Bernoulli Equation Calculator – Venturi Meters, Pitot Tubes, Torricelli Efflux and the Pressure–Velocity Trade-off
Squeeze a garden hose and the jet shoots further. Blow across the top of a sheet of paper and it lifts. Both are the same statement: in a moving fluid, where the speed goes up the pressure comes down. This Bernoulli equation calculator puts numbers on that trade-off, solving the steady-flow energy balance for whichever quantity you do not know and coupling it to continuity so a change of pipe diameter alone is enough to fix both velocities.
The equation, and what each term is
Along a single streamline in a steady, incompressible, frictionless flow, three quantities trade against one another but always sum to the same total:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
P is the static pressure— what a flush tap in the pipe wall reads. ½ρv² is the dynamic pressure, the kinetic term, which only shows up when you bring the flow to rest. ρgh is the hydrostatic term for height above your chosen datum. Their sum is the total pressure, and it is the invariant the whole tool rests on. Add the three terms at each point and the two totals must match; the results panel prints them side by side with the residual, so the conservation is something you can check rather than something you are told.
Take water at 200 kPa gauge moving at 3 m/s, accelerating to 6 m/s while climbing 2 m. The speed-up costs 13.5 kPa and the climb costs 19.6133 kPa, leaving 166.8867 kPa downstream. Both totals come to 204500 Pa exactly.
Continuity: why a narrower pipe means faster flow
Mass cannot pile up in a rigid pipe, so A₁v₁ = A₂v₂ = Q. Halve the bore and the area falls by four, so the speed quadruples. Run 10 L/s through an 80 mm pipe narrowing to 40 mm and the flow goes from 1.9894 m/s to 7.9577 m/s, costing 29.6839 kPa of static pressure. That is the whole mechanism behind a Venturi meter: measure the pressure drop across a known constriction and you have the flow rate without putting anything in the stream.
The four classic applications
A Venturi meter inverts the balance to give velocity from a measured drop: v₁ = √(2ΔP / (ρ((A₁/A₂)² − 1))). A Pitot tube uses the two-term form along a stagnation streamline, so v = √(2(P₀ − P)/ρ) — in sea-level air, 850 Pa of dynamic pressure is 37.25 m/s, or 134.1 km/h. Torricelli efflux is the tank case, where both ends sit at atmospheric and the pressure terms cancel to leave v = √(2gh); three metres of head gives 7.6707 m/s. And the general solve handles any single unknown among the eight state quantities.
Static, dynamic and stagnation pressure
These three get confused constantly. Static pressure is what the fluid exerts sideways on the pipe wall. Stagnation (or total) pressure is P + ½ρv²— what a probe facing into the flow reads, because it stops the fluid and converts all the kinetic energy back into pressure. The difference between them is the dynamic pressure, and that difference is precisely what an airspeed indicator measures. Every result here breaks the total into its three parts for both points, so which term paid for which is never a guess.
Where Bernoulli stops being true
Four other regimes break it. Above roughly Mach 0.3 the constant-density assumption fails, which is why the tool reports a Mach number rather than leaving you to check. Downstream of a sudden expansion the flow separates and there is no single well-defined streamline, so a diffuser raises a warning. Unsteady flow— a tank that is visibly draining — is quasi-steady at best. And any pump, turbine or heater between the two points adds or removes energy and puts the problem outside Bernoulli entirely.
Discharge coefficients are not fudge factors
A sharp-edged orifice passes far less than the ideal figure — typically about 62 %, so 3.7654 L/s becomes 2.3345 L/s. That gap is the vena contracta and viscous loss, both real physics the frictionless model omits, and the tool labels it as a model correction rather than filing it with the rounding noise. The residuals it reports for its own second routes are of order 10⁻¹² Pa; a 38 % discharge correction is a different kind of thing entirely, and the two should never be read in the same register.
Negative gauge pressure is a feature
Push 8 L/s through a 100 mm pipe necking to 40 mm and the throat lands at −4745.47 Pa gauge. That is not an error — it is suction, and it is exactly how a carburettor lifts fuel and an eductor pulls water. The tool reports the absolute pressure alongside, checks it against the fluid's vapour pressure, and warns only when the liquid would actually cavitate.