# Step-by-Step Solution: Francis Turbine Design ## **Given Data** - Inlet blade angle, \( \beta_1 = 60^\circ \) - Outlet blade angle, \( \beta_2 = 90^\circ \) - Inlet radius, \( r_1 = 5\,\text{m} \) - Outlet radius, \( r_2 = 3\,\text{m} \) - Width of runner, \( B = 1\,\text{m} \) - Discharge, \( Q = 126\,\text{m}^3/\text{s} \) - Rotational speed, \( N = 60\,\text{rpm} \) ## **1. Velocity Diagram at Inlet** ### **a. Angular Velocity** \[ \omega = \frac{2\pi N}{60} = \frac{2\pi \times 60}{60} = 2\pi\,\text{rad/s} \] ### **b. Peripheral (Tangential) Velocity at Inlet** \[ u_1 = \omega r_1 = 2\pi \times 5 = 10\pi \approx 31.416\,\text{m/s} \] ### **c. Discharge and Flow Velocity** Area at inlet: \[ A_1 = 2\pi r_1 B = 2\pi \times 5 \times 1 = 10\pi \approx 31.416\,\text{m}^2 \] Radial (flow) velocity at inlet: \[ V_{f1} = \frac{Q}{A_1} = \frac{126}{31.416} \approx 4.012\,\text{m/s} \] ### **d. Relative Velocity and Velocity Components** From velocity diagram (see figure below): - \( V_{f1} \): Flow velocity (axial/radial) - \( u_1 \): Peripheral velocity - \( V_{w1} \): Whirl velocity (tangential component) - \( V_{r1} \): Relative velocity to the blade \[ \tan \beta_1 = \frac{V_{f1}}{u_1 - V_{w1}} \] But at inlet, whirl velocity is generally unknown. However, for no separation, the **absolute velocity at entry should be tangent to the blade**, i.e., entrance angle \( \alpha_1 = \alpha \). From velocity triangle: \[ \tan \alpha_1 = \frac{V_{f1}}{V_{w1}} \] But, \[ V_{w1} = u_1 - V_{r1} \cos \beta_1 \] \[ V_{r1} = \frac{V_{f1}}{\sin \beta_1} \] So, \[ V_{r1} = \frac{4.012}{\sin 60^\circ} = \frac{4.012}{.866} \approx 4.633\,\text{m/s} \] Now, \[ V_{w1} = u_1 - V_{r1} \cos \beta_1 = 31.416 - 4.633 \times .5 = 31.416 - 2.3165 = 29.099\,\text{m/s} \] Now, entrance angle \( \alpha \) is: \[ \tan \alpha = \frac{V_{f1}}{V_{w1}} = \frac{4.012}{29.099} = .1378 \] \[ \alpha = \arctan(.1378) \approx 7.87^\circ \] --- ## **2. Maximum Power and Torque** ### **a. Work Done per Second (Power)** \[ \text{Power} = \rho Q (u_1 V_{w1} - u_2 V_{w2}) \] Assume water density \( \rho = 100\,\text{kg/m}^3 \). #### **At Outlet** - For Francis turbines, \( \beta_2 = 90^\circ \) means flow leaves radially (\( V_{w2} = \)). \[ u_2 = \omega r_2 = 2\pi \times 3 = 6\pi \approx 18.850\,\text{m/s} \] \[ \text{Thus,} \quad \text{Power}_{\text{max}} = 100 \times 126 \times 31.416 \times 29.099 \] Calculate: \[ 31.416 \times 29.099 = 915.475 \] \[ \text{Power}_{\text{max}} = 100 \times 126 \times 915.475 = 115,350,000\,\text{W} = 115.35\,\text{MW} \] ### **b. Torque** \[ \text{Torque},\ T = \frac{\text{Power}}{\omega} \] \[ T = \frac{115,350,000}{2\pi} = \frac{115,350,000}{6.283} \approx 18,368,000\,\text{Nm} \] --- ## **3. Suggested Improvement** For the same discharge and runner size: - The absolute velocity at entrance (\( \alpha \approx 7.87^\circ \)) is quite small, indicating a **very sharp entry**. This may make the turbine sensitive to flow disturbances and could risk separation at off-design conditions. - **Improvement suggestion:** - **Increase the inlet blade angle** (\( \beta_1 \)) to allow a higher entrance angle (\( \alpha \)), making the turbine less sensitive to minor flow variations and reducing risk of separation. - Alternatively, **increase the number of smaller blades** or use **curved blades** optimized for variable load conditions to reduce the risk of separation and improve efficiency. --- ## **Summary Table** | Parameter | Value | |--------------------------|------------------| | Entrance angle \( \alpha \) | \( 7.87^\circ \) | | Max power | 115.35 MW | | Max torque | 18,368,000 Nm | --- ## **Velocity Diagram**  *Alt text: Velocity triangle at the inlet of a Francis turbine showing the relationships between u, V_f, V_w, V_r, and entrance angle alpha.* --- ## **Final Answers** - **Entrance angle \( \alpha \) to prevent separation:** \( \boxed{7.87^\circ} \) - **Maximum power generated:** \( \boxed{115.35\,\text{MW}} \) - **Maximum torque generated:** \( \boxed{18,368,000\,\text{Nm}} \) - **Improvement suggestion:** Increase blade entry angle or use more, smaller, or curved blades to reduce separation risk and improve efficiency.