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.

The Bernoulli equation in fluid mechanics

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:

P+ρV22+ρgh=constant\small P + \dfrac{\rho V^2}{2} + \rho g h = \text{constant}

where:

  • PP — Static pressure of the fluid;
  • ρ\rho — Density of the fluid;
  • VV — Velocity of the fluid;
  • gg — Gravitational acceleration(on Earth typically taken as 9.80665 m/s²); and
  • hh — 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.

💡 To study how viscosity affects a fluid's pressure drop in a pipe, you can check the Poiseuille's law calculator.

Bernoulli's equation to study two points

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:

P1+ρV122+ρgh1=P2+ρV222+ρgh2\small P_1 + \cfrac{\rho V^2_1}{2} + \rho gh_1 = P_2 + \cfrac{\rho V^2_2}{2} + \rho gh_2

where:

  • P1P_1, ρV122\cfrac{\rho V^2_1}{2} and ρgh1\rho g h_1 — Values at point 1; and
  • P2P_2, ρV222\cfrac{\rho V^2_2}{2} and ρgh2\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:

m˙1=m˙2ρ1A1V1=ρ2A2V2\small \begin{aligned} \dot{m}_1 &= \dot{m}_2 \\ \rho_1 A_1 V_1 &= \rho_2 A_2 V_2 \end{aligned}

Where A1A_1 and A2A_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 ρ1=ρ2\rho_1 =\rho_2, then:

A1V1=A2V2V2=A1A2V1\small \begin{aligned} A_1 V_1 &= A_2 V_2 \\ V_2 &= \cfrac{A_1}{A_2} V_1 \end{aligned}

Here we can see that if the area reduces from point 1 to 2, A2<A1A_2 < A_1, the velocity of the fluid increases, V2>V1V_2 > V_1.

If we know the areas, we can determine the velocity at point 2. Thus, the volumetric flowrate at point 2 (q2q_2) as:

q2=V2A2\small q_2 = V_2 A_2

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

m˙2=ρq2=ρV2A2\small \dot{m}_2 = \rho q_2 = \rho V_2 A_2

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

Bernoulli's principle — Bernoulli equation energy form to explain it

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:

Pρ+V22+gh=constant\small \cfrac{P}{\rho} + \cfrac{V^2}{2} + gh = \text{constant}

where:

  • Pρ\cfrac{P}{\rho} — This term is associated with the pressure energy;
  • V22\cfrac{V^2}{2} — This relates to the kinetic energy of the fluid; and
  • ghgh — 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:

P1ρ+V122+gh1=P2ρ+V222+gh2\small \cfrac{P_1}{\rho} + \cfrac{V^2_1}{2} + gh_1 = \cfrac{P_2}{\rho} + \cfrac{V^2_2}{2} + gh_2

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

P1ρ+V122=P2ρ+V222\small \cfrac{P_1}{\rho} + \cfrac{V^2_1}{2}= \cfrac{P_2}{\rho} + \cfrac{V^2_2}{2}

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 V2>V1V_2 > V_1, then P2<P1P_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:

Pρg+V22g+h=constant\small \cfrac{P}{\rho g} + \cfrac{V^2}{2g} + h = \text{constant}

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

Form

Equation

Bernoulli energy equation

Pρ+V22+gh=constant\cfrac{P}{\rho} + \cfrac{V^2}{2} + gh = \text{constant}

Bernoulli pressure equation

P+ρV22+ρgh=constantP + \dfrac{\rho V^2}{2} + \rho g h = \text{constant}

Bernoulli head equation

Pρg+V22g+h=constant\cfrac{P}{\rho g} + \cfrac{V^2}{2g} + h = \text{constant}

💡 We can eliminate the pressure terms when evaluating two points for open channel flows since they have the same value.

Using the Bernoulli equation calculator

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.
Gabriela Diaz
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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