THESE de DOCTORAT - cerfacs

136 Chapter A: About the π ′ c = 0 assumption
w 2
U
v 2
w 1 v θ,2
v θ,1 v 1
U
Figure A.1: Velocity triangles
Considering now U 1 = U 2 = U for an axial compressor and ρ 1 u 1 = ρ 2 u 2 due to mass conservation,
it is possible to express W c as function of U and v θ ,
W
ρ 1 u 1
= h t,2 − h t,1 = U(v θ,2 − v θ,1 )
(A.5)
and introducing now π T
h t,1 (π T − 1) = U(v θ,2 − v θ,1 ) (A.6)
Let us now define a parameter ζ = Uv θ
h t
. The variable π T is then expressed as
h t,1 (π T − 1) = ζ 1 h t,1 − ζ 2 h t,2 = ζ 1 h t,1 − ζ 2 π T h t,1
(A.7)
(π T − 1) = ζ 1 − ζ 2 π T (A.8)
Finally, resolving for π T , it yields
π T = ζ 1 + 1
ζ 2 + 1
(A.9)
Applying the differential of the logarithm to Eq. (A.9) results in
π ′ T
¯π T
= ζ′ 1
¯ζ 1 + 1 − ζ′ 2
¯ζ 2 + 1
(A.10)