Low-Head Hydropower Turbine Selection Calculator

Head Loss:

\\[ h_{\text{loss}} = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g}, \quad v = \frac{Q}{\pi (D/2)^2} \\]

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: Head 2–30 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:

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

\\[ v = \frac{2}{\pi (1/2)^2} \approx 2.55 \ \text{m/s} \\] \\[ h_{\text{loss}} = 0.02 \cdot \frac{100}{1} \cdot \frac{2.55^2}{2 \cdot 9.81} \approx 0.66 \ \text{m} \\] \\[ h_{\text{net}} = 5 – 0.66 = 4.34 \ \text{m} \\] \\[ P = 0.85 \cdot 0.90 \cdot 1000 \cdot 9.81 \cdot 4.34 \cdot 2 \approx 65162.13 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{65162.13}{1000} \approx 65.16 \ \text{kW} \\]

Turbine: Kaplan (Head 2–30 m, Flow > 1 m³/s)

\\[ v = \frac{0.5}{\pi (0.5/2)^2} \approx 2.55 \ \text{m/s} \\] \\[ h_{\text{loss}} = 0.03 \cdot \frac{50}{0.5} \cdot \frac{2.55^2}{2 \cdot 9.81} \approx 0.99 \ \text{m} \\] \\[ h_{\text{net}} = 3 – 0.99 = 2.01 \ \text{m} \\] \\[ P = 0.75 \cdot 0.85 \cdot 1000 \cdot 9.81 \cdot 2.01 \cdot 0.5 \approx 6292.41 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{6292.41}{1000} \approx 6.29 \ \text{kW} \\]

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

\\[ v = \frac{5}{\pi (2/2)^2} \approx 1.59 \ \text{m/s} \\] \\[ h_{\text{loss}} = 0.015 \cdot \frac{200}{2} \cdot \frac{1.59^2}{2 \cdot 9.81} \approx 0.19 \ \text{m} \\] \\[ h_{\text{net}} = 20 – 0.19 = 19.81 \ \text{m} \\] \\[ P = 0.80 \cdot 0.92 \cdot 1000 \cdot 9.81 \cdot 19.81 \cdot 5 \approx 715284.67 \ \text{W} \\] \\[ P_{\text{kW}} = \frac{715284.67}{1000} \approx 715.28 \ \text{kW} \\]

Turbine: Francis (Head 10–30 m, Flow 0.5–10 m³/s)