## Wednesday, June 16, 2010

### Bernoulli’s theorem:

Bernoulli’s theorem:

Introduction:

Bernoulli’s theorem: Bernoulli’s theorem was proposed by Daniel Bernoulli at 1738. Bernoulli’s theorem is mainly for the flow of liquid. It is based on the conservation law of energy.

The most commonly asked question is What does Bernoulli's theorem state?

Statement of Bernoulli’s theorem: Bernoulli’s theorem states that when a liquid is flow, the total of the pressure energy, kinetic energy and potential energy per unit mass should be constant.

Let us now learn about the Equation of the Bernoulli's Theorem:

Equation of Bernoulli's Theorem:

Consider the mass m which is to be passed into the pipe, then the equation becomes as follows.

a1v1 [rho] =a2v2 [rho] =m

a1v1=a2v2= [m/rho] =V

Force acting on liquid at X=P1a1

Force acting on the liquid at Y=P2a2

Work done per second at X=P1a1 [xx] v1=P1v

Work done at Y=P2a2 [xx] v2=P2V

Total work performed per second on the liquid by the pressure starting from X to Y =P1V-P2V

If the mass of the liquid is m, then the potential energy per second on the liquid from X to Y is equal to mgh2-mgh1

Now the increase of kinetic energy per second= [1/2] mv22 - [1/2] mv12

The work energy principle states that work done per second by pressure is equal to adding the increase in potential energy per second with the increase in kinetic energy per second.

P1V - P2V = (mgh2 -mgh1 )+ ( [1/2] mv22 - [1/2] mv12)

P1V+mgh1+ [1/2] mv12 =P2V +mgh2 + [1/2] mv22

[(P_1V)/m] +gh1 + [1/2] v12 = [ (P_2 V)/m] +gh2 + [1/2] v22

[(P_1)/rho] + gh1 + [1/2] v12 = [(P_2)/rho] +gh2 + [1/2] v22

[P/rho] +gh + [1/2] v2 =constant

When dividing by g the term becomes

[P/(rhog)] + [(v^2)/(2g)] +h = constant

Here the Pressure is [P/(rhog)]

Velocity is [(v^2)/(2g)] and