Question 14.6: An axial turbine stage is to be designed by a constant nozzl......

(a) Constant nozzle angle design
The axial turbine stage has a constant nozzle outlet angle and zero exit swirl. Since ζ = 0.75,

\frac{r_{\text{m}}}{r_{\text{t}}} =\frac{U_{\text{m}}}{U_{\text{t}}} =\frac{1+\zeta}{2} =0.875 \quad \quad \quad (\text{a})

and

\frac{r_{\text{m}}}{r_{\text{t}}} =\frac{U_{\text{m}}}{U_{\text{t}}} =\frac{1+\zeta}{2\zeta} =1.166

The velocity triangle are

\frac{C_2}{C_{\text{2m}}} =\frac{C_{\text{a2}}}{C_{\text{a2m}}} =\frac{C_{\text{u2}}}{C_{\text{u2m}}} =\left(\frac{r_{\text{m}}}{r} \right) ^{(\sin \alpha_2)^2} \\ (C_{\text{a3}})^2=(C_{\text{a3m}})^2+2U_{\text{m}} \times C_{\text{u2m}}\left(1-\left(\frac{r}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right)

The specific work is given by the relation

Cp\Delta T_0=U_{\text{m}}\Delta C_{\text{um}}=U_{\text{m}}C_{\text{u2m}} \\ C_{\text{u2m}}=\frac{Cp\Delta T_0}{U_{\text{m}}} =492 \text{ m/s} \\ C_{\text{a2m}}=C_{\text{u2m}}\cot \alpha_2 \\ C_{\text{a2m}}=C_{\text{a3m}}=284 \text{ m/s} \\ \frac{C_{\text{a2}}}{C_{\text{a2m}}} =\frac{C_{\text{u2}}}{C_{\text{u2m}}} =\left(\frac{r_{\text{m}}}{r}\right) ^{(\sin \alpha_2)^2}

At hub section

C_{\text{a2h}}=C_{\text{a2m}}\left(\frac{r_{\text{m}}}{r_{\text{h}}} \right) ^{(\sin \alpha_2)^2}=318.8 \text{ m/s} \\ C_{\text{u2h}}=C_{\text{u2m}}\left(\frac{r_{\text{m}}}{r_{\text{h}}} \right)^{(\sin \alpha_2)^2} =552.2 \text{ m/s}

At the tip section

C_{\text{a2t}}=C_{\text{a2m}}\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) ^{(\sin \alpha_2)^2}=256.93 \text{ m/s} \\ C_{\text{u2t}}=C_{\text{u2m}}\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) ^{(\sin \alpha_2)^2}=446.6 \text{ m/s}

At the rotor outlet 

(C_{\text{a3}})^2-(C_{\text{a3m}})^2=2U_{\text{m}} \times C_{\text{u2m}}\left(1-\left(\frac{r}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right) . \\ (C_{\text{a3t}})=\sqrt{(C_{\text{a3m}})^2+2U_{\text{m}}\times C_{\text{u2m}}\left(1-\left(\frac{r_{\text{t}}}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right) }=262.1 \text{ m/s} \\ (C_{\text{a3h}})=\sqrt{(C_{\text{a3m}})^2+2U_{\text{m}}\times C_{\text{u2m}}\left(1-\left(\frac{r_{\text{h}}}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right) }=306 \text{ m/s}

(b) Approximates constant nozzle (\alpha_2 = constant, rC_{\text{u2}} = constant)
The governing equations are listed here:

r_{\text{m}}C_{\text{u2m}}=r_{\text{h}}C_{\text{u2h}}=r_{\text{t}}C_{\text{u2t}} \\ C_{\text{u2h}}=\frac{r_{\text{m}}C_{\text{u2m}}}{r_{\text{h}}} \\ C_{\text{a2h}}=C_{\text{u2h}}\cot \alpha_2 \\ C_{\text{u2t}}=\frac{r_{\text{m}}C_{\text{u2m}}}{r_{\text{t}}} \\ C_{\text{a2t}}=C_{\text{u2t}} \cot \alpha_2 \\ C_{\text{u3h}}=C_{\text{u3t}}=C_{\text{u3m}}=0 \\ C^2_{\text{a3}}=C^2_{\text{a3m}}+2U_{\text{m}}C_{\text{u2m}}\left[1-\left(\frac{r}{r_{\text{m}}} \right)^{\cos^2 \alpha_2} \right] \\ (C_{\text{a3t}})=\sqrt{(C_{\text{a3m}})^2+2U_{\text{m}}\times C_{\text{u2m}}\left(1-\left(\frac{r_{\text{t}}}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right) } \\ (C_{\text{a3h}})=\sqrt{(C_{\text{a3m}})^2+2U_{\text{m}}\times C_{\text{u2m}}\left(1-\left(\frac{r_{\text{h}}}{r_{\text{m}}} \right)^{(\cos \alpha_2)^2} \right) }

(c) Free vortex design

(rC_{\text{u}})_2 =\text{constant} \\ C_{\text{a}}= \text{ constant at inlet and outlet to rotor}

(d) Summary of the velocity variations using different design methods