Bernoulli Calculator

Bernoulli Calculator: Pressure, Velocity, and Height

This free Bernoulli Calculator solves ideal fluid-flow problems for pressure at a second point, velocity at a second point, or elevation at a second point. Enter density, pressure, velocity, height, and gravity values, then view a complete worked solution. The calculator uses SI units and clearly separates pressure energy, kinetic energy, and gravitational potential energy. Its live Venturi diagram follows the classic representation: a wide pipe narrows into a throat, streamlines accelerate, and pressure columns reveal the pressure difference.

It is useful for physics homework, fluid mechanics revision, Venturi-meter examples, pipe-flow demonstrations, aircraft discussions, and introductory engineering calculations. The chart updates with your values: P1 and P2 appear beside their pressure tubes, v1 and v2 appear inside the pipe, the manometer height difference is shown between the columns, and density plus elevation appear in a live data card.

Bernoulli Equation Formula

Bernoulli's equation is P + 1/2 rho v2 + rho g h = constant. Between two points along one streamline, it is written P1 + 1/2 rho v1² + rho g h1 = P2 + 1/2 rho v2² + rho g h2. P is static pressure in pascals, rho is fluid density in kg/m3, v is velocity in m/s, g is gravitational acceleration, and h is elevation in metres.

The equation describes conservation of mechanical energy per unit volume in ideal fluid flow. Pressure energy, kinetic energy, and elevation energy can change into one another. If velocity rises in a level pipe, the kinetic term rises; pressure usually falls to balance the equation. If a fluid rises to a greater height, some pressure or kinetic energy is converted into gravitational potential energy. Rearrange the formula to calculate whichever variable is unknown.

How to Use the Bernoulli Calculator

Select the downstream pressure, velocity, or height that you need. Enter the known values in SI units. Water is often approximated as 1000 kg/m3, although its exact density depends on temperature. Pressure must be in pascals: 1 kPa equals 1000 Pa. Then read the highlighted answer and the four calculation steps. The diagram is especially helpful for a Venturi-style question because it displays the high-pressure, low-speed wide section and the lower-pressure, higher-speed throat together.

For a horizontal water pipe, set h1 and h2 equal. With P1 = 180,000 Pa, v1 = 2 m/s, v2 = 8 m/s, and rho = 1000 kg/m3, the second pressure is P2 = 180,000 + 0.5 x 1000 x (4 - 64) = 150,000 Pa. The 30 kPa pressure reduction becomes kinetic energy as the fluid speeds up through the narrow section.

Venturi Effect and Pressure Tubes

A Venturi tube is a pipe that narrows smoothly and then expands. For steady incompressible flow, continuity requires the fluid to move faster through the smaller cross-sectional throat. Bernoulli's principle links that higher velocity to lower static pressure. The vertical tubes in the diagram are piezometer tubes: a higher liquid column indicates higher static pressure. The height difference between columns is related to pressure difference by Delta P = rho g Delta H when the measurement fluid matches the flowing fluid.

Venturi meters use this pressure difference to estimate flow rate. Measure the pressure upstream and at the throat, combine it with the pipe geometry and continuity equation, and calculate velocity or flow. Real meters use calibration coefficients because viscosity, turbulence, installation effects, and energy losses make flow non-ideal. The image-like diagram shows the physical reason behind the reading: the narrow region has fast flow and a smaller pressure column.

Bernoulli Principle and Continuity

Bernoulli's equation and the continuity equation work together but answer different questions. Continuity for incompressible flow says A1v1 = A2v2. It links pipe area and velocity. Bernoulli's equation links pressure, velocity, and height. In a narrowing pipe, continuity says v2 is greater than v1; Bernoulli then predicts that P2 is lower than P1 when the pipe is level and losses are small.

Do not assume that faster flow always means lower pressure in every real situation. The familiar statement is correct along a streamline in the ideal Bernoulli context, but pumps, friction, turbulence, curved paths, and moving boundaries can change the result. Use the full equation and define the two points carefully. A separate Flow Rate Calculator can help find velocity from area and volumetric flow before applying Bernoulli's equation.

Real Life Applications of Bernoulli's Principle

Aircraft wings are a well-known application. Wing shape and angle create a pressure distribution around the wing; lower pressure over parts of the upper surface contributes to lift, alongside the downward deflection of air. The simplified 'air travels farther and must meet at the same time' explanation is incomplete. A more accurate account uses pressure fields, circulation, and momentum, but Bernoulli's relation remains a useful way to connect velocity changes and pressure differences in the surrounding flow.

Venturi meters, carburettors, perfume atomisers, paint sprayers, Bunsen burners, aspirators, and medical nebulisers use low pressure created by a fast stream to draw in another fluid. A shower curtain can move inward when flowing water changes air pressure. Chimneys and ventilation systems can be influenced by moving air. In every application, a pressure difference drives the secondary motion, while the complete design must also consider losses, geometry, and operating conditions.

Bernoulli Applications in Pipes and Industry

Engineers use Bernoulli analysis as a starting point for pumps, nozzles, valves, pipe transitions, water networks, fuel delivery, and process equipment. A nozzle converts pressure into velocity to form a jet. A diffuser aims to reduce velocity and recover static pressure. Flow meters compare upstream and throat pressure. In water systems, elevation differences are important: water flowing uphill loses pressure head, while water flowing downhill can gain pressure.

In HVAC and environmental systems, the equation helps explain duct velocity, intake effects, and pressure measurement. In medicine, it helps discuss blood flow through narrowed vessels, although blood is pulsatile and viscous, so a clinical model needs more than ideal Bernoulli. In motorsport and architecture, airflow around surfaces can create pressure effects. The calculator is an educational ideal-flow solver, not a replacement for validated design analysis.

Worked Bernoulli Equation Examples

Consider water flowing horizontally from a wide pipe section at 3 m/s and 220 kPa to a throat at 9 m/s. With rho = 1000 kg/m3 and equal heights, P2 = 220,000 + 0.5 x 1000 x (3² - 9²) = 184,000 Pa, or 184 kPa. The 36 kPa drop is associated with the increase in velocity. The pressure tube on the throat side would be lower than the upstream tube.

For an elevation example, water has P1 = 300 kPa, P2 = 250 kPa, v1 = 2 m/s, and v2 = 4 m/s. The elevation difference can be calculated from the rearranged equation. The pressure difference provides roughly 5.1 m of water head, while the velocity increase consumes about 0.61 m of head; the remaining energy can raise the fluid. Showing each term in compatible units makes these problems easier to check.

Assumptions and Limits of Bernoulli's Equation

The basic Bernoulli equation assumes steady flow, an incompressible fluid, negligible viscosity, and analysis along one streamline. It works best when friction and energy losses are small. Real pipe systems lose energy through rough walls, bends, fittings, valves, filters, pumps, and turbulence. Engineers commonly add a head-loss term and pump or turbine energy terms to the extended energy equation.

Gases can be treated as incompressible only when density changes are small. At high speeds, strong pressure changes, or large temperature changes, compressibility matters. Time-varying flow, cavitation, multiphase mixtures, and rotating machinery also need more advanced methods. A calculated negative absolute pressure or an impossible square-root result is a sign that inputs or assumptions should be checked. Use measured data and appropriate standards for practical systems.

Units, Accuracy, and Safety

Keep units consistent: pascals for pressure, kg/m3 for density, m/s for velocity, metres for height, and m/s2 for gravity. Because pressure values are often stated in kPa, convert before using the equation. The calculator shows pressure head in metres as a helpful physical check. For water, a 9.81 kPa pressure difference is about one metre of water head.

Do not use an educational result to alter pressurised pipes, fuel systems, medical devices, gas equipment, aircraft equipment, or industrial machinery. Pressure and fluid systems can be hazardous. Follow manufacturer instructions, local regulations, and qualified engineering practice. This calculator explains the law, conversions, and ideal calculation steps, but it cannot determine structural safety, pump suitability, or real-system losses.

Bernoulli Calculator FAQ Summary

Bernoulli's equation is P + 1/2 rho v2 + rho gh = constant for ideal steady flow along a streamline. It explains the relationship between pressure, speed, and height. Use it with continuity to understand Venturi tubes, nozzles, wings, sprayers, meters, and pipe flow. Faster fluid in a level narrowing normally has lower static pressure, but real flows require loss, viscosity, and equipment effects to be considered.