Good morning, I welcome you all to this session
of fluid mechanics. In the last we started
the discussion on flow meters and we, if we
recall we discussed that there are three types
of flow meters namely venturi meter, orifice
meter and flow nozzles which work on the same
basic principle. Flow meter means if you want
to measure the flow rate of a fluid flowing
through a fluid circuit.
Well to measure the flow rate this three flow
meters venturi meter, orifice meter and flow
nozzle, they work on the same basic principle
that these meters provide a geometrical change
to the flow of fluid usually a coaxial contraction
to the path of the fluid flow. So, that the
pressure drop is registered in the flow of
fluid between two sections and at those two
sections the pressure drop is measured by
a pressure measuring instrument which is usually
a manometer. Then by the straight forward
application of Bernoulli’s equation, we
derive a equation in mathematical expression
relating the flow rate and the pressure drop
and flow rate is found out. So, let us discuss
one by one these flow meters as the applications
of fluid flow.
So, let us discuss first the venturi meter
venturi meter venturi meter. This name of
the venturi meter comes from the name of an
Italian scientist Venturi. Venturi is the
name of the scientist. So, a venturi meter
looks like this, well it looks like this.
If you draw a venturi meter it will look like
this, it is like this.
That means a venturi meter consist of a convergent
and a divergent. A convergent that means two
conical part rather it is better to tell two
conical part connected with a straight portion.
Now, the characteristic feature is that one
conical part is of shorter length that means
this one is of shorter length, this one is
of shorter length, this part whereas, another
conical part is of higher length that means
this cone angle is high whereas, this cone
angle is low, of high length, higher length.
Usually this venturi meter is installed in
the fluid flow path in this way that the flow
direction is like this, this is the direction
of flow. That means the shorter length part
higher convergence angle is a convergent dart.
Higher cone angle conical part it becomes
a convergent dart that means when fluid flows
these acts as a convergent data whereas the
longer length part that is a conical part
with a lower cone angle acts as a divergent
dart. Why it is made? It is very important
sometimes, it is asked in many interviews
that if you are given a venturi meter two
conical parts which way will you place. You
will place in such a way that the fluid flow
direction is such that it first goes through
the shorter length that is the higher cone
angle conical part and then lower cone angle
conical part or the vice versa it is done
like that.
This is because in this case when the flow
takes place through a convergent data whether
the angle of convergence is high or low, it
is an accelerating flow. Why? The velocity
increases, the area decreases in the direction
of flow. Therefore, from the continuity the
velocity increases that means this is an accelerating
flow as a consequence of Bernoulli’s equation
pressure decreases. Therefore, always pressure
at the upstream is higher than the downstream
which means the fluid is flowing in a, with
a favorable pressure gradient that means the
pressure force to any fluid element, this
is always in the direction of the flow. In
this case the separation loss as we discussed
earlier does not take place that means the
fluid particle near to the wall cannot flow
back.
Whereas in the divergent dart when the fluid
flow takes place you see from the continuity
since the area increases what happens the
velocity decreases and hence the pressure
increases. Therefore, in the direction of
flow there is an increase in pressure which
means the fluid is flowing against a, against
an adverse pressure gradient which means that
in a fluid element the pressure forces is
always in the direction opposite to the flow,
but still the fluid flows because of the energy
gradient.
So, in this case the fluid particles very
near to the wall which looses their kinetic
energy, which lose their kinetic energy because
of the friction, fluid friction then cannot
overcome these adverse pressure hill and they
flow back with the favorable pressure forces.
That means the pressure force is in this direction
that is the separation loss we discussed yesterday.
So therefore, the separation loss is avoided
or it is minimized if the angle of divergence
in the divergent part of the flow is made
very less. Usually in all design the divergence
angle is made between 8 to 10 degrees. Therefore,
you can make a very high convergence angle,
but we cannot make a very high divergence
angle whenever there is a divergent dart in
a hydraulic circuit or a fluid circuit the
angle of divergent should be limited to 8
to 10 degree, you understand 8 to 10 degree,
so that the separation loss is minimized.
This is the reason for which the upstream
part corresponds to the rapid higher cone
angle or a shorter length and the downstream
part corresponds to a lower cone angle higher
length.
Now, what happens throat is this, this section
is known as throat throat T H R O A T throat,
this is upstream, this is downstream section.
This is throat T H R O A T throat. Throat
is the minimum section, minimum section means
sorry minimum area minimum cross section,
minimum area section. Now, as the fluid flows
through this in this direction the minimum
maximum velocity occurs at the throat that
is the minimum area part. This is a very short
straight portion which connects these two
conical part.
Similarly, the pressure here is the minimum
velocity here is the maximum according to
continuity pressure here is the minimum. So,
if we measure the delta p between these two
there is a pressure difference between these
two section, let this section is 1, let throat
section is 2, p 1 is greater than p 2. We
measured the pressure difference and find
out the flow rate. How to do it? Let us consider
a practical installation. Now, let us consider
this venturi meter is installed in a pipe
line where the fluid is flowing, venturi meter
as a instrument is installed with the flinch
it is connected in the pipe line, this is
a practical thing.
Now, let us consider the pipe is inclined
in a vertical plane, in a general. Let us
consider the most general case let this be
the 
let this be the pipe. This is the pipe, let
this be the pipe, let this be the pipe and
this part is the venturi meter which is inclined
to the which is sorry installed in the pipe.
Pipe is inclined in a vertical plane. Therefore,
this is the section 1 of the venturi meter
upstream where the fluid flow is uniform then
it comes to the convergent dart, this is the
direction of the flow, then it attains a minimum
velocity. This is the throat, let this be
the axis and this is the downstream part.
So, venturi meter is installed within in the
pipeline which is an inclined pipeline transporting
water in this direction in a vertical plane.
Now, what happens if this section is denoted
as 1 and if this section is denoted as 2.
Then we can write from the Bernoulli’s equation
between these two section let the pressure
is p 1 velocity is V 1 the throat section
let the pressure is p 2 and the velocity is
V 2 and if we consider the z 2 as the elevation
head and here z 1 as the elevation head from
any reference datum then I can write the Bernoulli’s
equation p 1 by rho plus V 1 square by 2 rather
p 1 by rho g we write in terms of the head
plus z 1. Considering the in viscid fluid
I can write p 2 by rho g plus V 2 square by
2 g plus z 2 without any loss that means considering
the fluid to be in viscid Bernoulli’s equation
between this two point 1 and 2.
Then what we can write, now after this what
we can write V 1 and V 2 may be connected
through the continuity equation. Let us write
that V 1 A 1 is equal to V 2 A 2. We can write
the continuity equation where area is A 1
that is the area of the pipe which matches
this base diameter of the venturi meter, the
same area here A 1, but the throat area of
the venturi meter is A 2. It is very simple
V 1 A 1 is V 2 A 2, it is from the continuity
under steady state the same flow rate passes
through all sections, then if I just eliminate
V 2 then what we will get V 2 is V 1 A 1 by
A 2 that means we can write V V 2 is V V 1
is V 2 A 2 by A 1. Therefore, we can write
V 2 square by 2, g V 2 square by 2 g we take
here 1 minus what is V 1? A 2 square by A
1 square that is V 1 is equal to P 1 by rho
g plus z 1, let this minus p 2 by rho g plus
z 2.
Let us define this p 1 by plus z 1 as h 1
star and this is defined as h 2 star. So,
we can write V 2 square is h 1 star minus
h 2 star, this is h 1 star, this is h 2 star,
2 g of course, 2 g into 1 minus A 2 square
by A 1 square, alright or we can write V 2
is equal to root over 2 g h 1 star minus h
2 star divided by root over 1 minus A 2 square
by A 1 square. This is very simple. Now, this
is very simple algebraic steps, but this h
1 star and h 2 star represents the pressure
head plus the datum head at the inlet and
this is the pressure head plus the datum head.
This as a whole is known as piezometric pressure
head.
Piezometric pressure we have already discussed
what is piezometric pressure that takes care
of both the pressure plus the corresponding
equivalent pressure due to a static pressure
due to the height of the liquid. That means
in terms of head, it is the pressure head,
pressure energy per unit weight plus the datum
head that means due to the height of the liquid
from any arbitrary datum.
So, combination of p 1 by rho g plus z 1 is
the piezometric pressure head. So, we can
write in terms of piezometric pressure head.
Why we write in terms of this? There is a
meaning to it, mathematically here it may
not be very meaningful that why unnecessarily
we are writing h 1 let us better recognize
p 1 by rho g p 2 by rho g separately with
z 1 z 2 the elevation at 1 and 2, no, it should
be written in terms of this. This is because
in fact what is done we attach a manometer
between this two points to to register the
pressure difference.
Now, let us consider any manometric liquid
since p 1 is greater than p 2. Therefore,
the manometric liquid will manometric liquid
will take a position like that where this,
this is the deflection delta h. Now, if I
write the manometric equation and consider
the manometer, manometric fluid density as
rho m, consider the manometric fluid density
as rho m, alright. Consider this as rho m
then we can write the manometric equation
considering here the pressure p 1 now we write
again if we recall the hydrostatics that p
1 plus here the rho, rho is the density of
the working fluid that the fluid flowing through
the pipe and the venturi meter g into z 1.
Let we consider this level as the level 0
0 where or O 0 0 where we are finding out
the pressure from both the sides and equating
it and let the elevation is z 0, it is easy
if we just define the elevations like that
from a reference datum to find out the height.
That means this height is z 1 minus z 0. So,
this is the pressure from this side. What
is the pressure from this side is equal to
p 2 plus rho g z 2 minus z 0 minus delta h,
z 2 minus the elevation head minus z 0 elevation
head this one minus delta h minus delta h.
So, this is the pressure head due to this
plus this one plus rho m g delta h. Well,
if you now write it, it will be like this
p 1 minus p 2 is equal to so rho g z 0 and
rho g z 0 will be cancelled. So, rather not
p 1 minus p 2, I am very sorry my entire objective
is different. So, it will be p 1 plus rho
g z 1, I am sorry. So, it will be now p 1
plus rho g z 1. So, rho g z 0 rho g z 0 cancels
minus p 2 plus rho g z 2 is equal to this
and here also rho g delta h that means rho
m, this already we recognized earlier delta
h that means p 1 plus rho g z 1 and p 2 plus
rho g z 2 these are the piezometric pressures
not the static pressure.
This is the static pressure or the pressure
this is the piezometric pressure plus the
equivalent pressure for the height z 1 z 2
measured from any arbitrary reference datum.
So, if you divide it by rho g in terms of
head that is p 1 by rho g sorry plus z 1 minus
p 2 plus p 2 by rho g sorry p 2 plus rho g
plus z 2 is equal to then rho m by rho minus
1 delta h which means that the difference
in piezometric pressure head is nothing but
rho m by rho minus delta h. So, now this rho
m minus rho by delta h is ultimately replaced
here h 1 star minus h 2 star, that means if
we write, if you replace this, if you just
recall this equation…
This becomes V 2 is equal to root over 2 g,
again I am writing this h 1 star minus h 2
star by root over 1 minus A 2 square by A
1 square. We write V 2 is equal to root over
2 g h 1 star minus h 2 star is rho m by rho
minus 1 into delta h and they are root over
1 minus A 2 star by A 2 star.
Now, thing is that when the venturi meter
is horizontal that means this pipe is horizontal
and venturi meter is horizontal, in that case
what will happen h 1 star minus h 2 star is
h 1 minus h 2 where h 1 is equal to p 1 by
rho g and h 2 is equal to p 2 by rho g. That
means z 1 z 2 is different and in that case
delta h into rho m by rho minus 1 will give
you the value of h 1 minus h 2 which means
that if you write the equation for V 2 in
terms of the deflection of the manometer then
it is whether the venturi meter is inclined
or horizontal.
This is a very important point, but if you
write the equation in terms of p and z 1 then
it is very important because whether z 1 z
2 will be cancelled or not it is very important,
if you write here instead of h 1 star h 2
as p 1 by rho g plus z 1 p 2 by g because
in that case h 1 star minus h 2 star is not
always p 1 by rho g minus p 2 by rho g, if
it is a horizontal h 1 star minus h 2 star
is h 1 minus h 2 then p 1 by rho g minus p
2 by rho g.
Otherwise, it will be p 1 by rho g plus h
1 minus p 2 by rho g plus h 2, you have to
be very careful whether it is inclined or
horizontal, but if you straight forward write
the V 2 formula in terms of the deflection
of the mercury then it is immaterial because
the manometric equation which we discussed
earlier is such that if it is written in terms
of the deflection of the manometric fluid
then these equation represents the difference
of piezometric pressure between the two points
between which the manometer ends are connected.
So, whether there is the horizontal plane
or is displaced vertically it does not matter.
Ultimately these value gives the difference
in the piezometric pressure that means difference
in static pressure plus the equivalent pressure
due to the elevation head. Therefore, it is
immaterial whether the venturi meter is horizontal
or not pipeline is horizontal or not if you
use this equation, alright. Now, if I write
these equation again.
What we get? We get V 2 is equal to root over
2 g. So, this is a very very important conclusion,
sometimes it is asked even in the interviews
that will you be very careful if the venturi
meter installation is not perfectly horizontal.
The answer will be no because if I write the
expression in terms of the mercury deflection
in a manometer, if the mercury is the manometric
fluid deflection of the manometric fluid it
becomes the same equation.
So, now this is the value of the V 2 therefore,
the Q will be flow rate will be V 2 into A
2 if you recall the A 2 is the area where
the V 2 was found out, that means this is
the throat area A 2 where the velocity is
V 2 of flow rate is velocity into the area.
So, that means A 1 A 2 if I simplify then
root over 2 g into rho m by rho minus 1 into
delta h divided by root over A 1 square minus
A 2 square.
Now, see that I can find out the flow rate
if I know the area of the venturi meter. That
is venturi meter inlet and throat diameters
are known because it is known from the geometry
of the venturi meter and if I know the deflection
of the manometer that means if a manometer
is connected between two points. So, the measuring
value is delta h which is put into these equation
to find out the flow rate, but this equation
will always overestimates the actual flow
rate.
If one measures the actual flow rate for example,
there is a opportunity, there is an opportunity
to measuring the actual flow rate by collecting
the fluid then you will find that by utilizing
this equation with the measured values of
delta h from the manometer deflection will
always overestimate this flow rate. The reason
is such that these equation has been developed
considering the fluid to be in viscid.
So therefore, this pressure difference delta
h manometer difference or the pressure difference
is the pressure difference for the in viscid
fluid, but the actual pressure difference
which is registered in the manometer is more
than that of an in viscid fluid. Now, try
to understand this physically. Why there is
a pressure difference for an in viscid fluid?
This is because in of the difference in velocity.
So, velocity difference because there is a
difference in area. So, due to the change
in area, there is the difference in velocity
or change in momentum.
So, due to this change in momentum or change
in velocity there is a change in pressure
is a mutual conversion between pressure energy
to kinetic energy or kinetic energy to pressure
energy, whatever you call, but in that means
for example, here the kinetic energy is increased.
Therefore, the kinetic energy is decreased
increased at the expense of pressure energy,
pressure energy is decreased.
So therefore, the decrease in pressure energy
that is p 1 minus p 2 will depend upon the
increase in kinetic energy, but for real fluid
over and above there will be an additional
pressure drop because of the fluid friction.
Friction between the fluid and the solid surface
therefore, the pressure drop p 1 minus p 2
will be more. The p 2 will be still lower
than p 1 not only because of increase in velocity,
but also because of the fluid friction loss.
Therefore, actual pressure drop in a real
fluid in this situation is more than that
what is estimated by a, by an in viscid fluid.
So therefore, delta h is more than actually
what it could have been for the use of this
formula. This formula is determined by considering
the fluid to be in viscid. Therefore, Q is
giving a higher value.
So, for that we write that Q actual, this
is adjusted by multiplying this with a coefficient
that is known as coefficient of discharge.
So, coefficient of discharge is now very clear.
So, this is known as that means if we calculate
from this formula, the flow rate, this gives
you a higher flow rate; this is sometimes
known as theoretical flow rate. That means
flow rate which is calculated by considering
the fluid to be in viscid, but substituting
the pressure drop from the real fluid measurement.
So therefore, this gives a higher flow rate.
So, this is multiplied by a coefficient of
discharge C d and this coefficient of discharge
depends upon the flow rate depends upon the
area A 1 and A 2 of the venturi meter, that
means venturi meter geometry and flow rate,
but this dependency is very very weak and
at a higher flow rate usually C d becomes
constant and it depends upon mainly the Reynolds
number of flow which I will tell you afterwards,
you do not know this thing, when you will
start the viscous flow Reynolds number, you
will see that there is a number known as Reynolds
number, dimensionless number, characterizing
the viscous flow depends upon mainly the Reynolds
number.
But you must know at this stage that higher
flow rate for all venturi meter and within
all ranges of flow C d is virtually constant
and lies between a value of 0.95 to 0.98 that
means it is very very high. Usually it is
made of brass the venturi meter and surface
is so polished that the friction between fluid
surface and the solid is so low so 0.95 to
0.98, this value of C d.
Now, the only job for measuring the fluid
flow by a venturi meter is to calibrate the
venturi meter. What is the calibration? Calibration
means to know accurately the value of C d.
How do you know the value of C d? I give you
a venturi meter and tell you use the venturi
meter, the venturi meter C d value is 0.99,
you may doubt actual the venturi meter which
I have given so rough surface and so badly
designed it is value of C d is 0. 7 so there
will be a tremendous discrepancy in the result
if we use 0.9 as C d and 0.7. So, what you
do?
You calibrate the venturi meter. What is meant
by calibration? You find out the value of
C d that means you find the flow rate by any
other measuring device which is more accurate
than the venturi meter which is usually a
direct collection of flow in case of an hydraulic
circuit, then you measure the flow rate and
you find out the value of C d with known values
of A 1 A 2 and delta h and you find out that
in a range of flow rate where you measure
your flow rate what is the value of C d and
then you establish a relationship between
C d and flow rate which is usually done for
C d and Reynolds number, this relationship
to establish is known as calibration of venturi
meter, which is very important. That means
to find out the exact or accurate value of
C d to use the venturi meter to find the flow
rate from this equation, alright.
Now, let us see the next one is the orifice
meter. Now, when the venturi meter is well
understood orifice meter will not be a problem.
Orifice meter is like this, a orifice meter
is a orifice plate, that is a plate with a
hole at concentric hole at the centre. Now,
we consider a pipe, the orifice meter is like
that. Let me draw a orifice meter with a,
orifice meter is like this a little beveled
side. So, this is an orifice meter. So, orifice
meter if you see so this will be a circular
plate. So, this orifice meter with a concentric
hole.
So, this is simply a circular plate which
have a thickness which is very thin and a
sharp edged orifice with little beveling like
this which is known as orifice meter, this
is orifice meter. It is very simple in seeing
you see a plate, a plate circular plate, a
thin circular plate with a concentric hole
within it, a sharp edged concentric hole which
is known as orifice. So, this sharp edged
hole is known as orifice. So, this is an orifice.
So, this is placed in a pipe. So, this gives
when it is placed in the pipe where there
is a flow and flow has to be measured what
happens? This gives an obstruction to the
flow. So, what happens is the fluid streamline
behaves like this and there is a vena contracta
as I have already told you so these are the
streamlines. These are the streamlines. So,
this is one streamline centre of so these
are the streamlines.
So, what happens streamlines contracts because
now there is an area contraction that means
this placing this orifice meter by flinch
we place the orifice meter between the pipeline
connections to pipeline connection then what
happens these provides a coaxial contraction
to the path of the fluid then fluid streamlines
of the fluid flow just contracts and then
they form a vena contracta, that is the area
minimum cross sectional area section which
is very close to the orifice orifice plate
that means flows downstream. So, this is the
vena contracta vena contracta.
Now, what happens if we take a section here
and if we take a section here, if this section
is 1 and let this section is 2. Let us consider
the area of the pipe is A 1, let us consider
the area of the orifice in the orifice meter
plate is A 0 A 0. Now, this area let us consider
A C. A C is less than A 0 this is a vena contracta
area. Now, you see that V 2 will be V 2 or
V C, this section 2 V is V C. V C is greater
than V 1 or less than V 1? V C is greater
than V 1 because this is the maximum velocity
with the minimum area and p 1 is greater than
p C, p C is less than P 1. So, if we attach
a manometer within this then the manometric
fluid will behave like this.
So, you take this as the delta h. Here also
now if you write the Bernoulli’s equation
between these two points you can write that
p 1 by rho g again if there is an elevation
head at 1 and 2 p 1 by rho g plus z 1 plus
V 1 square by 2 g is equal to, if I write
now one tapping is made at this upstream portion
where the fluid is almost uniform fluid flow
that means usually this is made 1 diameter
upstream if D is the diameter of the pipe,
1 diameter upstream from the orifice plate
and usually this is made, this should be made
theoretically at the vena contracta section,
but it is very difficult to realize the vena
contracta section.
Usually it is made at D by 2 this tends downstream
from the orifice plate. It should not be far
away from the orifice plate, in that case
a delta p, a high value of delta p is very
difficult to be registered because fluid flow
area is again expanding. Therefore, the pressure
is again increasing, you understand.
So, this part is the A D portion as you know,
when the expansion is taking place fluid is
expanding, that is the diverging, streamlines
are diverging there are the flow reversal
zones. Therefore, you now recognize that this
portion if we can attach the manometer at
the vena contrata section we can get the maximum
pressure drop. If we attach the manometer
here we will get a pressure drop which is
equivalent to frictional pressure drop because
pressure drop pressure drop due to momentum
change will not be able to capture.
Therefore, it should be at the vena contracta
section. So, another connection, but it is
difficult to realize where the vena contracta
takes place. So, it is usually made at D by
2 distance downstream from the orifice plate.
So, let this section is 2 so this two sections
better I replace as C section because this
is the contracted section p C by rho g plus
V C square by 2 g rather z C z C if there
is a elevation head difference z C plus V
C square by 2 g considering the fluid to be
in viscid. So, from this we can find out that
V 1 minus V C again the same thing is there.
If you write V 1 A 1 is equal to V C A C.
So, you just write in terms of V C then we
can write it here that V C square by 2 g V
1 square by 2 g V C, V 1 is V 1 A 1 V C A
C by A 1 that means V c square by 2 g similar
way 1 minus V 1 is A C. A C square by A 1
square you follow it with the similar philosophy
is equal to 2 g into p 1 rho g that means
I write h 1 star minus h C star, what is this?
This is the piezometric pressure head at the
section 1, this is the piezometric pressure
head at the section C.
So, that means V C is equal to root over sorry
root over 2 g h 1 star minus h C star divided
by root over 1 minus A C square by A 1 square,
alright. Now, again h 1 star minus h C star
from this manometer equation if rho m is the
density of the manometric fluid will be equal
to root over what is this 2 g h 1 star 2 g
the same thing rho m by rho minus 1 into delta
h that means in terms of the deflection of
the manometric fluid in the manometer, this
will measure the h 1 star minus h C star.
If there is a, if this is horizontal so z
1 and z C equal so h 1 minus h 1 star minus
h C star simply h 1 minus h C that means oh
sorry, I am sorry, so, these two that means
this is p 1 by rho g minus p C by rho g. It
is as simple as that divided by root over
1 minus A C square by A 1 square.
Now, this is V C, alright. Now, one step further
that again with the same logic here itself
I multiply with a coefficient, why? Because
if I substitute delta h from the practical
measurements that measurement for a real fluid
then this is more than what one can expect
for an ideal fluid. Therefore, V C estimated
is more, it is over estimated.
So, V C is corrected with a coefficient. So,
this is known as coefficient of velocity that
means C V into root over 2 g rho m by rho
minus 1 into delta h root over 1 minus A C
square by A 1 square.
So, this is the velocity, expression of the
velocity. Now, if you find out Q? Q is equal
to now this area. Now, you see V C is this
area that is the A C that is the area of contraction
that is Q is equal to A C into C V into root
over 2 g rho m by rho minus 1 into delta h
divided by root over 1 minus A C square by
A 1 square, alright.
Now, A C we can replace as what is A C? C
C into A 0 because we define the coefficient
of contraction at A C by A 0 that means as
you know earlier also we define a coefficient
of contraction that is the ratio of the area
at the vena contracta to the geometrical area
of the orifice or the aperture. So, this is
A C by A 0. So, this can be written as all
now I express in terms of the geometrical
area which we can measure, we cannot measure
A C, but we know the orifice plate diameter
that means diameter of the orifice in the
orifice plate. So, C C A 0 times C V into
root over 2 g rho m by rho minus 1 delta h
into root over 1 minus A C is C C square A
0 square by A 1 square. So, this is the simple
equation, alright.
Then C C into C V we replace as C d. So, we
define C d as C C into C V. C d is the coefficient
of discharge, C d is equal to coefficient
of discharge, discharge. So, if I write this
as C d then we can write this I can write
now you see this.
Q is equal to C d A 0 into root over 2 g rho
m root over 2 g rho m by rho minus 1 divided
by root over 1 minus C C square A 0 square
by A 1 square rather this part I take out
into root over rho m by rho minus 1 into delta
h, alright. Now, this part is fixed for a
orifice meter if I know it’s value of C
d, if I know the value of A 0 that is the
orifice diameter of the orifice meter that
the diameter of the central hole of the orifice
meter 2 g 1 minus C C, if I know the value
of C C for this orifice meter A 0 A 1 if we
know, this can be written as a constant, all
this thing times this one.
This one is varying depending upon the manometer
that means depending upon the manometric fluid
the rho m value will depend upon the manometric
fluid used and the deflection of delta h that
means this C is known as constant of the meter,
constant of orifice meter which is a function
of the geometry of the orifice meter. And
its value C d C C which is again a function
of A 0 A 1 and the flow rate. A 0 A 1 and
the flow rate that means diameter of the orifice
meter, diameter of the orifice of the orifice
meter, hole that means diameter of the orifice
in the orifice meter, diameter of this that
means the area of the pipe in which the fluid
is flowing and the values of C d C C.
So, this is a purely constant for a given
orifice meter of a given pipe diameter and
in a given range of flow usually at a higher
range of flow it is independent of the flow
rate so if I know the value of C that means
the constant of the meter then I can find
out. So, now in using an orifice meter the
main or the pivotal job is to find out accurately
the value of C at different flow rate. So,
that if I use orifice meter I can immediately
find these value from my manometer, if I know
the manometric fluid I can find out rho m,
but when I will find out Q from here I must
know the value.
If the orifice meter constant is given I do
not bother with all these equations root over
2 g by root over 1 minus C C square I simply
multiply it, but what I must know? I must
know very accurately the value of C and to
know very accurately the value of C is or
to establish an accurate value of C within
a range of flow is known as calibration of
orifice meter. So, this is very important
a calibration of orifice meter before it use.
So, usually the value of C for orifice meter,
actually, it is value of C d for orifice meter
varies between whether we can write the value
of C considering this things A 0 A 1 usually
in the range varies from 0.75 to sorry 0.65
0.60 sorry to 0.65, 0.60 to 0.65. This is
the value of C d for the orifice meter. Alright,
now we come to flow nozzle well.
Now, we come to flow nozzle. Flow nozzle is
a intermediate thing between these two. Now,
flow nozzle is like that, it is like a nozzle
that means in the pipeline you just provide
a nozzle like this, you just provide a nozzle
like this that means if you see the sectional
view it will be like this. A nozzle, a nozzle
is attached to the flow that means what happens
it is unlike the orifice meter the, there
is a converging part. So, streamlines are
smoothly converging but what happens unlike
the venturi meter also this part is not a
diverging one so it cannot smoothly diverge.
So, streamlines goes like this. So, so streamline
goes like this that means this is the streamline
pattern and there will be eddy formations,
more eddy formation, more losses will be there.
So, this is a flow nozzle. So, if we make
a connection just at the downstream of the
nozzle and somewhere here in the upstream
we get a pressure drop delta p. Let this section
is 1, this section is 2. So, delta p is p
1 minus p 2, if this is registered by a manometer
same thing delta h we can find out the flow
as a function of delta h, same philosophy.
But here you see the flow nozzle the losses
are more for the venturi meter, why? Because
this part there is no, in this section there
is no divergent part. So, fluid stream after
converging smoothly along the surface they
diverge like this. So, that the formation
of eddies takes place due to separation losses.
Therefore, in this case the total energy loss
in case of this that means if we have a total
energy here, the total energy loss will be
more, but this is counterweighed by its lower
cost whereas, in case of a venturi meter,
if you recollect in case of a venturi meter
if you go like this, the energy loss from
a upstream section to downstream section is
very low because usually the separation loss
is minimized. And this angle is very small.
So, only frictional losses predominates and
that too the surface is so polished so the
friction loss is very small. Nevertheless,
there will be losses, but H 2 is very less
than, the difference between H 2 minus H 1
is very small.
So, energy loss is very small. Wherever, whereas,
in case of an orifice meter the energy loss
is tremendous because it abruptly contracts
and makes the vena contracta and then again
it diverges. So, the energy loss that H 2
H 1 and H 2 is maximum in case of a orifice
meter, but it is the cheapest. So, it is the
costliest, it is the cheapest, energy loss
is minimum, energy loss is maximum and this
falls in between the two.
That means if we compare the instrument you
see the meters, then you cause the accuracy
and then you tell the cost, then you tell
the loss of total energy, loss of total energy
and then the value of C d. Then first you
write the venturi meter. Venturi meter accuracy
is maximum, cost is also not maximum you write
high, cost is also high to make a venturi
meter and the length is very high in the divergent
part made of brass. So therefore, it is cost
is very high, loss of energy is very low and
C d value 0.95 to 0.98.
Similarly, orifice meter is other end, other
extreme. Accuracy is very low, cost is also
low. So, these two are compatible. So, loss
of energy is very high and these value is
0.65 to 0.6 and flow nozzle or simply nozzle,
flow nozzle this is intermediate between this
two, intermediate between this two. That means
I can write intermediate both all these three
columns intermediate between intermediate
between venturi meter and orifice meter and
venturi meter and orifice meter I am sorry
for handwriting is bad and the value of this
also C d lies between 0.75 to 0.80, alright.
Thank you for today.
