TECHSHORE INSPECTION SERVICES :BERNOULLI'S EQUATION

Bernoulli’s equation is related with pressure, velocity and elevation changes of a fluid in motion. it may be stated as follows ”In an ideal gas or fluid when the flow is steady and continuous,the sum of pressure energy, kinetic energy, potential energy is constant along a streamline.”

Proof:

Consider steady flow of incompressible liquid through anon uniform pipe lying entirely in x-y plane.the following assumptions is made in derivation of Bernoulli’s equation.

The liquid is ideal and incompressible

The flow is steady and continuous

The velocity is uniform over t6he cross section and equal to mean velocity

The only force acting on the liquid are the gravity force and the pressure force

All frictional forces are neglected

We examine a fluid section of mass m traveling to the right as shown in the above figure. The total work done in moving the fluid is

Where F denotes the force and an x a displacement. The second term picked up its negative sign because the displacement and force are in opposite directions.

Pressure is the force exerted over the cross-sectional area, or P = F/A. Rearranging this as F = PA and substitute  into Eq.(1) we can  find that

The displaced fluid volume V is the cross-sectional area A times the thickness x. This volume will be  constant for an incompressible fluid, so

Using Eq.(3) in Eq.(2) we have

Since work had been done, there will be a change in the mechanical energy of the fluid segment. This energy change is found with the help of the next diagram.

The change in energy between the initial and final positions is given by

Here, the kinetic energy K = mv²/2 where  v is the speed of the fluid and m is the fluid mass. The potential energy U = mgh where g is the acceleration of gravity, and h is average height of the fluid.

The work-energy theorem says that the amount net work done is equal to the change in the system energy. This can be expressed as

Substitution of Eq.(4) and Eq.(5) into Eq.(6) yields

Dividing Eq.(7) by the fluid volume, V gives us

is the mass density of the fluid . To complete our derivation, we reorganize Eq.(8).

Applications of Bernoulli’s equation

Even though Bernoulli's equation is mainly applied in incompressible flow problems, it is applied in involvement of energy also. The application of bernoulli's equation in measuring devices as follows

Venturimeter

Orificemeter

Pitot tube

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