Flow Velocity:

\\[ v = \frac{Q}{\pi (D/2)^2} \\]

Head Loss:

\\[ h_{\text{loss}} = \left( f \cdot \frac{L}{D} \cdot \frac{v^2}{2g} \right) + \left( K \cdot \frac{v^2}{2g} \right) \\]

Net Head:

\\[ h_{\text{net}} = h_{\text{gross}} – h_{\text{loss}} \\]

Power Output:

\\[ P = \eta_{\text{turbine}} \cdot \eta_{\text{generator}} \cdot \rho \cdot g \cdot h_{\text{net}} \cdot Q \\]

Power in Kilowatts:

\\[ P_{\text{kW}} = \frac{P}{1000} \\]

Turbine Selection:

• Kaplan (Tubular/Bulb): Head 2–20 m, Flow > 1 m³/s
• Francis: Head 10–30 m, Flow 0.5–10 m³/s
• Crossflow: Head 2–30 m, Flow < 1 m³/s

Where:

• \\( v \\): Flow velocity (m/s)
• \\( Q \\): Flow rate (m³/s)
• \\( D \\): Conduit diameter (m)
• \\( h_{\text{loss}} \\): Head loss (m)
• \\( f \\): Friction factor
• \\( L \\): Conduit length (m)
• \\( K \\): Minor loss coefficient
• \\( g \\): Gravitational acceleration (9.81 m/s²)
• \\( h_{\text{net}} \\): Net head (m)
• \\( h_{\text{gross}} \\): Gross head (m)
• \\( P \\): Power output (W)
• \\( \eta_{\text{turbine}} \\): Turbine efficiency
• \\( \eta_{\text{generator}} \\): Generator efficiency
• \\( \rho \\): Water density (kg/m³)

\\[ v = \frac{1.5}{\pi (0.8/2)^2} \approx 2.98 \ \text{m/s} \\] \\[ h_{\text{loss}} = \left( 0.02 \cdot \frac{200}{0.8} \cdot \frac{2.98^2}{2 \cdot 9.81} \right) + \left( 0.5 \cdot \frac{2.98^2}{2 \cdot 9.81} \right) \approx 2.72 \ \text{m} \\] \\[ h_{\text{net}} = 10 – 2.72 = 7.28 \ \text{m} \\] \\[ P = 0.85 \cdot 0.90 \cdot 1000 \cdot 9.81 \cdot 7.28 \cdot 1.5 \approx 81962.51 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{81962.51}{1000} \approx 81.96 \ \text{kW} \\]

Turbine: Kaplan (Head 2–20 m, Flow > 1 m³/s)

\\[ v = \frac{0.3}{\pi (0.4/2)^2} \approx 2.39 \ \text{m/s} \\] \\[ h_{\text{loss}} = \left( 0.03 \cdot \frac{100}{0.4} \cdot \frac{2.39^2}{2 \cdot 9.81} \right) + \left( 0.3 \cdot \frac{2.39^2}{2 \cdot 9.81} \right) \approx 2.48 \ \text{m} \\] \\[ h_{\text{net}} = 5 – 2.48 = 2.52 \ \text{m} \\] \\[ P = 0.75 \cdot 0.85 \cdot 1000 \cdot 9.81 \cdot 2.52 \cdot 0.3 \approx 4728.01 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{4728.01}{1000} \approx 4.73 \ \text{kW} \\]

Turbine: Crossflow (Head 2–30 m, Flow < 1 m³/s)

\\[ v = \frac{3}{\pi (1.2/2)^2} \approx 2.65 \ \text{m/s} \\] \\[ h_{\text{loss}} = \left( 0.015 \cdot \frac{300}{1.2} \cdot \frac{2.65^2}{2 \cdot 9.81} \right) + \left( 0.7 \cdot \frac{2.65^2}{2 \cdot 9.81} \right) \approx 1.74 \ \text{m} \\] \\[ h_{\text{net}} = 15 – 1.74 = 13.26 \ \text{m} \\] \\[ P = 0.80 \cdot 0.92 \cdot 1000 \cdot 9.81 \cdot 13.26 \cdot 3 \approx 287574.85 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{287574.85}{1000} \approx 287.57 \ \text{kW} \\]

Turbine: Kaplan (Head 2–20 m, Flow > 1 m³/s)