# Bernoulli Equation Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jul 12, 2022

If you're interested in fluid mechanics, you will definitely find this Bernoulli equation calculator extremely helpful. It is a tool that allows you to compare two points along a streamline and determine their elevation, flow speed, and pressure.

Additionally, you can use the Bernoulli calculator to determine the flow rate of the analyzed fluid. This way, you can choose proper pipe diameters to ensure a steady flow.

Please keep reading to learn more about the Bernoulli equation, or take a look at our buoyancy calculator!

## 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 environmental changes.

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

$\small p + \frac{1}{2}\rho v^2 + \rho h g = \text{constant}$

where:

• $p$Pressure at the chosen point;
• $\rho$Density of the fluid (constant over time);
• $v$ – Flow speed at the given point;
• $h$ – Elevation of the chosen point; and
• $g$ – Acceleration due to gravity (on Earth typically taken as 9.80665 m/s²).

The term $\frac{1}{2}\rho v^2$ corresponds to kinetic energy per unit volume, and $\rho h g$ 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:

$\footnotesize p_1\! +\! \frac{1}{2}\rho v_1^2\! +\! \rho h_1 g = p_2\! +\! \frac{1}{2}\rho v_2^2\! +\! \rho h_2 g$

It means that once you know five of the following values: $p_1$, $v_1$, $h_1$, $p_2$, $v_2$, and $h_2$, 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 $\rho = 1000\ \text{kg/m}³$.

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 at 2 meters per second.

3. Choose two out of three fluid properties at 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:

$p_1 = 1000\ \text{Pa}$
$p_2 = 1200\ \text{Pa}$
$h_1 = h_2 = 3\ \text m$
$v_1 = 2\ \text{m/s}$

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

$\scriptsize \qquad p_1\! +\! \frac{1}{2}\rho v_1^2\! +\! \rho h_1 g = p_2\! +\! \frac{1}{2}\rho v_2^2\! +\! \rho h_2 g$

$1000 + 0.5 \times 1000 \times 2^2 + 1000 \times 3 \times 9.80665 = 1200 + 0.5 \times 1000 \times v_2^2 + 1000 \times 3 \times 9.80665$

1. Simplify and solve the equation:
\footnotesize \qquad \begin{align*} 1000 + 2000 &= 1200 + 500 \times v_2^2\\[0.5em] 6 &= 2.4 + v_2^2\\[0.5em] v_2^2 &= 3.6\\[0.5em] v_2 &= 1.897\ \text{m/s} \end{align*}
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:

\footnotesize \qquad \begin{align*} \Delta p &= p_2 - p_1\\ &= 1200 - 1000 = 200\ \text{Pa} \end{align*}

## Flow rate

You can also use the Bernoulli equation calculator to determine your fluid's volumetric and mass flow rate. 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:

$\small q = \pi (d/2)^2v\times 3600$

where:

• $q$ is the volumetric flow rate in m³/h;
• $d$ is the pipe diameter in meters; and
• $v$ 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 $q_1 = q_2$, or:

$\small \pi (d/2)^2v_1 = \pi (d/2)^2v_2$

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

$\small m = q\rho$

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

## Incompressible and compressible fluids

As mentioned before, you can only use this Bernoulli equation calculator to analyze the flow of an incompressible fluid. Real-life applications use the Bernoulli equation to design water pumping systems where you must control pressure variation at the suction of the pump to avoid cavitation.

What you might know as a compressible gas can become an incompressible fluid at lower temperatures. It means 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 elevation change is omitted. However, the flow is then dependent on an additional value – the fluid's specific heat. To check out an application of the Bernoulli equation to incompressible flow, please check out our Magnus force calculator.

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