Bernoulli Equation Calculator

The Bernoulli equation states that along a streamline, the total head stays constant: P/(rho*g) + v^2/(2g) + z = constant, where P is pressure, rho is fluid density, v is flow velocity, z is elevation, and g is gravity. Each term is a head in metres: pressure head, velocity head, and elevation head. An optional manual head-loss term h_L can be added to the downstream side.

Four classic assumptions: (1) steady flow (nothing changes with time at a point), (2) incompressible flow (density constant — fine for liquids and for gases at low speed), (3) inviscid flow (friction neglected — real viscous losses are lumped into the manual head-loss input h_L), and (4) the equation applies along one streamline. Break any of these and the results are approximate. This calculator states these limits openly instead of hiding them.

No. Head loss h_L is a manual input here — the tool does not compute it. It simply shifts the energy balance: total head at point 1 equals total head at point 2 plus h_L. To estimate head loss from pipe friction (Darcy-Weisbach with the Colebrook-White friction factor), use the companion pipe-flow calculator, then enter its result here.

Take water (rho = 1000 kg/m^3) with P1 = 200 kPa, v1 = 2 m/s, z1 = 0 m, P2 = 150 kPa, z2 = 0 m and no head loss. The head breakdown is: pressure heads 20.3943 m and 15.2957 m, velocity head at point 1 = 0.2039 m, so total head H1 = 20.5983 m. Setting H2 = H1 gives velocity head at point 2 = 5.3025 m, hence v2 = sqrt(2*g*5.3025) = 10.1980 m/s (36.7129 km/h, 33.4581 ft/s). These numbers are computed live by the calculator above.

For a horizontal venturi meter with no head loss, continuity (A1*v1 = A2*v2) combines with Bernoulli to give v2 = sqrt(2*dP / (rho*(1 - beta^4))), where dP = P1 - P2 and beta = D2/D1 (throat diameter over pipe diameter). The tool also reports the inlet velocity v1 and the volume flow rate Q = A2*v2. Example: dP = 20 kPa, D1 = 0.2 m, D2 = 0.1 m in water gives v2 = 6.5320 m/s, v1 = 1.6330 m/s, Q = 0.051302 m^3/s (51.302 L/s).

A differential manometer converts a height reading into a pressure difference via dP = (rho_mano - rho_fluid)*g*dh. The tool offers mercury (13595 kg/m^3) as the manometer fluid. Example: a 0.05 m mercury reading against water gives dP = 6175.74 Pa = 6.17574 kPa, which you can feed straight into the venturi mode.

Only at low speeds where compressibility is negligible (roughly below 30% of the speed of sound, Mach < 0.3). The air preset (rho = 1.225 kg/m^3) covers duct and ventilation problems: for example, air at P1 = 101325 Pa accelerating from 10 m/s to 20 m/s in a horizontal duct drops to P2 = 101141.25 Pa, a 183.75 Pa drop. For high-speed gas flow use compressible-flow relations instead.

That happens when the inputs are physically inconsistent: the velocity head required would be negative, i.e. v^2/(2g) < 0. For instance, asking for v2 when point 2 has much higher pressure and elevation than the energy available at point 1 leaves no energy for motion. Check that your pressures, elevations and head loss are consistent — energy cannot be created.

Pressures: Pa, kPa, psi, bar (1 psi = 6894.757293168361 Pa; 1 bar = 100000 Pa). Velocities: m/s, km/h, ft/s (1 ft/s = 0.3048 m/s; 72 km/h = 20 m/s exactly). Elevations and all head terms are in metres. The equation is evaluated internally in SI units and converted back to your chosen display units.

Skip Bernoulli for: compressible high-speed gas flow (use isentropic relations), flows with large viscous losses you cannot quantify (the manual h_L cannot rescue a bad estimate), unsteady or pulsating flow, flow across a pump or turbine (add/subtract shaft head explicitly), and across different streamlines in rotational flow. When in doubt, the head-loss and pump-head physics live in the pipe-flow calculator, not here.