Page 137 - Fluid Mechanics and Thermodynamics of Turbomachinery

118 Fluid Mechanics, Thermodynamics of Turbomachinery
elongation strain at the allowable stress and temperature of the blade. To enable
operation at high temperatures and for long life of the blades, the creep strength
criterion is the one usually applied by designers.
An estimate of the average rotor blade temperature T b can be made using the
approximation,
2
T b D T 2 C 0.85w /.2C p /, (4.31)
2
i.e. 85% temperature recovery of the inlet relative kinetic energy.

EXAMPLE 4.4. Combustion gases enter the first stage of a gas turbine at a stag-
nation temperature and pressure of 1200 K and 4.0 bar. The rotor blade tip diameter
is 0.75 m, the blade height is 0.12 m and the shaft speed is 10 500 rev/min. At the
mean radius the stage operates with a reaction of 50 per cent, a flow coefficient of
0.7 and a stage loading coefficient of 2.5.
Determine:

(1) the relative and absolute flow angles for the stage;
(2) the velocity at nozzle exit;
(3) the static temperature and pressure at nozzle exit assuming a nozzle efficiency
of 0.96 and the mass flow;
(4) the rotor blade root stress assuming the blade is tapered with a stress taper factor
2
K of 2/3 and the blade material density is 8000 kg/m ;
(5) the approximate mean blade temperature;
(6) taking only the centrifugal stress into account suggest a suitable alloy from the
information provided which could be used to withstand 1000 hr of operation.

Solution. (1) The stage loading is
2
D h 0 /U D .w y3 C w y2 //U D .tan ˇ 3 C tan ˇ 2 /.
From eqn. (4.20) the reaction is
tan ˇ 2 //2.
R D .tan ˇ 3
Adding and subtracting these two expressions, we get
tan ˇ 3 D . /2 C R// and tan ˇ 2 D . /2 R// .

Substituting values of , and R into the preceding equations we obtain
°
ˇ 3 D 68.2 ,ˇ 2 D 46.98 °
and for similar triangles (i.e. 50% reaction)

˛ 2 D ˇ 3 and ˛ 3 D ˇ 2
(2) At the mean radius, r m D .0.75 0.12//2 D 0.315 m, the blade speed is
U m D r m D .10500/30/ ð ð 0.315 D 1099.6 ð 0.315 D 346.36 m/s. The axial
velocity c x D U m D 0.5 ð 346.36 D 242.45 m/s and the velocity of the gas at
nozzle exit is, c 2 D c x / cos ˛ 2 D 242.45/ cos 68.2 D 652.86 m/s.