Penstock Loss Calculator

Penstock Loss Calculator estimates frictional head loss in a hydropower penstock using the Darcy-Weisbach equation, supporting single or multiple flow rates, with detailed calculations and results table.

Flow Rate(s) (m³/s, comma-separated, e.g., 1.2, 1.5, 0.8):

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

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

Pipe Roughness (m, e.g., 0.00015 for steel):

Gross Head (m, e.g., 50):

Kinematic Viscosity (m²/s, e.g., 0.000001):

Formulas Used in Penstock Loss Calculator

The calculator uses the following formulas to estimate head loss:

Flow Velocity:

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

Reynolds Number:

\\[ Re = \frac{v D}{\nu} \\]

Friction Factor (approximated):

\\[ f \approx 0.0055 \left(1 + \left(20000 \frac{k}{D} + \frac{10^6}{Re}\right)^{1/3}\right) \\]

Head Loss:

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

Percentage Head Loss:

\\[ h_{\text{percent}} = \frac{h_f}{h_{\text{gross}}} \cdot 100 \\]

Where:

\\( v \\): Flow velocity (m/s)
\\( Q \\): Flow rate (m³/s)
\\( D \\): Pipe diameter (m)
\\( Re \\): Reynolds number
\\( \nu \\): Kinematic viscosity (m²/s)
\\( f \\): Darcy friction factor
\\( k \\): Pipe roughness (m)
\\( h_f \\): Head loss (m)
\\( L \\): Pipe length (m)
\\( g \\): Gravitational acceleration (9.81 m/s²)
\\( h_{\text{gross}} \\): Gross head (m)
\\( h_{\text{percent}} \\): Percentage head loss (%)

Example Calculations

Example 1: Small Flow

Input: Flow = 0.5 m³/s, Pipe Diameter = 0.3 m, Pipe Length = 50 m, Pipe Roughness = 0.00015 m, Gross Head = 10 m, Kinematic Viscosity = 0.000001 m²/s

\\[ v = \frac{0.5}{\frac{\pi \cdot 0.3^2}{4}} \approx 7.073 \ \text{m/s} \\] \\[ Re = \frac{7.073 \cdot 0.3}{0.000001} \approx 2,121,900 \\] \\[ f \approx 0.0055 \left(1 + \left(20000 \cdot \frac{0.00015}{0.3} + \frac{10^6}{2121900}\right)^{1/3}\right) \approx 0.0173 \\] \\[ h_f = 0.0173 \cdot \frac{50}{0.3} \cdot \frac{7.073^2}{2 \cdot 9.81} \approx 7.36 \ \text{m} \\] \\[ h_{\text{percent}} = \frac{7.36}{10} \cdot 100 \approx 73.60\% \\]

Result: Velocity = 7.073 m/s, Reynolds Number = 2,121,900, Friction Factor = 0.0173, Head Loss = 7.36 m, Percentage Head Loss = 73.60%

Example 2: Medium Flow

Input: Flow = 2.0 m³/s, Pipe Diameter = 0.5 m, Pipe Length = 100 m, Pipe Roughness = 0.00015 m, Gross Head = 50 m, Kinematic Viscosity = 0.000001 m²/s

\\[ v = \frac{2.0}{\frac{\pi \cdot 0.5^2}{4}} \approx 10.186 \ \text{m/s} \\] \\[ Re = \frac{10.186 \cdot 0.5}{0.000001} \approx 5,093,000 \\] \\[ f \approx 0.0055 \left(1 + \left(20000 \cdot \frac{0.00015}{0.5} + \frac{10^6}{5093000}\right)^{1/3}\right) \approx 0.0162 \\] \\[ h_f = 0.0162 \cdot \frac{100}{0.5} \cdot \frac{10.186^2}{2 \cdot 9.81} \approx 17.14 \ \text{m} \\] \\[ h_{\text{percent}} = \frac{17.14}{50} \cdot 100 \approx 34.28\% \\]

Result: Velocity = 10.186 m/s, Reynolds Number = 5,093,000, Friction Factor = 0.0162, Head Loss = 17.14 m, Percentage Head Loss = 34.28%

Example 3: Multiple Flows

Input: Flows = 3.0, 2.0, 1.0 m³/s, Pipe Diameter = 0.8 m, Pipe Length = 200 m, Pipe Roughness = 0.00015 m, Gross Head = 100 m, Kinematic Viscosity = 0.000001 m²/s

For \\( Q = 3.0 \\):

\\[ v = \frac{3.0}{\frac{\pi \cdot 0.8^2}{4}} \approx 5.968 \ \text{m/s} \\] \\[ Re = \frac{5.968 \cdot 0.8}{0.000001} \approx 4,774,600 \\] \\[ f \approx 0.0055 \left(1 + \left(20000 \cdot \frac{0.00015}{0.8} + \frac{10^6}{4774600}\right)^{1/3}\right) \approx 0.0159 \\] \\[ h_f = 0.0159 \cdot \frac{200}{0.8} \cdot \frac{5.968^2}{2 \cdot 9.81} \approx 7.22 \ \text{m} \\] \\[ h_{\text{percent}} = \frac{7.22}{100} \cdot 100 \approx 7.22\% \\]

(Similarly for \\( Q = 2.0 \\), \\( Q = 1.0 \\))

Result: For \\( Q = 3.0 \\): Head Loss = 7.22 m, 7.22%; \\( Q = 2.0 \\): Head Loss = 3.21 m, 3.21%; \\( Q = 1.0 \\): Head Loss = 0.80 m, 0.80%

Formulas Used in Penstock Loss Calculator

Example Calculations

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