TITLE.PM5

COMPRESSIBLE FLOW 875

dharm

\M-therm\Th16-1.pm5

Stagnation density, ρs :

ρs =

p

RT

s

s

= ×

×

131.27 10

287 358.4

3

= 1.276 kg/m^3 (Ans.)

Compressibility factor at M = 0.8 :

Compressibility factor = 1 +

M 02

4

2

24

+ −γ M 04 + ...

= 1 +

08

4

214

24

..^2

+ − × 0.8^4 = 1.1702 (Ans.)

Example 16.10. Air at a pressure of 220 kN/m^2 and temperature 27°C is moving at a
velocity of 200 m/s. Calculate the stagnation pressure if
(i)Compressibility is neglected ; (ii)Compressibility is accounted for.
For air take R = 287 J/kg K, γ = 1.4.
Sol. Pressure of air, p 0 = 200 kN/m^2
Temperature of air, T 0 = 27 + 233 = 300 K
Velocity of air, V 0 = 200 m/s
Stagnation pressure, ps :
(i) Compressibility is neglected :

ps = p 0 +

ρ 002

2

V

where ρ 0 =

p

RT

0

0

220 10^3

287 300

= ×

× = 2.555 kg/m

3

∴ ps = 220 +

2 555 200

2

. ×^2
× 10–3 (kN/m^2 ) = 271.1 kN/m^2. Ans.
(ii) Compressibility is accounted for :
The stagnation pressure, when compressibility is accounted for, is given by,

ps = p 0 +

ργ 002 02

2 1 4 04

2

24

VMF ++− M +

HG

I

KJ

... ...[Eqn. (16.19)]

Mach number, M 0 =

V

C RT

0

0 0

200 200

1 4 287 300

==

γ. ××

= 0.576

Whence, ps = 220 +

2 555 200

2

10 1 0 576

4

214

24

0 576

..^232. 4

× ×++− ×.

F

HG

I

KJ

−

or, ps = 220 + 51.1 (1 + 0.0829 + 0.00275) = 275.47 kN/m^2 (Ans.)
Example 16.11. In aircraft flying at an altitude where the pressure was 35 kPa and
temperature – 38 °C, stagnation pressure measured was 65.4 kPa. Calculate the speed of the
aircraft. Take molecular weight of air as 28. (UPSC, 1998)
Sol. Pressure of air, p 0 = 35 kPa = 35 × 10^3 N/m^2
Temperature of air, T 0 = – 38 + 273 = 235 K
Stagnation pressure, ps = 65.4 kPa = 65.4 × 10^3 N/m^2