Bernoulli's theorem

Theorem :

-  when the principal of conservation of energy is applied to the flow of fluid , the resulting equation is called Bernoulli's theorem .

- the law of conservation of energy states that energy can neither be created nor be destroyed , but can be changed from one form to another  .

- Bernoulli's theorem states that , in a steady flow , the sum of all forms of energy ( Pressure energy , kinetic energy , Potential energy ) in a fluid along a streamline is the same at all point on that streamline .

At Point A :

Pressure Energy  =  Pa / g(rhoa)

Pa = Pressure at point a .

G =  gravity 

(rhoa)  = density of liquid 

Potential energy is considered at  a hight of  Xa

Potential Energy  =  Xa 

Kinetic Energy passed by the body of its motion at fixed velocity .

Kinetic Energy =  u * A * A /  2g

Total Energy =  Pressure energy + potential energy + kinetic energy

Pa / g (rhoa) + Xa + u*A*A/2g  = constant

At Point B :

Pb / g (rhob) + Xb + u*B*B/2g =  constant

Input =  Output

In the transportation of fluid the pump has added certain amount of energy .

In the transportation of fluid the pump has added certain amount of energy .

Energy added by the pump = +w

During transportation of the liquid some energy is converted to heat due to frictional forces .

Loss of Energy =  -E

Bernoulli's Equation :

Pa / g(rhoa) + Xa + u*A*A/2g - E +W =  Pb / g(rhob) +Xb + u*B*B/2g

Applications:

Measurement of rate of flow .