Bernoulli's Equation Calculator
Compute the total mechanical energy per unit volume of a flowing fluid at a point by summing static pressure, kinetic pressure, and elevation head. Useful for analyzing pipe flow energy, sizing nozzles, computing venturi meter response, and explaining lift on airfoils.
Last updated: May 2026
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About this calculator
Bernoulli's equation expresses conservation of mechanical energy along a streamline for an incompressible, inviscid, steady flow: P + ½ρv² + ρgh = constant. The three terms are: P (static pressure in Pa), the actual thermodynamic pressure exerted on surfaces parallel to flow; ½ρv² (dynamic pressure in Pa), the kinetic energy per unit volume from the fluid's motion; and ρgh (hydrostatic pressure in Pa), the potential energy per unit volume from elevation above a reference datum. This calculator computes the sum at one point: E₁ = P₁ + ½ × ρ × v₁² + ρ × g × h₁, where ρ is fluid density in kg/m³, v is velocity in m/s, h is elevation in meters above the chosen reference datum, and g = 9.81 m/s². The result has units of Pa (Pascals) and represents total mechanical energy per unit volume. Variables and edge cases: Bernoulli's equation as stated assumes (1) incompressible flow (constant density — valid for liquids always, gases only at Mach < 0.3); (2) inviscid flow (no friction — invalid in real pipe flow; use extended Bernoulli with head-loss term for friction-affected systems); (3) steady flow (time-invariant — invalid in pulsating or unsteady systems); (4) flow along a single streamline (valid in idealized flow but rotational flow requires using stream function or full Navier-Stokes). For real pipe flow with friction, the extended Bernoulli equation is: P₁/ρg + v₁²/(2g) + h₁ = P₂/ρg + v₂²/(2g) + h₂ + h_loss, where h_loss is friction head loss (computed via Darcy-Weisbach). The conservation form is most useful when comparing two points along a streamline. The trade-off between static and dynamic pressure is the source of many phenomena: airfoil lift, venturi metering, carburetor function, atomizer sprays.
How to use
Example 1 — point analysis in a water pipe. Water (ρ = 1,000 kg/m³) flowing at v₁ = 3 m/s, pressure P₁ = 200,000 Pa, elevation h₁ = 5 m. Step 1: static term = 200,000 Pa. Step 2: dynamic term = 0.5 × 1,000 × 3² = 4,500 Pa. Step 3: elevation term = 1,000 × 9.81 × 5 = 49,050 Pa. Step 4: E₁ = 200,000 + 4,500 + 49,050 = 253,550 Pa. Verify: total head equivalent = E₁/(ρg) = 253,550 / (1,000 × 9.81) = 25.85 m of water column — physically reasonable for typical building water systems. Example 2 — two-point comparison along a streamline. Water flows from point 1 to point 2 in a frictionless pipe. Point 1: P₁ = 250,000 Pa, v₁ = 2 m/s, h₁ = 10 m. Point 2: pipe contracts to half diameter, h₂ = 5 m, find P₂. Step 1: by continuity, A₁v₁ = A₂v₂; if A₂ = A₁/4, then v₂ = 4v₁ = 8 m/s. Step 2: E₁ = 250,000 + 0.5 × 1000 × 4 + 1000 × 9.81 × 10 = 250,000 + 2,000 + 98,100 = 350,100 Pa. Step 3: by Bernoulli E₁ = E₂: 350,100 = P₂ + 0.5 × 1000 × 64 + 1000 × 9.81 × 5 = P₂ + 32,000 + 49,050. Step 4: P₂ = 350,100 − 32,000 − 49,050 = 269,050 Pa. Verify: pressure increased at point 2 because elevation drop (gaining ρgh = 49,050) plus pressure gain exceeded the kinetic-energy increase (paying 30,000 in dynamic pressure for higher velocity). This is the trade-off Bernoulli describes — static pressure falls when velocity rises (venturi effect), and rises when velocity falls.
Frequently asked questions
What are the assumptions for Bernoulli's equation and when do they fail?
Bernoulli's equation has four key assumptions: (1) Incompressible flow — density is constant. Valid for all liquids under normal pressures; valid for gases only at Mach < 0.3 (below ~100 m/s for air at sea level). At higher Mach numbers, compressibility matters and the compressible Bernoulli equation (with ½v² + ∫dp/ρ form) is needed. (2) Inviscid flow — no viscous friction. Invalid in real pipe flow, where friction always exists. For real flows, use the extended Bernoulli equation with a head-loss term h_loss computed via Darcy-Weisbach or a similar method. The simple Bernoulli equation is exact for short distances or over a boundary layer not yet developed. (3) Steady flow — properties don't change with time. Invalid for pulsating flows (pumps near vibration), water hammer events, or oscillating systems; use unsteady Bernoulli equation or full Navier-Stokes. (4) Flow along a single streamline — applies along a fluid path, not across streamlines. For irrotational flow, the constant is the same on all streamlines and the equation applies globally. For rotational flow (e.g., behind a propeller), the constant varies between streamlines and Bernoulli can't be applied across them. In practical engineering, Bernoulli is most useful as a starting point and conceptual tool; for accurate analysis include friction, transition effects, and possibly compressibility.
How does Bernoulli's equation explain how airplane wings generate lift?
An airfoil produces lift because the airflow over the top surface is faster than over the bottom surface — this difference is driven by the shape (camber and angle of attack) which forces air to take a longer or more curved path over the top. By Bernoulli's equation, faster flow has lower static pressure. The pressure difference between bottom (higher pressure) and top (lower pressure) integrated over the wing area equals the lift force. Quantitatively: lift coefficient C_L = lift / (½ρv²S) where v is freestream velocity and S is wing area. For a typical commercial aircraft wing at cruise, C_L ≈ 0.3–0.5, with the pressure difference dominated by the top-surface acceleration. There's a common misconception that air takes the same time to traverse top and bottom — this is wrong; the air above the wing actually moves much faster and traverses faster. Lift is produced by the pressure-velocity trade-off Bernoulli predicts, supplemented by viscous effects and circulation theory (Kutta condition). The Bernoulli explanation is partial — the full picture also requires Newton's third law (air pushed downward by the wing produces upward reaction on the wing) and circulation theory.
How are static, dynamic, and stagnation pressure measured and used in practice?
Static pressure (P) is what a pressure gauge mounted flush in a pipe wall measures — the pressure exerted by the fluid on surfaces parallel to flow direction. Use sidewall taps with the opening perpendicular to flow to measure static pressure correctly. Dynamic pressure (½ρv²) is computed from velocity and density; it cannot be measured directly but appears as the difference between stagnation and static pressure. Stagnation pressure (P + ½ρv²) is measured by a Pitot tube — a small open tube pointed directly into the flow that captures all kinetic energy as static pressure. Pitot-static tubes measure both stagnation pressure (from forward-facing port) and static pressure (from side-facing ports), and the difference gives dynamic pressure, from which velocity is calculated: v = √(2 × (P_stagnation − P_static) / ρ). This is how aircraft airspeed indicators work: a Pitot-static system feeds dynamic pressure to a calibrated dial showing indicated airspeed. Pitot tubes are also used in HVAC duct measurements, wind tunnels, and process control. For accurate measurements: the Pitot tube must be parallel to flow direction (5° error in alignment produces 1–2% velocity error); the static port must be unaffected by turbulence (smooth pipe sidewall); and for compressible flow, the simple Bernoulli formula requires compressibility correction.
What are common mistakes when applying Bernoulli's equation?
The most common mistake is applying Bernoulli to a frictional pipe flow without including a head-loss term — produces wildly wrong pressure drop predictions. Use extended Bernoulli with h_loss = friction loss + minor losses for real piping. Applying Bernoulli across a pump or turbine without including the pump head (energy added) — pumps violate Bernoulli unless you include their energy input. Treating gas flow at high Mach numbers (>0.3) as incompressible — Bernoulli fails because density varies along the streamline; use compressible-flow Bernoulli or full Navier-Stokes. Mixing units: P in psi, v in ft/s, ρ in slugs/ft³ requires careful unit tracking (English-engineering system has odd unit conversions). Mixing absolute and gauge pressures — Bernoulli works with either as long as consistency is maintained, but mixing them produces atmospheric-pressure errors. Forgetting that elevation reference is arbitrary; you can set h = 0 anywhere as long as both points use the same reference. Applying Bernoulli across streamlines in rotational flow (where it doesn't hold) — only valid along a streamline. Confusing dynamic pressure (½ρv²) with stagnation pressure (P + ½ρv²) — they're different quantities. Finally, ignoring entrance and exit effects in short tubes — Bernoulli requires fully-developed flow, which may not exist in short flow paths.
When should I NOT use this calculator?
Skip Bernoulli's equation for any pipe-flow problem where friction matters — long pipes, rough pipes, fittings, and any system with significant velocity, viscosity, or pipe length need the extended Bernoulli with friction-loss terms (Darcy-Weisbach). Do not use it for high-Mach (>0.3) gas flow where compressibility matters — use isentropic flow equations or full Navier-Stokes. Avoid it for unsteady flow problems (water hammer, oscillating pumps, pulsation) — use unsteady Bernoulli or transient flow analysis. The formula does not apply to two-phase flow (gas-liquid mixtures, slurries), non-Newtonian fluids, or rotating reference frames without modification. For real pumps and turbines, you cannot just compute Bernoulli on both sides — the pump head must be added to the equation. For airfoil lift in real flow, Bernoulli explains the pressure-velocity trade-off but cannot predict actual lift without circulation theory and viscous-flow corrections; use CFD or wind-tunnel data for engineering aircraft design. For very short flow paths where boundary layers haven't developed and entrance effects dominate, Bernoulli underestimates losses. For supercritical fluids, mixtures with varying composition, or any non-equilibrium thermodynamic state, integral energy balance equations are needed instead. Bernoulli remains valuable as a conceptual tool and starting analysis, but real engineering systems usually require more complete equations.