Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

The flow angle j at optimum power conditions is found from eqns. (10.50) and
(10.55),
2 1
tan f =
1
1 3 1
2
4 1
1
tanf
1 3 1
¢ - ( )
( + ¢ ) =
a a
a a
( - a) ( - a)
a ( a-)
\ =
a
( - a) ( - a)
(10.58)
Again, at optimum conditions, we can determine the blade loading coefficient l in
terms of the flow angle j. Starting with eqn. (10.55), we substitute for a¢ and a using
eqns. (10.36a) and (10.35a). After some simplification we obtain
2 2 l = sin f -2lcosf
Solving this quadratic equation we obtain a relation for the optimum blade loading coefficient
as a function of the flow angle j,
l = 1-cosf
∫
(10.59)
Returning to the general conditions, from eqn. (10.51) together with eqns. (10.35a) and
(10.36a), we obtain
( - )
tanf =
tan f
( + ¢ )
tanf
=
11 a 1 Ê a ˆ 2
j 1 a jËa¢
¯
Ê a ˆ
\ j =
Ë a¢
¯
Rewriting eqns. (10.35a) and (10.36a) in the form
1 1 1
= 1 +
1
a a¢
= -
sinftanf and cosf
l
l
1
and substituting into eqn. (10.60) we get
Ê cosf
- l ˆ
j = sinf
Ë lcosf+ sin f¯
2
Reintroducing optimum conditions with eqn. (10.59), then
j
=
ZlC
8pr
sinf( 2cosf - 1)
2
( 1 - cosf) cosf + sin f
sinf( 2cosf - 1)
\ j =
( 1+ 2cosf) ( 1-cosf)
sinf( 2cosf -1)
\ jl =
1+ 2cosf
L
[ ]
05 .
la
Wind Turbines 359
(10.60)
(10.61)
(10.62)
(10.63)
Some values of l are shown in Table 10.8. Equation (10.62) enables j to be calculated
directly from j. The above equations also allow the optimum blade layout in terms