Net Head:

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

Power Output:

\\[ P = \eta \cdot \rho \cdot g \cdot Q \cdot h_{\text{net}} / 1000 \\]

Specific Speed:

\\[ N_s = \frac{N \sqrt{P}}{h_{\text{net}}^{5/4}} \\]

Suitability Score: Based on head and specific speed ranges for each turbine type.

Where:

• \\( h_{\text{net}} \\): Net head (m)
• \\( h_{\text{gross}} \\): Gross head (m)
• \\( h_{\text{loss}} \\): Head loss (m)
• \\( P \\): Power output (kW)
• \\( \eta \\): Turbine efficiency (0 to 1)
• \\( \rho \\): Water density (1000 kg/m³)
• \\( g \\): Gravitational acceleration (9.81 m/s²)
• \\( Q \\): Flow rate (m³/s)
• \\( N_s \\): Specific speed (metric, rpm)
• \\( N \\): Rotational speed (rpm)

\\[ h_{\text{net}} = 200 – 10 = 190 \ \text{m} \\] \\[ P = 0.88 \cdot 1000 \cdot 9.81 \cdot 0.5 \cdot 190 / 1000 \approx 820.91 \ \text{kW} \\] \\[ N_s = \frac{1500 \sqrt{820.91}}{190^{5/4}} \approx 30.24 \ \text{rpm} \\]

Pelton (head 50–1000 m, \\( N_s \\) 10–50): Score = 92 (in range).

Result: Recommended Turbine = Pelton, Efficiency = 0.85–0.90, Power = 820.91 kW.

\\[ h_{\text{net}} = 50 – 3 = 47 \ \text{m} \\] \\[ P = 0.90 \cdot 1000 \cdot 9.81 \cdot 2.0 \cdot 47 / 1000 \approx 829.43 \ \text{kW} \\] \\[ N_s = \frac{1500 \sqrt{829.43}}{47^{5/4}} \approx 141.62 \ \text{rpm} \\]

Francis (head 10–400 m, \\( N_s \\) 50–250): Score = 95 (in range).

Result: Recommended Turbine = Francis, Efficiency = 0.88–0.93, Power = 829.43 kW.

\\[ h_{\text{net}} = 10 – 0.5 = 9.5 \ \text{m} \\] \\[ P = 0.92 \cdot 1000 \cdot 9.81 \cdot 5.0 \cdot 9.5 / 1000 \approx 429.03 \ \text{kW} \\] \\[ N_s = \frac{1500 \sqrt{429.03}}{9.5^{5/4}} \approx 435.72 \ \text{rpm} \\]

Kaplan (head 2–70 m, \\( N_s \\) 250–1000): Score = 90 (in range).

Result: Recommended Turbine = Kaplan, Efficiency = 0.90–0.95, Power = 429.03 kW.