Bernoulli Equation Calculator

The Bernoulli equation is derived from the principle of conservation of energy for flowing fluids. For any two points along a streamline in an inviscid, steady, incompressible flow, the total mechanical energy remains constant. The core equation is expressed as: P₁ + ½ ρ v₁² + ρ g h₁ = P₂ + ½ ρ v₂² + ρ g h₂, where P represents static pressure, ρ is fluid density, v is flow velocity, g is gravitational acceleration, and h denotes elevation height. Additionally, the continuity equation links flow velocities to pipe diameters through mass conservation, stating that volume flow rate remains constant: Q = A₁ v₁ = A₂ v₂, where A is the cross-sectional area determined by pipe diameter.

Consider water flowing through a horizontal pipe that constricts from a larger diameter to a smaller one. Let fluid density (ρ) be 1000 kg/m³, gravitational acceleration (g) be 9.81 m/s², and height remain constant (h₁ = h₂ = 0), eliminating potential energy terms. If the initial velocity (v₁) is 2 m/s at a starting pressure (P₁) of 200,000 Pa, and the pipe narrows such that the second velocity (v₂) accelerates to 5 m/s, we can find the new pressure (P₂). Using the Bernoulli equation: 200,000 + 0.5(1000)(2)² = P₂ + 0.5(1000)(5)². Simplifying the kinetic energy terms yields 200,000 + 2000 = P₂ + 12,500, which results in a final pressure P₂ = 189,500 Pa. The pressure drops as the fluid speeds up through the constriction.

Always ensure your input units are standardized in the SI system (meters, kilograms, seconds, Pascals) to avoid conversion errors during calculation. Remember that the standard Bernoulli equation assumes an incompressible, non-viscous fluid with steady, laminar flow; applying it to turbulent systems or highly compressible gases like high-pressure steam requires significant empirical correction factors. Finally, double-check your diameter inputs and area calculations, as velocity squares in the kinetic energy term, meaning minor measurement errors in pipe sizing can drastically alter your pressure outputs.