Page 132 - Fluid Mechanics and Thermodynamics of Turbomachinery

Axial-flow Turbines: Two-dimensional Theory 113
the value of ˛ 3 being the optimum flow outlet angle from the stage when R and
tan ˛ 3 //2 which when substituted into
are specified. From eqn. (4.26), k D .tan 2
eqn. (4.27) and simplified gives
tan ˛ 3 D cot ˛ 2 D tan. /2 ˛ 2 /.
Hence, the exact result that

˛ 3 D /2 ˛ 2 .

The corresponding idealised ts max and R are
ts max D [1 C . /2/ cot ˛ 2 ] 1 .4.28a/
R D 1 .tan ˛ 2 cot ˛ 2 //2.

(ii) To find the optimum ts when ˛ 2 and are specified

Differentiating eqn. (4.25b) with respect to tan ˛ 3 and equating the result to zero,
2
tan ˛ 3 C 2 tan ˛ 2 tan ˛ 3 1 D 0.
Solving this quadratic, the relevant root is
tan ˛ 2 .
tan ˛ 3 D sec ˛ 2
Using simple trignometric relations this simplifies still further to
˛ 3 D . /2 ˛ 2 //2.

Substituting this expression for ˛ 3 into eqn. (4.25b) the idealised maximum ts
is obtained
1
ts max D [1 C .sec ˛ 2 tan ˛ 2 /] . (4.28b)
The corresponding expressions for the degree of reaction R and stage loading coef-
2
ficient W/U are
R D 1 .tan ˛ 2 1 sec ˛ 2 /
2
W c 2
D sec ˛ 2 D . .4.29/
U 2 U
It is interesting that in this analysis the exit swirl angle ˛ 3 is only half that of the
constant reaction case. The difference is merely the outcome of the two different
sets of constraints used for the two analyses.
For both analyses, as the flow coefficient is reduced towards zero, ˛ 2 approaches
/2 and ˛ 3 approaches zero. Thus, for such high nozzle exit angle turbine stages,
the appropriate blade loading factor for maximum ts can be specified if the reaction
is known (and conversely). For a turbine stage of 50% reaction (and with ˛ 3 !
2
0 deg) the appropriate velocity diagram shows that W/U + 1 for maximum ts .
Similarly, a turbine stage of zero reaction (which is an impulse stage for ideal,
2
reversible flow) has a blade loading factor W/U + 2 for maximum ts .