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Bernoulli's principle

كلية الهندسة

22-10-2007, 08:02 PM
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Bernoulli's principle

Bernoulli's equation redirects here, see Bernoulli differential equation for an unrelated topic in ordinary differential equations.
Bernoulli's principle states that in fluid flow, an increase in velocity occurs simultaneously with decrease in pressure. This principle is a simplification of Bernoulli's equation which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path. It is named after the Dutch/Swiss mathematician/scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others. In a fluid flow with no viscosity, and therefore one in which a pressure difference is the only accelerating force, it is alent to Newton's laws of motion. It is important to note that the only cause of the change in fluid velocity is the difference in pressures either side of it. It is very common for the Bernoulli effect to be quoted as if it states that a change in velocity causes a change in pressure. The Bernoulli principle does not make this statement and it is not the case.


Examples used to strate the effect
One common and correct way of understanding how an airfoil develops lift relies upon the pressure differential above and below a wing. In this model the pressures can be calculated by finding the velocities around the wing and using Bernoulli's equation. However, this explanation often uses false information, such as the incorrect assumption that the two parcels of air which separate at the leading edge of a wing must meet again at the trailing edge, and the assumption that it is the difference in air speed that causes the changes in pressure.

Venturis
A common model used to strate the Bernoulli effect is a convergent, divergent nozzle also called a venturi. This is simply a large diameter tube feeding into a smaller diameter tube and then further feeding into another larger tube. Venturis are easier to understand when considering a gas rather than a liquid, but the functions for either are much the same. In order for any gas flow to occur it is essential that the exit pressure is lower than the entry pressure for this system. This pressure difference causes the fluid to accelerate from the intake larger tube into the smaller tube. The stored spring energy available to the fluid because of the pressure difference results in the fluid not only expanding as it goes from higher to lower pressure, but effectively overshooting in its expansion as a result of the mass of the gas particles and compressibility of the gas, springing apart beyond the point where all the forces would be balanced. Before the fluid can spring back, there is more fluid behind it, also at this lower pressure. This first sample of fluid then has no pressure difference either side of it to cause it to spring back. This part of the fluid then remains at a lower pressure until it merges with the slower fluid in the exit tube. The pressure in the exit tube will be higher than that in the smaller middle tube, and so the fluid moving from the smaller to larger tube is slowed down by this pressure difference.

Venturi effect and carburetors
Bernoulli's principle can be used to analyze the venturi effect that is used in carburetors and elsewhere. In a carburetor, air is passed through a Venturi tube to increase its speed and by the mechanisms explained above, decrease its pressure. The low pressure air is routed over a tube leading to a fuel bowl. The low pressure sucks the fuel into the airflow so that the combined fuel and air can be sent to the engine. The pressure reduction is proportional to the rate of air flow squared, so that more fuel is sucked in as the air flow increases, and the fuel:air mixture remains roughly the same proportion over a wide range of speeds. The pressure reduction effect can be observed by blowing over the top end of a straw with the bottom of the straw in a container of water; the water level will rise in the straw as the flow over the top of the straw increases in speed.

Bernoulli equations
There are typically two different formulations of the equations; one applies to incompressible flow and the other applies to compressible flow.

Incompressible flow
The original form, for incompressible flow in a uniform gravitational field (such as on Earth), is:

Bernoulli's principle
v = fluid velocity along the streamline
g = acceleration due to gravity on Earth
h = height from an arbitrary point in the direction of gravity
p = pressure along the streamline
ρ = fluid density
These assumptions must be met for the equation to apply:

Inviscid flow − viscosity (internal friction) = 0
Steady flow
Incompressible flow − ρ = constant along a streamline. Density may vary from streamline to streamline, however.
Generally, the equation applies along a streamline. For constant-density potential flow, it applies throughout the entire flow field.
The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.

Compressible flow
A second, more general form of Bernoulli's equation may be written for compressible fluids, in which case, following a streamline:


Bernoulli's principle
Bernoulli's principle = gravitational potential energy per unit mass, Bernoulli's principle in the case of a uniform gravitational field
Bernoulli's principle = fluid enthalpy per unit mass, which is also often written as Bernoulli's principle which conflicts with the use of Bernoulli's principle in this article for "height"). Note that Bernoulli's principle here is the fluid thermodynamic energy per unit mass, also known as the specific internal energy or "sie".
The constant on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When shock waves are present, in a reference frame moving with a shock, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy

Derivation

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.

Bernoulli's principle

Applying conservation of energy we find that:

the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy.
The work done by the forces is

Bernoulli's principle
The decrease of potential energy is
Bernoulli's principle

The increase in kinetic energy is

Bernoulli's principle

Putting these together,

Bernoulli's principle
or
Bernoulli's principle

After dividing by Δt, ρ and A1v1 (= rate of fluid flow = A2v2 as the fluid is incompressible):

Bernoulli's principle

or, as stated in the first paragraph:

Bernoulli's principle

Further division by g implies
Bernoulli's principle

A free falling mass from a height h (in vacuum), will reach a velocity

Bernoulli's principle

The hydrostatic pressure or static head is defined as

Bernoulli's principle

A way to see how this relates to conservation of energy directly is to multiply by density and by unit volume (which is allowed since both are constant) yielding:

Bernoulli's principle

Compressible fluids
The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A1 is equal to the amount of mass passing outwards through the boundary defined by the area A2:

Bernoulli's principle

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A1 and A2 is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,

Bernoulli's principle

where ΔE1 and ΔE2 are the energy entering through A1 and leaving through A2, respectively.

The energy entering through A1 is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical work:

Bernoulli's principle

A similar expression for ΔE2 may easily be constructed. So now setting 0 = ΔE1 − ΔE2:

Bernoulli's principle

which can be rewritten as:

Bernoulli's principle

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

Bernoulli's principle

which is the Bernoulli equation for compressible flow.

المصدر
http://en.wikipedia.org

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Bernoulli's principle

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