Energy quations and its application
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This document outlines a presentation on fluid mechanics concepts including the momentum equation, Bernoulli's theorem, and applications. It discusses the momentum equation, force exerted by fluid flow on pipe bends using a 2D flow equation, Euler's equation of motion, Bernoulli's theorem, and applications like Pitot tubes, Venturimeters, and orifice meters. It also provides an elementary introduction to notches and weirs.
Energy quations and its application
- 1. 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
- 2. Chapter Outline Momentum equation Force exerted by the fluid flow on pipe bend(2-D Dimensional flow equation) Euler’s equation of motion Bernoulli’s theorm Applications of Bernoulli’s theorm Pitot tube Venturimeter Orifice meter Elementary introduction of Notches and Weirs
- 3. 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
- 4. Conti… Now, a=dv/dt F=m(dv/dt) =d(mv)/dt Equation 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.
- 5. Conti…
- 6. 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).
- 7. 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 cosθ Similarly the momentum equation in Y-direction gives 0-P2A2 sinθ-Fy=density.Q(-V2 sinθ-0) Fy=density.Q(-V2 sinθ)-P2A2 sinθ Now the resultant force (Fr) acting on the bend and angle made by the F with X =
- 8. Euler’s Equation Of Motion Let us consider a steady flow of an ideal fluid along a streamline and small element AB of the flowing fluid as shown in figure.
- 9. 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 element We know that the external forces tending to accelerate the fluid element in the direction of the streamline
- 10. 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,
- 11. Mass of the fluid element = We see that the acceleration of the fluid elementt
- 12. 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
- 13. BERNOULLI’S THEOREMBERNOULLI’S THEOREM Bernoulli’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 an approximate equation that is valid only in in viscid regions of flow where net viscous forces are negligibly small compared to inertial, gravitational, or pressure forces. Such regions occur outside of boundary layers and wakes.
- 14. BERNOULLI’S EQUATIONBERNOULLI’S EQUATION If a section of pipe is as shown above, then Bernoulli’s Equation can be written as;
- 15. 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.
- 16. Limitations on the Use of the Bernoulli Equation Steady flow and incompressible flow No heat transfer into and out of the fluid Constant internal energy (constant temperature) Assumptions made in Bernoulli’s equation: Ideal fluid Stream lined flow Irrotational flow The gravity force and pressure force are only considered
- 27. NOTCHES AND WEIRS Notches 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.
- 28. Weirs It is a concrete or masonary structure which is placed in an open channel over which the flow occurs.
- 29. THANK YOU.