Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

358 Fluid Mechanics, Thermodynamics of Turbomachinery
From eqn. (10.52), after differentiating and some simplification, we find
2 da¢
j ( 1+ 2a¢ ) = 1-2a da
Substituting eqn. (10.53) into eqn. (10.54) we get
2 j ( 1+ 2a¢ ) a¢= ( 1-2a) ( 1-a)
Combining this equation with eqn. (10.51) we obtain
1+ 2 ¢ 1-2 =
1+
¢ a a
.
a a
Solving this equation for a¢,
\ ¢ = - 1 3a
a
4a-1 .
(10.54)
(10.55)
Substitute eqn. (10.55) back into eqn. (10.52) and using 1 + a¢ =a/(4a - 1), we get
2 aj ¢ = ( 1-a) ( 4a-1. )
(10.56)
Equations (10.53) and (10.55) can be used to determine the variation of the interference
factors a and a¢ with respect to the coordinate j along the turbine blade length.
After combining eqn. (10.55) with (10.56) we obtain
j = ( 4a-1) 1-
a
1-3a (10.57)
Equation (10.57), derived for these ideal conditions, is valid only over a very narrow
range of a, i.e. 1 – 4 < a < 1 – 3. It is important to keep in mind that optimum conditions are
much more restrictive than general conditions. Table 10.8 gives the values of a¢ and j
for increments of a in this range (as well as j and l). It will be seen that for large
values of j the interference factor a is only slightly less than 1 – 3 and a¢ is very small.
Conversely, for small values of j the interference factor a approaches the value 1 – 4 and
a¢ increases rapidly.
TABLE 10.8. Relationship between a¢, a, j, j and l at
optimum conditions
a a¢ j f deg l
0.260 5.500 0.0734 57.2 0.4583
0.270 2.375 0.157 54.06 0.4131
0.280 1.333 0.255 50.48 0.3637
0.290 0.812 0.374 46.33 0.3095
0.300 0.500 0.529 41.41 0.2500
0.310 0.292 0.753 35.33 0.1842
0.320 0.143 1.150 27.27 0.1111
0.330 0.031 2.63 13.93 0.0294
0.333 0.003 8.574 4.44 0.0030