.0409 0.2784 7.4501

3.62 0.0111 0.0401 0.2762 7.5891

Table B.1 (Cont.)

Isentropic Flow of

a Perfect Gas,

k 1.4

Ma p / p0 / 0 T / T 0 A / A*

1.48 0.2804 0.4032 0.6954 1.1629

1.5 0.2724 0.3950 0.6897 1.1762

1.52 0.2646 0.3869 0.6840 1.18991.54 0.2570 0.3789 0.6783 1.2042

1.56 0.2496 0.3710 0.6726 1.2190

1.58 0.2423 0.3633 0.6670 1.2344

1.6 0.2353 0.3557 0.6614 1.2502

1.62 0.2284 0.3483 0.6558 1.2666

1.64 0.2217 0.3409 0.6502 1.2836

1.66 0.2151 0.3337 0.6447 1.3010

1.68 0.2088 0.3266 0.6392 1.3190

1.7 0.2026 0.3197 0.6337 1.3376

1.72 0.1966 0.3129 0.6283 1.35671.74 0.1907 0.3062 0.6229 1.3764

1.76 0.1850 0.2996 0.6175 1.3967

1.78 0.1794 0.2931 0.6121 1.4175

1.8 0.1740 0.2868 0.6068 1.4390

1.82 0.1688 0.2806 0.6015 1.4610

1.84 0.1637 0.2745 0.5963 1.4836

1.86 0.1587 0.2686 0.5910 1.5069

1.88 0.1539 0.2627 0.5859 1.5308

1.9 0.1492 0.2570 0.5807 1.5553

1.92 0.1447 0.2514 0.5756 1.58041.94 0.1403 0.2459 0.5705 1.6062

1.96 0.1360 0.2405 0.5655 1.6326

1.98 0.1318 0.2352 0.5605 1.6597

2.0 0.1278 0.2300 0.5556 1.6875

2.02 0.1239 0.2250 0.5506 1.7160

2.04 0.1201 0.2200 0.5458 1.7451

2.06 0.1164 0.2152 0.5409 1.7750

2.08 0.1128 0.2104 0.5361 1.8056

2.1 0.1094 0.2058 0.5313 1.8369

2.12 0.1060 0.2013 0.5266 1.8690

2.14 0.1027 0.1968 0.5219 1.9018

2.16 0.0996 0.1925 0.5173 1.9354

2.18 0.0965 0.1882 0.5127 1.9698

2.2 0.0935 0.1841 0.5081 2.0050

2.22 0.0906 0.1800 0.5036 2.0409

2.24 0.0878 0.1760 0.4991 2.0777

2.26 0.0851 0.1721 0.4947 2.1153

2.28 0.0825 0.1683 0.4903 2.1538

2.3 0.0800 0.1646 0.4859 2.1931

2.32 0.0775 0.1609 0.4816 2.2333

2.34 0.0751 0.1574 0.4773 2.2744

2.36 0.0728 0.1539 0.4731 2.3164

2.38 0.0706 0.1505 0.4688 2.3593

2.4 0.0684 0.1472 0.4647 2.4031

2.42 0.0663 0.1439 0.4606 2.4479

2.44 0.0643 0.1408 0.4565 2.4936

2.46 0.0623 0.1377 0.4524 2.5403

2.48 0.0604 0.1346 0.4484 2.5880

2.5 0.0585 0.1317 0.4444 2.6367

2.52 0.0567 0.1288 0.4405 2.6865

2.54 0.0550 0.1260 0.4366 2.7372

Compressible-Flow Tables 775


8/26

776 Appendix B

Table B.1

(Cont.) Isentropic

Flow of a Perfect

Gas, k 1.4

Ma p / p0 / 0 T / T 0 A / A*

3.64 0.0108 0.0393 0.2740 7.7305

3.66 0.0105 0.0385 0.2718 7.8742

3.68 0.0102 0.0378 0.2697 8.02043.7 0.0099 0.0370 0.2675 8.1691

3.72 0.0096 0.0363 0.2654 8.3202

3.74 0.0094 0.0356 0.2633 8.4739

3.76 0.0091 0.0349 0.2613 8.6302

3.78 0.0089 0.0342 0.2592 8.7891

3.8 0.0086 0.0335 0.2572 8.9506

3.82 0.0084 0.0329 0.2552 9.1148

3.84 0.0082 0.0323 0.2532 9.2817

3.86 0.0080 0.0316 0.2513 9.4513

3.88 0.0077 0.0310 0.2493 9.62373.9 0.0075 0.0304 0.2474 9.7990

3.92 0.0073 0.0299 0.2455 9.9771

3.94 0.0071 0.0293 0.2436 10.1581

3.96 0.0069 0.0287 0.2418 10.3420

3.98 0.0068 0.0282 0.2399 10.5289

4.0 0.0066 0.0277 0.2381 10.7188

4.02 0.0064 0.0271 0.2363 10.9117

4.04 0.0062 0.0266 0.2345 11.1077

4.06 0.0061 0.0261 0.2327 11.3068

4.08 0.0059 0.0256 0.2310 11.50914.1 0.0058 0.0252 0.2293 11.7147

4.12 0.0056 0.0247 0.2275 11.9234

4.14 0.0055 0.0242 0.2258 12.1354

4.16 0.0053 0.0238 0.2242 12.3508

4.18 0.0052 0.0234 0.2225 12.5695

4.2 0.0051 0.0229 0.2208 12.7916

4.22 0.0049 0.0225 0.2192 13.0172

4.24 0.0048 0.0221 0.2176 13.2463

4.26 0.0047 0.0217 0.2160 13.4789

4.28 0.0046 0.0213 0.2144 13.7151

4.3 0.0044 0.0209 0.2129 13.9549

4.32 0.0043 0.0205 0.2113 14.1984

Ma p / p0 / 0 T / T 0 A / A*

4.34 0.0042 0.0202 0.2098 14.4456

4.36 0.0041 0.0198 0.2083 14.6965

4.38 0.0040 0.0194 0.2067 14.95134.4 0.0039 0.0191 0.2053 15.2099

4.42 0.0038 0.0187 0.2038 15.4724

4.44 0.0037 0.0184 0.2023 15.7388

4.46 0.0036 0.0181 0.2009 16.0092

4.48 0.0035 0.0178 0.1994 16.2837

4.5 0.0035 0.0174 0.1980 16.5622

4.52 0.0034 0.0171 0.1966 16.8449

4.54 0.0033 0.0168 0.1952 17.1317

4.56 0.0032 0.0165 0.1938 17.4228

4.58 0.0031 0.0163 0.1925 17.71814.6 0.0031 0.0160 0.1911 18.0178

4.62 0.0030 0.0157 0.1898 18.3218

4.64 0.0029 0.0154 0.1885 18.6303

4.66 0.0028 0.0152 0.1872 18.9433

4.68 0.0028 0.0149 0.1859 19.2608

4.7 0.0027 0.0146 0.1846 19.5828

4.72 0.0026 0.0144 0.1833 19.9095

4.74 0.0026 0.0141 0.1820 20.2409

4.76 0.0025 0.0139 0.1808 20.5770

4.78 0.0025 0.0137 0.1795 20.91794.8 0.0024 0.0134 0.1783 21.2637

4.82 0.0023 0.0132 0.1771 21.6144

4.84 0.0023 0.0130 0.1759 21.9700

4.86 0.0022 0.0128 0.1747 22.3306

4.88 0.0022 0.0125 0.1735 22.6963

4.9 0.0021 0.0123 0.1724 23.0671

4.92 0.0021 0.0121 0.1712 23.4431

4.94 0.0020 0.0119 0.1700 23.8243

4.96 0.0020 0.0117 0.1689 24.2109

4.98 0.0019 0.0115 0.1678 24.6027

5.0 0.0019 0.0113 0.1667 25.0000

Ma n1 Ma n2 p2 / p1 V 1 / V 2 2 / 1 T 2 / T 1 p02 / p01 A*2 / A*1

1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.02 0.9805 1.0471 1.0334 1.0132 1.0000 1.0000

1.04 0.9620 1.0952 1.0671 1.0263 0.9999 1.0001

1.06 0.9444 1.1442 1.1009 1.0393 0.9998 1.0002

1.08 0.9277 1.1941 1.1349 1.0522 0.9994 1.0006

1.1 0.9118 1.2450 1.1691 1.0649 0.9989 1.0011

1.12 0.8966 1.2968 1.2034 1.0776 0.9982 1.0018

1.14 0.8820 1.3495 1.2378 1.0903 0.9973 1.0027

1.16 0.8682 1.4032 1.2723 1.1029 0.9961 1.0040

1.18 0.8549 1.4578 1.3069 1.1154 0.9946 1.0055

1.2 0.8422 1.5133 1.3416 1.1280 0.9928 1.0073

1.22 0.8300 1.5698 1.3764 1.1405 0.9907 1.0094

1.24 0.8183 1.6272 1.4112 1.1531 0.9884 1.0118

Table B.2 Normal-Shock Relations

for a Perfect Gas, k 1.4


9/26


Table B.2 (Cont.) Normal-Shock

Relations for a Perfect Gas, k 1.4 Ma n1 Ma n2 p2 / p1 V 1 / V 2 2 / 1 T 2 / T 1 p02 / p01 A*2 / A*1

1.26 0.8071 1.6855 1.4460 1.1657 0.9857 1.0145

1.28 0.7963 1.7448 1.4808 1.1783 0.9827 1.0176

1.3 0.7860 1.8050 1.5157 1.1909 0.9794 1.02111.32 0.7760 1.8661 1.5505 1.2035 0.9758 1.0249

1.34 0.7664 1.9282 1.5854 1.2162 0.9718 1.0290

1.36 0.7572 1.9912 1.6202 1.2290 0.9676 1.0335

1.38 0.7483 2.0551 1.6549 1.2418 0.9630 1.0384

1.4 0.7397 2.1200 1.6897 1.2547 0.9582 1.0436

1.42 0.7314 2.1858 1.7243 1.2676 0.9531 1.0492

1.44 0.7235 2.2525 1.7589 1.2807 0.9476 1.0552

1.46 0.7157 2.3202 1.7934 1.2938 0.9420 1.0616

1.48 0.7083 2.3888 1.8278 1.3069 0.9360 1.0684

1.5 0.7011 2.4583 1.8621 1.3202 0.9298 1.07551.52 0.6941 2.5288 1.8963 1.3336 0.9233 1.0830

1.54 0.6874 2.6002 1.9303 1.3470 0.9166 1.0910

1.56 0.6809 2.6725 1.9643 1.3606 0.9097 1.0993

1.58 0.6746 2.7458 1.9981 1.3742 0.9026 1.1080

1.6 0.6684 2.8200 2.0317 1.3880 0.8952 1.1171

1.62 0.6625 2.8951 2.0653 1.4018 0.8877 1.1266

1.64 0.6568 2.9712 2.0986 1.4158 0.8799 1.1365

1.66 0.6512 3.0482 2.1318 1.4299 0.8720 1.1468

1.68 0.6458 3.1261 2.1649 1.4440 0.8639 1.1575

1.7 0.6405 3.2050 2.1977 1.4583 0.8557 1.16861.72 0.6355 3.2848 2.2304 1.4727 0.8474 1.1801

1.74 0.6305 3.3655 2.2629 1.4873 0.8389 1.1921

1.76 0.6257 3.4472 2.2952 1.5019 0.8302 1.2045

1.78 0.6210 3.5298 2.3273 1.5167 0.8215 1.2173

1.8 0.6165 3.6133 2.3592 1.5316 0.8127 1.2305

1.82 0.6121 3.6978 2.3909 1.5466 0.8038 1.2441

1.84 0.6078 3.7832 2.4224 1.5617 0.7948 1.2582

1.86 0.6036 3.8695 2.4537 1.5770 0.7857 1.2728

1.88 0.5996 3.9568 2.4848 1.5924 0.7765 1.2877

1.9 0.5956 4.0450 2.5157 1.6079 0.7674 1.3032

1.92 0.5918 4.1341 2.5463 1.6236 0.7581 1.3191

1.94 0.5880 4.2242 2.5767 1.6394 0.7488 1.3354

1.96 0.5844 4.3152 2.6069 1.6553 0.7395 1.3522

1.98 0.5808 4.4071 2.6369 1.6713 0.7302 1.3695

2.0 0.5774 4.5000 2.6667 1.6875 0.7209 1.3872

2.02 0.5740 4.5938 2.6962 1.7038 0.7115 1.4054

2.04 0.5707 4.6885 2.7255 1.7203 0.7022 1.4241

2.06 0.5675 4.7842 2.7545 1.7369 0.6928 1.4433

2.08 0.5643 4.8808 2.7833 1.7536 0.6835 1.4630

2.1 0.5613 4.9783 2.8119 1.7705 0.6742 1.4832

2.12 0.5583 5.0768 2.8402 1.7875 0.6649 1.5039

2.14 0.5554 5.1762 2.8683 1.8046 0.6557 1.5252

2.16 0.5525 5.2765 2.8962 1.8219 0.6464 1.5469

2.18 0.5498 5.3778 2.9238 1.8393 0.6373 1.5692

2.2 0.5471 5.4800 2.9512 1.8569 0.6281 1.5920

2.22 0.5444 5.5831 2.9784 1.8746 0.6191 1.6154

2.24 0.5418 5.6872 3.0053 1.8924 0.6100 1.6393

2.26 0.5393 5.7922 3.0319 1.9104 0.6011 1.6638

2.28 0.5368 5.8981 3.0584 1.9285 0.5921 1.6888

2.3 0.5344 6.0050 3.0845 1.9468 0.5833 1.7144

2.32 0.5321 6.1128 3.1105 1.9652 0.5745 1.7406


10/26



2.34 0.5297 6.2215 3.1362 1.9838 0.5658 1.7674

2.36 0.5275 6.3312 3.1617 2.0025 0.5572 1.7948

2.38 0.5253 6.4418 3.1869 2.0213 0.5486 1.82282.4 0.5231 6.5533 3.2119 2.0403 0.5401 1.8514

2.42 0.5210 6.6658 3.2367 2.0595 0.5317 1.8806

2.44 0.5189 6.7792 3.2612 2.0788 0.5234 1.9105

2.46 0.5169 6.8935 3.2855 2.0982 0.5152 1.9410

2.48 0.5149 7.0088 3.3095 2.1178 0.5071 1.9721

2.5 0.5130 7.1250 3.3333 2.1375 0.4990 2.0039

2.52 0.5111 7.2421 3.3569 2.1574 0.4911 2.0364

2.54 0.5092 7.3602 3.3803 2.1774 0.4832 2.0696

2.56 0.5074 7.4792 3.4034 2.1976 0.4754 2.1035

2.58 0.5056 7.5991 3.4263 2.2179 0.4677 2.13812.6 0.5039 7.7200 3.4490 2.2383 0.4601 2.1733

2.62 0.5022 7.8418 3.4714 2.2590 0.4526 2.2093

2.64 0.5005 7.9645 3.4937 2.2797 0.4452 2.2461

2.66 0.4988 8.0882 3.5157 2.3006 0.4379 2.2835

2.68 0.4972 8.2128 3.5374 2.3217 0.4307 2.3218

2.7 0.4956 8.3383 3.5590 2.3429 0.4236 2.3608

2.72 0.4941 8.4648 3.5803 2.3642 0.4166 2.4005

2.74 0.4926 8.5922 3.6015 2.3858 0.4097 2.4411

2.76 0.4911 8.7205 3.6224 2.4074 0.4028 2.4825

2.78 0.4896 8.8498 3.6431 2.4292 0.3961 2.52462.8 0.4882 8.9800 2.6636 2.4512 0.3895 2.5676

2.82 0.4868 9.1111 3.6838 2.4733 0.3829 2.6115

2.84 0.4854 9.2432 3.7039 2.4955 0.3765 2.6561

2.86 0.4840 9.3762 3.7238 2.5179 0.3701 2.7017

2.88 0.4827 9.5101 3.7434 2.5405 0.3639 2.7481

2.9 0.4814 9.6450 3.7629 2.5632 0.3577 2.7954

2.92 0.4801 9.7808 3.7821 2.5861 0.3517 2.8436

2.94 0.4788 9.9175 3.8012 2.6091 0.3457 2.8927

2.96 0.4776 10.0552 3.8200 2.6322 0.3398 2.9427

2.98 0.4764 10.1938 3.8387 2.6555 0.3340 2.9937

3.0 0.4752 10.3333 3.8571 2.6790 0.3283 3.0456

3.02 0.4740 10.4738 3.8754 2.7026 0.3227 3.0985

3.04 0.4729 10.6152 3.8935 2.7264 0.3172 3.1523

3.06 0.4717 10.7575 3.9114 2.7503 0.3118 3.2072

3.08 0.4706 10.9008 3.9291 2.7744 0.3065 3.2630

3.1 0.4695 11.0450 3.9466 2.7986 0.3012 3.3199

3.12 0.4685 11.1901 3.9639 2.8230 0.2960 3.3778

3.14 0.4674 11.3362 3.9811 2.8475 0.2910 3.4368

3.16 0.4664 11.4832 3.9981 2.8722 0.2860 3.4969

3.18 0.4654 11.6311 4.0149 2.8970 0.2811 3.5580

3.2 0.4643 11.7800 4.0315 2.9220 0.2762 3.6202

3.22 0.4634 11.9298 4.0479 2.9471 0.2715 3.6835

3.24 0.4624 12.0805 4.0642 2.9724 0.2668 3.7480

3.26 0.4614 12.2322 4.0803 2.9979 0.2622 3.8136

3.28 0.4605 12.3848 4.0963 3.0234 0.2577 3.8803

3.3 0.4596 12.5383 4.1120 3.0492 0.2533 3.9483

3.32 0.4587 12.6928 4.1276 3.0751 0.2489 4.0174

3.34 0.4578 12.8482 4.1431 3.1011 0.2446 4.0877

3.36 0.4569 13.0045 4.1583 3.1273 0.2404 4.1593

3.38 0.4560 13.1618 4.1734 3.1537 0.2363 4.2321

3.4 0.4552 13.3200 4.1884 3.1802 0.2322 4.3062

778 Appendix B


11/26




3.42 0.4544 13.4791 4.2032 3.2069 0.2282 4.3815

3.44 0.4535 13.6392 4.2178 3.2337 0.2243 4.4581

3.46 0.4527 13.8002 4.2323 3.2607 0.2205 4.53613.48 0.4519 13.9621 4.2467 3.2878 0.2167 4.6154

3.5 0.4512 14.1250 4.2609 3.3151 0.2129 4.6960

3.52 0.4504 14.2888 4.2749 3.3425 0.2093 4.7780

3.54 0.4496 14.4535 4.2888 3.3701 0.2057 4.8614

3.56 0.4489 14.6192 4.3026 3.3978 0.2022 4.9461

3.58 0.4481 14.7858 4.3162 3.4257 0.1987 5.0324

3.6 0.4474 14.9533 4.3296 3.4537 0.1953 5.1200

3.62 0.4467 15.1218 4.3429 3.4819 0.1920 5.2091

3.64 0.4460 15.2912 4.3561 3.5103 0.1887 5.2997

3.66 0.4453 15.4615 4.3692 3.5388 0.1855 5.39183.68 0.4446 15.6328 4.3821 3.5674 0.1823 5.4854

3.7 0.4439 15.8050 4.3949 3.5962 0.1792 5.5806

3.72 0.4433 15.9781 4.4075 3.6252 0.1761 5.6773

3.74 0.4426 16.1522 4.4200 3.6543 0.1731 5.7756

3.76 0.4420 16.3272 4.4324 3.6836 0.1702 5.8755

3.78 0.4414 16.5031 4.4447 3.7130 0.1673 5.9770

3.8 0.4407 16.6800 4.4568 3.7426 0.1645 6.0801

3.82 0.4401 16.8578 4.4688 3.7723 0.1617 6.1849

3.84 0.4395 17.0365 4.4807 3.8022 0.1589 6.2915

3.86 0.4389 17.2162 4.4924 3.8323 0.1563 6.39973.88 0.4383 17.3968 4.5041 3.8625 0.1536 6.5096

3.9 0.4377 17.5783 4.4156 3.8928 0.1510 6.6213

3.92 0.4372 17.7608 4.5270 3.9233 0.1485 6.7348

3.94 0.4366 17.9442 4.5383 3.9540 0.1460 6.8501

3.96 0.4360 18.1285 4.5494 3.9848 0.1435 6.9672

3.98 0.4355 18.3138 4.5605 4.0158 0.1411 7.0861

4.0 0.4350 18.5000 4.5714 4.0469 0.1388 7.2069

4.02 0.4344 18.6871 4.5823 4.0781 0.1364 7.3296

4.04 0.4339 18.8752 4.5930 4.1096 0.1342 7.4542

4.06 0.4334 19.0642 4.6036 4.1412 0.1319 7.5807

4.08 0.4329 19.2541 4.6141 4.1729 0.1297 7.7092

4.1 0.4324 19.4450 4.6245 4.2048 0.1276 7.8397

4.12 0.4319 19.6368 4.6348 4.2368 0.1254 7.9722

4.14 0.4314 19.8295 4.6450 4.2690 0.1234 8.1067

4.16 0.4309 20.0232 4.6550 4.3014 0.1213 8.2433

4.18 0.4304 20.2178 4.6650 4.3339 0.1193 8.3819

4.2 0.4299 20.4133 4.6749 4.3666 0.1173 8.5227

4.22 0.4295 20.6098 4.6847 4.3994 0.1154 8.6656

4.24 0.4290 20.8072 4.6944 4.4324 0.1135 8.8107

4.26 0.4286 21.0055 4.7040 4.4655 0.1116 8.9579

4.28 0.4281 21.2048 4.7135 4.4988 0.1098 9.1074

4.3 0.4277 21.4050 4.7229 4.5322 0.1080 9.2591

4.32 0.4272 21.6061 4.7322 4.5658 0.1062 9.4131

4.34 0.4268 21.8082 4.7414 4.5995 0.1045 9.5694

4.36 0.4264 22.0112 4.7505 4.6334 0.1028 9.7280

4.38 0.4260 22.2151 4.7595 4.6675 0.1011 9.8889

4.4 0.4255 22.4200 4.7685 4.7017 0.0995 10.0522

4.42 0.4251 22.6258 4.7773 4.7361 0.0979 10.2179

4.44 0.4247 22.8325 4.7861 4.7706 0.0963 10.3861

4.46 0.4243 23.0402 4.7948 4.8053 0.0947 10.5567

4.48 0.4239 23.2488 4.8034 4.8401 0.0932 10.7298


12/26


Relations for a Perfect Gas, k 1.4

780 Appendix B

Ma n1 Ma n2 p2 / p1 V 1 / V 2 2 / 1 T 2 / T 1 p02 / p01 A*2 / A*1

4.5 0.4236 23.4583 4.8119 4.8751 0.0917 10.9054

4.52 0.4232 23.6688 4.8203 4.9102 0.0902 11.0835

4.54 0.4228 23.8802 4.8287 4.9455 0.0888 11.26434.56 0.4224 24.0925 4.8369 4.9810 0.0874 11.4476

4.58 0.4220 24.3058 4.8451 5.0166 0.0860 11.6336

4.6 0.4217 24.5200 4.8532 5.0523 0.0846 11.8222

4.62 0.4213 24.7351 4.8612 5.0882 0.0832 12.0136

4.64 0.4210 24.9512 4.8692 5.1243 0.0819 12.2076

4.66 0.4206 25.1682 4.8771 5.1605 0.0806 12.4044

4.68 0.4203 25.3861 4.8849 5.1969 0.0793 12.6040

4.7 0.4199 25.6050 4.8926 5.2334 0.0781 12.8065

4.72 0.4196 25.8248 4.9002 5.2701 0.0769 13.0117

4.74 0.4192 26.0455 4.9078 5.3070 0.0756 13.21994.76 0.4189 26.2672 4.9153 5.3440 0.0745 13.4310

4.78 0.4186 26.4898 4.9227 5.3811 0.0733 13.6450

4.8 0.4183 26.7133 4.9301 5.4184 0.0721 13.8620

4.82 0.4179 26.9378 4.9374 5.4559 0.0710 14.0820

4.84 0.4176 27.1632 4.9446 5.4935 0.0699 14.3050

4.86 0.4173 27.3895 4.9518 5.5313 0.0688 14.5312

4.88 0.4170 27.6168 4.9589 5.5692 0.0677 14.7604

4.9 0.4167 27.8450 4.9659 5.6073 0.0667 14.9928

4.92 0.4164 28.0741 4.9728 5.6455 0.0657 15.2284

4.94 0.4161 28.3042 4.9797 5.6839 0.0647 15.46724.96 0.4158 28.5352 4.9865 5.7224 0.0637 15.7902

4.98 0.4155 28.7671 4.9933 5.7611 0.0627 15.9545

5.0 0.4152 29.0000 5.0000 5.8000 0.0617 16.2032

Table B.3 Adiabatic Frictional

Flow in a Constant-Area Duct for

k 1.4

Ma f L*/ D p / p* T / T * */ V / V * p0 / p*0

0.0 1.2000 0.0

0.02 1778.4500 54.7701 1.1999 0.0219 28.9421

0.04 440.3520 27.3817 1.1996 0.0438 14.4815

0.06 193.0310 18.2508 1.1991 0.0657 9.6659

0.08 106.7180 13.6843 1.1985 0.0876 7.2616

0.1 66.9216 10.9435 1.1976 0.1094 5.8218

0.12 45.4080 9.1156 1.1966 0.1313 4.8643

0.14 32.5113 7.8093 1.1953 0.1531 4.1824

0.16 24.1978 6.8291 1.1939 0.1748 3.6727

0.18 18.5427 6.0662 1.1923 0.1965 3.2779

0.2 14.5333 5.4554 1.1905 0.2182 2.9635

0.22 11.5961 4.9554 1.1885 0.2398 2.7076

0.24 9.3865 4.5383 1.1863 0.2614 2.4956

0.26 7.6876 4.1851 1.1840 0.2829 2.3173

0.28 6.3572 3.8820 1.1815 0.3043 2.1656

0.3 5.2993 3.6191 1.1788 0.3257 2.0351

0.32 4.4467 3.3887 1.1759 0.3470 1.9219

0.34 3.7520 3.1853 1.1729 0.3682 1.8229

0.36 3.1801 3.0042 1.1697 0.3893 1.7358

0.38 2.7054 2.8420 1.1663 0.4104 1.6587

0.4 2.3085 2.6958 1.1628 0.4313 1.5901

0.42 1.9744 2.5634 1.1591 0.4522 1.5289


13/26


Table B.3 (Cont.) Adiabatic

Frictional Flow in a Constant-Area

Duct for k 1.4

Ma f L*/ D p / p* T / T * */ V / V * p0 / p*0

0.44 1.6915 2.4428 1.1553 0.4729 1.4740

0.46 1.4509 2.3326 1.1513 0.4936 1.4246

0.48 1.2453 2.2313 1.1471 0.5141 1.38010.5 1.0691 2.1381 1.1429 0.5345 1.3398

0.52 0.9174 2.0519 1.1384 0.5548 1.3034

0.54 0.7866 1.9719 1.1339 0.5750 1.2703

0.56 0.6736 1.8975 1.1292 0.5951 1.2403

0.58 0.5757 1.8282 1.1244 0.6150 1.2130

0.6 0.4908 1.7634 1.1194 0.6348 1.1882

0.62 0.4172 1.7026 1.1143 0.6545 1.1656

0.64 0.3533 1.6456 1.1091 0.6740 1.1451

0.66 0.2979 1.5919 1.1038 0.6934 1.1265

0.68 0.2498 1.5413 1.0984 0.7127 1.10970.7 0.2081 1.4935 1.0929 0.7318 1.0944

0.72 0.1721 1.4482 1.0873 0.7508 1.0806

0.74 0.1411 1.4054 1.0815 0.7696 1.0681

0.76 0.1145 1.3647 1.0757 0.7883 1.0570

0.78 0.0917 1.3261 1.0698 0.8068 1.0471

0.8 0.0723 1.2893 1.0638 0.8251 1.0382

0.82 0.0559 1.2542 1.0578 0.8433 1.0305

0.84 0.0423 1.2208 1.0516 0.8614 1.0237

0.86 0.0310 1.1889 1.0454 0.8793 1.0179

0.88 0.0218 1.1583 1.0391 0.8970 1.0129

0.9 0.0145 1.1291 1.0327 0.9146 1.0089

0.92 0.0089 1.1011 1.0263 0.9320 1.0056

0.94 0.0048 1.0743 1.0198 0.9493 1.0031

0.96 0.0021 1.0485 1.0132 0.9663 1.0014

0.98 0.0005 1.0238 1.0066 0.9833 1.0003

1.0 0.0000 1.0000 1.0000 1.0000 1.0000

1.02 0.0005 0.9771 0.9933 1.0166 1.0003

1.04 0.0018 0.9551 0.9866 1.0330 1.0013

1.06 0.0038 0.9338 0.9798 1.0492 1.0029

1.08 0.0066 0.9133 0.9730 1.0653 1.0051

1.1 0.0099 0.8936 0.9662 1.0812 1.0079

1.12 0.0138 0.8745 0.9593 1.0970 1.0113

1.14 0.0182 0.8561 0.9524 1.1126 1.0153

1.16 0.0230 0.8383 0.9455 1.1280 1.0198

1.18 0.0281 0.8210 0.9386 1.1432 1.0248

1.2 0.0336 0.8044 0.9317 1.1583 1.0304

1.22 0.0394 0.7882 0.9247 1.1732 1.0366

1.24 0.0455 0.7726 0.9178 1.1879 1.0432

1.26 0.0517 0.7574 0.9108 1.2025 1.0504

1.28 0.0582 0.7427 0.9038 1.2169 1.0581

1.3 0.0648 0.7285 0.8969 1.2311 1.0663

1.32 0.0716 0.7147 0.8899 1.2452 1.0750

1.34 0.0785 0.7012 0.8829 1.2591 1.0842

1.36 0.0855 0.6882 0.8760 1.2729 1.0940

1.38 0.0926 0.6755 0.8690 1.2864 1.1042

1.4 0.0997 0.6632 0.8621 1.2999 1.1149

1.42 0.1069 0.6512 0.8551 1.3131 1.1262

1.44 0.1142 0.6396 0.8482 1.3262 1.1379

1.46 0.1215 0.6282 0.8413 1.3392 1.1501

1.48 0.1288 0.6172 0.8344 1.3520 1.1629

1.5 0.1361 0.6065 0.8276 1.3646 1.1762


14/26



Duct for k 1.4

782 Appendix B

Ma f L*/ D p / p* T / T * */ V / V * p0 / p*0

1.52 0.1433 0.5960 0.8207 1.3770 1.1899

1.54 0.1506 0.5858 0.8139 1.3894 1.2042

1.56 0.1579 0.5759 0.8071 1.4015 1.21901.58 0.1651 0.5662 0.8004 1.4135 1.2344

1.6 0.1724 0.5568 0.7937 1.4254 1.2502

1.62 0.1795 0.5476 0.7869 1.4371 1.2666

1.64 0.1867 0.5386 0.7803 1.4487 1.2836

1.66 0.1938 0.5299 0.7736 1.4601 1.3010

1.68 0.2008 0.5213 0.7670 1.4713 1.3190

1.7 0.2078 0.5130 0.7605 1.4825 1.3376

1.72 0.2147 0.5048 0.7539 1.4935 1.3567

1.74 0.2216 0.4969 0.7474 1.5043 1.3764

1.76 0.2284 0.4891 0.7410 1.5150 1.39671.78 0.2352 0.4815 0.7345 1.5256 1.4175

1.8 0.2419 0.4741 0.7282 1.5360 1.4390

1.82 0.2485 0.4668 0.7218 1.5463 1.4610

1.84 0.2551 0.4597 0.7155 1.5564 1.4836

1.86 0.2616 0.4528 0.7093 1.5664 1.5069

1.88 0.2680 0.4460 0.7030 1.5763 1.5308

1.9 0.2743 0.4394 0.6969 1.5861 1.5553

1.92 0.2806 0.4329 0.6907 1.5957 1.5804

1.94 0.2868 0.4265 0.6847 1.6052 1.6062

1.96 0.2929 0.4203 0.6786 1.6146 1.6326

1.98 0.2990 0.4142 0.6726 1.6239 1.6597

2.0 0.3050 0.4082 0.6667 1.6330 1.6875

2.02 0.3109 0.4024 0.6608 1.6420 1.7160

2.04 0.3168 0.3967 0.6549 1.6509 1.7451

2.06 0.3225 0.3911 0.6491 1.6597 1.7750

2.08 0.3282 0.3856 0.6433 1.6683 1.8056

2.1 0.3339 0.3802 0.6376 1.6769 1.8369

2.12 0.3394 0.3750 0.6320 1.6853 1.8690

2.14 0.3449 0.3698 0.6263 1.6936 1.9018

2.16 0.3503 0.3648 0.6208 1.7018 1.9354

2.18 0.3556 0.3598 0.6152 1.7099 1.9698

2.2 0.3609 0.3549 0.6098 1.7179 2.0050

2.22 0.3661 0.3502 0.6043 1.7258 2.0409

2.24 0.3712 0.3455 0.5989 1.7336 2.0777

2.26 0.3763 0.3409 0.5936 1.7412 2.1153

2.28 0.3813 0.3364 0.5883 1.7488 2.1538

2.3 0.3862 0.3320 0.5831 1.7563 2.1931

2.32 0.3911 0.3277 0.5779 1.7637 2.2333

2.34 0.3959 0.3234 0.5728 1.7709 2.2744

2.36 0.4006 0.3193 0.5677 1.7781 2.3164

2.38 0.4053 0.3152 0.5626 1.7852 2.3593

2.4 0.4099 0.3111 0.5576 1.7922 2.4031

2.42 0.4144 0.3072 0.5527 1.7991 2.4479

2.44 0.4189 0.3033 0.5478 1.8059 2.4936

2.46 0.4233 0.2995 0.5429 1.8126 2.5403

2.48 0.4277 0.2958 0.5381 1.8192 2.5880

2.5 0.4320 0.2921 0.5333 1.8257 2.6367

2.52 0.4362 0.2885 0.5286 1.8322 2.6865

2.54 0.4404 0.2850 0.5239 1.8386 2.7372

2.56 0.4445 0.2815 0.5193 1.8448 2.7891

2.58 0.4486 0.2781 0.5147 1.8510 2.8420


15/26




Duct for k 1.4

Ma f L*/ D p / p* T / T * */ V / V * p0 / p*0

2.6 0.4526 0.2747 0.5102 1.8571 2.8960

2.62 0.4565 0.2714 0.5057 1.8632 2.9511

2.64 0.4604 0.2682 0.5013 1.8691 3.00732.66 0.4643 0.2650 0.4969 1.8750 3.0647

2.68 0.4681 0.2619 0.4925 1.8808 3.1233

2.7 0.4718 0.2588 0.4882 1.8865 3.1830

2.72 0.4755 0.2558 0.4839 1.8922 3.2440

2.74 0.4791 0.2528 0.4797 1.8978 3.3061

2.76 0.4827 0.2498 0.4755 1.9033 3.3695

2.78 0.4863 0.2470 0.4714 1.9087 3.4342

2.8 0.4898 0.2441 0.4673 1.9140 3.5001

2.82 0.4932 0.2414 0.4632 1.9193 3.5674

2.84 0.4966 0.2386 0.4592 1.9246 3.63592.86 0.5000 0.2359 0.4552 1.9297 3.7058

2.88 0.5033 0.2333 0.4513 1.9348 3.7771

2.9 0.5065 0.2307 0.4474 1.9398 3.8498

2.92 0.5097 0.2281 0.4436 1.9448 3.9238

2.94 0.5129 0.2256 0.4398 1.9497 3.9993

2.96 0.5160 0.2231 0.4360 1.9545 4.0763

2.98 0.5191 0.2206 0.4323 1.9593 4.1547

3.0 0.5222 0.2182 0.4286 1.9640 4.2346

3.02 0.5252 0.2158 0.4249 1.9686 4.3160

3.04 0.5281 0.2135 0.4213 1.9732 4.3989

3.06 0.5310 0.2112 0.4177 1.9777 4.4835

3.08 0.5339 0.2090 0.4142 1.9822 4.5696

3.1 0.5368 0.2067 0.4107 1.9866 4.6573

3.12 0.5396 0.2045 0.4072 1.9910 4.7467

3.14 0.5424 0.2024 0.4038 1.9953 4.8377

3.16 0.5451 0.2002 0.4004 1.9995 4.9304

3.18 0.5478 0.1981 0.3970 2.0037 5.0248

3.2 0.5504 0.1961 0.3937 2.0079 5.1210

3.22 0.5531 0.1940 0.3904 2.0120 5.2189

3.24 0.5557 0.1920 0.3872 2.0160 5.3186

3.26 0.5582 0.1901 0.3839 2.0200 5.4201

3.28 0.5607 0.1881 0.3807 2.0239 5.5234

3.3 0.5632 0.1862 0.3776 2.0278 5.6286

3.32 0.5657 0.1843 0.3745 2.0317 5.7358

3.34 0.5681 0.1825 0.3714 2.0355 5.8448

3.36 0.5705 0.1806 0.3683 2.0392 5.9558

3.38 0.5729 0.1788 0.3653 2.0429 6.0687

3.4 0.5752 0.1770 0.3623 2.0466 6.1837

3.42 0.5775 0.1753 0.3594 2.0502 6.3007

3.44 0.5798 0.1736 0.3564 2.0537 6.4198

3.46 0.5820 0.1718 0.3535 2.0573 6.5409

3.48 0.5842 0.1702 0.3507 2.0607 6.6642

3.5 0.5864 0.1685 0.3478 2.0642 6.7896

3.52 0.5886 0.1669 0.3450 2.0676 6.9172

3.54 0.5907 0.1653 0.3422 2.0709 7.0471

3.56 0.5928 0.1637 0.3395 2.0743 7.1791

3.58 0.5949 0.1621 0.3368 2.0775 7.3135

3.6 0.5970 0.1616 0.3341 2.0808 7.4501

3.62 0.5990 0.1590 0.3314 2.0840 7.5891

3.64 0.6010 0.1575 0.3288 2.0871 7.7305

3.66 0.6030 0.1560 0.3262 2.0903 7.8742


16/26



Duct for k 1.4

784 Appendix B

Ma f L*/ D p / p* T / T * */ V / V * p0 / p*0

3.68 0.6049 0.1546 0.3236 2.0933 8.0204

3.7 0.6068 0.1531 0.3210 2.0964 8.1691

3.72 0.6087 0.1517 0.3185 2.0994 8.32023.74 0.6106 0.1503 0.3160 2.1024 8.4739

3.76 0.6125 0.1489 0.3135 2.1053 8.6302

3.78 0.6143 0.1475 0.3111 2.1082 8.7891

3.8 0.6161 0.1462 0.3086 2.1111 8.9506

3.82 0.6179 0.1449 0.3062 2.1140 9.1148

3.84 0.6197 0.1436 0.3039 2.1168 9.2817

3.86 0.6214 0.1423 0.3015 2.1195 9.4513

3.88 0.6231 0.1410 0.2992 2.1223 9.6237

3.9 0.6248 0.1397 0.2969 2.1250 9.7990

3.92 0.6265 0.1385 0.2946 2.1277 9.97713.94 0.6282 0.1372 0.2923 2.1303 10.1581

3.96 0.6298 0.1360 0.2901 2.1329 10.3420

3.98 0.6315 0.1348 0.2879 2.1355 10.5289

4.0 0.6331 0.1336 0.2857 2.1381 10.7188

Table B.4 Frictionless Duct Flow

with Heat Transfer for k 1.4 Ma T 0 / T *0 p / p* T / T * */ V / V * p0 / p*0

0.0 0.0 2.4000 0.0 0.0 1.2679

0.02 0.0019 2.3987 0.0023 0.0010 1.2675

0.04 0.0076 2.3946 0.0092 0.0038 1.2665

0.06 0.0171 2.3800 0.0205 0.0086 1.2647

0.08 0.0302 2.3787 0.0362 0.0152 1.2623

0.1 0.0468 2.3669 0.0560 0.0237 1.2591

0.12 0.0666 2.3526 0.0797 0.0339 1.2554

0.14 0.0895 2.3359 0.1069 0.0458 1.2510

0.16 0.1151 2.3170 0.1374 0.0593 1.2461

0.18 0.1432 2.2959 0.1708 0.0744 1.2406

0.2 0.1736 2.2727 0.2066 0.0909 1.2346

0.22 0.2057 2.2477 0.2445 0.1088 1.2281

0.24 0.2395 2.2209 0.2841 0.1279 1.2213

0.26 0.2745 2.1925 0.3250 0.1482 1.2140

0.28 0.3104 2.1626 0.3667 0.1696 1.2064

0.3 0.3469 2.1314 0.4089 0.1918 1.1985

0.32 0.3837 2.0991 0.4512 0.2149 1.1904

0.34 0.4206 2.0657 0.4933 0.2388 1.1822

0.36 0.4572 2.0314 0.5348 0.2633 1.1737

0.38 0.4935 1.9964 0.5755 0.2883 1.1652

0.4 0.5290 1.9608 0.6151 0.3137 1.1566

0.42 0.5638 1.9247 0.6535 0.3395 1.1480

0.44 0.5975 1.8882 0.6903 0.3656 1.1394

0.46 0.6301 1.8515 0.7254 0.3918 1.1308

0.48 0.6614 1.8147 0.7587 0.4181 1.1224

0.5 0.6914 1.7778 0.7901 0.4444 1.1141

0.52 0.7199 1.7409 0.8196 0.4708 1.1059

0.54 0.7470 1.7043 0.8469 0.4970 1.0979

0.56 0.7725 1.6678 0.8723 0.5230 1.0901

0.58 0.7965 1.6316 0.8955 0.5489 1.0826

0.6 0.8189 1.5957 0.9167 0.5745 1.0753


17/26


Table B.4 (Cont.) Frictionless

Duct Flow with Heat Transfer for

k 1.4

Ma T 0 / T *0 p / p* T / T * */ V / V * p0 / p*0

0.62 0.8398 1.5603 0.9358 0.5998 1.0682

0.64 0.8592 1.5253 0.9530 0.6248 1.0615

0.66 0.8771 1.4908 0.9682 0.6494 1.05500.68 0.8935 1.4569 0.9814 0.6737 1.0489

0.7 0.9085 1.4235 0.9929 0.6975 1.0431

0.72 0.9221 1.3907 1.0026 0.7209 1.0376

0.74 0.9344 1.3585 1.0106 0.7439 1.0325

0.76 0.9455 1.3270 1.0171 0.7665 1.0278

0.78 0.9553 1.2961 1.0220 0.7885 1.0234

0.8 0.9639 1.2658 1.0255 0.8101 1.0193

0.82 0.9715 1.2362 1.0276 0.8313 1.0157

0.84 0.9781 1.2073 1.0285 0.8519 1.0124

0.86 0.9836 1.1791 1.0283 0.8721 1.00950.88 0.9883 1.1515 1.0269 0.8918 1.0070

0.9 0.9921 1.1246 1.0245 0.9110 1.0049

0.92 0.9951 1.0984 1.0212 0.9297 1.0031

0.94 0.9973 1.0728 1.0170 0.9480 1.0017

0.96 0.9988 1.0479 1.0121 0.9658 1.0008

0.98 0.9997 1.0236 1.0064 0.9831 1.0002

1.0 1.0000 1.0000 1.0000 1.0000 1.0000

1.02 0.9997 0.9770 0.9930 1.0164 1.0002

1.04 0.9989 0.9546 0.9855 1.0325 1.0008

1.06 0.9977 0.9327 0.9776 1.0480 1.0017

1.08 0.9960 0.9115 0.9691 1.0632 1.0031

1.1 0.9939 0.8909 0.9603 1.0780 1.0049

1.12 0.9915 0.8708 0.9512 1.0923 1.0070

1.14 0.9887 0.8512 0.9417 1.1063 1.0095

1.16 0.9856 0.8322 0.9320 1.1198 1.0124

1.18 0.9823 0.8137 0.9220 1.1330 1.0157

1.2 0.9787 0.7958 0.9118 1.1459 1.0194

1.22 0.9749 0.7783 0.9015 1.1584 1.0235

1.24 0.9709 0.7613 0.8911 1.1705 1.0279

1.26 0.9668 0.7447 0.8805 1.1823 1.0328

1.28 0.9624 0.7287 0.8699 1.1938 1.0380

1.3 0.9580 0.7130 0.8592 1.2050 1.0437

1.32 0.9534 0.6978 0.8484 1.2159 1.0497

1.34 0.9487 0.6830 0.8377 1.2264 1.0561

1.36 0.9440 0.6686 0.8269 1.2367 1.0629

1.38 0.9391 0.6546 0.8161 1.2467 1.0701

1.4 0.9343 0.6410 0.8054 1.2564 1.0777

1.42 0.9293 0.6278 0.7947 1.2659 1.0856

1.44 0.9243 0.6149 0.7840 1.2751 1.0940

1.46 0.9193 0.6024 0.7735 1.2840 1.1028

1.48 0.9143 0.5902 0.7629 1.2927 1.1120

1.5 0.9093 0.5783 0.7525 1.3012 1.1215

1.52 0.9042 0.5668 0.7422 1.3095 1.1315

1.54 0.8992 0.5555 0.7319 1.3175 1.1419

1.56 0.8942 0.5446 0.7217 1.3253 1.1527

1.58 0.8892 0.5339 0.7117 1.3329 1.1640

1.6 0.8842 0.5236 0.7017 1.3403 1.1756

1.62 0.8792 0.5135 0.6919 1.3475 1.1877

1.64 0.8743 0.5036 0.6822 1.3546 1.2002

1.66 0.8694 0.4940 0.6726 1.3614 1.2131

1.68 0.8645 0.4847 0.6631 1.3681 1.2264

786 di


18/26



k 1.4

786 Appendix B

Ma T 0 / T *0 p / p* T / T * */ V / V * p0 / p*0

1.7 0.8597 0.4756 0.6538 1.3746 1.2402

1.72 0.8549 0.4668 0.6445 1.3809 1.2545

1.74 0.8502 0.4581 0.6355 1.3870 1.26921.76 0.8455 0.4497 0.6265 1.3931 1.2843

1.78 0.8409 0.4415 0.6176 1.3989 1.2999

1.8 0.8363 0.4335 0.6089 1.4046 1.3159

1.82 0.8317 0.4257 0.6004 1.4102 1.3324

1.84 0.8273 0.4181 0.5919 1.4156 1.3494

1.86 0.8228 0.4107 0.5836 1.4209 1.3669

1.88 0.8185 0.4035 0.5754 1.4261 1.3849

1.9 0.8141 0.3964 0.5673 1.4311 1.4033

1.92 0.8099 0.3895 0.5594 1.4360 1.4222

1.94 0.8057 0.3828 0.5516 1.4408 1.4417

1.96 0.8015 0.3763 0.5439 1.4455 1.4616

1.98 0.7974 0.3699 0.5364 1.4501 1.4821

2.0 0.7934 0.3636 0.5289 1.4545 1.5031

2.02 0.7894 0.3575 0.5216 1.4589 1.5246

2.04 0.7855 0.3516 0.5144 1.4632 1.5467

2.06 0.7816 0.3458 0.5074 1.4673 1.5693

2.08 0.7778 0.3401 0.5004 1.4714 1.5924

2.1 0.7741 0.3345 0.4936 1.4753 1.6162

2.12 0.7704 0.3291 0.4868 1.4792 1.6404

2.14 0.7667 0.3238 0.4802 1.4830 1.6653

2.16 0.7631 0.3186 0.4737 1.4867 1.6908

2.18 0.7596 0.3136 0.4673 1.4903 1.7168

2.2 0.7561 0.3086 0.4611 1.4938 1.7434

2.22 0.7527 0.3038 0.4549 1.4973 1.7707

2.24 0.7493 0.2991 0.4488 1.5007 1.7986

2.26 0.7460 0.2945 0.4428 1.5040 1.8271

2.28 0.7428 0.2899 0.4370 1.5072 1.8562

2.3 0.7395 0.2855 0.4312 1.5104 1.8860

2.32 0.7364 0.2812 0.4256 1.5134 1.9165

2.34 0.7333 0.2769 0.4200 1.5165 1.9476

2.36 0.7302 0.2728 0.4145 1.5194 1.9794

2.38 0.7272 0.2688 0.4091 1.5223 2.0119

2.4 0.7242 0.2648 0.4038 1.5252 2.0451

2.42 0.7213 0.2609 0.3986 1.5279 2.0789

2.44 0.7184 0.2571 0.3935 1.5306 2.1136

2.46 0.7156 0.2534 0.3885 1.5333 2.1489

2.48 0.7128 0.2497 0.3836 1.5359 2.1850

2.5 0.7101 0.2462 0.3787 1.5385 2.2218

2.52 0.7074 0.2427 0.3739 1.5410 2.2594

2.54 0.7047 0.2392 0.3692 1.5434 2.2978

2.56 0.7021 0.2359 0.3646 1.5458 2.3370

2.58 0.6995 0.2326 0.3601 1.5482 2.3770

2.6 0.6970 0.2294 0.3556 1.5505 2.4177

2.62 0.6945 0.2262 0.3512 1.5527 2.4593

2.64 0.6921 0.2231 0.3469 1.5549 2.5018

2.66 0.6896 0.2201 0.3427 1.5571 2.5451

2.68 0.6873 0.2171 0.3385 1.5592 2.5892

2.7 0.6849 0.2142 0.3344 1.5613 2.6343

2.72 0.6826 0.2113 0.3304 1.5634 2.6802

2.74 0.6804 0.2085 0.3264 1.5654 2.7270

2.76 0.6781 0.2058 0.3225 1.5673 2.7748

C ibl Fl T bl 787


19/26




k 1.4

Ma T 0 / T *0 p / p* T / T * */ V / V * p0 / p*0

2.78 0.6761 0.2030 0.3186 1.5693 2.8235

2.8 0.6738 0.2004 0.3149 1.5711 2.8731

2.82 0.6717 0.1978 0.3111 1.5730 2.92372.84 0.6696 0.1953 0.3075 1.5748 2.9752

2.86 0.6675 0.1927 0.3039 1.5766 3.0278

2.88 0.6655 0.1903 0.3004 1.5784 3.0813

2.9 0.6635 0.1879 0.2969 1.5801 3.1359

2.92 0.6615 0.1855 0.2934 1.5818 3.1914

2.94 0.6596 0.1832 0.2901 1.5834 3.2481

2.96 0.6577 0.1809 0.2868 1.5851 3.3058

2.98 0.6558 0.1787 0.2835 1.5867 3.3646

3.0 0.6540 0.1765 0.2803 1.5882 3.4245

3.02 0.6522 0.1743 0.2771 1.5898 3.4854

3.04 0.6504 0.1722 0.2740 1.5913 3.5476

3.06 0.6486 0.1701 0.2709 1.5928 3.6108

3.08 0.6469 0.1681 0.2679 1.5942 3.6752

3.1 0.6452 0.1660 0.2650 1.5957 3.7408

3.12 0.6435 0.1641 0.2620 1.5971 3.8076

3.14 0.6418 0.1621 0.2592 1.5985 3.8756

3.16 0.6402 0.1602 0.2563 1.5998 3.9449

3.18 0.6386 0.1583 0.2535 1.6012 4.0154

3.2 0.6370 0.1565 0.2508 1.6025 4.0871

3.22 0.6354 0.1547 0.2481 1.6038 4.1602

3.24 0.6339 0.1529 0.2454 1.6051 4.2345

3.26 0.6324 0.1511 0.2428 1.6063 4.3101

3.28 0.6309 0.1494 0.2402 1.6076 4.3871

3.3 0.6294 0.1477 0.2377 1.6088 4.4655

3.32 0.6280 0.1461 0.2352 1.6100 4.5452

3.34 0.6265 0.1444 0.2327 1.6111 4.6263

3.36 0.6251 0.1428 0.2303 1.6123 4.7089

3.38 0.6237 0.1412 0.2279 1.6134 4.7929

3.4 0.6224 0.1397 0.2255 1.6145 4.8783

3.42 0.6210 0.1381 0.2232 1.6156 4.9652

3.44 0.6197 0.1366 0.2209 1.6167 5.0536

3.46 0.6184 0.1351 0.2186 1.6178 5.1435

3.48 0.6171 0.1337 0.2164 1.6188 5.2350

3.5 0.6158 0.1322 0.2142 1.6198 5.3280

3.52 0.6145 0.1308 0.2120 1.6208 5.4226

3.54 0.6133 0.1294 0.2099 1.6218 5.5188

3.56 0.6121 0.1280 0.2078 1.6228 5.6167

3.58 0.6109 0.1267 0.2057 1.6238 5.7162

3.6 0.6097 0.1254 0.2037 1.6247 5.8173

3.62 0.6085 0.1241 0.2017 1.6257 5.9201

3.64 0.6074 0.1228 0.1997 1.6266 6.02473.66 0.6062 0.1215 0.1977 1.6275 6.1310

3.68 0.6051 0.1202 0.1958 1.6284 6.2390

3.7 0.6040 0.1190 0.1939 1.6293 6.3488

3.72 0.6029 0.1178 0.1920 1.6301 6.4605

3.74 0.6018 0.1166 0.1902 1.6310 6.5739

3.76 0.6008 0.1154 0.1884 1.6318 6.6893

3.78 0.5997 0.1143 0.1866 1.6327 6.8065

3.8 0.5987 0.1131 0.1848 1.6335 6.9256

3.82 0.5977 0.1120 0.1830 1.6343 7.0466

3.84 0.5967 0.1109 0.1813 1.6351 7.1696

788 Appendix B


20/26



k 1.4

788 Appendix B

Ma T 0 / T *0 p / p* T / T * */ V / V * p0 / p*0

3.86 0.5957 0.1098 0.1796 1.6359 7.2945

3.88 0.5947 0.1087 0.1779 1.6366 7.4215

3.9 0.5937 0.1077 0.1763 1.6374 7.55053.92 0.5928 0.1066 0.1746 1.6381 7.6816

3.94 0.5918 0.1056 0.1730 1.6389 7.8147

3.96 0.5909 0.1046 0.1714 1.6396 7.9499

3.98 0.5900 0.1036 0.1699 1.6403 8.0873

4.0 0.5891 0.1026 0.1683 1.6410 8.2269

Table B.5 Prandtl-Meyer

Supersonic Expansion Function for

k 1.4

Ma , deg Ma , deg Ma , deg Ma , deg

1.00 0.0

1.05 0.49 3.05 50.71 5.05 77.38 7.05 91.23

1.10 1.34 3.10 51.65 5.10 77.84 7.10 91.49

1.15 2.38 3.15 52.57 5.15 78.29 7.15 91.75

1.20 3.56 3.20 53.47 5.20 78.73 7.20 92.00

1.25 4.83 3.25 54.35 5.25 79.17 7.25 92.24

1.30 6.17 3.30 55.22 5.30 79.60 7.30 92.49

1.35 7.56 3.35 56.07 5.35 80.02 7.35 92.73

1.40 8.99 3.40 56.91 5.40 80.43 7.40 92.97

1.45 10.44 3.45 57.73 5.45 80.84 7.45 93.21

1.50 11.91 3.50 58.53 5.50 81.24 7.50 93.44

1.55 13.38 3.55 59.32 5.55 81.64 7.55 93.67

1.60 14.86 3.60 60.09 5.60 82.03 7.60 93.90

1.65 16.34 3.65 60.85 5.65 82.42 7.65 94.12

1.70 17.81 3.70 61.60 5.70 82.80 7.70 94.34

1.75 19.27 3.75 62.33 5.75 83.17 7.75 94.56

1.80 20.73 3.80 63.04 5.80 83.54 7.80 94.78

1.85 22.16 3.85 63.75 5.85 83.90 7.85 95.00

1.90 23.59 3.90 64.44 5.90 84.26 7.90 95.21

1.95 24.99 3.95 65.12 5.95 84.61 7.95 95.42

2.00 26.38 4.00 65.78 6.00 84.96 8.00 95.62

2.05 27.75 4.05 66.44 6.05 85.30 8.05 95.83

2.10 29.10 4.10 67.08 6.10 85.63 8.10 96.03

2.15 30.43 4.15 67.71 6.15 85.97 8.15 96.23

2.20 31.73 4.20 68.33 6.20 86.29 8.20 96.43

2.25 33.02 4.25 68.94 6.25 86.62 8.25 96.63

2.30 34.28 4.30 69.54 6.30 86.94 8.30 96.82

2.35 35.53 4.35 70.13 6.35 87.25 8.35 97.01

2.40 36.75 4.40 70.71 6.40 87.56 8.40 97.20

2.45 37.95 4.45 71.27 6.45 87.87 8.45 97.39

2.50 39.12 4.50 71.83 6.50 88.17 8.50 97.57

2.55 40.28 4.55 72.38 6.55 88.47 8.55 97.762.60 41.41 4.60 72.92 6.60 88.76 8.60 97.94

2.65 42.53 4.65 73.45 6.65 89.05 8.65 98.12

2.70 43.62 4.70 73.97 6.70 89.33 8.70 98.29

2.75 44.69 4.75 74.48 6.75 89.62 8.75 98.47

2.80 45.75 4.80 74.99 6.80 89.90 8.80 98.64

2.85 46.78 4.85 75.48 6.85 90.17 8.85 98.81

2.90 47.79 4.90 75.97 6.90 90.44 8.90 98.98

2.95 48.78 4.95 76.45 6.95 90.71 8.95 99.15

3.00 49.76 5.00 76.92 7.00 90.97 9.00 99.32



21/26

Compressible Flow Tables 789

Fig. B.1 Mach number downstream

of an oblique shock for k 1.4.

4.0

3.0

2.0

1.0

02.0 3.0 4.01.0

M a 2

Ma1

Mach line

= 0°

Weak

shock

Strong

shock

=β

20°

25

30

35

40

45

55

50

60

65

7075

859080

Normal shock

5

10

15

20

25

30

35

θ

β

Ma1

Ma2

θ

θ

790 Appendix B


22/26

pp

Fig. B.2 Pressure ratio downstream

of an oblique shock for k 1.4.

θ

Weak shock

Strong shock

β

80°

Normal shock

5

10

15

20

25

θ

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

Ma1, p 1 p 2

7090 75 65 60 55 50 45

40

35

30

25

20

β

30°θ =

=

p 2

p 1

1.01.0

Ma1

2.0 3.0 4.0


23/26

791

During this period of transition there is a constant need for conversions between BG

and SI units (see Table 1.2). Some additional conversions are given here. Conversion

factors are given inside the front cover.

Length Volume

1 ft 12 in 0.3048 m 1 ft3 0.028317 m3

1 mi 5280 ft 1609.344 m 1 U.S. gal 231 in3 0.0037854 m3

1 nautical mile (nmi) 6076 ft 1852 m 1 L 0.001 m3 0.035315 ft3

1 yd 3 ft 0.9144 m 1 U.S. fluid ounce 2.9574 E-5 m3

1 angstrom (Å) 1.0 E-10 m 1 U.S. quart (qt) 9.4635 E-4 m3

Mass Area

1 slug 32.174 lbm 14.594 kg 1 ft2 0.092903 m2

1 lbm 0.4536 kg 1 mi2 2.78784 E7 ft2 2.59 E6 m2

1 short ton 2000 lbm 907.185 kg 1 acre 43,560 ft2 4046.9 m2

1 tonne

1000 kg 1 hectare (ha)

10,000 m

2

Velocity Acceleration

1 ft/s 0.3048 m/s 1 ft/s2 0.3048 m/s2

1 mi/h 1.466666 ft/s 0.44704 m/s

1 kn 1 nmi/h 1.6878 ft/s 0.5144 m/s

Mass flow Volume flow

1 slug/s 14.594 kg/s 1 gal/min 0.002228 ft3 /s 0.06309 L/s

1 lbm/s 0.4536 kg/s 1 106 gal/day 1.5472 ft3 /s 0.04381 m3 /s

Pressure Force

1 lbf/ft2 47.88 Pa 1 lbf 4.448222 N 16 oz

1 lbf/in2 144 lbf/ft2 6895 Pa 1 kgf 2.2046 lbf 9.80665 N

1 atm 2116.2 lbf/ft2 14.696 lbf/in2 1 U.S. (short) ton 2000 lbf

101,325 Pa 1 dyne 1.0 E-5 N

1 inHg (at 20°C) 3375 Pa 1 ounce (avoirdupois) (oz) 0.27801 N1 bar 1.0 E5 Pa

Appendix C

Conversion Factors

792 Appendix C


24/26

Energy Power

1 ft lbf 1.35582 J 1 hp 550 ft lbf/s 745.7 W

1 Btu 252 cal 1055.056 J 778.17 ft lbf 1 ft lbf/s 1.3558 W

1 kilowatt hour (kWh) 3.6 E6 J

Specific weight Density

1 lbf/ft3 157.09 N/m3 1 slug/ft3 515.38 kg/m3

1 lbm/ft3 16.0185 kg/m3

1 g/cm3 1000 kg/m3

Viscosity Kinematic viscosity

1 slug/(ft s) 47.88 kg/(m s) 1 ft2 /h 0.000025806 m2 /s

1 poise (P) 1 g/(cm s) 0.1 kg/(m s) 1 stokes (St) 1 cm2 /s 0.0001 m2 /s

Temperature scale readings

T F 95T C 32 T C

59(T F 32) T R T F 459.69 T K T C 273.16

where subscripts F, C, R, and K refer to readings on the Fahrenheit, Celsius, Kelvin, and Rankine scales,respectively

Specific heat or gas constant* Thermal conductivity*

1 ft lbf/(slug °R) 0.16723 N m/(kg K) 1 Btu/(h ft °R) 1.7307 W/(m K)1 Btu/(lb °R) 4186.8 J/(kg K)

*Although the absolute (Kelvin) and Celsius temperature scales have different starting points, the intervals are the same

size: 1 kelvin 1 Celsius degree. The same holds true for the nonmetric absolute (Rankine) and Fahrenheit scales: 1Rankine degree 1 Fahrenheit degree. It is customary to express temperature differences in absolute-temperature units.


25/26

The equations of motion of an incompressible newtonian fluid with constant , k , and

c p are given here in cylindrical coordinates (r , , z), which are related to cartesian co-

ordinates ( x, y, z) as in Fig. 4.2:

x r cos y r sin z z (D.1)

The velocity components are r , , and z. The equations are:

Continuity:

1

r

r (r r )

1

r

( )

z ( z) 0 (D.2)

Convective time derivative:

V r

r

1

r

z

z

(D.3)

Laplacian operator:

2

1

r

r r

r r

12

2

2

z

2

2 (D.4)

The r -momentum equation:

t r (V ) r

1

r

2

1

p

r gr 2 r r

2

r

r

22

(D.5)

The -momentum equation:

t (V )

1

r r

1

r

p

g 2 r

2 r

22

r (D.6)

The z-momentum equation:

t z (V ) z

1

p

z g z

2 z (D.7)

Appendix D

Equations of Motion inCylindrical Coordinates

793

794 Appendix D


26/26

The energy equation:

c pT

t (V )T k 2T [2( 2rr 2 2 zz) 2 z 2rz 2r ] (D.8)

where rr

r r

1r

r

zz

z z z

1

r

z

z (D.9)

rz

zr

r z r

1

r

r

r

Viscous stress components:

rr 2 rr 2 zz 2 zz(D.10)

r r z z rz rz

Angular-velocity components:

r 1

r

z

z

zr

r z

(D.11)

z 1

r

r (r )

1

r

r

Documents

Fluid Mechanics F.M. White Apendices a-D