Page 124 - Fluid Mechanics and Thermodynamics of Turbomachinery

Axial-flow Turbines: Two-dimensional Theory 105
(ii) At nozzle exit the Mach number is
1/2
M 2 D c 2 /.
RT 2 /
and it is necessary to solve the velocity diagram to find c 2 and hence to determine
T 2 .
As c y3 D 0, W D Uc y2
W 276 ð 10 3
c y2 D D D 552 m/s
U 500
c 2 D c y2 / sin ˛ 2 D 588 m/s.
1 2
Referring to Figure 4.2, across the nozzle h 01 D h 02 D h 2 C c , thus
2 2
1 2
T 2 D T 01 c /C p D 973 K.
2 2
Hence, M 2 D 0.97 with
R D .
1/C p .
(iii) The axial velocity, c x D c 2 cos ˛ 2 D 200 m/s.
1 2
(iv) tt D W/.h 01 h 3ss c /.
2 3
After some rearrangement,
1 1 c 2 3 1 200 2
D D D 1.0775.
tt ts 2W 0.87 2 ð 276 ð 10 3
Therefore tt D 0.93.
(v) Using eqn. (4.22a), the reaction is

1
R D .c x /U/.tan ˇ 3 tan ˇ 2 /.
2
From the velocity diagram, tan ˇ 3 D U/c x and tan ˇ 2 D tan ˛ 2 U/c x
R D 1 1 .c x /U/ tan ˛ 2 D 1 200 ð 0.2745/1000
2
D 0.451.

EXAMPLE 4.3. Verify the assumed value of total-to-static efficiency in the above
example using Soderberg’s correlation method. The average blade aspect ratio for
the stage H/b D 5.0, the maximum blade thickness chord ratio is 0.2 and the
5
average Reynolds number, defined by eqn. (4.14), is 10 .
Solution. The approximation for total-to-static efficiency, eqn. (4.10a), is used
and can be rewritten as
2
2
1 R .w 3 /U/ C N .c 2 /U/ C .c x /U/ 2
D 1 C .
ts 2W/U 2
The loss coefficients R and N , uncorrected for the effects of blade aspect ratio,
are determined using eqn. (4.12) which requires a knowledge of flow turning angle
for each blade row.
For the nozzles, ˛ 1 D 0 and ˛ 2 D 70 deg, thus N D 70 deg.
Ł
2
D 0.04.1 C 1.5 ð 0.7 / D 0.0694.
N