In our previous lecture we were discussing
about the Venturimeter, and we will continue
with that as an example.
So if you recall; what was the purpose of
the Venturimeter; the purpose was to measure
the flow rate volume flow rate through a pipe.
Now for that we utilize this type of an
arrangement where you have a converging section,
which is sometimes also called as a
converging cone then you have a throat where
the area of cross section of the entire
arrangement is a minimum and then a diverging
section. And the purpose is obvious that
this part has to fit with the pipe. So, if
you have reduced the cross section area somehow
you have to increase it so that it again matches
with the pipe.
So, this black coloured portion is like a
fitting, which is fitted with the pipe to
measure
the flow which is occurring through it. The
question is why you have such a converging
section we have seen that, it gives rise to
an accelerated flow. So, it gives rise to
a high
velocity and therefore, a change in pressure;
and we have seen that it is not just the
pressure that is important, but the sum of
the pressure on the elevation head together
that
is P by rho g plus Z that term together is
something which is changing, because of the
change in the kinetic energy head and we gave
it a name called as Piezometric head. So,
the difference in the Piezometric head is
reflected by the difference in reading or
difference the reading in the difference in
the heights of the liquid columns in the 2
limns, and that delta h therefore, we will
be an indicator of the rate of flow which
we
found out by utilizing the Bernoulli’s equation.
The question is that is it a very reliable
way of finding out the flow rate. The answer
is
straight forward in some sense that it is
not reliable, the reason is that we have utilized
an
ideal type of equation this all the assumptions
which are there inbuilt with the
Bernoulli’s equation in steady form or inbuilt
with this and therefore, all the ideal
idealities which are also inbuilt with this
form of equation, those are assume to hold
at
the ame time we understand that in practice
such idealities do not hold true; what is
the
biggest deviation in practice it is never
a friction less flow. So, you have viscous
effects
and because of the viscous effects something
happens or quite a few things happen.
Now 1 of the important things is because of
the viscous effects if you have these are
the
total head or expressed in terms of different
units total energy, represented it is a
representative of the total energy at section
one; and this is a representative of the total
energy when we say energy here we only mean
the mechanical form of energy at section
2. We assume these 2 are equal because there
is no loss because it is a frictionless flow,
in practice there is a loss. So, you expect
that if you call this as say E 1 and if you
call
this as E 2 you expect that there is which
1 is more E 1 is more or E 2 is more.
E 1.
E 1 is more because you expect that there
may be a loss of energy and the loss of energy
will be because of the travel of the fluid
from the initial point to the final point.
So, when
travelling from 1 to 2 it will have a loss
so; that means, E 1 will be more so that when
you come to the section 2, it is E 2 plus
some losses; the losses which have taken
between 1 and 2 in 1 of our later chapters
we will try to characterise these losses in
a
more formal way, but we will just keep in
mind that there are certain losses because
of
viscous effects; and these losses will give
you a guideline of like what is the direction
of
flow. So, say you know nothing you are given
some E 1 you are given some E 2, if E 2 is
greater than E 1 you must be assured that
the flow is taking place from 2 to 1 not from
1
to 2. So, it is basically taking place from
a high head to a low head, it cannot be the
other
way because where will the head come from.
Now, here because of this loss what will happen?
See this eventually will boil down to a
large larger drop in the Piezometric head.
So, if it is flowing from 1 to 2 the Piezometric
head which is coming into the picture, the
Piezometric head drop that is the difference
between the 2 terms present there in the bracket,
this you expect to be more. Because you
expect a more severe drop in pressure, because
of the overcoming the frictional
resistance effects; that means, whatever delta
h you read here it is not the ideal delta
h,
see in this formula what delta h you put to
get the value of ideal Q this delta h is what
you experimentally observed. We have to keep
one thing in mind what is the basic
principle of measurements that we use in experiments,
that we have an expression in
which we have certain measurable quantities,
certain easily measurable quantities and we
express some more difficult to measure quantities
in terms of the easily measurable
quantities.
So, here delta h is something which you measure
easily, flow rate you do not measure
directly, but use this formula to write flow
rate express flow rate in terms of the measure
delta h. So, here may be to get Q ideal, it
would have been better if you put delta h
ideal;
but that you cannot do because delta h is
what you reading from the practical thing.
So, it
is giving the delta h which has got manifested
because of by considering all
practicalities. So, this delta h also considers
the practicality that there is some loss of
energy to overcome the fluid friction effects
; that means, this delta h is higher than
what
delta h would have been if it were a friction
less flow.
But, you cannot help this is what you read
experimentally and it is what you put here;
that means, even in terms of theoretical flow
rate it is not giving the correct theoretical
flow rate it is over estimating that; because
you are putting an over estimated delta h
the
reason is straight forward because experimentally
you cannot reveal an idealist picture
experimentally you reveal the real picture
where the delta h is much more severe than
what delta h you could get in a friction less
condition.
So, one important thing we realize that if
you consider no other effect, this particular
effect alone should give you that Q actual
should be less than Q theoretical, which you
calculate by using this formula, but there
are other important reasons also what is the
other important reason see when you have written
V 1 square by 2, actually we were
bothered about the velocity at the point 1.
But the velocity of the point 1 how did we
evaluate? We evaluate by using this Q equal
to A 1 V 1 V 1 equal to A 2 V 2. So, by that
we implicitly presume that V 1 and V 2 are
like same as the average velocities over the
section, that is possible only when the
velocity profiles are uniform over the sections
1 and 2, but because of the viscous effects
we have seen that those are not uniform. So,
there is a non ideality because of this
viscous effect not only in terms of the frictional
resistance, but also in terms of putting
the V square by 2 term. So, there is also
some error in that. So, these 2 errors are
very
very significant, 1 error is the frictional
resistance another error is the like miscalculating
the velocity expression or misinterpreting
the velocity expression.
So, when you put the velocity here ideally
you should have put velocity here in such
a
way that this could have been represented
the kinetic energy across the section one.
Again in our later chapter we will see that
how to exactly put that, but here we will
just
appreciate that we have not put it correctly.
So, whenever we make a mistake in writing
something, the first and foremost thing is
to appreciate that we have made a mistake.
So,
let us appreciate that this is not correct;
there is some error in it.
Now incidentally engineers are such classes
of people who are happy to get the final
result disregarding may be some mistakes which
have already been done, and then to
adjust that mistake let us say that some adjustment
factor is put, let us say we call a new
coefficient C d as Q actual y Q theoretical.
So, if somehow this coefficient is known to
us whatever by magic we will see what magic
will tell us this number, but if somehow
you get this value of C d then you straight
away multiply that with the Q theoretical
that
you get from here to get what is the Q actual;
if it is very close to the final result that
you
want in many practical engineering applications
people are happy.
So, we have to see that what is there in the
C d which will try to make a more and more
happy, and this coefficient is known as coefficient
of discharge. So, this takes into
account that we have realized that if it is
an ideal case all together this would have
equal
to 1. So, deviation of this form 1 represents
the extent of non ideality in the flow, and
not
only non ideality in the flow in general,
but more specifically how that non ideality
has
got manifested in the prediction of flow rate.
So that means, what is the total influence
of
these frictions in terms of the delta h and
what is the influence of the inaccuracy in
the
velocity distribution that has already got
inbuilt in 
the corresponding expression for
energy.
To look into that we will not going to all
sorts of detail, but we will just consider
one
thing that see at the section 1 say we have
sort 
of uniform velocity profile, somehow we
have maintained. The again it is we will later
on see that it is not easy to maintain that,
but fortunately we will be easily maintaining
such a situation when the flow is turbulent
and when will be discussing turbulent flows
we will see that, turbulence is a kind of
situation which will create almost a uniform
velocity profile over the section. So, that
will in some way take care some of our acts
of ignorance in writing or describing the
current velocity profile, but even then let
us try to see that at which section the error
will
be more severe at section 1 or section 2.
To do that let us say that we consider 2 stream
lines which are very close to each other;
say you have a stream line like this and another
stream line which is very close to each
other, both stream lines are connecting the
sections contained by 1 and 2. So, here 1
and
2 are points, but let us say that these are
sections which are contain the points 1 and
2.
So, we have 1 stream line and we have another
stream line.
Now, these 2 stream lines are very close.
So, close that let us say that here the velocity
is
u 1 here the velocity is u 2 here the velocity
is u 1 plus delta u 1 and here it is u 2 plus
delta u 2; where delta is a small change in
comparison to the other value. Now if these
stream lines are very close to each other
what will happen? There will be negligible
difference in pressure between these 2 stream
lines.
Always remember that there is a difference
in pressure between the stream lines because
of the curvature effects of the stream lines,
but stream if they are very close that effect
is
negligibility. So, if these 2 are very close
there is negligible difference in the pressure
head between these 2 stream lines at 1 and
at 2, and if they are very close there is
negligible difference in the height also that
is the Z coordinate. So, what we can say is
that the difference in kinetic energy heads
between the 2 points, it remains same for
the
stream line above and the stream line below
because the other terms they do not change.
So, how we may reflect that in our analysis
let us try to do that.
So, what we want intend to write is u 1 square
minus u 2 square is equal to u 1 plus delta
u 1 square minus u 2 plus delta u 2 square.
So, let us try to simplify it keeping in mind
that delta u 1 and delta u 2 are much smaller
as compared to u 1 and u 2 respectively. So,
you are clear that why such an equation has
come because other terms like P by rho g
and Z those are the same. So, they have cancelled
out when you consider the Bernoulli’s
equation for the stream line above and stream
line below; and when you have subtracted
that they that effect have gone. So, only
these terms remain basically u square by 2
and
all those things, but that division by 2 get
cancelled out that is how these terms come
out.
So, if you now write it like this. So, you
will. So, what you have here. So, you can
write
this as I mean u 1 plus delta u 1 square that
you can bring in one side, u 1 plus delta
u 1
square minus u 1 square is equal to u 2 plus
delta u 2 square minus u 2 square. So, you
can write it as 2 u 1 into delta u 1. So,
when you write this like a square minus b
square
formula when you write a plus b it is 2 u
1 plus delta u 1, delta u 1 is much smaller
as
compared to 2 u 1. So, only 2 u 1, and the
difference only delta u 1 is there, so that
is
equal to 2 u 2 delta u 2. We are interested
to express delta u 1 by u 1 in terms of delta
u 2
by u 2; to see that what is the relative error
in relative change between the velocities
into
adjacent stream lines. So, you will have delta
u 1 by u 1, as if you had dividing both
sides by u 1 square. So, you write as u 2
by u 1 square delta u 2. So, here you can
write
this as u 2s by u 1 whole square into delta
u 2 by u 2 right.
That means delta u 2 by u 2 is equal to u
1 by u 2 whole square, into delta u 1 by u
1.
Now which velocity you expect to be more u
1 or u 2. Look at the sections 1 and 2 here
the area is large, so the velocity will be
less. So, u 1 by u 2 more is this reduction
in area
u 1 by u 2 will be lesser and lesser. Square
of that will be small so from this our
conclusion is that this delta u 2 by u 2 is
expected to be much much less than delta u
1 by
u 1, provided there is a greater reduction
in section; that means, if it is approximately
uniform at section 1 it will be even better
at section 2. Because the non-uniformity is
much less, what this represents a non uniformity
when you go to a different stream line
along the same section, you expect the velocity
to be different and that difference give
rise to a non uniformity.
So, when you have this non uniformity, but
again see this an estimation, because for
estimating the non uniformity we have again
utilised the ideal equation which is like
the
Bernoulli’s equation, but what we have consider
that even for a non ideal case this is not
very very invalid, because whatever is the
frictional effect that also has cancelled
out
when you subtracted the Q equation; assuming
that the frictional effects are also same
as
the fluid flows from 1 to 2, along the 2 stream
lines above and below.
So, even if frictional effects are considered
and they these the Bernoulli’s equation
for
the 2 or the modified Bernoulli’s equations
considering the frictional effects they are
cancelled, or they are subtracted one from
the other that effect will cancelled. So,
this is
not a bad estimation. So, this estimation
shows that if the if the velocity is such
that you
are going towards a cross section of reduce
size, if at the bigger cross section the velocity
was more or less uniform, the smaller section
it is expected to even more uniform. The
reason is quite clear that if there were stream
lines like these, stream lines will more
converge to each other because they are now
confined to be there within a very small
space as compared to how they were earlier.
So, if the stream lines were are quite large
distances apart. So, if the stream lines were
like this.
Now, when the stream lines are confined; so
what will happen all the stream lines will
try to converge. So, when the stream lines
try to converge, you see the distance between
the stream lines corresponding stream lines
become smaller and smaller; and eventually
different stream lines represent the sort
of like different states of flow. So, if you
have
them quite close to each other and almost
parallel to each other, that non uniformity
in
the velocity is almost like it is not like
totally nullified, but it becomes a better
situation.
So, by having a section 2 like this which
is like a convergence sections it is not bad,
it
sort of eliminates one non ideality. The other
non-ideality because of negligibility
friction that may be reduced to some extent
by what; by minimizing the length travelled
between 1 and 2 because the frictional resistance
will be related to how much length the
fluid has travelled against the viscous effect.
So, how do you reduce that? One of the ways
is like is you have this angle this cone
angle quite large. So, that it converges quickly
to a small section, in practice this angle
is
like typically kept as like 20 degree or so.
These are like designed considerations of
this
device, it is not that it is 20 degree is
a magic number, and it is always kept like
that I am
just giving you a rough idea of what is the
range in which it is kept in practice.
Now, there are different issues like you cannot
make it as large angle as you like there
are issues of manufacturing the device and
so on. It is not that whatever angle you want
and you propose one has to also fabricate
it and put it in practice. One particular
aspect
on which one may not make a compromise which
is like by putting by locating this
section 1, where you are having this manometer
leem. It should be preferable somewhat
away from the place where the reduction has
started. So, that this disturbance is not
influencing the velocity at this point significantly,
and that is why it is kept little bit away
from this one and roughly it may be a if the
diameter of the pipe is d it is roughly like
distance d away I mean it is it is again a
rough estimate there are more accurate estimate
for each device.
So, the connections of the pressure tappings
are also very important, that is where are
they to be put. So, one is here then it is
roughly like 20 degree and this creates a
good
accelerated flow if you if you achieve it
in a very small or a short distance it is
good, you
have less frictional resistance; and smaller
the cross section you expect that more will
be
your resolution in terms of these delta h.
So, the experimental objective is the delta
h is if
the delta h is more it is better because that
is you reading. If it is very small your error
in
resolution will be will be effecting your
result significantly, but if the reliability
of this is
good.
Then the error corresponding error is less
that is why you are trying to trying desperately
to reduce the cross section area, so that
there is a change in the kinetic energy head
very
severely which is manifested in terms of this
delta h. Now after this section has come and
then what you have to do, then you have to
revert back to the pipe diameter again. So,
you have diffuser which is like a diverging
section. So, you have a converging cone you
have the throat where you have the minimum
area and then you again have a diffuser.
The question is what should be the angle of
this diffuser? I mean do not get confused
with the sketch that I have drawn in the board,
it is just because of lack of space that I
have drawn it not to scale. So, this does
not this angle does not represent what is
there in
reality it just represents the shape, but
not really the sense of the angle. So, what
should
be this angle?
Now, again there are 2 conflicting requirements:
engineering is such an area where when
you want to design something there are 2 aspects
that you have to keep in mind, one is it
should satisfy the fundamental scientific
requirements, so that the device is based
on a
thorough scientific principle. The other important
thing is that it must optimally satisfy
the performance requirements. So, what are
the corresponding in influencing parameters
always you will see that there will be opposing
parameters. So, opposing parameter
means if you increase this angle then something
good will happen and something bad
will also happen. So, let us see that if we
increase this angle of the diverging section,
first
let us see that what good thing will happen
that is very obvious.
So, if we increase the angle of the diverging
section what good thing will happen? Yes if
you increase the angle of this what is the
good effect of that? The portion will decrease
in
length. So, in a relatively short length this
device will merge with the pipe. So, the loss
due to frictional resistances will be less.
So, just like I mean what would have been
a
good effect of making this angle large the
same logic hold there also, but one of the
logics that does not hold is that there is
a great difference between and accelerating
and
decelerating section this is an accelerating
section, but this is a decelerating section
why
this is a decelerating section.
So, if you see the area of cross section is
increasing. So, you expect the kinetic energy
head to reduce; that means, if you expect
the kinetic energy head to reduce; that means,
P
by rho g plus Z that Piezometric head will
increase to compensate for that. So, if you
say
let us say that you are having a horizontal
venturimeter. So, if you having horizontal
venturimeter Z 1 and Z 2 are the same or may
be 2 and 3 here you consider another point
3 Z 2 and Z 3 are the same.
So, then if you go from a point 2 to say a
point 3, you expect what. If these 2 are located
at the same height, then what you expect you
expect that pressure will increase or
decrease pressure will increase. So, when
pressure increases; that means if you consider
the direction of the flow as x. So, you can
write the d P d x as the rate of change of
pressure with respect to x. In the converging
section d p d x was what less than 0, but
here in this particular section d p d x is
will be greater than 0 because pressure is
increasing with x, what is the consequence?
The consequence is see you expect that if
the pressure is decreasing we take that is
fine, because then a higher pressure is creating
a drive for you. That if you if pressure is
decreasing with x; that means, P 1 is greater
than P 2, and that is the in some way it is
trying to create a driving force for you.
On the other hand here from 2 to 3 the pressure
is opposing you, because as you are
moving from 2 to 3 you are experiencing higher
and higher pressure so; that means, it is
a sort of effect that tries to inhibit the
motion of the flow. So, that is why this type
of
pressure gradient is called as adverse pressure
gradient. So, this type pressure gradient
is
called as a favourable pressure gradient.
So, favourable and adverse the English names
are quiet clear, favourable means which
favours the flow and adverse means which is
not good for the flow. So, when you have
an adverse pressure gradient which is like
this d p d x is greater than 0 what happens?
The flow has a tendency to be decelerated
because of that kind of a pressure gradient.
So,
if you try to sketch that what happens to
the stream line in such a case? So, the flow
tries
to move like this, but because of the deceleration
effect and the deceleration effects are
more severe close to the wall.
Why because viscous effects propagates from
the wall. So, at those locations what will
happen is the flow may not be capable enough
of being dragged with the main or the
core flow. Because it is slowed down so severely
that it just creates a local rotation, but
it
does not contribute to the main flow. So,
that type of thing is called as a flow separation;
that means, you have a main flow like this,
now the flow or the fluid particles this poor
guys close to the wall they are so severely
disturbed because of the adverse special
gradient which are acting on them, that they
really cannot maintain the flow and they
might even reverse their direction of flow.
So, local vertices are created close to the
wall.
How do these vertices contribute? They contribute
in a sort of negative way, see these
vertices by virtue of rotation have some energy,
but that energy is not contributed to the
main flow. The main flow is like this which
is moving; now here this energy which is
there because of the rotation of this vertices
because of flow separation, that does not
contribute. So, effectively as if some energy
is taken away from the main flow to sustain
the rotation of these vertices. So, effectively
there is a kind of loss of energy of the main
flow, and that loss has been created because
of this flow separation; and this flow
separation effect is stronger more is this
angle of diffuse diffuser. The reason is more
is
this more severe will be the adverse special
gradient, because more severe will be the
pressure increase over a given length the
length becomes smaller. So, this is a conflict
with the requirement of the frictional resistance.
So, we have seen that if we increase the length
of this one or may be reduce this angle
then this effect will be less. So, the adverse
pressure gradient effect will be less if you
make this angle quite small. So, that this
length is large, but if this length is large
the
direct frictional resistance will be more.
So, these 2 are 2 conflicting parameters in
the
design, that is where you have to come to
an optimal design where you cannot keep this
angle may be as large as this, and the common
optimization is that this is typically like
5
degrees, 6 degrees like; much less than the
angle of the converging section.
So, this is something we have to understand
very carefully that why in the practical
design the diffuser angle is much much less
than the converging section angle. When you
have a converging section you do not have
such a case of flow separation. So, only the
frictional resistance is because of the length
is the only important resistance, because
flow separation will be there when the flow
is decelerated, but in the converging section
the flow is accelerated.
So, it does not suffer from a resistance because
of adverse pressure gradient. In fact, the
pressure gradient here is favourable which
makes it move in a much more convenient
manner. So, the design aspects are quite clear
that why you should have different angles
for converging and diverging sections, and
what are the parameters which should decide
the range of these angles.
And keeping these things in mind if one may
designs this device quite well by
minimizing the losses, then the coefficient
of discharge which is the ratio between the
actual flow rate and the ideal flow rate,
it is actually very close to 1; 0.98 - 0.97
like that.
So, somehow if the device is very cleverly
designed, some of the non idealities are taken
care of in some way not that it becomes ideal,
but our ignorance about non ideality does
not get manifested so much. The reason is
that one is you are using a like a continuously
converging section in this way, and the diffusing
section is also say properly well
designed.
Now this venturimeter is therefore a very
common device which may be used in a pipe
line to measure the flow rate. At the same
time this is not a very inexpensive device.
It is
not very highly expensive, but at the same
time for very routine applications one might
look for some cheaper devices which are broadly
following the similar principles. And
let us see one such device that device we
call as orifice meter.
