Page 260 - Fluid Mechanics and Thermodynamics of Turbomachinery

Radial Flow Gas Turbines 241
Basic design of the rotor

Each term in eqn. (8.4a) makes a contribution to the specific work done on the
2
rotor. A significant contribution comes from the first term, namely 1 .U 2 U /,
2 2 1
and is the main reason why the inward flow turbine has such an advantage over
the outward flow turbine where the contribution from this term would be negative.
For the axial flow turbine, where U 2 D U 1 , of course no contribution to the specific
work is obtained from this term. For the second term in eqn. (8.4a) a positive
contribution to the specific work is obtained when w 3 >w 2 . In fact, accelerating
the relative velocity through the rotor is a most useful aim of the designer as this
is conducive to achieving a low loss flow. The third term in eqn. (8.4a) indicates
that the absolute velocity at rotor inlet should be larger than at rotor outlet so as to
increase the work input to the rotor. With these considerations in mind the general
shape of the velocity diagram shown in Figure 8.3 results.
Nominal Design

The nominal design is defined by a relative flow of zero incidence at rotor inlet
(i.e. w 2 D c r2 / and an absolute flow at rotor exit which is axial (i.e. c 3 D c x3 /. Thus,
from eqn. (8.4), with c 3 D 0 and c 2 D U 2 , the specific work for the nominal design
is simply

2
W D U . (8.4b)
2
EXAMPLE 8.1. The rotor of an IFR turbine, which is designed to operate at the
nominal condition, is 23.76 cm in diameter and rotates at 38 140 rev/min. At the
design point the absolute flow angle at rotor entry is 72 deg. The rotor mean exit
diameter is one half of the rotor diameter and the relative velocity at rotor exit is
twice the relative velocity at rotor inlet.
Determine the relative contributions to the specific work of each of the three
terms in eqn. (8.4a).

Solution. The blade tip speed is U 2 D ND 2 /60 D ð 38 140 ð 0.2376/60 D
474.5 m/s.
ReferringtoFigure 8.3,w 2 DU 2 cot ˛ 2 D154.17 m/s,andc 2 DU 2 sin ˛ 2 D498.9 m/s.

2
2
2
2
1
2
c D w 2 U D .2 ð 154.17/ 2 . ð 474.5/ D 38 786 m /s .
3 3 3 2
2
2
2
2
2
2
2
Hence, .U 2 U /DU .1 1/4/D168 863 m /s , w 2 w D3 ð w D71 305 m /s 2
2 3 2 3 2 2
2
2
2
and c 2 c D 210 115 m /s . Thus, summing the values of the three terms and
2 3
2
2
dividing by two, we get W D 225 142 m /s .
2
The fractional inputs from each of the three terms are: for the U terms, 0.375;
2
2
for the w terms, 0.158; for the c terms, 0.467.
2
2
Finally, as a numerical check, the specific work is, W D U D 474.5 D 225 150
2
2
2
m /s which, apart from some rounding erors, agrees with the above computations.