Bernoulli Equation Calculator
Bernoulli's equation calculator. Solve for downstream pressure given upstream conditions + velocity + elevation changes. Conservation of fluid energy.
Bernoulli Calculator
Energy decomposition (pressure terms in kPa)
| Term | Point 1 | Point 2 |
|---|---|---|
| Static pressure P | — | — |
| Dynamic pressure ½ρv² | — | — |
| Hydrostatic pressure ρgz | — | — |
| Total pressure | — | — |
How to use Bernoulli's equation calculator
Enter fluid density
Water: 1000 kg/m³. Air at sea level: 1.225. Engine oil: 870. Mercury: 13600. Density is the key fluid property in Bernoulli's equation (along with viscosity, but Bernoulli assumes inviscid flow).
Enter Point 1 (upstream) conditions
Pressure in Pa (1 atm = 101325 Pa). Velocity in m/s — bulk flow speed at this cross-section. Elevation in m — height above reference datum (could be ground level, sea level, etc.). Point 1 is your "known" upstream state.
Enter Point 2 (downstream) conditions
Velocity v₂ — typically known from continuity equation (A₁v₁ = A₂v₂) if the cross-section is changing. Elevation z₂ — height at the downstream point. The pressure P₂ is what we're solving for.
Read pressure at Point 2
The calculator outputs P₂ in both Pa and kPa. The pressure drop P₁ − P₂ tells you whether the fluid speeds up (drop) or slows down (gain). Energy decomposition table shows how pressure, kinetic, and potential energy trade off.
Verify total pressure conservation
In ideal Bernoulli flow, total pressure is constant: static + dynamic (½ρv²) + hydrostatic (ρgz) = same at both points; divide each by ρg for the equivalent head in metres. Any difference in the table is rounding only. Real flows have head loss due to friction; Bernoulli is the ideal-fluid limit.
Bernoulli's equation — conservation of fluid energy
Bernoulli's equation, published by Daniel Bernoulli in his 1738 work Hydrodynamica, is the single most-used equation in undergraduate fluid mechanics. It states that for a steady, incompressible, inviscid flow along a streamline: P + ½ρv² + ρgz = constant. Three energy terms: pressure (static), kinetic (dynamic), gravitational (potential). When fluid accelerates (velocity increases), pressure must decrease — the principle behind Venturi flow meters, airplane wings, carburetors, sailboat aerodynamics. When fluid rises (elevation increases), pressure must decrease — the principle behind water towers + plumbing systems. Bernoulli ties pressure, velocity, and elevation in a single equation.
Assumptions and limitations
Bernoulli applies under four assumptions: (1) steady flow — no time-varying parameters. (2) incompressible flow — constant density. Valid for liquids universally + gases at Mach number < 0.3. (3) inviscid flow — no viscosity (no friction losses). Real flows always have some friction; the extended Bernoulli with head loss covers this. (4) along a streamline — equation applies between two points on the same streamline; across streamlines requires additional conditions. Violations: pumps add energy, friction removes energy, compressible flow has different equation, unsteady flow needs different treatment.
Bernoulli's equation is the foundation. Real engineering adds the friction term (head loss via Darcy-Weisbach), the pump term (head added), and the turbine term (head extracted). The "extended Bernoulli" handles real pipe systems; the original Bernoulli reveals the underlying physics.
Practical applications
The Venturi tube uses Bernoulli to measure flow rate — narrow throat causes velocity increase + pressure drop, which is measured. Airplane lift uses faster air over the wing's curved top + slower below = pressure difference. Pitot tubes measure airspeed via stagnation pressure (velocity reduces to zero, pressure rises). Sprayers + perfume bottles use the Venturi effect to draw liquid into an airstream. Roof damage during high winds — fast wind over roof creates low pressure, lifting roof up. All these systems are quantifiable via Bernoulli's equation given the right boundary conditions.
ASEAN engineering examples
HVAC systems in Singapore high-rises: ductwork pressure calculations via Bernoulli + Darcy-Weisbach head loss. Water supply in hilly Hong Kong: pressure increases at lower elevations follow Bernoulli's elevation term (ρgh ≈ 9.81 kPa per meter of head). Indonesia's rapid rivers exhibit Bernoulli effects in channel constrictions — velocity rises, water depth drops. Palm oil pipelines in Malaysia use Bernoulli for compressor sizing decisions.
Bernoulli's four assumptions, and where Mach 0.3 ends them
P + ½ρv² + ρgz = const along a streamline. Daniel Bernoulli, Hydrodynamica 1738.
Assumptions: steady, incompressible, inviscid, along streamline.
Pressure ↓ when velocity ↑ — the Venturi effect.
Pressure ↑ when elevation ↓ by ρgh ≈ 9.81 kPa per meter of water column.
Aerodynamic lift = pressure difference over curved wing surfaces (Bernoulli simplification).
Pitot tubes measure airspeed via Bernoulli: v = √(2(P_stag − P_static)/ρ).
Mach > 0.3: compressibility breaks Bernoulli. Different equation needed for supersonic.
"Head" = pressure / (ρg). Total head in meters of fluid column is common engineering currency.
Extended Bernoulli with friction loss + pumps is the workhorse of real pipe-system design.
Daniel Bernoulli was 38 when Hydrodynamica was published. The equation has held for nearly 300 years.
Frequently asked questions
Compressible flow (Mach > 0.3), viscous flow with friction losses, unsteady flow, flows with pumps/turbines, flows with heat transfer. Real engineering uses the extended Bernoulli with friction + pump terms.
Air over curved top of wing travels faster than air under flat bottom. Bernoulli: faster air = lower pressure. Pressure difference between top + bottom = lift. The full picture involves circulation theory + Kutta-Joukowski; Bernoulli is the simplified explanation.
Pressure when velocity is brought to zero. Pitot tube measures this. Stagnation = static + dynamic = P + ½ρv². Pitot static tubes measure both stagnation + static separately, subtract to get dynamic, then airspeed = √(2·dynamic/ρ).
Gravity converts elevation head to kinetic energy. From Bernoulli, ½v₂² = ½v₁² + g(z₁ − z₂). The narrowing stream (water "necking") reflects acceleration — mass conservation: smaller area = higher velocity. Surface tension breaks the narrowing into droplets eventually.
Bernoulli is conservation — energy is neither created nor destroyed. Pressure → kinetic when fluid accelerates through a constriction. Elevation → pressure when fluid descends. No external source; all internal conversion within the flow.
No. All inputs stay in your browser.
Below Mach 0.3, density changes are <5% — Bernoulli works. Above Mach 0.3, compressible Bernoulli: ½v² + γ/(γ−1) · p/ρ = const (for isentropic flow). Above Mach 1, normal shocks require Rankine-Hugoniot relations.
For short, smooth pipes with low velocity: very accurate. For long pipes or fittings: friction dominates and you need extended Bernoulli with f·(L/D)·(v²/2g) head loss term + minor losses for fittings (K factors).
SI: Pressure in Pa or kPa, velocity in m/s, elevation in m, density in kg/m³. US: psi, ft/s, ft, lbm/ft³. This calculator uses SI. Convert: 1 psi = 6895 Pa; 1 ft = 0.3048 m.
Bernoulli D., Hydrodynamica (1738). Fox + McDonald or Cengel + Cimbala fluid mechanics textbooks. Crane Technical Paper No. 410 for pipe flow practical engineering.
Method & sources
How it computes
Bernoulli's equation between two points on a streamline for steady, incompressible, inviscid flow, solved for the downstream pressure: P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(z₁ − z₂), with g = 9.81 m/s² and the density you enter (default 1 000 kg/m³, water). The static, dynamic (½ρv²) and hydrostatic (ρgz) terms are reported both as pressures in kPa and as heads in metres (P/ρg, v²/2g, z).
What this tool implements
- Steady, incompressible, inviscid flow along one streamline, with no pump, turbine or loss between the two points — the textbook restrictions on the equation.
- Friction and minor losses are neglected, so the computed downstream pressure is an upper bound; for a real pipe run take the loss from the Darcy–Weisbach pipe-flow calculator on this site.
- The incompressible form is adequate for liquids and for gases below about Mach 0.3, where the density change is under about 5%.
- One gravitational constant, g = 9.81 m/s² (standard gravity 9.806 65 m/s² rounded), is used for every head term.
Sources
- White FM. Fluid Mechanics. 8th ed. New York: McGraw-Hill Education; 2016. Chapter 3, the Bernoulli equation and its restrictions; Chapter 9 on the Mach 0.3 incompressibility limit.
- Munson BR, Okiishi TH, Huebsch WW, Rothmayer AP. Fundamentals of Fluid Mechanics. 7th ed. Hoboken, NJ: John Wiley & Sons; 2013. Chapter 3, Elementary fluid dynamics — the Bernoulli equation.
- Bernoulli D. Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Strasbourg: Johann Reinhold Dulsecker; 1738 — the original statement of the energy relation.
- Thompson A, Taylor BN. Guide for the Use of the International System of Units (SI). NIST Special Publication 811, 2008 edition — standard acceleration of gravity g_n = 9.806 65 m/s². https://www.nist.gov/pml/special-publication-811
What can make this go out of date
- None at runtime — no tabulated or published data; density is a user input.
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