# Bernoulli Equation Calculator

By Bogna Szyk
Last updated: Mar 19, 2019

If you're interested in fluid mechanics, you will definitely find this Bernoulli equation calculator to be extremely helpful. It is a tool that allows you to compare two points along a streamline and determine the elevation, flow speed and pressure at each of them. Additionally, you can use the Bernoulli calculator to find out what is the flow rate of the analyzed fluid. This way, you can choose proper pipe diameters to ensure that the flow is steady.

## Bernoulli equation

The Bernoulli equation describes a steady flow of an incompressible fluid. It means that the fluid doesn't change its properties (for example density) over time. According to the Bernoulli principle, the total pressure of such fluid (both static and dynamic) remains constant along the streamline, regardless of the changes in the environment.

This principle can be formulated in the form of an equation:

`p + 0.5ρv² + ρhg = constant`

where:

• p is the pressure at the chosen point;
• ρ is the density of the fluid (constant over time),
• v is the flow speed at the given point,
• h is the elevation of the chosen point, and
• g is the acceleration due to gravity (on Earth typically taken as 9.80665 m/s²).

The term `0.5ρv²` corresponds to kinetic energy per unit volume and `ρhg` is the hydrostatic pressure.

## Comparing two points along a streamline

You can use this Bernoulli equation calculator to compare two points along the same streamline. Since you know that the total pressure of the fluid is constant, you can write that

`p₁ + 0.5ρv₁² + ρh₁g = p₂ + 0.5ρv₂² + ρh₂g`

It means that once you know five of the following values: p₁, v₁, h₁, p₂, v₂, and h₂, you can easily calculate the sixth with the help of our calculator.

If you'd like to perform these calculations by hand, simply follow the steps below:

1. Choose the density of the fluid. We can assume `ρ = 1000 kg/m³`.

2. Determine the properties of the fluid at the initial point. Let's assume that the fluid is under the pressure of 1000 Pa, at the height of 3 meters, and flows with the speed of 2 meters per second.

3. Choose two out of three properties of the fluid in the second point. We can say that the pressure increased to 1200 Pa without a change in elevation.

4. Write down all of the variables:

`p₁ = 1000 Pa`

`p₂ = 1200 Pa`

`h₁ = h₂ = 3 m`

`v₁ = 2 m/s`

1. Now, input all of the variables into the Bernoulli equation:

`p₁ + 0.5ρv₁² + ρh₁g = p₂ + 0.5ρv₂² + ρh₂g`

`1000 + 0.5 * 1000 * 2² + 1000 * 3 * 9.80665 = 1200 + 0.5 * 1000 * v₂² + 1000 * 3 * 9.80665`

1. Simplify and solve the equation:

`1000 + 2000 = 1200 + 500 * v₂²`

`6 = 2.4 + v₂²`

`v₂² = 3.6`

`v₂ = 1.897 m/s`

1. You have found the new flow speed of the fluid. It is equal to 1.897 m/s.

2. You can also calculate the change in pressure:

`Δp = p₂ - p₁ = 1200 - 1000 = 200 Pa`

## Flow rate

You can also use the Bernoulli equation calculator to determine the volumetric and mass flow rate of your fluid. A flow rate describes how many cubic meters (in the case of volumetric flow rate) or how many kilograms (in the case of mass flow rate) flow through one point on the streamline during one hour.

In order to calculate the flow rate, you need to know the area of the cross-section the fluid is flowing through. As you will typically use pipes, all you need to know is the diameter of such a pipe. Then, you can calculate the volumetric flow rate with the use of the following formula:

`q = π * (d/2)² * v * 3600`

where

• q is the volumetric flow rate in m³/h,
• d is the pipe diameter in meters,
• v is the flow speed in m/s.

The flow rate is constant along the streamline. It means that, when comparing two points, you can assume `q₁ = q₂`, or

`π * (d₁/2)² * v₁ = π * (d₂/2)² * v₂`

To calculate the mass flow rate, simply multiply the volumetric flow rate by the fluid density:

`m = q * ρ`

The mass flow rate is one of the main specifications quoted for fans, turbines...

## Incompressible and compressible fluids

As mentioned before, this Bernoulli equation calculator can only be used to analyze the flow of an incompressible fluid. It means that the fluid has a constant density and cannot be compressed under pressure. Still, it is possible to develop a similar equation for compressible fluids. In such a case, the influence of the change in elevation is omitted. However, the flow is then dependent on an additional value - the specific heat of the fluid.

Bogna Szyk
Gravitational acceleration
g
Fluid density
kg/m³
Position 1
Pressure
bar
Height
in
Speed
ft/s
Pipe diameter
mm
Position 2
Pressure
bar
Height
in
Speed
ft/s
Pipe diameter
mm
Results
Pressure change
bar
Volume flow rate
/h
Mass flow rate
lb
/h
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