Hydropower Head Loss Calculator

Hydropower Head Loss Calculator estimates the head loss in a hydropower penstock due to friction using the Darcy-Weisbach equation, with a detailed breakdown and step-by-step calculations.

Flow Rate (m³/s, e.g., 1):

Pipe Diameter (m, e.g., 0.5):

Pipe Length (m, e.g., 100):

Pipe Material:

Water Temperature (°C, e.g., 20):

Formulas Used in Hydropower Head Loss Calculator

The calculator uses the following formulas to estimate head loss in a hydropower penstock:

Head Loss (Darcy-Weisbach):

\\[ h_f = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g} \\]

Flow Velocity:

\\[ v = \frac{Q}{A} = \frac{Q}{\frac{\pi D^2}{4}} \\]

Reynolds Number:

\\[ Re = \frac{\rho v D}{\mu} \\]

Friction Factor (Laminar, Re < 2000):

\\[ f = \frac{64}{Re} \\]

Friction Factor (Turbulent, Colebrook-White):

\\[ \frac{1}{\sqrt{f}} = -2 \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) \\]

Where:

\\( h_f \\): Head loss (m)
\\( f \\): Friction factor (dimensionless)
\\( L \\): Pipe length (m)
\\( D \\): Pipe diameter (m)
\\( v \\): Flow velocity (m/s)
\\( g \\): Gravitational acceleration (9.81 m/s²)
\\( Q \\): Flow rate (m³/s)
\\( A \\): Pipe cross-sectional area (m²)
\\( Re \\): Reynolds number (dimensionless)
\\( \rho \\): Water density (1000 kg/m³)
\\( \mu \\): Dynamic viscosity (Pa·s, temperature-dependent)
\\( \epsilon \\): Pipe roughness (m)

Example Calculations

Example 1: Small-Scale Penstock

Input: Flow Rate = 1 m³/s, Diameter = 0.3 m, Length = 50 m, Material = Steel (ε = 0.000045 m), Temperature = 20°C

\\[ v = \frac{1}{\frac{\pi \cdot 0.3^2}{4}} \approx 14.147 \ \text{m/s} \\] \\[ \mu = 0.00179 \cdot e^{-0.0256 \cdot 20} \approx 0.00107 \ \text{Pa·s} \\] \\[ Re = \frac{1000 \cdot 14.147 \cdot 0.3}{0.00107} \approx 3,963,551 \\] \\[ f \approx 0.0193 \ \text{(iterative solution)} \\] \\[ h_f = 0.0193 \cdot \frac{50}{0.3} \cdot \frac{14.147^2}{2 \cdot 9.81} \approx 32.85 \ \text{m} \\]

Result: Velocity = 14.147 m/s, Reynolds Number = 3,963,551, Friction Factor = 0.0193, Head Loss = 32.85 m

Example 2: Medium-Scale Penstock

Input: Flow Rate = 2.265 m³/s, Diameter = 0.8 m, Length = 200 m, Material = PVC (ε = 0.0000015 m), Temperature = 15°C

\\[ v = \frac{2.265}{\frac{\pi \cdot 0.8^2}{4}} \approx 4.503 \ \text{m/s} \\] \\[ \mu = 0.00179 \cdot e^{-0.0256 \cdot 15} \approx 0.00122 \ \text{Pa·s} \\] \\[ Re = \frac{1000 \cdot 4.503 \cdot 0.8}{0.00122} \approx 2,952,459 \\] \\[ f \approx 0.0137 \ \text{(iterative solution)} \\] \\[ h_f = 0.0137 \cdot \frac{200}{0.8} \cdot \frac{4.503^2}{2 \cdot 9.81} \approx 3.54 \ \text{m} \\]

Result: Velocity = 4.503 m/s, Reynolds Number = 2,952,459, Friction Factor = 0.0137, Head Loss = 3.54 m

Example 3: Large-Scale Penstock

Input: Flow Rate = 11.992 m³/s, Diameter = 2 m, Length = 500 m, Material = Concrete (ε = 0.0003 m), Temperature = 10°C

\\[ v = \frac{11.992}{\frac{\pi \cdot 2^2}{4}} \approx 3.816 \ \text{m/s} \\] \\[ \mu = 0.00179 \cdot e^{-0.0256 \cdot 10} \approx 0.00139 \ \text{Pa·s} \\] \\[ Re = \frac{1000 \cdot 3.816 \cdot 2}{0.00139} \approx 5,496,403 \\] \\[ f \approx 0.0178 \ \text{(iterative solution)} \\] \\[ h_f = 0.0178 \cdot \frac{500}{2} \cdot \frac{3.816^2}{2 \cdot 9.81} \approx 3.30 \ \text{m} \\]

Result: Velocity = 3.816 m/s, Reynolds Number = 5,496,403, Friction Factor = 0.0178, Head Loss = 3.30 m

Formulas Used in Hydropower Head Loss Calculator

Example Calculations

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