# Bernoulli's Equation

^{[1]}Bernoulli's Equation is one of the most fundamental equations in Fluid Mechanics. It is an **approximate** relation between a fluid's pressure, velocity and elevation and is only valid in regions of **steady, incompressible flow**, where friction forces are negligible. Care must be taken in applying Bernoulli's equation, as it is only an approximate value and can only be applied in regions **outside** of boundary layers and wakes.

Simply speaking, the Bernouli Equation is given as:

- Where:
- P = Pressure (Pa)
- ρ = Density (kg/m³)
- V = Velocity (m/s)
- g = Acceleration due to gravity (ms
^{-2}) - z = Elevation (m)

## Contents |

## Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 197 - 205.

## Derivation

*The derivation is added here in full even though it will never be examined, however explaining parts of it may be'*.

^{[2]}Consider a fluid particle in a flow field undergoing **steady flow**. The forces acting on this particle along a streamline can be combined under Newton's Second Law (in the **s** direction). Assuming frictional forces to be negligible, the only forces acting on the particle are pressure (F = P.A) and its own weight. Since pressure acts in both the positive and negative **s** directions, two terms are included: P and -(P+dP). Since the weight acts in a perpendicular direction, its term is W.sinθ. Adding it all up we get:

Now, since m = ρ.dA.ds (density times volume), W = mg = ρ.g.dA.ds while sinθ = dz/ds (elevation over horizontal). By substitution:

By cancelling out dA from all terms and simplifying, we get:

And by noting that V dv = (1/2)d(V^{2} and dividing by ρ gives:

Now be integration, and noting that the last two terms are exact differentials:

Under incompressible flow (density is constant so the integral becomes the integral of 1/ρ dP), the first term also becomes an exact differential, yielding Bernoulli's equation:

In addition, Bernoulli's equation can be used between two points on the same streamline by:

## Limitations

^{[3]}Bernoulli's equations is limited to a situation in which all of the following properties apply:

- Steady flow
- Incompressible flow
- Negligible viscous effects
- Negligible heat transfer
- Irrotational flow (no vorticity)
- No shaft work

## Static, Dynamic and Stagnation Pressures

^{[4]}Bernoulli's equation, under inspection, can be stated as the sum of kinetic, potential and **flow** energies of a fluid. In effect, this means that a fluid can convert between pressure, velocity and elevation. For clarity, multiply each term in Bernoulli's equation by the density to get:

Now each term has units of pressure (Pa or kPa) so that they can be defined as:

- P =
**Static Pressure**- the actual thermodynamic pressure of the fluid, without taking into account any dynamic effects - (1/2)ρ V
^{2}=**Dynamic Pressure**- represents the pressure rise when the fluid in motion is brought to a stop isentropically (i.e. in an adiabatic and reversible manner) - ρgz =
**Hydrostatic Pressure**- accounts for elevation effect though it is not in itself a pressure measurement since its value is dependant on the reference height - P
_{stag}= P + (1/2)ρ V^{2}=**Stagnation Pressure**- pressure at a point when the fluid is brought to a complete stop isentropically

When static and stagnation pressures are measured at a specific location, the velocity at that location is given by:

## Heads

^{[5]}Similarly to before, dividing Bernoulli's equation by g yields a useful equation whose terms are all in units of meters (m). This is useful since it gives a measure of the different energies in terms of heights. Bernoulli's equation can be written as:

Where H is that **total head** for the flow. The equation can then be split up into the components:

- P/ρg =
**Pressure Head**- height of fluid column that produces the static pressure P - V
^{2}/2g =**Velocity Head**- elevation needed for fluid to reach a velocity during freefall - z =
**Elevation Head**- potential energy of fluid

## References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).