Speed a fluid up and its pressure falls; slow it down and its pressure rises. That simple, almost counter-intuitive trade-off is the heart of the Bernoulli equation — the single most useful relationship in fluid mechanics. It links pressure, velocity and elevation between any two points along a streamline, and it explains everything from how a venturi meter works to why an aircraft wing lifts. This free Bernoulli Equation Calculator solves for the unknown pressure, velocity or height between two points, with an optional head-loss term for real, frictional flow.
Table of Contents
The Bernoulli Equation Calculator
Enter the known values at point 1 and point 2, choose what to solve for, and add head loss if the flow has friction. Consistent SI units (Pa, m/s, m).
Bernoulli Equation Calculator
The Bernoulli Equation
Along a streamline in an ideal fluid, the total mechanical energy is conserved:
The three terms are the static pressure, the dynamic pressure (½ρv²), and the hydrostatic pressure (ρgz). Their sum — the total pressure — stays constant. Divide through by ρg and you get the equally common head form.
Understanding Head
Engineers often express Bernoulli in head (metres of fluid), because it's intuitive and matches pump curves:
| Term | Name | Meaning |
|---|---|---|
| p/(ρg) | Pressure head | Energy from pressure |
| v²/(2g) | Velocity head | Kinetic energy |
| z | Elevation head | Potential energy |
The velocity head links directly to the pipe flow velocity in a system, and the total head is what a pump must supply — see the pump power calculator.
Assumptions & Limits
- Steady flow (not changing with time)
- Incompressible fluid (liquids, or gas below Mach ~0.3)
- Frictionless (ideal) — unless you add head loss
- Along a single streamline
- No pump or turbine between the points (unless added as work terms)
Adding Head Loss (Real Flow)
Real pipes have friction, so the extended energy equation subtracts a head loss term downstream:
Find hL first with the pressure drop & head loss calculator, then apply Bernoulli. The flow regime (from the Reynolds number) sets the friction factor behind that loss.
Real-World Uses
- Venturi & orifice meters — velocity change → measurable pressure drop
- Pitot tubes — measure airspeed from dynamic pressure
- Aircraft lift — faster flow over the wing = lower pressure
- Carburettors & spray nozzles — low pressure at a throat draws in fluid
- Tank draining (Torricelli) — v = √(2gh)
Worked Example
Water flows up a tapering pipe. Point 1: p₁ = 200 kPa, v₁ = 2 m/s, z₁ = 5 m. At point 2 the pipe narrows (v₂ = 4 m/s) and drops to z₂ = 0:
- p₂ = p₁ + ½ρ(v₁² − v₂²) + ρg(z₁ − z₂)
- p₂ = 200000 + 500×(4 − 16) + 1000×9.81×5
- p₂ = 200000 − 6000 + 49050 = 243050 Pa ≈ 243 kPa
The pressure rose despite the fluid speeding up — because the large drop in elevation added more than the velocity increase removed.
Common Mistakes
- Ignoring friction in long pipes — add head loss for realism.
- Mixing gauge and absolute pressure — be consistent at both points.
- Using it across a pump or turbine without adding the work term.
- Applying it to compressible high-speed gas (Mach > 0.3).
- Inconsistent units — keep Pa, m/s, m, kg/m³.
- Different streamlines — the basic form is along one streamline.
Frequently Asked Questions
What is the Bernoulli equation?
A statement of energy conservation along a streamline: pressure + ½ρv² + ρgz is constant for ideal, incompressible, frictionless flow. Faster flow means lower pressure.
What are the assumptions of the Bernoulli equation?
Steady, incompressible, frictionless flow along one streamline with no pump or turbine. An extended form adds head loss and pump/turbine work.
What is head in the Bernoulli equation?
Energy as an equivalent height of fluid (m): pressure head p/(ρg), velocity head v²/(2g), and elevation head z. Their sum is the total head.
Why does pressure drop when velocity increases?
Because total energy is conserved: if elevation is constant, a rise in velocity head forces a fall in pressure head. This drives venturis, pitot tubes and wing lift.
Can the Bernoulli equation include friction?
Yes — the extended form adds a head loss term downstream. This calculator has an optional head-loss input for real pipe flow.
Conclusion
The Bernoulli equation captures a profound idea in one line: pressure, velocity and elevation trade off while total energy stays constant. Use it to find any one of them between two points, add head loss for real pipes, and you have the workhorse tool of practical fluid mechanics. Try the calculator above for instant, validated answers.
For more fluid mechanics, pipe-flow and CFD tutorials plus free engineering calculators, explore Free CFD Tutorial. If this tool helped you, please share it with your fellow engineers and students.
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