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Bernoulli Principle

TABLE OF CONTENTS

1)	    Introduction 2)	     Derivation 3)	    Use Of Bernoulli Equation 4)	     Real World Application 5)	    Bernoulli Principal Demonstration 6)	     Limitations Of Bernoulli Principle 7)	    Reference

Introduction In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers. Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere. Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.[ Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

Derivation of Bernoulli Equation

Bernoulli equation for incompressible fluids The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects. The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe. Define a parcel of fluid moving through a pipe with cross-sectional area "A", the length of the parcel is "dx", and the volume of the parcel A dx. If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρ A dx. The change in pressure over distance dx is "dp" and flow velocity v = dx / dt. Apply Newton's Second Law of Motion Force =mass×acceleration and recognizing that the effective force on the parcel of fluid is -A dp. If the pressure decreases along the length of the pipe, dp is negative but the force resulting in flow is positive along the x axis.

In steady flow the velocity field is constant with respect to time, v = v(x) = v(x(t)), so v itself is not directly a function of time t. It is only when the parcel moves through x that the cross sectional area changes: v depends on t only through the cross-sectional position x(t). With density ρ constant, the equation of motion can be written as by integrating with respect to x where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa. In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's principle was inherently derived by a simple manipulation of the momentum equation.

A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (s), and cross-sectional area. Note that in this figure elevation is denoted as h, contrary to the text where it is given by z. Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that the change in the kinetic energy Ekin of the system equals the net work W done on the system; Therefore, the work done by the forces in the fluid = increase in kinetic energy. The system consists of the volume of fluid, initially between the cross-sections A1 and A2. In the time interval Δt fluid elements initially at the inflow cross-section A1 move over a distance s1 = v1 Δt, while at the outflow cross-section the fluid moves away from cross-section A2 over a distance s2 = v2 Δt. The displaced fluid volumes at the inflow and outflow are respectively A1 s1 and A2 s2. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so ρ A1 s1 and ρ A2 s2. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm:

The work done by the forces consists of two parts: •	The work done by the pressure acting on the areas A1 and A2 •	The work done by gravity: the gravitational potential energy in the volume A1 s1 is lost, and at the outflow in the volume A2 s2 is gained. So, the change in gravitational potential energy ΔEpot,gravity in the time interval Δt is Now, the work by the force of gravity is opposite to the change in potential energy, Wgravity = −ΔEpot,gravity: while the force of gravity is in the negative z-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δz = z2 − z1, while the corresponding potential energy change is positive.[23] So: And the total work done in this time interval is The increase in kinetic energy is Putting these together, the work-kinetic energy theorem W = ΔEkin gives: or After dividing by the mass Δm = ρ A1 v1 Δt = ρ A2 v2 Δt the result is

or, as stated in the first paragraph: (Eqn. 1), Which is also Equation (A) Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle: (Eqn. 2a) The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed when arriving at elevation z = 0. Or when we rearrange it as a head: The term v2 / (2 g) is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion. The hydrostatic pressure p is defined as, with p0 some reference pressure, or when we rearrange it as a head: The term p / (ρg) is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head. (Eqn. 2b) If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms: (Eqn. 3)

We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system

Use of Bernoulli Principle

Real-world application In 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 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. 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 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. To understand why, it is helpful to understand circulation, the Kutta condition, and the Kutta–Joukowski theorem. •	The Dyson Bladeless Fan (or Air Multiplier) is an implementation that takes advantage of the Venturi effect, Coandă effect and Bernoulli's Principle. •	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. •	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.

•	In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its extension were recently (2009) developed. It was proved that the depth-averaged specific energy reaches a minimum in converging accelerating free-surface flow over weirs and flumes (also). Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics.

Bernoulli’s Principal Demonstration Materials: •	Two ping pong balls •	String •	scissors •	tape •	table or other flat surface •	one straw for each scout Instructions: 1.	Cut two pieces of string, each about six inches long 2.	Using the tape, attach one end a piece of string to a ping pong ball. Repeat with the second piece of string and the second ping pong ball. 3.	Tape the loose end of each string to the table top so that the ping pong balls are hanging next to each other. The gap between the balls should be about one centimeter (half inch). 4.	Read Bernoulli’s principal: “The pressure of a moving gas decreases as its speed increases.” 5.	State the problem being explored. “What will happen if you blow between the ping pong balls?” 6.	Have the scouts make a hypothesis for the problem. Will the balls move further apart, closer together, or not move at all? 7.	Using their straws, have them blow a stream of air between the balls one at a time. The balls should move closer together and bounce off of each other. Explanation The air being blown between the ping pong balls is moving faster than the air on the opposite sides of the balls. So the air pressure between the balls is less that the air pressure on the outside. This difference in pressure causes a force on the ping pong balls which causes them to move toward each other. This is the same principal which causes lift in an airplane wing.

Limitation of Bernoulli's Equation The Velocity of Liquid particle in the centre of a pipe is maximum and gradually decreases towards the walls of the pipe due to friction. Thus while using Bernoulli's Equation,only the Mean Velocity of the Liquid should be taken into account because the Velocity of Liquid particles is not uniform.As per assumption it is not practicle. There are always some external Forces acting on the Liquid,which affects the Flow of Liquid.Thus while using Bernoulli's Equation,all such external forces are neglected which is not happened in actual practise.If some Energy is supplied to or extracted from the Flow,same should also taken into account.

In Turbulent Flow some Kinetic Energy is converted into Heat Energy and in a Viscous Flow some Energy is lost due to Shear Forces.Thus while using Bernoulli's Equation all such losses should be neglected,which is not happened in actual practise.

If the Liquid is Flowing through curved path,the Energy due to Centrifugal Forces should also be taken into account.

Reference www.princeton.edu wikipedia.org/wiki/Bernoulli%27s_principle easycalculation.com/theorems/bernoulli.php www.meritnation.com/applications-of-bernoullis-theorem-explain-in-detail-plzzz www.scribd.com/doc/Full-Report-Bernoulli-Experiment-Tiqa www.helpwithassignment.com/blog/bernoulli-theorem-in-chemical-engineering-from-helpwithassignment-com/

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