What does the Bernoulli equation calculate?

It describes the relationship between pressure, velocity and height of a fluid at two points along a streamline. If five of these quantities are input, the sixth can be calculated, and changes in pressure, volumetric flow rate and mass flow rate through the pipe can also be determined.

Why does pressure decrease when a fluid accelerates?

The total pressure along a streamline is constant. As the flow speed increases, the dynamic pressure rises and to keep the total pressure constant, the static pressure must decrease. This inverse relationship is at the core of Bernoulli's principle.

What assumptions are behind this expression?

The fluid must be incompressible and inviscid, the flow must be steady, and the two points must lie on a streamline. Since real fluids only approximately fulfill these conditions, small deviations due to friction and turbulence are expected.

How is flow rate calculated?

If you switch to the Flow mode and enter the diameter of a pipe and the velocity of the fluid, the calculator will give the volumetric flow rate and mass flow rate. If you enter a second diameter, it uses the continuity equation to show the downstream velocity.

Can this equation be used for air?

Yes, this formula can only be used if the airflow is slow enough to be considered incompressible. A rule of thumb is less than about one third the speed of sound. At faster flows compressibility effects become important and a different equation must be used.

Bernoulli Equation Calculator

Fluid & mode

What do you want to calculate?

Compare two points

Flow rate & pipe sizing

Fluid density

The two points along the streamline

Gravitational acceleration

m/s2

Pressure at point 1

Fluid speed at point 1

Elevation at point 1

Pressure at point 2

Fluid speed at point 2

Elevation at point 2

Fill in five of the six values across the two points, then leave the one you want blank. The calculator solves for it in place.

Please input values into the fields

How the pressures balance

Along a streamline the three pressures trade off but their sum stays constant. Where the fluid speeds up, static pressure drops; where it climbs, hydrostatic pressure rises.

The Bernoulli Equation Calculator is used to compare two points in a fluid along the same streamline. By entering the pressure, velocity and height at each point as well as one value in any field, the calculator will calculate the missing values. It can also calculate the volumetric flow rate and mass flow rate from the pipe diameter and flow velocity which is helpful when determining pipe sizes or checking the velocity of a fluid passing through an orifice.

What is the Bernoulli equation?

The Bernoulli equation describes the steady flow of incompressible fluids. It shows that the total pressure of a fluid remains constant along a streamline even if the fluid is accelerating, decelerating, rising or falling.

This principle can be expressed by the following formula:

p + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}

In this equation p represents the static pressure at that point, ρ is the density of the fluid, v is the flow velocity, h is the height and g is the gravitational acceleration which on Earth is about 9.80665 m/s2.

The middle term represents the dynamic pressure, or kinetic energy per unit volume of fluid. The last term ρ g h is the hydrostatic pressure due to the height of the fluid. The sum of static pressure, dynamic pressure and hydrostatic pressure gives the total pressure, which remains constant along a streamline.

Comparison of two points on a streamline

Since the total pressure is equal at any point along a streamline, then the sum of individual terms for one point can be set equal to the sum of individual terms for another point.

p_1 + \tfrac{1}{2}\rho v_1^2 + \rho g h_1 = p_2 + \tfrac{1}{2}\rho v_2^2 + \rho g h_2

If five of the six variables are known - either p1, v1, h1, p2, v2 or h2 - then the remaining variable can be calculated from a formula. This is what a calculator does. Enter any five known variables and leave the unknown one free.

Example calculation:

Let's say we have water with a density of 1000 kg/m3. At the first point, the pressure is 1000 Pa, the height is 3 m and the flow velocity of the fluid is 2 m/s. At a downstream point at the same elevation, the pressure has increased to 1200 Pa. To calculate the new velocity, we substitute these values into the equation for both points.

1000 + \tfrac{1}{2}(1000)(2^2) + \rho g h = 1200 + \tfrac{1}{2}(1000)v_2^2 + \rho g h

Since the terms for height are equal, they cancel each other out. Solving for the remaining part results in a downstream velocity v2 of about 1.897 m/s. As the pressure of the fluid increases, its speed decreases - this is Bernoulli's principle in action. The change in pressure is 200 Pa and is calculated as the difference between 1200 and 1000.

Symbol	Meaning	Example
p	Static pressure	1000 Pa
rho	Fluid density	1000 kg/m3
v	Flow speed	2 m/s
h	Elevation	3 m
g	Gravity	9.80665 m/s2

Flow through pipelines

The calculation tools also calculate the flow rate. When a liquid flows through a pipe with diameter d at speed v, the volume flow is the product of cross-sectional area and velocity.

q = \pi \left(\tfrac{d}{2}\right)^2 v

The mass flow rate is the product of volumetric flow and density; m is the result of multiplying q by rho. Since the same amount of fluid must pass through every point, the flow rate remains constant along a streamline, so that q1 equals q2. This means that a liquid has to move faster when it's in a narrower pipe. If you squeeze one end of a hose with your thumb, water squirts out further, and this is why.

Application of the Bernoulli equation

Bernoulli's principle can explain many phenomena in daily life and technology.

It can explain the lift produced by an airplane wing. The flow speed of air on the curved upper side is higher than that on the lower side, resulting in a lower pressure and creating an upward force. It also underlies the Venturi principle in carburetors and flow meters, Pitot tubes for measuring relative wind speeds, and the Magnus effect that deflects spinning soccer balls. Engineers use this principle when designing pumping systems to ensure sufficient suction head pressure to prevent cavitation.

Assumptions and limitations

The Bernoulli equation is based on four assumptions. It assumes that the flow is incompressible, meaning that the density remains constant. The flow is inviscid, which means that viscosity can be neglected. The flow is steady and does not change with time. Comparisons are made along the same streamline.

Since liquids such as water are nearly incompressible, this equation works well. Gases can only be considered incompressible at low speeds. At high speeds gases compress, so a modified formula that takes into account the specific heat capacity must be used.

This calculator is for general educational purposes and engineering estimates only. Real flows have friction, turbulence and losses that are not accounted for in the idealized Bernoulli equation. Therefore, results should be considered as a starting point rather than final design values.

Frequently asked questions

What does the Bernoulli equation calculate?: It describes the relationship between pressure, velocity and height of a fluid at two points along a streamline. If five of these quantities are input, the sixth can be calculated, and changes in pressure, volumetric flow rate and mass flow rate through the pipe can also be determined.
Why does pressure decrease when a fluid accelerates?: The total pressure along a streamline is constant. As the flow speed increases, the dynamic pressure rises and to keep the total pressure constant, the static pressure must decrease. This inverse relationship is at the core of Bernoulli's principle.
What assumptions are behind this expression?: The fluid must be incompressible and inviscid, the flow must be steady, and the two points must lie on a streamline. Since real fluids only approximately fulfill these conditions, small deviations due to friction and turbulence are expected.
How is flow rate calculated?: If you switch to the Flow mode and enter the diameter of a pipe and the velocity of the fluid, the calculator will give the volumetric flow rate and mass flow rate. If you enter a second diameter, it uses the continuity equation to show the downstream velocity.
Can this equation be used for air?: Yes, this formula can only be used if the airflow is slow enough to be considered incompressible. A rule of thumb is less than about one third the speed of sound. At faster flows compressibility effects become important and a different equation must be used.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

NASA Glenn Research Center: Bernoulli's Equation
Derivation and aerodynamic context for Bernoulli's equation.
Britannica: Bernoulli's theorem
Overview of the principle and its applications.