Page 196 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 177
The axial velocity must change in crossing the rotor row so that eqn. (6.17) is only
a crude approximation at the best. Just how crude is this approximation will be
indicated below.
Assuming constant stagnation enthalpy at entry to the stage, integrating
eqn. (6.6a), the axial velocity distributions before and after the rotor are
1
c 2 D constant 4a. ar 2 b ln r/, .6.18a/
x1 2
1
2
c 2 D constant 4a. ar C b ln r/, .6.18b/
x2 2
More conveniently, these expressions can be written non-dimensionally as,
" #

2 2 2
c x1 2a 1 r b r
D A 1 2 ln , .6.19a/
U t  2 r t ar r t
t
" #
2 2 2
c x2 2a 1 r b r
D A 2 C 2 ln , .6.19b/
U t  2 r t ar t r t
in which U t D r t is the tip blade speed. The constants A 1 , A 2 are not entirely
arbitrary as the continuity equation, eqn. (6.12), must be satisfied.
EXAMPLE 6.2. As an illustration consider a single stage of an axial-flow air
compressor of hub-tip ratio 0.4 with a nominally constant reaction (i.e. according to
eqn. (6.17)) of 50%. Assuming incompressible, inviscid flow, a blade tip speed of
300 m/s, a blade tip diameter of 0.6 m, and a stagnation temperature rise of 16.1 ° C,
determine the radial equilibrium values of axial velocity before and after the rotor.
The axial velocity far upstream of the rotor at the casing is 120 m/s. Take C p for
air as 1.005 kJ/(kg ° C).

Solution: The constants in eqn. (6.19) can be easily determined. From eqn. (6.17)
2a/ D 2.1 R/ D 1.0.

Combining eqns. (6.14) and (6.17)
b W C p Ð T 0
D D
2
2
ar 2 2 .1 R/r 2 2U .1 R/
t t t
1005 ð 16.1
D D 0.18.
300 2
The inlet axial velocity distribution is completely specified and the constant A 1
solved. From eqn. (6.19a)
2

c x1 1 2
D A 1 [ .r/r t / 0.18 ln.r/r t /].
2
U t
At r D r t , c x1 /U t D 0.4 and hence A 1 D 0.66.
Although an explicit solution for A 2 can be worked out from eqn. (6.19b) and
eqn. (6.12), it is far speedier to use a semigraphical procedure. For an arbitrarily
selected value of A 2 , the distribution of c x2 /U t is known. Values of .r/r t / Ð .c x2 /U t /