Bernoulli Equation Calculator (Pressure, Velocity & Height)
By Santaji GadepatilinUpdated November 2, 2026
Bernoulli's equation says that along a streamline the sum of pressure, kinetic energy per volume and potential energy per volume stays constant for a steady, frictionless, incompressible flow. This Bernoulli equation calculator finds the pressure, velocity or height at a second point from conditions at the first — see Bernoulli's equation and its formula for the derivation.
How to use the Bernoulli equation calculator
Enter the fluid density, the pressure, velocity and elevation at point 1, and the velocity and elevation at point 2. The calculator returns the pressure at point 2 and the total head of the flow. To find a velocity or height instead, enter P2 and clear the field you want. Use absolute or gauge pressure consistently at both points.
Bernoulli equation formula
P1 + ½ρv1² + ρgh1 = P2 + ½ρv2² + ρgh2
H = P ÷ (ρg) + v² ÷ (2g) + h
Example
Water (1,000 kg/m³) flows at 2 m/s and 200 kPa at ground level, then rises 3 m into a narrower pipe where it moves at 5 m/s:
- Velocity term: ½ × 1000 × (2² − 5²) = −10.5 kPa
- Elevation term: 1000 × 9.80665 × (0 − 3) = −29.42 kPa
- P2 = 200 − 10.5 − 29.42 = 160.08 kPa
- Total head H = 20.394 + 0.204 + 0 = 20.598 m
Dynamic pressure and velocity head
| Velocity | Dynamic pressure (water) | Velocity head v²/2g | Dynamic pressure (air, 1.2 kg/m³) |
|---|---|---|---|
| 0.5 m/s | 0.125 kPa | 0.013 m | 0.1 Pa |
| 1 m/s | 0.500 kPa | 0.051 m | 0.6 Pa |
| 2 m/s | 2.000 kPa | 0.204 m | 2.4 Pa |
| 3 m/s | 4.500 kPa | 0.459 m | 5.4 Pa |
| 5 m/s | 12.500 kPa | 1.275 m | 15.0 Pa |
| 10 m/s | 50.000 kPa | 5.099 m | 60.0 Pa |
| 20 m/s | 200.000 kPa | 20.394 m | 240.0 Pa |
| 30 m/s | 450.000 kPa | 45.887 m | 540.0 Pa |
Practical notes
- Real pipe flow loses head to friction and fittings: P2 is lower than this ideal value by ρg·hL. Estimate hL with the Darcy–Weisbach calculator.
- Flow meters such as the venturi and the pitot tube are direct applications of Bernoulli’s equation.
- If P2 falls below the liquid’s vapour pressure, the liquid flashes or cavitates and the equation no longer applies.
Frequently asked questions
What is Bernoulli's equation?
P + ½ρv² + ρgh = constant along a streamline. P is static pressure, ½ρv² the dynamic pressure and ρgh the hydrostatic term, all in pascals (J/m³).
Why does pressure drop when velocity increases?
Energy is conserved. If the fluid speeds up through a narrower section, its kinetic energy rises, so its pressure energy must fall — the principle behind venturi meters, carburettors and aspirators.
When is Bernoulli's equation not valid?
It assumes steady, incompressible, inviscid flow with no pumps or turbines between the points. For real pipes add the friction head loss, and for gases above about Mach 0.3 use compressible-flow relations.
What is total head?
The Bernoulli sum expressed as a height of fluid — pressure head P/ρg plus velocity head v²/2g plus elevation z. Without losses it is the same at every point.