Bernoulli Equation Calculator
Enter the conditions at two points along a streamline - pressure, fluid speed, and elevation - and this calculator applies the Bernoulli equation to find the unknown quantity. It also computes the pressure change, volumetric flow rate, and mass flow rate when you supply the pipe diameters. Works in metric (SI) or imperial units and lets you pick the fluid or enter a custom density.
Formula
Worked example
Water (rho = 998 kg/m^3) flows through a horizontal pipe that narrows from 0.10 m diameter to 0.05 m diameter. At point 1, P1 = 200 000 Pa and v1 = 2 m/s (h1 = h2 = 0). Continuity: A1*v1 = A2*v2, so v2 = (0.10/0.05)^2 * 2 = 8 m/s. Bernoulli: P2 = P1 + rho*(v1^2 - v2^2)/2 = 200 000 + 998*(4 - 64)/2 = 200 000 - 29 940 = 170 060 Pa. The pressure drops by about 29 940 Pa as the fluid speeds up, which is the Venturi effect.
What is the Bernoulli equation?
The Bernoulli equation is a statement of the conservation of mechanical energy for a steady, inviscid (frictionless), incompressible fluid flowing along a streamline. It was derived by the Swiss mathematician Daniel Bernoulli in 1738 and published in his work Hydrodynamica. The equation says that the sum of three energy terms is constant at every point on a streamline: static pressure (P), dynamic pressure (one half times density times velocity squared), and hydrostatic pressure (density times gravity times height). In symbols: P + (1/2)*rho*v^2 + rho*g*h = constant. The practical consequence is that when fluid speeds up it must exert less sideways pressure, and when it slows down pressure rises - the mechanism behind aircraft lift, Venturi meters, carburettors, and the spray from a garden hose nozzle.
How to use this calculator
Enter the pressure, velocity, and elevation at points 1 and 2 along a streamline. The calculator solves for the unknown velocity at point 2 using the Bernoulli equation, or, if you supply pipe diameters, it first applies the continuity equation (A1*v1 = A2*v2) so you only need to know the upstream velocity. It then reports the pressure change, volumetric flow rate (Q), mass flow rate, dynamic pressures, and total hydraulic heads at both points. Choose your unit system (SI or imperial) and fluid from the preset list, or enter a custom density. Gravity defaults to the standard Earth value of 9.80665 m/s^2 but you can change it for other planets or non-standard conditions.
Bernoulli equation and the Venturi effect
The most famous application of Bernoulli is the Venturi effect: when an incompressible fluid flows through a constriction, its speed increases and its static pressure falls. This is precisely the operating principle of the Venturi meter, a device that infers flow rate from the measured pressure drop across a throat section. Engineers use the same principle in carburettors (to atomise fuel), in ejector pumps (to move fluid without a moving pump), in aircraft pitot tubes (to measure airspeed), and in medical devices such as the peak-flow meter. The chart in this calculator shows how velocity and pressure at point 2 change as you vary the ratio of the pipe diameters, illustrating the Venturi relationship directly.
Hydraulic head and the head form of Bernoulli
Dividing each term of the Bernoulli equation by the specific weight (rho*g) gives the head form: P/(rho*g) + v^2/(2*g) + h = constant (metres of fluid). Each term is a "head": pressure head, velocity head, and elevation head. Their sum is the total hydraulic head H. For ideal flow, H is the same at every point on the streamline. In real pipe systems, friction causes H to decrease in the direction of flow, which is quantified by the Darcy-Weisbach head-loss term. Hydraulic head is the fundamental design variable for pumping systems, gravity-feed water supplies, and dam spillways.
When does the Bernoulli equation apply?
Bernoulli assumes four conditions: the flow is steady (nothing changes with time at a given point), the fluid is incompressible (constant density - valid for liquids and low-speed gases), viscosity is negligible (frictionless streamlines), and all points considered lie on the same streamline. For gases travelling faster than about Mach 0.3 the compressibility correction is significant and the isentropic Bernoulli equation should be used instead. For viscous pipe flow the Darcy-Weisbach equation or the extended Bernoulli with head losses is more appropriate. For turbulent or unsteady flows, computational fluid dynamics (CFD) is usually required.
Common fluid densities at atmospheric pressure and ~20 degrees C
| Fluid | Density (kg/m^3) | Typical use |
|---|---|---|
| Air (20 C, 1 atm) | 1.204 | Aerodynamics, HVAC |
| Water (20 C) | 998 | Hydraulics, plumbing, general reference |
| Sea water | 1025 | Marine engineering |
| Light mineral oil | 870 | Hydraulic systems, lubrication |
| Gasoline | 720 | Fuel systems, carburettors |
| Ethanol | 789 | Chemical processing |
| Mercury | 13550 | Manometry, barometers |
Approximate values - actual density varies with temperature and pressure.
Frequently asked questions
What does the Bernoulli equation calculate?
Given five of the six Bernoulli variables (P1, v1, h1, P2, v2, h2) plus the fluid density, it finds the sixth. This calculator also computes the pressure change (P1 - P2), volumetric flow rate, mass flow rate, dynamic pressures, and total hydraulic heads at both points.
Why does pressure drop when velocity increases?
The Bernoulli equation says the sum of pressure energy, kinetic energy (dynamic pressure), and potential energy is constant. When a fluid speeds up its kinetic energy rises, so its pressure energy must fall to keep the total constant. This trade-off between speed and pressure is the Venturi effect and is responsible for lift on an aircraft wing.
What is the continuity equation and how does it relate to Bernoulli?
For an incompressible fluid in a pipe, mass must be conserved: the flow rate entering a cross-section equals the flow rate leaving it. This is written A1*v1 = A2*v2, where A is the cross-sectional area. When you enter pipe diameters, this calculator uses continuity to find v2 before applying Bernoulli to find the pressure change. Together they let you solve real pipe-flow problems from geometry alone.
Can I use this calculator for gases like air?
Yes for low-speed air flow (below about Mach 0.3 or roughly 100 m/s at sea level). At those speeds air behaves nearly incompressibly and Bernoulli applies well. Above Mach 0.3, density changes with velocity and the isentropic compressible-flow equations are more accurate. Select "Air at 20 C" from the fluid menu for convenience.
What is hydraulic head?
Total hydraulic head is the Bernoulli constant expressed in metres (or feet) of fluid column: H = P/(rho*g) + v^2/(2*g) + h. It represents the total mechanical energy per unit weight of fluid. In ideal flow H is the same at every point on a streamline. In real systems, friction and fittings cause head loss, reducing H in the flow direction.
What are the limitations of the Bernoulli equation?
Bernoulli assumes steady, incompressible, inviscid (frictionless) flow along a single streamline. It does not account for viscous losses, heat transfer, turbulence, flow separation, or compressibility at high speeds. For engineering pipe design, the extended Bernoulli with Darcy-Weisbach head-loss terms is used instead. For high-speed gas flow, compressible isentropic relations are required.
How do I convert between Pa and psi?
1 psi = 6894.757 Pa, so to convert psi to Pa multiply by 6894.757, and to convert Pa to psi divide by 6894.757. For example, 200 000 Pa is about 29.0 psi. This calculator handles the conversion automatically when you select imperial units.