Page 216 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 197
free vortex swirl at this speed. You may assume that losses are absent. Comment upon the
results you obtain.
(Take C p D 1.148 kJ(kg ° C) and
D 1.33.)
3. Gas enters the nozzles of an axial flow turbine stage with uniform total pressure at a
uniform velocity c 1 in the axial direction and leaves the nozzles at a constant flow angle ˛ 2
to the axial direction. The absolute flow leaving the rotor c 3 is completely axial at all radii.
Using radial equilibrium theory and assuming no losses in total pressure show that
2
" #
cos ˛ 2
r
2
.c 2 c //2 D U m c m2 1
3 1
r m
where U m is the mean blade speed,
c m2 is the tangential velocity component at nozzle exit at the mean radius r D r m .
(Note: The approximate c 3 D c 1 at r D r m is used to derive the above expression.)
4. Gas leaves an untwisted turbine nozzle at an angle ˛ to the axial direction and in radial
equilibrium. Show that the variation in axial velocity from root to tip, assuming total pressure
is constant, is given by
c x r sin 2 ˛ D constant.
Determine the axial velocity at a radius of 0.6 m when the axial velocity is 100 m/s at a
radius of 0.3 m. The outlet angle ˛ is 45 deg.
5. The flow at the entrance and exit of an axial-flow compressor rotor is in radial equilib-
rium. The distributions of the tangential components of absolute velocity with radius are:
c 1 D ar b/r, before the rotor,
c 2 D ar C b/r, after the rotor,

where a and b are constants. What is the variation of work done with radius? Deduce expres-
sions for the axial velocity distributions before and after the rotor, assuming incompressible
flow theory and that the radial gradient of stagnation pressure is zero.
2
At the mean radius, r D 0.3 m, the stage loading coefficient, D W/U is 0.3, the
t
reaction ratio is 0.5 and the mean axial velocity is 150 m/s. The rotor speed is 7640 rev/min.
Determine the rotor flow inlet and outlet angles at a radius of 0.24 m given that the hub tip
ratio is 0.5. Assume that at the mean radius the axial velocity remained unchanged .c x1 D c x2
at r D 0.3 m).
(Note: W is the specific work and U t the blade tip speed.)
6. An axial-flow turbine stage is to be designed for free-vortex conditions at exit from
the nozzle row and for zero swirl at exit from the rotor. The gas entering the stage has a
stagnation temperature of 1000 K, the mass flow rate is 32 kg/s, the root and tip diameters
are 0.56 m and 0.76 m respectively, and the rotor speed is 8000 rev/min. At the rotor tip the
stage reaction is 50% and the axial velocity is constant at 183 m/s. The velocity of the gas
entering the stage is equal to that leaving.
Determine:
(i) the maximum velocity leaving the nozzles;
(ii) the maximum absolute Mach number in the stage;
(iii) the root section reaction;
(iv) the power output of the stage;
(v) the stagnation and static temperatures at stage exit.
(Take R D 0.287 kJ/(kg ° C) and C p D 1.147 kJ/(kg ° C).)