Bernoulli's Equation

Bernoulli’s Equation

It is based on law of conservation of energy. This equation is applicable when it is assumed that

• Flow is steady and irrotational
• Fluid is ideal (non-viscous)
• Fluid is incompressible

It states in a steady, ideal flow of an incompressible fluid, the total energy at any point of the fluid is constant.

The total energy consists of pressure energy, kinetic energy and potential energy or datum energy. These energies per unit weight of the fluid are:

• Pressure energy

                             

• Kinetic energy

                             

• Datum energy = z

Bernoulli’s theorem is written as:

 

• Bernoulli’s equation can be obtained by Euler’s equation

 

As fluid is incompressible, ρ = constant

where, 

• Restrictions inthe application of Bernoulli’s equation
  • Flow is steady
  • Density is constant (incompressible)
  • Friction losses are negligible
  • It relates the states at two points along a single streamline, (not conditions on two different streamlines)

The Bernoulli equation is applied along streamlines like that joining points 1 and 2 

 

Total head at 1 = Total head at 2

This equation assumes no energy losses (e.g. from friction) or energy gains (e.g. from a pump) along the streamline. It can be expanded to include these simply, by adding the appropriate energy terms

Note Point: 

The Bernoulli equation is often combined with the continuity equation to find velocities and pressures at points in the flow connected by a streamline.

Kinetic Energy Correction Factor (α)

In a real fluid flowing through a pipe or over a solid surface, the velocity will be zero at the solid boundary and will increase as the distance from the boundary increases. The kinetic energy per unit weight of the fluid will increase in a similar manner.

The kinetic energy in terms of average velocity V at the section and a kinetic energy correction factor α can be determined as:

In which m = ρAVdt is the total mass of the fluid flowing across the cross-section during dt. By comparing the two expressions for kinetic energy, it is obvious that,

The numerical value of α will always be greater than 1, taking kinetic energy correction factor, α, as