Library / Fluid Mechanics / FLU-02
Fluid Mechanics
Total head (Bernoulli)
Energy of a flowing liquid expressed as a head in metres.
- Ref
- FLU-02
- Rev
- v1
- Field
- Fluid Mechanics
- Result
- H [m]
Steady incompressible flow. α corrects for the velocity profile (1 for uniform flow, 2 for laminar pipe flow). Ideal Bernoulli along a streamline uses the local speed with no machines or losses. Gauge pressure is acceptable if used throughout.
The energy of a flowing liquid can be written as three heads: pressure, speed and height, all in metres of liquid.
Calculator
- Pressure head15.3 m
- Velocity head0.4589 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For water, 1 bar corresponds to about 10.2 m of head.
- Along a frictionless streamline H is constant; with friction it falls in the flow direction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z Elevation | −20 to 100 m | −4.245 | 115.8 | 120 | +10% | +2.41 % |
| p Pressure | −50 to 1,000 kPa | 0.3603 | 107.4 | 107.1 | +10% | +7.37 % |
| V Mean speed | 0 to 20 m/s | 20.3 | 40.69 | 20.39 | +10% | +0.464 % |
| ρ Density | 600 to 1,500 kg/m³ | 15.66 | 30.95 | 15.3 | +10% | −6.7 % |
| α Velocity-profile factor | 1 to 2 | 20.75 | 21.21 | 0.4589 | +undefined% | — |
Sweep of Mean speed
| Mean speed (m/s) | Total head (m) |
|---|---|
| 0 | 20.3 |
| 2.5 | 20.61 |
| 5 | 21.57 |
| 7.5 | 23.16 |
| 10 | 25.39 |
| 12.5 | 28.26 |
| 15 | 31.77 |
| 17.5 | 35.91 |
| 20 | 40.69 |
Sweep one variable
Variables and how each one contributes
Total head
What the equation gives you. Everything else feeds it.
Pressure
Gauge pressure at the point. Directly proportional.
Density
Density of the liquid. A denser liquid gives less head for the same pressure.
Mean speed
Mean flow speed. Squared in the velocity head.
Elevation
Height of the point above the datum. Adds directly.
Velocity-profile factor
1 for uniform flow, about 1.05 for turbulent and 2 for laminar pipe flow. Scales the velocity head.
Worked example with the values above
- Pressure head: p/(ρg) = 150,000 / (1,000 × 9.807) = 15.3 m.
- Velocity head: αV²/(2g) = 1 × 3² / 19.61 = 0.4589 m.
- H = 15.3 + 0.4589 + 5 = 20.75 m.
Assumptions and limits
- Steady incompressible flow.
- Single streamline or uniform profile.
Common mistakes
- Mixing gauge and absolute pressures at two points.
- Using the same elevation datum inconsistently.
Statistics and uncertainty
Pressure readings are good to about 1% but the elevation datum and velocity profile factors add further error.
Where it is used
- Pipe network and pump head calculations.
- Venturi and orifice meters.
Maths you need
- Pressure: Pa = N/m²
- Density: kg/m³