Modern Engineering Thermodynamics

686 CHAPTER 16: Compressible Fluid Flow
5. The velocity of sound c in an ideal gas:
c ideal
gas
p
= ffiffiffiffiffiffiffiffiffiffiffiffi
kg c RT
6. The stagnation state properties in terms of the Mach number:
T os
T
= 1 + k−1
2 M2 ,
p
os
p = 1 + k − 1 k/ðk−1Þ, ρ
os
2 M2
ρ = 1 + k−1 1/ðk−1Þ
2 M2
7. The properties at the throat of a choked flow nozzle (denoted by an *):
8. The choked flow mass flow rate equations:
k/ðk–1Þ
T 2
= T os , p 2
= p os , ρ 2
= ρ
k + 1
k + 1
os
k + 1
1/2
_m max /A kg
= p c k + 1
os
2
RT os
ðk + 1Þ/2ð1 − kÞ
and, for air (k = 1.40) in the Engineering English units system,
ð _m max /A Þ air = 0:532 lbm pffiffiffi
. R
os
pffiffiffiffiffiffi
lbf .s T os
1/ðk−1Þ
where p ffiffiffi
_m max is in lbm/s, A* isinin 2 , p os is in psia, T os is in R, and the constant 0.532 has units of
lbm. R /ðlbs .sÞ: For air in the SI units system,
ð _m max /A Þ air
= 0:0404 kg pffiffiffi
. K
os
pffiffiffiffiffiffi
N.s T os
where p ffiffiffi
_m max is in kg/s, A* isinm 2 , p os is in N/m 2 , T os is in K, and the constant 0.0404 has units of
kg. K /ðN .sÞ:
9. The general cross-sectional area ratio for a supersonic nozzle at its maximum flow rate:
and, for air (k = 1.40), this reduces to
A/A = 1 M
2
k + 1
1 + k−1
2 M2
ðk+1Þ/2ðk−1Þ
ðA/A Þ air
= 1
1 + 0:2M 2 3
M 1:2
10. Thermodynamic property relations across a shock wave, where the subscript x denotes the upstream (M x >1)
and the subscript y denotes downstream conditions (M y < 1):
x M x
p ffiffiffiffiffi = yM y
p ffiffiffiffi ,
T x
T y
M 2 y = ðk−1ÞM2 x + 2
2kM 2 x + 1 − k
1 + k−1
T x
=
2 M2 y
T y
1 + k−1 ,
2 M2 x
ðfor M x ≥ 1Þ
p x
= 1 + kM2 y
p y 1 + kM 2 ,
x
In the last expression M x and M y can be interchanged if M y is known and M x is the unknown.
11. The nozzle efficiency η N , velocity coefficient C v , and discharge coefficient C d :
ðk − 1Þ
ðV exit /c inlet Þ 2
η N =
2
1 − ðp exit /p inlet Þ , C p ðk−1Þ/k v = ffiffiffiffiffi η N,
Cd =
12. The diffuser efficiency η D and pressure recovery coefficient C p :
η D =
_m actual
_m isentropic
=
ðρAVÞ
actua1
ðρAVÞ isentropic
1 + k − 1
2 M2 inlet ðp ose /p osi Þ ðk−1Þ/k − 1 ð
ðk−1ÞM 2 inlet /2 , C p = p exitÞ actual
– p inlet
p osi – p inlet