Bernoulli Equation Calculator
Solve Bernoulli's equation to find downstream pressure in a flowing fluid given upstream conditions, elevation change, and system losses. Used in pipe system analysis, nozzle design, and pump selection.
About this calculator
Bernoulli's equation expresses conservation of energy per unit volume along a streamline: P₁ + ½ρV₁² + ρgz₁ = P₂ + ½ρV₂² + ρgz₂ + h_L, where P is pressure (Pa), ρ is fluid density (kg/m³), V is velocity (m/s), g = 9.81 m/s², z is elevation (m), and h_L represents head losses. This calculator solves for P₂: P₂ = P₁ + ½ρV₁² + ρg(z₁ − z₂) − K × ½ρV₁², where K is the dimensionless loss coefficient covering friction, fittings, and bends. A K of 0 represents an ideal, lossless system. The equation assumes steady, incompressible, inviscid flow along a streamline, so results are most accurate for liquids at moderate velocities well below the speed of sound.
How to use
Water (ρ = 1000 kg/m³) flows at V₁ = 3 m/s, P₁ = 200,000 Pa, z₁ = 5 m, z₂ = 2 m, loss coefficient K = 0.2. P₂ = 200,000 + ½ × 1000 × 3² + 1000 × 9.81 × (5 − 2) − 0.2 × ½ × 1000 × 3². Compute each term: ½ρV₁² = 4,500 Pa; ρg Δz = 29,430 Pa; loss term = 0.2 × 4,500 = 900 Pa. P₂ = 200,000 + 4,500 + 29,430 − 900 = 233,030 Pa ≈ 233 kPa. The elevation drop increases pressure while the loss coefficient reduces it.
Frequently asked questions
What assumptions does Bernoulli's equation make and when do they break down?
Bernoulli's equation assumes steady, incompressible, inviscid (frictionless) flow along a single streamline. It breaks down for compressible flows such as high-speed gases near sonic conditions, highly viscous fluids like heavy oils, or flows with significant turbulence and mixing between streamlines. It also fails across rotating machinery such as pump impellers, where energy is added to the fluid. In practice, the loss coefficient K is introduced as a correction to account for real friction and minor losses, extending the equation's usefulness to most engineering pipe flow problems.
How do I choose the right loss coefficient for a pipe system in Bernoulli calculations?
The loss coefficient K represents all friction and minor losses expressed as multiples of the dynamic pressure ½ρV². For straight pipes, K = f × L/D from the Darcy-Weisbach equation, where f is the friction factor and L/D is the pipe length-to-diameter ratio. Minor losses from fittings have tabulated K values: a fully open gate valve ≈ 0.1, a 90° elbow ≈ 0.9, a sudden pipe expansion ≈ 1.0. The total system K is the sum of all individual contributions. Handbook references such as Crane Technical Paper 410 provide extensive K tables for standard fittings and valves.
What is the difference between Bernoulli's equation and the energy equation for pipe flow?
Bernoulli's equation is a simplified form that assumes no energy is added or removed and no losses occur, applicable along a single streamline in ideal flow. The general energy equation for pipe flow extends this by explicitly including pump head H_p added to the fluid and turbine head H_t extracted, giving: P₁/ρg + V₁²/2g + z₁ + H_p = P₂/ρg + V₂²/2g + z₂ + H_t + h_L. This form is essential for pump and turbine system design where Bernoulli alone would give dangerously incorrect results. In this calculator, losses are lumped into the K coefficient, making it suitable for passive pipe networks without rotating machinery.