Formula & Calculator
Bernoulli's Equation
Energy conservation in fluid flow (steady, incompressible, inviscid).
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P | Pressure | Pa |
| ρ | Density | kg/m³ |
| v | Velocity | m/s |
| g | Gravitational acceleration | m/s² |
| h | Height | m |
What it means
Bernoulli’s equation is a statement of conservation of mechanical energy for a fluid flowing along a streamline, assuming steady, incompressible, and inviscid flow. It states that the sum of the pressure energy (P), kinetic energy per unit volume (½ρv²), and potential energy per unit volume (ρgh) is constant. This equation is widely used to analyse fluid flow in pipes, nozzles, venturis, and around airfoils. It explains phenomena such as the lift on an aeroplane wing and the operation of a carburettor. In engineering, it is used to calculate flow rates from pressure measurements (e.g., pitot tubes), to design piping systems, and to estimate pump head requirements. The equation is derived from Newton’s second law and is valid for frictionless flow. In practice, corrections for friction losses (via the Darcy‑Weisbach equation) and minor losses are added. Bernoulli’s equation is a cornerstone of fluid dynamics and is taught in all introductory fluid mechanics courses.
Worked example
Bernoulli's Equation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| P | 101,325 Pa |
| ρ | 1.225 kg/m³ |
| v | 0 m/s |
| g | 9.81 m/s² |
| h | 0 m |
| Parameter | Value |
|---|---|
| P | 100,000 Pa |
| v | 20 m/s |
| ρ | 1.225 kg/m³ |
Common mistakes
- Assumptions: Bernoulli applies to steady, inviscid, incompressible flow along a streamline. Do not use it for compressible, unsteady, or viscous flows without correction.
- Units: All terms must have the same units (pressure: Pa, velocity term: Pa, elevation term: Pa). Use consistent units.
- Reference points: The equation is valid between two points on the same streamline; ensure both points are on the same streamline.
- Friction losses: For real fluids, add a head loss term; Bernoulli alone is an idealisation.
- Elevation term: Use the same datum for both points.
Applications
Bernoulli's equation, P + ½ρv² + ρgh = constant, expresses the conservation of mechanical energy along a streamline for steady, inviscid, incompressible flow. It relates pressure, velocity, and elevation, making it a fundamental tool in fluid mechanics. Engineers use it to design piping systems, to measure flow rates using venturi meters and pitot tubes, and to analyse the performance of pumps, turbines, and aircraft wings. In hydraulic engineering, it helps calculate water levels in reservoirs and flow through sluices. The equation also explains phenomena such as the lift on an airfoil (where higher velocity reduces pressure) and the entrainment of air in aspirators. By applying Bernoulli's principle, engineers can predict pressure drops, identify energy losses, and optimise system performance.
- Design of flow measurement devices (venturi, orifice, pitot tube)
- Analysis of pipe networks and hydraulic systems
- Performance prediction of pumps and turbines
- Aerodynamic design of aircraft wings and vehicle bodies
- Sizing of nozzles, diffusers, and ejectors