898 ENGINEERING THERMODYNAMICS
dharm
\M-therm\Th16-2.pm5(ii) The mass rate of flow is given by :m = A 2 2
1 112
12
2
11
γ
γ
ρ γγ
γ
−F
HGI
KJF
HGI
KJ−F
HGI
KJL
N
M
M
MO
Q
P
P
P+
p p
pp
p(iii) Value ofp
p2
1F
HGI
KJ for maximum value of mass rate of flow is given by :p
p2
1F
HGI
KJ =2
11
γγ
γ
+F
HGI
KJ− = 0.528 (when γ = 1.4)(iv) Value of V 2 for maximum rate of flow of liquid is given as,V 2 =2
11
1γ
γρ+F
HGI
KJp
(= C 2 )(v) Maximum rate of flow of fluid through nozzle,mmax = A 22
12
12(^111)
2
1
1
γ 1
γ ρ
γ
γγ
γ
γ
γ
−
F
HG
I
KJ +
F
HG
I
KJ
−F +
HG
I
KJ
L
N
M
M
M
O
Q
P
P
P
−
- p −
For air, substituting γ = 1.4, we get
mmax = 0.685 A 2 p 11 ρ
If the pressure ratio is less than 0.528, the mass rate of flow of the fluid is always corresponding to the
pressure ratio of 0.528. But if the pressure ratio is more than 0.528, the mass rate of flow of fluid is
corresponding to the given pressure ratio.
- Whenever a supersonic flow (compressible) changes to subsonic flow, a shock wave (analogous to hydrau-
lic jump in an open channel) is produced, resulting in a sudden rise in pressure, density, temperature and
entropy.
p 1 + ()ρ
ρ
112
1V = p
2 +()ρ
ρ222
2V ... Ranking Line Equation(^) γ−γ 1
p 1
ρ 1
F
HG
I
KJ +
()ρ
ρ
112
212
V = γ
γ− 1
p^2
ρ 2
F
HG
I
KJ
()ρ
ρ
222
222
V ... Fanno line Equation
p
p
2
1
γ
γ
ρ
ρ
γ
γ
ρ
ρ
−
F
HG
I
KJ
−
−
F
HG
I
KJ
−
1
1
1
1
1
2
1
2
1
...Rankinge-Hugoniot Equations
ρρ^2
1
V
V
1
2
1 1
1
1
1
2
1
2
1
- −
F
HG
I
KJ
- −
−
F
HG
I
KJ
γ
γ
γ
γ
p
p
p
p
One can also express pp^2
1
, VV^2
1
, ρρ^2
1
and TT^2
1
in terms of Mach number as follows :
p
p
2
1
21
1
γγ 12
γ
M −−
()
...(i)
U
V
|
|
|
|
||
W
| | | | | |