Question 14.5: (A) Two design methods are used in an axial turbine design. ......

(a)
(1) For general power with N = 1 at radius (r), the whirl velocity is C_{\text{U}} .

C_{\text{u}}=C_{\text{um}}\left(\frac{r}{r_{\text{m}}} \right) ^N

For N = 1

C_{\text{u}}=C_{\text{um}}\left(\frac{r}{r_{\text{m}}} \right)

(2) For free vortex design at radius (r), the whirl velocity is C^\prime_{\text{u}} , where

rC^\prime_{\text{u}}=\text{constant}\quad \text{or}\quad rC^\prime_{\text{u}}=r_{\text{m}}C_{\text{um}} \\ C^\prime_{\text{u}}=\frac{r_{\text{m}}}{r} C_{\text{um}}

Then

\frac{C_{\text{u}}}{C^\prime_{\text{u}}} =\frac{r}{r_{\text{m}}} \times \frac{r}{r_{\text{m}}} =\frac{r^2}{r^2_{\text{m}}}

(3) The value of the constant \lambda=1/r^2_{\text{m}} .
(4) The relation between the axial velocities.

(i) For general power with N = 1 at radius (r), the whirl velocity is C_{\text{a}}

\frac{C_{\text{a}}}{C_{\text{am}}} =\left\{1-\tan^2 \alpha_{\text{m}}\left(\frac{N+1}{N}\right) \left[\left(\frac{r}{r_{\text{m}}} \right)^{\text{2N}} -1\right] \right\} ^{1/2}

For N = 1

\frac{C_{\text{a}}}{C_{\text{am}}} =\left\{1-2\tan^2\alpha_{\text{m}} \left[\left(\frac{r}{r_{\text{m}}} \right)^2-1 \right] \right\} ^{1/2}

(ii) For free vortex design at radius (r), the whirl velocity is C^\prime_{\text{a}}

C^\prime_{\text{a}}=C_{\text{am}} \quad \text{or} \quad \frac{C^\prime_{\text{a}}}{C_{\text{am}}} =1

Then

\frac{C_{\text{a}}}{C^\setminus _{\text{a}}} =\left\{1-2\tan^2 \alpha_{\text{m}}\left[\left(\frac{r}{r_{\text{m}}} \right)^2-1 \right] \right\} ^{1/2}

B
(i) At the mean (Figure 14.25)

U_{\text{m}}=340 \text{ m/s} \quad \text{ and } \quad \frac{U_{\text{m}}}{C_{\text{a2}}} =1.25, \text{then} \\ \frac{340}{C_{\text{a2}}} =1.25,\\ C_{\text{a2m}}=C_{\text{a3m}}=272 \text{ m/s} \\ C_{\text{u2}}=C_{\text{a2}}\tan \alpha_2 \\ C_{\text{u2}}=272 \tan (58.5), \quad C_{\text{u2m}}=443.86 \text{ m/s} \\ C_{\text{u3}}=C_{\text{a3}}\tan \alpha_3 \\ C_{\text{u3}}=272 \tan(10), \quad C_{\text{u3m}}=47.96 \text{ m/s}

(ii) At the root (Figure 14.26)

\alpha_{2\text{r}}=\alpha_{\text{2m}}=58.5^\circ \\ \frac{U_{\text{m}}}{U_{\text{r2}}} =\left(\frac{r_{\text{m}}}{r_{\text{r}}} \right) _2 \\ \frac{340}{U_{\text{r2}}} =1.16, \quad U_{\text{r2}}=293.1 \text{ m/s} \\ \frac{U_{\text{m}}}{U_{\text{r3}}} =\left(\frac{r_{\text{m}}}{r_{\text{r}}} \right) _3 \\ \frac{340}{U_{\text{r3}}} =1.22 , \quad U_{\text{r3}}=278.69 \text{ m/s} \\ C_{\text{a2r}}=C_{\text{a2m}}\left(\frac{r_{\text{m}}}{r_{\text{r}}} \right) _2^{\sin^2 \alpha_2} \\ C_{\text{a2r}}=272(1.16)^{\sin^2(58.5)}=302.99 \text{ m/s} \\ C_{\text{u2r}}=C_{\text{u2m}}\left(\frac{r_{\text{m}}}{r_{\text{r}}} \right) _2^{\sin^2\alpha_2} \\ C_{\text{u2r}}=443.86(1.16)^{\sin^2(58.5)}=494.43 \text{ m/s} \\ \tan \alpha_2-\tan \beta_{2\text{r}}=\frac{U_{\text{2r}}}{C_{\text{a2r}}} \\ \tan (58.5)-\tan \beta_{\text{2r}}=\frac{293.1}{302.99} , \quad \beta_{\text{2r}}=33.6 \\ C^2_{\text{a3r}}=C^2_{\text{a3m}}+2U_{\text{m}}C_{\text{u2m}}\left[1-\left(\frac{r_{\text{r}}}{r_{\text{m}}} \right)_3^{\cos^2 \alpha_2} \right] \\ C^2_{\text{a3r}}=(272)^2+2 \times 340 \times 443.86\left[1-\left(\frac{1}{1.22} \right)^{\cos^2(58.5)} \right] , \quad C_{\text{a3r}}=299.89 \text{ m/s} \\ (rC_{\text{u}})_3=\text{constant} \\ (r_{\text{r}}C_{\text{ur}})_3=(r_{\text{m}}C_{\text{um}})_3 \\ C_{\text{u3r}}=C_{\text{u3m}}\left(\frac{r_{\text{m}}}{r_{\text{r}}} \right) _3 \\ C_{\text{u3r}}=47.96 \times 1.22=58.511 \text{ m/s} \\ \tan \alpha_3=\frac{C_{\text{u3}}}{C_{\text{a3}}} =\frac{58.11}{299.89} , \quad \alpha_3=11.04^\circ \\ \tan \beta_3-\tan \alpha_3=\frac{U_3}{C_{\text{a3}}} \\ \tan \beta_3-\tan (11.04)=\frac{278.69}{299.89} , \quad \beta_3=48.35^\circ

(iii) At the tip (Figure 14.27)

\alpha_{\text{2t}}=\alpha_{\text{2m}}=58.5^\circ \\ \frac{U_{\text{m}}}{U_{\text{t2}}} =\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) _2 \\ \frac{340}{U_{\text{t2}}} =0.88, \quad U_{\text{t2}}=386.36 \text{ m/s} \\ \frac{U_{\text{m}}}{U_{\text{t3}}} =\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) _3 \\ \frac{340}{U_{\text{t3}}} =0.85, \quad U_{\text{t3}}=400 \text{ m/s} \\ C_{\text{a2t}}=C_{\text{a2m}}\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) _2^{\sin^2 \alpha_2} \\ C_{\text{a2t}}=272(0.88)^{\sin^2 (58.5)}=247.86 \text{ m/s} \\ C_{\text{u2t}}=C_{\text{2um}}\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) _2^{\sin^2\alpha_2} \\ C_{\text{u2t}}=443.86(0.88)^{\sin^2(58.5)}=404.46 \text{ m/s} \\ \tan \alpha_{2\text{t}}- \tan \beta_{\text{2t}}=\frac{U_{\text{2t}}}{C_{\text{a2t}}} \\ \tan (58.5)-\tan \beta_{\text{2t}}=\frac{386.36}{247.86} , \quad \beta_{\text{2t}}=3.87^\circ \\ C^2_{\text{a3t}}=C^2_{\text{a3m}}+2U_{\text{m}}C_{\text{u2m}}\left[1-\left(\frac{r_{\text{t}}}{r_{\text{m}}} \right)_3^{\cos^2 \alpha_2} \right] \\ C^2_{\text{a3t}}=(272)^2+2 \times 340 \times 443.86\left[1-\left(\frac{1}{0.85} \right)^{\cos^2(58.5)} \right] \\ C_{\text{a3t}}=245.54 \text{ m/s} \\ (rC_{\text{u}})_3=\text{constant} \\ (r_{\text{t}}C_{\text{ut}})_3=(r_{\text{m}}C_{\text{um}})_3 \\ C_{\text{u3t}}=C_{\text{u3m}}\left(\frac{r_{\text{m}}}{r_{\text{t}}} \right) _3 \\ C_{\text{u3t}}= 47.96 \times 0.85=40.766 \text{ m/s} \\ \tan \alpha_{\text{3t}}=\frac{C_{\text{u3t}}}{C_{\text{a3t}}} =\frac{40.766}{245.54} , \quad \alpha_{\text{3t}}=9.43 \\ \tan \beta_{\text{3t}}-\tan \alpha_{\text{3t}}=\frac{U_{\text{3t}}}{C_{\text{a3t}}} \\ \tan \beta_{\text{3t}}-\tan (9.43)=\frac{400}{245.54} , \quad \beta_{\text{3t}}=60.88^\circ