An inward flow reaction turbine operates at a rotational speed of 200 rpm. The turbine's runner has an external diameter of 0.9 m and an internal diameter of 0.45 m, with an inlet width of 200 mm. The velocity of flow through the runner remains constant at 1.8 m/s. The guide blade is positioned at an angle of 10° to the tangent of the wheel, and the turbine discharges radially at the outlet. Required Analysis: Velocity Triangles: Construct the inlet and outlet velocity triangles. Velocity Components at the Inlet: Determine the absolute velocity at the inlet of the runner. Calculate the velocity of whirl at the inlet. Find the relative velocity at the inlet. Blade and Flow Characteristics: Determine the runner blade angles. Compute the width of the runner at the outlet. Hydraulic and Performance Parameters: Calculate the mass flow rate of water through the runner per second. Determine the head at the inlet of the turbine. Evaluate the power efficiency of the turbine.

 To solve the problem of the inward flow reaction turbine, we start by analyzing the given data and constructing the velocity triangles. Here are the key steps and results:

Given Data

• Speed: 200 RPM

• Inlet width: 200 mm (0.2 m)

• External diameter (D1): 0.9 m

• Internal diameter (D2): 0.45 m

• Constant flow velocity (Vf1 = Vf2): 1.8 m/s

• Guide blade angle (α1): 10°

• Radial discharge at outlet (α2 = 90°)

1. Absolute Velocity at Inlet (V1)

Using trigonometry with the guide blade angle:

$$V_{w1}=\frac{V_{f1}}{\tan (10^{\circ })}=\frac{1.8}{0.1763}\approx 10.21\text{ m/s}$$$$V1=\sqrt{V_{w1}^2+V_{f1}^2}=\sqrt{10.21^2+1.8^2}\approx 10.37\text{ m/s}$$

2. Velocity of Whirl at Inlet (Vw1)

Already calculated as:

$$V_{w1}\approx 10.21\text{ m/s}$$

3. Relative Velocity at Inlet (Vr1)

Peripheral velocity at inlet (U1):

$$U1=\frac{\pi D1N}{60}=\frac{\pi \times 0.9\times 200}{60}\approx 9.425\text{ m/s}$$$$V_{r1}=\sqrt{(V_{w1}-U1{)}^2+V_{f1}^2}=\sqrt{(10.21-9.425{)}^2+1.8^2}\approx 1.96\text{ m/s}$$

4. Runner Blade Angles

• Inlet (β1):

  $$\beta 1=\arctan \left(\frac{V_{f1}}{V_{w1}-U1}\right)=\arctan \left(\frac{1.8}{0.785}\right)\approx 66.5^{\circ }$$

• Outlet (β2):
  Peripheral velocity at outlet (U2):

  $$U2=\frac{\pi D2N}{60}=\frac{\pi \times 0.45\times 200}{60}\approx 4.712\text{ m/s}$$$$\beta 2=\arctan \left(\frac{V_{f2}}{U2}\right)=\arctan \left(\frac{1.8}{4.712}\right)\approx 21^{\circ }$$

5. Width of the Runner at Outlet (B2)

Using continuity equation:

$$Q=2\pi r1B1V_{f1}=2\pi r2B2V_{f2}$$$$B2=\frac{r1B1}{r2}=\frac{0.45\times 0.2}{0.225}=0.4\text{ m}$$

6. Mass Flow Rate

$$\dot{m}=\rho Q=1000\times 2\pi \times 0.45\times 0.2\times 1.8\approx 1017\text{ kg/s}$$

7. Head at Inlet (H)

Using Euler's equation:

$$H=\frac{V_{w1}U1}{g}=\frac{10.21\times 9.425}{9.81}\approx 9.81\text{ m}$$

8. Power and Efficiency

• Power (P):

  $$P=\dot{m}(V_{w1}U1)=1017\times 10.21\times 9.425\approx 97.87\text{ kW}$$

• Efficiency:
  Theoretical efficiency is 100% due to ideal conditions.

Final Answers

1. Absolute velocity at inlet: $\boxed{{\displaystyle 10.37\text{ m/s}}}$

2. Velocity of whirl at inlet: $\boxed{{\displaystyle 10.21\text{ m/s}}}$

3. Relative velocity at inlet: $\boxed{{\displaystyle 1.96\text{ m/s}}}$

4. Runner blade angles: Inlet $\boxed{{\displaystyle 66.5^{\circ }}}$, Outlet $\boxed{{\displaystyle 21^{\circ }}}$

5. Width of the runner at outlet: $\boxed{{\displaystyle 0.4\text{ m}}}$

6. Mass flow rate: $\boxed{{\displaystyle 1017\text{ kg/s}}}$

7. Head at inlet: $\boxed{{\displaystyle 9.81\text{ m}}}$

8. Power and efficiency: Power $\boxed{{\displaystyle 97.87\text{ kW}}}$, Efficiency $\boxed{{\displaystyle 100\mathrm{\%}}}$ (Theoretical)