This notes sheet introduces the Bernoulli equation used to calculate properties of fluids at two different points on the same streamline.

## The Bernoulli Equation

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

- See the derivation of the Bernoulli Equation here.

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) – each term represents a different energy:

| Pressure potential energy |

| Kinetic energy per unit mass |

| Gravitational energy per unit mass |

### 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 acted upon by the viscous force.

## Conservation of Energy for Steady Flow

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

We can substitute this into the Reynolds transport theorem, where is the total energy per unit mass:

And for steady flow, the time derivative 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 left does not include pressure work (this is in the integral on the right).

For a simple control volume with one inlet and one outlet, the Bernoulli equation becomes:

## The Pipe Flow Energy Equation

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

is the lost energy

, the pump work

The pipe flow energy equation thus becomes:

Often, it is more 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

- The Bernoulli equation applies between two points:
- The flow must be
**steady and inviscid, with constant density**, where**the two point lie on the same streamline**, or the**Bernoulli constant of each is the same:**

- The flow must be
- The
**Pipe Flow Energy Equation (PFEE)**is the fluid equivalent of the SFEE in thermodynamics:- This only applies in steady, adiabatic, incompressible, uniform velocity flow between a single inlet and outlet