# The Bernoulli Equation

In this notes sheet...

## The Bernoulli Equation

Generally, the Bernoulli Equation is expressed between two points, 1 and 2:

See the derivation of the Bernoulli Equationhere.

This applies to **steady, inviscid flow with constant density**, where **the two point lie on the same streamline**, or the **Bernoulli constant of each is the same:**

We can use the Bernoulli equation to model the conservation of mechanical energy (when there is no friction).

### Stagnation Point Flow

In an ideal situation, when inviscid, incompressible steady flow with uniform velocity and pressure hits a wall, the central streamline will not deflect but become stationary.

This is known as the stagnation point.

In this instance, the Bernoulli equation simplifies to:

All the mechanical energy is in terms of pressure potential energy: the maximum possible pressure in a given flow (stagnation pressure).

In reality, this does not occur, as close to the wall the fluid is very viscous.

## Conservation of Energy for Steady Flow

The first law of thermodynamics is the conservation of energy in a system:

Putting this in the Reynolds transport theorem, where η is the total energy per mass:

And for steady flow, where the d/dt term on the right disappears:

And for steady flow, where the d/dt term disappears:

Here, the work done by the pressure force is included in the work term on the left. This is not helpful to us, so we want to extract it.

Therefore:

The shaft work on the right does not include pressure work (this is in the integral on the left).

For a simple control volume with one inlet and one outlet:

## The Pipe Flow Energy Equation

This is a simple conservation of energy equation for adiabatic flow. This means the Q term disappears, however there is a small amount of heat lost through friction. Therefore, we take into account energy losses:

Therefore, the pipe flow energy equation becomes:

Often, it is most helpful to give values in terms of height – from this the pressure can easily be calculated. We call these ‘heads’, and so the lost work and pump work become lost head and pump head respectively:

This only applies in steady, adiabatic, incompressible, uniform velocity flow between a single inlet and outlet

As you can see then, it’s a pretty niche equation.

The pipe flow energy equation is the adiabatic equivalent of the SFEE in thermodynamics but for incompressible fluids, not compressible gasses.