Bernoulli Equation

The Bernoulli Equation:-

is basically conservation of energy along a pipe. It can be written in different ways by converting energy to head (Kinsky) or pressure, etc.

Bernoulli's equation says in an ideal situation, the TOTAL HEAD = constant.

There are assumptions; An ideal fluid (no friction) flowing steadily:-

• The fluid is incompressible and nonviscous.
• There is no energy loss due to friction between the fluid and the wall of the pipe.
• There is no heat energy transferred across the boundaries of the pipe to the fluid as either a heat gain or loss.
• There are no pumps in the section of pipe under consideration.
• The fluid flow is laminar and steady state.

PRESSURE DENSITY :-

Pressure in a fluid is like the energy per unit volume (energy density). From the definition of pressure:

Here is the Bernoulli equation in terms of energy per unit volume;

Note that this correponds exactly with the conservation of energy equation. Where pressure = spring energy, velocity = kinetic energy and height = potential energy.

SE₁ + KE₁ + PE₁= SE₂ + KE₂ + PE₂

Where: PE₁ = mgh₁ and KE₁ = 0.5mv₁² and SE₁ = 0.5kx₁²

TOTAL HEAD:-

Total Head (also called Total Dynamic Head) is the sum of the three components; pressure head, velocity head and potential head. From the above equation, divide by g. Head is a simple way to think of each term (pressure, velocity and height) in terms of the number of METRES of that fluid it is equivalent to.

 

• All terms are head (m), which is measured in the working fluid. 
• You can use gauge or absolute pressure throughout, but gauge is normal
• Continuity equation is also true (and often needed to solve the question)
• Ideal fluid is assumed (no friction)

Bernouli's Equation requires velocity. This is usually found using the continuity equation.

 = V / t  =   A

Since most pipes are round, this gives;

v₁d²/4 = v₂d²/4

or