Bernoulli's law describes the behavior of a fluid under varying ...

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24 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Bernoulli’s equation

Bernoulli's equation

describes the behavior of a fluid under varying
conditions of flow and height. It states


(1)

where P is the
static pressure

(in Newtons per square meter),


is the fluid
density (in kg
per cubic meter),
v

is the velocity of fluid flow (in meters per
second) and
h

is the height
above a reference surface. The second term in
this equation
,
,

is
also
known as the
dynamic pressure
.


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.


Streamlines are the family of curves that are instantaneously tangent t
o the
velocity vector of the flow. This means that if a point is picked
,

then at that
point
,

the flow moves in a certain direction. Moving a small distan
ce along
this direction and

finding out where the flow now points would draw out a
streamline.


Let’s consider

a pipe through which and ideal fluid is flowing at a steady
rate. Let
W

denote the work done by applying a pressure
P

over an area
A
,
producing an offset
, or volume change

. Let a subscript 1 denote
fluid
element
s at an initial

point down the pipe, and a subscript 2 denote
fluid
element
s further down the pipe. Then the work done by pressure force


(2)

at points 1 and 2 is




(3)




(4)


and the difference is


(5)


Equating this with the change in total energy (written as the sum of kinetic
and potential

energies gives










(6)


Equating (6) and (5
),


(7)

which, upon rearranging, gives


(8)

so writing the density a
s


=

m/

V
then gives



(9)

This quantity is constant for
all points along the streamline.

This

is an expression of the law of conservation of mechanical energy in

a
form more convenient for fluid mechanics.




Venturi Meter



The Venturi meter is a device for
measuring flowrate in a pipe. It
consists of a rapidly converging
section which increases the
velocity of flow and hence reduces
the pressure. It then retu
rns to the
original dimensions of the pipe by
a gently diverging 'diffuser'
section. By measuring the pressure
differences the flowrate can be
calculated. This is a particularly
accurate method of flow
measurement as energy losses are very small.


Applying

Bernoulli along the streamline from point 1 to point 2 in the
narro
w
throat
of th
e Venturi meter we have


By the using the continuity equation we can eliminate the velocity u
2
,



Substituting this into
and rearranging the Bernoulli equation we get



To get the theoretical flowrate this is multiplied by
the area. To get the
actual flowrate taking in to account the losses due to friction, we include a
coefficient of discharge
.
A d
ischarge coefficient

of 0.975 may be taken as
standard, but the value varies noticeably at low values of the Reynolds
number.




This can also be expressed in terms of the manometer readings


Thus the flowrate can be expressed in terms of the manometer reading:



Notice how this expression does not include any terms for the elevation or
orientation (
z
1

or

z
2
)

of
the Venturi meter. This means that the meter can be
at any convenient angle to function.


The purpose of the diffuser in a Venturi meter is to assure gradual and
steady deceleration after the throat. This is designed to ensure that the
pressure rises agai
n to something near to the original value before the
Venturi meter. The angle of the diffuser is usually between 6 and 8 degrees.
Wider than this and the flow might separate from the walls resulting in
increased friction and energy and pressure loss. If th
e angle is less than this
the meter becomes very long and pressure losses again become significant.
The efficiency of the diffuser of increasing pressure back to the original is
rarely greater than 80%.