Bernoulli Equation Calculator

Understanding the Bernoulli equation calculator — concept and principle

The Bernoulli equation links pressure, speed and height at two points along the same flow. Give it the pressure, velocity and elevation at point 1, plus the velocity and elevation at point 2, and it returns the pressure at point 2. It is used to estimate pressures in pipes that narrow or climb and to see why fast-moving fluid has lower pressure.

Faster and higher water pays for it with pressure: 200 kPa becomes 154.6 kPa.

How it is calculated

The calculator applies p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂ and solves for p₂:

p₂ = p₁ + ½ρ(v₁² − v₂²) + ρg(z₁ − z₂)

p is static pressure (entered in kPa, converted to Pa internally), v is mean velocity in m/s, z is elevation in m above any common reference.
ρ is the fluid density in kg/m³ (water ≈ 1000) and g = 9.80665 m/s², standard gravity.
Use gauge or absolute pressure consistently; p₂ comes out in the same type.

The principle

Bernoulli's equation is conservation of energy for a moving fluid, written per unit volume. ½ρv² is kinetic energy, ρgz is potential energy, and p is the work that pressure can do (flow work). If no energy is added by a pump, removed by a turbine or lost to friction, their sum stays constant along a streamline. So when the fluid speeds up or rises, its pressure must fall by exactly the same amount. It assumes steady, incompressible, frictionless flow.

Worked example

Inputs: p₁ = 200 kPa, v₁ = 2 m/s, z₁ = 0 m, v₂ = 6 m/s, z₂ = 3 m, ρ = 1000 kg/m³.
Velocity term: ½ × 1000 × (2² − 6²) = 500 × (4 − 36) = −16,000 Pa.
Height term: 1000 × 9.80665 × (0 − 3) = −29,419.95 Pa.
p₂ = 200,000 − 16,000 − 29,419.95 = 154,580.05 Pa.
Result: p₂ ≈ 154.58 kPa (shown as 154.58005).

Each point is the calculator re-run with only “p₁ (kPa)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

p₁ (kPa)	p₂ (kPa)
50	4.58005
100	54.58005
150	104.58005
200	154.58005
250	204.58005
300	254.58005
400	354.58005

How to read the result

The drop of about 45.4 kPa splits into 16 kPa used to accelerate the water and 29.4 kPa used to lift it 3 m. As a rule of thumb, each metre of rise in water costs about 9.8 kPa. If p₂ falls below the liquid's vapour pressure (absolute), cavitation would occur first.

Common mistakes

Mixing gauge pressure at one point with absolute pressure at the other.
Entering diameters instead of velocities; get v from the continuity equation first (v = Q/A).
Using the equation across a pump, valve or long pipe without adding pump head or friction losses.

Keep in mind: This is the ideal equation. Real pipes lose energy to friction, fittings and turbulence, so actual p₂ is lower than shown. For real systems add a head-loss term (for example from Darcy-Weisbach) and have designs checked by a qualified engineer against local codes.

Frequently asked questions

Why does pressure drop when velocity increases?

Total energy is fixed, so energy that goes into motion must come out of pressure. Venturi meters and spray nozzles rely on this.

Can I use the Bernoulli equation for air?

Yes, as long as speeds are low enough that density barely changes, roughly below Mach 0.3 (about 100 m/s at sea level). Use ρ ≈ 1.2 kg/m³.

What is the head form of the equation?

Divide every term by ρg: p/ρg + v²/2g + z = constant. Each term is then a height in metres, called pressure head, velocity head and elevation head.

Sources: D. Bernoulli, Hydrodynamica (1738) · F. M. White, Fluid Mechanics, McGraw-Hill (chapter on the Bernoulli equation) · B. R. Munson et al., Fundamentals of Fluid Mechanics, Wiley

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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