Energy along a streamline
Bernoulli’s principle, published by Daniel Bernoulli in 1738, is a statement of energy conservation for a moving fluid. For a steady, incompressible flow without friction, the following sum stays the same at every point along a streamline:
p + ½ × ρ × v² + ρ × g × h = constant
where p is the pressure, ρ the fluid density, v the velocity, g gravity and h the height. The three terms are the pressure energy, the kinetic energy and the potential energy of each unit of volume.
What the calculator does
You enter the pressure and velocity at point 1, the pipe diameters at both points and the two heights. The calculator first uses continuity, the fact that the same volume passes every section:
v₂ = v₁ × (d₁ ÷ d₂)²
It then solves Bernoulli’s equation for the pressure at point 2, and gives the flow rate and the dynamic pressure ½ρv² at point 2. Pressures are gauge pressures, measured relative to the atmosphere.
Worked example
Water at 30 psi flows at 6 ft/s in a 2-inch pipe that narrows to 1 inch and rises 10 feet:
| Result | Value |
|---|---|
| Velocity at point 2 | 24 ft/s |
| Pressure at point 2 | 22.03 psi |
| Pressure change | −7.97 psi |
| Flow rate | 58.75 gal/min |
| Dynamic pressure at point 2 | 3.88 psi |
Halving the diameter quadruples the velocity. About 3.6 psi of the drop goes into speeding the water up, and about 4.3 psi into lifting it 10 feet (each foot of water column is worth 0.433 psi).
Where Bernoulli shows up
- Venturi meters measure flow from the pressure drop at a narrow throat.
- Pitot tubes on aircraft compare total and static pressure to find airspeed.
- Carburetors and sprayers use the low pressure in a fast jet to draw in fuel or liquid.
- Water towers convert height into pressure: every 2.31 feet of height gives 1 psi.
Limits of the ideal equation
Real pipes lose energy to friction, bends, valves and sudden changes in section. The real pressure downstream is therefore lower than the ideal value, sometimes by a lot over long runs; add a head-loss estimate (Darcy–Weisbach or Hazen–Williams) for design work. The equation also assumes incompressible flow, which is fine for liquids and for air below about 30% of the speed of sound.
If the computed gauge pressure approaches −14.7 psi, the liquid would reach a vacuum and start to boil at room temperature. This cavitation limits how fast water can be pushed through a constriction.