BERNOULLIS THEOREM

BERNOULLIS THEOREM MADE BY-GAURAV YADAVXi a 10

BERNOULLIS THEOREM Daniel BernoulliSwiss physicist (17001782)Bernoulli made important discoveriesin fluid dynamics. Bernoullis mostfamous work, Hydrodynamica, waspublished in 1738; it is both a theoretical and a practical study of equilibrium,pressure, and speed in fluids. He showedthat as the speed of a fluid increases,its pressure decreases. Referred to asBernoullis principle, Bernoullis workis used to produce a partial vacuum inchemical laboratories by connecting avessel to a tube through which water isrunning rapidly.

INTRODUCTION In fluid dynamics, Bernoulli's principle states that for an inviscid flow of a non-conducting fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.[3]

BERNOLLIS EQUATION

DIMENSIONS

DERIVATION

Energy form:HEAD FORM:Pressure form:

APPLICATIONSIn modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid[25] and a small viscosity often has a large effect on the flow.

Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lifting force.[26][27] Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equations[28] established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli's principle does not explain why the air flows faster past the top of the wing and slower past the underside. See the article on aerodynamic lift for more info.

The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli's principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator, which determines the dynamic pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure.[29]The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, showing that Torricelli's law is compatible with Bernoulli's principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice.[30]The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

Condensation visible over the upper surface of anAirbus A340wing caused by the fall in temperature accompanyingthe fall in pressure, both due to acceleration of the air.

APPLICATION IN PUMPS: Volute in the casing of centrifugal pumps converts velocity of fluid into pressure energy by increasing area of flow.The conversion of kinetic energy into pressure is according to the Bernoulli equation.

SIZING OF PUMPS:

EJECTORS:Ejectors are designed to convert the pressure energy of a motivating fluid to velocity energy to entrain suction fluid and then to recompress the mixed fluids by converting velocity energy back into pressure energy. Ejectors are composed of three basic parts: a nozzle, a mixing chamber and a diffuser. The diagram illustrates a typical ejector

PITOT TUBE

CARBURETOR:The carburetor works onBernoulli principle: the faster air moves, the lower itsstatic pressure, and the higher itsdynamic pressure.Thethrottle(accelerator) linkage does not directly control the flow of liquid fuel. Instead, it actuates carburetor mechanisms which meter the flow of air being pulled into the engine. The speed of this flow, and therefore its pressure, determines the amount of fuel drawn into the airstream.

SIPHON

VIDEO Lecture

NUMERICAL Problem # 1:Water at a gauge pressure of 3.8 atm at street level flows in to an office building at a speed of 0.06 m/s through a pipe 5.0 cm in diameter. The pipes taper down to 2.6cm in diameter by the top floor, 20 m above. Calculate the flow velocity and the gauge pressure in such a pipe on the top floor. Assume no branch pipe and ignore viscosity.

Solution:

By continuity equation:

v2= (A1v1) / A2= ( (5.0 / 2)2(0.60) ) / ( (2.6 / 2)2)

v2= 2.2 m/s

By Bernoullis Equation:

P1+ gh1+ (v1)2= P2+ gh2+ (v2)2 (Po = atmospheric pressure)

P2= (3.8 x Po) + Po + (1000)(0.6)2 (1000)(9.8)(20) (1000)(2.2)2

P2= 2.8 x 105Pa

#2 ProblemThe diameter of a pipe changes from 200mm at a section 5m above datum to 50mm at a section 3m above datum. The pressure of water at first section is 500kPa. If the velocity of the flow at the first section is 1m/s, determine the intensity of pressure at the second section.

Given,

Let, = Velocity of flow at section(2), and = Pressure at section(2).We know that area of the pipe at section(1),

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