Bernoulli Equation Calculator

What this calculator does

Applies the steady incompressible energy equation between two points. Solve for pressure, velocity, elevation, or required pump head while accounting for head loss and turbine extraction.

Inputs

Solve for

Pressure P₁ kPa

Velocity V₁ m/s

Elevation z₁ m

Pressure P₂ kPa

Velocity V₂ m/s

Elevation z₂ m

Pump head added m

Turbine head removed m

Head loss m

Fluid density kg/m³

Check the inputs. Velocity solutions require a non-negative squared velocity.

Results

Solved value

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Point 1 total head

—

Point 2 total head

—

Energy residual

—

Energy balance: —

What the equation represents

Bernoulli’s equation is a mechanical-energy balance per unit weight of fluid. Pressure head, velocity head and elevation head can exchange with one another as fluid moves through a system.

Why it is useful

It links conditions at two locations and can solve for an unknown pressure, velocity, elevation or ideal pump head. It is widely used for nozzles, tanks, siphons, pipe systems and preliminary pump studies.

How to define the two points

Choose two stations along the same flow path. Enter pressure P, mean velocity V and elevation z at each station using one common elevation datum and one common pressure reference—both gauge or both absolute.

Pumps, turbines and losses

Pump head adds mechanical energy. Turbine head removes useful mechanical energy. Head loss represents irreversible dissipation from pipe friction, fittings, valves, entrances and exits.

Assumptions and limits

The implemented form assumes steady, incompressible, one-dimensional flow with kinetic-energy correction factor equal to one. Large density changes, unsteady transients, multiphase flow and major heat effects need a more complete control-volume energy model.

Worked example — water rising through a contraction

Water at 998.2 kg/m³ enters at 300 kPa, 1 m/s, at datum level; it leaves 5 m higher at 3 m/s, with 1 m of friction head lost between the points.

Step 1 — work in heads. Divide every energy term by ρg and each becomes a length in metres:

p/ρg + v²/2g + z + h_pump = p/ρg + v²/2g + z + h_turbine + h_L

Step 2 — total head at point 1.

Pressure head  300 000 / (998.2 × 9.80665) = 30.647 m
Velocity head  1² / (2 × 9.80665) = 0.051 m
Elevation            = 0.000 m
H₁ = 30.698 m

Step 3 — account for what happens in between. The flow climbs 5 m, accelerates, and loses 1 m to friction. What remains is pressure head:

p₂/ρg = 30.698 − 0.459 − 5 − 1 = 24.240 m
p₂ = 24.240 × 998.2 × 9.80665 = 237.27 kPa

Look at the relative sizes. Of the 62.7 kPa total drop, elevation accounts for about 49 kPa, friction 9.8 kPa and acceleration only 4 kPa. Velocity head is almost always the smallest term in a liquid system at ordinary pipe velocities — 1 m/s is worth just 51 mm of head. That is why static lift and friction dominate pump sizing.

Where the equation stops being valid

Textbook Bernoulli assumes steady, incompressible, inviscid flow along a streamline. The h_L, h_pump and h_turbine terms turn it into the engineering energy equation, which is what makes it usable — but the remaining assumptions still bite.

Both points must lie on the same streamline, or the flow must be irrotational. Applying it between the free stream and a separated wake or recirculation zone is meaningless. This is the most frequent misuse.

Steady flow only. Water hammer, pump start-up and valve closure are unsteady, and the pressures can be many times the steady values. A rapid closure can generate a Joukowsky surge well above 10 bar in a system running at 3 bar — nothing here predicts it.

Incompressible only. For gases the constant-density assumption holds to a few percent up to about Mach 0.3. Above that use the compressible energy equation.

h_L is an input, not a result. The calculator accepts a head loss; it does not compute one. That comes from Darcy-Weisbach with a friction factor from Colebrook-White or the Moody chart, plus minor losses. Entering zero gives the frictionless ideal — useful as a bound, wrong as a design value.

Uniform velocity is assumed. Real pipe flow has a profile, so true kinetic energy exceeds v²/2g from mean velocity. The correction factor α is about 1.05 in turbulent flow and 2.0 in laminar — negligible in the first case, significant in the second.

The check worth doing every time: cavitation

Because the equation gives the pressure at a point, it also tells you whether that pressure is physically possible.

If the calculated pressure reaches the fluid's vapour pressure, the liquid boils. Water at 20 °C has a vapour pressure of about 2.34 kPa absolute; at 80 °C it is 47.4 kPa. A result below that is not a low pressure — it is cavitation, and the flow the equation assumed no longer exists.

The locations are predictable: pump suction, the crest of a siphon, just downstream of a throttling valve, and the inlet of a contraction. Note that pressure at a valve's vena contracta is lower than the recovered downstream pressure, so a valve can cavitate while both gauge readings look comfortable.

On a pump suction the formal version of this check is NPSH available against NPSH required — the same balance with vapour pressure subtracted.

References and what to do next

Head loss: Crane Technical Paper No. 410 is the standard practical reference for pipe friction and fitting losses. Colebrook-White (or the Moody chart) gives the friction factor; Swamee-Jain is an explicit approximation within about 1%.

Flow measurement: ISO 5167 covers orifice plates, nozzles and Venturi tubes — Bernoulli applied across a known contraction, with a discharge coefficient absorbing the real-fluid effects.

Pump systems: ANSI/HI standards for NPSH margin and system design.

Next steps: compute h_L properly rather than assuming it — in most real systems it decides the answer. Check the pressure at every high and low point against vapour pressure at the operating temperature, not at 20 °C. For gases, confirm Mach below 0.3 before trusting the incompressible form. And if anything can close or start quickly, treat this steady result as a baseline for a separate surge analysis, not as the worst case.