Bernoulli Equation Calculator
Pressure and speed at two points along a flow from Bernoulli's equation — and what a change in pipe diameter does to both, which is how a venturi meter measures flow.
Along a streamline in a steady, incompressible flow, pressure plus kinetic energy per volume plus potential energy per volume is constant.
How the bernoulli equation calculator works
Along a streamline in a steady, incompressible flow, pressure plus kinetic energy per volume plus potential energy per volume is constant. Where the fluid speeds up, the pressure drops.
With continuity — the same volume passes every section — a narrowing pipe forces the speed up by the ratio of areas, and the pressure falls by ½ρ(v₂² − v₁²). That pressure drop, measured, gives the flow rate: a venturi meter is nothing more.
Formula: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂; A₁v₁ = A₂v₂
Worked examples
| Inputs | Pressure at point 2 (kPa) | Note |
|---|---|---|
| 100 mm to 50 mm at 2 m/s | 270 | speed quadruples, pressure drops 30 kPa |
| A gentle taper | 297.117 | a small drop |
| Rizing 10 m | 201.934 | pressure falls by ρgh |
FAQFrequently asked questions
What is Bernoulli's equation?
Energy conservation for a flowing fluid: pressure plus kinetic plus potential energy per unit volume is constant along a streamline, for steady, frictionless, incompressible flow.
Why does pressure drop when the fluid speeds up?
Because the energy has to come from somewhere. The fluid trades pressure for speed in the narrow section and gets it back when the pipe widens.
What is a venturi meter?
A pipe with a narrowing. Measure the pressure drop across it and Bernoulli plus continuity give the flow rate — no moving parts.
When does it not apply?
Viscous flow in long pipes, turbulence, compressible gases near the speed of sound, and anywhere energy is lost to friction. Real pipes need a loss term added.
What is dynamic pressure?
Half rho v squared — the pressure a moving fluid would exert if brought to rest. It is what a pitot tube measures to find airspeed.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.