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.
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
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
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%