Energy quations and its application

In partial fulfillment of the subject

Fluid Mechanics

Submitted by:

Sagar Damani 130120119041 (MECH:4-A{A3}) Sanket Chopde 130120119039

2141906

GANDHINAGAR INSTITUTE OF TECHNOLOGY

PRESENTATION ON

The Energy Equation and its Application

Chapter OutlineMomentum equationForce exerted by the fluid flow on pipe bend(2-D

Dimensional flow equation)Euler’s equation of motionBernoulli’s theormApplications of Bernoulli’s theorm

Pitot tube VenturimeterOrifice meter

Elementary introduction of Notches and Weirs

THE MOMENTUM EQUATION • It is based on the law of conservation of momentum or on

the momentum principle,

Which states,• “the net force acting on a fluid mass is equal to the

change in momentum of flow per unit time in that direction”.

The force acting on a fluid mass’m’is given by the Newton’s second law of motion.

F=m*a

Conti…Now, a=dv/dt

F=m(dv/dt)

=d(mv)/dtEquation is known as the momentum principle.Equation can also be written as

F.dt=d(mv)

Which is known as the impulse-momentum equation and states that the impulse of a force F acting on a fluid o mass m in a short interval of time dt is equal to the change of momentum d(mv) in the direction of force.

Conti…

Force exerted by a flowing fluid on a pipe-bend

Consider two sections(1) and(2),as shown in fig.Let v1=vel.of flow at section(1) p1=pressure intensity at section(1) A1=area of cross-section of pipe at section(1)V2,P2,A2=corresponding values at section (2).

Conti…Let Fx and Fy be the components of the forces exerted by the flowing fluid on

the bend in x and y respectively.Then the force exerted by the bend on the fluid in the direction of x and y will be Fx and Fy but in opp.direction.

Net force acting on fluid in the x direction=Rate of change of momentum in x

direction.

P1A1-P2A2 cosθ-Fx=M.dv

=density.Q(V2 cosθ-V1)

Fx= density.Q(V2 cosθ-V1)+P1A1-P2A2 cosθSimilarly the momentum equation in Y-direction gives

0-P2A2 sinθ-Fy=density.Q(-V2 sinθ-0)

Fy=density.Q(-V2 sinθ)-P2A2 sinθNow the resultant force (Fr) acting on the bend and angle made by the F with X

=

Euler’s Equation Of MotionLet us consider a steady flow of an ideal fluid along a

streamline and small element AB of the flowing fluid as shown in figure.

Conti…Let,•dA = Cross-sectional area of the fluid element•ds = Length of the fluid element•dW = Weight of the fluid element•P = Pressure on the element at A•P+dP = Pressure on the element at B•v = velocity of the fluid elementWe know that the external forces tending to accelerate the fluid element in the direction of the streamline

We also know that the weight of the fluid element,

From the geometry of the figure, we find that the component of the weight of the fluid element in the direction of flow,

Mass of the fluid element =

We see that the acceleration of the fluid element t

Now, as per Newton's second law of motion, we know that Force = Mass *Acceleration

Dividing both sides by

,

Or

This is the required Euler’s equation of motion for a fluid

BERNOULLI’S THEOREMBERNOULLI’S THEOREMBernoulli’s theorem which is also known as Bernoulli’s principle, states that an increase in the speed of moving air or a flowing fluid is accompanied by a decrease in the air or fluid’s pressure or sum of the kinetic (velocity head), pressure(static head) and Potential energy of the fluid at any point remains constant, provided that the flow is steady, irrotational, and frictionless and the fluid is incompressible.

The Bernoulli equation is anapproximate equation that is validonly in in viscid regions of flow where net viscous forces are negligibly small compared to inertial, gravitational, or pressure forces. Such regions occuroutside of boundary layers and wakes.

BERNOULLI’S EQUATIONBERNOULLI’S EQUATION

If a section of pipe is as shown above, then Bernoulli’s Equation can be written as;

BERNOULLI’S EQUATIONBERNOULLI’S EQUATION

Where (in SI units)

P= static pressure of fluid at the cross section;ρ= density of the flowing fluid in; g= acceleration due to gravity;v= mean velocity of fluid flow at the cross section in;h= elevation head of the center of the cross section with

respect to a datum.

Limitations on the Use of the Bernoulli Equation

Steady flow and incompressible flowNo heat transfer into and out of the fluid

Constant internal energy (constant temperature)

Assumptions made in Bernoulli’s equation: Ideal fluid

Stream lined flowIrrotational flow

The gravity force and pressure force are only considered

NOTCHES AND WEIRSNotches

It is defined as a device which measures the flow rate of a liquid through a small channel or tank.

It is an opening in the side of a tank or a small channel in such a way that the surface of liquid in the tank or small channel is below the top edge of opening.

WeirsIt is a concrete or masonary structure which is placed

in an open channel over which the flow occurs.

THANK YOU.