WEBVTT
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We are talking about three-dimensional flows
in actual flow compressors. In the last class,
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we had a look at various aspects of physics
of the three-dimensional flow in actual flow
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compressors, and its various ramifications
of the understanding of how actual flow compressor
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works, and how it impacts on the performance
of actual flow compressor. In today's class,
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we will try to capture much of the three-dimensional
flow in mathematical form. These mathematical
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forms are very useful in the sense that eventually,
they would be used for various computational
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analysis purposes, which aids the design process,
and in the process of this coupling between
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design and analysis, the design time is substantially
cut down.
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We do not have to go through a very costly
experimental analysis to finalize the design.
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So, the design time has indeed been cut down
by years with the help of these analytical
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tools with the help of CFD. And in today's
class will try to capture some of the mathematical
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form of the physics of the flow that we have
done in the last class. So, let us take a
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look at some of these issues that we are about
to capture in mathematical form. So, today's
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class is about three-dimensional flow analysis
in actual flow compressors.
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Now, as we have seen many of the blade theories
are indeed actually based on two-dimensional
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understanding, and those two-dimensional understanding
actually ignore the effect of three-dimensionality
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of flow while the fluid is passing through
the blade passage. Now, the picture below
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we have seen in the last lecture and it gives
us a fairly detailed idea about the three-dimensional
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nature of the flow. There are so many things
going on inside the blades, specially inside
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the rotating a row of blades that it is really
a not a right thing to do is ignore some of
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these three-dimensionalities; so, we will
today try to capture some of these things
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in mathematical form. So, that much of it
is actually included in the design and analysis
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process.
So, we will keep this picture in mind which
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we have done in the last class, and try to
move forward to see what all things in some
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summary form in a overall form is captured
in mathematical formation.
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Now, we see that one of the things that is
normally ignored or assume to be not present
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is radial component of the flow, and we had
a quick look at a simple three-dimensional
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radial equilibrium, what it actually means
is that the radial motion of the fluid particles
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inside the blade is sort of assumed to be
or pretended to be not present at all. Now,
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we see what are the reasons why radial flow
would almost invariably be present in actual
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flow compressors. Then, the first reason is
a centrifugal action of the rotational motion
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which is passed on to the fluid, and this
centrifugal action would invariably try to
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impart certain amount of radial flow to the
fluid as it is passing through the blade specially
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the rotating blade.
Then the convergence of the annulus flow track,
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we will you know discuss a lot of things about
the flow track little later in this lecture
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series. But, the flow track in the modern
compressors is quite often converging type
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and especially in the highly loaded model
aircraft, engine compressors and this kind
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of annulus flow track are geometry introduces
the radiality anyway, then of course, we have
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the twist and the taper and various three-dimensional
blade shapes, the solid body of the blade
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often has highly three-dimensional shape these
days they are. In fact, becoming more and
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more three-dimensional and as a result of
which are certain radial component of the
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fluid is invariably introduced into the fluid
motion as it passes through the blades.
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And then of course, we have the tip clearance
effects. Now, the tip clearance effects we
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have studied in the last lectures, and we
know that there is a substantial cross flow
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through the open tip of the rotor and this
introduces a three-dimensional radial flow
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and definitely introduce a certain amount
of radial component to the fluid flow.
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Then, if we look at the fact that you know
in the first picture that you we looked at
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even today that a passage vortex is formed
inside the blades. Now, this formation of
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passage vortex creates radial flow a strong
component of radial flow in the main flow
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itself, which is passing through the blade
passage. So, as a result that is another contribution
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from the various things that are happening
to the introduction of radial flow. Then,
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through the blades quite often, we assume
in our theories that the temperature and then
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the enthalpy and then entropy gradients are
you know zero.
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Now, that is an assumption in a real blade
quite often, we will find that there is a
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certain gradient in the rear stages of a multistage
axial flow compressors, we would find a temperature
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gradient from root to tip or even from blade
to blade, there would be enthalpy gradient,
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we assume that the enthalpy is a uniform along
the length of the blade from hub to tip, and
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we are also assume that the work done is constant
from hub to tip in the earlier simple theories
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and then of course, there is a general assumption
in various theories that we have propounded
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that the entropy gradient in the radial direction
is absent; that means, there is no gradient.
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Now, these are assumptions in reality some
kind of gradient would invariably appear as
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the flow goes through the various stages,
and those will have to be taken into account
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when you are analyzing fluid flow. Now, taking
them into account means bringing in certain
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amount of thermodynamics and those thermodynamic
issues would invariably be present in the
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real flow as mentioned specially in the latest
stages of a actual flow compressor, where
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the temperature is higher, the flow has become
more three-dimensional and as a result temperature,
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enthalpy and entropy gradient start appearing
in the fluid flow passing through the blade
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passage.
Then, we have the blade solid body thickness
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which is a blockage; now, this includes that
camber and the stagger you know more the camber
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higher the thickness, higher the stagger,
this blockage to the main flow is likely to
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increase, usually the flow is basically an
actual flow and the moment you have blade
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there, 8 solid blade it obviously has a blockage,
and this blockage will change with these essentially
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with these three parameters, the thickness
of the blade the camber and the stagger, now,
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you see we have already noticed that the thickness
the camber and stagger, all three of them
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actually vary from root to the tip of a blade.
Now, if they are varying from root to the
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tip of the blade,. it it stands to reason
that the blockage effect would also vary from
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root to the tip of the blade. And this variation
would also then introduce on other aspect
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of radial flow or three-dimensionality into
the flow. And then of course, we have the
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end wall boundary layers. And then, now these
boundary layers are developed, because of
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the flow essentially being in a adverse pressure
gradient in a compressor and these boundary
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layers over the casing and the hub internal
surfaces actually create a certain amount
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of blockage, these blockage is a fluid mechanic
blockage and this blockage often has a tendency
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to deflect the main flow in wards.
From the casing and from the hub and this
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inward deflection of course, introduces three-dimensionality,
it introduces a radial flow in addition to
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the fact that it has a tendency this blockage
has a tendency to reduce the main flow rate
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at in the actual flow rate that you would
see would be less than what was assumed initially
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probably during the design. So, these are
some of the effects of actual realistic flow
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on the formation of radial and three-dimensional
flow features inside actual flow compressor
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blading.
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Now, we have created a radial equilibrium
theory earlier a simpler version; now, this
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is based on the premise or assumption that
the radial gradient of the forces which are
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experienced by the fluid, and we had taken
a fluid element earlier of an appropriate
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shape and size and that a fluid element experiences
radial gradients, and those things contribute
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to the radial movement of the flow, and if
they are to be checked; that thing the radial
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movement of the flow or not to be allowed,
it it becomes incumbent that those forces
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must be balanced by the static forces exerted
by the pressure gradient which also created
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inside the flow.
So, that at any instant of time the fluid
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system or the fluid particle or the fluid
element is in radial balance of forces that
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is in radial equilibrium. It is this balance
of forces between the dynamic and the static
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forces that we would move forward take, forward
today to create a more comprehensive radial
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equilibrium theory, much different from much
more complex and much more involved than the
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simple radial equilibrium theory that we had
created in the last class. So, this balance
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of forces is one of the issues that axial
compressor, turbo machinery designers quite
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often need to do during the process of design
and analysis.
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Now, let us try to lay down a few simple conditions
mathematical and geometrical conditions that
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are going to help us in formulating the mathematical
theory of three-dimensional flow. We assume
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that we have motion of a particle with respect
to 2 coordinate systems, and this particle
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P is moving in an arbitrary path within the
2 coordinate systems. Now, the two coordinate
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systems are XYZ. And then with origin o and
then x y and z with origin o prime, so, we
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have two origins and two coordinate systems
and the small xyz coordinate system is a moving
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coordinate system.
In fact, with respect axial flow compressor,
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it would be a rotating coordinate system which
means it is fixed on the rotor body. So, it
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is a it is a body fix coordinate system which
rotates with the rotor whereas, we have capital
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XYZ which is a fixed coordinate system, and
static coordinate system and the stated analysis
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could be done with respect to that coordinate
system. But the rotor needs to be analyze
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with respect to the rotating or body fix coordinate
system.
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So, we do have 2 coordinate systems to deal
with in this theoretical formulation, and
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we need to keep that in mind. The moving coordinate
system is at a distance R from the origin
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of the fix coordinate system, and the particle
P is at a distance rho dashed or rho prime
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from the moving coordinate system origin and
is at a distance small r from the fix coordinate
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system origin. So, we have 3 distances over
here. One is this coordinate system is at
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a distance capital R from the fix coordinate
system, and then the particle p itself here
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a distance a rho dashed from the moving coordinate
system origin and is at a distance small r
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from the origin of the fix coordinate system.
So, this is how we have put together the particle
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that is moving in an arbitrary path.
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Now if we move forward, the two reference
systems have relative motion which is represented
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by r bar r dash r bar, and that is it is the
motion of the position vector r with respect
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to fix origin o, omega is the rotation of
the particle with respect to the moving access
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system xyz and V xyz is the translational
motion of the particle with respect to the
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moving it access system xyz. Hence, the velocity
of the particle p with respect to the fixed
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access system capital XYZ based on this figure
would be v, xyz equal to d r bar d t of xyz
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capital XYZ. So, this is the basic definition
or description of the fluid particle that
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we are tracking in a motion involving 2 coordinate
systems.
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Now, velocity of the particle p with respect
to small xyz that is the moving x y, z is
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V small xyz and that would be d rho dashed
d t as per the diagram that we had given.
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Now, the vectorially; the motion of particle
P is summation of the motion of the moving
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coordinate system with respect to the fix
coordinate system, and the particle P with
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respect to the moving system. So, vectorially
r bar would be equal to capital R bar which
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is the distance of the moving coordinate system
to the fix coordinate system, and rho bar
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which is the distance of the particle with
respect to the moving coordinate system.
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So, from which we can say that d r bar d t,
xyz that is capital XYZ would be equal to
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d r d capital R bar d t plus d rho dashed
bar d t and these are of course, the velocities
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of the respective coordinate system. So, we
can write that down that the velocity of p
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with respect to the fix system would then
be capital V, X capital XYZ that would be
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equal to r dot plus v, xyz that is small xyz
plus omega that is the angular velocity into
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rho dashed. So, that gives us the victorial
summation of the velocity field that we have
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in front of us for the tracking of the fluid
particle through the two coordinate systems.
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Now, if we define the accelerations, the acceleration
of P with respect to fix coordinate system
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that is capital XYZ is given as a capital
XY Z equal to d V capital XYZ d t with respect
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to xyz and the acceleration of p with respect
to the rotating or moving coordinate system
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that is small xyz and that is a small xyz
equal to d V small xyz d t with respect to
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the small xyz coordinate system. These are
the two accelerations which we will be contending
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with, because we have seen before the acceleration
is is what contributes to the creation of
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dynamic forces that is mass into acceleration.
So, we need to write down what the acceleration
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terms are going to be for this three-dimensional
fluid flow.
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So, let us move forward, then we can write
down that the total acceleration of P with
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respect to the fix coordinate system capital
XYZ would be a capital XYZ and that is d V
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xyz d t with respect to the capital XYZ and
that would then be equal to d V small xyz
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d t plus r double dot, this is the acceleration
term the earlier one r dot was the velocity
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term of the position vector r and then d of
omega rho dashed d t; now, this is of course,
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the due to the rotational component of the
coordinate system is small xyz itself with
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respect to capital XYZ. So, this is acceleration
term that we can write down with respect to
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the coordinate system that we have defined
earlier.
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Now, the acceleration with respect to the
body fitted rotating coordinate system small
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xyz. Now, this is a summation of its translation
and rotating motion at the moment, we are
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assuming that it has translational motion
and rotating motions; and this we can write
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down in terms of d V small xyz d t and that
would be equal to our d V small xyz d t plus
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omega into V small xyz. Now, you see the rotating
component is the the second term over here.
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Now, if we write that down over here, we see
here d omega rho dashed d t xyz would be can
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be split up in two terms that is omega into
d rho dashed d t and d rho d t into omega
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dashed.
Now in the first term d rho dashed d t which
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is a differential of the position vector rho
dashed can be written down in terms of d rho
20:33.210 --> 20:40.210
of dashed d t is equal to d rho dashed d t
into small that is of the small xyz plus omega
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into rho dashed. So, rotation in xyz can now
be captured in the form of these equations
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with respect to the position vectors that
we have defined before.
20:56.680 --> 21:03.680
Now, acceleration of the particle P with respect
to the fixed origin o may be written down,
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now we have seen that we need to write down
the acceleration terms and these lines be
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edited in terms of a xyz that will be equal
to a small xyz plus r double dot plus twice
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of omega into V xyz plus omega into omega
into a rho dashed dot plus rho dashed rho
21:29.780 --> 21:36.780
dot plus omega dot omega dot into rho dash
dot. Now, if we assume, now this is the full
21:39.250 --> 21:44.260
the first gives the full equation that we
get of the acceleration term with respect
21:44.260 --> 21:49.650
to the fix coordinate system.
Now, if we assume that the angular velocity
21:49.650 --> 21:56.650
omega is constant, and it does not change
then the rho omega dot term would of course,
21:58.490 --> 22:04.680
immediately go and as a result of which we
get a little simpler a version of that acceleration
22:04.680 --> 22:11.680
term and that is a capital XYZ and that would
be equal to a small xyz plus twice omega V
22:13.870 --> 22:20.870
xyz plus omega into omega rho dashed. So,
this is the terminology that we finally, get
22:22.470 --> 22:28.760
for acceleration assuming that the flow is
a constant angular velocity which is very
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fair, because most of the time the axial flow
compressors are rotating with constant angular
22:35.150 --> 22:42.150
velocity except when they are in transition
during acceleration or deceleration of the
22:43.200 --> 22:44.030
machine.
22:44.030 --> 22:50.870
Now, if we look at for a compressor blade
passage, the flow velocities that we are looking
22:50.870 --> 22:57.870
at are V small, xyz and now this is the relative
velocity you remember, we had two velocities
22:59.040 --> 23:06.040
that we had defined one was relative velocity,
and another is the absolute velocity. Now,
23:06.550 --> 23:13.250
we can relate those two known velocities with
respect to what we are doing right now today.
23:13.250 --> 23:20.250
So, V small xyz is the relative velocity and
V capital XYZ is the absolute velocity C.
23:21.690 --> 23:28.180
We have seen earlier that sometimes the relative
velocity is the larger component, and that
23:28.180 --> 23:34.720
is typically at the inlet to a rotor sometimes
the absolute velocities the larger component
23:34.720 --> 23:41.720
and that could be true at the exit of the
rotor. So, V capital XYZ that is absolute
23:41.970 --> 23:48.710
velocity is of course, a summation of the
other two vectors that is V small xyz plus
23:48.710 --> 23:55.110
omega rho dashed.
Now, this can be written in terms of V plus
23:55.110 --> 24:01.730
omega r that we had written earlier r being
the radius at which the fluid particle is
24:01.730 --> 24:07.030
rotating with respect to the rotating coordinate
system, and this earlier we had written down
24:07.030 --> 24:14.030
as u. So, V plus u then becomes the victorial
summation which gives us C and this is what
24:14.510 --> 24:21.510
we had done earlier very quickly in our two-dimensional
flow theories. So, we come back to what we
24:22.010 --> 24:26.650
had done in two-dimensional flow theories,
in the sense we were nothing particularly
24:26.650 --> 24:31.730
wrong with them except that they were two-dimensional
in their nature.
24:31.730 --> 24:38.730
Now, if we differentiate this situation what
we get is acceleration terms, and now we see
24:40.860 --> 24:47.860
that a capital XYZ would be equal to d V capital
XYZ d t and then d omega rho dashed d t with
24:53.230 --> 25:00.020
respect to capital XYZ. Now, the first term
then is the translational motion and the second
25:00.020 --> 25:05.870
term is due to the rotational motion. So,
the both the terms would need to be taken
25:05.870 --> 25:12.870
into account for finding the total acceleration.
And then the final acceleration then is a
25:13.070 --> 25:20.070
capital XYZ is equal to a small xyz and then
twice of omega V xyz plus omega into omega
25:23.590 --> 25:28.060
rho dashed, now this is what we have done
earlier if you remember in the last slide.
25:28.060 --> 25:34.780
So, we get back. So, this equation with constant
angular velocity is what we again get back
25:34.780 --> 25:41.530
here using the concepts that we had used earlier
in our two-dimensional flow theories.
25:41.530 --> 25:48.530
So, we get the same acceleration term, a combination
of acceleration with respect to rotating motion,
25:50.570 --> 25:57.570
and combination of the two and the translational
motion. Now, the third term is often sometimes
25:57.800 --> 26:03.140
referred to has carioles force on carioles
acceleration, and we have done that before
26:03.140 --> 26:10.140
also, and this is what typically comes out
of a full mathematical formulation that we
26:10.940 --> 26:13.340
are attempting in this class today.
26:13.340 --> 26:20.340
Now, if we consider that equilibrium of forces
along arbitrary flow direction s. we get between
26:23.440 --> 26:30.440
any two actual stations separated by a small
distance delta S where A i is constant, A
26:32.310 --> 26:39.310
i being that area at that particular station.
So, if we take two stations of an axial flow
26:40.760 --> 26:47.760
compressor longitudinally. The during which
we assume that A i is constant. A i being
26:49.050 --> 26:55.710
the area annulus area at that particular station.
And we assume that is constant and in this
26:55.710 --> 27:02.710
longitudinal flow field which is a small distance
delta S. And in this we try to track what
27:04.430 --> 27:10.720
is happening in the flow field in this small
distance delta S that we have defined here.
27:10.720 --> 27:16.690
Now, over this distance what we see here is
the pressure force that is working, this is
27:16.690 --> 27:22.980
of course, the static pressure force delta
p in to A i which is constant now, over this
27:22.980 --> 27:29.980
flow field, and then we have A i into rho,
rho now is density please understand we have
27:31.550 --> 27:37.410
always used rho as density and that is why
we used rho dashed earlier has a position
27:37.410 --> 27:44.410
vector. So, this rho is density and delta
S which is the distance. So, A into this is
27:44.670 --> 27:49.770
the volume and rho is the density. So, that
is the mass and this is acceleration. So,
27:49.770 --> 27:56.770
mass into acceleration is the force that is
the dynamic force that has been created and
27:57.000 --> 28:02.420
this is the axial force that has been created
by delta p and that is the pressure differential
28:02.420 --> 28:09.420
or pressure will change over this distance
small distance delta S.
28:09.470 --> 28:16.470
If we resolve this, from this what we get
is one by rho delta p by delta S equal to
28:17.510 --> 28:24.510
acceleration a xyz. So, this is now the pressure
gradient along the distance delta s. Now,
28:25.970 --> 28:32.970
we can write down that the acceleration equation
from the last slide, now can be written down.
28:34.380 --> 28:41.380
In terms of this acceleration term can now
be replaced by one by rho delta p, and then
28:43.530 --> 28:50.530
we get the D V D t plus omega square r position
vector rho dashed is now replaced by r which
28:51.730 --> 28:58.620
we are familiar with r being the radial distance
of a particle from the axis of rotation of
28:58.620 --> 29:05.620
the moving body that is the axial flow compressor
access, and twice omega V which we had called
29:07.090 --> 29:11.120
sometimes in some books is referred to as
cordials force.
29:11.120 --> 29:16.290
So, this is what the equation we get now.
So, acceleration term has now been replaced
29:16.290 --> 29:23.110
by pressure terminology or pressure term,
pressure gradient along the distance s or
29:23.110 --> 29:24.980
[de\delta] delta s.
29:24.980 --> 29:31.410
So, with this we can now write down that as
the flow in compressor blade is a diffusing
29:31.410 --> 29:38.410
flow, we need to think that D V D t is likely
to be negative; that means, the flow to the
29:39.130 --> 29:45.910
blade passage is actually decelerating flow.
So, the D V D t is likely to be a negative
29:45.910 --> 29:52.910
in nature, and as a result of which the equation
can be slightly recast as minus 1 by rho into
29:53.470 --> 30:00.470
delta p equal to a D V D t minus omega square
r plus two twice two cross omega cross v.
30:03.290 --> 30:09.610
So, that is here in Victoria representation
of the corialise force.
30:09.610 --> 30:16.610
So, we get compressor blade; now, positioned
signifying that D V D t is most likely to
30:18.880 --> 30:20.990
be negative in nature.
30:20.990 --> 30:27.990
Now, this gives us a situation that we now
need to look at representing the flow in a
30:28.590 --> 30:35.590
manner that gives us a generalized flow path
of the particle, while the position of the
30:36.980 --> 30:43.980
particle is being tracked, it as its rotational
motion captured in the form of omega, and
30:44.590 --> 30:50.910
then we have the three coordinate system which
we are familiar with the radial coordinate
30:50.910 --> 30:57.140
system is r the axial coordinate system is
a and a tangential or the peripheral coordinate
30:57.140 --> 31:03.200
system as w which we call the whirl component.
Now, this is what we have been doing earlier
31:03.200 --> 31:08.370
in our two-dimensional flow theories where
we had the radial component, we had the actual
31:08.370 --> 31:13.990
component and we had the whirl component,
and we call the three velocities for example,
31:13.990 --> 31:20.990
V a, V r, and V w. Now, V r had been neglected
earlier in all two-dimensional flow theories,
31:22.380 --> 31:28.530
but we are going to bring that back here,
and V w remains the whirl component or the
31:28.530 --> 31:34.560
peripheral component, and we would now go
back to those terminologies which we had used
31:34.560 --> 31:41.430
earlier, and are somewhat familiar to us from
the earlier theorizing that we have done.
31:41.430 --> 31:48.430
So, as we can see there is nothing particularly
different about the notation that we had used
31:48.810 --> 31:55.810
earlier, and the notation that we have introduced
in today's lecture; you can find very easily
31:56.540 --> 32:02.310
the parallels between the two, and the equality
between the two sets of notations. So, you
32:02.310 --> 32:07.840
may have to really just sit down and look
at these notations and find that these things
32:07.840 --> 32:14.730
actually mean one and the same thing. So,
with this we look at the new axis of notations
32:14.730 --> 32:15.090
system.
32:15.090 --> 32:22.090
Now, the assumptions made here are that the
fluid is frictionless, the rotor is rigid
32:24.100 --> 32:29.140
and rotates with constant angular velocity
an assumption we made just a little while
32:29.140 --> 32:36.140
earlier; the flow is steady relative to the
rotor, we assume that the flow is not experiencing
32:39.420 --> 32:46.080
any unsteadiness. We will formalize it in
our mathematical formulation and the radial
32:46.080 --> 32:52.800
variation of density is neglected. Now, this
is something which is an assumption in many
32:52.800 --> 32:59.800
flows that may not actually be true, but radial
variation of density to begin with for design
32:59.820 --> 33:06.820
purposes, and for design analysis look that
assumption is reasonable and a valid assumption
33:07.080 --> 33:13.300
and maybe proceeded with.
Now, this leaves is still enough scope for
33:13.300 --> 33:20.250
formation of viscosity, then entropy radiands
and stagnation enthalpy gradients in the flow
33:20.250 --> 33:27.250
field from hub to the tip of a typical compressor
blade. So, these gradients and the formation
33:28.840 --> 33:35.100
of the passage vortex, and the tip vortex,
and the trailing edge vortex all these things
33:35.100 --> 33:42.100
would indeed bring in the three-dimensionality
which actually impacts the real flow, and
33:43.070 --> 33:50.020
we are trying to see how much of these things
can be captured a priory in a mathematical
33:50.020 --> 33:56.760
formulation, so that it can aid the design
and analysis process of axial flow compressor.
33:56.760 --> 34:03.760
So, these are those assumption that we have
to live within our mathematical formulation.
34:05.370 --> 34:10.129
The other things we are going to bring in
are the definition of the unit vectors which
34:10.129 --> 34:17.129
we have written down here are in terms of
D i r i being the unit vector in the radial
34:18.050 --> 34:25.050
direction. And then D i w which is one in
the whirl component or the whirl direction,
34:26.200 --> 34:33.200
so the overall V is a V r into unit vector
i r V w into unit vector i w and V a into
34:36.849 --> 34:43.849
unit vector, i a those in part the directionality
with which is the flow is proceeding in or
34:45.409 --> 34:52.409
the flow has three components in radial whirl
component and the actual component.
34:53.190 --> 35:00.089
The other thing we are in to do is look at
the fact that a capital DDT is a total operator.
35:00.089 --> 35:06.500
Now, from your basic mathematics you know
that the total operator can be split up in
35:06.500 --> 35:13.500
two terms, one is the operator with respect
to the space coordinate where s is the length
35:13.710 --> 35:20.710
in any direction; so, the same term can be
written down in terms of total operator DDS
35:22.019 --> 35:29.019
and that plus d s d t, now the d s d t of
course, is the unsteady terminology, t being
35:30.529 --> 35:36.910
the time; now, as a result of which d s d
t is an unsteady terminology if the flows
35:36.910 --> 35:42.339
are shown to be steady and at the moment we
are assuming the flow to be steady, then the
35:42.339 --> 35:49.150
d s d t is equal to zero. Hence, that term
would be neglected and as a result will be
35:49.150 --> 35:56.150
left with only the total operator with respect
to the space of the distance s, and the with
35:56.829 --> 36:00.630
respect to time it is considered to be zero.
36:00.630 --> 36:07.630
So, the equation for the flow inside a compressor,
we would now like to recast using r theta
36:09.079 --> 36:15.859
z coordinate system, because simply because
it is a more appropriate for a rotating coordinate
36:15.859 --> 36:22.720
system also, because of the fact that. There
is a variation of parameter in the circumferential
36:22.720 --> 36:28.619
direction and from blade to blade, and this
blade to blade circumferential variation is
36:28.619 --> 36:35.619
quite often an important variation. We know
that the two surfaces of the blades are indeed
36:36.180 --> 36:41.470
actually dissimilar, one is a suction surface
and on[e] one is a pressure surface.
36:41.470 --> 36:48.470
So, theta variation from suction to pressure
surface is quite often an important variation
36:48.609 --> 36:53.529
it sometimes it is caught a lot in highly
loaded, and we shall we be know that that
36:53.529 --> 36:58.220
variation also changes from hub to the tip
of the blade. So, variation from one blade
36:58.220 --> 37:04.480
surface to the other in the passage is an
important variation, it everything varies
37:04.480 --> 37:10.359
the pressure varies, the velocity varies,
the temperature varies, the enthalpy varies
37:10.359 --> 37:17.359
from one surface to the other in one passage.
So, that can be tracked by variation of theta
37:18.119 --> 37:23.839
that is the angular circumferential variation
from one surface to the other, the other is
37:23.839 --> 37:29.190
of course, from root to tip, and the third
is in the z direction that is axial coordinate,
37:29.190 --> 37:35.500
axial distance.
So, if we use r theta z coordinate system,
37:35.500 --> 37:42.500
we see that DV Dt can be written down, now
in terms of unit vector i r into D V r D t
37:43.589 --> 37:50.589
V w d theta d t plus unit vector i w DV w
Dt plus V r d theta d t plus unit vector i
37:53.809 --> 38:00.809
a DV a Dt. Now, these coordinate systems and
the flow velocities in the relative frame
38:02.529 --> 38:09.529
[weto\we] we always have considered as V v
its components are being shown again as V
38:11.250 --> 38:18.250
a, V r and V w as we have done before in a
2-D flow theories, where we have used use
38:19.130 --> 38:26.130
V a and V w we did not use V r as at that
time, we assume that V r does not exist. So,
38:27.829 --> 38:34.579
we see here that the acceleration term can
be split up in three components.
38:34.579 --> 38:41.579
Now the equation may be resolved in its three
components using the r, theta z coordinate
38:41.809 --> 38:48.680
system that we have brought in now; now the
what it shows is that the three components
38:48.680 --> 38:55.680
are 1 by rho del p del r and that would be
equal to V d V r d s minus V w plus omega
38:58.509 --> 39:05.509
r square by r that is equation S. Now, that
is in the radial direction; so, the left hand
39:06.930 --> 39:13.930
term is the pressure gradient, and the right
hand term is the dynamic acceleration terminology
39:15.039 --> 39:22.039
of the terms; the second equation is in the
circumferential direction theta coordinate
39:22.890 --> 39:29.890
and that is 1 by rho del p del theta, 1 by
rho 1 by r del p del theta. So, that is the
39:31.950 --> 39:38.890
pressure gradient in the theta direction and
that would be equal to V into DV a DS plus
39:38.890 --> 39:45.890
V w into V r by r plus twice omega V r and
that is the equation two that is in the circumferential
39:49.440 --> 39:54.009
direction.
In the actual direction it is 1 by rho del
39:54.009 --> 40:01.009
p del z equal to V DV a DS and that is in
the actual direction. So, again the left hand
40:03.390 --> 40:09.880
term is the pressure gradient and the right
hand term is the acceleration term, which
40:09.880 --> 40:15.720
gives rise to the dynamic or the kinematic
forces. So, these are the three components
40:15.720 --> 40:21.779
of the equation that we had written down in
the earlier slide, and three components are
40:21.779 --> 40:27.849
in the radial circumferential and in actual
directions.
40:27.849 --> 40:34.849
We can now denote these components in a very
simple vector diagram, in which all the velocity
40:36.569 --> 40:43.029
components are shown here, we see the velocity
velocity vector V r has now appeared, we are
40:43.029 --> 40:50.029
tracking fluid flow particle and this now
has three components V r, V a which is actual
40:51.940 --> 40:58.940
velocity. We are looking at a particular plane
on which we have the fluid particle, and this
41:00.180 --> 41:07.180
fluid particle is experiencing a rotating
motion, omega it is at a distance r from the
41:07.269 --> 41:14.269
centre of rotation and it is at a [ang\angular]
at a angular distance theta from the reference
41:15.170 --> 41:22.170
axis. So, at this position it has three velocity
components V a that is axial velocity, V r
41:23.720 --> 41:29.099
that is the radial velocity, and V w that
is the peripheral velocity.
41:29.099 --> 41:35.309
A combination of all the three is of course,
the overall relative velocity with respect
41:35.309 --> 41:40.880
to this coordinate system which write in the
beginning in today's lecture, we have seen
41:40.880 --> 41:46.599
itself may be moving with respect to a fixed
coordinate system. But let us look at this
41:46.599 --> 41:53.599
moving coordinate system analysis right now.
Now, we see that a combination of V r which
41:53.779 --> 42:00.779
is the radial component and V a the axial
component also gives a velocity which can
42:00.980 --> 42:07.980
be now described as V m, and these V m is
velocity which is a resultant of V r and V
42:09.430 --> 42:16.430
a and is on a particular plane of this plane
is a radial, and axial plane a combination
42:18.039 --> 42:23.299
of radial and axial plane. In this plane,
which is perpendicular to the plane we are
42:23.299 --> 42:30.299
looking at which is the radial and peripheral
plane and, V m is on that radial axial plane
42:33.339 --> 42:37.970
and this V m is referred to as meredional
velocity.
42:37.970 --> 42:42.900
We will come to this definition of meredional
velocity. So, the velocity triangle for this
42:42.900 --> 42:49.900
flow gives us that the C w would be equal
to V w plus omega r, what we see now is that
42:51.049 --> 42:58.049
the velocity triangles which we are all familiar
with actually gave as C w that is equal to
42:58.140 --> 43:05.140
the axial velocity equal to V w plus omega
r omega being the angular velocity, and in
43:06.499 --> 43:13.499
the earlier coordinate system that we have
just seen we get a V w which is the peripheral
43:13.980 --> 43:20.150
component a relative peripheral component
with respect to the rotating coordinate system.
43:20.150 --> 43:27.150
So, we come back to our C w which is vectorially
has to be equal to V w plus omega, r in this
43:27.680 --> 43:32.230
coordinate system, you remember this whole
thing is rotating or the particle is rotating
43:32.230 --> 43:38.630
with respect to this with angular velocity
omega. So, this entire coordinate system is
43:38.630 --> 43:45.630
a rotating with angular velocity omega. Now,
the equation a and b from the two slides earlier,
43:46.220 --> 43:52.700
we had three components from the slide 21,
let us go back very quickly what we see is
43:52.700 --> 43:59.700
we have two components of the full equation
in two directions the radial and the peripheral.
44:01.019 --> 44:08.019
If we bring forth, then we have the radial
component can be recast as one by minus one
44:09.109 --> 44:16.109
by rho d p d r equal to V into d V r D s minus
C w square r by r and this rewrite as a equation
44:23.329 --> 44:30.329
d, and the second equation is now one by rho
r into the del p del theta equal to V by r
44:31.920 --> 44:38.920
into D r C w D s. So, what we have done is
we have replaced the V w is now with C w,
44:40.890 --> 44:47.890
because that is the axial velocity component
that the flow is actually having a with respect
44:47.890 --> 44:54.180
to the fixed coordinate system. So, the equations
have been slightly recast.
44:54.180 --> 45:01.180
If we now look at the flow, we can now use
the kinematic relations as V D capital operator
45:03.119 --> 45:10.119
D D S equal to V a capital operator D V a
of any parameter where V a and a are the axial
45:13.150 --> 45:20.150
components of V and S. S being the distance
and a is the axial distance and V a is the
45:20.839 --> 45:27.839
axial velocity and V o is the overall velocity.
We can now define a meredional velocity to
45:28.240 --> 45:35.240
begin with let us define a meredional direction,
and this meredional direction is D m into
45:35.829 --> 45:42.829
unit vector i m equal to D r into unit vector
i r plus D a into unit vector i a and this
45:44.359 --> 45:49.890
is what I was talking about just a little
while earlier that this meredional flow is
45:49.890 --> 45:54.029
in a plane that is composed of radial and
axial plane.
45:54.029 --> 46:01.029
So, in this plane we can have a meredional
direction defined, and this meredional direction
46:01.269 --> 46:08.269
of a flow or a particular flow path a flow
particle taking a particular path would have
46:08.490 --> 46:15.460
would create an angle phi with respect to
the axial direction. So, this phi is the angle
46:15.460 --> 46:22.410
with which the meredional flow is proceeding
in the meredional direction, which is not
46:22.410 --> 46:29.099
the same as axial direction. Now, the equation
d from the last slide can be rewritten as
46:29.099 --> 46:35.930
minus 1 by rho del p del r equal to V m we
have brought in the meredional velocity V
46:35.930 --> 46:42.930
m and that is d V r D m and d m is the meredional
direction minus C w square by r.
46:45.190 --> 46:50.849
Now, C w of course, we are familiar with is
the whirl component of the absolute velocity
46:50.849 --> 46:57.849
are being the distance of the fluid particle
with respect to the axial coordinate system.
46:58.150 --> 47:03.359
As a result of which we get a for motion again
on the left hand side, we have the pressure
47:03.359 --> 47:09.769
gradient on the right hand side, we have the
acceleration terms due to the dynamics of
47:09.769 --> 47:11.440
the flow.
47:11.440 --> 47:17.660
This gives us the fact that in the meredional
direction, if we complete the definition of
47:17.660 --> 47:24.660
meredional direction 10 phi is V r by V z
or V a and V is V of meredional direction,
47:27.640 --> 47:34.640
V m into sign phi and phi is defined here
and as a result of which we can write down
47:34.749 --> 47:41.749
that the force balance equation only in the
radial direction can be recast or rewritten
47:42.069 --> 47:49.069
as one by rho del p del del p del rho is equal
to C w square by r minus V m square d sin
47:55.259 --> 48:02.259
phi d m equal to minus V m sin phi d V m D
m. Now, this is the full equation in the radial
48:04.289 --> 48:11.289
direction. Now, by our definition V D D s
the total operator is equal to V m D D m,
48:14.249 --> 48:21.249
this is mathematically approved and as a result
of which we can replace with respect to the
48:21.410 --> 48:27.009
meredional velocity and using the meredional
direction in this equation.
48:27.009 --> 48:34.009
So, if we look at the flow as we [desc\describing]
what we are describing is that we have other
48:34.230 --> 48:41.230
meredional plane over here now; and this is
the direction of the fluid particle of fluid
48:42.890 --> 48:49.890
path given as phi, V r is the radial flow
component, V a is the axial flow component
48:50.619 --> 48:57.609
and V m here is the meredional flow component
in a fluid path. And this is what we are talking
48:57.609 --> 49:03.119
about an V w of course, remains the peripheral
component or the whirl component that we have
49:03.119 --> 49:09.349
talked about which means this total V would
be the full complement of all the three components
49:09.349 --> 49:16.349
put together omega of course, is the rotational
angular velocity that the entire flow is experiencing
49:16.749 --> 49:18.880
with the rotating blade.
49:18.880 --> 49:25.880
So, given this model, given this model we
can now write down that D sin phi D m would
49:27.549 --> 49:34.549
then be equal to cos phi D phi D m, and D
phi D m will be equal to minus one by r m.
49:37.079 --> 49:43.579
Now, minus one by r m is indeed the radius
of curvature of the meridional flow, and this
49:43.579 --> 49:49.390
radius of curvature of the meridional flow
is due to the fact that the meridional path
49:49.390 --> 49:55.170
may not be linear, it may not be a curved
path and if it is a curved path the radius
49:55.170 --> 50:02.170
of curvature of that path. In the meridional
direction would create a certain centrifugal
50:03.130 --> 50:09.119
action, and hence the meridional radius of
curvature of the meridional flow is being
50:09.119 --> 50:13.450
brought into the picture.
Now, this negative sign is a somewhat arbitrary,
50:13.450 --> 50:20.160
but the axial flow compressor the flow track
inside generally moves towards a lesser value
50:20.160 --> 50:27.160
of phi as it moves actually forward that is
the r m becomes higher and higher which means
50:28.069 --> 50:34.660
the flow tends to become a later on flatter
and flatter. So, it moves from a high r m
50:34.660 --> 50:40.799
to low r m as it moves forward through the
blade passage. Hence, the full radial equilibrium
50:40.799 --> 50:47.799
that we are looking at now is 1 by rho del
p del r equal to C w square by r plus cos
50:49.799 --> 50:56.799
phi into V m square by r m minus V r D V m
D m. And this is the full radial equilibrium
51:00.890 --> 51:07.619
equation which in the say is a circumferential
average from blade to blade flow properties
51:07.619 --> 51:12.920
inside a termination blade row.
Now, this radial equilibrium equation captures
51:12.920 --> 51:18.799
the all the components of the dynamics of
the flow, in terms of all the acceleration
51:18.799 --> 51:25.799
terms that come about due to three-dimensionality
of the flow. Now, this we can see is has a
51:27.499 --> 51:34.499
more terms than what we had seen in the simple
radial equilibrium equation, we had done couple
51:34.599 --> 51:41.499
of classes earlier which tells us that more
of the dynamics or kinematics of the flow
51:41.499 --> 51:45.739
have been captured in our mathematical formulation.
51:45.739 --> 51:52.739
And this tells us that if we now simplify
backwards; that means, if we consider that
51:54.769 --> 52:01.769
the flow is actually more or less actual then
the last term is actually eliminated in the
52:02.700 --> 52:08.630
very early designs, it was a done thing that
the flow path was considered linear, and the
52:08.630 --> 52:15.630
second term is also then neglected or it vanishes
and which gives us back the simple radial
52:16.670 --> 52:23.670
equilibrium equation which we had formulated
earlier in terms of one by rho del p del r
52:24.039 --> 52:29.890
equal to c w square by r.
Now, this was the simple radial equilibrium
52:29.890 --> 52:36.690
equation; what we need to look at now is a
full radial equilibrium equation and we see
52:36.690 --> 52:43.329
now that a the simple radial equilibrium equation
was somewhat inadequate, and it becomes necessary
52:43.329 --> 52:50.329
to utilize the full equilibrium in design
and analysis of modern axial flow compressors.
52:51.930 --> 52:57.160
Now, if we go back and summarize some of the
things that we have done, we can say that
52:57.160 --> 53:03.640
wherever the flow is not experiencing the
centrifugal force the radial equilibrium cannot
53:03.640 --> 53:10.640
be applied, and with with this understanding
it was at one time assume that radial equilibrium
53:11.769 --> 53:18.769
is not applicable between the rotor and the
stator where no blades exist. However, experience
53:19.289 --> 53:24.890
experiments have shown that in between the
blade rows in the axial gap between the rotor
53:24.890 --> 53:30.489
and the stator there could be a radial shift
of the meridional path, and hence for accurate
53:30.489 --> 53:37.009
design analysis, radial equilibrium full radial
equilibrium equation should be used in that
53:37.009 --> 53:42.930
flow path also between the rotor and the stator.
For the computational purpose is further steps
53:42.930 --> 53:49.920
need to be taken from this mathematical form,
the radial equilibrium equation is transformed
53:49.920 --> 53:56.920
into a form that contains partial derivatives
of all parameters with respect to radius and
53:58.039 --> 54:05.039
a peripheral coordinate theta, because all
the parameters like pressure, and density
54:05.440 --> 54:11.210
and enthalpy are likely to vary from root
to the tip of the blade and also likely to
54:11.210 --> 54:17.759
vary from blade to blade. So, this blade to
blade variation and root to tip variation
54:17.759 --> 54:24.470
need to be captured, I will in the next slide
how the physical model or mathematical model
54:24.470 --> 54:28.450
attempts to capture it in its computational
form.
54:28.450 --> 54:35.450
The next circumferential average of those
parameters is then integrated over the theta
54:35.730 --> 54:41.049
from the pressure side of one blade to the
suction side of the other blade. And then
54:41.049 --> 54:47.819
the flow is analyzed and various axial stations
with energy equation, continuity equation
54:47.819 --> 54:54.599
and the radial equilibrium equation. So, you
need all three of them to analyze the flow.
54:54.599 --> 55:00.180
Now, this is what I was talking about that,
you have a model in which you have two blades
55:00.180 --> 55:07.180
in between the two blades, you need to create
these surfaces, these are known as S 1 surfaces
55:08.690 --> 55:15.400
which if you very S 1 from root to the tip
of the blade, you have the variation of various
55:15.400 --> 55:20.589
parameters from root to the tip of the blade,
base two surfaces from blade to blade that
55:20.589 --> 55:25.569
is from pressure surface to suction surface
of the other blade. So, if you take the variation
55:25.569 --> 55:31.739
of parameters on is two surface, and then
vary them circumferentially in the theta direction
55:31.739 --> 55:38.739
from one surface to the other, you have the
variation of all the parameters in the circumferential
55:38.819 --> 55:42.710
direction.
After you integrate on the S two surface and
55:42.710 --> 55:49.549
on the S one surface you get integrated parameters
of all the parameters. So, this model has
55:49.549 --> 55:56.549
been used by the computational people to transform
radial equilibrium equation into a computational
55:56.710 --> 56:03.039
tool for analyzing the flow through action
flow compressors.
56:03.039 --> 56:08.400
It is necessary that the flow properties obtained
in this manner and various actual stations
56:08.400 --> 56:14.170
be consistent with one another as the flow
properties are evaluated from up to the tip
56:14.170 --> 56:19.720
at each station, that mean that the radial
acceleration of the fluid particle is to be
56:19.720 --> 56:26.470
accounted for through radial equilibrium equation,
which is the additional equation, you need
56:26.470 --> 56:31.349
to use in addition to the energy equation
and the continuity equation or continuity
56:31.349 --> 56:37.989
condition for analyzing flow through action
flow compressors. Now, this can be achieved
56:37.989 --> 56:44.489
by assuming shapes for the meridional streamlines.
So, you need to assume the shapes of the meridional
56:44.489 --> 56:50.509
streamlines, consistent with the continuity
condition, which expresses the radial acceleration
56:50.509 --> 56:57.509
in terms of the streamlined slow and the streamlined
curvature. So, you got to find the meridional
56:57.529 --> 57:04.529
path of the meridional streamlined along which
you can apply the radial equilibrium equation
57:04.729 --> 57:11.200
that meridional path can be found by using
the continuity condition along the meridional
57:11.200 --> 57:15.380
path.
This of course, implies that unique may need
57:15.380 --> 57:21.559
to hydrate the method of solution, because
paraphrasing to begin with you may not know
57:21.559 --> 57:27.729
the meridional path, you may not know that
streamlines that create the meridional path.
57:27.729 --> 57:33.960
This method in which surfaces are used to
build up the flow inside to missionary blade
57:33.960 --> 57:40.259
has been widely used and this wide you widely
used formulation is being used for last twenty
57:40.259 --> 57:45.799
twenty five years for analyzing flow through
action flow compressor from blade to blade
57:45.799 --> 57:52.799
surface and those on the meridional plane,
and they need to be solve separately to account
57:52.839 --> 57:59.839
for all aspects of the three-dimensionality
of the flow through axial flow compressors.
58:00.470 --> 58:05.299
The 3-D flow contributions has provided huge
assistance to the Indian designers of the
58:05.299 --> 58:10.769
compressor design is more specifically, it
is cut down the design time and it has reduced
58:10.769 --> 58:17.059
dependence on costly experimental analysis,
experimental analysis making the blade testing
58:17.059 --> 58:22.630
it in the real is a very costly business and
time consuming business; competition is cut
58:22.630 --> 58:29.630
it down substantially. The 3-D methods have
helped understand various flow phenomena such
58:29.779 --> 58:36.499
as the secondary flow development, choking
in the stages, as an when the occur aspects
58:36.499 --> 58:43.279
end of wall flows, not to speak of problems
like stalling, and those can be then sorted
58:43.279 --> 58:48.349
out in the design itself, before the blade
is actually fabricated.
58:48.349 --> 58:54.720
However the designers use these solutions
quite often in conjunction, even today in
58:54.720 --> 59:00.819
conjunction with empirical relations specially
the losses that occur are very difficult to
59:00.819 --> 59:07.440
capture in CFD, and hence quite often empirical
relations do again come back into picture,
59:07.440 --> 59:14.440
and experimental data to account for some
of the losses, and thereby predict the performance
59:14.819 --> 59:21.599
of the design action flow compressor blade,
which means that there is still a scope for
59:21.599 --> 59:28.099
improvement of these methods. So that the
empiricism, the empirical formulations are
59:28.099 --> 59:33.450
reduced further to the minimum, some of it
may have to be done in the last stage of the
59:33.450 --> 59:40.450
design before the blades are actually fabricated
and tested; however the dependence on the
59:41.390 --> 59:47.279
empirical formula can be reduced, if this
is the method is taking forward. So that more
59:47.279 --> 59:52.470
and more of the three-dimensionality that
we have seen right in the first slide is captured
59:52.470 --> 59:58.229
in the formulation.
So, today's class we have done lots of attempt
59:58.229 --> 1:00:03.989
to capture a lots of three-dimensionality
in the mathematical form and finally, we have
1:00:03.989 --> 1:00:10.410
got a full radial equilibrium equation, that
is substantially different and for more complex
1:00:10.410 --> 1:00:16.470
in Bob Anderson radial equilibrium equation
that we had done earlier. This allows us to
1:00:16.470 --> 1:00:23.470
go forward and do an analysis of a modern
action flow compressor and eight in the process
1:00:23.890 --> 1:00:30.630
of the design of this modern axial flow compressor.
So, that brings us to the end of today's lecture
1:00:30.630 --> 1:00:36.489
of creating a mathematical formulation of
flow through a three-dimensional flow through
1:00:36.489 --> 1:00:43.289
axial flow compressor. In the next class,
we will look at and tent to solve some of
1:00:43.289 --> 1:00:48.660
the problems of simple 3-D flow theory is
that we have done earlier, and we will try
1:00:48.660 --> 1:00:54.319
to use those theories to solve some of the
problems, use the free vortex theory and other
1:00:54.319 --> 1:01:00.299
vortex theory to solve some of the typical
problems, and show how the numbers actually
1:01:00.299 --> 1:01:03.539
come up in solution of axial flow compressors.