We continue with the example that we were
discussing in the last lecture that that there
is a tank and through the bottom of the tank
the water is being drained out and the height
of water in the tank is therefore changing
with time. Our objective is to find out how
the height changes with time. Now we were
discussing about what is the significance
or impact of the unsteady term that is retained
or that should not be retained or should be
retained is our doubt in the Bernoulli equation.
Now if you try to approximate it in some way.
See in engineering we try to get a feel of
the order of magnitude. So we may try to approximate
it by a certain term which should be like
our derivative of velocity with respect to
time. Time sum height . So let us say that
if this dvdt was a constant if it was a constant
not that it is a constant. If it was a constant
it could have come out of the integral and
then it would have been sum equivalent constant
dvdt times s2-s1.
So s2-s1 maybe roughly like the height if
you take the streamline which is just along
the axis then it is exactly=h, but you cannot
just write it as sum equivalent dvdt*h because
V is changing with time in an unknown way
so you do not have really an equivalent constant
dvdt, but you may make a kind of approximation.
You can say that I approximate this dvdt with
dv1dt.
Why if you see that except for very close
to the outlet the streamlines are almost parallel
to each other and when the streamlines are
almost parallel to each other it represents
a case when that v is not varying very much.
See why v is varying see there is a flow rate
confined between these. So when the streamline
the distance between these two streamline
remains the same you have to say A1 V1=A2
V2.
So A1 is like this A2 is like this both are
like as the cross sectional area with the
streamline as envelope. In fact you can have
a large number of streamlines their envelope
will look like an imaginary pipe or a tube
that is known as a stream tube. So it is a
collection of streamlines making an imaginary
tube within with the fluid is flowing. So
if you consider such a tube you can always
see that the extent of that tube that remains
almost the same till you come to the exit
where it is really accelerating.
Because now the area available to it is so
small that it has to get adjusted to itself.
So when the area is very small and it has
to get adjusted to itself that is only a small
portion in comparison to the tank extent.
So if you approximate this dvdt with dv1dt
it is wrong, but it will give us some picture
some idea what is the effect of what is the
impact of this term? So if you make an approximation
that this is=dv1 dt times h.
You have to remember that both are functions
of time h is a function of time v1 is also
a function of time. So if you write this equation
in a bit different way you can write say v2
square-v1 square/2+ or – g v1-v2= - of and
z1-z2=h which is itself a function of time.
So these are valid locally at each and every
interval of time at that time you have dvdt
and you have an h.
Now if you try to compare different term say
we want to compare these term with these term.
So if these 2 terms are compared then let
us say this is term A and this is term B.
So when can you neglect term B in comparison
to term A when you have this mode of this/
mode of gh when this is much, much <1 then
B is much < A. So if the condition well h
is something which you do not consider locally
because this is like h is always a local constant.
That means whatever h is a function of time
here in term a same h is there in term B.
So only that means you are comparing dvdt
with g. So the rate at which the change of
velocity of the free surface it is there that
is it is sort of acceleration if it is comparable
with the acceleration due to gravity then
you cannot drop this term and then you should
retain this term at least frame a differential
equation it cannot be solved analytically.
But if this is the case which is true for
most of the practical cases then it is possible
to drop this term. The second important point
is irrespective of whether you drop this term
or not A1 V1= A2 V2 is what you are always
using. The reason is straight forward the
origin of these does not come from steady
flow. Although this is valid for steady flow
it does not mean that it cannot be used for
cases when the flow is unsteady.
Because the fundamental way in which it was
derived from what from a continuity equation.
First by dropping the partial derivative of
rho with respect to time=0. So if rho is a
constant partial derivative of rho with respect
to time is 0. It may still be unsteady flow
because the velocity may be function of time,
but rho not being a function of time was the
first thing to drop the first term in the
continuity equation the derivative with respect
to time.
For the other terms then how we came up with
this we integrated this that differential
form of the continuity equation and then if
there say rho at the inlet and the exit sections
are equal again if rho= constant that is valid
then you have A1 V1=A2 V2. So a very important
thing is for A1 V1 for A2 V2 to be satisfied
it is not necessary that it has to be a steady
flow only thing rho should not change that
is a very important thing that we have to
keep in mind.
So even when it is varying with time you can
use that. Now let us say that this is the
case so that we can drop the term B. So if
we can drop the term B then you can write
V2 square- V1 square/2=g h. Now what is V2
or you can express V2 in terms of V1 so V2
is V1* capital D square/ small d square. So
it is V 1 square capital D square/small d
square -1/ 2=gh. And the remaining work is
very straight forward.
You can find out so V1 is of the form sum
constant *root 2 gh where that constant is
basically D square/d square-1 by that okay
square root of that. See this gives a contradiction
what is the contradiction? When small d is
very small you consider the limit as small
d/ capital D tends to 0 that it is a very
big tank of a large cross section area and
there is a very small hole through which the
water is coming down.
Then how does this work? Yes how does this
work? C is almost if c is almost 0 then V1
is almost 0 I mean practically it is true
that if it is a tank of very large area and
if there is a small hole the velocity at which
the free surface is coming down is not perceptible
it is very small so that is okay. Let us not
bother about that too much. Let us just try
to complete this one by writing this as –dhdt=
c root 2 gh.
Now if you integrate with respect to time
you can find out how h varies with t this
is a very simple work. Now try to relate this
with a kind of again formula that you have
used earlier in your studies. So let us think
that this hole is not located here, but located
at the side. This is a different example just
I am drawing in the same figure to save the
effort. So let us say that now this height
is h which is changing with time.
So there is no hole here, but there is some
hole here. There is a nozzle that is fitted
and water is coming out. So when you are doing
that the way in which most of you have done
is like you have assume the velocity that
which the jet is coming out is root 2 gh.
This is known as Torricelli's formula. So
how you have arrived at that equation. You
have used Bernoulli equation between 1 and
2 at that time you are not very careful about
whether they are along same stream line or
not just out of pleasure you are applied between
2 points.
And then when you applied between 2 points
you put V1=0 you put p1=p2 the different between
the 2 height h and so V2 will come root 2
gh. So what are the assumptions under which
that is validated that is not a very bad formula.
Torricelli's derived it long back I mean in
a historical perspective it is a great development
because nowadays we can speak these big words
but the subject when it was fundamentally
developed this itself was not a very trivial
matter to resolve.
So then when Torricelli's found out this expression
what are the assumptions in which these expression
you expect to work still. So one of the things
was taken as V1=0 that means V1=0 when capital
D is much, much greater than small d. So V1
is approximately tending to 0 the other approximations
are that you are having a streamline like
this with respect to which you have the points
1 and 2.
And the unsteady term does not appear in that
analysis and it is assumed to be an inviscid
flow. The greatest deviation from realities
is because of the assumption of the inviscid
flow. So that is one of the very important
features that we have to keep in mind. So
with that assumption this formula is not illogical,
but a very important thing is we must keep
in mind that some of those assumptions are
to be questioned.
One of the important assumption is like capital
d is much, much greater than small d which
is true if it is a very large tank and from
that there is a small hole through which water
is coming out and dropping of the unsteady
term and we have discussed that how this unsteady
term this particular term in what condition
it may be dropped or not. So this is a very
simple problem.
But if you try to look into this problem very
carefully it will give you a lot of insight
on the use of Bernoulli equation under different
condition. And I would encourage you to think
about it more deeply under what conditions
different terms are important in different
ways not just satisfied with finding h of
the function of time, but to write the differential
equation of maybe say V1 as a function of
time in a very simple case and in the most
general case.
And try to compare them that what are the
terms that are making them to be different.
We will consider another example in the unsteady
Bernoulli equation in the use of the unsteady
Bernoulli equation that is given by the next
problem.
Let us say that you have 2 plates these are
circular plates. We have solved problems with
rectangular plates just for a change let us
consider that it is a circular plate. So this
is 
like this plate is coming down with a uniform
velocity V. And this is a circular plate.
The radius of the plate is r and say we are
considering a coordinate system the local
coordinate as small r. So small r is the local
coordinate at a radius r.
Now with this we are interested to see so
the bottom plate is stationary. There is some
water with rho=constant and when this plate
is coming down what is happening water is
squeezed out of the plates because whatever
water was there say originally this was B0.
So B= B0 at time=0, but as this is coming
down this B is changing B is decreasing. So
where will that water go that water will be
squeezed out radial to make sure that the
continuity is maintained.
So we are interested to find out how the pressure
varies with R. Assume inviscid flow and flow
is constant that we have already defined or
we have already assumed. So as we have seen
that in all these cases it is important to
get a feel of the velocity profile. So if
it is an inviscid flow the velocity variation
over the section is not there so that velocity
is uniform over each section, but these uniform
velocities is changing with radius.
So how you can find out it you have to think
that what is the rate at which this is pulling
water downwards with the same rate at which
it is being squeezed out. So if you consider
a local radius R what is the rate at which
this is coming down. So when you write A1
V1= A2 V2 question is how do you write V1
V2 A 1 and A2. What is V1? V1 is the rate
at which so it is like an artificial flow
imposed by the movement of the top plate.
So that flow velocity is given by V1. So what
is that A1 *. So what is A1? So if you consider
only up to a local radius of small R. So A1
is pi* small R square. So A1 is pi* smaller
square what is V1? V1 is V because this is
a uniform rate. This is uniform this is not
a function of time this is constant= what
is A2 2 pi r*B. B is a function of time * V2
or V as a function of R. Let us write Vr just
to emphasize that it is v at a radius R.
So you can write V at a radius R= V divided
by Vr/2v. Now so this is a velocity at a radius
R. Next we are interested to find out the
pressure. So if we are satisfied with inviscid
flow and flow= constant we can consider a
streamline that connects two points. Any 2
points say 1 and 2. So the streamlines how
the streamlines will look. So the streamlines
will virtually look like this.
So the flow is being squeezed out in this
way so that is how a streamline will look.
So let us take any 2 points located on the
streamline and write the Bernoulli equation
between those 2 points located on these identified
streamlines.
But because it is an unsteady flow we need
to retain the unsteady term in the Bernoulli
equation. So p1/rho + V1 square/2+gz 1= p2/rho+V2
square/2+g z2+ let us say that we apply that
between 2 points. One point is located at
R= small r and another point 2 is located
at R= capital R. So when you have such a case
you are getting rid of many things. One is
between the points 1 and 2 there is no difference
in height.
So of course if this gap B itself is narrow
then even if there was a change in height
because of taking the points 1 and 2 not exactly
along the same line that term itself is not
that large, but like if you take them along
the same horizontal lines they are identically
the same. Then you are interested to find
out p1 and p2 you know p2 is the atmospheric
pressure. So because it is at the exit plane.
So you are interested to write p1-say p2 is
p atmospheric p1-p atmospheric/ rho= now v2
squares- v1 square/2 so V2 square- v1 square/2
is V2/ 4 b square * capital r square-small
r square because V is having only this component.
Then + this term so by 2 will be there. 8
b square then + let us calculate the third
term.
So what is the partial derivative of V with
respect to t that is the partial derivative
of Vr with respect to t that is the only v
component that is there which is a function
of t here. B is a function of t here. So this
will be- Vr/2 b square * db dt and –dbdt=V.
So –dbdt=v. Just like the previous tank
problem that we were considering. So this
term becomes v square r/ 2 b square.
So that you can substitute here and ds will
be dr because you have chosen your streamline
in such a way that the change in s is like
change in r.
So this is from integration from small r to
capital R v square r/2 b square dr very straight
forward to complete it. It becomes v square/
4 b square* capital R square- small r square.
So at a given instant you can see the pressure
at the radius small r is varying with time
because b is a function of time. So this only
a given instant you can say. So at different
instance you have different values of b.
And you can find out what is the value of
b at a given time how because you know dbdt
is –v. So b= b0-vt. So if you are given
a particular time so this will give you b=b0-vt
So if you are given a particular time you
can find out what is the value of b at that
time then you may substitute the value of
b at that particular time to get the pressure
at a radius. So you can clearly see that the
unsteady Bernoulli equation how it can be
utilized.
Now the next topic that we are going to discuss
in the context of these Bernoulli equation
is the use of such equations. See the Bernoulli
equation has been one of the very popular
equations in fluid mechanics not just because
of its simplicity, but because of its applicability
in an approximate sense in terms of quantifying
the nature or the principle of working of
many engineering devices.
And we will look into such examples of applications
of Bernoulli equations. So some of the examples
we will not detail very much, but we will
only get the essence the details of most of
these examples are uploaded in a course website
note on the application of the Bernoulli equation.
So if you go through that in details you will
get all the detail picture because we are
going to discuss subsequently about certain
devices.
These devices have certain intricacies and
we will only highlight the major or important
features, but for the other detail feature
you should refer to those notes. Now before
coming to any device of very great engineering
application we may come up with a sort of
a very primitive device which you have already
heard of something called as a siphon. So
if you have say water in a tank like this.
And you are having a bent tube which is sort
of sucking water and ejecting water to a different
place from the tank so this is called as a
siphon. The apparent amazing features of the
siphon is out of nothing it is pulling the
water in a upward direction that is the apparent
amazing feature, but if you look into it a
bit carefully it is not at all any amazing
feature because eventually when it is discharged
it is discharged at a level below.
So the actual head difference which is working
on it is this one which is a favorable one
because effectively it is coming from this
elevation to this elevation and this net elevation
difference is actually giving it a velocity.
So with that velocity the water is being sucked.
So the fact that is going up is nothing very
special because eventually it comes down and
it gets ejected from height which is less
than or below the level of the tank.
But the good thing is that while doing it
can traverse a vertically upward distance.
Question is how much distance it can vertically
traverse? So what should be this say if you
call this as h then what is this h max? This
is given by a practical consideration. Let
us try to identify a streamline which connects
the points say streamlines will be bent like
this, but let us just consider a streamline
which is confined between that points 1 and
2 which are almost like located on a vertical
line.
So if we are interested to write the Bernoulli
Equation we can write p1/ rho+v1 square/2.
So every time whenever we are writing the
Bernoulli Equation we are not repeating the
assumptions, but you should keep in mind that
what are the assumptions on the basis of which
we are writing it. So p1/ rho+ v1 square/2+
gz1= p2/rho +v2 square/2+ . Now you can clearly
see that at the level 1 you have pressure
as the atmospheric pressure.
So this is p atmospheric. V1 is approximately=0
just like the Torricelli's equation because
the area here is so large that the velocity
with respect to which this level is changing
is very small as compared to the velocity
here v2 is same as the velocity at which the
jet is ejected here if the area of cross section
remains the same. So V1 is small because A1
is large as compared to the area available
at 2.
Then V2= Vj that is the velocity at which
the jet is coming out if the cross section
is same and you can find it out that Vj is
nothing, but approximately root 2 g * capital
H by writing the Bernoulli equation between
2 points on the same streamline whether if
you continue with that streamline it goes
like that and comes out. So the net elevation
difference will remain this one.
If you write the Bernoulli equation of a streamline
between say point 1 and say point j which
is located here. Now when you write that one
what you will get? You will get p2/rho= g*
z1- z2. So g* z1-z2 is –gh- vj square/2.
So you can clearly see that if you take the
atmospheric pressure as 0 reference. So this
is written by taking atmospheric pressure
as 0 reference. So this is a like a expression.
So when you take the atmospheric pressure
as 0 reference then p2 is negative because
h is positive vj square is positive. That
means the pressure at this point is below
atmospheric. So if it is below atmospheric
it may come to a state when it comes to the
local vapor pressure. So when the pressure
falls below the local vapor pressure then
what happens then vapor bubbles are formed.
So when the vapor bubbles are formed. So when
the vapor bubbles are formed it is nothing
very special that vapor bubbles are formed,
but what is special is that when this vapor
bubbles are transported or moved to a different
place where the pressure is again higher they
will collapse again to formal liquid and once
they collapse what happens basically then
they were occupying a large volume, but when
they collapse again to be converted to liquid
again there is a volume change.
So it creates an unsteadiness in the flow
and it can create a lot of vibration and noise
and that is not so good for the flow and that
type of phenomenon is known as gravitation.
We will see in details what is cavitation
when we will be discussing about the fluid
machinery which will be our last chapter in
this particular course. So we will not go
into the details of like what is cavitation
at this stage.
But we have to keep in mind that it is better
if we keep the pressure at 2 below the local
vapor pressure that is below the vapor pressure
which should be there at that corresponding
temperature so that vapor is not formed. So
that means we are keeping a restriction that
p2 must be less than the vapor pressure at
that local temperature of the fluid. So then
you can see that you get I h max from that.
And that is the maximum h with respect to
which you should design your system. So that
you do not have a problem with formation of
vapor. So the siphon in principle maybe designed
to be very like tall in height in terms of
this vent tube, but in practice one should
not make it too tall because if you make it
too tall it is possible that the pressure
is so low that vapors are formed and that
can create other disadvantages in terms of
operation of the device.
The next application when we consider we will
keep in mind that now whatever applications
we are going to study our objective will be
to have a Bernoulli equation utilized in devices
through which we are interested to measure
the velocity or flow rate in a pipeline.
So let us take an example let us say that
you have a pipe like this a horizontal pipe.
Now water is flowing and you make certain
holes in the pipeline. What holes you make?
So first you make a hole like this so when
you make such a hole what will happen the
water will rise and it will come to a height.
The height with respect to which the water
rises will be an indicator of the local pressure
at that location.
Pressure at we are interested about the central
line. So if we are interested about a point
in the central line what we are doing we are
sacrificing one thing we are not able to exactly
probe at the central line at the same actual
location we are proving at a point which is
different from the central line and we know
that it is very much possible that the pressure
at the central line should be different in
general from pressure at this.
When they are different when you have a curvature
of the streamline. We have just in the previous
lecture seen that if you have the gradient
of pressure in the direction of n you will
only when the streamlines have a radius of
curvature which is non infinity, but here
if we consider that the streamlines are parallel
to each other then you do not have that effect
of the stream line curvature in terms of the
pressure gradient.
So whatever is the pressure here should be
the same as the pressure here. So then this
is an indicator of the local pressure. Now
say we are interested to have an indication
of the velocity so for that what we can do
we can have another tube where we make a penetration
in the wall, but before that we have the tube
directly confronting with the flow. So this
tube and this tube is different this is not
directly interfering with the flow.
But this is directly interfering with the
flow. When it is directly interfering with
the flow it is bringing the flow to a standstill
or a dead stop. So it is creating like a stagnation
point where the flow comes to a dead stop
it cannot go further. So whatever water was
coming here it comes to a dead stop what it
will do it will it tube and the question is
will the rise will be greater than this one
or less than this one. See this rise was the
function of the pressure.
Now the entire energy which was there in the
flow if we assume that assumption on the Bernoulli
equation those are valid. Now we have made
the kinetic energy to 0 so the entire energy
now contribution of pressure term plus the
kinetic energy term will be successful to
make it go further up because where will that
energy go you have made the fluid to a dead
top you are assuming that is a frictionless
flow.
Then where will that energy go. It will obviously
make the fluid rise to a greater height and
the difference between these 2 heights is
if this points are very close to each other
the pressure almost the same the difference
between these 2 heights is just v square/
2 g. So from this principle V is the velocity
of flow at this point. So from this principle
it is possible to make an estimation of the
velocity and if you know the estimation of
the velocity.
And if you assume it to be uniform then you
can also have an estimate of the flow rate.
Now if it is not uniform you can keep it at
different radial locations and you can even
find out how velocity varies radially because
these tube you can put at different radial
locations. So this is put at r=0 at the central
line, but you can also keep it away from the
central line. So at different radius if you
put it will give you a picture of velocity
at different radius.
So it is possible even to get a velocity profile
if this is quite accurate of course there
are many doubts about the accuracy of such
a simple arrangement, but it gives as a conceptual
understanding.
So the device is based on this conceptual
understanding is known as a Pitot Tube. So
the last t is silent so it is pronounced as
Pitot Tube. So this of course is to honor
the name of inventor of this device and it
is a very simple device and the working principle
of this device is based on 2 important definition
which we will tell now. One is known as static
pressure. So what is a static pressure?
Static pressure is the pressure which is there
because of the intermolecular collisions so
that means if one is moving with the flow
then what is the pressure felt because of
just moving with the flow is the static pressure.
So this is a pressure experienced in moving
with the flow. So this is the result of the
intermolecular collision and this is the pressure
that we fundamentally define.
Now we are also going to define something
called as stagnation pressure. So what is
the stagnation pressure? Stagnation pressure
is the pressure that is there at a point if
the fluid is subjected to 0 velocity at that
point in a reversible and adiabatic manner.
So pressure at a point at which fluid is subjected
to rest in a reversible and adiabatic. We
will not go into the details of the reversible
and adiabatic processes.
Because these you will learn more in details
in the thermodynamic course that we will have
subsequently, but important understanding
in our context is that one of the important
requirements of this is it is a frictionless
flow. So that means when the fluid is subjected
to rest at a point you have to make sure that
it is subjected to rest in a frictionless
manner. So whatever is the pressure that these
tube is getting is the stagnation pressure.
So this is also known as stagnation tube because
its reading gives an indication of the stagnation
pressure and this is known as the static tube.
So if you want to write the Bernoulli equations
between 2 points 1 and 2 which are located
in such a closed manner that point 1 if you
have pressure as Ps or say p1/rho+v1 square/2.
We are not writing g z1 and g z2 they are
so close that the difference in height is
negligible= p2/rho+ v2 square/2+ g z2 we are
not writing again.
So what is V2 V2 is 0 because it is a stagnation
point. So the definition of the stagnation
point is velocity is 0. So you can see that
you can write p2 which is the stagnation pressure
as p1 which is the static pressure. This is
same as p1 this is p static+ 1/2 rho v1 square
that means stagnation pressure is a sort of
property of the flow if you know the velocity
of the flow.
But you have to keep in mind that this equation
is derived by considering a frictionless condition.
And frictionless condition is valid when you
are subjecting the flow to rest in a reversible
and adiabatic process. So the definition of
the stagnation pressure is to be kept in mind.
Stagnation pressure is not just pressure at
a stagnation point. What is a stagnation point?
The stagnation point is a point where you
have 0 velocity, but it does not mean that
pressure at that point is a stagnation point.
Pressure at that point will be a stagnation
pressure only if the flow is subjected to
rest in a frictionless manner because the
stagnation pressure is defined in that way.
It is not just sufficient it is necessary
that you must have the velocity to be 0 at
that point so that the pressure is a stagnation
pressure, but at the same time it is not velocity
subjected to 0 in any way, but it is subjected
to 0 in a frictionless way.
The second important thing is since these
points are very close to each other and you
can just say stagnation point stagnation pressure
at a point just as a property which is dependent
on the local velocity. So stagnation pressure
did not always be measured through a stagnation
point. So if you want to say find out stagnation
pressure at a point you can simply say that
it is the static pressure which is the regular
of the normal pressure+ 1/2 rho v 2 that is
a definition.
So the stagnation pressure does not mean that
you have to bring the fluid to rest at that
point to get a pressure. It is like how you
physically conceive that pressure not that.
So it should not give you the false idea that
whenever the velocity is non 0 stagnation
pressure is not defined. It is definitely
defined. It is just a physical way of looking
into it interpretation. Now the next we will
discuss 1 or 2 important flow measuring devices.
And the first device that we will discuss
is known as a venturimeter. So what is a venturimeter?
Say you have a pipeline and you are interested
to measure flow through a pipeline. So what
you are trying to do say you have a pipeline
like this you want to measure what is the
rate of flow through the pipe? So how will
you do it there are many ways in which it
can be done is one of the ways is by utilizing
a device called as venturimeter.
So what is done a part of the pipe is like
replaced with a device. What is that device?
The device is like this so you have an accelerating
section by having a converging cone and then
you have a zone of uniform cross section and
then 
you will again come back to the pipe dimension.
So this is known as diffuser. This is a known
as a throat and this is a converging section.
So what is the objective? The objective is
by this way you are reducing the cross sectional
area. So to maintain the continuity in a steady
state what you are doing. So if you consider
now the points let us say that you consider
points 1 and 2. The point 1 was having the
velocity as same as that of the velocity of
flow in a pipe. Now at the point 2 the velocity
will be more or less. It will be more because
the area of cross section has reduced.
So since the velocity is more. Now if you
write the Bernoulli equation assume that it
is a friction less flow. Then p/rho+ g z that
term will be what that term will be changing
and if we can find out a measure of that change
then it is possible to find out the velocity
through the Bernoulli equation how we do that?
Now let us say that you make a tapping of
a manometer that means let us say that you
consider a hole in the pipeline and a hole
here.
And connecting that with a manometer. So when
you are connecting that with a manometer see
we have not taken the point 1 at the inlet
of the converging section, but at some location
which is sufficiently away from that because
here the streamline curvature effect will
tend to become more and more dominant. So
you want to take it away from such a place
where the streamlines are almost parallel
to each other.
So pressure at these point and maybe pressure
at these points should not be very different
because of the streamline curvature effect.
So we are having a manometer in which we have
a fluid. Now in which limb the fluid height
will be more or in which limb it will be less.
Let us write the equation the Bernoulli equation
along the streamline between the points connecting
the points 1 and 2.
So let us say you have a streamline that connects
1 and 2. So you can write p1/rho+v1 square/2+g
z1= p2/rho+v2 square/2+ . At the point 1 if
you have this as the height of the limb and
at the point 2 if you have this as the height
of the limb. Now I have drawn it in this way.
Do you accept that it should be like this
let us a fluid a marker is there as a manometric
fluid here.
We call it rho m the density of the mercury.
Now is it an acceptable sketch in this case
the remaining is filled up with water. So
if water is flowing through this tube let
us say this is filled with water. The same
fluid which it is flowing here. So is this
acceptable? By this you are expecting that
pressure at 1 is greater than pressure at
2. We will see that may not be correct also
let us see.
But this figure is correct? How that is possible
let us see. So let us say that this is the
difference in height that we measure so that
is= delta h. So when you measure this height
delta h then from that delta h it is possible
to write the equation of the manometric principle
that is you can write that if you have 2 points
A and B at the same horizontal level you have
PA= PB.
So when you write PA= PB. Let us say that
you are writing say this is your reference
for measuring z1 and z2 in the Bernoulli equation.
So this is your z1 and this is your z2. You
can use any datum, but this is a convenient
datum.
So you can write p1+rho g z1 that is= pressure
at 8 where rho is the density of the water
that is flowing through the pipe=P2+ rho g
z2- delta h+ rho mg delta h. So when you are
finding out the difference in p1 and P2 p1-p2
you see that you can clean up the expression
by noting that it is not just p1-p2 that is
important. You have p1+rho g z1- p2+ rho g
z2 that is what is going to be important.
So if you write p1+ rho g z1- p2+ rho g z2
then that is rho M-rho * g * delta h. So in
this figure you are expecting that delta h
is positive. Isn’t it? This is just the
dimension. Rho m say this is mercury so we
know that it is much heavier than water. So
rho m-rho is positive that means we are expecting
this to be positive. So what this reading
gives us. This reading gives us not the difference
in p1 and p2, but the difference in sum of
p1+rho g z1 and p2+ rho g z2. So it is not
giving us the pressure difference. So what
it is giving us.
So let us write this in a bit more explicit
way. So let us write it as p1/rho g+z1. So
we are dividing it by rho g. So p2/rho g because
we know that in this process we will get something
called as head which we use as a terminology
for this calculation. So this is rho m/rho-1*
delta h. This delta h is very important because
this is what experimentally you can read.
So when you read experimentally delta h you
see that it is an indicator of not just the
difference in pressure, but the difference
in pressure head+ the elevation.
So when it is flowing from 1 to 2 it is possible
that p1  p2/rho
g+z2. So this flow is taking place from a
higher value of this collected term to a lower
value of this collected term. This collected
term which is given by p/rho g +z is known
as a piezometric head. So p/rho g +z this
is called as a piezometric head. Why it is
called as a piezometric head?
The reason is that if you say have a pipe
and if you puncture the pipe or if you penetrate
the pipe say and if you have a tube through
which the water goes up. It is just like that
static tube that we considered in the previous
example then the elevation that it assumes
here is the elevation because of its vertical
location+ because of the pressure at static
pressure at that point and this tube is commonly
known as a piezometer tube.
So that is why the name piezometric head.
So in the manometer in this kind of an example
we do not measure the pressure difference,
but we measure piezometric pressure or piezometric
head difference. In terms of head it is called
as piezometric head. If you express in terms
of pressure units it is called as piezometric
pressure. So always keep in mind. In this
case manometer is not measuring pressure difference.
It is measuring piezometric pressure difference.
These are very, very fundamental mistake that
people make. See as I told in a very introductory
class that we are bound with certain intuitions
that it will flow from high pressure to low
pressure and you can clearly see that with
a very simple example where it is not actually
a practical example because we have considered
a frictionless flow, but even that it gives
a very important insight that it did not be
from a higher pressure to low pressure.
It is basically from a high piezometric pressure
to a low piezometric pressure. Now fortunately
what is important for this equation is only
the piezometric pressure because if you see
like if you write it in this form you will
get p1/rho g+z1-p2/rho g+z2 that is= v2 square-v1
square/2 g. So this is something which is
a very simple term for us now because from
the manometer we have got an explicit expression
for that is rho m/rho-1 * delta h.
So this we can write as rho m/rho-1* delta
h and this is= now you can express V2 and
V1 in terms of the volume flow rate. So if
Q is the volume flow rate.
\
Then you can write as Q=A1 V1=A2 V2. Again
what are the assumptions? Rho is constant
and it is a uniform velocity profile over
the section that is inviscid flow. Viscous
effects are not there. So you can write V1
as Q/ A1 and V2 as Q/ A2. So if you substitute
that in this expression.
It is possible to express V2 square-v1 square
as Q square/2 g* no this g is there so because
of division by( 58:24). So Q2/2g* 1/A2 square-
1/A1 square= rho m/rho-1* delta h. So from
here you can solve for what is Q? Remember
it is very theoretical. Why it is theoretical
because it has considered many idealization
which do not actually occur in practice. So
we will keep this in mind and in the last
class we will try to identify that what are
the idealization which were here which need
to be rectified.
And what are the important design considerations
that should go with this device matching with
the non idealization. That we will discuss
in the next class, but if it was ideal just
by getting the delta h reading you could get
what is the flow rate through the pipe because
A1 and A2 you know are the areas of cross
sections of 1 and 2 which are given geometrical
parameters. So we stop here today we will
continue with that in the next class.
