Bernoulli Equation Calculator

The Bernoulli Equation Calculator solves the famous fluid-dynamics relationship between pressure, flow velocity, and elevation along a streamline. Enter what you know at two points on the streamline, pick the one quantity you want to find, and the tool returns it — together with an animated pipe diagram, an energy grade line chart that visually proves energy is conserved, and a full step-by-step solution. It is the calculator of choice for students, engineers, and anyone working with pipes, nozzles, Venturi meters, or aerodynamics.

What Is the Bernoulli Equation?

Daniel Bernoulli's 1738 principle is a statement of conservation of energy for a moving fluid. For steady, incompressible, frictionless flow along a single streamline, the sum of the pressure energy, kinetic energy, and potential energy per unit volume stays constant from one point to the next.

Here \(P\) is the static pressure, \(\rho\) (rho) is the fluid density, \(v\) is the flow speed, \(g\) is gravitational acceleration (9.81 m/s²), and \(h\) is the elevation. The three terms represent, in order, pressure energy, kinetic (velocity) energy, and potential (elevation) energy per unit volume.

The Head Form and the Energy Grade Line

Dividing every term by \(\rho g\) rewrites the equation in heads — each term becomes a height in metres of fluid. This is the form the calculator visualizes:

The three heads are the pressure head \(P/\rho g\), the velocity head \(v^2/2g\), and the elevation head \(h\). Their sum is the total head \(H\), which is constant along the streamline for ideal flow — this constant level is called the energy grade line (EGL). The two stacked bars in the result are always the same total height, which is the clearest possible picture of Bernoulli's principle: energy simply shifts between pressure, speed, and height while the total stays fixed.

How This Calculator Solves for Any Term

The Bernoulli equation links six quantities across the two points (\(P_1, v_1, h_1, P_2, v_2, h_2\)). If you know five of them, the equation can be rearranged to find the sixth:

• Solving for pressure: \(P = E - \tfrac{1}{2}\rho v^{2} - \rho g h\), where \(E\) is the total energy taken from the fully-known point.
• Solving for velocity: \(v = \sqrt{\dfrac{2\,(E - P - \rho g h)}{\rho}}\). If the known values would require a negative number under the square root, no real flow can satisfy them and the tool reports it.
• Solving for elevation: \(h = \dfrac{E - P - \tfrac{1}{2}\rho v^{2}}{\rho g}\).

Worked Example: Flow Through a Narrowing Pipe

Water (\(\rho = 998\) kg/m³) flows in a horizontal pipe. At Point 1 the pressure is 200 kPa and the speed is 2 m/s. Downstream the pipe narrows and the pressure drops to 180 kPa. What is the new speed?

1. Total energy at Point 1: \(E = 200{,}000 + \tfrac{1}{2}(998)(2)^2 = 201{,}996\) Pa.
2. Solve for \(v_2\): \(v_2 = \sqrt{2(201{,}996 - 180{,}000)/998} \approx 6.64\) m/s.

The fluid speeds up from 2 to about 6.6 m/s as the pipe narrows, and its pressure falls — exactly what Bernoulli predicts and what a Venturi meter measures.

Real-World Applications of Bernoulli's Principle

Assumptions and Limitations

The Bernoulli equation is exact only under ideal conditions. Keep these limits in mind:

• Steady flow — conditions at each point do not change with time.
• Incompressible fluid — constant density, a good assumption for liquids and for air below roughly Mach 0.3.
• Negligible friction — no viscous or turbulent losses. Real pipes lose head to friction, so the downstream total head is slightly lower than the ideal value.
• Along one streamline — the two points must lie on the same streamline.
• No pumps or turbines between the points, which would add or remove energy.

How to Use This Calculator

1. Choose what to solve for: pick the unknown — pressure, velocity, or elevation at Point 1 or Point 2 — from the Solve for menu. That field is greyed out as the answer.
2. Select the fluid and units: choose water, air, seawater, oil, or a custom density, and the pressure unit (Pa, kPa, bar, psi, or atm).
3. Enter the known values at both points for the remaining five terms.
4. Click Calculate to get the unknown value, the animated streamline diagram, the energy grade line chart, the head breakdown table, and the step-by-step solution.

Frequently Asked Questions

What is the Bernoulli equation?

The Bernoulli equation states that for steady, incompressible, frictionless flow along a streamline, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant: P + ½ρv² + ρgh = constant. It expresses conservation of energy for a flowing fluid.

What can this Bernoulli calculator solve for?

It can solve for any one of six terms: the pressure, velocity, or elevation at either of two points on the same streamline. Choose the unknown from the Solve for menu, enter the other five values, and the calculator returns the missing one.

What is the head form of the Bernoulli equation?

Dividing every term by ρg converts the equation into heads measured in metres of fluid: pressure head P/(ρg), velocity head v²/(2g), and elevation head h. Their sum is the total head, which stays constant along the streamline for ideal flow. This is what the energy grade line chart shows.

What assumptions does the Bernoulli equation make?

It assumes steady flow, an incompressible fluid of constant density, negligible friction or viscous losses, flow along a single streamline, and no energy added or removed by pumps or turbines. Real systems with friction lose head, so the downstream total head is slightly lower than the ideal value.

Why does pressure drop when a fluid speeds up?

Because total energy is conserved. When a fluid accelerates — for example through a narrowing pipe — its velocity head increases, so the pressure head must decrease to keep the total head constant. This inverse relationship between speed and pressure is the core of Bernoulli's principle and explains lift on a wing and flow through a Venturi.

What units should I use?

Velocity is entered in metres per second and elevation in metres. Pressure can be entered in pascals, kilopascals, bar, psi, or atmospheres, and the result is shown in the same unit. Density is in kilograms per cubic metre, with presets for common fluids.

Additional Resources

• Bernoulli's Principle - Wikipedia
• Hydraulic Head & Energy Grade Line - Wikipedia