Bernoulli's Equation

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[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:

Bernouli.png
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:

BernouliDerivation1.png

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

BernouliDerivation2.png

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

BernouliDerivation3.png

And by noting that V dv = (1/2)d(V2 and dividing by ρ gives:

BernouliDerivation4.png

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

BernouliDerivation5.png

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:

Bernouli.png

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

Bernouli2Points.png

Limitations

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

  1. Steady flow
  2. Incompressible flow
  3. Negligible viscous effects
  4. Negligible heat transfer
  5. Irrotational flow (no vorticity)
  6. 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:

Bernoulli2.png

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

  1. P = Static Pressure- the actual thermodynamic pressure of the fluid, without taking into account any dynamic effects
  2. (1/2)ρ V2= 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)
  3. 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
  4. Pstag = P + (1/2)ρ V2 = 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:

StagnationVelocity.png

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:

BernoulliHeads.png

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

  1. P/ρg = Pressure Head- height of fluid column that produces the static pressure P
  2. V2/2g = Velocity Head- elevation needed for fluid to reach a velocity during freefall
  3. 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).

  1. Textbook p. 197
  2. Textbook pp.198-199
  3. Textbook pp. 202 - 203
  4. Textbook pp. 200-201
  5. Textbook pp 203-204
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