Bernoulli & Fluid Pressure Calculator
Work out the pressure under a depth of liquid and how pressure, speed and height trade off in a flowing pipe. Solve p = ρgh, Bernoulli’s equation for any unknown, Venturi-meter flow and Torricelli’s outlet speed.
How to Use
- Pick a mode: Pressure at depth (p = ρgh), Bernoulli between two points, Venturi meter, or Torricelli for a tank emptying through a hole.
- Choose what to solve for in that mode, for example the depth that gives a pressure, or the speed at the second point of a pipe.
- Enter the liquid density: 1,000 kg/m³ for fresh water, about 1,025 kg/m³ for seawater, or your own.
- Fill in the other values with their units: kPa, bar, psi or atm for pressure, m or ft for heights, m/s for speeds, mm for pipe sizes.
- Read the answer in its highlighted field and the first readout, with absolute pressure, speeds or flow beside it, and see the step-by-step working below.
Worked Example
Pressure on a diver. At 30 m in seawater (1,025 kg/m³) the water adds p = ρgh = 1,025 × 9.80665 × 30 = 301.6 kPa of gauge pressure. Add the 101.325 kPa of air above the sea and the absolute pressure is 402.9 kPa, 3.976 atmospheres.
A pipe that rises and narrows. Water at 300 kPa and 2 m/s climbs 5 m and speeds up to 4 m/s. Bernoulli’s total at point 1 is 300,000 + ½ × 1,000 × 2² + 0 = 302,000 Pa. At point 2 the speed takes ½ × 1,000 × 4² = 8,000 Pa and the height takes 1,000 × 9.80665 × 5 = 49,033 Pa, leaving p₂ = 244,967 Pa (245.0 kPa).
The common mistake: mixing gauge and absolute pressure. Boyle’s law works only with absolute pressure. A diver’s lungful of air at 30 m is squeezed to 101.325 ÷ 402.9 = 25.15% of its surface volume. Using the gauge pressure, 301.6 kPa, gives 33.6%, a third too much. In Bernoulli’s equation either kind works, as long as both points use the same one.
Show Work
Formulas
Torricelli, Pascal and Bernoulli
Evangelista Torricelli, Galileo’s successor in Florence, made the first mercury barometer in 1643, showing that the air’s weight holds up a column of mercury about 76 cm tall. In the same years he worked out that liquid leaves a hole at the speed it would have falling from the surface, Torricelli’s law. In 1648 Blaise Pascal had his brother-in-law Florin Périer carry a barometer up the Puy de Dôme, and the column fell with height, proving that pressure comes from the weight of what lies above.
Daniel Bernoulli connected pressure and speed in Hydrodynamica, published in 1738, and Leonhard Euler wrote the relationship as the equation used today in the 1750s. Giovanni Battista Venturi studied the pressure drop in a narrowing tube in 1797, and the American engineer Clemens Herschel turned it into a practical flow meter for water mains in the 1880s, naming it after Venturi.
About This Tool
This calculator covers the basic sums of fluids at rest and in motion. Pressure at depth gives gauge and absolute pressure, or the depth or density that produce a pressure. Bernoulli solves any one of the six values between two points of a pipe. The Venturi mode turns a pressure difference into a flow rate, and the Torricelli mode gives the speed of a jet from a tank, the flow through the hole and how long the tank takes to drain. It assumes an incompressible liquid with no friction losses; for flow rate, pipe sizing and the Reynolds number, see the Fluid Flow Calculator.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Fluid Flow Calculator, Buoyancy Calculator, and Density, Mass & Volume Calculator.
Frequently Asked Questions
How do you calculate water pressure at a depth?
Multiply density, gravity and depth: p = ρgh. Under 10 m of fresh water (1,000 kg/m³) the pressure is 1,000 × 9.80665 × 10 = 98,066.5 Pa, or 98.07 kPa, almost one atmosphere. Each metre adds 9.807 kPa (1.422 psi), whatever the shape of the container.
What is the difference between gauge and absolute pressure?
Gauge pressure counts only what is above the air pressure; absolute pressure adds the atmosphere on top. At 30 m in seawater (1,025 kg/m³) the gauge pressure is 301.6 kPa and the absolute pressure is 301.6 + 101.325 = 402.9 kPa, about 3.976 atmospheres.
What does Bernoulli’s equation say?
Along a steady flow with no friction, p + ½ρv² + ρgh stays the same: pressure, speed and height trade against each other. Water at 300 kPa moving at 2 m/s that rises 5 m and speeds up to 4 m/s arrives at 300,000 + 2,000 − 8,000 − 49,033 = 244,967 Pa, about 245 kPa.
How does a Venturi meter measure flow?
It narrows the pipe and measures how much the pressure drops in the throat. With a 100 mm pipe, a 50 mm throat, a 15 kPa drop and a discharge coefficient of 0.98, the flow is Q = 0.98 × 0.001963 × √(2 × 15,000 ÷ (1,000 × 0.9375)) = 10.89 L/s, or 653.1 L/min.
How fast does water come out of a hole in a tank?
Torricelli’s law says v = √(2gh), the speed of something dropped from the water surface. With the surface 2 m above the hole, v = √(2 × 9.80665 × 2) = 6.263 m/s. A 20 mm sharp-edged hole (Cd ≈ 0.61) passes 1.2 L/s at first and empties a 1 m wide tank down to the hole in 43.62 minutes.
How do I use the Bernoulli & Fluid Pressure Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Diving
At 30 m in seawater the absolute pressure is 3.976 atm, so a breath of air squeezes to 25.15% of its surface volume.
Water supply
A water tower 30 m above a tap gives 294.2 kPa (42.67 psi) of static pressure, before any flow losses.
Barometers
Solve for density: one atmosphere holding up a 0.76 m column gives 13,595 kg/m³, the density of mercury used to define the mmHg.
Flow measurement
A Venturi meter with a 100 mm to 50 mm throat reading 12.66 kPa means 10 L/s of water is passing.
Airspeed
A pitot tube reading 500 Pa of dynamic pressure in air (1.225 kg/m³) means v = √(2 × 500 ÷ 1.225) = 28.57 m/s.
Draining tanks
A tank 1 m across with 2 m of water above a 20 mm hole empties to the hole in 26.61 minutes for an ideal nozzle and 43.62 minutes for a sharp-edged hole.
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