Bernoulli Equation Calculator

Background.

Water speeding up through a narrower section of pipe loses pressure, which is the opposite of what most people expect when they picture fluid being squeezed. Bernoulli's equation is the bookkeeping behind that result. Along a streamline in steady flow, with friction ignored and density constant, the sum P + ½ρv² + ρgh stays fixed: static pressure, plus the kinetic energy carried by each cubic metre of fluid, plus its potential energy from height. Whatever one term gains, the other two must give up.

Give the pressure, speed and height at one point, two of the three at a second point, and the fluid's density, and the equation returns the missing one. Solving for P₂ tells you what a gauge downstream should read. Solving for v₂ gives the speed that a pressure drop or a fall in height can produce, which is how a Pitot tube turns a pressure reading into airspeed and how a draining tank's jet speed follows from its depth. Solving for h₂ gives the elevation consistent with the other readings, which turns out to be a neat way to measure friction losses. Alongside the answer come the total head at each point, the same energy written as a height of fluid in metres, and the change in dynamic pressure between them.

The arithmetic takes three lines; deciding whether a problem fits the equation takes more care. It describes an ideal fluid: no friction at the walls, no pump or turbine between the two points, no change in density, nothing varying with time. Short contractions, nozzles, free jets and tanks come close. A long pipe does not, because friction steadily drains head that the ideal equation insists is still there. Air qualifies only while it moves slowly compared with the speed of sound, roughly below a third of it. And a computed pressure below the liquid's vapour pressure means the real liquid would flash into vapour bubbles at that point, something the ideal equation knows nothing about. The calculator accepts speeds up to 100 m/s and refuses a solved pressure below zero, which no fluid can have; a solved pressure that is positive but below the vapour pressure is still returned, so compare it with the vapour pressure yourself.

Everything is in SI units: pascals, metres per second, metres and kilograms per cubic metre.

What is bernoulli equation calculator?

Bernoulli's equation is conservation of mechanical energy for a moving fluid, written per unit volume: P + ½ρv² + ρgh is constant along a streamline. P is the static pressure, the pressure a gauge travelling with the fluid would read; ½ρv² is the dynamic pressure, the kinetic energy per cubic metre; and ρgh is the potential energy per cubic metre above a chosen datum. Each term is in pascals, which are the same as joules per cubic metre. Divide through by ρg and every term becomes a length — pressure head, velocity head and elevation head — whose sum is the total head. Daniel Bernoulli's Hydrodynamica of 1738 set out the trade between pressure and speed; the equation as written today comes from integrating Euler's equation of motion along a streamline for steady, inviscid, constant-density flow. Where the flow is also irrotational, the constant is the same everywhere, and the equation can link any two points rather than only points on one streamline.

How to use this calculator.

1. 01Pick the quantity to find in Solve for unknown: P₂, v₂ or h₂. Whatever is in that quantity's own field is ignored, so you do not need to clear it.
2. 02Enter Fluid density (ρ) in kg/m³: about 1,000 for fresh water (998 at 20 °C), about 1,025 for seawater, and 1.225 for air at sea level in the standard atmosphere.
3. 03Enter P₁, v₁ and h₁ for the first point, then the two known values at point 2. Pressures are in pascals: multiply kPa by 1,000, bar by 100,000 and psi by 6,894.76. Use gauge or absolute pressure at both points, never a mix, and switch to absolute if either point is below atmospheric, because the pressure fields take no negative values; the standard atmosphere is 101,325 Pa.
4. 04Measure both heights from the same datum, which can be anywhere; points below it take negative heights. Only the difference h₁ − h₂ affects a solved pressure or speed, while the total heads shift with the datum you choose.
5. 05Leave Gravitational acceleration at 9.80665 m/s², the standard value, unless a problem specifies otherwise. Real gravity at the Earth's surface only ranges from about 9.78 to 9.83 m/s², so it rarely matters.
6. 06Check the answer against what the equation cannot see: a computed pressure below the fluid's vapour pressure (about 2.34 kPa absolute for water at 20 °C) means the ideal result no longer describes the real flow. Speeds above 100 m/s are refused, because beyond roughly Mach 0.3 in air the density no longer stays constant, and a solved pressure below zero is refused as impossible.

The formula.

Rearranged for each unknown, the equation gives P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂); v₂ = √[v₁² + 2(P₁ − P₂)/ρ + 2g(h₁ − h₂)]; and h₂ = h₁ + (P₁ − P₂)/(ρg) + (v₁² − v₂²)/(2g).

The pressure form shows both trades at once. Speeding up costs pressure in proportion to the difference of the squared speeds, scaled by density: for water, going from 2 to 4 m/s costs 500 × (16 − 4) = 6,000 Pa. Dropping in height adds pressure at ρg per metre, 9,806.65 Pa per metre for water at 1,000 kg/m³, which is why every 10 m of depth adds roughly one atmosphere. With both speeds zero the equation reduces to the hydrostatic law, P₂ = P₁ + ρg(h₁ − h₂). Because speed enters squared, it dominates at high velocity and barely registers at low: at 1 m/s water's velocity head is about 5 cm.

In the velocity form, the quantity under the root is the speed squared that the available energy can buy. If it comes out negative, point 2 would need more energy than point 1 supplies and there is no real answer. That usually means the pressures or heights have been swapped, or something such as a pump is adding energy between the points.

The total head, H = P/(ρg) + v²/(2g) + h, states the same energy in metres of the fluid. The solver enforces the equation exactly, so the heads at the two points always come out equal; they are there to show the energy line in the units pump curves and tank levels use. The dynamic pressure difference, ½ρ(v₂² − v₁²), is the part of the pressure change due to speed alone, positive when point 2 is faster.

The derivation assumes constant density, no viscosity, steady flow, and points on one streamline or in irrotational flow. Velocity in a real pipe runs from zero at the wall to a maximum at the centre, so using the average speed understates the kinetic-energy term, by a factor of two in slow laminar flow and only by a few per cent in turbulent flow. Engineers handle wall friction by adding a head-loss term h_L on the downstream side, usually from the Darcy–Weisbach equation; nothing of the kind is subtracted here.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Bernoulli, D. (1738). Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii. Dulsecker.
2. [2]Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 978-0-521-66396-0
3. [3]White, F.M. (2016). Fluid Mechanics, 8th ed. McGraw-Hill. ISBN 978-1-259-69587-0
4. [4]Anderson, J.D. (2017). Fundamentals of Aerodynamics, 6th ed. McGraw-Hill. ISBN 978-1-259-12991-9
5. [5]CODATA (2018). Recommended Values of the Fundamental Physical Constants.