Bernoulli Equation Calculator

This fluid mechanics Bernoulli equation calculator will help you quickly find a fluid's pressure, velocity, or elevation. You can also use the Bernoulli equation calculator to find a fluid's volumetric and mass flow rates.

Keep reading to learn:

• What Bernoulli's equation is;

• The Bernoulli equation energy form to understand Bernoulli's principle; and

• Bernoulli's equation's pressure, energy, and head forms.

In fluid mechanics, the Bernoulli equation is a tool that helps us understand a fluid's behavior by relating its pressure, velocity, and elevation. According to Bernoulli's equation, the pressure of a flowing fluid along a streamline remains constant, as shown below:

where:

• $P$ — Static pressure of the fluid;
• $\rho$ — Density of the fluid;
• $V$ — Velocity of the fluid;
• $g$ — Gravitational acceleration(on Earth typically taken as 9.80665 m/s²); and
• $h$ — Elevation of the fluid, its height above a reference level.

Usually, we find that Bernoulli's equation is applied to flowmeter devices such as orifice flow meters, Venturi tube, and Pitot meter.

When applying Bernoulli's equation, keep in mind that it's valid under certain assumptions:

• Incompressible fluid: the fluid's density does not change. This is the case for liquids and some gases at low temperatures.
• Steady flow: flow rate doesn't change over time.
• Laminar flow: this means no vortex or swirling present in the fluid.
• Inviscid fluid: shear forces caused by viscosity are so small that they can be ignored.
• Flow along a streamline: when evaluating two points using Bernoulli's equation, we assume they are on the same streamline.

More complex versions of Bernoulli's equation can account for viscosity forces or unsteady and compressible flows.

Given that Bernoulli's equation represents a constant, we can use it to evaluate two points in the same streamline. By choosing two points, we can write the following expression:

where:

• $P_1$, $\cfrac{\rho V^2_1}{2}$ and $\rho g h_1$ — Values at point 1; and
• $P_2$, $\cfrac{\rho V^2_2}{2}$ and $\rho g h_2$ — Values at point 2.

This means that knowing the values of five of these variables allows you to calculate the remaining one. You can use this Bernoulli equation calculator to

When studying fluids with Bernoulli's equation, we'll often find ourselves also using the continuity equation, or mass conservation principle, to calculate volumetric and mass flow rates. This one is expressed as:

Where $A_1$ and $A_2$ represent the cross-sectional areas of the enclosed where the fluid is flowing, e.g., a pipe.

For an incompressible fluid densities are the same $\rho_1 =\rho_2$, then:

Here we can see that if the area reduces from point 1 to 2, $A_2 < A_1$, the velocity of the fluid increases, $V_2 > V_1$.

If we know the areas, we can determine the velocity at point 2. Thus, the volumetric flowrate at point 2 ($q_2$) as:

And the mass flow rate at point 2 ($\dot{m}_2$):

With this Bernoulli equation solver, you'll also be able to determine the fluid's volumetric and mass flow rates.

We've seen some of the math related to Bernoulli's principle, but what exactly is Bernoulli's principle? Well, here's what it states:

An increase in fluid's velocity always occurs accompanied by a decrease in fluid's pressure.

To better understand Bernoulli's principle, it is usually worth presenting Bernoulli's equation energy form:

where:

• $\cfrac{P}{\rho}$ — This term is associated with the pressure energy;
• $\cfrac{V^2}{2}$ — This relates to the kinetic energy of the fluid; and
• $gh$ — This one to the potential energy.

You can think of this form of Bernoulli's equation as an energy balance equation, in which the total energy always remains the same, meaning that if one of the energies rises, the others must inevitably decrease.

When we apply it to two points in the same streamline:

If there're no changes in elevation, we can eliminate the potential energy terms and simplify the equation as follows:

For this Bernoulli equation example, suppose that we are studying a fluid flowing in a pipe with a decrease in diameter. From continuity, we know that if the area decreases, the velocity rises. Notice then that in order for $V_2 > V_1$, then $P_2 < P_1$ for the equality to remain true.

According to the law of conservation of energy, if kinetic energy grows, other types of energy must decrease, namely the potential and pressure energies. Since we don't have the potential energy term in our Bernoulli equation example, only the pressure energy is the one transforming into kinetic energy.

Although we have presented Bernoulli's in its pressure and energy forms, for some problems, you may find it more convenient to use its head form instead:

In the following table, we've summarized the different forms of Bernoulli's equation. The Bernoulli energy equation, the pressure, and the head form:

Let's see how to use the Bernoulli equation calculator to study a fluid's properties between two points:

1. Indicate the Gravitational acceleration and Fluid density.
2. Enter the known conditions for Position 1. These are: Pressure, Height, Speed, and Pipe diameter.
3. Proceed to indicate the knowns for Position 2; the calculator will determine the remaining one.
4. The calculator will also show the fluid's Pressure change, Volume flow rate, and Mass flow rate.