Apply the steady, incompressible mechanical-energy balance between two points in a flow and solve for the downstream pressure P₂. The calculator accounts for changes in velocity head, elevation head, and friction (head loss) between the two stations, all in SI units.
Bernoulli's equation is the energy bookkeeping of fluid flow: along a streamline, the sum of pressure, kinetic, and potential energy per unit weight stays constant for an ideal fluid, and decreases by the friction loss for a real one. This calculator applies the steady, incompressible mechanical-energy balance between an upstream point 1 and a downstream point 2 and solves for the downstream pressure, showing how each form of head contributes.
In head form, P₁/(ρg) + v₁²/2g + z₁ = P₂/(ρg) + v₂²/2g + z₂ + h_loss. Each term has units of metres of fluid: P/(ρg) is the pressure head, v²/2g the velocity head, and z the elevation head. Rearranging for the downstream pressure gives P₂ = P₁ + ρg(z₁ − z₂) + (ρ/2)(v₁² − v₂²) − ρg·h_loss. Multiplying the head form by ρg recovers the pressure form, and multiplying by g gives the energy-per-mass form — all three are the same equation.
The frictionless Bernoulli equation assumes steady flow, incompressible fluid (constant ρ), flow along a single streamline, and no shaft work (no pump or turbine) between the two points. It also neglects viscous dissipation. These assumptions are reasonable for short, smooth runs of liquid or low-speed gas. When they fail — long pipes, fittings, pumps, or compressible high-speed gas — you must use the extended form or a full compressible analysis.
The three heads trade off against one another. Where a pipe narrows, velocity head rises and pressure head must fall (the Venturi effect); where the line climbs, elevation head rises at the expense of pressure or velocity. The total head is conserved only when there is no loss. Reading the breakdown at point 1 makes it clear which form of energy dominates and how the redistribution drives the downstream pressure.
Real systems lose mechanical energy to friction, so the engineering form adds a head-loss term h_loss (and, when present, a pump head h_pump and turbine head h_turbine): P₁/(ρg) + v₁²/2g + z₁ + h_pump = P₂/(ρg) + v₂²/2g + z₂ + h_turbine + h_loss. The head loss combines major (straight-pipe, Darcy-Weisbach) and minor (fittings, K·v²/2g) contributions. This calculator includes h_loss directly, so you enter the friction head from a pressure-drop calculation and get the realistic downstream pressure.
The classic frictionless form assumes steady, incompressible flow along a streamline with no shaft work and negligible viscous losses. It works well for short, smooth liquid runs and low-speed gas. For long pipes, fittings, pumps, turbines, or high-speed compressible gas, use the extended energy equation with head-loss, pump, and turbine terms instead.
It uses the extended Bernoulli form, subtracting a head-loss term h_loss (in metres of fluid) on the downstream side. You supply h_loss from a separate pipe pressure-drop or fittings calculation. Setting h_loss to zero recovers the ideal, frictionless Bernoulli result for comparison.
Total head is shared among pressure, velocity, and elevation heads. In a contraction the fluid speeds up, so velocity head rises; with elevation fixed and total head conserved, pressure head must fall to compensate. This is the principle behind Venturi meters, carburetors, and aspirators — faster flow means lower static pressure.
Yes, as long as the gas density is essentially constant — typically when the pressure change is under about 10% of the absolute pressure and the Mach number is low (below ~0.3). For larger pressure changes or high speeds, density varies significantly and you need a compressible-flow energy balance rather than the incompressible Bernoulli equation.
Add the pump head h_pump to the upstream side of the energy equation: P₁/(ρg) + v₁²/2g + z₁ + h_pump = P₂/(ρg) + v₂²/2g + z₂ + h_loss. The pump head is the mechanical energy added per unit weight of fluid, read from the pump curve. This calculator solves the no-pump balance; to include one, add its head to P₁/(ρg) before entering, or use the NPSH and pump-sizing tool.