Basics of Fluid Mechanics, 2014a

11.7. FANNO FLOW 443
pressure on the Mach number. Dividing equation (11.176) in equation (11.178) yields
dP
P
1+(k − 1M
dM 2 = − (
2
2 M 2 1+ k − 1 ) dM 2 (11.191)
M
M 2
2
2
The symbol “*” denotes the state when the flow is choked and Mach number is equal
to 1. Thus, M =1when P = P ∗ equation (11.191) can be integrated to yield:
Mach–Pressure Ratio
P
P ∗ = 1 k +1
2
M
√
1+ k − 1
2
M 2
In the same fashion the variables ratios can be obtained
The density ratio is
The velocity ratio is
Temperature Ratio
T
T ∗ = c2
k+1
c = 2
∗2
1+ k−1
2 M 2
Density Ratio
ρ
ρ ∗ = 1 1+ k − 1
2
M
√ k +1
2
Velocity Ratio
( ) −1 U ρ
U ∗ = ρ ∗ = M
√
M 2
k +1
2
1+ k − 1
2
The stagnation pressure decreases and can be expressed by
M 2
(11.192)
(11.193)
(11.194)
(11.195)
P 0
P 0
∗ =
(1+ 1−k
2 M 2 )
k−1
k
{}}{
P 0
P
P 0
∗
P ∗
}{{}
( 2
k+1) k
k−1
P
P ∗ (11.196)