About the Bernoulli's Equation
The Bernoulli's Equation calculator finds the fluid pressure at a second point in a flow system, given the pressure and velocity at a first point along with any change in elevation. It's built for the classic pipe-narrowing or airfoil scenario, where a fluid speeds up and its pressure correspondingly drops. Engineers, students, and hobbyist plumbers use it to see how much pressure is gained or lost as flow conditions change between two points on the same streamline.
How It Works
Enter the pressure and velocity at point 1, the velocity at point 2, and optionally the height difference between the two points and the fluid's density (both tucked under advanced fields, defaulting to a level pipe carrying water). The calculator applies Bernoulli's equation, which balances pressure, kinetic energy, and gravitational potential energy per unit volume between the two points, and solves directly for the unknown pressure at point 2.
Formula & Methodology
To work this out by hand, first compute the kinetic energy difference term as one-half times fluid density times the difference between the squares of the two velocities, not the square of the difference. Then compute the elevation term as fluid density times gravitational acceleration times the height change. Add the first term and subtract the second term from the known pressure at point 1 to get the pressure at point 2.
Examples
Water Speeding Up Through a Nozzle
Water at atmospheric pressure (101,325 Pa) moves at 2 m/s at point 1 and speeds up to 8 m/s at a narrower point 2, with no change in height. The calculator finds P2 = 101,325 + 0.5 x 1000 x (2^2 - 8^2) = 71,325 Pa, a drop of 30,000 Pa from the increase in speed alone.
Flow Rising Through a Vertical Pipe
Water at 200,000 Pa and 1 m/s at the base of a pipe accelerates to 3 m/s while rising 2 meters to point 2. P2 = 200,000 + 0.5 x 1000 x (1^2 - 3^2) - 1000 x 9.81 x 2 = 176,380 Pa, reflecting pressure lost to both the speed increase and the climb in elevation.
Advantages
- Handles the height-change term that's easy to forget when working the pressure-velocity relationship by hand.
- Separates the two effects, speed change and elevation change, into distinct terms so it's clear which one is driving the pressure result.
- Uses default values (atmospheric pressure, water density) that match a common real-world setup, making it quick to adapt for pipe or nozzle problems.
Common Mistakes
- Forgetting to set the sign of the height difference correctly when fluid flows downward instead of upward, which flips whether that term adds to or subtracts from the pressure result.
- Mixing units, such as entering pressure in psi or kPa while the calculator expects pascals.
- Applying the formula across a pump, valve, or long rough pipe section, where Bernoulli's frictionless, no-external-work assumption no longer holds.
Edge Cases to Watch For
- If velocity at point 2 is higher than at point 1 and the two points are at the same height, pressure at point 2 always comes out lower than at point 1, and can go negative for large speed increases starting from a low base pressure, which in a real system would signal cavitation risk rather than a physically meaningful negative pressure.
- The height difference is entered as a single signed number (point 2 minus point 1), so a fluid flowing upward should be given a positive value, which subtracts from the resulting pressure.
- The formula assumes steady, incompressible, frictionless flow along a single streamline with no pump or turbine between the two points, so it does not account for pipe friction losses over long runs or compressibility effects in gases at high speed.
Common Use Cases
- Fluid mechanics students verifying pressure-velocity tradeoff problems from a textbook.
- Plumbing and irrigation designers estimating pressure changes across a narrowing or rising section of pipe.
- Hobbyists and engineers sanity-checking airfoil or venturi meter behavior before more detailed analysis.