Njuguna Kimani: Bernoulli's principle (The velocity and pressure)

Bernoulli's principle states that as the velocity of a fluid increases, the pressure exerted by that fluid decreases

Its application is wide:

a) Airplanes get their lift by taking advantage of Bernoulli's principle.

b) Race cars employ Bernoulli's principle to keep their rear wheels on the ground while traveling at high speeds.

The Continuity Equation relates the speed of a fluid moving through a pipe to the cross sectional area of the pipe. It says that as a radius of the pipe decreases the speed of fluid flow must increase and visa-versa.

In fluid dynamics, Bernoulli's principle argues that for an inviscid flow, increase in speed of fluid will simultaneously cause a decrease in pressure or a decrease in the fluid's potential energy.Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.

Incompressible flow equation

Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation – in its incompressible flow form – can not be assumed to be valid. However if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas. According to the gas law, an isobaric (An isobaric process is a thermodynamic process in which the pressure stays constant) or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature, however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

In most flows of liquids, and of gases at low Mach number (

Ma

M

) is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure. It is commonly used to represent the speed of an object when it is traveling close to or above the speed of sound.

$\ M = \frac {{V}}{{a}}$

where

$\ M$ is the Mach number

$\ V$ is the relative velocity of the source to the medium and

$\ a$ is the speed of sound in the medium

the mass density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. For this reason the fluid in such flows can be considered to be incompressible and these flows can be described as incompressible flow. According to Bernoulli's experiments on liquids his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline where gravity is constant, is:

${v^2 \over 2}+gz+{p\over\rho}=\text{constant}$

where:

$v\,$ is the fluid flow speed at a point on a streamline,

$g\,$ is the acceleration due to gravity,

$z\,$ is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,

$p\,$ is the pressure at the chosen point, and

$\rho\,$ is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:

${v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}$

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply:

the fluid must be incompressible – even though pressure varies, the density must remain constant along a streamline;
friction by viscous forces has to be negligible.

By multiplying with the fluid density

ρ

, equation (A) can be rewritten as:

$\tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\, \text{constant}\,$

or:

$q\, +\, \rho\, g\, h\, =\, p_0\, +\, \rho\, g\, z\, =\, \text{constant}\,$

where:

$q\, =\, \tfrac12\, \rho\, v^2$ is dynamic pressure,

$h\, =\, z\, +\, \frac{p}{\rho g}$ is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head) and

$p_0\, =\, p\, +\, q\,$ is the total pressure (the sum of the static pressure p and dynamic pressure q).

The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head H:

$H\, =\, z\, +\, \frac{p}{\rho g}\, +\, \frac{v^2}{2\,g}\, =\, h\, +\, \frac{v^2}{2\,g},$

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

The Real-world application of the Bernoulli's Principle

In modern 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 you know the behavior of the fluid flow in the vicinity of the foil. 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 lift force.Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated 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.

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.

In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its extension were recently developed. It was proved that the depth-averaged specific energy reaches a minimum in converging accelerating free-surface flow over weirs and flumes. 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.

The principle also makes it possible for sail-powered craft to travel faster than the wind that propels them . If the wind passing in front of the sail is fast enough to experience a significant reduction in pressure, the sail is pulled forward, in addition to being pushed from behind. Although boats in water must contend with the friction of the water along the hull, ice sailing and land sailing vehicles can travel faster than the wind .

Sources of Information;

Daniel Bernoulli (1738). Hydrodynamica.

Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations.

Hydrodynamica. Britannica Online Encyclopedia. Retrieved 2011-01-28

Njuguna Kimani

Thursday, January 27, 2011

Bernoulli's principle (The velocity and pressure)

Incompressible flow equation

The Real-world application of the Bernoulli's Principle

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