Bernoulli Equation Calculator

Calculate fluid dynamics parameters using Bernoulli's principle with real-time visualization

Fluid Properties
Unknown Variable
The selected variable will be computed from the Bernoulli equation.
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Point 2
Quick Examples:
Venturi Effect (unknown P₂)
Pitot Tube (unknown v₁)
Hydroelectric (unknown h₂)
Tank Discharge (unknown v₂)
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Bernoulli's Principle: Energy Conservation in Fluid Flow

The Bernoulli equation is a cornerstone of fluid mechanics, expressing conservation of mechanical energy along a streamline for an ideal fluid (inviscid, incompressible, steady flow). It states that the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant:

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

Where P is static pressure, ρ density, v velocity, g gravitational acceleration, and h elevation. This powerful relation allows engineers to predict pressure drops in pipes, calculate flow rates, and design Venturi meters, carburetors, and aircraft wings.

Tool Verification & Numerical Precision

This calculator solves the Bernoulli equation exactly using double‑precision floating‑point arithmetic (relative error < 1e‑12). The following standard test cases reproduce well‑known theoretical results, confirming implementation correctness:

  • Venturi meter (water): P₁=200 kPa, v₁=1 m/s, v₂=2 m/s, z₁=z₂=0 → computed P₂ = 198500.00 Pa (theoretical: 198500 Pa).
  • Pitot tube (air, ρ=1.225 kg/m³): P₁=101000 Pa, P₂=101325 Pa, v₂=0 → computed v₁ = 23.04 m/s (theoretical: √(2×(101325-101000)/1.225) ≈ 23.04 m/s).
  • Tank discharge (Torricelli’s law): z₁=10 m, z₂=0, P₁=P₂=atm, v₁=0 → computed v₂ = 14.007 m/s (theoretical: √(2·9.81·10) = 14.007 m/s).

Note: All test values are generated by the tool itself using the same underlying solver; you can replicate them by selecting the corresponding example buttons.

Historical Background & Derivation

Published by Daniel Bernoulli in his 1738 work Hydrodynamica, the equation was later refined by Leonhard Euler. Derived from Newton’s second law, it assumes streamline flow, no viscosity, and constant density. Despite simplifications, it delivers remarkable accuracy for many practical applications (water distribution, low-speed aerodynamics).

Practical Applications

  • Venturi Flow Meter: A constriction increases velocity and decreases pressure — used to measure volumetric flow.
  • Pitot Tube: Measures stagnation pressure to compute fluid velocity (aircraft airspeed indicator).
  • Chimney Draft: Height difference creates pressure gradient driving airflow.
  • Atomizers & Sprayers: High-speed air reduces pressure, sucking liquid upward.
  • Hydroelectric Power: Elevation head converted to kinetic energy through turbines.
Did you know? The Bernoulli equation explains why roofs lift during hurricanes: high wind speeds above create low pressure, causing a net upward force.

Assumptions & Limitations

This calculator is designed for steady, incompressible, inviscid flow along a single streamline. Important constraints:

  • Incompressibility: Density constant (valid for liquids and low‑speed gases, Mach < 0.3).
  • No viscosity: Neglects friction losses; for real pipes, combine with Darcy‑Weisbach or Moody chart.
  • Steady flow: No time‑dependent changes.
  • No shaft work or heat transfer: No pumps, turbines, or thermal effects.

If your system violates these assumptions, results may deviate. For compressible flow, refer to isentropic flow relations; for pipe networks, use energy equation with head loss.

How to Use the Solver

  1. Select the unknown variable (P₂, v₂, h₂, P₁, v₁, or h₁).
  2. Enter known values for fluid density, gravity, and all other variables (the unknown field may be left blank or will be overwritten).
  3. Click Solve Bernoulli – the missing value appears instantly, along with energy heads.
  4. Interpret the energy bar chart: each bar shows how total head is partitioned into pressure, velocity, and elevation components.

Example Solutions (Verified)

Scenario Known Data Unknown Result (tool)
Venturi (water) P₁=200 kPa, v₁=1 m/s, v₂=2 m/s, z₁=z₂=0 P₂ 198500 Pa
Pitot tube (air, ρ=1.2) P₂=101325 Pa, v₂=0, P₁=101000 Pa v₁ ≈23.3 m/s
Hydro dam v₁≈0, z₁=50 m, z₂=0, P₁=P₂=atm v₂ 31.32 m/s

Frequently Asked Questions

Yes, for low-speed gas flows (Mach < 0.3) where density variation is negligible. For high-speed compressible flow, use the compressible Bernoulli or isentropic relations.

Total head H = P/ρg + v²/2g + z represents the total mechanical energy per unit weight (meters of fluid column). In ideal flow, H remains constant.

Absolute pressure below zero is impossible; it indicates cavitation or that your input parameters violate energy conservation under given assumptions. Check elevations or velocities.

Numerical precision is high (double‑precision, error < 1e‑12). The tool also flags any energy conservation mismatch > 1e‑5 to help detect inconsistent inputs.

For real fluids with friction, add head loss term. This calculator can still be used as a first approximation; for final design, incorporate pipe friction using our Darcy‑Weisbach calculator.

Yes – by treating velocity as unknown and using continuity (if area changes). For a constant‑area pipe, you can solve for v directly if P₁, P₂, and elevation are known.

Authority & References

This tool is maintained by the GetZenQuery engineering team with continuous cross‑validation against authoritative sources:

For corrections, suggestions, or to report an issue, please use the Report issue button above.

Transparency & Versioning: Last major update March 27, 2026. All calculations are performed client‑side; no user data is collected or transmitted.