Question 7.12: The following particulars relate to a small inward flow radi......
The following particulars relate to a small inward flow radial gas turbine.
Rotor inlet tip diameter 92 mm
Rotor outlet tip diameter 64 mm
Rotor outlet hub diameter 26 mm
Ratio C_3/C_0 0.447
Ratio U_2/C_0 (ideal) 0.707
Blade rotational speed 30,500 rpm
Density at impeller exit 1.75 kg/m³
Determine
(1) The dimensionless specific speed of the turbine.
(2) The volume flow rate at impeller outlet.
(3) The power developed by the turbine.
(1) Dimensionless specific speed is
N_s = 0.336 \left(\frac{C_3} {C_0}\right)^{\frac{1}{2}}\left(\frac{A_3} {A_d}\right)^{\frac{1}{2}} , rev
Now
A_3 = \frac{\pi (D^2_{3t} – D^2_{3h})}{4}
= \frac{\pi(0.064^2 – 0.026^2)}{4} = (2.73)(10^{-3}) m²
A_d = \frac{\pi D_2^2} {4} = \frac{\pi}{4} (0.092^2) = (6.65)(10^{-3}) m²
Dimensionless specific speed
N_s = 0.336 =\left(\frac{[0.447][2.73]} {6.65}\right)^{\frac{1}{2}}
= 0.144 rev
= 0.904 rad
(2) The flow rate at outlet for the ideal turbine is given by Eq. (7.54).
N_s = 0.18\left(\frac{Q_3} {ND^3_2}\right)^{1/2} , rev (7.54)
N_s = 0.18\left(\frac{Q_3} {ND^3_2}\right)^{1/2}
0.144 = 0.18\left(\frac{[Q_3][60]}{[30,500][0.092^3]}\right)^{1/2}
Hence
Q_3 = 0.253 m³ /s
(3) The power developed by the turbine is given by
W_t = \dot{m}U^2_3
= ρ_3Q_3U^2_3
= 1.75 × 0.253 × \left(\frac{\pi ND_2}{60}\right)^2
= 1.75 × 0.253 × \left(\frac{[\pi][30,500][0.092]}{60}\right)^2
= 9.565 kW