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Bernoulli Equation Calculator

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Bernoulli equationComparing two points along a streamlineFlow rateIncompressible and compressible fluidsFAQs

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:

p+12ρv2+ρhg=constant\small p + \frac{1}{2}\rho v^2 + \rho h g = \text{constant}

where:

  • pp — Pressure at the chosen point. To learn more about pressure, visit our pressure conversion.
  • ρ\rho — Density of the fluid (constant over time). Learn more about density in our density calculator.
  • vv — Flow speed at the given point;
  • hh — Elevation of the chosen point; and
  • gg — Acceleration due to gravity (on Earth, typically taken as 9.80665 m/s²).

The term 12ρv2\frac{1}{2}\rho v^2 corresponds to kinetic energy per unit volume, and ρhg \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:

p1+12ρv12+ρh1g=p2+12ρv22+ρh2g\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: p1p_1, v1v_1, h1h_1, p2p_2, v2v_2, and h2h_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 ρ=1000 kg/m³\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:

    p1=1000 Pap_1 = 1000\ \text{Pa}
    p2=1200 Pap_2 = 1200\ \text{Pa}
    h1=h2=3 mh_1 = h_2 = 3\ \text m
    v1=2 m/sv_1 = 2\ \text{m/s}

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

p1+12ρv12+ρh1g=p2+12ρv22+ρh2g\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+0.5×1000×22+1000×3×9.80665=1200+0.5×1000×v22+1000×3×9.806651000 + 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:
1000+2000=1200+500×v226=2.4+v22v22=3.6v2=1.897 m/s\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:

Δp=p2p1=12001000=200 Pa\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:

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

where:

  • qq — Volumetric flow rate in m³/h;
  • dd — Pipe diameter in meters; and
  • vv — Flow speed in m/s.

The flow rate is constant along the streamline. It means that, when comparing two points, you can assume q1=q2q_1 = q_2, or:

π(d/2)2v1=π(d/2)2v2\small \pi (d/2)^2v_1 = \pi (d/2)^2v_2

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

m=qρ\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.

FAQs

What does the Bernoulli equation calculate?

The Bernoulli equation calculates the pressure change, volume flow, and mass flow of a fluid along a streamline.

To compute these, you must know the following variables:

  • The density of the fluid;
  • Its speed;
  • Its pressure;
  • Its height, and
  • The diameter of the pipe.

Bernoulli's equation is a relationship between the pressure of a fluid in a container, its kinetic energy, and its gravitational potential energy.

What is the average flow rate of a kitchen faucet?

The average flow rate for kitchen and bathroom faucets in the United States is between 1.0 and 2.2 gallons per minute (GPM) at 60 pounds per inch (psi).

What is the Bernoulli equation for flow rate?

Bernoulli's principle implies that in the flow of a fluid, such as a liquid or a gas, an acceleration coincides with a decrease in pressure.

As seen above, the equation is:

  • q = π(d/2)2v × 3600

The flow rate is constant along the streamline.

For instance, when an incompressible fluid reaches a narrow section of pipe, its velocity increases to maintain a constant volume flow. This is why water gushes out of a garden hose faster when a narrow nozzle is fitted.

What is the Bernoulli equation used for?

The Bernoulli equation explains many phenomena, particularly in aerodynamics:

  • The Magnus effect;
  • The Venturi effect;
  • The Pitot tube; and
  • The law of hydrostatics.

Let's examine Bernoulli's principle with a concrete example: the Magnus effect, which can be observed during soccer games.

As the ball spins, it drags the air along with it, all the more so as it is irregular. This rotational movement causes a difference in pressure between the two sides of the ball, which is then displaced to the side with the lower pressure and thus affects the ball's trajectory.

What are the four assumptions of the Bernoulli equation?

The four assumptions of the Bernoulli equation, are as follows:

  • Fluid is incompressible;
  • Fluid is inviscid, i.e., having no or negligible viscosity;
  • Flow is steady; and
  • Flow is along a streamline.

A fluid is a perfectly deformable material medium. Fluids include liquids, gases, and plasmas. Gases and plasmas are highly compressible, while liquids are only slightly so (barely more than solids).

How does Bernoulli's principle apply to airplanes?

Bernoulli's principle explains how differences in fluid pressure acting on an object can make it move. Let's see how it applies to airplanes:

Under the plane's wing, air speed is unaffected by the wing's flat shape. On the other side, the wing is curved, which causes the air to travel a greater distance, increasing its speed. This increase in speed creates a low-pressure zone.

This pressure difference (normal pressure under the wing and low pressure above the wing) causes an upward force, creating lift. This applies not only to airplane wings but also to bird wings!

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