Bernoulli Equation Calculator
Solve Bernoulli's equation P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for the pressure, speed or height at a second point of a flow, with the speed from pipe diameters if you like.
Result…
How it's calculated
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ Continuity: A₁v₁ = A₂v₂, so v₂ = v₁ × (D₁ ÷ D₂)² Total head = P ÷ (ρg) + v² ÷ (2g) + h
Bernoulli's equation is conservation of energy for a moving fluid: pressure energy, kinetic energy ½ρv² and potential energy ρgh per unit volume add to the same total at every point on a streamline. Enter everything at point 1 and two of the three quantities at point 2 to find the third. The continuity equation (what flows in must flow out) gives the speed at point 2 from the two pipe diameters. The calculator works in SI units inside; unit factors are exact: 1 psi = 6894.757… Pa, 1 ft = 0.3048 m, 1 lb/ft³ = 16.0185… kg/m³.
Your numbers, step by step
About the Bernoulli Equation calculator
This Bernoulli equation calculator finds the pressure, speed or height at a second point of a flowing liquid or gas from what you know at the first point, using P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. When the pipe changes size, it can work out the speed at point 2 from the two inside diameters with the continuity equation A₁v₁ = A₂v₂, and it then gives the flow rate too. Work in metric (kPa, m/s, m, kg/m³, mm) or imperial (psi, ft/s, ft, lb/ft³, in); the conversions use exact factors. It suits fluid mechanics homework, a quick check on a pipe reducer, a venturi or a water tank outlet. Every step is shown with your numbers, along with the dynamic pressure at each point and the total head, so you can see where the energy goes.
How to use it
- Choose what to “Solve for”: the pressure, speed or height at point 2.
- Pick “Units” and enter the “Fluid density” for your fluid.
- Enter the pressure, speed and height at point 1, then the two known values at point 2.
- For a pipe that narrows or widens, set “Speed at point 2 from” to pipe diameters and enter both diameters.
- Read the answer, the pressure change, the dynamic pressures and the total head below it.
Worked example
Water (1000 kg/m³) flows at 1.5 m/s and 300 kPa in a 100 mm pipe, which narrows to 50 mm and rises 3 m. Continuity gives v₂ = 1.5 × (100 ÷ 50)² = 6 m/s. Bernoulli then gives P₂ = 300 + 1.125 − 18 − 29.42 ≈ 253.7 kPa, a drop of about 46.3 kPa. The flow rate is about 11.8 L/s.
Updated
Questions people ask
How do I use Bernoulli's equation to find pressure?
Add up the pressure, ½ρv² and ρgh at the first point, then subtract ½ρv² and ρgh at the second point. What is left is the pressure there, in pascals if you work in SI units.
Why does pressure drop when a pipe gets narrower?
The same flow has to pass through a smaller area, so it speeds up. The extra kinetic energy comes out of the pressure term, which is how a venturi meter measures flow.
Should I use gauge or absolute pressure?
Either works, as long as both points use the same kind. Only the difference between the points matters. Switch to absolute pressure if you want to check for cavitation, because that depends on the real pressure compared with the vapour pressure.
What does total head mean?
Total head is the energy per unit weight of fluid, written as a height: P ÷ (ρg) + v² ÷ (2g) + h. In ideal Bernoulli flow it stays the same along the streamline. For flow from speed and pipe size alone, try the flow rate calculator.
When does Bernoulli's equation not apply?
It assumes steady, incompressible, frictionless flow with no pump or turbine between the points. Long pipes, valves, bends and fast gas flows break those assumptions, so real pressures will be lower. Pipe sizing for real systems needs friction-loss methods and an engineer's check.