COMPRESSIBLE FLOW 871

dharm

\M-therm\Th16-1.pm5

For adiabatic process, the sonic velocity is given by,

C = γγRT ρ

p

= Q

p RT

ρ

=

F

HG

I

KJ

For point O, C 0 = γρ

p 0

0

or C 02 = γ

p 0

ρ 0

Substituting the value of

γ

ρ

p 0

0

= C 02 in eqn. (iii), we get

1 +

V 02

2

(γ – 1) ×^1

02 0

1

C

p

p

=F s

HG

I

KJ

−γ

γ

or, 1 +

V

C

02

0

2 2 (γ – 1) =

p

p

s

0

1

F

HG

I

KJ

−γ

γ

1 +

M 02

2 (γ – 1) =

p

p

s

0

1

F

HG

I

KJ

−γ

γ

Q V

C

(^0) M
2
02
= 02
F
HG
I
KJ
or,
p
p
s
0
1
F
HG
I
KJ
−γ
γ
=^1
1
2 0
+F −^2
HG
I
KJ
L
N
M
O
Q
P
γ M
or,
p
p
s
0
=^1
1
2 0
+F −^21
HG
I
KJ
L
N
M
O
Q
P
γ −
γ
γ
M ...(iv)
or, ps = p 0 1 1
2 0
+F −^21
HG
I
KJ
L
N
M
O
Q
P
γ −
γ
γ
M ...(16.17)
Eqn. (16.17) gives the value of stagnation pressure.
Compressibility correction factor :
If the right hand side of eqn. (16.17) is expanded by the binomial theorem, we get
ps = p 0 1
28
2
(^048)
2
0
4
0
L +++− 6
NM
O
QP
γγγγMM()M
= p 0 1
2
1
4
2
24
0
22
0
+++F −^4 +
HG
I
KJ
L
N
M
M
O
Q
P
P
γγMM M ...
or, ps = p 0 +
pM 002 M (^02) M
2 1 4 04
2
24
γγF ++− +
HG
I
KJ
... ...(16.18)
But, M 02 =
V
C
V
p
V
p
0
2
0
2
0
2
0
0
0
2
0
0

F
HG
I
KJ

γ
ρ
ρ
γ
Q C
p
0
2 0
0

F
HG
I
KJ
γ
ρ