Question 15.1: A small radial inflow turbine develops 60 kW power when its ......
A small radial inflow turbine develops 60 kW power when its rotor is running at 60,000 rpm. The pressure ratio P_{01}/P_3 is 2.0. The inlet total temperature is 1200 K. The rotor has an inlet tip diameter of 12 cm and rotor exit tip diameter of 7.5 cm. The hub–tip ratio at exit is 0.3. The mass flow rate of gases is 0.35 kg/s. They enter the rotor radially and leave axially and have the following inlet and outlet angles \alpha_2 = 70° and \beta_3 = 40° . The nozzle loss coefficient (\lambda_{\text{N}}) = 0.07.
It is required to calculate
1. The isentropic efficiency (\eta_{\text{ts}})
2. The rotor loss coefficient (\lambda_{\text{R}})
The rotor tip rotational speed is U_2=\pi \times D_2 \times N=377 \text{ m/s}.
From the inlet velocity triangle; Figure 15.3 with \beta_2=0, then
\sin \alpha_2=\frac{U_2}{C_2}
The absolute velocity at inlet is C_2=U_2 \times \text{cosec } \alpha_2=401.185 \text{ m/s}.
The static temperature at inlet is T_2=T_{02}-(C_2^2/2Cp)=1130 \text{ K}
T_{01}-T^\prime _3=T_{01}\left[1-\frac{T^\prime _3}{T_{01}}\right] =T_{01}\left[1-\left(\frac{P_3}{P_{01}}\right)^{(\gamma-1/\gamma)} \right] =190.92 \text{ K}
The turbine power is \mathscr{P} =\dot{m} \times Cp \times ( T_{01}-T_{03})
T_{o1}-T_{o3}=\frac{60,000}{0.35 \times 1,148} =149.328 \text{ K}
Since \eta_{\text{ts}}=(T_{01}-T_{03})/(T_{01}-T^\prime_3)=0.782
\frac{r_3}{r_2} =\frac{D_{3\text{h}}+D_{3\text{s}}}{2D_2} =\frac{\xi \times D_{3\text{s}}+D_{3\text{s}}}{2 \times D_2} =0.40625
From Equation 15.13
\frac{T^\prime_3}{T^\prime_2} =1-\frac{U^2_2}{2 \times Cp \times T_2} \left[1+\left(\frac{r_3}{r_2} \right)^2\left\{(1+\lambda_{\text{R}})\text{cosec}^2 \beta_3-1\right\}- \cot^2 \alpha_2 \right] \\ \frac{T^\prime_3}{T_2^\prime} =0.9396-0.02187 \times \lambda_{\text{R}}
Substituting the value of T^\prime_3/T^\prime_2 in Equation 15.12b
\eta_{\text{ts}}=\left[1+0.5\left\{\left(\frac{r_3}{r_2}\right)^2\left(\cot^2 \beta_3+\lambda_{\text{R}}\cos ec^2 \beta_3\right) + \lambda_{\text{N}}\frac{T^\prime_3}{T^\prime_2} \cos ec^2 \alpha_2 \right\} \right] ^{-1} \\ (0.782)^{-1}=1.1544+0.2005 \times \lambda_{\text{R}} \\ \lambda_{\text{R}}=0.62
