The Bernoulli Equation

The Bernoulli equation is a useful way of approximating the state at a point in a streamline given the state at another point.  Sometimes we have a situation where it's easy to measure or observe the state of a point in a fluid system, but we want to know the state of the system at another point where it's not as easy to measure.  The Bernoulli equation, a.k.a. the Bernoulli principle, gives an easy method to translate the properties at one point to a second point.

The Bernoulli Equation

Bernoulli's principle states that a fluid particle's energy remains constant as it travels along a streamline.  Any change in potential, kinematic or pressure energy is compensated by a change in the other two energy values.  The equation is:

The usefulness of Bernoulli's principle is when we know the state at one point in the streamline, and we know two of the energy values at a second point.  Using this information we can find the state at the second point.  Since energy is constant, we can equate the two points and solve for the unknown value.

Limiting Assumptions

The use of Bernoulli's principle assumes:

  • The fluid is incompressible.
  • No fluid friction.
  • Adiabatic process, i.e. no heat or mass transfer.

Example

Consider the water tank shown below.  The surface of the water at point A is open to the atmosphere.  Water is discharging from the tank through a pipe at point B.  It's assumed that the water volume in the tank is so large that the elevation of point A is not changing in a significant manner.

At point A:

velocity = 0

pressure = 0

elevation = 8 meters

At point B:

velocity = unknown

pressure = 0

elevation = 0

The discharge velocity at point B is unknown.  By using the equation for two points and dropping zero terms we get the following equation.

Entering the values in the equation, and solving for velocity at B:

Using the MEboost Bernoulli Tool

MEboost has a calculator for the Bernoulli equation that easily solves for the unknown value.  The Bernoulli form is shown below with values from the example earlier.