Productive ToolboxBernoulli Equation Calculator
Apply the Bernoulli principle to calculate fluid pressure, velocity, and elevation. Solve for any unknown variable with step-by-step explanations and unit conversion.
Bernoulli Equation Calculator
Select the unknown variable, enter all known values, and instantly solve using the Bernoulli principle: P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂
Result
Settings & Actions
Solve For
Pressure
Velocity
Elevation
Fluid Properties
Standard: 9.81 m/s² (Earth)
Moon: 1.62 m/s²
Related tools
What is the Bernoulli Equation?
The Bernoulli Equation is a fundamental principle in fluid mechanics that describes the conservation of energy in a flowing fluid. It states that the total mechanical energy — the sum of pressure energy, kinetic energy, and potential energy — remains constant along a streamline for an ideal, incompressible fluid.
The equation is expressed as: P + ½ρV² + ρgh = constant, or in its two-point form: P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂. This allows engineers to relate pressure, velocity, and elevation at any two points in a flow system.
Named after Swiss mathematician Daniel Bernoulli who published it in 1738, this equation underpins the design of aircraft wings, venturi meters, carburetors, nozzles, and countless other engineering systems where fluid flow is involved.
How to Use This Calculator
Step-by-Step Guide
- 1Choose the unknown: Select what to solve for: pressure P₁ or P₂, velocity V₁ or V₂, or height h₁ or h₂. That field is calculated; the others are inputs.
- 2Enter the known values: Type the pressures, velocities and heights at points 1 and 2, each in its own unit: Pa, kPa, bar or psi; m/s or ft/s; m or ft.
- 3Set the fluid: Pick water, seawater, air, oil or gasoline, or type a custom density in kg/m³ or lb/ft³. Gravity defaults to 9.81 m/s².
- 4Read the result: The answer updates as you type, with the rearranged formula, the step-by-step working and the pressure, kinetic and potential energy terms at each point.
- 5Copy, save or export: Copy the result, save it to the history in this browser, or export the calculation as a text file.
Key Features
- ✓Solve for any of 6 variables (P₁, P₂, V₁, V₂, h₁, h₂)
- ✓Real-time calculation as you type
- ✓Multi-unit support — Pa, kPa, bar, psi, m/s, ft/s
- ✓Automatic SI unit conversion
- ✓Step-by-step formula substitution
- ✓Energy terms breakdown table
- ✓Fluid presets: Water, Air, Oil, Gasoline, Seawater
- ✓Calculation history
- ✓Export results as TXT file
- ✓Swap inputs between Point 1 and Point 2
Example Calculations
| Scenario | Known Values | Result |
|---|---|---|
| Pipe constriction (water) | P₁=200 kPa, V₁=2 m/s, h₁=0, V₂=5 m/s, h₂=3 m, ρ=1000 | P₂ ≈ 170.9 kPa |
| Venturi meter | P₁=150 kPa, P₂=120 kPa, V₁=1 m/s, h₁=h₂=0, ρ=1000 | V₂ ≈ 7.76 m/s |
| Elevation change | P₁=P₂=101325 Pa, V₁=V₂=3 m/s, h₁=0, ρ=1000 | h₂ = 0 m |
| Nozzle exit velocity | P₁=300 kPa, P₂=101.3 kPa, V₁=0.5 m/s, h₁=h₂=0, ρ=1000 | V₂ ≈ 19.9 m/s |
| Height difference | P₁=200 kPa, P₂=180 kPa, V₁=2 m/s, V₂=2 m/s, h₁=0, ρ=1000 | h₂ ≈ 2.04 m |
Common Fluid Densities
| Fluid | Density (kg/m³) | Temperature | Common Use |
|---|---|---|---|
| Water | 1000 | 20°C | Plumbing, hydraulics |
| Seawater | 1025 | 20°C | Marine engineering |
| Air | 1.225 | 15°C | HVAC, aerodynamics |
| Oil (SAE 30) | 876 | 40°C | Lubrication systems |
| Gasoline | 720 | 20°C | Fuel systems |
| Mercury | 13,546 | 20°C | Manometers |
Real-World Applications
Aerodynamics
Aircraft wing lift is generated by pressure differences explained by Bernoulli's principle.
Pipe Flow Systems
Engineers use Bernoulli to size pipes, predict pressure changes, and design water supply networks.
Venturi Meters
Flow measurement devices use the pressure-velocity relationship to calculate volumetric flow rate.
Nozzle Design
Nozzle exit velocities and pressures are calculated directly from the Bernoulli equation.
HVAC Systems
Duct pressure analysis and fan sizing rely on Bernoulli to ensure proper airflow distribution.
Carburetors
Fuel-air mixing in carburetors uses the Venturi effect, a direct application of Bernoulli's principle.
Frequently Asked Questions
What is the Bernoulli equation?
P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂. Along a streamline, the sum of the pressure, the kinetic energy per volume (½ρV²) and the potential energy per volume (ρgh) stays constant. P is pressure in Pa, ρ density in kg/m³, V velocity in m/s, g gravity in m/s² and h height in m.
What are its assumptions?
Steady, incompressible flow along one streamline, with no friction (viscous) losses and no pump or turbine between the two points. For real pipes, add a head-loss term from the Darcy–Weisbach equation, and a pump or turbine head where there is one.
Why does pressure drop when the fluid speeds up?
Because the total energy is constant. Where a pipe narrows, the fluid must speed up to carry the same flow, so its kinetic energy rises and its pressure falls. Venturi meters, carburetors and atomizers use this effect.
Can I use it for air or other gases?
Yes, as long as the flow is slower than about Mach 0.3, roughly 100 m/s (330 ft/s) in air, where the density changes by less than about 5%. Faster gas flows need the compressible form of the energy equation.
Should I use gauge or absolute pressure?
Either, as long as both points use the same one. The equation only depends on the pressure difference, so gauge pressures (such as psig) work as well as absolute ones.
Why does the calculator say a velocity would be imaginary?
Solving for a velocity takes a square root. If the numbers you entered would need a negative kinetic energy, for example a higher pressure and height downstream with nothing driving the flow, there is no real solution. Check the signs, units and which point is upstream.
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