870 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-1.pm5

and ps, Vs, ρs and Ts are corresponding values of
pressure, velocity density, and temperature at point
S.
Applying Bernoulli’s equation for adiabatic
(frictionless) flow at points O and S, (given by
eqn. 16.7), we get

γ

γρ−

F

HG

I

KJ

+

12

0

0

0

p^2

g

V

g

+ z 0 =

γ

γρ−

F

HG

I

KJ

+

12

p^2

g

V

g

s

s

s + z

s

But z 0 = zs ; the above equation reduces to

γ

γρ−

F

HG

I

KJ

+

12

0

0

0

p^2

g

V

g

=

γ

γρ−

F

HG

I

KJ

+

12

p^2

g

V

g

s

s

s

Cancelling ‘g’ on both the sides, we have

γ

γρ−

F

HG

I

KJ

+

12

0

0

0

pV^2

=

γ

γρ−

F

HG

I

KJ

+

12

pV^2

s

s

s

At point S the velocity is zero, i.e., Vs = 0 ; the above equation becomes

γ

γρρ−

F

HG

I

KJ

−

F

HG

I

KJ

=−

12

0

0

pps V 02

s

or,

γ

γρ ρ

ρ

−

F

HG

I

KJ

−×

F

HG

I

(^1) KJ
(^01)
0
0
0
pp
p
s
s
= –
V 02
2
or,
γ
γρ
ρ
− ρ
F
HG
I
KJ
−×
F
HG
I
(^1) KJ
(^01)
00
pp 0
p
s
s
= – V^0
2
2
...(i)
For adiabatic process : p^0
ρ 0
γ =
ps
ρsγ
or
p
ps
(^0) = ρ
ρ
γ
γ
0
s
or
ρ
ρ
0
s

p
ps
0
1
F
HG
I
KJ
γ
...(ii)
Substituting the value of ρ
ρ
0
s
in eqn. (i), we get
γ
γρ
γ
−
F
HG
I
KJ
−×
F
HG
I
KJ
L
N
M
M
M
O
Q
P
P
P
=−
1
1
2
0
00
0
1
0
pp^2
p
p
p
s V
s
or, γ
γρ
γ
−
F
HG
I
KJ
−
F
HG
I
KJ
R
S
|
T
|
U
V
|
W
|
−
1
(^01)
00
1 1
pp
p
s = – V^0
2
2
or, 1
0
1
−
F
HG
I
KJ
L
N
M
M
M
O
Q
P
P
P
−
p
p
s
γ
γ
= – V^0
2
2
γ
γ
F ρ−
HG
I
KJ
(^10)
p 0
or, 1 + V^0
2
2
γ
γ
F ρ−
HG
I
KJ
(^10)
p 0
= p
p
s
0
1
F
HG
I
KJ
−γ
γ
...(iii)
O
Streamlines
S (stagnation point)
Body
Fig. 16.4. Stagnation properties.