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Preface to the Fifth Edition

The data given in the previous edition has been thoroughly checked. Errors occurring in somepages have been corrected.

This little book has been the companion volume of the author’s book—Fundamentals of Com-pressible Flow. They together will continue to serve the needs of students, scientists and engineersin areas of aerospace, mechanical, chemical and gas engineering.

It is hoped that the new format of this edition will be more convenient to use.

New Delhi S.M. YAHYA

July, 2006

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Preface to the First Edition

There was a long-felt need for gas tables to support the courses in Gas Dynamics, Turbomachineryand Aerodynamics taught at different levels.

These tables have been prepared for compressible fluids of common use. Long and detailedtables in a wide range covering a number of values of γ = cp /cv besides being expensive are notfully used by a majority of students.

Various formulas which form the basis of the tables are also included which will be useful forready reference.

With numerical work of this magnitude it is highly probable that some errors may have creptin; reference to these errors and suggestions for improvement will be highly appreciated.

The ranges of various tables have been chosen according to their importance and frequencyin use.

New Delhi S.M. YAHYA

July, 1978

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ContentsPreface to the Fifth Edition v

Preface to the First Edition vii

Notation 1

Introduction 2

1. Gas Dynamic Properties of Some Gases and Vapours

1.1 Air 141.2 Hydrogen 151.3 Carbon Dioxide 161.4 Steam 171.5 Freon–21 18

2. Properties of Standard Atmosphere 19

3. Isentropic Flow of Perfect Gases

3.1 γ = 1.3 213.2 γ = 1.4 283.3 γ = 1.67 40

4. Normal Shocks in Perfect Gases

4.1 γ = 1.3 474.2 γ = 1.4 524.3 γ = 1.67 57

5. Oblique Shocks in Perfect Gases

5.1 Wave Angles for Given Deflections, γ = 1.4 625.2 Upstream and Downstream Mach Numbers and Deflections from

Normal Shock Relations, γ = 1.4 71

6. Flow of Perfect Gases with Friction (Fanno Flow)

6.1 γ = 1.3 756.2 γ = 1.4 816.3 γ = 1.67 87

7. Isothermal Flow of Perfect Gases with Friction

7.1 γ = 1.3 937.2 γ = 1.4 99

8. Flow of Perfect Gases with Heat Transfer (Rayleigh Flow)

8.1 γ = 1.3 1058.2 γ = 1.4 1118.3 γ = 1.67 1178.4 Maximum Heat Transfer in Perfect Gases 123

9. Prandtl-Meyer Function with Mach Angles

9.1 α and ω (M) as functions of M 1259.2 M and α as functions of ω (M) 130

10. Approximate Velocities and Mach Numbers of AirCorresponding to Dynamic Heads, and Pressure Ratios

10.1 Air velocities and Mach numbers as functions ofdynamic heads in mm W.G. 134

10.2 Air velocities and Mach numbers as functions ofdynamic heads in mm Hg. 135

10.3 Mach Numbers as functions of pressure ratios (γ = 1.4) 136

11. Manometer Inclinations Required for the Desired Magnificationof the Deflection 138

12. Static Pressure Drop in Gas Pipe Lines 139

13. Isentropic Compressor Power 141

14. Pumping Power for Fans and Blowers 143

15. S.I, Metric and British Units

15.1 Principal dimensions and units 14715.2 Conversion of Units 148

x Contents

Introduction 1

Notation

a Velocity of sound A area of cross-section

c fluid velocity

cp specific heat at constant pressure

cv specific heat at constant volume D hydraulic mean diameter

f mean coefficient of friction F impulse function

h enthalpy, actual deflection k constant

L length of the pipe line

1 magnified value of manometer deflection

Lmax maximum pipe length required for choking conditions

m mass flow rate M Mach number

P power p pressure

∆p pressure rise or drop pr pressure ratio

Qmax maximum heat transfer Q volume flow rate

R Gas constant s entropy

t temperature in °C T temperature in K

v specific volume w mass flow rate

Z altitude ∆s change of entropy

α Mach angle, manometer inclination

γ = cp /cv σ wave angle

δ deflection or half wedge angle µ coefficient of viscosity

ρ density ω(M) Prandtl-Meyer function

η efficiency

Subscripts

o Stagnation state x upstream of normal shock

y downstream of normal shock l upstream of oblique shock

2 downstream of oblique shock s isentropic

Superscript

* corresponding to choking conditions (M = 1)

*t corresponding to M = 1

γ (isothermal flow)

1

2 Gas Tables for Compressible Flow Calculations

IntroductionThis section is designed to help the user to make the best use of the data given in the tables.Detailed explanation of the various theories behind the formulae mentioned in this section are notincluded; the user is expected to be aware of these while using the tables. However, some explana-tory notes and figures have been included.

1. Gas dynamic properties of some vapours and gasesIt is well known that the specific heats (cp and cv) of gases and vapours vary with temperature.Therefore, the values of the ratio of specific heats, Gas constant and the maximum mass flowparameter also vary with temperature. These have been tabulated for a range of temperatures atgiven values of pressure.

Only some of the commonly used gases and vapours have been chosen for these tables.

Formulae

γ = c cp v ...(1.1)

R c cp v= − ...(1.2)

a RT= γ ...(1.3)

w T

A P R0

0

1

1

1 2

2

1*=

+FHGIKJ

L

NMM

O

QPP

+−γ

γ

γγ

...(1.4)

2. Properties of the standard atmosphereValues of temperature, velocity of sound, pressure, density and dynamic viscosity of air are tabu-lated against altitude in the range of 0 to 20,000 meters.

The lapse rate in temperature is 0.65°C for each 100 meters of altitude upto an altitude of11000 metres.

Kinematic viscosity (µ/ρ) can be determined with the help of the last two columns.

3. Isentropic flow of perfect gasesSubsonic and supersonic isentropic flows in nozzles and diffusers are shown in figures 1 and 2respectively. Pressures corresponding to the choking and stagnation states in both the cases arealso shown.

Assumptions

(a) One dimensional isentropic flow

(b) Perfect gas with constant specific heats and molecular weight.

Introduction 3

Fig. 1. Isentropic Accelerating Flow

Fig. 2. Isentropic Decelerating Flow

S

h

p0

M = 0

M < 1

M = M* = 1

M > 1

p

p*

p

h

p0

M = 0

M < 1

M = M* = 1

M > 1

p

p*

p

S


Formulas

T T0 0= =* constant ...(3.1)

T

T M0 2

1

11

2

=+ −γ ...(3.2)

p

p

T

TM

0 0

1 2 11

1

2=FHGIKJ = + −FHG

IKJ

− −γ

γγ

γγ...(3.3)

ρρ

γγ γ

0 0

1

1 2

1

11

1

2=FHGIKJ = + −FHG

IKJ

− −T

TM ...(3.4)

Mc

c

M

M

**

= =

+

+ −

γ

γ

12

11

2

2

2

...(3.5)

A

A MM

*=

+FHGIKJ + −FHG

IKJ

LNM

OQP

+−1 2

11

1

22

1

2 1

γγ

γγb g ...(3.6)

F

F

M

MM

*= +

+ + −FHG

IKJ

1

2 1 11

2

2

2

γ

γ γb g...(3.7)

A

A

p

pMM

*× =

+FHGIKJ

+ −

+−

0

12 1

2

21

11

2

γγ

γγb g

...(3.8)

4. Normal shocks in perfect gasesA normal shock from state X to state Y is shown in figure 3.

Assumptions

(a) One dimensional flow with normal shocks(b) Constant area duct(c) Perfect gas with constant specific heats and molecular weight(d) Adiabatic flow

Introduction 5

Formulas

T Tx y0 0= ...(4.1)

MM

My

x

x

= −+

−−

L

N

MMMM

O

Q

PPPP

21

21

1

2

2

1 2

γγ

γ

...(4.2)

p

p

M

MM

y

x

x

yx= +

+=

+− −

+1

1

2

1

1

1

2

22γ

γγ

γγγ

...(4.3)

T

T

M

M

M M

M

y

x

x

y

x x

x

=+ −

+ −=

+ −FHG

IKJ −

−FHG

IKJ

+−

11

2

11

2

11

22

11

1

2 1

2

2

2 2

22

γ

γ

γ γγ

γγb gb g

...(4.4)

a

a

T

Ty

x

y

x

=FHGIKJ

1 2

...(4.5)

ρρ

y

x

y

x

x

y

p

p

T

T= ...(4.6)

p

pMx

xx

0

2 11

1

2= + −FHG

IKJ

−γγ

γ...(4.7)

p

pMy

yy

0 2 11

1

2= + −FHG

IKJ

−γγ

γ...(4.8)

p

p

p

p

p

p

p

py

x

y

y

y

x

x

x

0

0

0

0

= ...(4.9)

∆s

R

p

px

y

= ln 0

0...(4.10)

p

p

p

p

p

py

x

y

y

y

x

0 0= ...(4.11)


Fig. 3. Normal Shock, Fanno and Rayleigh Lines

5. Oblique shocks in perfect gasesAn oblique shock wave is inclined at an angle to the direction of upstream flow. Figure 4 shows anoblique shock wave at the corner of a wall. Supersonic flow passing through an oblique shock(wave angle = σ) is deflected through an angle (δ). The same picture is obtained when a supersonicflow is deflected through an angle (δ) either over a wedge or a wall corner.

Fig. 4. Flow Through Oblique Shock Wave

Figure 4 shows the upstream and downstream Mach numbers corresponding to a normal shock.

X

px

M > 1M > 1

M = 1

p*

M < 1

YFanno line

py

M < 1

M = 1

p*

p = constant

Rayleigh line

Nor

mal

shoc

k

M = 1/ �

h

s

My

Mx

Obliqueshock

M = c /a1 1 1

M = c /a2 2 2

�

�

Introduction 7

Assumptions

(a) Perfect gas, with constant specific heats and molecular weight,(b) adiabatic flow,(c) absence of body forces,(d) absence of external work.

Formulas

tansin cot

cosδ σ σ

γ σ= −

+ +M

M12

12

2 2

2 2b g ...(5.1)

M Mx = 1 sin σ ...(5.2)

M My = −2 sin σ δb g ...(5.3)

Table 5.1 is used to determine the wave angle for given values of upstream Mach numbers anddeflection.

Table 5.2 enables calculations of properties downstream of the oblique shock through normalshock tables. The gas property ratios corresponding to Mach number Mx can be found from Table4.2.

6. Flow of perfect gases with friction (Fanno Flow)A Fanno line is shown on h-s co-ordinates in figure 3.

Friction in the passage leads to increase in entropy and takes the process towards the chokingconditions (M = M* = 1).

Assumptions

(a) One dimensional flow with friction,(b) Constant area duct,(c) perfect gas with constant specific heats and molecular weight,(d) adiabatic flow.

Formulas

T T0 0= =* constant ...(6.1)

p

p MM

*= +

+ −FHG

IKJ

1 1

2 11

22

γγ ...(6.2)

c

cM

M*

*

= = +

+ −FHG

IKJ

ρρ

γγ

1

2 11

22 ...(6.3)


T

T M*

= +

+ −FHG

IKJ

γγ

1

2 11

22 ...(6.4)

p

p M

M0

0

2

12 1

12 1

12

1*=

+ −FHG

IKJ

+

L

N

MMMM

O

Q

PPPP

+−γ

γ

γγ

b g

b g

...(6.5)

F

F

M

M M*

= +

+ + −FHG

IKJ

LNM

OQP

1

2 1 11

2

2

21 2

γ

γ γb g ...(6.6)

41 1

2

1

2 11

2

2

2

2

2f

L

D

M

M

M

M

max ln= − + + +

+ −FHG

IKJ

γγ

γγ

γb g

...(6.7)

1*

22

2 2p

s s 1ln M

1c2 M 1 M

2

γ+γ− γ +⎡ ⎤= ⎢ ⎥γ −⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

...(6.8)

7. Isothermal flow of perfect gases with frictionThe combined effect of friction and heat transfer in long ducts can be considered to produce anisothermal flow model.

Here the reference values of the properties p c T pt t t t t* * * * *max, , , ,ρ 0 0 and Le j are taken at the

Mach number

M t* = 1

γ

Assumptions

(a) One dimensional flow with friction and heat transfer,

(b) Constant area duct,

(c) Perfect gas with constant specific heats and molecular weight,

(d) Isothermal flow.

Introduction 9

Formulas

p

p t M*= 1

γ ...(7.1)

c

c t

tM

*

*

= =ρρ

γ ...(7.2)

T

TM

t0

0

22

3 11

1

2*=

−+ −FHG

IKJ

γγ

γ...(7.3)

p

pM

t M0

0

2 11 2

3 11

1

2*=

−+ −FHG

IKJ

LNM

OQP

−

γγ

γγ

γγ

...(7.4)

41 2

22f

L

D

M

MMmax ln= − +γ

γγe j ...(7.5)

8. Flow of perfect gases with heat transfer (Rayleigh flow)A Rayleight line is shown in figure 3. Heat addition increases the entropy and takes the processtowards the choking conditions, (M = M* = 1). This behaviour imposes a limit to heat addition;therefore for a given initial Mach Number there is a fixed value of the maximum possible heattransfer (Qmax). This has been tabulated in Table 8.4.

Assumptions

(a) One dimensional flow with heat transfer,

(b) Constant area frictionless duct,

(c) Perfect gas with constant specific heats and molecular weight.

Formulas

p

p M*= +

+1

1 2

γγ ...(8.1)

12

0* 20

12 1 M

p 1 21p 1 M

γγ−⎡ ⎤γ −⎛ ⎞+⎜ ⎟⎢ ⎥+ γ ⎝ ⎠⎢ ⎥=

γ +⎢ ⎥+ γ ⎣ ⎦...(8.2)

T

T MM

*= +

+

FHG

IKJ

1

1 2

2

2γγ ...(8.3)


T

T

M M

M

0

0

2 2

2 2

2 1 11

2

1*

=+ + −F

HGIKJ

+

γ γ

γ

b g

e j...(8.4)

ρρ

γγ*

*

= = ++

c

c

M

M

1

1

2

2b g ...(8.5)

s s

cM

Mp

− = ++

FHG

IKJ

+*

ln 22

1

1

1

γγ

γγ

...(8.6)

F F= =* constant ...(8.7)

Q

c T

M

Mp

max =−

+

2 2

2

1

2 1

e jb gγ

...(8.8)

9. Prandtl-Meyer functions with Mach anglesSupersonic flow of a gas along a wall turning in the clock-wise direction is shown in figure 5.Mach wave and the Mach angle (α) are also shown.

Fig. 5. Prandtl-Meyer Expansion

Assumptions

(a) Perfect gas with constant specific heats and molecular weight,

(b) Isentropic flow.

�

Mach wave

M > 1

Wall

M + dM

Introduction 11

Formulas

sin α = 1

M...(9.1)

ω γγ

γγ

M M Mb g e j= +−

−+

− − −− −1

1

1

11 11 2 1 2tan tan ...(9.2)

10. Approximate velocities and Mach numbers of airAt low velocities (M < 0.30) correct values of the velocity of a gas can be determined by measur-

ing the dynamic head p p c021

2− =F

HGIKJρ . Conversely the value of the dynamic head required for

obtaining a given velocity can also be determined.

Knowledge of the temperature of the gas along with the dynamic head gives the value of MachNumber.

c RT p p= −20ρb g ...(10.1)

Mc

RT=

γb g1 2 ...(10.2)

For lower velocities dynamic head is measured in millimetres of water (mmW.G) on a verticalor inclined tube manometer. At higher velocities dynamic head is more conveniently measured inmillimeters of mercury (mm Hg).

Tables 10.1 and 10.2 give approximate velocities and Mach numbers at the prescribed valuesof pressure and temperature; the approximation is less realistic at higher Mach numbers on ac-count of increasing deviation of the actual flow from incompressible flow conditions. Therefore,at higher Mach numbers Table 10.2 gives only rough estimates of the velocity and Mach number.On account of this discrepancy the use of Table 10.3 is recommended at higher Mach numbers.

11. Measurement of low pressures on inclined manometers.Low pressures and velocities of gases can be measured by magnifying the actual manometerdeflection (h) many times. This is done by inclining the manometer tube/tubes by angles shown intable 11.

Actual (h) and magnified ( ) values of the deflection are shown in figure 6.

Fig. 6. Actual (h) & Magnified ( ) Values of Deflections in Inclined Tube Manometer

�

h

Horizontal plane

Plane of the manometer tubes

�


12. Static pressure drop (pr = outlet pressure/inlet pressure) in gas pipe linesCompressors are required to raise the pressure of gases to compensate for the pressure drop in longpipe lines. In these tables values of static pressure drop in the pipe lines have been tabulated withthe assumption of isothermal flow. The flow has been further assumed to start with a low Machnumber of M1 = 0.02. Pressure drop has been expressed as pressure ratio

p p pr = 2 1.

Following relations have been used to compute the values of the pressure ratio:

p M f L Dr = φ γ1, , , ...(12.1)

p f L Dr = −0 02 2500 4. γ ...(12.2)

Following values of the parameters have been covered:

γ = 1 3 1 4. , ,

f = 0 002 0 0025 0 0030 0 0035 0 0040 0 0050. , . , . , . , . , .

L D = − ×10 2 105

These values of the pressure drop can be employed to obtain the magnitudes of the compressorpower.

13. Isentropic compressor powerThe ideal power required to drive a compressor is given by:

′ = ′ − ′ =P m c T Tp s2 1c h Isentropic power

′ = ′ ′ −−P m c T pp r11 1γ γb g

...(13.1)

The pressure ratio is given by

p p p T Tr s= = −exit pressure entry pressure 2 1 2 1

1b g b gγ γ ...(13.2)

For the sake of generality following values have been used in the calculations:

′ =c kJ kg Kp 1 0.

′ =T1 300 K

The actual power (for a given pressure ratio) can be calculated by multiplying the values in thetable by the factor k given by:

k c T T c Tp p= × ′ = ×1 0 1 1 0300η η ...(13.3)

where T1 = actual temperature of the gas at entry

η0 = overall efficiency of the compressor and motor (prime mover) together.

Introduction 13

14. Approximate values of pumping power for fans and blowersFans and blower of various sizes are required to raise the pressure of a gas from a few millimetresof W.G. to large values. Table 14 presents approximate values of the power requirement of fansand blowers in a wide range.

Power requirement for a specific application corresponding to values of Q and ∆p which donot appear in the tables can also be found with the help of available data.

Following formulas have been used:

Q m= ν ...(14.1)

P Q p= ⋅ ∆ η0 ...(14.2)

Overall all efficiency η0 = 0.65 (assumed)

P Q p kW= ⋅ ∆ 66 261. ...(14.3)

Q is in m3/s ; ∆p is in mm W.G.

15. Dimensions and unitsPrincipal dimensions and units occurring in gas dynamics are given in Table 15.1. Table 15.2 givesthe conversion of some units.


TABLE 1.1

Gas dynamic properties of air at various temperatures

T cp

cv

γ R a w T A p0 0∗

K kJ/kg K kJ/kg K kJ/kg K m/s

273 1.004 0.716 1.402 0.288 332.0 0.0404

373 1.010 0.723 1.397 0.288 387.4 0.0403

473 1.025 0.737 1.390 0.288 435.3 0.0402

573 1.045 0.758 1.379 0.287 476.2 0.0400

673 1.069 0.782 1.367 0.287 513.8 0.0400

773 1.092 0.806 1.355 0.286 547.3 0.0400

873 1.115 0.828 1.347 0.287 580.9 0.0398

973 1.136 0.849 1.338 0.287 611.3 0.0396

1073 1.154 0.867 1.331 0.287 640.2 0.0396

1173 1.170 0.884 1.324 0.286 666.3 0.0397

1273 1.185 0.898 1.320 0.287 694.3 0.0395

1373 1.197 0.910 1.315 0.287 719.9 0.0395

1473 1.208 0.920 1.313 0.288 746.3 0.0394

1573 1.218 0.930 1.309 0.288 770.3 0.0393

1673 1.227 0.940 1.305 0.287 791.7 0.0394

Gas Dynamic Properties of Some Gases and Vapours 15

TABLE 1.2

Gas dynamic properties of hydrogen at various temperatures

T cp cv γ R a w T A p0 0∗


273 14.195 10.071 1.409 4.124 1260.0 0.01068

373 14.448 10.324 1.399 4.124 1467.2 0.01066

473 14.503 10.380 1.397 4.123 1650.7 0.1065

573 14.534 10.409 1.396 4.125 1816.7 0.01065

673 14.581 10.456 1.395 4.125 1967.9 0.01065

773 14.663 10.538 1.391 4.125 2106.4 0.01064

873 14.779 10.654 1.387 4.125 2235.0 0.01059

973 14.930 10.805 1.381 4.125 2355.0 0.0106

1073 15.115 10.991 1.375 4.124 2466.8 0.01059

1173 15.313 11.188 1.368 4.125 2573.4 0.01058

1273 15.518 11.393 1.362 4.125 2674.4 0.01055

1373 15.736 11.611 1.355 4.125 2770.5 0.01053

1473 15.950 11.825 1.349 4.125 2862.8 0.01052

1573 16.166 12.040 1.342 4.126 2952.0 0.0105

1673 16.370 12.245 1.337 4.125 3037.4 0.0105

1773 16.565 12.440 1.331 4.125 3120.7 0.01047


TABLE 1.3

Gas dynamic properties of carbon dioxide at various temperatures



273 0.815 0.626 1.302 0.189 259.2 0.0485

373 0.914 0.725 1.261 0.189 298.1 0.0481

473 0.993 0.804 1.237 0.189 332.6 0.0476

573 1.057 0.868 1.218 0.189 363.1 0.0473

673 1.110 0.921 1.205 0.189 391.5 0.0472

773 1.155 0.966 1.195 0.189 417.9 0.0470

873 1.192 1.003 1.188 0.189 442.8 0.0469

973 1.223 1.034 1.182 0.189 466.4 0.0469

1073 1.249 1.060 1.178 0.189 488.8 0.0467

1173 1.272 1.082 1.176 0.190 511.8 0.0466

1273 1.290 1.101 1.172 0.189 530.9 0.0467

1373 1.306 1.117 1.169 0.189 550.8 0.0466

1473 1.319 1.130 1.167 0.189 570.1 0.0466

1573 1.332 1.142 1.166 0.190 590.4 0.0465

1673 1.342 1.152 1.164 0.190 608.5 0.0465

1773 1.350 1.161 1.162 0.189 624.2 0.0465

1873 1.357 1.168 1.162 0.189 641.3 0.0465

1973 1.364 1.175 1.161 0.189 657.9 0.0466

2073 1.369 1.181 1.159 0.188 672.1 0.0465

2173 1.374 1.185 1.159 0.189 690.0 0.0464

Gas Dynamic Properties of Some Gases and Vapours 17

TABLE 1.4

Gas dynamic properties of superheated steam at p = 0.981 bar



413 1.997 1.482 1.347 0.515 535.0 0.0297433 1.985 1.485 1.336 0.500 537.8 0.0300453 1.975 1.490 1.325 0.485 539.6 0.0305473 1.980 1.500 1.320 0.480 547.4 0.0306493 1.985 1.505 1.318 0.480 558.5 0.0306

513 1.990 1.515 1.313 0.475 565.6 0.0307533 1.997 1.525 1.310 0.472 574.0 ‘0.0308553 2.000 1.536 1.302 0.464 578.0 0.0311573 2.013 1.548 1.300 0.465 588.6 0.0310593 2.025 1.556 1.300 0.469 601.6 0.0308

613 2.040 1.571 1.298 0.469 611.0 0.0308633 2.045 1.584 1.291 0.461 613.8 0.0310653 2.050 1.594 1.286 0.456 618.8 0.0311673 2.071 1.607 1.288 0.464 634.4 0.0308693 2.085 1.619 1.287 0.466 644.9 0.0308

At p = 9.81 bar

453 2.500 1.738 1.438 0.762 704.6 0.0250493 2.278 1.658 1.374 0.620 648.0 0.0273533 2.190 1.624 1.348 0.566 637.8 0.0283573 2.152 1.616 1.332 0.536 639.5 0.0291613 2.139 1.620 1.320 0.519 648.1 0.0294

653 2.135 1.633 1.307 0.502 654.6 0.0298693 2.144 1.650 1.299 0.494 667.0 0.0299733 2.156 1.671 1.290 0.485 677.2 0.0302773 2.173 1.691 1.285 0.482 691.9 0.0302813 2.194 1.717 1.278 0.477 703.9 0.0303

853 2.215 1.742 1.271 0.473 716.3 0.0304893 2.240 1.767 1.267 0.473 731.7 0.0304933 2.265 1.792 1.264 0.473 746.8 0.0304

TABLE 1.5

Gas dynamic properties of superheated Freon-21 at various temperatures



At p = 0.10 bar

243 0.531 0.447 1.187 0.084 155.7 0.0705283 0.578 0.496 1.165 0.082 164.4 0.0713323 0.622 0.540 1.152 0.082 174.7 0.0733363 0.661 0.579 1.142 0.082 184.3 0.0698403 0.695 0.615 1.130 0.080 190.9 0.0705

At p = 1.0 bar

293 0.616 0.521 1.182 0.095 181.4 0.0662313 0.630 0.539 1.168 0.091 182.5 0.0673333 0.646 0.557 1.159 0.089 185.4 0.0679353 0.662 0.575 1.151 0.087 188.0 0.0685373 0.678 0.593 1.143 0.085 190.4 0.0691

At p = 10.0 bar

333 0.812 0.644 1.261 0.168 265.6 0.0508383 0.799 0.645 1.239 0.154 270.3 0.0527393 0.792 0.647 1.224 0.145 264.1 0.0541403 0.786 0.651 1.207 0.135 256.3 0.0558413 0.783 0.655 1.195 0.128 251.4 0.0566423 0.782 0.658 1.188 0.124 249.7 0.0580433 0.782 0.663 1.179 0.119 246.5 0.0590443 0.782 0.667 1.172 0.115 244.4 0.0599453 0.783 0.672 1.165 0.111 242.1 0.0608463 0.785 0.676 1.161 0.109 242.1 0.0614473 0.786 0.681 1.154 0.105 239.4 0.0624

At p = 20.0 bar

403 0.984 0.703 1.396 0.281 397.6 0.0408423 0.909 0.697 1.304 0.212 342.0 0.0458443 0.872 0.696 1.252 0.176 312.5 0.0459463 0.852 0.699 1.219 0.153 293.8 0.0527473 0.846 0.701 1.207 0.145 287.7 0.0538


TABLE 2

Properties of standard atmosphere

Z t T a p ρ µ

m °C K m/s bar kg/m3 Ns

m2610×

0 15.00 288.15 340.5 1.0133 1.225 17.93

100 14.35 287.50 340.0 1.000 1.213 17.90

200 13.70 286.85 339.7 0.989 1.202 17.88

300 13.05 286.20 339.5 0.978 1.190 17.84

400 12.40 285.55 338.9 0.966 1.178 17.80

500 11.75 284.90 338.5 0.955 1.167 17.77

600 11.10 284.25 338.0 0.943 1.156 17.74

700 10.45 283.60 337.6 0.932 1.145 17.70

800 9.80 282.95 337.3 0.921 1.134 17.66

900 9.15 282.30 337.0 0.909 1.123 17.64

1000 8.50 281.65 336.6 0.898 1.111 17.61

1100 7.85 281.00 336.1 0.888 1.100 17.58

1200 7.20 280.35 335.7 0.877 1.090 17.55

1300 6.55 279.70 335.4 0.866 1.079 17.50

1400 5.90 279.05 335.0 0.856 1.070 17.48

1500 5.25 278.40 334.5 0.845 1.058 17.45

1600 4.60 277.75 334.1 0.835 1.048 17.40

1700 3.95 277.10 333.7 0.825 1.037 17.38

1800 3.30 276.45 333.4 0.815 1.027 17.35

1900 2.65 275.80 333.0 0.805 1.017 17.31

2000 2.00 275.15 332.6 0.795 1.006 17.28

2100 1.35 274.50 332.2 0.785 0.996 17.25

2200 0.70 273.85 332.0 0.775 0.986 17.22

2300 0.05 273.20 331.5 0.766 0.976 17.17

2400 −0.60 272.55 331.0 0.756 0.967 17.14

2500 −1.25 271.90 330.6 0.747 0.957 17.12

Properties of Standard Atmosphere 19

TABLE 2

Properties of standard atmosphere

Z t T a p ρ µ

m °C K m/s bar kg/m3 Ns

m2610×

2600 -1.90 271.25 330.2 0.737 0.948 17.092700 -2.55 270.60 329.7 0.727 0.938 17.052800 -3.20 269.95 329.4 0.719 0.929 17.012900 -3.85 269.30 329.0 0.710 0.920 16.983000 -4.50 268.65 328.7 0.701 0.909 16.95

3500 -7.75 265.40 326.7 0.657 0.863 16.804000 -11.00 262.15 324.7 0.616 0.819 16.624500 -14.25 258.90 322.7 0.577 0.777 16.455000 -17.50 255.65 320.7 0.540 0.736 16.27

5500 -20.75 252.40 318.7 0.505 0.697 16.106000 -24.00 249.15 316.5 0.472 0.660 15.946500 -27.25 245.90 314.5 0.440 0.624 15.767000 -30.50 242.65 312.4 0.411 0.589 15.60

7500 -33.75 239.40 310.03 0.383 0.557 15.428000 -37.00 236.15 308.2 0.356 0.525 15.248500 -40.25 232.90 306.0 0.331 0.495 15.069000 -43.50 229.65 304.0 0.307 0.466 14.89

9500 -46.75 226.40 301.8 0.285 0.439 14.7210000 -50.00 223.15 299.6 0.264 0.413 14.5411000 -56.50 216.65 295.2 0.226 0.364 14.1712000 -56.50 216.65 295.2 0.193 0.311 14.17

13000 -56.50 216.65 295.2 0.165 0.266 14.1714000 -56.50 216.65 295.2 0.141 0.227 14.1715000 -56.50 216.65 295.2 0.120 0.194 14.1716000 -56.50 216.65 295.2 0.103 0.165 14.17

17000 -56.50 216.65 295.2 0.0878 0.141 14.1718000 -56.50 216.65 295.2 0.0751 0.121 14.1719000 -56.50 216.65 295.2 0.0641 0.103 14.1720000 -56.60 216.65 295.2 0.0548 0.088 14.17


TABLE 3.1

Isentropic flow of perfect gases (γ =γ =γ =γ =γ = 1.3)

M M* T/T0 p/p0 A/A* F/F*A

A

p

p*0

0.00 0.000 1.000 1.000 ∞ ∞ ∞0.02 0.0214 0.999 0.999 29.268 23.324 29.2610.04 0.0429 0.999 0.998 14.644 11.679 14.6290.06 0.064 0.999 0.997 9.774 7.805 9.7510.08 0.086 0.999 0.995 7.342 5.874 7.3120.10 0.107 0.998 0.994 5.885 4.720 5.848

0.12 0.1285 0.998 0.991 4.917 3.954 4.8720.14 0.150 0.997 0.987 4.228 3.410 4.1740.16 0.171 0.996 0.984 3.712 3.005 3.6510.18 0.193 0.995 0.979 3.312 2.693 3.2430.20 0.214 0.994 0.975 2.994 2.445 2.917

0.22 0.235 0.993 0.969 2.735 2.245 2.6510.24 0.256 0.991 0.963 2.520 2.079 2.4280.26 0.277 0.989 0.957 2.340 1.941 2.2390.28 0.298 0.988 0.951 2.186 1.824 2.0780.30 0.319 0.987 0.944 2.054 1.724 1.938

0.32 0.341 0.985 0.936 1.939 1.638 1.8740.34 0.362 0.983 0.928 1.838 1.564 1.7590.36 0.383 0.981 0.920 1.750 1.498 1.6100.38 0.403 0.979 0.911 1.672 1.442 1.5230.40 0.424 0.977 0.902 1.602 1.391 1.446

0.42 0.444 0.974 0.893 1.540 1.347 1.3430.44 0.465 0.972 0.883 1.484 1.308 1.2810.46 0.486 0.969 0.873 1.434 1.273 1.2250.48 0.506 0.967 0.863 1.388 1.241 1.1730.50 0.526 0.964 0.853 1.348 1.213 1.149

0.52 0.547 0.961 0.842 1.310 1.188 1.1030.54 0.567 0.958 0.831 1.277 1.165 1.0610.56 0.587 0.955 0.819 1.246 1.145 1.0210.58 0.607 0.952 0.807 1.218 1.127 0.9840.60 0.627 0.948 0.796 1.193 1.111 0.950

Isentropic Flow of Perfect Gases 21

TABLE 3.1

Isentropic flow of perfect gases (γγγγγ = 1.3)

M M* T/T0

p/p0

A/A* F/F*A

A

p

p*0

0.62 0.646 0.945 0.784 1.170 1.097 0.9180.64 0.666 0.942 0.772 1.149 1.084 0.8870.66 0.686 0.938 0.760 1.130 1.072 0.8590.68 0.705 0.935 0.748 1.113 1.062 0.8320.70 0.725 0.932 0.735 1.097 1.052 0.807

0.72 0.744 0.928 0.723 1.083 1.044 0.7830.74 0.763 0.924 0.710 1.070 1.037 0.7600.76 0.782 0.920 0.697 1.059 1.030 0.7380.78 0.800 0.916 0.685 1.048 1.025 0.7180.80 0.820 0.912 0.673 1.039 1.020 0.698

0.82 0.838 0.908 0.659 1.032 1.015 0.6800.84 0.857 0.904 0.647 1.025 1.012 0.6630.86 0.875 0.900 0.634 1.018 1.009 0.6450.88 0.893 0.896 0.621 1.013 1.006 0.6290.90 0.911 0.892 0.608 1.009 1.004 0.614

0.92 0.929 0.887 0.596 1.006 1.003 0.5990.94 0.947 0.883 0.583 1.003 1.002 0.5850.96 0.965 0.878 0.571 1.001 1.0006 0.5710.98 0.983 0.874 0.558 1.0004 1.0002 0.5581.00 1.000 0.870 0.546 1.000 1.000 0.546

1.02 1.017 0.865 0.533 1.0003 1.0001 0.5341.04 1.035 0.860 0.521 1.001 1.0006 0.5221.06 1.052 0.856 0.509 1.003 1.001 0.5111.08 1.068 0.851 0.497 1.005 1.002 0.5001.10 1.085 0.846 0.485 1.008 1.003 0.489

1.12 1.102 0.842 0.474 1.012 1.005 0.4791.14 1.118 0.837 0.462 1.016 1.006 0.4691.16 1.135 0.832 0.451 1.021 1.008 0.4601.18 1.151 0.827 0.439 1.026 1.010 0.4501.20 1.167 0.822 0.428 1.032 1.012 0.442


TABLE 3.1


M M* T/T0

p/p0

A/A* F/F*A

A

p

p*0

1.22 1.183 0.817 0.418 1.038 1.014 0.4341.24 1.198 0.813 0.407 1.045 1.016 0.4251.26 1.214 0.807 0.396 1.053 1.019 0.4171.28 1.230 0.803 0.386 1.062 1.022 0.4091.30 1.245 0.798 0.376 1.070 1.024 0.402

1.32 1.260 0.793 0.365 1.080 1.027 0.3981.34 1.276 0.788 0.356 1.089 1.029 0.3871.36 1.291 0.783 0.346 1.100 1.033 0.3811.38 1.305 0.778 0.337 1.111 1.036 0.3741.40 1.320 0.773 0.327 1.123 1.039 0.367

1.42 1.334 0.768 0.318 1.135 1.042 0.3611.44 1.348 0.763 0.309 1.148 1.045 0.3521.46 1.363 0.758 0.300 1.161 1.048 0.3461.48 1.377 0.752 0.292 1.175 1.051 0.3401.50 1.391 0.748 0.284 1.189 1.055 0.337

1.52 1.405 0.743 0.275 1.205 1.058 0.3321.54 1.418 0.737 0.267 1.220 1.062 0.3261.56 1.432 0.732 0.259 1.237 1.065 0.3211.58 1.445 0.727 0.252 1.254 1.068 0.3161.60 1.458 0.723 0.245 1.271 1.072 0.310

1.62 1.472 0.717 0.237 1.289 1.075 0.3061.64 1.485 0.712 0.230 1.308 1.079 0.3011.66 1.497 0.707 0.223 1.328 1.083 0.2961.68 1.510 0.703 0.216 1.348 1.086 0.2921.70 1.523 0.698 0.210 1.369 1.089 0.288

1.72 1.535 0.693 0.203 1.391 1.093 0.2831.74 1.547 0.687 0.197 1.413 1.097 0.2791.76 1.559 0.683 0.191 1.436 1.100 0.2751.78 1.572 0.678 0.185 1.459 1.104 0.2711.80 1.584 0.673 0.180 1.484 1.108 0.267



TABLE 3.1



A

p

p*0

1.82 1.595 0.688 0.174 1.509 1.111 0.2631.84 1.607 0.663 0.168 1.535 1.115 0.2591.86 1.618 0.658 0.163 1.562 1.118 0.2551.88 1.629 0.654 0.158 1.589 1.122 0.2521.90 1.641 0.648 0.153 1.618 1.125 0.248

1.92 1.652 0.644 0.148 1.647 1.128 0.2451.94 1.663 0.639 0.144 1.678 1.132 0.2411.96 1.674 0.634 0.139 1.708 1.136 0.2381.98 1.685 0.629 0.135 1.740 1.139 0.2352.00 1.696 0.625 0.131 1.773 1.143 0.231

2.02 1.706 0.620 0.127 1.807 1.146 0.2282.04 1.717 0.616 0.122 1.842 1.150 0.2252.06 1.727 0.611 0.118 1.877 1.153 0.2222.08 1.737 0.606 0.114 1.914 1.156 0.2192.10 1.747 0.602 0.111 1.951 1.160 0.216

2.12 1.757 0.597 0.107 1.990 1.163 0.2132.14 1.767 0.593 0.104 2.029 1.166 0.2112.16 1.776 0.588 0.100 2.071 1.169 0.2082.18 1.786 0.583 0.097 2.133 1.173 0.2052.20 1.796 0.579 0.094 2.156 1.175 0.203

2.22 1.805 0.575 0.091 2.199 1.179 0.2002.24 1.815 0.571 0.088 2.245 1.183 0.1972.26 1.824 0.566 0.085 2.292 1.186 0.1952.28 1.833 0.562 0.082 2.339 1.189 0.1922.30 1.842 0.558 0.080 2.388 1.192 0.190

2.32 1.851 0.553 0.077 2.439 1.196 0.1872.34 1.859 0.549 0.074 2.490 1.198 0.1852.36 1.868 0.545 0.072 2.543 1.202 0.1832.38 1.877 0.541 0.069 2.598 1.205 0.1802.40 1.885 0.537 0.067 2.654 1.208 0.178


TABLE 3.1



A

p

p*0

2.42 1.894 0.532 0.065 2.711 1.211 0.1762.44 1.902 0.528 0.063 2.769 1.214 0.1742.46 1.910 0.524 0.061 2.829 1.217 0.1722.48 1.918 0.520 0.059 2.891 1.219 0.1702.50 1.926 0.516 0.057 2.954 1.223 0.168

2.52 1.934 0.512 0.055 3.019 1.226 0.1662.54 1.942 0.508 0.053 3.086 1.228 0.1642.56 1.950 0.504 0.051 3.154 1.231 0.1622.58 1.957 0.500 0.049 3.224 1.234 0.1602.60 1.965 0.496 0.048 3.295 1.237 0.158

2.62 1.972 0.493 0.046 3.370 1.240 0.1572.64 1.980 0.489 0.045 3.444 1.242 0.1552.66 1.987 0.485 0.043 3.521 1.245 0.1532.68 1.994 0.481 0.042 3.600 1.248 0.1512.70 2.001 0.478 0.041 3.681 1.250 0.150

2.72 2.008 0.474 0.039 3.764 1.253 0.1482.74 2.015 0.470 0.038 3.849 1.256 0.1462.76 2.022 0.467 0.037 3.936 1.258 0.1452.78 2.029 0.463 0.036 4.025 1.261 0.1432.80 2.036 0.460 0.034 4.116 1.264 0.142

2.82 2.042 0.456 0.033 4.210 1.266 0.1402.84 2.048 0.453 0.032 4.306 1.268 0.1382.86 2.055 0.449 0.031 4.404 1.271 0.1372.88 2.062 0.446 0.030 4.504 1.273 0.1352.90 2.068 0.442 0.029 4.607 1.275 0.134

2.92 2.074 0.439 0.028 4.713 1.278 0.1332.94 2.081 0.435 0.027 4.821 1.281 0.1312.96 2.087 0.432 0.026 4.931 1.283 0.1292.98 2.093 0.429 0.0256 5.044 1.285 0.1283.00 2.099 0.426 0.025 5.160 1.288 0.127


TABLE 3.1



A

p

p*0

3.05 2.113 0.417 0.0227 5.461 1.293 0.1243.10 2.127 0.409 0.0209 5.781 1.298 0.1213.15 2.141 0.402 0.0193 6.119 1.304 0.1183.20 2.155 0.394 0.0177 6.477 1.309 0.1153.25 2.168 0.387 0.0163 6.857 1.315 0.112

3.30 2.181 0.379 0.0151 7.258 1.319 0.1093.35 2.193 0.373 0.0139 7.684 1.325 0.1073.40 2.205 0.366 0.0128 8.133 1.329 0.1043.45 2.217 0.359 0.0118 8.608 1.334 0.1013.50 2.228 0.352 0.011 9.110 1.338 0.099

3.55 2.239 0.346 0.010 9.640 1.343 0.0973.60 2.250 0.339 0.0093 10.200 1.347 0.0953.65 2.261 0.334 0.0086 10.792 1.351 0.0933.70 2.271 0.327 0.0079 11.416 1.356 0.0923.75 2.281 0.322 0.0073 12.075 1.360 0.088

3.80 2.290 0.316 0.0068 12.769 1.363 0.0863.85 2.299 0.310 0.0063 13.502 1.367 0.0853.90 2.308 0.305 0.0058 14.274 1.371 0.0833.95 2.318 0.299 0.0054 15.087 1.375 0.0814.00 2.326 0.294 0.005 15.940 1.378 0.079

4.05 2.335 0.289 4.6 × 10−3 16.846 1.382 0.0774.10 2.343 0.284 4.27 × 10−3 17.796 1.385 0.0764.15 2.351 0.279 3.9 × 10−3 18.796 1.388 0.0754.20 2.359 0.274 3.68 × 10−3 19.850 1.391 0.0734.25 2.367 0.270 3.41 × 10−3 20.953 1.395 0.072

4.30 2.374 0.265 3.17 × 10−3 22.116 1.397 0.0704.35 2.381 0.261 2.94 × 10−3 23.338 1.400 0.0684.40 2.383 0.256 2.73 × 10−3 24.622 1.403 0.0674.45 2.395 0.252 2.54 × 10−3 25.971 1.406 0.0664.50 2.402 0.248 2.36 × 10−3 27.390 1.408 0.065


TABLE 3.1



A

p

p*0

4.55 2.408 0.243 2.2 × 10−3 28.874 1.412 0.063

4.60 2.415 0.240 2.05 × 10−3 30.434 1.414 0.062

4.65 2.421 0.235 1.91 × 10−3 32.070 1.417 0.061

4.70 2.427 0.232 1.77 × 10−3 33.786 1.420 0.060

4.75 2.433 0.228 1.65 × 10−3 35.585 1.422 0.059

4.80 2.438 0.224 1.54 × 10−3 37.472 1.424 0.058

4.85 2.444 0.221 1.44 × 10−3 39.448 1.427 0.057

4.90 2.450 0.217 1.34 × 10−3 41.518 1.429 0.055

4.95 2.455 0.214 1.25 × 10−3 43.687 1.431 0.0547

5.00 2.460 0.211 1.17 × 10−3 45.950 1.433 0.054

6.00 2.543 0.156 32.0 × 10−5 120.100 1.468 0.038

7.00 2.598 0.119 10.0 × 10−5 285.300 1.491 0.029

8.00 2.635 0.094 3.6 × 10−5 625.000 1.508 0.023

9.00 2.662 0.076 1.4 × 10−5 1275.000 1.520 0.018

10.00 2.680 0.063 0.6 × 10−5 2438.000 1.527 0.015

∞ 2.769 0 0 ∞ 1.565 0


TABLE 3.2



A

p

p*0

0.00 0.000 1.000 1.000 ∞ ∞ ∞0.01 0.011 0.999 0.999 57.874 45.65 57.8690.02 0.022 0.999 0.999 28.942 22.834 28.9340.03 0.033 0.999 0.999 19.300 15.232 19.2880.04 0.044 0.999 0.998 14.482 11.435 14.4650.05 0.0548 0.999 0.998 11.592 9.158 11.571

0.06 0.0657 0.999 0.997 9.666 7.643 9.6420.07 0.0766 0.999 0.996 8.292 6.562 8.2630.08 0.0876 0.998 0.995 7.262 5.753 7.2290.09 0.0985 0.998 0.994 6.461 5.125 6.4250.10 0.1094 0.998 0.993 5.822 4.624 5.781

0.11 0.1204 0.9976 0.992 5.299 4.215 5.2550.12 0.1313 0.9971 0.989 4.864 3.875 4.8150.13 0.1422 0.9966 0.988 4.497 3.588 4.4450.14 0.1531 0.996 0.986 4.182 3.343 4.1260.15 0.1640 0.995 0.984 3.910 3.132 3.849

0.16 0.1750 0.9949 0.982 3.673 2.947 3.6070.17 0.186 0.9943 0.980 3.464 2.786 3.3940.18 0.1965 0.994 0.978 3.278 2.642 3.2050.19 0.207 0.993 0.975 3.112 2.515 3.0350.20 0.218 0.992 0.973 2.964 2.400 2.882

0.21 0.229 0.991 0.9697 2.829 2.297 2.7440.22 0.239 0.990 0.967 2.707 2.205 2.6180.23 0.251 0.989 0.964 2.597 2.120 2.5030.24 0.261 0.988 0.961 2.495 2.043 2.3980.25 0.272 0.987 0.957 2.403 1.973 2.300

0.26 0.283 0.986 0.954 2.317 1.909 2.2110.27 0.294 0.9855 0.951 2.238 1.849 2.1280.28 0.304 0.985 0.947 2.166 1.795 2.0510.29 0.315 0.983 0.943 2.098 1.745 1.9790.30 0.325 0.982 0.939 2.035 1.698 1.912


TABLE 3.2



A

p

p*0

0.31 0.336 0.981 0.936 1.976 1.655 1.849

0.32 0.347 0.979 0.932 1.922 1.614 1.7900.33 0.357 0.978 0.927 1.871 1.577 1.7350.34 0.368 0.977 0.923 1.823 1.542 1.6830.35 0.378 0.976 0.918 1.778 1.509 1.634

0.36 0.389 0.975 0.914 1.735 1.479 1.5870.37 0.400 0.973 0.909 1.696 1.450 1.5430.38 0.410 0.972 0.905 1.658 1.424 1.5010.39 0.421 0.970 0.900 1.623 1.398 1.4620.40 0.431 0.969 0.895 1.590 1.375 1.424

0.41 0.442 0.967 0.891 1.558 1.353 1.3880.42 0.452 0.966 0.886 1.529 1.332 1.3540.43 0.463 0.964 0.881 1.500 1.312 1.3220.44 0.473 0.963 0.876 1.474 1.294 1.2900.45 0.483 0.961 0.870 1.448 1.276 1.261

0.46 0.494 0.959 0.865 1.425 1.259 1.2320.47 0.504 0.958 0.859 1.402 1.244 1.2050.48 0.514 0.956 0.854 1.380 1.230 1.1780.49 0.524 0.954 0.848 1.360 1.216 1.1540.50 0.534 0.952 0.843 1.340 1.203 1.129

0.51 0.545 0.951 0.837 1.321 1.190 1.1060.52 0.555 0.948 0.832 1.303 1.178 1.0840.53 0.565 0.947 0.826 1.287 1.167 1.0620.54 0.575 0.945 0.820 1.270 1.157 1.0420.55 0.585 0.943 0.814 1.255 1.147 1.022

0.56 0.595 0.941 0.808 1.240 1.138 1.0020.57 0.605 0.939 0.802 1.226 1.129 0.9840.58 0.615 0.937 0.796 1.213 1.121 0.9660.59 0.625 0.935 0.790 1.200 1.113 0.9480.60 0.635 0.933 0.784 1.188 1.105 0.932


TABLE 3.2



A

p

p*0

0.61 0.645 0.931 0.778 1.177 1.098 0.9150.62 0.655 0.928 0.772 1.166 1.091 0.9000.63 0.664 0.926 0.765 1.155 1.085 0.8850.64 0.674 0.924 0.759 1.145 1.078 0.8690.65 0.684 0.922 0.753 1.136 1.073 0.855

0.66 0.693 0.919 0.746 1.126 1.068 0.8410.67 0.703 0.917 0.740 1.118 1.063 0.8270.68 0.713 0.915 0.734 1.109 1.058 0.8140.69 0.722 0.913 0.727 1.102 1.053 0.8020.70 0.732 0.911 0.721 1.094 1.049 0.789

0.71 0.741 0.908 0.714 1.087 1.045 0.7770.72 0.751 0.906 0.708 1.081 1.041 0.7650.73 0.760 0.904 0.701 1.074 1.038 0.7530.74 0.769 0.901 0.695 1.068 1.035 0.7420.75 0.779 0.899 0.688 1.062 1.031 0.732

0.76 0.788 0.896 0.682 1.057 1.028 0.7210.77 0.797 0.894 0.676 1.052 1.026 0.7110.78 0.807 0.892 0.669 1.047 1.023 0.7000.79 0.816 0.889 0.663 1.043 1.021 0.6910.80 0.825 0.886 0.656 1.038 1.018 0.681

0.81 0.834 0.884 0.649 1.034 1.017 0.6720.82 0.843 0.881 0.643 1.031 1.015 0.6630.83 0.852 0.879 0.636 1.027 1.013 0.6540.84 0.861 0.876 0.630 1.024 1.011 0.6450.85 0.870 0.874 0.624 1.021 1.009 0.636

0.86 0.879 0.871 0.617 1.018 1.008 0.6280.87 0.888 0.868 0.610 1.015 1.007 0.6200.88 0.897 0.869 0.604 1.013 1.006 0.6120.89 0.906 0.863 0.598 1.011 1.005 0.6040.90 0.915 0.861 0.591 1.009 1.004 0.596


TABLE 3.2



A

p

p*0

0.91 0.923 0.858 0.585 1.007 1.003 0.5890.92 0.932 0.855 0.578 1.006 1.0025 0.5820.93 0.941 0.853 0.572 1.004 1.002 0.5750.94 0.949 0.849 0.566 1.003 1.0014 0.5670.95 0.958 0.847 0.559 1.002 1.0009 0.561

0.96 0.966 0.844 0.553 1.0014 1.0006 0.5540.97 0.975 0.842 0.547 1.0008 1.0003 0.5470.98 0.983 0.839 0.541 1.0003 1.0001 0.5410.99 0.992 0.836 0.534 1.0001 1.000 0.5351.00 1.000 0.834 0.528 1.000 1.000 0.528

1.01 1.0083 0.831 0.522 1.0001 1.000 0.5221.02 1.017 0.828 0.516 1.0003 1.0001 0.5161.03 1.025 0.825 0.510 1.0007 1.0003 0.5101.04 1.033 0.822 0.504 1.0013 1.0005 0.5051.05 1.041 0.819 0.498 1.002 1.0008 0.498

1.06 1.049 0.816 0.492 1.003 1.0012 0.4931.07 1.057 0.814 0.486 1.004 1.0016 0.4881.08 1.065 0.811 0.480 1.005 1.002 0.4831.09 1.073 0.808 0.474 1.006 1.0025 0.4771.10 1.081 0.805 6.468 1.008 1.003 0.472

1.11 1.089 0.802 0.463 1.009 1.0036 0.4671.12 1.097 0.799 0.457 1.011 1.004 0.4621.13 1.105 0.796 0.451 1.013 1.005 0.4571.14 1.113 0.794 0.445 1.015 1.0057 0.4521.15 1.120 0.791 0.439 1.017 1.006 0.447

1.16 1.128 0.788 0.434 1.019 1.007 0.4431.17 1.136 0.785 0.428 1.022 1.008 0.4381.18 1.143 0.782 0.423 1.025 1.009 0.4341.19 1.151 0.779 0.418 1.028 1.010 0.4291.20 1.158 0.776 0.412 1.030 1.011 0.425


TABLE 3.2



A

p

p*0

1.21 1.166 0.774 0.407 1.034 1.012 0.4211.22 1.173 0.771 0.402 1.037 1.013 0.4161.23 1.181 0.768 0.396 1.039 1.014 0.4121.24 1.188 0.765 0.391 1.043 1.015 0.4081.25 1.195 0.762 0.386 1.047 1.016 0.404

1.26 1.203 0.759 0.381 1.050 1.017 0.4001.27 1.210 0.756 0.376 1.054 1.018 0.3961.28 1.217 0.753 0.371 1.058 1.019 0.3921.29 1.224 0.750 0.366 1.062 1.021 0.3881.30 1.231 0.747 0.361 1.066 1.022 0.385

1.31 1.238 0.745 0.356 1.071 1.023 0.3811.32 1.245 0.742 0.351 1.075 1.024 0.3781.33 1.252 0.738 0.346 1.079 1.025 0.3741.34 1.259 0.736 0.342 1.084 1.027 0.3701.35 1.266 0.733 0.337 1.089 1.028 0.367

1.36 1.273 0.730 0.332 1.094 1.029 0.3641.37 1.280 0.727 0.328 1.099 1.031 0.3601.38 1.286 0.724 0.323 1.104 1.032 0.3571.39 1.293 0.721 0.318 1.110 1.033 0.3541.40 1.300 0.718 0.314 1.115 1.035 0.350

1.41 1.306 0.716 0.309 1.121 1.036 0.3471.42 1.313 0.713 0.305 1.126 1.037 0.3441.43 1.319 0.709 0.301 1.132 1.039 0.3411.44 1.326 0.706 0.297 1.138 1.040 0.3381.45 1.333 0.704 0.293 1.144 1.042 0.335

1.46 1.339 0.701 0.288 1.150 1.043 0.3321.47 1.345 0.698 0.285 1.156 1.044 0.3291.48 1.352 0.695 0.280 1.163 1.046 0.3261.49 1.358 0.693 0.276 1.170 1.047 0.3231.50 1.365 0.689 0.272 1.176 1.048 0.320


TABLE 3.2



A

p

p*0

1.51 1.371 0.687 0.268 1.183 1.050 0.3171.52 1.377 0.684 0.265 1.189 1.052 0.3151.53 1.383 0.681 0.261 1.197 1.053 0.3121.54 1.389 0.678 0.257 1.204 1.055 0.3091.55 1.395 0.675 0.253 1.211 1.056 0.307

1.56 1.402 0.673 0.249 1.219 1.057 0.3041.57 1.408 0.670 0.246 1.227 1.059 0.3021.58 1.414 0.667 0.242 1.234 1.061 0.3001.59 1.419 0.664 0.238 1.242 1.062 0.2971.60 1.425 0.661 0.235 1.250 1.063 0.294

1.61 1.431 0.658 0.232 1.258 1.065 0.2921.62 1.437 0.656 0.228 1.267 1.067 0.2891.63 1.443 0.653 0.225 1.275 1.068 0.2871.64 1.448 0.650 0.222 1.284 1.069 0.2851.65 1.454 0.647 0.218 1.292 1.071 0.282

1.66 1.460 0.645 0.215 1.301 1.073 0.2801.67 1.466 0.642 0.212 1.310 1.074 0.2781.68 1.471 0.639 0.208 1.319 1.076 0.2751.69 1.477 0.636 0.206 1.328 l.077 0.2731.70 1.483 0.634 0.203 1.337 1.078 0.271

1.71 1.485 0.631 0.200 1.347 1.080 0.2691.72 1.494 0.628 0.196 1.357 1.082 0.2671.73 1.499 0.626 0.194 1.367 1.083 0.2651.74 1.504 0.623 0.191 1.376 1.085 0.2621.75 1.509 0.620 0.187 1.387 1.086 0.260

1.76 1.515 0.617 0.185 1.397 1.087 0.2581.77 1.520 0.615 0.182 1.407 1.089 0.2561.78 1.526 0.612 0.179 1.418 1.091 0.2541.79 1.531 0.609 0.177 1.428 1.092 0.2521.80 1.536 0.607 0.174 1.439 1.094 0.250


TABLE 3.2



A

p

p*0

1.81 1.541 0.604 0.171 1.450 1.095 0.2481.82 1.546 0.602 0.168 1.461 1.096 0.2471.83 1.551 0.598 0.166 1.472 1.098 0.2451.84 1.556 0.596 0.164 1.484 1.099 0.2431.85 1.561 0.594 0.161 1.495 1.100 0.241

1.86 1.566 0.591 0.158 1.507 1.102 0.2391.87 1.571 0.588 0.156 1.518 1.104 0.2371.88 1.576 0.585 0.154 1.531 1.105 0.2351.89 1.581 0.583 0.152 1.543 1.107 0.2341.90 1.586 0.581 0.149 1.555 1.108 0.232

1.91 1.591 0.578 0.147 1.568 1.110 0.2301.92 1.596 0.575 0.145 1.580 1.111 0.2281.93 1.600 0.573 0.142 1.593 1.113 0.2271.94 1.605 0.570 0.140 1.606 1.114 0.2251.95 1.609 0.568 0.138 1.619 1.116 0.223

1.96 1.615 0.566 0.136 1.633 1.117 0.2221.97 1.619 0.563 0.134 1.646 1.118 0.2201.98 1.624 0.561 0.132 1.659 1.120 0.2191.99 1.628 0.558 0.129 1.674 1.121 0.2172.00 1.633 0.555 0.128 1.687 1.123 0.216

2.01 1.638 0.553 0.126 1.702 1.124 0.2142.02 1.642 0.551 0.114 1.716 1.126 0.2132.03 1.646 0.548 0.122 1.731 1.127 0.2112.04 1.651 0.546 0.120 1.745 1.128 0.2092.05 1.655 0.543 0.118 1.760 1.130 0.208

2.06 1.660 0.541 0.116 1.775 1.131 0.2072.07 1.664 0.538 0.115 1.790 1.133 0.2052.08 1.668 0.536 0.113 1.805 1.134 0.2042.09 1.673 0.534 0.111 1.821 1.135 0.2022.10 1.677 0.531 0.109 1.837 1.137 0.200


TABLE 3.2



A

p

p*0

2.11 1.681 0.529 0.108 1.853 1.138 0.1992.12 1.685 0.527 0.106 1.869 1.139 0.1982.13 1.689 0.524 0.1043 1.885 1.141 0.1972.14 1.694 0.522 0.1027 1.902 1.142 0.1952.15 1.698 0.519 0.1010 1.918 1.143 0.194

2.16 1.702 0.517 0.0996 1.935 1.145 0.1932.17 1.706 0.515 0.0980 1.953 1.146 0.1912.18 1.710 0.513 0.0965 1.969 1.147 0.1902.19 1.714 0.510 0.095 1.987 1.148 0.1882.20 1.718 0.508 0.0935 2.005 1.150 0.187

2.21 1.722 0.506 0.0921 2.023 1.151 0.1862.22 1.726 0.504 0.0906 2.041 1.153 0.1852.23 1.730 0.501 0.0892 2.059 1.154 0.1842.24 1.734 0.499 0.0878 2.078 1.155 0.1832.25 1.737 0.497 0.0865 2.096 1.156 0.181

2.26 1.741 0.495 0.0851 2.115 1.158 0.1802.27 1.745 0.492 0.0838 2.135 1.159 0.1792.28 1.749 0.490 0.0825 2.154 1.160 0.1782.29 1.753 0.488 0.0812 2.173 1.162 0.17652.30 1.756 0.486 0.0800 2.193 1.163 0.175

2.31 1.760 0.484 0.0787 2.213 1.164 0.1742.32 1.764 0.482 0.0775 2.233 1.165 0.1732.33 1.767 0.479 0.0763 2.254 1.167 0.1722.34 1.771 0.477 0.0751 2.274 1.168 0.1712.35 1.775 0.475 0.0740 2.295 1.169 0.169

2.36 1.778 0.473 0.0728 2.316 1.170 0.1682.37 1.782 0.471 0.0717 2.338 1.172 0.1672.38 1.785 0.469 0.0706 2.359 1.173 0.1662.39 1.789 0.467 0.0695 2.381 1.174 0.1652.40 1.792 0.464 0.0684 2.403 1.175 0.164


TABLE 3.2



A

p

p*0

2.41 1.796 0.463 0.0673 2.425 1.176 0.1632.42 1.799 0.461 0.0663 2.448 1.178 0.1622.43 1.803 0.458 0.0653 2.471 1.179 0.1612.44 1.806 0.456 0.0643 2.494 1.180 0.1602.45 1.809 0.454 0.0633 2.517 1.181 0.159

2.46 1.813 0.452 0.0623 2.540 1.182 0.1582.47 1.816 0.450 0.0613 2.564 1.183 0.1572.48 1.819 0.448 0.0604 2.588 1.184 0.1562.49 1.823 0.446 0.0595 2.612 1.186 0.1552.50 1.826 0.444 0.0585 2.637 1.187 0.154

2.51 1.829 0.442 0.0576 2.662 1.188 0.1532.52 1.832 0.441 0.0567 2.686 1.189 0.1522.53 1.835 0.438 0.0558 2.712 1.190 0.1512.54 1.838 0.436 0.0550 2.737 1.191 0.1502.55 1.842 0.434 0.0542 2.763 1.192 0.149

2.56 1.845 0.433 0.0533 2.789 1.193 0.14872.57 1.848 0.431 0.0525 2.815 1.195 0.14752.58 1.851 0.429 0.0517 2.842 1.196 0.1472.59 1.854 0.427 0.0509 2.869 1.197 0.1462.60 1.857 0.425 0.0501 2.896 1.198 0.145

2.61 1.860 0.423 0.0494 2.923 1.199 0.1442.62 1.863 0.421 0.0486 2.951 1.200 0.1432.63 1.866 0.419 0.0478 2.980 1.201 0.14252.64 1.869 0.417 0.0471 3.007 1.202 0.1422.65 1.872 0.415 0.0464 3.306 1.203 0.141

2.66 1.875 0.414 0.0457 3.065 1.204 0.1402.67 1.878 0.412 0.0449 3.094 1.205 0.1392.68 1.881 0.410 0.0443 3.123 1.206 0.1382.69 1.884 0.408 0.0436 3.153 1.207 0.13752.70 1.886 0.407 0.043 3.183 1.208 0.137


TABLE 3.2



A

p

p*0

2.71 1.889 0.405 0.0423 3.213 1.209 0.1362.72 1.892 0.403 0.0417 3.244 1.210 0.1352.73 1.895 0.402 0.0410 3.275 1.211 0.1342.74 1.898 0.399 0.0404 3.306 1.212 0.13352.75 1.900 0.398 0.0398 3.337 1.213 0.133

2.76 1.903 0.396 0.0392 3.369 1.214 0.1322.77 1.906 0.395 0.0386 3.402 1.215 0.1312.78 1.908 0.393 0.0379 3.434 1.216 0.1302.79 1.911 0.391 0.0374 3.467 1.217 0.12972.80 1.914 0.389 0.0369 3.500 1.218 0.129

2.81 1.915 0.388 0.0363 3.534 1.219 0.1282.82 1.919 0.386 0.0357 3.567 1.220 0.12752.83 1.922 0.384 0.0352 3.602 1.221 0.1272.84 1.925 0.382 0.0347 3.636 1.222 0.1262.85 1.927 0.381 0.0342 3.670 1.223 0.125

2.86 1.929 0.379 0.0336 3.706 1.224 0.12452.87 1.932 0.378 0.0331 3.741 1.225 0.1242.88 1.935 0.376 0.0326 3.777 1.226 0.1232.89 1.937 0.375 0.0321 3.813 1.227 0.12252.90 1.940 0.373 0.0317 3.850 1.228 0.122

2.91 1.942 0.371 0.0312 3.887 1.229 0.1212.92 1.945 0.369 0.0307 3.924 1.230 0.1202.93 1.947 0.368 0.0303 3.961 1.2305 0.11982.94 1.950 0.367 0.0298 4.000 1.231 0.1192.95 1.952 0.365 0.0294 4.037 1.232 0.1185

2.96 1.955 0.363 0.0289 4.076 1.233 0.1182.97 1.957 0.362 0.0285 4.115 1.234 0.1172.98 1.959 0.360 0.0281 4.155 1.235 0.1172.99 1.962 0.358 0.0276 4.195 1.236 0.1163.00 1.964 0.357 0.0272 4.235 1.237 0.115


TABLE 3.2



A

p

p*0

3.05 1.976 0.349 0.0253 4.441 1.241 0.1123.10 1.987 0.342 0.0235 4.657 1.246 0.1093.15 1.997 0.335 0.0218 4.884 1.249 0.1063.20 2.008 0.328 0.0202 5.121 1.253 0.1043.25 2.018 0.321 0.0188 5.369 1.257 0.101

3.30 2.028 0.315 0.0175 5.628 1.260 0.0983.35 2.037 0.308 0.0163 5.900 1.264 0.0963.40 2.047 0.302 0.0151 6.184 1.268 0.09353.45 2.056 0.296 0.0141 6.480 1.272 0.0913.50 2.064 0.289 0.0131 6.789 1.274 0.089

3.55 2.073 0.284 12.2 × 10−3 7.113 1.278 0.0873.60 2.081 0.278 11.38 × 10−3 7.450 1.281 0.0853.65 2.089 0.273 10.62 × 10−3 7.802 1.284 0.0833.70 2.095 0.268 9.9 × 10−3 8.169 1.287 0.0813.75 2.104 0.262 9.24 × 10−3 8.552 1.289 0.079

3.80 2.111 0.257 8.63 × 10−3 8.950 1.292 0.0773.85 2.118 0.252 8.06 × 10−3 9.366 1.295 0.0753.90 2.125 0.247 7.53 × 10−3 9.800 1.298 0.07383.95 2.132 0.243 7.04 × 10−3 10.249 1.300 0.0724.00 2.138 0.238 6.592 × 10−3 10.719 1.303 0.071

4.05 2.145 0.234 6.16 × 10−3 11.207 1.305 0.0694.10 2.151 0.229 5.77 × 10−3 11.715 1.308 0.06764.15 2.156 0.235 5.4 × 10−3 12.243 1.310 0.0664.20 2.162 0.221 5.06 × 10−3 12.792 1.312 0.06484.25 2.168 0.217 4.74 × 10−3 13.362 1.315 0.0635

4.30 2.173 0.213 4.45 × 10−3 13.955 1.317 0.0624.35 2.178 0.209 4.17 × 10−3 14.571 1.319 0.06084.40 2.184 0.205 3.92 × 10−3 15.210 1.321 0.0604.45 2.188 0.202 3.68 × 10−3 15.872 1.323 0.0584.50 2.194 0.198 3.46 × 10−3 16.562 1.325 0.0575.00 2.236 0.167 1.89 × 10−3 25.000 1.341 0.047


TABLE 3.2



A

p

p*0

6.00 2.295 0.122 63.3 × 10-5 53.180 1.365 0.033

7.00 2.333 0.093 24.2 × 10−5 104.143 1.381 0.025

8.00 2.359 0.072 10.2 × 10−5 190.110 1.392 0.0195

9.00 2.377 0.058 4.74 × 10−5 327.189 1.399 0.0156

10.00 2.390 0.047 2.36 × 10−5 536.00 1.404 0.0126

∞ 2.45 0 0 ∞ 1.429 0

TABLE 3.3



A

p

p*0

0.00 0.000 1.000 1.000 ∞ ∞ ∞0.02 0.023 0.999 0.999 28.132 21.662 28.1130.04 0.046 0.999 0.998 14.078 10.851 14.0540.06 0.069 0.998 0.997 9.397 7.255 9.3660.08 0.092 0.997 0.995 7.061 5.464 7.0210.10 0.115 0.997 0.992 5.661 4.392 5.615

0.12 0.138 0.995 0.988 4.733 3.686 4.6750.14 0.161 0.994 0.984 4.070 3.184 4.0030.16 0.184 0.992 0.979 3.576 2.809 3.4990.18 0.207 0.989 0.973 3.193 2.522 3.1070.20 0.229 0.987 0.967 2.886 2.293 2.793

0.22 0.252 0.984 0.961 2.640 2.110 2.5350.24 0.275 0.981 0.954 2.435 1.958 2.3200.26 0.297 0.978 0.945 2.262 1.833 2.1380.28 0.319 0.975 0.937 2.115 1.725 1.9820.30 0.342 0.971 0.928 1.989 1.635 1.848

0.32 0.363 0.967 0.919 1.880 1.558 1.7280.34 0.385 0.963 0.909 1.784 1.491 1.6230.36 0.407 0.958 0.899 1.700 1.432 1.5290.38 0.428 0.954 0.889 1.626 1.381 1.4450.40 0.450 0.949 0.878 1.560 1.336 1.370

0.42 0.471 0.945 0.867 1.502 1.297 1.3000.44 0.493 0.939 0.855 1.448 1.262 1.2380.46 0.513 0.934 0.843 1.401 1.231 1.1810.48 0.534 0.928 0.831 1.358 1.203 1.1280.50 0.555 0.923 0.818 1.320 1.178 1.080

0.52 0.575 0.917 0.806 1.286 1.157 1.0350.54 0.595 0.911 0.793 1.254 1.137 0.9940.56 0.615 0.905 0.780 1.225 1.120 0.9550.58 0.635 0.899 0.767 1.200 1.105 0.9190.60 0.655 0.893 0.753 1.176 1.091 0.885


TABLE 3.3



A

p

p*0

0.62 0.674 0.886 0.739 1.154 1.085 0.8540.64 0.693 0.879 0.726 1.135 1.068 0.8240.66 0.712 0.873 0.712 1.118 1.058 0.7960.68 0.731 0.866 0.698 1.102 1.049 0.7690.70 0.750 0.859 0.685 1.087 1.042 0.745

0.72 0.768 0.853 0.671 1.075 1.035 0.7210.74 0.786 0.845 0.657 1.063 1.029 0.6980.76 0.804 0.838 0.644 1.053 1.024 0.6770.78 0.821 0.831 0.630 1.043 1.019 0.6570.80 0.839 0.823 0.616 1.035 1.016 0.638

0.82 0.856 0.817 0.603 1.028 1.012 0.6190.84 0.873 0.809 0.589 1.022 1.009 0.6020.86 0.889 0.802 0.570 1.016 1.007 0.5850.88 0.906 0.795 0.563 1.011 1.005 0.5690.90 0.922 0.787 0.550 1.008 1.003 0.555

0.92 0.938 0.779 0.537 1.005 1.002 0.5390.94 0.954 0.772 0.524 1.003 1.001 0.5250.96 0.969 0.765 0.511 1.001 1.0005 0.5120.98 0.985 0.757 0.499 1.0003 1.0001 0.4991.00 1.000 0.749 0.487 1.000 1.000 0.487

1.02 1.015 0.742 0.475 1.0003 1.0001 0.4751.04 1.029 0.735 0.463 1.001 1.0004 0.4631.06 1.044 0.727 0.451 1.003 1.001 0.4521.08 1.058 0.720 0.440 1.005 1.0015 0.4421.10 1.072 0.712 0.428 1.007 1.002 0.432

1.12 1.086 0.705 0.418 1.010 1.003 0.4211.14 1.100 0.698 0.407 1.014 1.005 0.4121.16 1.113 0.690 0.396 1.018 1.006 0.4021.18 1.126 0.682 0.385 1.022 1.007 0.3931.20 1.139 0.675 0.375 1.027 1.008 0.385


TABLE 3.3



A

p

p*0

1.22 1.152 0.668 0.365 1.032 1.010 0.3761.24 1.164 0.661 0.355 1.038 1.011 0.3681.26 1.176 0.654 0.346 1.044 1.013 0.3611.28 1.188 0.647 0.336 1.051 1.015 0.3531.30 1.200 0.638 0.327 1.057 1.017 0.346

1.32 1.212 0.632 0.318 1.065 1.018 0.3381.24 1.224 0.625 0.309 1.073 1.021 0.3321.36 1.235 0.618 0.300 1.081 1.022 0.3251.38 1.246 0.612 0.293 1.089 1.024 0.3181.40 1.257 0.604 0.284 1.098 1.026 0.312

1.42 1.268 0.598 0.276 1.108 1.028 0.3061.44 1.278 0.591 0.268 1.117 1.030 0.3001.46 1.289 0.585 0.261 1.127 1.032 0.2941.48 1.299 0.578 0.253 1.138 1.035 0.2891.50 1.308 0.570 0.246 1.148 1.036 0.283

1.52 1.319 0.565 0.239 1.160 1.038 0.2781.54 1.329 0.558 0.233 1.171 1.041 0.2731.56 1.338 0.552 0.226 1.183 1.043 0.2671.58 1.348 0.545 0.220 1.195 1.045 0.2631.60 1.356 0.538 0.214 1.207 1.047 0.258

1.62 1.365 0.533 0.208 1.220 1.049 0.2531.64 1.375 0.527 0.202 1.234 1.051 0.2481.66 1.384 0.521 0.196 1.247 1.053 0.2441.68 1.393 0.515 0.191 1.261 1.055 0.2391.70 1.400 0.508 0.185 1.275 1.057 0.236

1.72 1.409 0.503 0.179 1.290 1.060 0.2321.74 1.417 0.498 0.175 1.305 1.062 0.2281.76 1.425 0.492 0.169 1.320 1.064 0.2241.78 1.433 0.486 0.165 1.336 1.065 0.2201.80 1.440 0.480 0.160 1.351 1.067 0.216


TABLE 3.3



A

p

p*0

1.82 1.448 0.475 0.155 1.368 1.069 0.2131.84 1.456 0.469 0.151 1.385 1.072 0.2091.86 1.464 0.464 0.147 1.402 1.073 0.2061.88 1.471 0.459 0.143 1.419 1.075 0.2021.90 1.477 0.453 0.138 1.435 1.077 0.199

1.92 1.485 0.448 0.135 1.455 1.079 0.1961.94 1.492 0.443 0.131 1.474 1.081 0.1931.96 1.498 0.438 0.127 1.493 1.083 0.1891.98 1.505 0.433 0.124 1.511 1.085 0.1872.00 1.511 0.428 0.120 1.530 1.086 0.184

2.02 1.518 0.424 0.117 1.551 1.088 0.1812.04 1.525 0.418 0.114 1.571 1.090 0.1782.06 1.531 0.414 0.110 1.592 1.092 0.1752.08 1.537 0.405 0.107 1.613 1.094 0.1732.10 1.542 0.404 0.104 1.632 1.095 0.170

2.12 1.548 0.400 0.101 1.656 1.097 0.1682.14 1.554 0.396 0.098 1.678 1.099 0.1652.16 1.560 0.391 0.096 1.700 1.101 0.1632.18 1.566 0.387 0.093 1.723 1.102 0.1602.20 1.570 0.380 0.091 1.745 1.104 0.158

2.22 1.577 0.378 0.088 1.769 1.106 0.1562.24 1.582 0.374 0.085 1.793 1.107 0.1532.26 1.587 0.370 0.083 1.817 1.108 0.1512.28 1.592 0.366 0.081 1.842 1.110 0.1492.30 1.596 0.361 0.078 1.865 1.111 0.147

2.32 1.603 0.358 0.077 1.893 1.113 0.1452.34 1.607 0.354 0.075 1.918 1.115 0.1432.36 1.612 0.350 0.073 1.945 1.116 0.1412.38 1.617 0.346 0.071 1.971 1.118 0.1392.40 1.620 0.341 0.068 1.995 1.120 0.137


TABLE 3.3



A

p

p*0

2.42 1.626 0.338 0.067 2.025 1.121 0.1352.44 1.631 0.335 0.065 2.053 1.122 0.1332.46 1.635 0.331 0.063 2.081 1.123 0.1312.48 1.639 0.328 0.062 2.109 1.124 0.1292.50 1.642 0.323 0.060 2.135 1.125 0.128

2.52 1.648 0.321 0.058 2.168 1.127 0.1262.54 1.652 0.317 0.057 2.197 1.128 0.1242.56 1.656 0.314 0.055 2.228 1.130 0.1232.58 1.660 0.311 0.053 2.258 1.131 0.1212.60 1.663 0.306 0.052 2.285 1.132 0.120

2.62 1.668 0.304 0.051 2.321 1.134 0.1182.64 1.672 0.301 0.049 2.353 1.135 0.1172.66 1.676 0.298 0.048 2.385 1.136 0.1152.68 1.680 0.295 0.047 2.417 1.137 0.1142.70 1.682 0.291 0.046 2.445 1.138 0.112

2.72 1.687 0.288 0.045 2.484 1.139 0.1112.74 1.690 0.285 0.044 2.518 1.141 0.1092.76 1.694 0.282 0.042 2.552 1.142 0.1082.78 1.697 0.279 0.041 2.587 1.143 0.1072.80 1.700 0.276 0.040 2.615 1.145 0.106

2.82 1.704 0.274 0.039 2.660 1.146 0.1042.84 1.707 0.271 0.038 2.694 1.147 0.1032.86 1.710 0.268 0.037 2.731 1.148 0.1012.88 1.713 0.265 0.036 2.768 1.149 0.1002.90 1.714 0.262 0.035 2.800 1.150 0.099

2.92 1.720 0.260 0.034 2.843 1.151 0.0982.94 1.723 0.258 0.0337 2.881 1.152 0.0972.96 1.726 0.225 0.033 2.920 1.153 0.0962.98 1.728 0.252 0.032 2.960 1.154 0.0953.00 1.730 0.250 0.031 2.990 1.155 0.094


TABLE 3.3



A

p

p*0

3.05 1.739 0.244 0.0294 3.100 1.157 0.09083.10 1.746 0.238 0.0276 3.205 1.159 0.0883.15 1.752 0.232 0.0259 3.312 1.162 0.0863.20 1.760 0.226 0.0245 3.423 1.164 0.0833.25 1.765 0.221 0.0230 3.536 1.166 0.081

3.30 1.771 0.216 0.0217 3.653 1.168 0.0793.35 1.776 0.211 0.0204 3.772 1.169 0.077

.3.40 1.782 0.206 0.0193 3.895 1.172 0.0753.45 1.787 0.201 0.0182 4.022 1.173 0.0733.50 1.790 0.196 0.0170 4.135 1.175 0.071

3.55 1.797 0.192 16.21 × 10−3 4.284 1.177 0.06933.60 1.802 0.188 15.32 × 10−3 4.421 1.178 0.06763.65 1.807 0.184 14.5 × 10−3 4.559 1.180 0.06593.70 1.811 0.179 13.7 × 10−3 4.704 1.182 0.06433.75 1.815 0.176 12.97 × 10−3 4.849 1.183 0.063

3.80 1.819 0.172 12.28 × 10−3 5.000 1.185 0.0613.85 1.824 0.168 11.6 × 10−3 5.154 1.186 0.05983.90 1.827 0.165 11.02 × 10−3 5.311 1.187 0.05843.95 1.832 0.161 10.45 × 10−3 5.472 1.188 0.0574.00 1.834 0.157 9.94 × 10−3 5.607 1.189 0.056

4.05 1.838 0.154 9.4 × 10−3 5.806 1.191 0.0544.10 1.842 0.151 8.93 × 10−3 5.978 1.193 0.0534.15 1.845 0.148 8.48 × 10−3 6.155 1.194 0.0524.20 1.848 0.145 8.06 × 10−3 6.336 1.195 0.0514.25 1.852 0.142 7.66 × 10−3 6.519 1.196 0.049

4.30 1.855 0.139 7.28 × 10−3 6.708 1.197 0.04874.35 1.858 0.137 6.93 × 10−3 6.900 1.198 0.04774.40 1.861 0.134 6.6 × 10−3 7.097 1.199 0.04674.45 1.864 0.132 6.28 × 10−3 7.298 1.200 0.04574.50 1.865 0.128 6.00 × 10−3 7.455 1.201 0.045



TABLE 3.3



A

p

p*0

4.55 1.869 0.126 5.7 × 10−3 7.712 1.202 0.044

4.60 1.871 0.124 5.4 × 10−3 7.925 1.203 0.043

4.65 1.874 0.122 5.2 × 10−3 8.143 1.204 0.042

4.70 1.876 0.119 4.9 × 10−3 8.365 1.205 0.041

4.75 1.878 0.117 4.7 × 10−3 8.591 1.206 0.040

4.80 1.881 0.115 4.5 × 10−3 8.822 1.206 0.039

4.85 1.883 0.113 4.3 × 10−3 9.058 1.207 0.0389

4.90 1.885 0.111 4.1 × 10−3 9.298 1.208 0.0381

4.95 1.887 0.109 3.9 × 10−3 9.543 1.208 0.0374

5.00 1.888 0.107 3.7 × 10−3 9.720 1.209 0.037

6.00 1.918 0.076 165.0 × 10−5 15.680 1.220 0.026

7.00 1.937 0.057 81.0 × 10−5 23.850 1.228 0.019

8.00 1.950 0.045 43.0 × 10−5 34.600 1.232 0.015

9.00 1.960 0.036 24.5 × 10−5 48.250 1.234 0.012

10.00 1.965 0.029 14.5 × 10−5 65.200 1.238 0.009

∞ 1.995 0 0 ∞ 1.248 0

Normal Shocks in Perfect Gases 47

TABLE 4.1

Normal shocks in perfect gases (γ γ γ γ γ = 1.3)

Mx My py /px Ty /Tx p0y /p0x p0y /px

1.00 1.000 1.000 1.000 1.000 1.8321.02 0.981 1.046 1.010 0.999 1.8751.04 0.962 1.092 1.021 0.999 1.9181.06 0.944 1.139 1.031 0.999 1.9631.08 0.927 1.185 1.041 0.999 2.0101.10 0.911 1.237 1.051 0.999 2.058

1.12 0.896 1.287 1.061 0.998 2.1071.14 0.881 1.338 1.070 0.997 2.1571.16 0.867 1.391 1.080 0.995 2.2091.18 0.853 1.444 1.089 0.994 2.2621.20 0.840 1.497 1.099 0.993 2.316

1.22 0.828 1.552 1.109 0.990 2.3721.24 0.816 1.608 1.119 0.987 2.4281.26 0.804 1.664 1.128 0.985 2.4861.28 0.793 1.722 1.138 0.982 2.5451.30 0.783 1.780 1.148 0.978 2.605

1.32 0.772 1.839 1.158 0.975 2.6671.34 0.762 1.899 1.167 0.971 2.7281.36 0.753 1.960 1.177 0.966 2.7921.38 0.744 2.022 1.187 0.961 2.8561.40 0.735 2.085 1.197 0.956 2.922

1.42 0.726 2.149 1.207 0.951 2.9881.44 0.714 2.214 1.217 0.945 3.0561.46 0.709 2.279 1.227 0.939 3.1251.48 0.702 2.346 1.237 0.933 3.1951.50 0.694 2.413 1.247 0.926 3.265

1.52 0.687 2.481 1.257 0.919 3.3371.54 0.680 2.551 1.268 0.912 3.4101.56 0.673 2.621 1.278 0.905 3.4841.58 0.666 2.692 1.288 0.897 3.5591.60 0.660 2.764 1.300 0.889 3.635


TABLE 4.1



1.62 0.654 2.836 1.310 0.881 3.7121.64 0.647 2.910 1.320 0.873 3.7911.66 0.642 2.985 1.331 0.864 3.8701.68 0.636 3.060 1.342 0.856 3.9911.70 0.630 3.137 1.353 0.847 4.031

1.72 0.625 3.214 1.364 0.837 4.1551.74 0.620 3.292 1.375 0.828 4.1961.76 0.615 3.371 1.386 0.819 4.2811.78 0.609 3.451 1.397 0.810 4.3661.80 0.605 3.532 1.407 0.800 4.452

1.82 0.600 3.614 1.420 0.790 4.5391.84 0.596 3.697 1.432 0.781 4.6281.86 0.591 3.780 1.443 0.771 4.7171.88 0.587 3.865 1.455 0.761 4.8071.90 0.583 3.950 1.466 0.751 4.898

1.92 0.578 4.037 1.479 0.741 4.9911.94 0.574 4.124 1.491 0.731 5.0841.96 0.570 4.212 1.503 0.721 5.1781.98 0.567 4.301 1.515 0.711 5.2742.00 0.563 4.390 1.527 0.700 5.370

2.02 0.559 4.482 1.539 0.690 5.4672.04 0.556 4.574 1.552 0.680 5.5652.06 0.552 4.667 1.565 0.670 5.6652.08 0.548 4.760 1.578 0.660 5.7652.10 0.546 4.856 1.590 0.650 5.867

2.12 0.542 4.950 1.600 0.639 5.9682.14 0.539 5.046 1.617 0.629 6.0712.16 0.536 5.144 1.630 0.619 6.1762.18 0.533 5.242 1.643 0.609 6.2812.20 0.530 5.341 1.656 0.600 6.387


TABLE 4.1



2.22 0.527 5.441 1.669 0.590 6.4942.24 0.524 5.542 1.683 0.580 6.6032.26 0.522 5.643 1.697 0.571 6.7122.28 0.519 5.746 1.711 0.561 6.8222.30 0.516 5.849 1.725 0.552 6.934

2.32 0.514 5.954 1.738 0.542 7.0452.34 0.511 6.059 1.753 0.533 7.1582.36 0.508 6.166 1.767 0.523 7.2722.38 0.506 6.273 1.781 0.515 7.3872.40 0.505 6.380 1.797 0.505 7.504

2.42 0.502 6.489 1.810 0.496 7.6212.44 0.499 6.599 1.825 0.487 7.7402.46 0.497 6.711 1.840 0.478 7.8572.48 0.495 6.822 1.854 0.469 7.9772.50 0.493 6.935 1.869 0.460 8.098

2.52 0.491 7.050 1.885 0.452 8.2202.54 0.489 7.163 1.899 0.444 8.3432.56 0.487 7.278 1.915 0.436 8.4672.58 0.485 7.394 1.930 0.427 8.5922.60 0.483 7.512 1.945 0.420 8.715

2.62 0.481 7.629 1.962 0.411 8.8452.64 0.479 7.748 1.977 0.404 8.9732.66 0.477 7.865 1.993 0.396 9.1012.68 0.475 7.988 2.009 0.388 9.2312.70 0.474 8.770 2.025 0.381 9.361

2.72 0.472 8.233 2.042 0.374 9.4942.74 0.470 8.351 2.058 0.366 9.6262.76 0.468 8.481 2.074 0.359 9.7602.78 0.467 8.606 2.091 0.352 9.8952.80 0.466 8.733 2.107 0.345 10.031


TABLE 4.1



2.82 0.464 8.859 2.124 0.338 10.1672.84 0.462 8.987 2.141 0.332 10.3052.86 0.461 9.116 2.158 0.325 10.4442.88 0.459 9.246 2.175 0.318 10.5832.90 0.458 9.380 2.194 0.313 10.725

2.92 0.456 9.508 2.210 0.306 16.8652.94 0.455 9.641 2.227 0.300 11.0082.96 0.454 9.774 2.245 0.294 11.1512.98 0.452 9.908 2.263 0.288 11.2953.00 0.451 10.041 2.280 0.282 11.450

3.05 0.448 10.385 2.325 0.268 11.7353.10 0.445 10.733 2.371 0.255 12.1833.15 0.442 11.086 2.418 0.242 12.5633.20 0.439 11.445 2.465 0.229 13.0273.25 0.436 11.809 2.513 0.218 13.421

3.30 0.434 12.180 2.561 0.207 13.7403.35 0.431 12.556 2.611 0.196 14.1453.40 0.428 12.937 2.661 0.186 14.5563.45 0.426 13.325 2.711 0.178 14.9723.50 0.424 13.720 2.762 0.168 15.391

3.55 0.422 14.116 2.815 0.159 15.8233.60 0.420 14.520 2.868 0.151 16.2593.65 0.418 14.930 2.922 0.143 16.7003.70 0.416 15.345 2.976 0.136 17.1473.75 0.414 15.766 3.031 0.129 17.600

3.80 0.412 16.193 3.087 0.122 18.0603.85 0.411 16.625 3.144 0.116 18.5263.90 0.409 17.063 3.201 0.110 18.9973.95 0.407 17.507 3.259 0.105 19.4754.00 0.406 17.960 3.317 0.099 19.961


TABLE 4.1



4.05 0.404 18.411 3.378 0.094 20.4494.10 0.403 18.872 3.438 0.089 20.9454.15 0.401 19.338 3.498 0.085 21.4474.20 0.400 19.810 3.561 0.081 21.9564.25 0.398 20.288 3.623 0.077 22.470

4.30 0.397 20.771 3.686 0.073 22.9994.35 0.396 21.260 3.750 0.069 23.5164.40 0.395 21.755 3.815 0.066 24.0494.45 0.394 22.255 3.880 0.062 24.5884.50 0.393 22.760 3.945 0.0594 25.181

4.55 0.392 23.272 4.013 0.056 25.6834.60 0.391 23.789 4.081 0.054 26.2414.65 0.389 24.312 4.149 0.051 26.8044.70 0.388 24.841 4.218 0.048 27.3734.75 0.387 25.375 4.287 0.046 27.948

4.80 0.387 25.915 4.358 0.044 28.5294.85 0.386 26.460 4.429 0.042 29.1174.90 0.385 27.011 4.502 0.039 29.7104.95 0.384 27.568 4.574 0.038 30.3095.00 0.383 28.130 4.646 0.036 30.901

6.00 0.370 40.570 6.271 0.014 44.3207.00 0.363 55.261 8.189 0.0061 60.1408.00 0.357 72.220 10.401 0.00283 78.4009.00 0.354 91.430 12.907 0.0014 99.101

10.00 0.351 112.911 15.710 0.00074 122.25

∞ 0.340 ∞ ∞ 0 ∞


TABLE 4.2



1.00 1.000 1.000 1.000 1.000 1.8931.02 0.981 1.047 1.013 0.999 1.9381.04 0.962 1.095 1.026 0.999 1.9841.06 0.945 1.144 1.039 0.999 2.0321.08 0.928 1.194 1.052 0.999 2.0821.10 0.911 1.245 1.065 0.998 2.133

1.12 0.897 1.297 1.078 0.998 2.1851.14 0.882 1.349 1.090 0..997 2.2381.16 0.868 1.403 1.103 0.996 2.2941.18 0.855 1.458 1.115 0.994 2.3501.20 0.842 1.513 1.128 0.993 2.407

1.22 0.830 1.570 1.141 0.991 2.4661.24 0.818 1.627 1.153 0.988 2.5261.26 0.807 1.685 1.166 0.985 2.5871.28 0.796 1.745 1.178 0.983 2.6501.30 0.786 1.805 1.191 0.979 2.714

1.32 0.776 1.866 1.204 0.976 2.7781.34 0.766 1.928 1.216 0.972 2.8451.36 0.757 1.991 1.229 0.967 2.9111.38 0.748 2.055 1.242 0.963 2.9791.40 0.740 2.120 1.255 0.958 3.049

1.42 0.731 2.186 1.267 0.953 3.1201.44 0.723 2.253 1.281 0.947 3.1921.46 0.716 2.320 1.294 0.942 3.2651.48 0.708 2.389 1.307 0.936 3.3381.50 0.701 2.458 1.320 0.930 3.413

1.52 0.694 2.529 1.334 0.923 3.4891.54 0.687 2.600 1.347 0.917 3.5671.56 0.681 2.673 1.361 0.909 3.6451.58 0.675 2.746 1.374 0.903 3.7251.60 0.668 2.820 1.388 0.895 3.805


TABLE 4.2



1.62 0.663 2.895 1.402 0.887 3.8861.64 0.657 2.971 1.416 0.880 3.9691.66 0.651 3.048 1.430 0.872 4.0531.68 0.646 3.126 1.444 0.864 4.1381.70 0.641 3.205 1.458 0.856 4.224

1.72 0.635 3.285 1.473 0.847 4.3151.74 0.631 3.336 1.487 0.839 4.4431.76 0.626 3.447 1.502 0.830 4.5331.78 0.621 3.530 1.517 0.822 4.5781.80 0.616 3.613 1.532 0.813 4.670

1.82 0.612 3.698 1.547 0.804 4.7621.84 0.608 3.781 1.562 0.795 4.8551.86 0.604 3.870 1.578 0.786 4.9471.88 0.599 3.957 1.593 0.776 5.0431.90 0.596 4.045 1.608 0.767 5.142

1.92 0.592 4.134 1.624 0.758 5.2391.94 0.588 4.224 1.639 0.749 5.3381.96 0.584 4.315 1.656 0.739 5.4381.98 0.581 4.407 1.671 0.730 5.5382.00 0.577 4.500 1.687 0.721 5.641

2.02 0.574 4.594 1.704 0.712 5.7432.04 0.571 4.688 1.720 0.702 5.8472.06 0.567 4.784 1.737 0.693 5.9532.08 0.564 4.881 1.754 0.684 6.0582.10 0.561 4.978 1.770 0.674 6.166

2.12 0.558 5.077 1.787 0.665 6.2742.14 0.556 5.176 1.805 0.656 6.3832.16 0.553 5.276 1.822 0.646 6.4932.18 0.550 5.378 1.839 0.637 6.6042.20 0.547 5.480 1.857 0.628 6.716


TABLE 4.2



2.22 0.545 5.583 1.875 0.619 6.8292.24 0.542 5.687 1.892 0.610 6.9452.26 0.539 5.792 1.910 0.601 7.0602.28 0.537 5.898 1.928 0.592 7.1762.30 0.534 6.005 1.947 0.583 7.294

2.32 0.532 6.113 1.965 0.574 7.4122.34 0.529 6.222 1.984 0.566 7.5322.36 0.527 6.331 2.003 0.557 7.6532.38 0.525 6.442 2.021 0.548 7.7752.40 0.523 6.553 2.040 0.540 7.897

2.42 0.521 6.666 2.059 0.532 8.0272.44 0.519 6.780 2.079 0.523 8.1452.46 0.517 6.894 2.098 0.515 8.2712.48 0.515 7.008 2.118 0.507 8.3982.50 0.513 7.125 2.138 0.499 8.526

2.52 0.511 7.242 2.157 0.491 8.6552.54 0.509 7.360 2.177 0.483 8.7852.56 0.507 7.479 2.197 0.475 8.9162.58 0.506 7.599 2.218 0.467 9.1452.60 0.504 7.720 2.238 0.460 9.181

2.62 0.502 7.842 2.259 0.453 9.3152.64 0.500 7.965 2.279 0.445 9.4512.66 0.498 8.088 2.300 0.438 9.5872.68 0.497 8.213 2.322 0.431 9.7242.70 0.496 8.338 2.343 0.423 9.863

2.72 0.494 8.465 2.364 0.416 10.0022.74 0.493 8.592 2.386 0.409 10.1422.76 0.491 8.721 2.407 0.403 10.2832.78 0.490 8.849 2.429 0.396 10.4262.80 0.488 8.980 2.451 0.389 10.569


TABLE 4.2



2.82 0.487 9.111 2.473 0.383 10.7142.84 0.485 9.243 2.495 0.376 10.8592.86 0.484 9.376 2.518 0.370 11.0062.88 0.483 9.510 2.540 0.364 11.1542.90 0.481 9.645 2.563 0.358 11.302

2.92 0.480 9.781 2.586 0.352 11.4522.94 0.478 9.917 2.609 0.346 11.6032.96 0.477 10.055 2.632 0.339 11.7542.98 0.476 10.194 2.656 0.334 11.9073.00 0.475 10.333 2.679 0.328 12.061

3.05 0.472 10.686 2.738 0.315 12.4503.10 0.469 11.045 2.798 0.301 12.8463.15 0.467 11.409 2.860 0.288 13.2473.20 0.464 11.780 2.922 0.276 13.6563.25 0.462 12.156 2.985 0.264 14.071

3.30 0.459 12.538 3.049 0.253 14.4923.35 0.457 12.926 3.114 0.243 14.9203.40 0.455 13.320 3.180 0.232 15.3543.54 0.453 13.791 3.247 0.222 15.7953.50 0.451 14.125 3.315 0.213 16.242

3.55 0.449 14.536 3.384 0.204 16.7003.60 0.447 14.953 3.454 0.195 17.1563.65 0.445 15.376 3.525 0.187 17.7153.70 0.444 15.805 3.596 0.179 18.0953.75 0.442 16.239 3.669 0.172 18.575

3.80 0.441 16.680 3.743 0.164 19.0603.85 0.439 17.126 3.817 0.157 19.5533.90 0.438 17.578 3.893 0.151 20.0513.95 0.436 18.036 3.969 0.145 20.5564.00 0.435 18.500 4.047 0.138 21.068


TABLE 4.2



4.05 0.434 18.969 4.125 0.133 21.5864.10 0.432 19.445 4.205 0.127 22.1114.15 0.431 19.926 4.285 0.122 22.6424.20 0.430 20.413 4.367 0.117 23.1794.25 0.429 20.906 4.449 0.113 23.723

4.30 0.428 21.405 4.532 0.108 24.2734.35 0.427 21.909 4.616 0.104 24.8294.40 0.426 22.420 4.702 0.099 25.3934.45 0.425 22.936 4.788 0.095 26.0784.50 0.424 23.458 4.875 0.0917 26.539

4.55 0.423 23.986 4.963 0.088 27.1204.60 0.422 24.520 5.052 0.085 27.7104.65 0.421 25.059 5.142 0.081 28.3054.70 0.420 25.605 5.233 0.078 28.9074.75 0.419 26.156 5.325 0.075 29.516

4.80 0.418 26.713 5.418 0.072 30.1304.85 0.417 27.276 5.512 0.069 30.7514.90 0.4167 27.845 5.607 0.067 31.3794.95 0.416 28.419 5.703 0.0642 32.0135.00 0.415 29.000 5.800 0.0617 32.654

6.00 0.404 41.833 7.941 0.0296 46.8157.00 0.397 57.000 10.469 0.0154 63.5528.00 0.393 74.500 13.387 0.0085 82.8659.00 0.389 94.333 16.693 0.005 104.753

10.000 0.388 116.500 20.388 0.003 129.217

∞ 0.378 ∞ ∞ 0 ∞


TABLE 4.3



1.00 1.000 1.000 1.000 1.000 2.0551.02 0.981 1.051 1.020 0.999 2.1071.04 0.962 1.102 1.040 0.999 2.1611.06 0.945 1.155 1.059 0.999 2.2171.08 0.928 1.208 l.078 0.999 2.2741.10 0.913 1.262 1.098 0.999 2.333

1.12 0.898 1.318 1.118 0.998 2.3931.14 0.884 1.375 1.137 0.997 2.4551.16 0.871 1.432 1.157 0.996 2.5181.18 0.858 1.490 1.176 0.995 2.5841.20 0.846 1.551 1.195 0.993 2.649

1.22 0.835 1.611 1.215 0.991 2.7181.24 0.824 1.672 1.234 0.989 2.7871.26 0.813 1.735 1.253 0.987 2.8571.28 0.805 1.798 1.273 0.984 2.9291.30 0.794 1.863 1.293 0.981 3.002

1.32 0.784 1.928 1.312 0.978 3.0761.34 0.775 1.995 1.332 0.975 3.1521.36 0.767 2.062 1.352 0.971 3.2291.38 0.758 2.131 1.372 0.967 3.3081.40 0.751 2.201 1.394 0.963 3.388

1.42 0.743 2.271 1.412 0.958 3.4681.44 0.736 2.342 1.433 0.953 3.5511.46 0.729 2.415 1.453 0.948 3.6341.48 0.723 2.488 1.475 0.943 3.7191.50 0.716 2.562 1.497 0.938 3.805

1.52 0.709 2.638 1.516 0.933 3.8921.54 0.703 2.715 1.537 0.927 3.9811.56 0.697 2.792 1.560 0.921 4.0711.58 0.692 2.871 1.580 0.915 4.1621.60 0.687 2.950 1.605 0.908 4.255


TABLE 4.3

Normal shocks in perfect gases (γγγγγ = 1.67)


1.62 0.681 3.031 1.624 0.902 4.3471.64 0.676 3.112 1.646 0.896 4.4421.66 0.671 3.195 1.668 0.889 4.5381.68 0.667 3.278 1.691 0.882 4.6351.70 0.662 3.365 1.715 0.875 4.734

1.72 0.657 3.448 1.736 0.868 4.8331.74 0.653 3.535 1.760 0.861 4.9331.76 0.649 3.622 1.783 0.854 5.0361.78 0.645 3.711 1.806 0.847 5.1391.80 0.641 3.800 1.832 0.840 5.244

1.82 0.637 3.891 1.853 0.832 5.3491.84 0.633 3.982 1.878 0.824 5.4561.86 0.630 4.075 1.903 0.817 5.5641.88 0.626 4.168 1.927 0.809 5.6731.90 0.623 4.266 1.954 0.802 5.785

1.92 0.619 4.358 1.976 0.794 5.8951.94 0.616 4.454 2.002 0.786 6.0081.96 0.613 4.552 2.027 0.778 6.1221.98 0.610 4.651 2.053 0.771 6.2372.00 0.607 4.753 2.080 0.763 6.355

2.02 0.604 4.851 2.105 0.756 6.4722.-04 0.601 4.952 2.131 0.747 6.5002.06 0.598 5.055 2.157 0.740 6.7112.08 0.596 5.158 2.184 0.732 6.8322.10 0.594 5.266 2.213 0.725 6.953

2.12 0.591 5.368 2.238 0.717 7.0782.14 0.588 5.475 2.266 0.709 7.2032.16 0.586 5.582 2.293 0.701 7.3282.18 0.584 5.891 2.321 0.694 7.4562.20 0.582 5.805 2.354 0.687 7.586


TABLE 4.3



2.22 0.579 5.911 2.378 0.678 7.7142.24 0.577 6.022 2.405 0.671 7.8452.26 0.575 6.135 2.435 0.664 7.9772.28 0.573 6.248 2.464 0.656 8.1102.30 0.571 6.367 2.500 0.649 8.245

2.32 0.569 6.478 2.523 0.641 8.3792.34 0.567 6.595 2.553 0.634 8.5172.36 0.565 6.713 2.583 0.627 8.6542.38 0.563 6.831 2.613 0.619 8.7942.40 0.562 6.953 2.651 0.614 8.935

2.42 0.560 7.071 2.674 0.605 9.0042.44 0.558 7.193 2.705 0.598 9.2182.46 0.556 7.315 2.736 0.591 9.3622.48 0.555 7.438 2.767 0.584 9.5072.50 0.554 7.567 2.805 0.578 9.651

2.52 0.552 7.689 2.831 0.571 9.8002.54 0.550 7.815 2.863 0.564 9.9492.56 0.548 7.943 2.895 0.557 10.0982.58 0.547 8.071 2.928 0.551 10.2492.60 0.546 8.205 2.968 0.545 10.401

2.62 0.544 8.333 2.994 0.538 10.5562.64 0.543 8.463 3.027 0.531 10.7092.66 0.541 8.595 3.061 0.525 10.8652.68 0.540 8.729 3.094 0.518 11.0222.70 0.539 8.870 3.138 0.514 11.180

2.72 0.538 8.995 3.163 0.506 11.3392.74 0.536 9.135 3.197 0.500 11.5002.76 0.535 9.273 3.232 0.494 11.6622.78 0.534 9.411 3.267 0.488 11.8242.80 0.533 9.557 3.311 0.483 12.000


TABLE 4.3



2.82 0.532 9.691 3.337 0.476 12.1532.84 0.531 9.833 3.373 0.471 12.3192.86 0.529 9.976 3.409 0.465 12.4872.88 0.528 10.118 3.445 0.459 12.6562.90 0.527 10.270 3.492 0.455 12.831

2.92 0.526 10.408 3.518 0.448 12.9972.94 0.525 10.555 3.555 0.443 13.1692.96 0.524 10.703 3.593 0.438 13.3422.98 0.523 10.851 3.630 0.433 13.5173.00 0.523 11.010 3.680 0.428 13.690

3.05 0.520 11.379 3.763 0.415 14.1373.10 0.518 11.763 3.860 0.402 14.5893.15 0.516 12.154 3.958 0.390 15.0483.20 0.514 12.551 4.053 0.378 15.5153.25 0.511 12.954 4.160 0.367 15.989

3.30 0.509 13.363 4.262 0.357 16.4713.35 0.508 13.779 4.367 0.346 16.9603.40 0.506 14.201 4.473 0.336 17.4563.45 0.505 14.620 4.581 0.326 17.9593.50 0.504 15.075 4.705 0.318 18.471

3.55 0.502 15.504 4.800 0.309 18.9883.60 0.500 15.951 4.912 0.298 19.5143.65 0.498 16.404 5.026 0.290 20.0473.70 0.497 16.864 5.141 0.282 20.5883.75 0.496 17.330 5.258 0.274 21.135

3.80 0.495 17.801 5.376 0.266 21.6913.85 0.494 18.279 5.496 0.258 22.2533.90 0.493 18.764 5.618 0.251 22.8233.95 0.492 19.255 5.741 0.244 23.4004.00 0.491 19.761 5.886 0.239 23.990


TABLE 4.3



4.05 0.489 20.255 5.992 0.231 24.5774.10 0.488 20.764 6.119 0.225 25.1754.15 0.487 21.280 6.249 0.218 25.7834.20 0.486 24.802 6.379 0.213 26.3974.25 0.485 22.330 6.511 0.207 27.018

4.30 0.485 22.864 6.645 0.201 27.6474.35 0.484 23.405 6.781 0.196 28.2834.40 0.483 23.952 6.918 0.191 28.9264.45 0.482 24.505 7.057 0.186 29.5784.50 0.482 25.081 7.222 0.181 30.250

4.55 0.481 25.630 7.338 0.176 30.9014.60 0.480 26.202 7.482 0.171 31.5744.65 0.479 26.780 7.626 0.167 32.2544.70 0.479 27.365 7.773 0.163 32.9424.75 0.478 27.955 7.921 0.158 33.637

4.80 0.478 28.552 8.070 0.155 34.3404.85 0.477 29.155 8.221 0.151 35.0494.90 0.476 29.765 8.373 0.147 35.7664.95 0.476 30.380 8.528 0.143 36.4915.00 0.476 31.021 8.700 0.140 37.230

6.00 0.467 44.780 12.165 0.089 53.4007.00 0.462 61.050 16.250 0.058 72.5308.00 0.459 79.810 20.960 0.041 94.6009.00 0.457 101.080 26.300 0.029 119.600

10.00 0.456 124.85 32.260 0.022 147.50

∞ 0.448 ∞ ∞ 0 ∞


TABLE 5.1

Oblique shocks in perfect gases

Wave angles for given deflection (γ γ γ γ γ = 1.4, δδδδδ = 2°)

M1 σ M1 σ M1 σ

1.02 83.49 1.62 40.15 2.22 28.321.04 80.93 1.64 39.50 2.24 28.051.06 79.05 1.66 39.00 2.26 27.801.08 77.56 1.68 38.45 2.28 27.541.10 76.30 1.70 37.93 2.30 27.30

1.12 75.22 1.72 37.45 2.32 27.051.14 68.71 1.74 36.93 2.34 26.811.16 65.55 1.76 36.45 2.36 26.511.18 63.10 1.78 36.00 2.38 26.351.20 61.10 1.80 35.50 2.40 26.12

1.22 59.24 1.82 35.10 2.42 25.901.24 57.50 1.84 34.70 2.44 25.701.26 56.15 1.86 34.25 2.46 25.501.28 54.75 1.88 33.85 2.48 25.251.30 53.50 1.90 33.45 2.50 25.05

1.32 52.30 1.92 33.05 2.52 24.851.34 51.15 1.94 32.70 2.54 24.651.36 50.10 1.96 32.35 2.56 24.451.38 49.12 1.98 32.00 2.58 24.251.40 48.15 2.00 31.65 2.60 24.06

1.42 47.27 2.02 31.30 2.62 23.901.44 46.42 2.04 30.95 2.64 23.701.46 45.60 2.06 30.65 2.66 23.521.48 44.82 2.08 30.35 2.68 23.351.50 44.05 2.10 30.03 2.70 23.15

1.52 43.35 2.12 29.74 2.72 23.001.54 42.65 2.14 29.40 2.74 22.831.56 42.00 2.16 29.15 2.76 22.651.58 41.35 2.18 28.85 2.78 22.501.60 40.75 2.20 28.59 2.80 22.35

Oblique Shocks in Perfect Gases 63

TABLE 5.1



M1 σ M1 σ M1 σ

2.82 22.20 4.05 15.632.84 22.05 4.10 15.452.86 21.88 4.15 15.252.88 21.74 4.20 15.102.90 21.58 4.25 14.94

2.92 21.45 4.30 14.752.94 21.30 4.35 14.622.96 21.15 4.40 14.452.98 21.00 4.45 14.303.00 20.85 4.50 14.15

3.05 20.53 4.55 14.033.10 20.20 4.60 13.883.15 19.90 4.65 13.753.20 19.60 4.70 13.603.25 19.29 4.75 13.48

3.30 19.00 4.80 13.353.35 18.75 4.85 13.233.40 18.45 4.90 13.103.45 18.21 4.95 12.983.50 17.95 5.00 12.85

3.55 17.70 5.50 11.803.60 17.50 6.00 10.913.65 17.25 6.50 10.163.70 17.05 7.00 9.553.75 16.81 7.50 9.00

3.80 16.60 8.00 8.503.85 16.40 8.50 8.083.90 16.20 9.00 7.703.95 16.00 9.50 7.354.00 15.81 10.00 7.00


TABLE 5.1



M1 σ M1 σ M1 σ

1.02 1.62 44.65 2.22 31.701.04 1.64 44.00 2.24 31.421.06 1.66 43.36 2.26 31.151.08 1.68 42.75 2.28 30.851.10 1.70 42.15 2.30 30.60

1.12 1.72 41.56 2.32 30.351.14 1.74 41.02 2.34 30.101.16 1.76 40.50 2.36 29.841.18 1.78 40.00 2.38 29.601.20 1.80 39.47 2.40 29.38

1.22 71.35 1.82 39.00 2.42 29.151.24 70.80 1.84 38.55 2.44 28.921.26 70.30 1.86 38.05 2.46 28.701.28 67.38 1.88 37.65 2.48 28.471.30 63.50 1.90 37.22 2.50 28.25

1.32 61.05 1.92 36.80 2.52 28.051.34 59.08 1.94 36.40 2.54 27.851.36 57.45 1.96 36.00 2.56 27.641.38 55.96 1.98 35.62 2.58 27.451.40 54.65 2.00 35.24 2.60 27.25

1.42 53.42 2.02 34.90 2.62 27.051.44 52.30 2.04 34.50 2.64 26.851.46 51.24 2.06 34.16 2.66 26.661.48 50.25 2.08 33.83 2.68 26.501.50 49.35 2.10 33.50 2.70 26.30

1.52 48.45 2.12 33.20 2.72 26.131.54 47.62 2.14 32.88 2.74 25.951.56 46.81 2.16 32.60 2.76 25.801.58 46.05 2.18 32.28 2.78 25.611.60 45.35 2.20 31.97 2.80 25.46


TABLE 5.1



M1 σ M1 σ M1 σ

2.82 25.30 4.05 18.602.84 25.15 4.10 18.452.86 25.00 4.15 18.272.88 24.81 4.20 18.102.90 24.67 4.25 17.95

2.92 24.32 4.30 17.782.94 24.38 4.35 17.652.96 24.21 4.40 17.452.98 24.05 4.45 17.303.00 23.95 4.50 17.17

3.05 23.60 4.55 17.003.10 23.30 4.60 16.903.15 22.95 4.65 16.753.20 22.63 4.70 16.623.25 22.32 4.75 16.50

3.30 22.05 4.80 16.363.35 21.76 4.85 16.253.40 21.50 4.90 16.103.45 21.23 4.95 16.003.50 21.00 5.00 15.89

3.55 20.73 5.50 14.843.60 20.50 6.00 14.003.65 20.25 6.50 13.273.70 20.00 7.00 12.703.75 19.80 7.50 12.16

3.80 19.60 8.00 11.703.85 19.40 8.50 11.343.90 19.20 9.00 11.003.95 19.00 9.50 10.704.00 18.80 10.00 10.45


TABLE 5.1



M1 σ M1 σ M1 σ

1.02 1.62 50.20 2.22 35.451.04 1.64 49.40 2.24 35.151.06 1.66 48.63 2.26 34.901.08 1.68 47.90 2.28 34.601.10 1.70 47.15 2.30 34.34

1.12 1.72 46.50 2.32 34.001.14 1.74 45.83 2.34 33.801.16 1.76 45.22 2.36 33.531.18 1.78 44.65 2.38 33.301.20 1.80 44.06 2.40 33.00

1.22 1.82 43.50 2.42 32.801.24 1.84 43.00 2.44 32.54

1..26 1.86 42.47 2.46 32.301.28 1.88 42.00 2.48 32.001.30 1.90 41.50 2.50 31.85

1.32 1.92 41.00 2.52 31.651.34 1.94 40.60 2.54 31.401.36 68.30 1.96 40.15 2.56 31.201.38 68.00 1.98 39.73 2.58 31.001.40 67.75 2.00 39.30 2.60 30.80

1.42 67.50 2.02 38.91 2.62 30.601.44 62.30 2.04 38.50 2.64 30.401.46 60.00 2.06 38.15 2.66 30.201.48 58.20 2.08 37.80 2.68 30.001.50 56.71 2.10 37.43 2.70 29.82

1.52 55.34 2.12 37.10 2.72 29.651.54 54.16 2.14 36.76 2.74 29.451.56 53.05 2.16 36.43 2.76 29.271.58 52.10 2.18 36.10 2.78 29.111.60 51.10 2.20 35.80 2.80 28.95


TABLE 5.1



M1 σ M1 σ M1 σ

2.82 28.78 4.05 22.042.84 28.60 4.10 21.902.86 28.44 4.15 21.702.88 28.30 4.20 21.542.90 28.15 4.25 21.36

2.92 28.00 4.30 21.252.94 27.81 4.35 21.052.96 27.70 4.40 20.902.98 27.55 4.45 20.783.00 27.35 4.50 20.61

3.05 27.00 4.55 20.503.10 26.70 4.60 20.353.15 26.36 4.65 20.213.20 26.10 4.70 20.103.25 25.80 4.75 19.95

3.30 25.45 4.80 19.853.35 25.16 4.85 19.723.40 24.90 4.90 19.603.45 24.65 4.95 19.503.50 24.38 5.00 19.40

3.55 24.14 5.50 18.403.60 23.90 6.00 17.603.65 23.66 6.50 16.953.70 23.45 7.00 16.403.75 23.25 7.50 15.93

3.80 23.00 8.00 15.533.85 22.80 8.50 15.203.90 22.60 9.00 14.903.95 22.42 9.50 14.654.00 22.25 10.00 14.45


TABLE 5.1



M1 σ M1 σ M1 σ

1.62 2.22 47.50 2.82 39.301.64 2.24 47.14 2.84 39.121.66 2.26 46.74 2.86 38.951.68 2.28 46.37 2.88 38.761.70 2.30 46.00 2.90 38.57

1.72 2.32 45.65 2.92 38.401.74 2.34 45.30 2.94 38.251.76 65.08 2.36 44.98 2.96 38.101.78 65.00 2.38 44.65 2.98 37.901.80 64.95 2.40 44.35 3.00 37.75

1.82 64.94 2.42 44.04 3.05 37.371.84 64.90 2.44 43.75 3.10 37.001.86 60.90 2.46 43.45 3.15 36.701.88 59.20 2.48 43.16 3.20 36.351.90 57.90 2.50 42.90 3.25 36.00

1.92 56.80 2.52 42.60 3.30 35.701.94 55.83 2.54 42.35 3.35 35.411.96 54.95 2.56 42.10 3.40 35.151.98 54.15 2.58 41.85 3.45 34.862.00 53.40 2.60 41.60 3.50 34.60

2.02 52.75 2.62 41.40 3.55 34.352.04 52.10 2.64 41.15 3.60 34.102.06 51.50 2.66 40.93 3.65 33.902.08 50.90 2.68 40.70 3.70 33.652.10 50.35 2.70 40.50 3.75 33.45

2.12 49.84 2.72 40.30 3.80 33.252.14 49.35 2.74 40.10 3.85 33.052.16 48.88 2.76 39.90 3.90 32.852.18 48.40 2.78 39.70 3.95 32.652.20 48.00 2.80 39.50 4.00 32.45


TABLE 5.1



M1 σ M1 σ M1 σ

4.05 32.304.10 32.124.15 31.964.20 31.804.25 31.64

4.30 31.504.35 31.354.40 31.204.45 31.054.50 30.95

4.55 30.804.60 30.704.65 30.564.70 30.454.75 30.33

4.80 30.224.85 30.114.90 30.004.95 29.905.00 29.80

5.50 28.936.00 28.256.50 27.707.00 27.297.50 26.92

8.00 26.628.50 26.369.00 26.159.50 25.98

10.00 25.80


TABLE 5.1



M1 σ M1 σ M1 σ

2.22 2.82 54.50 4.05 45.052.24 2.84 54.15 4.10 44.852.26 2.86 53.85 4.15 44.642.28 2.88 53.55 4.20 44.462.30 2.90 53.30 4.25 44.30

2.32 2.92 53.00 4.30 44.132.34 2.94 52.73 4.35 44.002.36 64.69 2.96 52.50 4.40 43.832.38 64.70 2.98 52.25 4.45 43.702.40 64.71 3.00 52.00 4.50 43.55

2.42 64.72 3.05 51.50 4.55 43.402.44 64.73 3.10 50.80 4.60 43.302.46 64.75 3.15 50.50 4.65 43.152.48 64.77 3.20 50.00 4.70 43.002.50 64.78 3.25 49.60 4.75 42.90

2.52 64.30 3.30 49.15 4.80 42.752.54 62.00 3.35 48.80 4.85 42.672.56 60.95 3.40 48.40 4.90 42.502.58 60.10 3.45 48.10 4.95 42.452.60 59.35 3.50 47.75 5.00 42.35

2.62 58.70 3.55 47.45 5.50 41.472.64 58.15 3.60 47.15 6.00 40.802.66 57.62 3.65 46.90 6.50 40.302.68 57.15 3.70 46.60 7.00 39.852.70 56.70 3.75 46.35 7.50 39.50

2.72 56.26 3.80 46.10 8.00 39.232.74 55.88 3.85 45.87 8.50 39.002.76 55.50 3.90 45.65 9.00 38.802.78 55.10 3.95 45.40 9.50 38.672.80 54.80 4.00 45.21 10.00 38.50


TABLE 5.2

Oblique shocks in perfect gases (γγγγγ = 1.4)Upstream and downstream Mach numbers and deflection from

normal shock relations

Mx σ M1 δ My M2

1.05 60 1.212 2.039 0.953 1.1241.10 60 1.270 4.018 0.912 1.1001.15 60 1.328 5.927 0.875 1.0811.20 60 1.385 7.761 0.842 1.0651.25 60 1.443 9.515 0.812 1.053

1.30 60 1.501 11.189 0.786 1.0441.35 60 1.559 12.780 0.762 1.0381.40 60 1.616 14.290 0.739 1.0331.45 60 1.674 15.720 0.719 1.0311.50 60 1.732 17.072 0.701 1.029

1.55 60 1.789 18.348 0.684 1.02931.60 60 1.847 19.553 0.668 1.0301.65 60 1.905 20.688 0.654 1.0321.70 60 1.963 21.758 0.641 1.0351.75 60 2.021 22.766 0.628 1.038

1.80 60 2.078 23.715 0.616 1.0421.85 60 2.136 24.609 0.605 1.0451.90 60 2.194 25.453 0.596 1.0501.95 60 2.252 26.246 0.586 1.0552.00 60 2.309 26.995 0.577 1.060

2.10 60 2.425 28.368 0.561 1.0702.20 60 2.540 29.591 0.547 1.0812.30 60 2.656 30.685 0.534 1.0912.40 60 2.771 31.664 0.523 1.1022.50 60 2.887 32.543 0.513 1.112

2.60 60 3.002 33.334 0.504 1.1232.70 60 3.117 34.049 0.495 1.1332.80 60 3.233 34.696 0.488 1.1422.90 60 3.348 35.283 0.481 1.1513.00 60 3.464 35.817 0.475 1.160


TABLE 5.2

Oblique shocks in perfect gases (γγγγγ = 1.4)

Mx σ M1 δ My M2

3.10 60 3.579 36.305 0.469 1.1683.20 60 3.695 36.750 0.464 1.1763.30 60 3.811 37.158 0.459 1.1843.40 60 3.926 37.533 0.455 1.1913.50 60 4.041 37.878 0.451 1.198

3.60 60 4.157 38.196 0.447 1.2053.70 60 4.272 38.490 0.444 1.2113.80 60 4.387 38.762 0.441 1.2173.90 60 4.503 39.015 0.438 1.2224.00 60 4.618 39.249 0.435 1.227

1.05 70 1.117 1.531 0.953 1.0251.10 70 1.171 3.050 0.912 0.9911.15 70 1.224 4.551 0.875 0.9621.20 70 1.277 6.027 0.842 0.9371.25 70 1.330 7.473 0.813 0.916

1.30 70 1.383 8.884 0.786 0.8971.35 70 1.437 10.258 0.762 0.8821.40 70 1.489 11.591 0.739 0.8681.45 70 1.543 12.881 0.719 0.8571.50 70 1.596 14.127 0.701 0.847

1.55 70 1.649 15.328 0.684 0.8381.60 70 1.703 16.483 0.668 0.8311.65 70 1.756 17.592 0.654 0.8251.70 70 1.809 18.656 0.641 0.8201.75 70 1.862 19.676 0.628 0.816

1.80 70 1.916 20.652 0.616 0.8131.82 70 1.968 21.586 0.606 0.8091.90 70 2.022 22.478 0.596 0.8071.95 70 2.075 23.331 0.586 0.8062.00 70 2.128 24.145 0.577 0.805


TABLE 5.2


Mx σ M1 δ My M2

2.10 70 2.235 25.664 0.561 0.80312.20 70 2.341 27.047 0.547 0.80282.30 70 2.447 28.308 0.534 0.80352.40 70 2.554 29.456 0.523 0.80472.50 70 2.660 30.503 0.513 0.806

2.60 70 2.767 31.459 0.504 0.80872.70 70 2.873 32.332 0.496 0.8112.80 70 2.979 33.132 0.488 0.8142.90 70 3.086 33.865 0.481 0.8163.00 70 3.193 34.537 0.475 0.819

3.10 70 3.299 35.156 0.469 0.8223.20 70 3.405 35.725 0.464 0.8253.30 70 3.512 36.251 0.459 0.8273.40 70 3.618 36.736 0.455 0.8303.50 70 3.725 37.185 0.451 0.833

3.60 70 3.831 37.602 0.447 0.8353.70 70 3.937 37.988 0.444 0.8373.80 70 4.044 38.347 0.441 0.8393.90 70 4.150 38.682 0.438 0.8424.00 70 4.257 38.993 0.435 0.844

1.05 80 1.066 0.821 0.953 0.9701.10 80 1.117 1.648 0.912 0.9311.15 80 1.168 2.478 0.875 0.8961.20 80 1.218 3.309 0.842 0.8651.25 80 1.269 4.138 0.812 0.838

1.30 80 1.320 4.963 0.786 0.8131.35 80 1.371 5.781 0.762 0.7911.40 80 1.422 6.591 0.739 0.7721.45 80 1.472 7.390 0.719 0.7541.50 80 1.523 8.177 0.701 0.738

TABLE 5.2


Mx σ M1 δ My M2

1.55 80 1.574 8.950 0.684 0.7231.60 80 1.625 9.710 0.668 0.7101.65 80 1.675 10.454 0.654 0.6981.70 80 1.726 11.182 0.641 0.6871.75 80 1.777 11.893 0.628 0.677

1.80 80 1.828 12.587 0.616 0.6681.85 80 1.878 13.263 0.605 0.6591.90 80 1.929 13.921 0.595 0.6521.95 80 1.980 14.561 0.586 0.6442.00 80 2.031 15.183 0.577 0.638

2.10 80 2.132 16.373 0.561 0.6262.20 80 2.234 17.491 0.547 0.6172.30 80 2.335 18.541 0.534 0.6082.40 80 2.437 19.525 0.523 0.6012.50 80 2.538 20.445 0.513 0.595

2.60 80 2.640 21.306 0.504 0.5892.70 80 2.741 22.110 0.496 0.5852.80 80 2.843 22.862 0.488 0.5812.90 80 2.945 23.564 0.481 0.5783.00 80 3.046 24.220 0.475 0.575

3.10 80 3.147 24.834 0.469 0.5723.20 80 3.249 25.408 0.464 0.5693.30 80 3.351 25.945 0.459 0.5673.40 80 3.452 26.447 0.455 0.5653.50 80 3.554 26.918 0.451 0.564

3.60 80 3.656 27.359 0.447 0.5633.70 80 3.757 27.773 0.444 0.5613.80 80 3.858 28.162 0.441 0.5603.90 80 3.960 28.527 0.438 0.5594.00 80 4.062 28.871 0.435 0.558


TABLE 6.1

Flow of perfect gases with friction (Fanno flow, γ = γ = γ = γ = γ = 1.3)

M p/p* c/c* = ρ*/ρ T/T* p0 /p0* F/F* 4 f L Dmax

0.02 53.617 0.021 1.150 29.268 23.324 1915.5100.04 26.806 0.043 1.150 14.644 11.679 474.4280.06 17.868 0.064 1.150 9.775 7.805 208.0520.08 13.398 0.086 1.149 7.342 5.874 115.0780.10 10.716 0.107 1.148 5.885 4.720 72.200

0.12 8.927 0.128 1.147 4.917 3.954 49.0200.14 7.648 0.150 1.146 4.228 3.410 35.1200.16 6.690 0.171 1.145 3.712 3.005 26.1570.18 5.943 0.193 1.144 3.312 2.693 20.0580.20 5.346 0.214 1.143 2.994 2.445 15.730

0.22 4.856 0.235 1.142 2.735 2.245 12.5620.24 4.450 0.256 1.140 2.520 2.071 10.1760.26 4.104 0.277 1.138 2.340 1.941 8.3410.28 3.807 0.298 1.136 2.186 1.825 6.9040.30 3.551 0.319 1.134 2.054 1.725 5.759

0.32 3.326 0.341 1.132 1.939 1.639 4.8370.34 3.127 0.362 1.130 1.839 1.564 4.0850.36 2.950 0.382 1.128 1.750 1.499 3.4650.38 2.792 0.403 1.126 1.672 1.442 2.9500.40 2.649 0.424 1.123 1.602 1.391 2.520

0.42 2.520 0.445 1.120 1.540 1.347 2.1570.44 2.403 0.465 1.118 1.484 1.308 1.8500.46 2.295 0.485 1.115 1.434 1.272 1.5880.48 2.196 0.506 1.111 1.389 1.241 1.3640.50 2.106 0.526 1.108 1.348 1.213 1.172

0.52 2.022 0.547 1.105 1.311 1.188 1.0070.54 1.944 0.567 1.102 1.277 1.166 0.8640.56 1.872 0.587 1.098 1.246 1.145 0.7410.58 1.804 0.607 1.095 1.219 1.127 0.6340.60 1.741 0.627 1.091 1.193 1.111 0.541

Flow of Perfect Gases with Friction 75

TABLE 6.1



0.62 1.682 0.646 1.087 1.170 1.097 0.4600.64 1.626 0.667 1.083 1.149 1.084 0.3900.66 1.574 0.686 1.079 1.130 1.072 0.3290.68 1.525 0.705 1.075 1.113 1.062 0.2760.70 1.479 0.725 1.071 1.097 1.052 0.231

0.72 1.435 0.744 1.067 1.083 1.044 0.1910.74 1.393 0.763 1.063 1.070 1.037 0.1570.76 1.354 0.782 1.058 1.059 1.030 0.1270.78 1.316 0.800 1.054 1.048 1.025 0.1020.80 1.280 0.820 1.049 1.039 1.020 0.081

0.82 1.246 0.838 1.045 1.032 1.016 0.0620.84 1.214 0.857 1.040 1.025 1.012 0.0470.86 1.183 0.875 1.035 1.019 1.009 0.0350.88 1.153 0.893 1.030 1.013 1.006 0.0240.90 1.125 0.911 1.025 1.009 1.004 0.016

0.92 1.098 0.929 1.020 1.006 1.003 0.0100.94 1.072 0.947 1.015 1.003 1.002 0.0050.96 1.047 0.965 1.010 1.001 1.001 0.00230.98 1.023 0.983 1.005 1.0004 1.0002 0.00051.00 1.000 1.000 1.000 1.000 1.000 0.000

1.02 0.978 1.017 0.995 1.0003 1.0001 0.00051.04 0.956 1.035 0.989 1.0014 1.0006 0.00201.06 0.936 1.051 0.984 1.003 1.001 0.00431.08 0.916 1.068 0.978 1.005 1.002 0.00741.10 0.897 1.085 0.973 1.008 1.003 0.0112

1.12 0.878 1.102 0.968 1.012 1.005 0.0151.14 0.861 1.118 0.962 1.016 1.006 0.0201.16 0.843 1.135 0.957 1.021 1.008 0.0261.18 0.827 1.151 0.951 1.026 1.010 0.0321.20 0.810 1.167 0.946 1.032 1.012 0.0382


TABLE 6.1



1.22 0.795 1.183 0.940 1.038 1.014 0.0451.24 0.780 1.198 0.934 1.046 1.017 0.0521.26 0.765 1.214 0.928 1.053 1.019 0.0591.28 0.751 1.230 0.923 1.062 1.022 0.0661.30 0.737 1.245 0.917 1.070 1.024 0.074

1.32 0.723 1.260 0.912 1.080 1.027 0.0821.34 0.710 1.275 0.906 1.089 1.030 0.0891.36 0.697 1.290 0.900 1.100 1.033 0.0981.38 0.685 1.305 0.894 1.111 1.036 0.1061.40 0.673 1.320 0.889 1.123 1.039 0.114

1.42 0.662 1.334 0.883 1.135 1.042 0.1231.44 0.650 1.349 0.877 1.147 1.045 0.1311.46 0.639 1.363 0.871 1.160 1.048 0.1391.48 0.629 1.377 0.866 1.175 1.052 0.1481.50 0.618 1.391 0.860 1.189 1.055 0.156

1.52 0.608 1.404 0.854 1.205 1.058 0.1651.54 0.598 1.418 0.848 1.220 1.062 0.1731.56 0.588 1.432 0.842 1.237 1.065 0.1821.58 0.579 1.445 0.837 1.254 1.068 0.1901.60 0.569 1.458 0.831 1.271 1.072 0.199

1.62 0.561 1.472 0.825 1.290 1.076 0.2071.64 0.552 1.485 0.819 1.308 1.079 0.2161.66 0.543 1.497 0.814 1.328 1.083 0.2241.68 0.535 1.510 0.808 1.348 1.086 0.2331.70 0.527 1.523 0.802 1.369 1.089 0.241

1.72 0.519 1.535 0.796 1.391 1.093 0.2491.74 0.511 1.547 0.791 1.413 1.097 0.2571.76 0.503 1.559 0.785 1.436 1.100 0.2651.78 0.496 1.572 0.780 1.459 1.104 0.2731.80 0.488 1.584 0.774 l.484 1.108 0.281


TABLE 6.1



1.82 0.482 1.595 0.768 1.509 1.111 0.2891.84 0.475 1.607 0.763 1.535 1.115 0.2971.86 0.467 1.618 0.757 1.562 1.118 0.3071.88 0.461 1.630 0.752 1.590 1.122 0.3131.90 0.455 1.641 0.746 1.618 1.125 0.320

1.92 0.448 1.652 0.741 1.647 1.129 0.3281.94 0.442 1.663 0.735 1.678 1.132 0.3351.96 0.436 1.674 0.729 1.708 1.136 0.3431.98 0.429 1.684 0.724 1.740 1.139 0.3502.00 0.424 1.696 0.719 1.773 1.143 0.357

2.02 0.418 1.706 0.713 1.807 1.146 0.3642.04 0.412 1.716 0.708 1.842 1.150 0.3722.06 0.407 1.727 0.703 1.878 1.153 0.3782.08 0.402 1.737 0.697 1.914 1.156 0.3862.10 0.396 1.747 0.692 1.951 1.160 0.393

2.12 0.391 1.757 0.687 1.990 1.163 0.3992.14 0.386 1.767 0.682 2.030 1.167 0.4062.16 0.381 1.777 0.676 2.070 1.169 0.4122.18 0.376 1.786 0.671 2.113 1.173 0.4192.20 0.371 1.796 0.667 2.156 1.176 0.426

2.22 0.366 1.805 0.661 2.200 1.179 0.4322.24 0.362 1.815 0.656 2.245 1.183 0.4382.26 0.357 1.824 0.651 2.292 1.186 0.4442.28 0.353 1.833 0.646 2.339 1.189 0.4512.30 0.348 1.842 0.641 2.388 1.192 0.457

2.32 0.344 1.851 0.636 2.438 1.196 0.4632.34 0.340 1.859 0.631 2.490 1.199 0.4692.36 0.335 1.868 0.627 2.543 1.202 0.4742.38 0.331 1.877 0.622 2.598 1.205 0.4802.40 0.327 1.885 0.617 2.654 1.209 0.486


TABLE 6.1



2.42 0.323 1.893 0.612 2.710 1.211 0.4922.44 0.319 1.902 0.608 2.770 1.214 0.4972.46 0.316 1.910 0.603 2.829 1.217 0.5032.48 0.312 1.918 0.598 2.891 1.220 0.5082.50 0.308 1.926 0.594 2.954 1.223 0.514

2.52 0.305 1.934 0.589 3.019 1.226 0.5192.54 0.300 1.942 0.585 3.086 1.228 0.5242.56 0.297 1.950 0.580 3.154 1.231 0.5292.58 0.294 1.957 0.575 3.224 1.234 0.5342.60 0.291 1.965 0.571 3.295 1.237 0.539

2.62 0.287 1.972 0.567 3.369 1.240 0.5442.64 0.284 1.980 0.562 3.443 1.243 0.5492.66 0.280 1.987 0.558 3.521 1.245 0.5542.68 0.278 1.994 0.554 3.601 1.248 0.5592.70 0.275 2.001 0.549 3.681 1.250 0.564

2.72 0.271 2.008 0.545 3.765 1.253 0.5682.74 0.268 2.015 0.541 3.849 1.256 0.5732.76 0.265 2.022 0.537 3.936 1.258 0.5782.78 0.263 2.029 0.533 4.025 1.261 0.5822.80 0.260 2.036 0.529 4.116 1.263 0.586

2.82 0.257 2.042 0.525 4.210 1.266 0.5912.84 0.254 2.048 0.520 4.306 1.268 0.5952.86 0.251 2.055 0.516 4.404 1.271 0.5992.88 0.248 2.062 0.512 4.505 1.273 0.6042.90 0.246 2.068 0.508 4.607 1.276 0.608

2.90 0.243 2.074 0.505 4.713 1.278 0.6112.94 0.241 2.080 0.500 4.820 1.281 0.6162.96 0.238 2.097 0.497 4.931 1.283 0.6192.98 0.235 2.093 0.493 5.045 1.285 0.6243.00 0.233 2.099 0.489 5.160 1.288 0.628


TABLE 6.1



3.10 0.221 2.127 0.471 5.781 1.299 0.6473.20 0.210 2.155 0.453 6.477 1.310 0.6643.30 0.200 2.181 0.437 7.259 1.319 0.6813.40 0.191 2.205 0.421 8.133 1.330 0.6973.50 0.182 2.228 0.405 9.110 1.338 0.711

3.60 0.174 2.250 0.391 10.200 1.347 0.7253.70 0.166 2.271 0.377 11.420 1.356 0.7383.80 0.159 2.290 0.363 12.790 1.363 0.7513.90 0.152 2.308 0.351 14.274 1.371 0.7624.00 0.145 2.326 0.338 15.940 1.378 0.773

4.10 0.139 2.343 0.327 17.796 1.385 0.7834.20 0.134 2.358 0.315 19.848 1.391 0.7924.30 0.128 2.374 0.305 22.116 1.398 0.8024.40 0.123 2.388 0.294 24.622 1.403 0.8114.50 0.118 2.402 0.285 27.390 1.409 0.819

4.60 0.114 2.415 0.276 30.433 1.414 0.8274.70 0.110 2.427 0.267 33.786 1.420 0.8344.80 0.105 2.438 0.258 37.472 1.424 0.8414.90 0.102 2.449 0.250 41.520 1.429 0.8485.00 0.098 2.460 0.242 45.950 1.433 0.854

6.00 0.071 2.543 0.179 120.1 1.468 0.9047.00 0.053 2.598 0.138 285.3 1.491 0.9368.00 0.041 2.635 0.108 625.2 1.507 0.9579.00 0.033 2.662 0.088 1275.0 1.519 0.972

10.00 0.0277 2.681 0.072 2438.0 1.527 0.983


TABLE 6.2



0.02 54.770 0.022 1.199 28.942 22.834 1778.4500.04 27.382 0.044 1.199 14.482 11.435 440.3520.06 18.251 0.066 1.199 9.666 7.643 193.0310.08 13.684 0.087 1.198 7.262 5.753 106.7180.10 10.943 0.109 1.197 5.822 4.624 66.922

0.12 9.116 0.131 1.196 4.864 3.875 45.4070.14 7.809 0.153 1.195 4.183 3.343 32.5110.16 6.830 0.175 1.194 3.673 2.950 24.1970.18 6.067 0.196 1.192 3.278 2.643 18.5430.20 5.455 0.218 1.190 2.964 2.400 14.533

0.22 4.955 0.239 1.188 2.708 2.205 11.5960.24 4.538 0.261 1.186 2.496 2.043 9.3870.26 4.185 0.283 1.184 2.317 1.909 7.6870.28 3.882 0.304 1.181 2.166 1.795 6.3570.30 3.619 0.326 1.178 2.035 1.698 5.299

0.32 3.388 0.347 1.176 1.922 1.615 4.4470.34 3.185 0.368 1.172 1.823 1.542 3.7520.36 3.004 0.389 1.169 1.736 1.479 3.1800.38 2.842 0.410 1.166 1.658 1.423 2.7050.40 2.695 0.431 1.163 1.590 1.375 2.308

0.42 2.563 0.452 1.159 1.529 1.332 1.9740.44 2.442 0.473 1.155 1.474 1.294 1.6920.46 2.333 0.494 1.151 1.425 1.260 1.4510.48 2.231 0.514 1.147 1.380 1.230 1.2450.50 2.138 0.534 1.143 1.340 1.203 1.069

0.52 2.052 0.555 1.138 1.303 1.179 0.9170.54 1.972 0.575 1.134 1.270 1.157 0.7870.56 1.897 0.595 1.129 1.240 1.138 0.6740.58 1.828 0.615 1.124 1.213 1.120 0.5750.60 1.763 0.635 1.119 1.188 1.105 0.491



TABLE 6.2



0.62 1.703 0.654 1.114 1.166 1.092 0.4170.64 1.645 0.674 1.109 1.145 1.079 0.3530.66 1.592 0.693 1.104 1.127 1.068 0.2980.68 1.541 0.713 1.098 1.109 1.058 0.2490.70 1.493 0.732 1.093 1.094 1.049 0.208

0.72 1.448 0.751 1.087 1.081 1.041 0.1720.74 1.405 0.770 1.082 1.068 1.035 0.1410.76 1.365 0.788 1.076 1.057 1.028 0.1140.78 1.326 0.807 1.069 1.047 1.023 0.0920.80 1.289 0.825 1.064 1.038 1.018 0.073

0.82 1.254 0.843 1.058 1.031 1.015 0.0560.84 1.221 0.861 1.051 1.024 1.011 0.0420.86 1.189 0.879 1.045 1.018 1.008 0.0310.88 1.158 0.897 1.039 1.013 1.006 0.0220.90 1.129 0.915 1.003 1.009 1.004 0.015

0.92 1.101 0.932 1.026 1.006 1.003 0.00890.94 1.074 0.949 1.020 1.003 1.001 0.0050.96 1.048 0.966 1.013 1.001 1.0006 0.00200.98 1.024 0.983 1.006 1.0003 1.0001 0.00051.00 1.000 1.000 1.000 1.000 1.000 0.000

1.02 0.977 1.016 0.993 1.0003 1.0001 0.000461.04 0.955 1.033 0.986 1.0013 1.0005 0.00181.06 0.933 1.049 0.979 1.003 1.0012 0.0041.08 0.913 1.065 0.973 1.005 1.002 0.00661.10 0.894 1.081 0.966 1.008 1.003 0.0099

1.12 0.875 1.097 0.959 1.011 1.004 0.01381.14 0.856 1.113 0.952 1.015 1.006 0.01821.16 0.838 1.128 0.945 1.020 1.007 0.0231.18 0.821 1.143 0.938 1.025 1.009 0.0281.20 0.804 1.158 0.932 1.030 1.011 0.034


TABLE 6.2


M p/p* c/c* = ρ*/ρ T/T* p0/p0* F/F* 4 f L Dmax

1.22 0.788 1.173 0.925 1.037 1.013 0.0391.24 0.773 1.188 0.918 1.043 1.015 0.0451.26 0.757 1.202 0.911 1.050 1.017 0.05171.28 0.743 1.217 0.903 1.058 1.019 0.05821.30 0.728 1.231 0.897 1.066 1.022 0.065

1.32 0.715 1.245 0.889 1.075 1.024 0.07161.34 0.701 1.259 0.883 1.084 1.027 0.07851.36 0.688 1.273 0.876 1.094 1.029 0.08551.38 0.676 1.286 0.869 1.104 1.032 0.0931.40 0.663 1.300 0.862 1.115 1.035 0.099

1.42 0.651 1.313 0.855 1.126 1.037 0.1071.44 0.640 1.326 0.848 1.138 1.040 0.1141.46 0.628 1.339 0.841 1.150 1.043 0.1211.48 0.617 1.352 0.834 1.163 1.046 0.1281.50 0.606 1.364 0.828 1.176 1.048 0.136

1.52 0.596 1.377 0.821 1.189 1.052 0.1431.54 0.586 1.389 0.814 1.204 1.055 0.1511.56 0.576 1.401 0.807 1.219 1.057 0.1581.58 0.566 1.413 0.800 1.234 1.061 0.1651.60 0.557 1.425 0.794 1.250 1.063 0.172

1.62 0.547 1.437 0.787 1.267 1.067 0.1791.64 0.538 1.448 0.780 1.284 1.069 0.1871.66 0.529 1.460 0.774 1.301 1.073 0.1941.68 0.521 1.471 0.767 1.319 1.075 0.2011.70 0.513 1.482 0.760 1.340 1.078 0.208

1.72 0.505 1.493 0.754 1.357 1.082 0.2151.74 0.496 1.504 0.747 1.376 1.085 0.2221.76 0.489 1.515 0.741 1.397 1.088 0.2281.78 0.481 1.525 0.735 1.418 1.091 0.2351.80 0.474 1.536 0.728 1.439 1.093 0.242


TABLE 6.2



1.82 0.467 1.546 0.722 1.461 1.096 0.2481.84 0.459 1.556 0.716 1.484 1.099 0.2551.86 0.453 1.566 0.709 1.507 1.102 0.2621.88 0.446 1.576 0.703 1.530 1.105 0.2681.90 0.439 1.586 0.697 1.555 1.108 0.274

1.92 0.433 1.596 0.691 1.580 1.111 0.2811.94 0.426 1.605 0.685 1.610 1.114 0.2871.96 0.420 1.615 0.678 1.633 1.117 0.2931.99 0.414 1.624 0.673 1.660 1.120 0.2992.00 0.408 1.633 0.667 1.687 1.123 0.305

2.02 0.402 1.642 0.661 1.716 1.126 0.3112.04 0.397 1.651 0.655 1.747 1.128 0.3172.06 0.391 1.659 0.649 1.775 1.131 0.3232.08 0.386 1.668 0.643 1.806 1.134 0.3282.10 0.380 1.677 0.637 1.837 1.137 0.334

2.12 0.375 1.685 0.632 1.870 1.139 0.3392.14 0.369 1.694 0.623 1.902 1.142 0.3452.16 0.365 1.702 0.621 1.935 1.145 0.3502.18 0.359 1.710 0.615 1.970 1.147 0.3562.20 0.355 1.718 0.609 2.005 1.150 0.361

2.22 0.350 1.726 0.604 2.041 1.153 0.3662.24 0.346 1.733 0.598 2.078 1.155 0.3712.26 0.341 1.741 0.594 2.115 1.158 0.3762.28 0.336 1.748 0.588 2.154 1.160 0.3812.30 0.332 1.756 0.583 2.193 1.163 0.386

2.32 0.327 1.764 0.578 2.233 1.165 0.3912.34 0.323 1.771 0.573 2.274 1.168 0.3962.36 0.319 1.778 0.567 2.316 1.170 0.4002.38 0.315 1.785 0.563 2.360 1.173 0.4052.40 0.311 1.792 0.557 2.403 1.175 0.409


TABLE 6.2



2.42 0.307 1.799 0.553 2.448 1.178 0.4142.44 0.303 1.806 0.548 2.494 1.179 0.4192.46 0.299 1.813 0.543 2.540 1.182 0.4232.48 0.296 1.819 0.538 2.588 1.185 0.4272.50 0.292 1.826 0.533 2.637 1.187 0.432

2.52 0.289 1.832 0.528 2.687 1.189 0.4362.54 0.285 1.838 0.524 2.737 1.191 0.4402.56 0.281 1.845 0.519 2.789 1.193 0.4452.58 0.278 1.851 0.515 2.842 1.195 0.4482.60 0.275 1.857 0.510 2.896 1.198 0.453

2.62 0.271 1.863 0.506 2.951 1.200 0.4572.64 0.268 1.869 0.501 3.007 1.202 0.4602.66 0.265 1.875 0.497 3.065 1.204 0.4642.68 0.262 1.881 0.493 3.123 1.206 0.4682.70 0..258 1.886 0.488 3.183 1.208 0.472

2.72 0.256 1.892 0.484 3.244 1.210 0.4762.74 0.253 1.898 0.479 3.306 1.212 0.4792.76 0.250 1.903 0.476 3.370 1.214 0.4832.79 0.247 1.908 0.471 3.435 1.216 0.4862.80 0.244 1.914 0.467 3.500 1.218 0.489

2.82 0.241 1.919 0.463 3.567 1.220 0.4932.84 0.238 1.925 0.459 3.636 1.222 0.4972.86 0.236 1.929 0.455 3.706 1.224 0.4992.88 0.233 1.935 0.451 3.780 1.226 0.5032.90 0.231 1.940 0.447 3.850 1.228 0.506

2.92 0.228 1.945 0.443 3.924 1.229 0.5092.94 0.226 1.949 0.439 4.000 1.231 0.5132.96 0.223 1.954 0.436 4.076 1.233 0.5162.98 0.220 1.959 0.432 4.155 1.235 0.5193.00 0.218 1.964 0.428 4.235 1.237 0.522


TABLE 6.2



3.10 0.207 1.987 0.411 4.657 1.245 0.5373.20 0.196 2.008 0.394 5.121 1.253 0.5503.30 0.186 2.028 0.377 5.628 1.260 0.5633.40 0.177 2.046 0.362 6.184 1.267 0.5753.50 0.168 2.064 0.348 6.789 1..274 0.586

3.60 0.160 2.081 0.334 7.450 1.281 0.5973.70 0.153 2.096 0.321 8.169 1.287 0.6073.80 0.146 2.111 0.308 8.951 1.293 0.6163.90 0.139 2.125 0.297 9.800 1.298 0.6254.00 0.134 2.138 0.286 10.720 1.303 0.633

4.10 0.128 2.150 0.275 11.715 1.308 0.6414.20 0.123 2.162 0.265 12.792 1.312 0.6484.30 0.117 2.173 0.255 13.955 1.317 0.6554.40 0.113 2.184 0.246 15.210 1.321 0.6624.50 0.108 2.194 0.237 16.562 1.325 0.668

4.60 0.104 2.203 0.229 18.018 1.330 0.6734.70 0.100 2.212 0.221 19.583 1.332 0.6784.80 0.096 2.220 0.214 21.264 1.335 0.6844.90 0.093 2.228 0.207 23.070 1.339 0.6895.00 0.089 2.236 0.200 25.000 1.342 0.694

6.00 0.064 2.295 0.146 53.180 1.366 0.7307.00 0.047 2.333 0.111 104.14 1.381 0.7538.00 0.037 2.359 0.087 190.11 1.392 0.7689.00 0.029 2.377 0.069 327.19 1.399 0.779

10.00 0.024 2.391 0.057 535.94 1.404 0.787


TABLE 6.3



0.02 57.753 0.023 1.333 28.132 21.662 1493.0720.04 28.862 0.046 1.332 14.078 10.851 369.4050.06 19.235 0.069 1.332 9.397 7.256 161.7620.08 14.420 0.092 1.331 7.061 5.465 89.3200.10 11.535 0.115 1.330 5.661 4.392 55.830

0.12 9.600 0.138 1.327 4.733 3.686 37.8920.14 8.220 0.161 1.325 4.071 3.183 27.0860.16 7.187 0.184 1.322 3.576 2.810 20.1250.18 6.381 0.207 1.319 3.193 2.532 15.3930.20 5.739 0.229 1.317 2.887 2.293 12.110

0.22 5.207 0.252 1.313 2.640 2.110 9.5890.24 4.766 0.275 1.308 2.435 1.959 7.7470.26 4.392 0.297 1.304 2.262 1.833 6.3320.28 4.071 0.319 1.300 2.115 1.726 5.2250.30 3.795 0.342 1.296 1.989 1.635 4.337

0.32 3.549 0.364 1.289 1.880 1.558 3.6390.34 3.333 0.385 1.284 1.785 1.490 3.0630.36 3.141 0.407 1.278 1.700 1.432 2.5910.38 2.968 0.429 1.273 1.626 1.381 2.2000.40 2.814 0.450 1.267 1.560 1.336 1.868

0.42 2.672 0.472 1.260 1.501 1.297 1.5970.44 2.545 0.493 1.253 1.450 1.262 1.3650.46 2.427 0.514 1.245 1.402 1.231 1.1680.48 2.318 0.534 1.238 1.359 1.203 1.00050.50 2.220 0.555 1.232 1.320 1.178 0.855

0.52 2.127 0.575 1.223 1.286 1.157 0.7340.54 2.042 0.596 1.215 1.255 1.138 0.6280.56 1.962 0.616 1.807 1.225 1.120 0.5360.58 1.888 0.635 1.200 1.200 1.105 0.4570.60 1.819 0.655 1.191 1.176 1.091 0.388


TABLE 6.3



0.62 1.754 0.674 1.182 1.155 1.079 0.3290.64 1.693 0.693 1.173 1.135 1.068 0.2780.66 1.635 0.712 1.165 1.118 1.058 0.2340.68 1.581 0.731 1.155 1.102 1.050 0.1960.70 1.530 0.750 1.146 1.087 1.042 0.163

0.72 1.481 0.768 1.137 1.075 1.035 0.1340.74 1.435 0.786 1.128 1.063 1.029 0.1100.76 1.391 0.804 1.118 1.053 1.024 0.0890.78 1.350 0.821 1.108 1.043 1.019 0.0710.80 1.311 0.839 1.099 1.035 1.016 0.056

0.82 1.273 0.856 1.089 1.028 1.012 0.0430.84 1.237 0.873 1.079 1.022 1.010 0.03250.86 1.203 0.889 1.070 1.016 1.007 0.02380.88 1.170 0.906 1.060 1.012 1.005 0.01670.90 1.139 0.922 1.050 1.008 1.003 0.011

0.92 1.108 0.938 1.040 1.005 1.002 0.0070.94 1.080 0.954 1.030 1.003 1.001 0.00370.96 1.052 0.969 1.020 1.0012 1.0005 0.001560.98 1.026 0.985 1.010 1.0003 1.0001 0.000371.00 1.000 1.000 1.000 1.000 1.000 0.000

1.02 0.975 1.015 0.990 1.0003 1.0001 0.000351.04 0.952 1.029 0.980 1.0012 1.0004 0.00131.06 0.929 1.044 0.970 1.0026 1.001 0.00291.08 0.907 1.058 0.960 1.005 1.0016 0.0051.10 0.886 1.072 0.950 1.007 1.002 0.0074

1.12 0.866 1.086 0.940 1.010 1.0034 0.0101.14 0.846 1.099 0.930 1.014 1.0045 0.01351.16 0.827 1.113 0.920 1.018 1.006 0.0171.18 0.809 1.126 0.911 1.022 1.007 0.0211.20 0.791 1.139 0.900 1.027 1.008 0.025


TABLE 6.3



1.22 0.774 1.151 0.891 1.032 1.010 0.02911.24 0.757 1.164 0.882 1.038 1.012 0.03351.26 0.741 1.176 0.872 1.044 1.013 0.0381.28 0.726 1.188 0.862 1.051 1.015 0.0431.30 0.710 1.200 0.852 1.057 1.017 0.047

J.32 0.696 1.211 0.843 1.065 1.019 0.0521.34 0.682 1.223 0.834 1.073 1.021 0.0571.36 0.668 1.235 0.825 1.081 1.022 0.0621.38 0.654 1.246 0.816 1.089 1.024 0.0671.40 0.641 1.257 0.806 1.098 1.026 0.072

1.42 0.629 1.268 0.797 1.108 1.028 0.0781.44 0.616 1.278 0.788 1.117 1.030 0.0831.46 0.605 1.288 0.780 1.127 1.032 0.0881.48 0.593 1.298 0.771 1.138 1.035 0.0931.50 0.582 1.309 0.761 1.148 1.036 0.0975

1.52 0.571 1.318 0.753 1.160 1.039 0.1031.54 0.560 1.328 0.745 1.171 1.041 0.1081.56 0.550 1.338 0.736 1.183 1.043 0.1131.58 0.540 1.347 0.728 1.195 1.045 0.1181.60 0.530 1.356 0.718 1.207 1.047 0.123

1.62 0.521 1.365 0.711 1.220 1.049 0.1281.64 0.511 1.374 0.703 1.234 1.051 0.1331.66 0.502 1.383 0.695 1.250 1.053 0.1381.68 0.493 1.391 0.687 1.261 1.056 0.1421.70 0.485 1.400 0.678 1.275 1.057 0.146

1.72 0.476 1.408 0.671 1.290 1.060 0.1521.74 0.468 1.416 0.664 1.305 1.0615 0.1561.76 0.460 1.425 0.656 1.320 1.0635 0.1611.78 0.452 1.432 0.648 1.336 1.065 0.1651.80 0.445 1.440 0.640 1.351 1.067 0.169


TABLE 6.3



1.82 0.437 1.448 0.634 1.370 1.070 0.1741.84 0.430 1.455 0.626 1.385 1.0715 0.1781.86 0.423 1.463 0.619 1.402 1.073 0.1831.88 0.416 1.469 0.612 1.419 1.075 0.1871.90 0.409 1.477 0.604 1.436 1.077 0.191

1.92 0.403 1.484 0.598 1.455 1.079 0.1951.94 0.396 1.491 0.591 1.474 1.081 0.1991.96 0.390 1.497 0.585 1.492 1.083 0.2031.98 0.384 1.504 0.578 1.511 1.085 0.2072.00 0.378 1.510 0.571 1.530 1.086 0.211

2.02 0.372 1.517 0.565 1.551 1.088 0.2152.04 0.366 1.523 0.558 1.571 1.090 0.2192.06 0.361 1.530 0.552 1.592 1.092 0.2222.08 0.355 1.535 0.545 1.613 1.094 0.2262.10 0.350 1.541 0.539 1.632 1.095 0.229

2.12 0.345 1.547 0.534 1.656 1.097 0.2332.14 0.339 1.553 0.527 1.678 1.099 0.2372.16 0.335 1.559 0.522 1.700 1.101 0.2402.18 0.329 1.565 0.516 1.723 1.102 0.2442.20 0.324 1.570 0.509 1.744 1.104 0.246

2.22 0.320 1.575 0.505 1.770 1.106 0.2502.24 0.315 1.581 0.499 1.793 1.107 0.2542.26 0.311 1.586 0.493 1.818 1.108 0.2572.28 0.306 1.591 0.487 1.842 1.110 0.2592.30 0.302 1.596 0.482 1.865 1.111 0.262

2.32 0.298 1.601 0.477 1.893 1.113 0.2652.34 0.294 1.606 0.472 1.920 1.115 0.2692.36 0.289 1.611 0.467 1.945 1.116 0.2722.38 0.285 1.615 0.461 1.971 1.118 0.2752.40 0.281 1.620 0.456 1.995 1.119 0.277


TABLE 6.3



2.42 0.278 1.625 0.452 2.025 1.121 0.2802.44 0.274 1.629 0.446 2.053 1.122 0.2832.46 0.270 1.634 0.442 2.081 1.123 0.2862.48 0.267 1.638 0.437 2.110 1.125 0.2882.50 0.263 1.642 0.432 2.135 1.126 0.290

2.50 0.260 1.646 0.428 2.168 1.127 0.2942.54 0.256 1.651 0.423 2.198 1.129 0.2962.56 0.253 1.655 0.418 2.228 1.130 0.2992.58 0.249 1.658 0.414 2.260 1.131 0.3012.60 0.246 1.663 0.409 2.285 1.132 0.303

2.62 0.243 1.667 0.405 2.321 1.134 0.3062.64 0.240 1.670 0.401 2.352 1.135 0.3082.66 0.236 1.674 0.397 2.385 1.136 0.3112.68 0.234 1.678 0.393 2.417 1.1375 0.3132.70 0.231 1.682 0.388 2.445 1.138 0.314

2.72 0.228 1.685 0.385 2.484 1.139 0.3172.74 0.225 1.688 0.381 2.518 1.141 0.3192.76 0.222 1.692 0.377 2.552 1.142 0.3222.78 0.219 1.694 0.373 2.587 1.143 0.3242.80 0.217 1.699 0.368 2.616 1.144 0.325

2.82 0.214 1.702 0.365 2.658 1.145 0.3282.84 0.211 1.705 0.361 2.694 1.146 0.3302.86 0.209 1.709 0.358 2.730 1.148 0.3322.88 0.207 1.712 0.354 2.767 1.1485 0.3342.90 0.204 1.715 0.350 2.797 1.149 0.335

2.92 0.202 1.718 0.347 2.843 1.150 0.3382.94 0.199 1.721 0.343 2.881 1.152 0.3402.96 0.197 1.724 0.340 2.920 1.153 0.3422.98 0.195 1.727 0.337 2.960 1.1537 0.3433.00 0.192 1.730 0.333 2.990 1.154 0.344

TABLE 6.3

Flow of perfect gases with friction (γ = γ = γ = γ = γ = 1.67)


3.10 0.182 1.744 0.317 3.205 1.159 0.3543.20 0.172 1.756 0.302 3.423 1.164 0.3623.30 0.163 1.768 0.288 3.653 1.168 0.3693.40 0.154 1.780 0.275 3.895 1.172 0.3763.50 0.146 1.790 0.262 4.134 1.174 0.381

3.60 0.139 1.800 0.251 4.420 1.178 0.3893.70 0.132 1.808 0.239 4.703 1.182 0.3943.80 0.126 1.817 0.229 5.000 1.185 0.4003.90 0.120 1.825 0.219 5.311 1.187 0.4044.00 0.1145 1.833 0.210 5.608 1.189 0.407

4.10 0.109 1.840 0.202 5.979 1.193 0.4134.20 0.105 1.846 0.194 6.336 1.195 0.4174.30 0.100 1.852 0.186 6.708 1.197 0.4214.40 0.096 1.858 0.178 7.098 1.199 0.42464.50 0.092 1.864 0.172 7.456 1.200 0.426

4.60 0.088 1.868 0.166 7.925 1.203 0.4314.70 0.085 1.873 0.159 8.365 1.205 0.4344.80 0.082 1.878 0.154 8.827 1.207 0.4374.90 0.079 1.883 0.150 9.300 1.208 0.4395.00 0.075 1.887 0.142 9.721 1.209 0.440

6.00 0.053 1.918 0.102 15.680 1.220 0.4597.00 0.040 1.938 0.077 23.850 1.227 0.4728.00 0.030 1.951 0.059 34.580 1.232 0.4799.00 0.024 1.960 0.047 48.240 1.235 0.485

10.00 0.0199 1.967 0.038 65.180 1.238 0.489


TABLE 7.1

Isothermal flow of perfect gases with friction (γ = γ = γ = γ = γ = 1.3)

M p/p*t c/c*t = ρ*t/ρ T0 /T0*t p0 /p0*

t 4 f L Dmax

0.02 43.853 0.0228 0.897 27.328 1914.5150.04 21.926 0.0456 0.897 13.674 473.5930.06 14.617 0.0684 0.897 9.128 207.3110.08 10.963 0.0912 0.897 6.858 114.4030.10 8.770 0.1140 0.897 5.499 71.580

0.12 7.308 0.1368 0.898 4.596 48.4400.14 6.264 0.1596 0.899 3.953 34.5760.16 5.481 0.1824 0.900 3.472 25.6450.18 4.872 0.2052 0.900 3.100 19.5740.20 4.385 0.2280 0.902 2.804 15.274

0.22 3.986 0.250 0.903 2.563 12.1270.24 3.654 0.273 0.904 2.363 9.7620.26 3.373 0.296 0.905 2.195 7.9470.28 3.132 0.319 0.907 2.053 6.5280.30 2.923 0.342 0.908 1.930 5.401

0.32 2.741 0.365 0.910 1.824 4.4950.34 2.579 0.387 0.912 1.731 3.7590.36 2.436 0.410 0.914 1.650 3.1540.39 2.308 0.433 0.916 1.578 2.6540.40 2.192 0.456 0.918 1.514 2.237

0.42 2.088 0.478 0.920 1.457 1.8880.44 1.993 0.501 0.923 1.405 1.5930.46 1.907 0.524 0.925 1.360 1.3440.48 1.827 0.547 0.927 1.319 1.1330.50 1.754 0.570 0.930 1.282 0.953

0.52 1.686 0.593 0.933 1.248 0.7990.54 1.624 0.615 0.936 1.218 0.6680.56 1.566 0.638 0.938 1.190 0.5550.58 1.512 0.661 0.942 1.166 0.4590.60 1.461 0.684 0.945 1.144 0.377

Isothermal Flow of Perfect Gases with Friction 93

TABLE 7.1



t 4 f L Dmax

0.62 1.415 0.707 0.948 1.124 0.3070.64 1.370 0.729 0.952 1.105 0.2480.66 1.329 0.753 0.955 1.089 0.1970.68 1.289 0.775 0.958 1.074 0.1540.70 1.253 0.798 0.962 1.061 0.119

0.72 1.218 0.821 0.966 1.049 0.0890.74 1.185 0.844 0.970 1.039 0.0650.76 1.154 0.866 0.974 1.031 0.0450.78 1.124 0.889 0.978 1.023 0.0290.80 1.096 0.912 0.983 1.016 0.018

0.82 1.069 0.935 0.987 1.010 0.00950.84 1.044 0.958 0.991 1.006 0.00380.86 1.019 0.980 0.996 1.002 0.00780.88 0.997 1.003 1.0007 0.999 0.22 × 10−4

0.90 0.974 1.026 1.005 0.998 0.0013

0.92 0.953 1.049 1.010 0.997 0.00440.94 0.933 1.072 1.015 0.997 0.00920.96 0.913 1.095 1.020 0.9975 0.0150.98 0.895 1.117 1.025 0.999 0.0231.00 0.877 1.140 1.031 1.001 0.0316

1.02 0.859 1.163 1.036 1.004 0.04131.04 0.843 1.185 1.042 1.008 0.0521.06 0.827 1.208 1.047 1.012 0.06351.08 0.812 1.231 1.053 1.017 0.0761.10 0.797 1.254 1.059 1.023 0.088

1.12 0.783 1.277 1.065 1.029 0.1021.14 0.769 1.299 1.071 1.037 0.1161.16 0.756 1.322 1.078 1.045 0.1311.18 0.743 1.345 1.083 1.053 0.1461.20 0.731 1.368 1.090 1.063 0.161


TABLE 7.1



t 4 f L Dmax

1.22 0.718 1.391 1.097 1.073 0.1771.24 0.707 1.414 1.103 1.083 0.1931.26 0.696 1.437 1.110 1.094 0.2091.28 0.685 1.459 1.116 1.106 0.2261.30 0.674 1.482 1.124 1.119 0.242

1.32 0.664 1.505 1.131 1.132 0.2591.34 0.654 1.528 1.138 1.146 0.2761.36 0.645 1.550 1.145 1.161 0.2931.38 0.635 1.573 1.152 1.176 0.3101.40 0.626 1.596 1.160 1.192 0.328

1.42 0.617 1.619 1.168 1.209 0.3451.44 0.609 1.642 1.175 1.227 0.3621.46 0.601 1.664 1.183 1.245 0.3801.48 0.592 1.687 1.191 1.264 0.3971.50 0.594 1.710 1.199 1.284 0.415

1.52 0.577 1.733 1.207 1.305 0.4331.54 0.569 1.756 1.215 1.327 0.4501.56 0.562 1.778 1.224 1.349 0.4681.58 0.555 1.801 1.232 1.372 0.4851.60 0.548 1.824 1.241 1.396 0.503

1.62 0.541 1.847 1.249 1.421 0.5201.64 0.534 1.869 1.258 1.447 0.5381.66 0.528 1.893 1.267 1.474 0.5551.68 0.522 1.915 1.276 1.502 0.5721.70 0.516 1.938 1.285 1.530 0.589

1.72 0.510 1.961 1.294 1.560 0.6071.74 0.504 1.984 1.303 1.591 0.6241.76 0.498 2.007 1.313 1.622 0.6411.78 0.493 2.029 1.323 1.655 0.6581.80 0.487 2.052 1.132 1.689 0.675


TABLE 7.1



t 4 f L Dmax

1.82 0.482 2.075 1.342 1.724 0.6921.84 0.476 2.098 1.352 1.760 0.7091.86 0.471 2.121 1.362 1.797 0.7261.88 0.466 2.143 1.372 1.836 0.7421.90 0.461 2.166 1.382 1.876 0.759

1.92 0.457 2.189 1.392 1.917 0.7751.94 0.452 2.212 1.403 1.959 0.7921.96 0.447 2.235 1.413 2.003 0.8081.98 0.443 2.257 1.424 2.048 0.8252.00 0.438 2.280 1.434 2.094 0.841

2.02 0.434 2.303 1.445 2.142 0.8572.04 0.430 2.326 1.456 2.191 0.8732.06 0.426 2.348 1.467 2.242 0.8892.08 0.422 2.372 1.478 2.294 0.9052.10 0.417 2.394 1.489 2.348 0.921

2.12 0.414 2.417 1.501 2.404 0.9362.14 0.409 2.440 1.512 2.462 0.9522.16 0.406 2.463 1.524 2.521 0.9672.18 0.402 2.485 1.536 2.581 0.9832.20 0.398 2.508 1.547 2.644 0.998

2.22 0.395 2.531 1.559 2.708 1.0132.24 0.392 2.554 1.571 2.775 1.0282.26 0.388 2.577 1.583 2.843 1.0432.28 0.385 2.599 1.596 2.914 1.0582.30 0.381 2.622 1.608 2.986 1.073

2.32 0.378 2.645 1.620 3.061 1.0882.34 0.375 2.668 1.633 3.138 1.1032.36 0.372 2.691 1.645 3.217 1.1172.38 0.368 2.714 1.658 3.298 1.1322.40 0.365 2.736 1.671 3.383 1.147


TABLE 7.1



t 4 f L Dmax

2.42 0.362 2.759 1.684 3.469 1.1612.44 0.359 2.782 1.697 3.557 1.1752.46 0.356 2.805 1.710 3.649 1.1892.48 0.354 2.827 1.724 3.743 1.2042.50 0.351 2.850 1.737 3.839 1.218

2.52 0.348 2.873 1.750 3.939 1.2322.54 0.345 2.896 1.764 4.042 1.2462.56 0.343 2.919 1.778 4.147 1.2592.58 0.340 2.942 1.792 4.255 1.2732.60 0.337 2.964 1.805 4.366 1.287

2.62 0.334 2.987 1.819 4.481 1.3012.64 0.332 3.010 1.834 4.599 1.3142.66 0.329 3.033 1.848 4.720 1.3282.68 0.327 3.055 1.862 4.845 1.3412.70 0.325 3.078 1.877 4.973 1.354

2.72 0.322 3.101 1.891 5.105 1.3672.74 0.320 3.124 1.906 5.240 1.3812.76 0.318 3.147 1.921 5.379 1.3942.78 0.315 3.169 1.936 5.522 1.4072.80 0.313 3.192 1.951 5.669 1.419

2.82 0.311 3.215 1.966 5.821 1.4322.84 0.309 3.238 1.981 5.976 1.4452.86 0.307 3.261 1.996 6.136 1.4582.88 0.304 3.283 2.012 6.300 1.4712.90 0.302 3.306 2.027 6.469 1.483

2.92 0.300 3.329 2.043 6.642 1.4952.94 0.298 3.352 2.059 6.821 1.5082.96 0.296 3.375 2.075 7.004 1.5212.98 0.294 3.398 2.091 7.192 1.5333.00 0.292 3.421 2.107 7.385 1.545


TABLE 7.1



t 4 f L Dmax

3.10 0.283 3.534 2.189 8.433 1.6053.20 0.274 3.648 2.274 9.631 1.6643.30 0.266 3.763 2.361 10.998 1.7213.40 0.258 3.876 2.451 12.555 1.7763.50 0.251 3.991 2.544 14.327 1.831

3.60 0.243 4.105 2.639 16.341 1.8833.70 0.237 4.218 2.737 18.626 1.9353.80 0.231 4.332 2.838 21.214 1.9853.90 0.225 4.447 2.942 24.142 2.0354.00 0.219 4.561 3.048 27.449 2.083

4.10 0.214 4.675 3.157 31.181 2.1304.20 0.209 4.789 3.269 35.384 2.1764.30 0.204 4.903 3.383 40.112 2.2214.40 0.199 5.017 3.500 45.423 2.2654.50 0.195 5.131 3.619 51.380 2.308

4.60 0.191 5.245 3.742 58.053 2.3514.70 0.187 5.359 3.867 65.517 2.3924.80 0.183 5.473 3.995 73.854 2.4334.90 0.179 5.587 4.125 83.155 2.4725.00 0.175 5.701 4.258 93.517 2.512

5.50 0.159 6.271 4.965 165.267 2.6976.00 0.146 6.841 5.738 283.672 2.8676.50 0.135 7.411 6.578 473.496 3.0247.00 0.125 7.981 7.486 769.836 3.1697.50 0.117 8.551 8.461 1221.350 3.306

8.00 0.109 9.121 9.503 1894.190 3.4338.50 0.103 9.691 10.613 2876.708 3.5539.00 0.097 10.262 11.789 4285.063 3.6679.50 0.092 10.831 13.033 6269.795 3.773

10.00 0.0877 11.402 14.344 9020.438 3.875


TABLE 7.2



t 4 f L Dmax

0.02 42.257 0.0237 0.8751 26.488 1777.2270.04 21.128 0.0473 0.8752 13.255 439.3270.06 14.086 0.0710 0.8756 8.849 192.1220.08 10.564 0.0946 0.8761 6.650 105.8920.10 8.451 0.1183 0.8770 5.333 66.159

0.12 7.043 0.142 0.8775 4.458 44.6990.14 6.037 0.165 0.8784 3.835 31.8470.16 5.282 0.189 0.8795 3.369 23.5730.18 4.695 0.213 0.881 3.010 17.9520.20 4.226 0.237 0.882 2.723 13.975

0.22 3.841 0.260 0.883 2.490 11.0660.24 3.521 0.284 0.885 2.297 8.8830.26 3.251 0.307 0.887 2.135 7.2080.28 3.018 0.331 0.888 1.997 5.9010.30 2.817 0.355 0.891 1.879 4.865

0.32 2.641 0.378 0.893 1.777 4.0330.34 2.486 0.402 0.895 1.687 3.3580.36 2.347 0.426 0.897 1.609 2.8050.38 2.224 0.449 0.900 1.539 2.3480.40 2.113 0.473 0.903 1.478 1.968

0.42 2.012 0.497 0.906 1.424 1.6510.44 1.921 0.521 0.909 1.375 1.3840.46 1.837 0.544 0.912 1.331 1.1590.48 1.761 0.568 0.915 1.292 0.9680.50 1.690 0.592 0.918 1.256 0.807

0.52 1.625 0.615 0.922 1.225 0.6700.54 1.565 0.639 0.926 1.196 0.5530.56 1.509 0.663 0.929 1.170 0.4540.58 1.457 0.686 0.934 1.147 0.3700.60 1.408 0.710 0.938 1.126 0.299


TABLE 7.2



t 4 f L Dmax

0.62 1.363 0.733 0.942 1.107 0.2380.64 1.320 0.757 0.947 1.090 0.1880.66 1.280 0.781 0.951 1.075 0.1450.68 1.243 0.805 0.956 1.061 0.10980.70 1.207 0.828 0.961 1.049 0.0808

0.72 1.174 0.852 0.966 1.039 0.05730.74 1.142 0.876 0.971 1.029 0.03860.76 1.112 0.899 0.976 1.022 0.02420.78 1.083 0.923 0.981 1.015 0.01350.80 1.056 0.946 0.987 1.009 62.5 × 10−4

0.82 1.031 0.970 0.993 1.004 18.6 × 10−4

0.84 1.006 0.994 0.998 1.001 0.75 × 10−4

0.86 0.983 1.017 1.004 0.998 6.0 × 10−4

0.88 0.960 1.041 1.010 0.996 31.8 × 10−4

0.90 0.939 1.065 1.017 0.995 75.8 × 10−4

0.92 0.918 1.088 1.023 0.995 1.36 × 10−2

0.94 0.899 1.112 1.029 0.9958 2.11 × 10−2

0.96 0.880 1.135 1.036 0.997 2.98 × 10−2

0.98 0.962 1.159 1.043 0.999 3.98 × 10−2

1.00 0.845 1.183 1.050 1.0025 5.07 × 10−−−−−2

1.02 0.828 1.207 1.057 1.0062 0.06261.04 0.813 1.230 1.064 1.0106 0.07531.06 0.797 1.254 1.072 1.0157 0.08871.08 0.783 1.278 1.079 1.0215 0.10281.10 0.768 1.302 1.087 1.028 0.1174

1.12 0.755 1.325 1.095 1.035 0.13251.14 0.741 1.349 1.102 1.043 0.14811.16 0.728 1.372 1.110 1.051 0.1641.18 0.716 1.396 1.118 1.061 0.1801.20 0.704 1.419 1.127 l.070 0.197


TABLE 7.2



t 4 f L Dmax

1.22 0.693 1.443 1.135 1.081 0.2141.24 0.681 1.467 1.144 1.092 0.2311.26 0.671 1.491 1.153 1.103 0.2481.28 0.660 1.515 1.162 1.116 0.2661.30 0.650 1.538 1.171 1.129 0.284

1.32 0.640 1.562 1.180 1.142 0.3021.34 0.631 1.585 1.189 1.157 0.3191.36 0.621 1.609 1.198 1.172 0.3371.38 0.612 1.633 1.208 1.187 0.3561.40 0.604 1.656 1.218 1.204 0.374

1.42 0.595 1.680 1.228 1.221 0.3921.44 0.587 1.704 1.238 1.238 0.4101.46 0.578 1.727 1.248 1.257 0.4281.48 0.571 1.751 1.258 1.276 0.4471.50 0.563 1.775 1.268 1.296 0.465

1.52 0.556 1.798 1.279 1.317 0.4831.54 0.549 1.822 1.290 1.338 0.5011.56 0.542 1.846 1.301 1.360 0.5191.58 0.535 1.869 1.312 1.383 0.5371.60 0.528 1.893 1.323 1.407 0.555

1.62 0.522 1.917 1.334 1.431 0.5731.64 0.515 1.940 1.345 1.457 0.5911.66 0.509 1.964 1.357 1.483 0.6091.69 0.503 1.987 1.368 1.510 0.6271.70 0.497 2.011 1.381 1.538 0.645

1.72 0.491 2.035 1.393 1.567 0.6631.74 0.486 2.058 1.405 1.596 0.6801.76 0.480 2.082 1.417 1.627 0.6971.78 0.475 2.106 1.429 1.658 0.7151.80 0.469 2.129 1.442 1.691 0.733


TABLE 7.2



t 4 f L Dmax

1.82 0.464 2.153 1.455 1.724 0.7491.84 0.459 2.177 1.467 1.758 0.7671.86 0.454 2.200 1.480 1.793 0.7841.88 0.449 2.224 1.493 1.830 0.8011.90 0.445 2.248 1.507 1.868 0.818

1.92 0.440 2.272 1.520 1.906 0.8351.94 0.435 2.295 1.534 1.946 0.8521.96 0.431 2.319 1.547 1.986 0.8681.98 0.427 2.343 1.561 2.029 0.8852.00 0.423 2.366 1.575 2.072 0.901

2.02 0.418 2.390 1.589 2.116 0.9182.04 0.414 2.414 1.603 2.162 0.9342.06 0.410 2.437 1.617 2.208 0.9502.08 0.406 2.461 1.632 2.257 0.9662.10 0.402 2.484 1.647 2.306 0.982

2.12 0.398 2.508 1.662 2.357 0.9982.14 0.395 2.532 1.676 2.409 1.0142.16 0.391 2.556 1.691 2.463 1.0292.18 0.387 2.579 1.707 2.517 1.0452.20 0.384 2.603 1.722 2.574 1.061

2.22 0.381 2.627 1.737 2.632 1.0762.24 0.377 2.650 1.753 2.691 1.0922.26 0.374 2.674 1.769 2.752 1.1072.28 0.370 2.697 1.785 2.815 1.1222.30 0.367 2.721 1.800 2.879 1.137

2.32 0.364 2.745 1.817 2.945 1.1522.34 0.361 2.768 1.833 3.013 1.1672.36 0.358 2.792 1.849 3.082 1.1822.38 0.355 2.816 1.866 3.153 1.1972.40 0.352 2.839 1.883 3.226 1.211


TABLE 7.2



t 4 f L Dmax

2.42 0.349 2.863 1.899 3.301 1.2262.44 0.346 2.887 1.917 3.378 1.2402.46 0.343 2.911 1.934 3.456 1.2552.48 0.341 2.934 1.951 3.537 1.2692.50 0.338 2.958 1.968 3.619 1.283

2.52 0.335 2.982 1.986 3.704 1.2972.54 0.333 3.005 2.004 3.791 1.3112.56 0.330 3.029 2.022 3.880 1.3252.58 0.327 3.053 2.039 3.971 1.3392.60 0.325 3.076 2.058 4.064 1.353

2.62 0.322 3.100 2.076 4.160 1.3672.64 0.320 3.123 2.094 4.258 1.3802.66 0.318 3.147 2.113 4.359 1.3942.68 0.315 3.171 2.132 4.462 1.4072.70 0.313 3.195 2.151 4.567 1.421

2.72 0.311 3.218 2.169 4.675 1.4342.74 0.308 3.242 2.189 4.785 1.4472.76 0.306 3.265 2.208 4.898 1.4612.78 0.304 3.289 2.227 5.015 1.4732.80 0.302 3.313 2.247 5.133 1.487

2.82 0.299 3.337 2.267 5.255 1.4992.84 0.297 3.360 2.286 5.379 1.5122.86 0.295 3.384 2.306 5.506 1.5252.88 0.293 3.407 2.327 5.637 1.5382.90 0.291 3.431 2.346 5.770 1.551

2.92 0.289 3.455 2.367 5.906 1.5632.94 0.287 3.478 2.387 6.046 1.5752.96 0.285 3.502 2.408 6.189 1.5882.98 0.283 3.526 2.429 6.335 1.6013.00 0.282 3.549 2.450 6.485 1.613



TABLE 7.2



t 4 f L Dmax

3.10 0.273 3.668 2.557 7.286 1.6733.20 0.264 3.786 2.667 8.182 1.7323.30 0.256 3.905 2.781 9.183 1.7903.40 0.248 4.023 2.898 10.229 1.8463.50 0.241 4.141 3.018 11.541 1.900

3.60 0.235 4.259 3.143 12.922 1.9533.70 0.228 4.378 3.271 14.454 2.0053.80 0.222 4.496 3.402 16.152 2.0563.90 0.217 4.614 3.537 18.029 2.1054.00 0.211 4.733 3.675 20.104 2.153

4.10 0.206 4.851 3.817 22.391 2.2014.20 0.201 4.969 3.962 24.911 2.2474.30 0.196 5.088 4.110 27.682 2.2924.40 0.192 5.206 4.263 30.725 2.3364.50 0.188 5.324 4.418 34.062 2.380

4.60 0.184 5.443 4.578 37.718 2.4224.70 0.179 5.561 4.741 41.716 2.4644.80 0.176 5.679 4.907 46.084 2.5054.90 0.172 5.798 5.077 50.850 2.5445.00 0.169 5.916 5.250 56.043 2.584

5.50 0.154 6.507 6.168 89.591 2.7696.00 0.141 7.099 7.175 139.367 2.9396.50 0.130 7.691 8.268 211.379 3.0977.00 0.121 8.282 9.450 313.219 3.2437.50 0.112 8.874 10.718 454.338 3.379

8.00 0.1056 9.466 12.075 646.325 3.5068.50 0.0994 10.057 13.518 903.222 3.6269.00 0.0939 10.649 15.050 1241.856 3.7399.50 0.0889 11.241 16.668 1682.182 3.847

10.00 0.0845 11.832 18.375 2247.66 3.948

Flow of Perfect Gases with Heat Transfer 105

TABLE 8.1

Flow of perfect gases with heat transfer (γ = γ = γ = γ = γ = 1.3)(Rayleigh flow)

M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

0.02 2.299 1.255 0.002 0.00184 0.0020.04 2.295 1.254 0.009 0.0073 0.00370.06 2.289 1.252 0.0189 0.0164 0.0080.08 2.281 1.250 0.0333 0.029 0.0150.10 2.270 1.247 0.0516 0.0449 0.0227

0.12 2.258 1.244 0.073 0.064 0.03250.14 2.243 1.239 0.098 0.086 0.0440.16 2.226 1.235 0.127 0.110 0.0570.18 2.207 1.230 0.158 0.138 0.0720.20 2.186 1.224 0.191 0.167 0.0875

0.22 2.164 1.128 0.277 0.198 1.1050.24 2.140 1.212 0.264 0.231 0.1230.26 2.114 1.205 0.302 0.265 0.1430.28 2.087 1.198 0.342 0.300 0.1630.30 2.059 1.191 0.382 0.336 0.185

0.32 2.030 1.183 0.422 0.373 0.2080.34 1.999 1.176 0.462 0.408 0.2310.36 1.968 1.167 0.502 0.445 0.2550.38 1.936 1.159 0.542 0.481 0.2790.40 1.904 1.152 0.580 0.516 0.305

0.42 1.871 1.143 0.617 0.551 0.3300.44 1.837 1.135 0.654 0.585 0.3560.46 1.804 1.127 0.688 0.618 0.3820.48 1.769 1.119 0.722 0.649 0.4080.50 1.736 1.111 0.753 0.680 0.434

0.52 1.702 1.103 0.783 0.708 0.4600.54 1.670 1.096 0.811 0.736 0.4860.56 1.634 1.088 0.837 0.762 0.5120.58 1.600 1.081 0.861 0.787 0.5380.60 1.567 1.074 0.884 0.810 0.564


TABLE 8.1


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

0.62 1.534 1.067 0.904 0.832 0.5890.64 1.500 1.061 0.923 0.852 0.6150.66 1.468 1.054 0.939 0.870 0.6390.68 1.436 1.048 0.954 0.887 0.6640.70 1.405 1.042 0.967 0.903 0.688

0.72 1.374 1.037 0.978 0.917 0.7120.74 1.344 1.032 0.989 0.930 0.7360.76 1.314 1.028 0.997 0.942 0.7580.78 1.284 1.023 1.003 0.952 0.7810.80 1.255 1.019 1.009 0.961 0.804

0.82 1.227 1.016 1.013 0.969 0.8250.84 1.199 1.012 1.015 0.976 0.8460.86 1.173 1.010 1.0172 0.983 0.8670.88 1.146 1.007 1.0173 0.987 0.8870.90 1.120 1.005 1.017 0.991 0.907

0.92 1.095 1.003 1.015 0.995 0.9270.94 1.070 1.002 1.012 0.997 0.9460.96 1.046 1.0007 1.009 0.998 0.9640.98 1.023 1.0002 1.005 0.999 0.9821.00 1.000 1.000 1.000 1.000 1.000

1.02 0.977 1.0002 0.994 0.999 1.0171.04 0.956 1.0008 0.988 0.998 1.0341.06 0.935 1.002 0.982 0.997 1.0501.08 0.914 1.003 0.975 0.995 1.0661.10 0.894 1.005 0.967 0.993 1.082

1.12 0.874 1.007 0.958 0.990 1.0971.14 0.855 1.009 0.950 0.987 1.1111.16 0.836 1.0127 0.942 0.984 1.1261.18 0.818 1.016 0.933 0.981 1.1391.20 0.800 1.020 0.924 0.976 1.153


TABLE 8.1


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

1.22 0.784 1.024 0.914 0.972 1.1661.24 0.767 1.028 0.904 0.967 1.1791.26 0.751 1.034 0.894 0.963 1.1921.28 0.735 1.039 0.885 0.958 1.2041.30 0.719 1.045 0.875 0.953 1.216

1.32 0.705 1.051 0.865 0.948 1.2271.34 0.689 1.058 0.854 0.943 1.2381.36 0.676 1.065 0.844 0.938 1.2491.38 0.662 1.073 0.834 0.932 1.2601.40 0.648 1.081 0.824 0.927 1.270

1.42 0.635 1.089 0.813 0.921 1.2811.44 0.622 1.098 0.803 0.915 1.2911.46 0.609 1.108 0.793 0.910 1.3001.48 0.598 1.117 0.783 0.904 1.3091.50 0.586 1.128 0.773 0.898 1.318

1.52 0.575 1.138 0.763 0.893 1.3271.54 0.563 1.149 0.753 0.887 1.3361.56 0.552 1.161 0.743 0.882 1.3441.58 0.542 1.173 0.733 0.876 1.3521.60 0.531 1.185 0.723 0.870 1.360

1.62 0.521 1.195 0.713 0.865 1.3681.64 0.512 1.213 0.704 0.859 1.3761.66 0.502 1.227 0.694 0.853 1.3831.68 0.492 1.241 0.685 0.847 1.3901.70 0.484 1.256 0.676 0.842 1.397

1.72 0.475 1.272 0.670 0.837 1.4041.74 0.465 1.288 0.657 0.831 1.4111.76 0.457 1.305 0.648 0.825 1.4171.78 0.449 1.322 0.639 0.821 1.4231.80 0.441 1.340 0.631 0.815 1.430


TABLE 8.1


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

1.82 0.433 1.358 0.622 0.810 1.4361.84 0.426 1.377 0.614 0.805 1.4421.86 0.418 1.397 0.606 0.799 1.4471.88 0.411 1.417 0.597 0.795 1.4531.90 0.404 1.438 0.589 0.790 1.459

1.92 0.397 1.451 0.581 0.785 1.4641.94 0.390 1.482 0.573 0.780 1.4691.96 0.383 1.505 0.565 0.775 1.4741.98 0.377 1.528 0.557 0.771 1.4792.00 0.371 1.552 0.551 0.766 1.484

2.02 0.365 1.576 0.543 0.761 1.4882.04 0.358 1.602 0.536 0.757 1.4932.06 0.353 1.628 0.528 0.752 1.4982.08 0.347 1.655 0.522 0.748 1.5022.10 0.341 1.683 0.515 0.744 1.506

2.12 0.336 1.711 0.508 0.739 1.5112.14 0.331 1.740 0.501 0.735 1.5152.16 0.325 1.770 0.495 0.731 1.5182.18 0.320 1.800 0.487 0.727 1.5232.20 0.315 1.832 0.482 0.723 1.527

2.22 0.310 1.865 0.475 0.718 1.5302.24 0.306 1.898 0.469 0.715 1.5342.26 0.301 1.932 0.463 0.711 1.5372.28 0.296 1.967 0.457 0.707 1.5412.30 0.292 2.003 0.451 0.703 1.545

2.32 0.287 2.040 0.445 0.699 1.5482.34 0.283 2.078 0.440 0.696 1.5512.36 0.279 2.117 0.434 0.693 1.5552.38 0.275 2.156 0.428 0.688 1.5572.40 0.271 2.197 0.423 0.686 1.561


TABLE 8.1


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

2.42 0.267 2.238 0.417 0.682 1.5642.44 0.263 2.281 0.412 0.678 1.5672.46 0.259 2.325 0.407 0.675 1.5692.48 0.255 2.370 0.402 0.672 1.5732.50 0.252 2.416 0.397 0.670 1.575

2.52 0.248 2.464 0.392 0.666 1.5782.54 0.245 2.512 0.387 0.663 1.5802.56 0.242 2.562 0.382 0.659 1.5832.58 0.238 2.612 0.378 0.657 1.5862.60 0.235 2.664 0.373 0.654 1.588

2.62 0.232 2.718 0.368 0.650 1.5912.64 0.228 2.772 0.364 0.648 1.5932.66 0.225 2.828 0.360 0.645 1.5962.68 0.222 2.885 0.356 0.642 1.5982.70 0.219 2.945 0.351 0.640 1.600

2.72 0.217 3.004 0.347 0.636 1.6032.74 0.214 3.065 0.343 0.634 1.6052.76 0.211 3.128 0.339 0.631 1.6072.78 0.208 3.192 0.335 0.629 1.6092.80 0.206 3.258 0.331 0.627 1.612

2.82 0.203 3.326 0.327 0.624 1.6132.84 0.200 3.395 0.323 0.622 1.6152.86 0.198 3.465 0.319 0.619 1.6172.88 0.195 3.537 0.316 0.617 1.6192.90 0.193 3.611 0.312 0.614 1.620

2.92 0.190 3.687 0.309 0.612 1.6232.94 0.188 3.765 0.305 0.609 1.6252.96 0.185 3.844 0.302 0.607 1.6262.98 0.183 3.925 0.298 0.605 1.6283.00 0.181 4.007 0.295 0.603 1.630

TABLE 8.1


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

3.10 0.170 4.451 0.279 0.593 1.6383.20 0.161 4.947 0.264 0.583 1.6453.30 0.152 5.500 0.251 0.574 1.6533.40 0.143 6.118 0.238 0.566 1.6593.50 0.136 6.806 0.226 0.558 1.665

3.60 0.128 7.571 0.215 0.551 1.6703.70 0.122 8.422 0.205 0.544 1.6753.80 0.116 9.365 0.195 0.538 1.6803.90 0.110 10.411 0.186 0.532 1.6844.00 0.106 11.570 0.178 0.527 1.688

4.10 0.100 12.850 0.170 0.521 1.6924.20 0.096 14.265 0.163 0.516 1.695-4.30 0.092 15.825 0.156 0.512 1.6984.40 0.088 17.545 0.149 0.508 1.7014.50 0.084 19.440 0.143 0.504 1.705

4.60 0.081 21.518 0.138 0.499 1.7074.70 0.077 23.803 0.132 0.496 1.7094.80 0.074 26.309 0.127 0.493 1.7134.90 0.071 29.056 0.122 0.489 1.7145.00 0.068 32.060 0.118 0.487 1.715

6.00 0.048 76.970 0.083 0.464 1.7327.00 0.035 191.300 0.062 0.450 1.7428.00 0.027 413.400 0.048 0.440 1.7489.00 0.0216 833.400 0.038 0.434 1.753

10.00 0.0175 1582.00 0.031 0.429 1.756


TABLE 8.2


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

0.02 2.398 1.268 0.0023 0.00192 0.000960.04 2.394 1.267 0.0092 0.0077 0.00380.06 2.388 1.265 0.021 0.017 0.00860.08 2.378 1.262 0.036 0.030 0.0150.10 2.367 1.259 0.056 0.0468 0.0237

0.12 2.353 1.255 0.079 0.066 0.0340.14 2.336 1.251 0.107 0.089 0.04570.16 2.317 1.246 0.137 0.115 0.0590.18 2.295 1.241 0.171 0.143 0.0740.20 2.273 1.235 0.207 0.174 0.091

0.22 2.248 1.228 0.245 0.206 0.1080.24 2.221 1.221 0.284 0.239 0.1280.26 2.193 1.214 0.325 0.274 0.1480.28 2.163 2.206 0.367 0.310 0.1690.30 2.131 1.198 0.409 0.347 0.192

0.32 2.099 1.190 0.451 0.384 0.2150.34 2.066 1.182 0.493 0.421 0.2390.36 2.031 1.174 0.535 0.457 0.2630.38 1.996 1.165 0.576 0.493 0.2880.40 1.961 1.157 0.615 0.529 0.314

0.42 1.925 1.148 0.653 0.564 0.3390.44 1.888 1.139 0.690 0.597 0.3650.46 1.852 1.131 0.725 0.630 0.3920.48 1.815 1.122 0.758 0.661 0.4180.50 1.778 1.114 0.790 0.691 0.444

0.52 1.741 1.106 0.819 0.720 0.4710.54 1.704 1.098 0.847 0.747 0.4970.56 1.668 1.090 0.872 0.773 0.5230.58 1.632 1.082 0.896 0.796 0.5480.60 1.596 1.075 0.917 0.819 0.574



TABLE 8.2


M p/p* p0/p0* T/T* T0/T0* c/c* = ρ*/ρ

0.62 1.560 1.068 0.935 0.839 0.5990.64 1.525 1.061 0.953 0.859 0.6250.66 1.490 1.055 0.968 0.877 0.6500.68 1.457 1.049 0.981 0.893 0.6730.70 1.423 1.043 0.993 0.908 0.697

0.72 1.391 1.037 1.002 0.922 0.7210.74 1.358 1.032 1.011 0.934 0.7440.76 1.327 1.028 1.017 0.945 0.7660.78 1.296 1.023 1.022 0.955 0.7880.80 1.266 1.019 1.025 0.964 0.810

0.82 1.236 1.015 1.027 0.972 0.8310.84 1.207 1.012 1.028 0.978 0.8520.86 1.179 1.010 1.028 0.984 0.8720.88 1.151 1.007 1.027 0.998 0.8920.90 1.125 1.0045 1.025 0.992 0.911

0.92 1.098 1.003 1.021 0.995 0.9300.94 1.073 1.002 1.017 0.997 0.9500.96 1.048 1.001 1.012 0.998 0.9660.98 1.023 1.0002 1.006 0.999 0.9831.00 1.000 1.000 1.000 1.000 1.000

1.02 0.977 1.0002 0.993 0.999 1.0161.04 0.955 1.0007 0.985 0.998 1.0321.06 0.933 1.002 0.978 0.997 1.0481.08 0.911 1.0033 0.969 0.996 1.0631.10 0.891 1.005 0.960 0.994 1.078

1.12 0.871 1.007 0.951 0.991 1.0921.14 0.851 1.009 0.942 0.988 1.1061.16 0.832 1.012 0.932 0.985 1.1201.18 0.814 1.016 0.922 0.982 1.1331.20 0.796 1.019 0.912 0.978 1.146


TABLE 8.2


M p/p* p0/p0* T/T* T0/T0* c/c* = ρ*/ρ

1.22 0.778 1.023 0.902 0.975 1.1581.24 0.761 1.028 0.891 0.971 1.1711.26 0.745 1.033 0.881 0.967 1.1821.28 0.728 1.038 0.869 0.962 1.1941.30 0.713 1.044 0.859 0.958 1.205

1.32 0.698 1.049 0.848 0.953 1.2161.34 0.683 1.056 0.837 0.948 1.2261.36 0.668 1.063 0.827 0.944 1.2371.38 0.654 1.070 0.816 0.939 1.2471.40 0.641 1.078 0.805 0.934 1.256

1.42 0.628 1.086 0.795 0.929 1.2661.44 0.615 1.094 0.784 0.924 1.2751.46 0.602 1.103 0.773 0.919 1.2841.48 0.590 1.111 0.763 0.914 1.2931.50 0.578 1.122 0.753 0.909 1.301

1.52 0.567 1.132 0.742 0.904 1.3091.54 0.556 1.142 0.732 0.899 1.3171.56 0.544 1.153 0.722 0.894 1.3251.58 0.534 1.164 0.711 0.889 1.3331.60 0.524 1.176 0.702 0.884 1.340

1.62 0.513 1.187 0.692 0.879 1.3471.64 0.503 1.200 0.682 0.874 1.3541.66 0.494 1.213 0.673 0.869 1.3621.68 0.484 1.226 0.663 0.865 1.3681.70 0.476 1.240 0.654 0.860 1.375

1.72 0.467 1.254 0.644 0.855 1.3811.74 0.458 1.269 0.635 0.850 1.3871.76 0.449 1.284 0.626 0.845 1.3931.78 0.442 1.299 0.617 0.841 1.3991.80 0.434 1.316 0.609 0.836 1.405


TABLE 8.2


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

1.82 0.425 1.332 0.600 0.832 1.4101.84 0.418 1.349 0.592 0.827 1.4151.86 0.411 1.367 0.584 0.823 1.4211.88 0.403 1.385 0.575 0.818 1.4261.90 0.396 1.403 0.567 0.814 1.431

1.92 0.389 1.422 0.559 0.810 1.4361.94 0.383 1.442 0.551 0.806 1.4411.96 0.376 1.462 0.544 0.801 1.4461.98 0.369 1.482 0.536 0.797 1.4502.00 0.364 1.503 0.529 0.793 1.455

2.02 0.357 1.525 0.522 0.789 1.4592.04 0.352 1.547 0.514 0.785 1.4632.06 0.345 1.569 0.507 0.782 1.4672.08 0.340 1.592 0.500 0.778 1.4712.10 0.334 1.616 0.494 0.774 1.475

2.12 0.329 1.640 0.487 0.770 1.4802.14 0.324 1.665 0.480 0.767 1.4832.16 0.318 1.690 0.474 0.763 1.4872.18 0.314 1.717 0.467 0.759 1.4902.20 0.308 1.743 0.461 0.756 1.494

2.22 0.304 1.771 0.455 0.750 1.4972.24 0.299 1.798 0.448 0.749 1.5002.26 0.294 1.827 0.443 0.746 1.5042.28 0.290 1.856 0.437 0.743 1.5072.30 6.285 1.886 0.431 0.739 1.510

2.32 0.281 1.916 0.426 0.736 1.5132.34 0.277 1.947 0.420 0.733 1.5162.36 0.273 1.980 0.415 0.730 1.5192.38 0.268 2.012 0.409 0.727 1.5222.40 0.265 2.045 0.404 0.724 1.525


TABLE 8.2


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

2.42 0.261 2.078 0.398 0.721 1.5282.44 0.257 2.114 0.393 0.718 1.5302.46 0.253 2.149 0.388 0.716 1.5332.48 0.249 2.185 0.383 0.713 1.5362.50 0.246 2.222 0.379 0.710 1.540

2.52 0.242 2.260 0.374 0.707 1.5422.54 0.239 2.298 0.369 0.704 1.5442.56 0.235 2.337 0.365 0.702 1.5462.58 0.232 2.377 0.360 0.699 1.5482.60 0.229 2.418 0.356 0.697 1.551

2.62 0.226 2.459 0.351 0.695 1.5532.64 0.223 2.502 0.347 0.692 1.5552.66 0.220 2.545 0.343 0.689 1.5572.68 0.217 2.589 0.338 0.687 1.5592.70 0.214 2.634 0.334 0.685 1.561

2.72 0.211 2.680 0.330 0.683 1.5632.74 0.208 2.727 0.326 0.680 1.5652.76 0.206 2.775 0.322 0.678 1.5672.78 0.203 2.823 0.318 0.676 1.5692.80 0.200 2.873 0.315 0.674 1.571

2.82 0.198 2.923 0.311 0.672 1.5722.84 0.195 2.975 0.307 0.669 1.5742.86 0.193 3.028 0.304 0.667 1.5762.88 0.190 3.081 0.300 0.665 1.5782.90 0.188 3.136 0.297 0.663 1.580

2.92 0.185 3.191 0.293 0.661 1.5822.94 0.183 3.248 0.290 0.659 1.5832.96 0.181 3.306 0.286 0.658 1.5852.98 0.178 3.365 0.284 0.655 1.5863.00 0.176 3.424 0.281 0.654 1.588

TABLE 8.2


M p/p* p0/p0* T/T* T0/T0* c/c* = ρ*/ρ

3.10 0.166 3.741 0.265 0.645 1.5953.20 0.156 4.097 0.251 0.637 1.6033.30 0.148 4.465 0.237 0.629 1.6093.40 0.130 4.878 0.225 0.622 1.6153.50 0.1275 5.328 0.214 0.616 1.620

3.60 0.125 5.817 0.204 0.609 1.6253.70 0.119 6.349 0.193 0.604 1.6303.80 0.113 6.925 0.185 0.598 1.6333.90 0.107 7.551 0.176 0.594 1.6374.00 0.1026 8.227 0.168 0.589 1.641

4.10 0.098 8.958 0.161 0.585 1.6454.20 0.093 9.747 0.154 0.581 1.6474.30 0.089 10.598 0.147 0.577 1.6534.40 0.085 11.515 0.141 0.573 1.6544.50 0.0818 12.502 0.135 0.569 1.656

4.60 0.078 13.563 0.130 0.567 1.6584.70 0.075 14.702 0.125 0.564 1.6614.80 0.072 15.923 0.120 0.561 1.6634.90 0.069 17.232 0.115 0.558 1.6655.00 0.0667 15.634 0.111 0.556 1.667

6.00 0.0467 38.946 0.0785 0.536 1.6817.00 0.0345 75.415 0.058 0.524 1.6908.00 0.0265 136.62 0.045 0.516 1.6959.00 0.021 233.88 0.036 0.511 1.699

10.00 0.017 381.62 0.029 0.507 1.702


TABLE 8.3


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

0.02 2.665 1.298 0.0025 0.0021 0.0010.04 2.659 1.297 0.0113 0.0085 0.0040.06 2.651 1.295 0.025 0.0190 0.0090.08 2.638 1.292 0.044 0.033 0.0170.10 2.626 1.289 0.069 0.052 0.026

0.12 2.605 1.284 0.097 0.074 0.0370.14 2.583 1.278 0.131 0.098 0.0510.16 2.558 1.273 0.167 0.127 0.0650.18 2.530 1.266 0.207 0.157 0.0820.20 2.503 1.259 0.251 0.190 0.100

0.22 2.468 1.251 0.294 0.225 0.1190.24 2.433 1.243 0.341 0.261 0.1400.26 2.397 1.234 0.388 0.298 0.1620.28 2.358 1.225 0.436 0.335 0.1850.30 2.320 1.215 0.485 0.374 0.209

0.32 2.278 1.207 0.531 0.412 0.2330.34 2.236 1.197 0.578 0.450 0.2580.36 2.193 1.187 0.623 0.487 0.2840.38 2.149 1.178 0.667 0.525 0.3100.40 2.107 1.168 0.710 0.561 0.337

0.42 2.061 1.158 0.749 0.595 0.3640.44 2.016 1.148 0.787 0.628 0.3900.46 1.971 1.138 0.822 0.660 0.4170.48 1.927 1.129 0.855 0.691 0.4450.50 1.884 1.120 0.887 0.720 0.471

0.52 1.838 1.111 0.914 0.747 0.4970.54 1.795 1.102 0.939 0.773 0.5240.56 1.751 1.094 0.962 0.797 0.5490.58 1.708 1.086 0.982 0.819 0.5750.60 1.667 1.078 1.001 0.840 0.600



TABLE 8.3


M p/p* p0/p0* T/T* T0/T0* c/c* = ρ*/ρ

0.62 1.625 1.070 1.015 0.859 0.6250.64 1.585 1.063 1.028 0.877 0.6490.66 1.545 1.056 1.039 0.893 0.6730.68 1.506 1.049 1.049 0.907 0.6970.70 1.468 1.044 1.056 0.921 0.719

0.72 1.431 1.038 1.061 0.933 0.7420.74 1.394 1.033 1.065 0.944 0.7640.76 1.358 1.028 1.067 0.954 0.7850.78 1.324 1.023 1.066 0.962 0.8060.80 1.291 1.019 1.066 0.970 0.826

0.82 1.257 1.016 1.063 0.976 0.8450.84 1.226 1.012 1.059 0.982 0.8650.86 1.194 1.010 1.055 0.986 0.8830.88 1.164 1.007 1.049 0.990 0.9020.90 1.135 1.005 1.043 0.994 0.919

0.92 1.106 1.003 1.036 0.996 0.9360.94 1.078 1.0017 1.028 0.998 0.9530.96 1.052 1.0008 1.019 0.999 0.9690.98 1.025 1.0002 1.009 0.999 0.9851.00 1.000 1.000 1.000 1.000 1.000

1.02 0.975 1.0002 0.989 0.999 1.0151.04 0.951 1.0007 0.979 0.999 1.0291.06 0.928 1.0017 0.968 0.998 1.0431.08 0.906 1.003 0.957 0.997 1.0561.10 0.884 1.0046 0.945 0.995 1.070

1.12 0.863 1.0066 0.934 0.993 1.0821.14 0.842 1.009 0.922 0.991 1.0951.16 0.822 1.011 0.910 0.988 1.1061.18 0.803 1.015 0.898 0.986 1.1181.20 0.784 1.018 0.886 0.983 1.130


TABLE 8.3


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

1.22 0.766 1.022 0.874 0.960 1.1401.24 0.748 1.026 0.861 0.978 1.1501.26 0.731 1.030 0.849 0.974 1.1611.28 0.715 1.035 0.837 0.971 1.1711.30 0.698 1.040 0.825 0.967 1.181

1.32 0.683 1.045 0.813 0.964 1.1891.34 0.668 1.051 0.801 0.960 1.1981.36 0.653 1.057 0.789 0.957 1.2081.38 0.639 1.063 0.778 0.953 1.2161.40 0.625 1.070 0.765 0.950 1.225

1.42 0.611 1.077 0.754 0.946 1.2331.44 0.598 1.084 0.743 0.942 1.2401.46 0.586 1.092 0.731 0.938 1.2481.48 0.573 1.099 0.720 0.935 1.2561.50 0.561 1.108 0.708 0.931 1.263

1.52 0.549 1.116 0.698 0.927 1.2691.54 0.538 1.125 0.688 0.923 1.2761.56 0.527 1.134 0.677 0.920 1.2831.58 0.517 1.144 0.667 0.916 1.2891.60 0.506 1.154 0.656 0.913 1.296

1.62 0.496 1.163 0.646 0.908 1.3021.64 0.486 1.174 0.636 0.905 1.3071.66 0.476 1.184 0.626 0.901 1.3131.68 0.467 1.195 0.617 0.898 1.3191.70 0.458 1.206 0.607 0.895 1.325

1.72 0.449 1.217 0.598 0.891 1.3291.74 0.441 1.229 0.589 0.887 1.3351.76 0.432 1.241 0.580 0.884 1.3401.78 0.425 1.254 0.571 0.881 1.3451.80 0.416 1.266 0.562 0.878 1.350


TABLE 8.3


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

1.82 0.409 1.280 0.554 0.875 1.3541.84 0.401 1.293 0.546 0.871 1.3581.86 0.394 1.306 0.537 0.868 1.3621.88 0.387 1.320 0.529 0.865 1.3671.90 0.380 1.334 0.521 0.862 1.371

1.92 0.373 1.348 0.514 0.858 1.3751.94 0.367 1.363 0.506 0.856 1.3791.96 0.360 1.378 0.498 0.853 1.3831.98 0.354 1.393 0.491 0.850 1.3872.00 0.348 1.408 0.483 0.848 1.391

2.02 0.342 1.425 0.477 0.844 1.3942.04 0.336 1.441 0.470 0.841 1.3972.06 0.330 1.457 0.463 0.838 1.4012.08 0.325 1.475 0.456 0.836 1.4042.10 0.319 1.490 0.450 0.834 1.408

2.12 0.314 1.509 0.443 0.831 1.4112.14 0.309 1.527 0.437 0.828 1.4142.16 0.304 1.545 0.431 0.826 1.4172.18 0.299 1.563 0.425 0.823 1.4192.20 0.294 1.580 0.418 0.821 1.423

2.22 0.289 1.600 0.413 0.818 1.4262.24 0.285 1.620 0.407 0.816 1.4282.26 0.280 1.640 0.401 0.814 1.4312.28 0.276 1.660 0.396 0.811 1.4332.30 0.272 1.678 0.390 0.810 1.435

2.32 0.267 1.700 0.385 0.807 1.4382.34 0.263 1.721 0.380 0.805 1.4412.36 0.259 1.742 0.375 0.803 1.4442.38 0.255 1.764 0.369 0.800 1.4462.40 0.251 1.783 0.364 0.800 1.450


TABLE 8.3


M p/p* p0 /p0* T/T* T0 /T0* c/c* = ρ*/ρ

2.42 0.248 1.807 0.359 0.796 1.4502.44 0.244 1.830 0.355 0.795 1.4532.46 0.241 1.853 0.350 0.793 1.4552.48 0.237 1.876 0.345 0.791 1.4572.50 0.233 1.897 0.341 0.789 1.459

2.52 0.230 1.923 0.336 0.786 1.4612.54 0.227 1.947 0.332 0.785 1.4602.56 0.223 1.971 0.328 0.783 1.4652.58 0.220 1.996 0.324 0.781 1.4672.60 0.217 2.018 0.319 0.780 1.469

2.62 0.214 2.050 0.315 0.778 1.4702.64 0.211 2.073 0.311 0.776 1.4722.66 0.208 2.098 0.307 0.774 1.4742.68 0.206 2.125 0.304 0.773 1.4762.70 0.203 2.148 0.299 0.772 1.477

2.72 0.200 2.179 0.296 0.769 1.4792.74 0.197 2.207 0.292 0.768 1.4812.76 0.195 2.234 0.288 0.767 1.4822.78 0.192 2.263 0.285 0.765 1.4842.80 0.189 2.287 0.281 0.764 1.485

2.82 0.187 2.320 0.278 0.762 1.4872.84 0.185 2.350 0.275 0.761 1.4882.86 0.182 2.379 0.272 0.759 1.4892.88 0.180 2.409 0.268 0.758 1.4912.90 0.178 2.435 0.265 0.757 1.492

2.92 0.175 2.471 0.262 0.755 1.4942.94 0.173 2.502 0.259 0.754 1.4952.96 0.171 2.533 0.256 0.753 1.4962.98 0.168 2.565 0.253 0.751 1.4983.00 0.167 2.587 0.250 0.750 1.500

TABLE 8.3


M p/p* p0 /p

0* T/T* T

0 /T

0* c/c* = ρ*/ρ

3.10 0.157 2.765 0.236 0.745 1.5053.20 0.148 2.941 0.223 0.738 1.5113.30 0.139 3.127 0.211 0.733 1.5153.40 0.132 3.324 0.200 0.728 1.5203.50 0.124 3.521 0.189 0.725 1.524

3.60 0.118 3.750 0.180 0.720 1.5283.70 0.112 3.980 0.172 0.716 1.5323.80 0.106 4.221 0.163 0.713 1.5353.90 0.101 4.473 0.156 0.709 1.5384.00 0.096 4.715 0.148 0.707 1.542

4.10 0.092 5.014 0.142 0.703 1.5444.20 0.087 5.303 0.136 0.700 1.5464.30 0.084 5.605 0.129 0.698 1.5484.40 0.080 5.918 0.124 0.695 1.5504.50 0.077 6.213 0.119 0.694 1.553

4.60 0.074 6.588 0.114 0.691 1.5554.70 0.070 6.944 0.109 0.689 1.5564.80 0.067 7.314 0.105 0.687 1.5584.90 0.065 7.698 0.101 0.685 1.5595.00 0.062 8.045 0.098 0.685 1.560

6.00 0.044 12.860 0.068 0.672 1.5737.00 0.032 19.440 0.051 0.664 1.5808.00 0.0248 28.070 0.039 0.660 1.5859.00 0.0190 39.051 0.031 0.656 1.587

10.00 0.0159 52.660 0.025 0.653 1.589


TABLE 8.4

Maximum heat transfer in perfect gases

M Qmax /cpT

γ = 1.3 γ = 1.4 γ = 1.67

0.00 ∞ ∞ ∞

0.10 21.300 20.410 18.4000.20 5.000 4.800 4.3070.30 2.000 1.920 1.7220.40 0.960 0.920 0.8250.50 0.490 0.468 0.421

0.60 0.2475 0.2368 0.21290.70 0.116 0.111 0.09940.80 0.044 0.0421 0.03780.90 0.0097 0.0093 0.008361.00 0.000 0.000 0.000

1.10 0.00791 0.0076 0.006821.20 0.0292 0.0280 0.02521.30 0.0615 0.0588 0.05281.40 0.102 0.0980 0.08801.50 0.1511 0.1450 0.130

1.60 0.2065 0.1980 0.1781.70 0.268 0.257 0.2311.80 0.337 0.323 0.2901.90 0.411 0.394 0.3542.00 0.489 0.468 0.421

2.10 0.574 0.550 0.4942.20 0.661 0.633 0.5692.30 0.756 0.725 0.6522.40 0.854 0.818 0.7362.50 0.958 0.918 0.826



TABLE 8.4

Maximum heat transfer in perfect gases

M Qmax /cpT

γ = 1.3 γ = 1.4 γ = 1.67

2.60 1.067 1.023 0.9202.70 1.179 1.130 1.0162.80 1.298 1.243 1.1182.90 1.419 1.360 1.2233.00 1.546 1.481 1.331

3.50 2.245 2.152 1.9344.00 3.060 2.930 2.6334.50 3.978 3.812 3.4275.00 5.000 4.800 4.3146.00 7.397 7.089 6.373

7.00 10.220 9.796 8.8058.00 13.480 12.919 11.6139.00 17.174 16.458 14.794

10.00 21.306 20.418 18.354∞ ∞ ∞ ∞

TABLE 9.1

Prandtl–Meyer functions and Mach angles

M α ω (M)

γ = 1.3 γ = 1.4 γ = 1.67

1.00 90.00 0.00 0.00 0.00

1.02 78.63 0.131 0.126 0.1131.04 74.06 0.367 0.351 0.3151.06 70.63 0.670 0.637 0.5701.08 67.81 1.014 0.968 0.8651.10 65.38 1.400 1.336 1.191

1.12 63.23 1.820 1.735 1.5431.14 61.30 2.270 2.160 1.9171.16 59.55 2.740 2.607 2.3101.18 57.94 3.233 3.075 2.7181.20 56.44 3.750 3.558 3.140

1.22 55.05 4.275 4.057 3.5731.24 53.75 4.820 4.569 4.0171.26 52.53 5.375 5.093 4.4691.28 51.38 5.944 5.627 4.9281.30 50.29 6.523 6.170 5.393

1.32 49.25 7.112 6.721 5.8641.34 48.27 7.710 7.280 6.3401.36 47.33 8.311 4.844 6.8201.38 46.44 8.923 8.413 7.3001.40 45.59 9.542 8.987 7.784

1.42 44.77 10.164 9.565 8.2701.44 43.98 10.791 10.147 8.7561.46 43.23 11.421 10.730 9.2441.48 42.51 12.056 11.317 9.7321.50 41.81 12.693 11.905 10.220

1.52 41.14 13.333 12.495 10.7081.54 40.50 13.975 13.086 11.1951.56 39.87 14.620 13.677 11.6811.58 39.27 15.263 14.268 12.1671.60 38.68 15.910 14.860 12.650

Prandtl–Meyer Functions and Mach Angles 125


TABLE 9.1


M α ω (M)

γ = 1.3 γ = 1.4 γ = 1.67

1.62 38.12 16.556 15.452 13.1311.64 37.57 17.203 16.043 13.6111.66 37.04 17.850 16.633 14.0891.68 36.53 18.500 17.222 14.5641.70 36.03 19.144 17.810 15.037

1.72 35.55 19.790 18.397 15.5081.74 35.08 20.435 18.982 15.9761.76 34.62 21.080 19.565 16.4411.78 34.18 21.723 20.146 16.9031.80 33.75 22.360 20.725 17.363

1.82 33.33 23.004 21.302 17.8201.84 32.92 23.644 21.877 18.2731.86 32.53 24.281 22.450 18.7231.88 32.13 24.916 23.020 19.1701.90 31.75 25.550 23.586 19.614

1.92 31.39 26.181 24.151 20.0541.94 31.03 26.805 24.712 20.4911.96 30.68 27.435 25.271 20.9251.98 30.34 28.060 25.827 21.3552.00 30.00 28.681 26.380 21.782

2.02 29.67 29.300 26.930 22.2062.04 29.35 29.915 27.476 22.6262.06 29.04 30.530 28.020 23.0422.08 28.74 31.120 28.561 23.4552.10 28.44 31.750 29.097 23.865

2.12 28.15 32.350 29.631 24.2712.14 27.86 32.952 30.161 24.6742.16 27.58 33.561 30.690 25.0732.18 27.31 34.150 31.212 25.4682.20 27.04 34.750 31.733 25.860

TABLE 9.1


M α ω (M)

γ = 1.3 γ = 1.4 γ = 1.67

2.22 26.77 35.332 32.250 26.2492.24 26.52 35.920 32.763 26.3642.26 26.26 36.503 33.273 27.0172.28 26.01 37.080 33.780 27.3952.30 25.77 37.660 34.283 27.771

2.32 25.54 38.233 34.783 28.1412.34 25.30 38.803 35.279 28.5102.36 25.07 39.372 35.771 28.8752.38 24.85 39.936 36.261 29.2362.40 24.63 40.495 36.746 29.594

2.42 24.41 41.052 37.229 29.5012.44 24.20 41.607 37.708 30.3012.46 23.99 42.158 38.183 30.6502.48 23.78 42.705 32.655 31.0002.50 23.58 43.250 39.124 31.337

2.52 23.38 43.790 39.589 31.5072.54 23.19 44.325 40.051 32.0122.56 22.99 44.860 40.509 32.3442.58 22.81 45.390 40.963 32.6742.60 22.62 45.920 41.415 33.000

2.62 22.44 46.440 41.863 33.3242.64 22.26 46.960 42.307 33.6452.66 22.08 47.477 42.749 33.9612.68 21.91 47.990 43.187 34.2772.70 21.74 48.500 43.622 34.588

2.72 21.57 49.008 44.053 34.8972.74 21.41 49.511 44.481 35.2032.76 21.25 50.011 44.906 35.5052.78 21.08 50.508 45.328 35.8062.80 20.93 51.002 45.746 36.104



TABLE 9.1


M α ω (M)

γ = 1.3 γ = 1.4 γ = 1.67

2.82 20.77 51.490 46.161 36.3982.84 20.62 51.980 46.573 36.6912.86 20.46 52.463 46.982 36.9802.88 20.32 52.943 47.388 37.2672.90 20.17 53.420 47.790 37.551

2.92 20.03 53.895 48.190 37.8322.94 19.89 54.365 48.586 38.1112.96 19.75 54.831 48.980 38.3872.98 19.61 55.300 49.370 38.6613.00 19.47 55.758 49.757 38.932

3.05 19.14 56.900 50.713 39.6003.10 18.82 58.020 51.650 40.2523.15 18.51 59.120 52.569 40.8903.20 18.21 60.202 53.470 41.5133.25 17.92 61.266 54.355 42.122

3.30 17.63 62.311 55.222 42.7203.35 17.37 63.341 56.073 43.3013.40 17.10 64.350 56.907 43.8713.45 16.85 65.341 57.726 44.4303.50 16.60 66.318 58.530 44.975

3.55 16.36 67.277 59.316 45.5103.60 16.13 68.220 60.092 46.0323.65 15.91 69.150 60.850 46.5443.70 15.68 70.060 61.595 47.0453.75 15.47 70.954 62.327 47.535

3.80 15.26 71.833 63.044 48.0153.85 15.05 72.700 63.748 48.4853.90 14.86 73.551 64.440 48.9463.95 14.67 74.388 65.118 49.3974.00 14.48 75.210 65.785 49.840

TABLE 9.1


M α ω (M)

γ = 1.3 γ = 1.4 γ = 1.67

4.05 14.30 76.020 66.439 50.2734.10 14.11 76.815 67.081 50.6984.15 13.94 77.600 67.713 51.1144.20 13.78 78.367 68.333 51.5224.25 13.61 79.123 68.943 51.923

4.30 13.45 79.868 69.541 52.3154.35 13.29 80.600 70.130 52.7004.40 13.14 81.321 70.706 53.0784.45 12.98 82.031 71.274 53.4494.50 12.84 82.730 71.832 53.812

4.55 12.69 83.413 72.380 54.1694.60 12.56 84.088 72.920 54.5204.65 12.42 84.751 73.450 54.8634.70 12.28 85.408 73.970 55.2014.75 12.15 86.051 74.480 55.532

4.80 12.03 86.685 74.990 55.8614.85 11.89 87.309 75.482 56.1784.90 11.77 87.923 75.971 56.4924.95 11.66 88.528 76.449 56.8005.00 11.54 89.123 76.920 57.104



TABLE 9.2

Mach numbers and Mach angles as functions of ωωωωω (M)

ω (M) γ = 1.3 γ = 1.4 γ = 1.67

M α M α M α

0 1.000 90.000 1.000 90.000 1.000 90.000

0.5 1.049 72.358 1.051 72.098 1.055 71.437

1.0 1.079 67.905 1.082 67.574 1.089 66.732

1.5 1.105 64.833 1.108 64.451 1.117 63.476

2.0 1.128 62.422 1.133 62.000 1.144 60.915

2.5 1.150 60.411 1.155 59.950 1.169 58.774

3.0 1.171 58.675 1.177 58.180 1.194 56.920

3.5 1.191 57.138 1.198 56.615 1.217 55.278

4.0 1.209 55.755 1.218 55.205 1.240 53.800

4.5 1.228 54.500 1.238 53.920 1.262 52.455

5.0 1.247 53.340 1.257 52.738 1.283 51.200

5.5 1.265 52.270 1.275 51.642 1.305 50.050

6.0 1.282 51.265 1.294 50.619 1.326 48.964

6.5 1.300 50.327 1.312 49.658 1.347 47.950

7.0 1.316 49.442 1.330 48.753 1.368 46.990

7.5 1.333 48.605 1.348 47.896 1.389 46.081

8.0 1.349 47.810 1.366 47.082 1.409 45.216

8.5 1.366 47.054 1.383 46.305 1.430 44.392

9.0 1.383 46.332 1.400 45.566 1.450 43.605

9.5 1.398 45.641 1.418 44.857 1.471 42.848

10.0 1.415 44.978 1.435 44.177 1.490 42.120

11.0 1.447 43.730 1.469 42.894 1.532 40.750

12.0 1.478 42.569 1.503 41.700 1.573 39.470

13.0 1.510 41.485 1.537 40.585 1.615 38.270

14.0 1.541 40.468 1.571 39.537 1.656 37.140

TABLE 9.2


ω (M) γ = 1.3 γ = 1.4 γ = 1.67

M α M α M α

15.0 1.572 39.508 1.605 38.547 1.696 36.071

16.0 1.603 38.602 1.638 37.611 1.740 35.060

17.0 1.634 37.740 1.673 36.721 1.784 34.089

18.0 1.665 36.923 1.707 35.874 1.828 33.165

19.0 1.695 36.140 1.741 35.065 1.873 32.281

20.0 1.727 35.394 1.775 34.290 1.918 31.433

21.0 1.758 34.679 1.810 33.548 1.964 30.615

22.0 1.789 33.993 1.845 32.835 2.010 29.830

23.0 1.820 33.333 1.879 32.148 2.058 29.073

24.0 1.851 32.700 1.915 31.486 2.107 28.339

25.0 1.883 32.084 1.950 30.847 2.157 27.629

26.0 1.915 31.493 1.986 30.229 2.207 26.941

27.0 1.945 30.921 2.023 29.632 2.259 26.273

28.0 1.978 30.367 2.059 29.053 2.313 25.624

29.0 2.010 29.831 2.096 28.491 2.366 25.000

30.0 2.043 29.310 2.134 27.945 2.423 24.377

31.0 2.075 28.805 2.172 27.415 2.480 23.777

32.0 2.108 28.315 2.210 26.900 2.540 23.192

33.0 2.142 27.838 2.249 26.397 2.600 22.620

34.0 2.175 27.373 2.289 25.908 2.663 22.061

35.0 2.209 26.921 2.328 25.430 2.727 21.515

36.0 2.243 26.480 2.369 24.965 2.793 20.980

37.0 2.278 26.050 2.411 24.510 2.861 20.456

38.0 2.312 25.630 2.452 24.066 2.932 19.942

39.0 2.347 25.222 2.495 23.631 3.005 19.437

40.0 2.383 24.820 2.538 23.206 3.081 18.942



TABLE 9.2


ω (M) γ = 1.3 γ = 1.4 γ = 1.67

M α M α M α

41.0 2.418 24.428 2.584 22.790 3.159 18.456

42.0 2.454 24.045 2.626 22.382 3.240 17.980

43.0 2.491 23.670 2.672 21.983 3.325 17.510

44.0 2.528 23.305 2.718 21.591 3.411 17.045

45.0 2.565 22.943 2.765 21.207 3.502 16.590

46.0 2.603 22.591 2.812 20.830 3.597 16.142

47.0 2.642 22.245 2.861 20.459 3.696 15.700

48.0 2.680 21.906 2.911 20.096 3.800 15.262

49.0 2.720 21.573 2.961 19.735 3.906 14.832

50.0 2.760 21.246 3.013 19.387 4.018 14.410

51.0 2.800 20.925 3.013 19.387 4.136 14.000

52.0 2.841 20.610 3.065 19.041 4.260 13.580

53.0 2.883 20.300 3.119 18.700 4.390 13.168

54.0 2.925 20.000 3.174 18.366 4.526 12.764

55.0 2.967 19.695 3.229 18.036 4.670 12.364

56.0 3.010 19.400 3.287 17.711 4.822 11.968

57.0 3.055 19.110 3.346 17.391 4.983 11.578

58.0 3.100 18.825 3.406 17.076 5.153 11.190

59.0 3.145 18.543 3.470 16.765 5.334 10.806

60.0 3.191 18.265 3.592 16.155 5.525 10.424

61.0 3.237 18.000 3.660 15.857 5.731 10.048

62.0 3.285 17.723 3.727 15.561 5.950 9.675

63.0 3.334 17.457 3.797 15.270 6.185 9.304

64.0 3.383 17.195 3.868 14.983 6.438 8.935

65.0 3.433 16.937 3.941 14.698 6.710 8.571

TABLE 9.2


ω (M) γ = 1.3 γ = 1.4 γ = 1.67

M α M α M α

66.0 3.484 16.682 4.016 14.417 7.000 8.20967.0 3.536 16.430 4.094 14.140 7.323 7.85068.0 3.588 16.182 4.173 13.965 7.670 7.49069.0 3.642 15.937 4.255 13.593 8.050 7.13570.0 3.697 15.694 4.340 13.325 8.468 6.782

71.0 3.753 15.455 4.426 13.059 8.928 6.43072.0 3.810 15.219 4.515 12.800 9.440 6.08173.0 3.868 14.985 4.608 12.535 10.010 5.73374.0 3.927 14.754 4.703 12.277 10.651 5.38775.0 3.987 14.525 4.801 12.021 11.378 5.043



TABLE 10.1

Approximate velocities and Mach numbers of air at p = 1.0133 bart = 25°C (T = 298.15 K and ρρρρρ = 1.183 kg/m3 ; a = 347.07 m/s)

Dynamic head Velocity (c) Mach number(mm W.G.) m/s (M)

0 0.00 0.0001 4.07 0.01172 5.76 0.01663 7.05 0.02034 8.14 0.02355 9.10 0.0262

10 12.87 0.037115 15.76 0.045420 18.20 0.052525 20.35 0.058630 22.29 0.0643

35 24.08 0.069440 25.74 0.074245 27.30 0.078750 28.78 0.082955 30.18 0.0270

60 31.53 0.090865 32.81 0.094570 34.05 0.098175 35.25 0.101580 36.40 0.1049

85 37.52 0.108190 38.61 0.111395 39.67 0.1143

100 40.70 0.1173

200 57.56 0.1658300 70.49 0.2032400 81.40 0.2346500 91.00 0.2623600 99.69 0.2873

TABLE 10.2

Approximate velocities and Mach numbers of air at p = 1.0133 bart = 25°C (T = 298.15 K and ρρρρρ = 1.183 kg/m3 ; a = 347.07 m/s)

Dynamic head Velocity (c) Mach number(mm Hg) m/s (M)

50 106.06 0.30660 116.19 0.33570 125.50 0.36280 134.16 0.387

90 142.30 0.410100 150.00 0.432110 157.30 0.453120 164.32 0.474

130 171.03 0.493140 177.48 0.511150 183.71 0.529160 189.74 0.547

170 195.50 0.564180 201.25 0.580190 206.76 0.596200 212.13 0.611

250 237.17 0.683300 259.81 0.748350 280.62 0.808400 300.00 6.865450 318.17 0.917500 335.47 0.966

Approximate Velocities and Mach Numbers of Air 135


TABLE 10.3

Mach numbers as functions of pressure ratios (γγγγγ = 1.4)

p/p0 M p/p0 M

1.00 0.000 0.74 0.6700.99 0.120 0.73 0.6860.98 0.170 0.72 0.7020.97 0.209 0.71 0.7170.96 0.242 0.70 0.7320.95 0.272

0.94 0.298 0.69 0.7480.93 0.324 0.68 0.7630.92 0.347 0.67 0.7780.91 0.370 0.66 0.7940.90 0.391 0.65 0.809

0.89 0.411 0.64 0.8250.88 0.431 0.63 0.8400.87 0.451 0.62 0.8550.86 0.469 0.61 0.8710.85 0.488 0.60 0.886

0.84 0.505 0.59 0.9020.83 0.523 0.58 0.9180.82 0.540 0.57 0.9330.81 0.557 0.56 0.9490.80 0.574 0.55 0.965

0.79 0.590 0.54 0.9810.78 0.607 0.53 0.9970.77 0.623 0.52 1.0140.76 0.639 0.51 1.0300.75 0.655 0.50 1.046

TABLE 10.3

Mach numbers as functions of pressure ratios (γγγγγ = 1.4)

p/p0 M p/p0 M

0.49 1.063 0.24 1.5870.48 1.080 0.23 1.6150.47 1.097 0.22 1.6450.46 1.114 0.21 1.6760.45 1.132 0.20 1.709

0.44 1.150 0.19 1.7420.43 1.168 0.18 1.7780.42 1.186 0.17 1.8150.41 1.204 0.16 1.8550.40 1.223 0.15 1.897

0.39 1.242 0.14 1.9410.38 1.262 0.13 1.9890.37 1.282 0.12 2.0400.36 1.302 0.11 2.0960.35 1.322 0.10 2.157

0.34 1.344 0.09 2.2250.33 1.365 0.08 2.3000.32 1.387 0.07 2.3850.31 1.410 0.06 2.4840.30 1.433 0.05 2.601

0.29 1.457 0.04 2.7460.28 1.481 0.03 2.9350.27 1.506 0.02 3.2080.26 1.532 0.01 3.6930.25 1.560 0.00 ∞

Pressure Ratios and Mach Numbers of Air 137


TABLE 11

Manometer inclinations required for the desired magnification of the deflection

Magnification required Inclination from the horizontal(1/h) α = (sin−1 h/1) degrees

2 30

3 19.47

4 14.47

5 11.53

6 9.59

7 8.21

8 7.18

9 6.38

10 5.74

TABLE 12.1

Static pressure drop (pr = outlet pressure/inlet pressure) in gas pipe lines(Isothermal flow, f = 0.002 −−−−− 0.005, γγγγγ = 1.3, M

1 = 0.02)

L/D Static pressure ratio, pr = p

2 /p

1

f = 0.0020 f = 0.0025 f = 0.0030 f = 0.0035 f = 0.0040 f = 0.0050

10 0.9999 0.9999 0.9999 0.9999 0.9999 0.999950 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997

100 0.9997 0.9997 0.9996 0.9996 0.9995 0.9995250 0.9995 0.9993 0.9992 0.9991 0.9989 0.9987400 0.9992 0.9989 0.9987 0.9985 0.9983 0.9979

500 0.9989 0.9986 0.9984 0.9982 0.9979 0.9974600 0.9987 0.9984 0.9981 0.9978 0.9975 0.9968700 0.9985 0.9981 0.9978 0.9975 0.9970 0.9963800 0.9983 0.9979 0.9975 0.9970 0.9967 0.9958900 0.9981 0.9976 0.9971 0.9967 0.9962 0.9953

1000 0.9979 0.9973 0.9968 0.9963 0.9958 0.99472000 0.9958 0.9947 0.9937 0.9927 0.9916 0.98953000 0.9937 0.9921 0.9905 0.9890 0.9874 0.98424000 0.9916 0.9895 0.9874 0.9853 0.9832 0.97895000 0.9895 0.9869 0.9842 0.9816 0.9789 0.9736

6000 0.9874 0.9842 0.9811 0.9779 0.9747 0.96837000 0.9853 0.9816 0.9779 0.9742 0.9704 0.96298000 0.9832 0.9789 0.9747 0.9704 0.9661 0.95758500 0.9821 0.9776 0.9731 0.9685 0.9639 0.95489000 0.9811 0.9763 0.9715 0.9667 0.9618 0.9520

1 × 104 0.9789 0.9736 0.9683 0.9629 0.9575 0.94661.5 × 104 0.9683 0.9602 0.9520 0.9438 0.9355 0.91872.0 × 104 0.9575 0.9465 0.9355 0.9243 0.9130 0.88992.5 × 104 0.9465 0.9327 0.9187 0.9044 0.8899 0.8602

3 × 104 0.9355 0.9186 0.9015 0.8841 0.8662 0.8294

4 × 104 0.9130 0.8899 0.8663 0.8419 0.8168 0.76425 × 104 0.8899 0.8602 0.8295 0.7975 0.7642 0.69286 × 104 0.8663 0.8295 0.7909 0.7505 0.7077 0.61327 × 104 0.8419 0.7975 0.7505 0.7003 0.6462 0.52158 × 104 0.8168 0.7642 0.7077 0.6462 0.5783 0.4099

1 × 105 0.7642 0.6928 0.6132 0.5215 0.4099 –2 × 105 0.4098 – – – – –

Static Pressure Drop in Gas Pipe Lines 139


TABLE 12.2

Static pressure drop (pr = outlet pressure/linlet pressure) in gas pipe lines(Isothermal flow, f = 0.002 −−−−− 0.005, γγγγγ = 1.4, M

1 = 0.02)

L/D Static pressure ratio, pr = p

2 /p

1

f = 0.0020 f = 0.0025 f = 0.0030 f = 0.0035 f = 0.0040 f = 0.0050

10 0.9999 0.9999 0.9999 0.9999 0.9999 0.999950 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997

100 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994250 0.9994 0.9993 0.9991 0.9990 0.9988 0.9986400 0.9991 0.9988 0.9986 0.9984 0.9982 0.9977

500 0.9988 0.9986 0.9983 0.9980 0.9977 0.9972600 0.9986 0.9983 0.9979 0.9976 0.9973 0.9966700 0.9984 0.9980 0.9976 0.9972 0.9968 0.9960800 0.9982 0.9977 0.9973 0.9968 0.9964 0.9955900 0.9979 0.9975 0.9969 0.9964 0.9959 0.9949

1000 0.9977 0.9972 0.9966 0.9961 0.9955 0.99442000 0.9955 0.9943 0.9932 0.9921 0.9910 0.98873000 0.9932 0.9915 0.9898 0.9882 0.9864 0.98304000 0.9910 0.9887 0.9864 0.9842 0.9819 0.97735000 0.9887 0.9859 0.9830 0.9802 0.9773 0.9716

6000 0.9864 0.9830 0.9796 0.9762 0.9727 0.96587000 0.9842 0.9802 0.9762 0.9721 0.9681 0.96008000 0.9819 0.9773 0.9727 0.9681 0.9635 0.95418500 0.9808 0.9759 0.9710 0.9661 0.9611 0.95129000 0.9796 0.9744 0.9692 0.9640 0.9588 0.9482

1 × 104 0.9773 0.9716 0.9658 0.9600 0.9541 0.94231.5 × 104 0.9658 0.9571 0.9482 0.9393 0.9303 0.9121

2 × 104 0.9541 0.9423 0.9303 0.9182 0.9059 0.88092.5 × 104 0.9423 0.9273 0.9121 0.8967 0.8809 0.8485

3 × 104 0.9304 0.9121 0.8935 0.8745 0.8551 0.8150

4 × 104 0.9060 0.8809 0.8551 0.8285 0.8010 0.74305 × 104 0.8809 0.8485 0.8149 0.7797 0.7430 0.66336 × 104 0.8551 0.8149 0.7725 0.7277 0.6800 0.57277 × 104 0.8285 0.7797 0.7277 0.6717 0.6106 0.46488 × 104 0.8010 0.7429 0.6800 0.6106 0.5322 0.3225

1 × 105 0.7429 0.6633 0.5727 0.4648 0.3225 –2 × 105 0.3225 – – – – –

TABLE 13.1

Isentropic compressor power in kW (cp =1.0 kJ/kg K, T1 = 300 K, γγγγγ = 1.3)

pr = p

2 /p

1

m = mass flow rate in kg/s

1.0 1.5 2.0 3.0 4.0 5.0

1.1 6.671 10.007 13.342 20.014 26.685 33.3561.2 12.891 19.337 25.782 38.673 51.564 64.4551.3 18.724 28.086 37.448 56.172 74.896 93.6201.4 24.221 36.332 48.443 72.664 96.886 121.1071.5 29.425 44.137 58.850 88.274 117.700 147.123

1.6 34.367 51.551 68.735 103.102 137.470 171.8371.7 39.078 58.617 78.156 117.234 156.312 195.3901.8 43.580 65.370 87.160 130.740 174.320 217.9001.9 47.894 71.840 95.787 143.681 191.574 239.4682.0 52.036 78.054 104.072 156.108 208.144 260.180

2.1 56.022 84.033 112.044 168.065 224.087 280.1092.2 59.864 89.796 119.729 179.593 239.457 299.3212.3 63.575 95.362 127.149 190.724 254.299 317.8732.4 67.163 100.744 134.326 201.489 268.652 335.8152.5 70.638 105.957 141.276 211.914 282.552 353.190

2.6 74.008 111.011 148.015 222.023 296.031 370.0382.7 77.279 115.919 154.558 231.837 309.116 386.3962.8 80.459 120.688 160.917 241.376 321.835 402.2932.9 83.552 125.328 167.104 250.656 334.208 417.7603.0 86.564 129.846 173.128 259.693 346.257 432.822

3.5 100.563 150.844 201.125 301.688 402.250 502.8134.0 113.098 169.646 226.195 339.293 452.390 565.4884.5 124.479 186.719 248.959 373.438 497.917 622.3975.0 134.926 202.389 269.852 404.779 539.705 674.6315.5 144.598 216.897 289.196 433.794 578.391 722.990

6.0 153.615 230.423 307.230 460.845 614.460 768.0757.0 170.041 255.062 340.083 510.124 680.165 850.2078.0 184.751 277.126 369.501 554.252 739.002 923.7539.0 198.107 297.160 396.213 594.320 792.426 990.533

10.0 210.365 315.548 420.731 631.096 841.462 1051.827

Isentropic Compressor Power 141


TABLE 13.2

Isentropic compressor power in kW (cp =1.0 kJ/kg K, T1 = 300 K, γγγγγ = 1.4)

pr = p

2 /p

1

m = mass flow rate in kg/s

1.0 1.5 2.0 3.0 4.0 5.0

1.1 8.281 12.422 16.563 24.844 33.124 41.4061.2 16.041 24.061 32.082 48.123 64.164 80.2051.3 23.351 35.027 46.703 70.054 93.406 116.7571.4 30.271 45.406 60.541 90.812 121.083 151.3531.5 36.845 55.268 73.690 110.536 147.381 184.227

1.6 43.114 64.671 86.228 129.342 172.456 215.5701.7 49.109 73.663 98.217 147.326 196.434 245.5431.8 54.856 82.285 109.713 164.569 219.425 274.2821.9 60.380 90.571 120.761 181.141 241.522 301.9022.0 65.700 98.551 131.401 197.101 262.802 328.502

2.1 70.834 106.251 141.668 212.501 283.335 354.1692.2 75.795 113.693 151.591 227.386 303.181 378.9772.3 80.598 120.898 161.197 241.795 322.393 402.9922.4 85.254 127.882 170.509 255.763 341.018 426.2722.5 89.774 134.661 179.548 269.322 359.095 448.869

2.6 94.166 141.249 188.332 282.498 376.664 470.8302.7 98.439 147.659 196.878 295.317 393.756 492.1952.8 102.600 153.901 205.201 307.801 410.402 513.0022.9 106.657 159.986 213.314 319.971 426.628 533.2863.0 110.615 165.922 221.230 331.845 442.460 553.075

3.5 129.103 193.654 258.206 387.309 516.412 645.5154.0 145.789 218.684 291.579 437.368 583.158 728.9474.5 161.046 241.569 322.092 483.137 644.183 805.2295.0 175.135 262.702 350.270 525.405 700.540 875.6755.5 188.251 282.376 376.501 564.7521 753.003 941.253

6.0 200.540 300.810 401.081 601.621 802.161 1002.7027.0 223.077 334.616 446.154 669.232 892.309 1115.3868.0 243.418 365.127 486.836 730.254 973.672 1217.0909.0 262.016 393.023 524.031 786.047 1048.062 1310.078

10.0 279.190 418.785 558.381 837.570 1116.761 1395.951

TABLE 14

Approximate values of pumping power (kW) for fans and blowers

Q m s( / )3 ∆p = Pressure rise in millimeters W.G.

1 2 3 4 5

0.1 0.0015 0.0030 0.0045 0.0060 0.00750.2 0.0030 0.0060 0.0091 0.0121 0.01510.3 0.0045 0.0091 0.0136 0.0181 0.02260.4 0.0060 0.0121 0.0181 0.0241 0.03020.5 0.0075 0.0151 0.0226 0.0302 0.0377

1.0 0.0151 0.0302 0.0453 0.0604 0.07551.2 0.0181 0.0362 0.0543 0.0724 0.09061.4 0.0211 0.0423 0.0634 0.0845 0.10561.6 0.0241 0.0483 0.0724 0.0966 0.12071.8 0.0272 0.0543 0.0815 0.1087 0.1358

2.0 0.0302 0.0604 0.0906 0.1207 0.15092.5 0.0377 0.0755 0.1132 0.1509 0.18863.0 0.0453 0.0906 0.1358 0.1811 0.22643.5 0.0528 0.1056 0.1585 0.2111 0.26414.0 0.0604 0.1207 0.1811 0.2415 0.3018

5.0 0.0755 0.1509 0.2264 0.3018 0.377310.0 0.1509 0.3018 0.4528 0.6037 0.754620.0 0.3018 0.6037 0.9055 1.2073 1.509250.0 0.7546 1.5092 2.2638 3.0184 3.7730100 1.5092 3.0184 4.5276 6.0367 7.5460

150 2.2638 4.5276 6.7913 9.0551 11.3189200 3.0184 6.0367 9.0551 12.0735 15.0918400 6.0367 12.0735 18.1102 24.1469 30.1837600 9.0551 18.1102 27.1653 36.2204 45.2755800 12.0735 24.1469 36.2204 48.2939 60.3673

1000 15.0918 30.1837 45.2755 60.3673 75.45921500 22.6378 45.2755 67.9133 90.5510 113.18882000 30.1837 60.3673 90.5510 120.7347 150.91832500 37.7296 75.4592 113.1888 150.9183 188.64793000 45.2755 90.5510 135.8265 181.1020 226.3775

Pumping Power for Fans and Blowers 143


TABLE 14



10 20 30 40 50

0.1 0.0151 0.0302 0.0453 0.0604 0.07550.2 0.0302 0.0604 0.0906 0.1207 0.15090.3 0.0453 0.0906 0.1358 0.1811 0.22640.4 0.0604 0.1207 0.1811 0.2415 0.30180.5 0.0755 0.1509 0.2264 0.3018 0.3773

1.0 0.1509 0.3018 0.4528 0.6037 0.75461.2 0.1811 0.3622 0.5433 0.7244 0.90551.4 0.2113 0.4226 0.6339 0.8451 1.05641.6 0.2415 0.4829 0.7244 0.9659 1.20731.8 0.2717 0.5433 0.8150 1.0866 1.3583

2.0 0.3018 0.6037 0.9055 1.2073 1.50922.5 0.3773 0.7546 1.1319 1.5092 1.88653.0 0.4528 0.9055 1.3583 1.8110 2.26383.5 0.5282 1.0564 1.5846 2.1129 2.64114.0 0.6037 1.2073 1.8110 2.4147 3.0184

5.0 0.7546 1.5092 2.2638 3.0184 3.773010.0 1.5092 3.0184 4.5276 6.0367 7.545920.0 3.0184 6.0367 9.0551 12.0735 15.091850.0 7.5459 15.0918 22.6378 30.1837 37.7296100 15.0918 30.1837 45.2755 60.3673 75.4592

150 22.6378 45.2755 67.9133 90.5510 113.1888200 30.1837 60.3673 90.5510 120.7347 150.9183400 60.3673 120.7347 181.1020 241.4693 301.8367600 90.5510 181.1020 271.6530 362.2040 452.7550800 120.7347 241.4693 362.2040 482.9387 603.6733

1000 150.9183 301.8367 452.7550 603.6733 754.59171500 226.3775 452.7550 679.1325 905.5100 1131.88762000 301.8367 603.6733 905.5100 1207.3467 1509.18332500 377.2958 754.5917 1131.8876 1509.1833 1886.47923000 452.7550 905.5100 1358.2650 1811.0200 2263.7751

TABLE 14



100 200 300 400 500

0.1 0.1509 0.3018 0.4528 0.6037 0.75460.2 0.3018 0.6037 0.9055 1.2073 1.50920.3 0.4528 0.9055 1.3583 1.8110 2.26380.4 0.6037 1.2073 1.8110 2.4147 3.01840.5 0.7546 1.5092 2.2638 3.0184 3.7730

1.0 1.5092 3.0184 4.5276 6.0367 7.54591.2 1.8110 3.6220 5.4331 7.2441 9.05511.4 2.1129 4.2257 6.3386 8.4514 10.56431.6 2.4147 4.8294 7.2441 9.6588 12.07351.8 2.7165 5.4331 8.1496 10.8661 13.5827

2.0 3.0184 6.0367 9.0551 12.0735 15.09182.5 3.7730 7.5459 11.3189 15.0918 18.86483.0 4.5276 9.0551 13.5827 18.1102 22.63783.5 5.2821 10.5643 15.8464 21.1286 26.41074.0 6.0367 12.0735 18.1102 24.1469 30.1837

5.0 7.5459 15.0918 22.6378 30.1837 37.729610.0 15.0918 30.1837 45.2755 60.3673 75.459220.0 30.1837 60.3673 90.5510 120.7347 150.918350.0 75.4592 150.9183 226.3775 301.8367 377.2958100 150.9183 301.8367 452.7550 603.6733 754.5917

150 226.3775 452.7550 679.1325 905.5100 1131.8876200 301.8367 603.6733 905.5100 1207.3467 1509.1833400 603.6733 1207.3467 1811.0200 2414.6934 3018.3667600 905.5100 1811.0200 2716.5300 3622.0400 4527.5503800 1207.3467 2414.6934 3622.0400 4829.3867 6036.7334

1000 1509.1833 3018.3667 4527.5503 6036.7334 7.546 MW1500 2263.7751 4527.5503 6791.3252 9055.1006 11.319 MW2000 3018.3667 6036.7334 9055.1006 12073.4668 15.092 MW2500 3772.9585 7545.9170 11318.8750 15091.8340 18.865 MW3000 4527.5503 9055.1006 13582.6504 18110.2012 22.638 MW

Pumping Power for Fans and Blowers 145


TABLE 14



600 700 800 1000 1500

0.1 0.9055 1.0564 1.2073 1.5092 2.26380.2 1.8110 2.1129 2.4147 3.0184 4.52760.3 2.7165 3.1693 3.6220 4.5276 6.79130.4 3.6220 4.2257 4.8294 6.0367 9.05510.5 4.5276 5.2821 6.0367 7.5459 11.3189

1.0 9.0551 10.5643 12.0735 15.0918 22.63801.2 10.8661 12.6771 14.4882 18.1102 27.16531.4 12.6771 14.7900 16.9029 21.1286 31.69301.6 14.4882 16.9029 19.3175 24.1469 36.22041.8 16.2992 19.0157 21.7322 27.1653 40.7480

2.0 18.1102 21.1286 24.1469 30.1837 45.27552.5 22.6378 26.4107 30.1837 37.7296 56.59443.0 27.1653 31.6929 36.2204 45.2755 67.91333.5 31.6929 36.9750 42.2571 52.8214 79.23214.0 36.2204 42.2571 48.2939 60.3673 90.5510

5.0 45.2755 52.8214 60.3673 75.4592 113.188810.0 90.5510 105.6428 120.7347 150.9183 226.378020.0 181.1020 211.2857 241.4693 301.8367 452.755050.0 452.7550 528.2142 603.6733 754.5917 1131.8876100 905.5100 1056.4283 1207.3467 1509.1833 2263.7751

150 1358.2650 1584.6426 1811.0200 2263.7751 3395.6626200 1811.0200 2112.8567 2414.6934 3018.3667 4527.5503400 3622.0400 4225.7134 4829.3867 6036.7334 9055.1006600 5433.0601 6338.5703 7244.0801 9055.1006 13582.6504800 7244.0801 8451.4268 9658.7734 12073.4668 18110.2012

1000 9055.1006 10564.2832 12073.4668 15.091 MW 22.637 MW1500 13582.6504 15846.4258 18110.2012 22.637 MW 33.957 MW2000 18110.2012 21128.5664 24146.9336 30.184 MW 45.275 MW2500 22637.7500 26410.7090 30183.6680 37.729 MW 56.594 MW3000 27165.3008 31692.8516 36220.4023 45.275 MW 67.913 MW

TABLE 15.1

S.I, Metric and British units

Principal dimensions and units

Quantity Dimensions S.I. Metric British

Length L m m ft

Mass M kg kgm lbm

Time T s sec sec

Temperature θ K °K °R

Area L2 m2 m2 ft2

Volume L3 m3 m3 ft3

Density M/L3 kg/m3 kgm/m3 Ibm/ft3

Force ML/T2 N = kg m/s2 kgf lbf

Torque ML2/T2 mN m kgf ft lbf

Velocity L/T m/s m/sec ft/sec

Acceleration L/T2 m/s2 m/s2 ft/sec2

Pressure M/LT2 Pa = N/m2 kgf/m2 lbf/ft2

Work and energy ML2/T2 J = Nm kgf m ft lbf

Heat ML2/T2 Joule = J cal Btu

Specific heat L2/T2θ J/kg K cal/kgm °K Btu/lbm °R

Power ML2/T3 Nm/s, J/s kgf m/s ft lbf/sec

Specific entropy L2/T2θ J/kg K cal/kgm °K Btu/lbm °R

Dynamic viscosity M/LT Ns/m2 kgf-sec/m2 lbf-sec/ft2

kg/ms

Kinematic viscosity L2/T m2/s m2/sec ft2/sec

Weight ML/T2 N = kg m/s2 kgf lbf

S.I. Metric and British Units 147


TABLE 15.2

Conversion of units

Length Pressure1m = 3.28 ft 1 bar = 105 N/m2 = 105 Pa = 100 kN/m2

1 cm = 0.3937 in = 0.1 MN/m2 = 1.0197 kgf/cm2

1 km = 0.6215 mile = 14.504 lbf/in2

1 ft = 30.48 cm 1 m bar = 10-3 bar = 100 N/m2 = 100 Pa1 in = 2.54 cm = 10.2 mm W.G.1 mile = 1.609 km 1 mm W.G. = 1 kgf/m2 = 9.807 N/m2 = 0.098 m bar1 nautical mile = 6076 ft = 1.853 km 1 torr = 1 mm Hg = 13.6 mm W.G. = 133.38N/m2

= 1.334 m barArea 1 atmosphere = 1.013 bar = 1.033 kgf/cm2

1 m2 = 10.765 ft2 = 14.696 lbf/in2 = 29.921 in Hg1 cm2 = 0.155 in2 = 760 mm Hg = 10.336 m W.G.1 ft2 = 0.0929 m2

1 in2 = 6.452 cm2 Density1 kg/m3 = 0.0624 lbm/ft3

Volume 1 lbm/ft3 = 16.025 kg/m3

1 m3 = 1000 litres = 35.315 ft2

1 litre = 1000 cc = 0.001 m3 = 0.0353 ft3 Work and Energy1 cc = 0.061 in3 1 Nm = 1 Joule = 107 ergs1 ft3 = 0.0283 m3 = 28.328 litres = 0.102 kgf m = 0.7374 ft lbf1 in3 = 16.387 cm3 1 kgf m = 9.807 Nm = 7.229 ft lbf1 gallon (British) = 8 pints = 4.546 litres 1 ft lbf = 0.1383 kgf m = 1.356 Nm = 1.356 J

= 0.160 ft3 = 277.3 in3

1 pint = 568.25 cc Heat1 kJ = 1000 J = 0.2388 k cal = 0.9478 Btu

Mass 1 k cal = 427 kgf m = 4.187 kJ = 3.968 Btu1 kgm = 2.204 lbm 1 Btu = 778 ft lbf = 1.055 kJ = 0.252 k cal1 lbm = 0.4537 kgm1 metric ton (tonne) = 0.984 ton Power

1 Nm/s = 1 J/s = 1 wattTemperature= 0.102 kgf m/sec = 0.7375 ft lb/sec

°C = 1 kW = 1000 J/s = 102 kgf m/sec = 737.5 ft lbf/sec

°F = 1.8 °C + 32 = 1.36 metric H.P. = 1.34 H.P. (FPS)°K = 273.15 + °C = Kelvins (in S.I. Units) 1 metric H.P. = 75 kgf m/sec = 0.735 kW = 0.985 H.P.°R = 459.4 + °F = 1.8 °K 1 H.P. (FPS) = 550 ft lb/sec = 1.015 metric H.P.

= 0.746 kWForce1 N = 105 dynes = 0.102 kgf = 0.2248 lbf Dynamic Viscosity1 kgf = 9.807 N = 2.204 lbf 1 Ns/m2 = 1 kg/ms = 0.102 kgf/m2

1 metric ton = 1000 kgf = 0.0209 lbf sec/ft2 = 10 poise1 lbf = 4.448 N = 0.4537 kgf 1 lbf sec/ft = 47.847 Ns/m2

Velocity1 m/s = 3.28 ft/sec Kinematic Viscosity1 km/hr = 0.621 mile/hr 1 m2/s = 10.76 ft2/sec = 104 cm2/sec = 104 Stokes1 mile/hr = 1.609 km/hr 1 ft2/sec = 0.0929 m2/s1 ft/sec = 0.305 m/s1 knot = 1.689 ft/sec = 0.515 m/s

o F 32

1.8

−

gas tables

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