Page 198 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 179
From the radial equilibrium solution in eqns. (6.19), after some rearrangement,
c 2 c 2 2a
r
x1 x2 D A 1 A 2 C ln ,
4b 2 t  r t
where
W C p T 0
t D D .
2 2
U 2  r
t t
1
In the above example, 1 a/ D , t D 0.18
2
0
R D 0.778 C ln.r/r t /.
The true reaction variation is shown in Figure 6.5 and it is evident that eqn. (6.17)
is invalid as a result of axial velocity changes.

The direct problem

The flow angle variation is specified in the direct problem and the radial equi-
librium equation enables the solution of c x and c to be found. The general radial
equilibrium equation can be written in the form

ds c 2 dc
dh 0
T D C c
dr dr r dr
2
2
c sin ˛ dc
D C c , .6.20/
r dr
as c D c sin ˛.
If both dh 0 /dr and ds/dr are zero, eqn. (6.20) integrated gives
Z
dr
2
log c D sin ˛ C constant
r
or, if c D c m at r D r m , then
Z r
c 2 dr
D exp sin ˛ . (6.21)
c m r
r m
If the flow angle ˛ is held constant, eqn. (6.21) simplifies still further,
2
sin ˛
c c x c r
D D D (6.22)
c m c xm c m r m
The vortex distribution represented by eqn. (6.22) is frequently employed in practice
as untwisted blades are relatively simple to manufacture.
The general solution of eqn. (6.20) can be found by introducing a suitable inte-
R 2
grating factor into the equation. Multiplying throughout by exp[2 sin ˛dr/r]it
follows that
Z Z
d 2 2 dh 0 ds 2
c exp 2 sin ˛dr/r D 2 T exp 2 sin ˛dr/r .
dr dr dr