Derive Bernoulli's Equation.

Principle of conservation of energy:

Since energy cannot be generated or destroyed, it can be converted into another form. The principle of energy conservation is what this is characterized as. This principle is the basis for Bernoulli’s theorem. According to the principle of conservation of energy, the total energy at any point remains constant.

Potential energy:

The energy possessed by a fluid by virtue of its position above or below the datum line is called the potential energy.

Potential energy=Wh=mgh=mgZ

Where,

h,Z-Height of fluid particle above datum line in m

m-Mass of fluid in kg

W-Weight of fluid in N

g-Acceleration due to gravity

Epotential in  J/kg

Pressure energy:

The energy due to fluid’s pressure is called the pressure energy.

Pressure energy=PUs

P-Pressure in N/m2

Us-Specific volume of fluid in m3/kg

Epressure in  J/kg

Us=1 ,

So, pressure energy=p in J/kg

Kinetic energy:

The energy of the fluid by virtue of its velocity is called kinetic energy.

Kinetic energy=12mV2

m-Mass of the fluid in kg

V-Velocity of the fluid in m/s

Ekinematics in  J/kg

According to the principle of conservation of energy, total energy of the fluid remains constant.

Total energy=Potential energy+Kinetic energy+Pressure energy

=gZ+V22+P

Dividing the above equation by g on both sides,

Total head or Total enegy=Z+V22g+Pρg

=Z+V22g+Pw

Since, total energy or total head of the fluid remains constant,

Z1+V122g+P1w=Z2+V222g+P2w

This is the Bernoulli’s equation