Page 265 - Fluid Mechanics and Thermodynamics of Turbomachinery

246 Fluid Mechanics, Thermodynamics of Turbomachinery
The lowest value of this relative velocity ratio occurs when r 3 is least, i.e. r 3 D
r 3h D .34.4 20.1//2 D 7.15 mm, so that

w 3 2 2 1/2
D 0.475 ð 2.904[0.415 C 0.7536 ] D 1.19.
w 2av
min
The relative velocity ratio corresponding to the mean exit radius is,
w 3av 2 1/2
D 0.475 ð 2.904[1 C 0.7536 ] D 1.73.
w 2av
It is worth commenting that higher total-to-static efficiencies have been obtained
in other small radial turbines operating at higher pressure ratios. Rodgers (1969) has
suggested that total-to-static efficiencies in excess of 90% for pressure ratios up to
five to one can be attained. Nusbaum and Kofskey (1969) reported an experimental
value of 88.8% for a small radial turbine (fitted with an outlet diffuser, admittedly!)
at a pressure ratio p 01 /p 4 of 1.763. In the design point exercise given above the high
rotor enthalpy loss coefficient and the corresponding relatively low total-to-static
efficiency may well be related to the low relative velocity ratio determined on the
hub. Matters are probably worse than this as the calculation is based only on a simple
one-dimensional treatment. In determining velocity ratios across the rotor, account
should also be taken of the effect of blade to blade velocity variation (outlined in this
chapter) as well as viscous effects. The number of vanes in the rotor (ten) may be
Ł
insufficient on the basis of Jamieson’s theory (1955) which suggests 18 vanes (i.e.
Z min D 2 tan ˛ 2 ). For this turbine, at lower nozzle exit angles, eqn. (8.13) suggests
that the relative velocity ratio becomes even less favourable despite the fact that the
Jamieson blade spacing criterion is being approached. (For Z D 10, the optimum
value of ˛ 2 is about 58 deg.)

Mach number relations
Assuming the fluid is a perfect gas, expressions can be deduced for the important
Mach numbers in the turbine. At nozzle outlet the absolute Mach number at the
nominal design point is,

c 2 U 2
M 2 D D cosec ˛ 2 .
a 2 a 2
2 1 2 2
Now, T 2 D T 01 c /.2C p / D T 01 U cosec ˛ 2 /C p .
2 2 2
∴ T 2 D 1 1 .
2 2
2 1/.U 2 /a 01 / cosec ˛ 2
T 01
where a 2 D a 01 .T 2 /T 01 / 1/2 . Hence,

U 2 /a 01
M 2 D 1 (8.14)
2
2
sin ˛ 2 [1 .
1/.U 2 /a 01 / cosec ˛ 2 ] 1/2
2
Ł Included in a later part of this Chapter.