Page 199 - Fluid Mechanics and Thermodynamics of Turbomachinery

180 Fluid Mechanics, Thermodynamics of Turbomachinery
After integrating and inserting the limit c D c m at r D r m , then
Z Z
r r m
2
2
2
2
c exp 2 sin ˛dr/r c exp 2 sin ˛dr/r
m
Z Z
r
dh 0 ds 2
D 2 T exp 2 sin ˛dr/r dr. .6.23/
dr dr
r m
Particular solutions of eqn. (6.23) can be readily obtained for simple radial distri-
butions of ˛, h 0 and s. Two solutions are considered here in which both 2dh 0 /dr D
2
kc /r m and ds/dr D 0,k being an arbitrary constant
m
R 2 a
2
(i) Let a D 2 sin ˛. Then exp[2 sin ˛dr/r] D r and, hence
" #
2
a 1Ca
c r k r
D 1 C 1 . (6.23a)
c m r m 1 C a r m
Equation (6.22) is obtained immediately from this result with k D 0.
2
(ii) Let br/r m D 2 sin ˛. Then,
r
Z
2
2
2
c exp.br/r m / c exp.b/ D .kc /r m / exp.br/r m /dr
m m
r m
and eventually,
2
c k k r
D C 1 exp b 1 . (6.23b)
c m b b r m
Compressible flow through a fixed blade row
In the blade rows of high-performance gas turbines, fluid velocities approaching,
or even exceeding, the speed of sound are quite normal and compressibility effects
may no longer be ignored. A simple analysis is outlined below for the inviscid flow
of a perfect gas through a fixed row of blades which, nevertheless, can be extended
to the flow through moving blade rows.
The radial equilibrium equation, eqn. (6.6), applies to compressible flow as well
as incompressible flow. With constant stagnation enthalpy and constant entropy, a
free-vortex flow therefore implies uniform axial velocity downstream of a blade row,
regardless of any density changes incurred in passing through the blade row. In fact,
for high-speed flows there must be a density change in the blade row which implies
a streamline shift as shown in Figure 6.1. This may be illustrated by considering
the free-vortex flow of a perfect gas as follows. In radial equilibrium,

1 dp c 2 K 2
D D with c D K/r.
dr r r 3
For reversible adiabatic flow of a perfect gas, D Ep 1/
, where E is
constant. Thus
Z Z
p 1/
dp D EK 2 r 3 dr C constant,