WEBVTT
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Hello and welcome to lecture number 20 of
this lecture series on Turbomachinery Aerodynamics.
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We have we have probably half way through
this course, and I
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guess you must have had some good idea about
what in, what is involved in turbomachinery
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analysis, and what is involved in design of
different types of turbo
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machines, especially the compressors. Now,
starting the last lecture onwards, we are
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now looking at the axial turbines, and of
course subsequently we were also
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in talking about the radial turbines and so
on.
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So, I think in the last class, you must have
had got some introduction to what axial turbines
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are, and what constitutes axial turbines and
so on. So, let us take that
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discussion little bit further, in today's
class where we will we will be talking about
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two-dimensional analysis of axial compressors,
well axial turbines. In a very
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similar fashion to what we had discussed for
axial compressors. If you remember during
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one of the initial lectures probably the lecture
- the second lecture or the
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third lecture, we had been talking about axial
compressors, and how one can analyze axial
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compressors in a two-dimensional sense. So,
we will we will carry out
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the similar analysis and discussion in today's
class about how the same thing can be carried
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out for turbines, axial turbines in particular.
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In today's class we basically being going
to talk about the following topics. We will
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initially discuss, have some introduction
to axial turbines, turbines in general.
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We will talk about impulse and reaction turbine
stages which are the two basic types of axial
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turbines. We will then talk about the work
and stage dynamics, how
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you can calculate work done by a turbine,
and how is it different for impulse and reaction
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turbines. We will then spend some time on
discussion about turbine
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blade cascade.
We will assume that the nomenclature we had
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used in the case of compressors, will still
be valid, but of course we will just highlight
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some simple differences
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between a compressor cascade, and a turbine
cascade, but the nomenclature remains the
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same in the sense that what we had called
as camber or stagger or
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incidence all that remains the same for a
turbine. So, I will probably spend lesser
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time discussing about those and take up some
more topics on cascade analysis
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which we had not really covered in detail
in compressors.
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Now, when we talk about turbines, you must
have had some discussion, some introduction
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to some the different types of turbines. As
we know the different
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types of compressors like axial and centrifugal;
similarly, we have different types of turbines
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as well. Now, in a turbine just like in a
compressor, we have
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different components. In a compressor, we
know that we have rotor followed by a stator.
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In the case of turbines we have a nozzle or
a stator which pre-seals a
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rotor. So, a nozzle or a stator guides and
accelerates the flow into the into a rotor,
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and of course, the work extraction takes place
in the rotor. And which is unlike
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in a compressor, where it is the rotor which
comes first and drives the flow, and then
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that goes into a stator which again turns
it back to the acceleration and so
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on, diffusion takes place in both the rotor
and the stator.
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In a turbine, as well you could have differential
amounts of acceleration or pressure drop taking
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place in the rotor and the stator. And there
are certain types of
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turbines where the entire pressure drop takes
place only in the stator, this rotor does
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not contribute to any pressure drop, it simply
deflects the flow, these are
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called impulse turbines. We will discuss that
in little more detail in some of these later
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slides.
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So, basically the flow in a turbine is accelerated
in nozzle or a stator. And it then passes
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through a rotor. In a rotor, the working fluid
basically imparts momentum
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to the rotor, and basically that converts
the kinetic energy to power output. Now, depending
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upon the power requirement, this process obviously
is repeated in
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multiple stages, you would have number of
stages which will generate the required work
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output, which is also similar to what you
have in a compressor, where
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you might have multiple stages which basically
are meant to give you the required pressure
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rise in typical axial compressor.
Now, we have seen this aspect in compressors
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as well that due to the motion of the rotor
blades, you have basically two distinct components
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or types of
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velocities. One is the absolute component
or type of velocity and the other is relative
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component or relative velocity. This was also
discussed in detail in
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compressors, and so you would in in a turbine
the analysis that we will do in by analyzing
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the velocity triangle. You will see that there
are these two distinct
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components which will become obvious, when
we take up the velocity triangles, and this
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is very similar to what you had discussed
in compressors. So, if you
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have understood velocity triangle construction
for an axial compressor, it is pretty much
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the same in the case of a turbine as well.
So, that will that is probably
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the reason why it will make it simpler for
you to understand the construction of the
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velocity triangle.
Now, the fundamental difference between compressor
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and the turbine is the fact that the compressor
is required to generate a certain pressure
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rise, there is a
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work input into the compressor, which is what
is used in increasing the pressure across
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the compressor. Compressor operates in an
adverse pressure gradient
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mode; that is the flow always sees an increasing
the pressure downstream. In the case of a
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turbine, it is not that case, it is the other
way round that the flow
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always sees a favorable pressure gradient,
because there is a pressure drop taking place
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in a turbine which leads to a which is how
the turbine extracts work from
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the flow. That is it converts part of the
kinetic energy which the flow has into work
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output.
And therefore, in a turbine the flow always
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sees a favorable pressure gradient, and that
is one fundamental difference between a turbine
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and a compressor. Now,
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because you have a favorable pressure gradient,
the the problems that we have seen in the
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case of compressor like, flow separation and
blade stall and seers and
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all that does not really effect turbine, because
a turbine the flows always in a accelerating
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mode, and so the problem of flow separation
does not really limit the
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performance of a turbine. So, it is possible
that we can extract lot more work per stage
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in a turbine as compare to that of a compressor.
And therefore, you would
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if you have noticed schematic of typical modern
day jet engine you will find that there are
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numerous stages of compressor may be 15 or
20, which are actually
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given by may be 2 or 3 stages of turbine.
So, each stage of a turbine can actually give
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you much greater pressure drop, then what
we can achieve or the kind of pressure rise
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we can achieve in one stage
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of a compressor, which is why a single stage
of a turbine can drive multiple stages of
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compressors. So, that is the very important
aspect that you need to
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understand, because the fundamental reason
for this being the fact that turbines operate
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in a favorable pressure gradient, compressors
operate in an adverse
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pressure gradient.
So, there are limitations in a compressor,
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which will prevent us from having very high
values of pressure rise per stage; that is
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not a limitation in a turbine; and
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that is why you have much greater pressure
drop taking place in a turbine as compared
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to that of the pressure rise, that you get
from one stage of a compressor.
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So, turbines like compressors can be of different
types; the compressors we have seen can be
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either axial or centrifugal. In the case of
turbines, you get in fact in
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the some literatures also says we could also
have mixed type of compressors axial and centrifugal
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mixed.
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Similar thing is also there in the case of
turbine, you could have an axial turbine or
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a radial turbine or a mixture or combination
of the two called mixed flow
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turbines. Axial turbines obviously can handle
large mass flows, and obviously are more efficient
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as very similar analogy we can take from compressors,
which
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have larger mass flow and are obviously more
efficient. And axial turbine main advantage
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is that it has the same frontal area of that
of a compressor. And also it
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is possible that we can use an axial turbine
with that of a centrifugal compressor. So,
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that is also an advantage.
And what is also seen is that efficiency of
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turbines are usually higher than that of compressors.
The basic reason again is related to the common
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timer earlier that
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turbines operate in a favorable pressure gradient,
and so the problems that flows sees in an
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adverse pressure gradients is not seen; there
are no problems of flow
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separation except in some rare cases. And
this also means that theoretically come turbines
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are easier to design, well easier is in-cord
and un-cord, well in the
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sense that you know compressors require little
more care in terms of aerodynamic design,
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but of course turbines have a different problem,
because of high
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temperatures and so turbine blade cooling
and associated problems; that is an entirely
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different problem altogether.
So, aerodynamically if you have to design
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a compressor and a turbine, turbines would
be as easier to design than compressors, just
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because of the fact that. You
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do not to really worry about the chances of
flow separation across a turbine, because
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it is always an accelerating flow. In the
case of compressors that is not the
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case and there is always a risk that a compressor
might enter into stall. So, let us now take
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a look at. Now, that I have spoken lot about
types of turbines and
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there functions and so on, let us take look
at a typical axial turbine stage.
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So, what is shown here is a simple schematic
of an axial turbine stage. So, an axial turbine
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stage consists of as I mentioned nozzle or
a stator followed by a rotor.
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So, this is just representing a nozzle through
which hot gasses from the combustion chamber
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are expanded and then that passes through
a rotor which is what
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gives us the power output. Rotor is mounted
on what is known as disc, and of course, the
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flow from the rotor is exhausted into either
a next stage or through the
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component downstream which could be a nozzle
in the case of a air craft engine.
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So, usually we would be denoting the stator
inlet as station1, stator exit as station2
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and rotor exit exist as station3. In some
of the earlier generation turbines, the
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disc was a separate entity, rotor was mounted
on slots which were provided on the disc,
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and so those separation separate mechanism
for mounting rotor blades
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on the disc. Some of the modern day... So,
it was very soon realized that having a separate
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disc and different blades in obviously will
increase in the number of
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parts. So, the part count will increase tremendously.
So, but with modern day manufacturing capabilities
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in terms of 5 axis and 7 axis, numerical machines
called CNC machine, computer guided machines.
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It is
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possible for us to make them out of a single
piece. And this is done in smaller sized engines
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now, and some of the companies have their
own names for that, for
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example, GE called such a disc which is combination
of disc and the blade as blisc. Blisc means
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blade and disc together machined out of a
single piece of metal.
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And similarly, their competitors also have
their own terminologies like paternity called
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Integrated Blade Rotor or IBR; where there
is no distinct root fixture for a
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blade, because blade and a disc are a single
component.
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The main advantage being that you have reduced
significantly the number of parts. Whereas
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you would have let us say, typical turbine
blade may have
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something like 70 to 80 blades or even more
of course, mounted on a disc. So, that is
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like 80 to 90 parts for one stage of a rotor.
Now, if you have a blisc, you
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have just a one component, because all the
blades have been mounted on one disc. That
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is the tremendous advantage for in terms of
maintenance aspect.
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But at the same time, the primary disadvantage
is the fact that - if there is one blade which
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gets damaged, in the earlier scenario you
have just to replace the
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blade, here it becomes impossible to replace
the blade, and so then of course, you will
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have to do rebalancing of the disc, and if
the damage is severe then the
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whole disc as to be replaced. Of course, there
are of having integrated blade rotor concept
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and of course there are lot of disadvantages
and advantages. But for at
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least smaller engines economically that is
in the long run that seems to be an advantage
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that you have a combination of the blade and
the the disc.
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So, having understood some of the fundamentals
of turbines, let us move on to the more important
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aspect of analysis - the two dimensional analysis;
that is to do
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with velocity triangles. I think we spent
quite some time discussing velocity triangles
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for compressors. So, I will assume that you
have understood the
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fundamentals of velocity triangles and try
to kind of move on to constructing the velocity
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triangles just like that, unlike in compressors
where I had done it step
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by step. The process is exactly the same as
what you have done for a compressor. But of
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course it being a turbine there are certain
differences which you need to
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understand.
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Now, velocity triangle analysis is an elementary
analysis, and this is elementary to axial
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turbines as well, just like in the case of
compressors. Now, the usual
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procedure for analysis is to carry out this
analysis at the mean blade height, and we
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will have blade speed at that height assuming
to be U capital U. Absolute
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component of velocity will be denote by C
and relative component we will denote by V.
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And the axial velocity the absolute component
of that is of denoted by
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C subscript a, just like in compressors, tangential
components will be denoted by a subscript
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w. So, C w is of absolute component of tangential
velocity, V w is
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the relative component in the tangential direction.
And regarding angles, alpha will denote the
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angle between the absolute velocity and the
axial direction, and beta denotes the corresponding
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angle for relative
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velocity. So, these are the terminologies,
nomenclature that we have used, even in a
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compressor we will follow exactly the same
nomenclature in the case of
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turbine as well. So, let us move directly
to a velocity triangle of a typical turbine
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stage.
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So, turbine stage as we have already seen
consists of a rotor, well a stator or a nozzle.
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It is usually refer to as nozzle in the case
of turbine, because the flow is
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accelerated in in an stator of a turbine,
and that is why it is called an nozzle, and
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then you have a rotor which follows a stator
or a nozzle. Now, inlet to stator is
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denoted as station 1, exit is denoted as station
2, exit of the rotor is station 3. So, let
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us say there is an inlet velocity which is
given by C 1 which is absolute
17:39.870 --> 17:46.870
velocity entering at an angle alpha 1, it
exists the stator or nozzle with a highly
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accelerated flow which is C 2, you can see
that C 2 is much higher than C 1, and
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that is exactly the reason why this is called
a nozzle.
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Now, at the rotor entry, we also have a blade
speed U; please note that the direction of
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this vector U is from the pressure surface
to the section surface, unlike in
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compressor where it was other way round. Here
the flow drives the blades and that is why
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you have the blade speed which is in this
direction. This is the
18:22.880 --> 18:29.880
absolute velocity entering the rotor and relative
velocity will be the vector some of these
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two or vector difference between these two
and that is given by V 2.
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Alpha 2 is the angle which C 2 makes with
the axial direction, beta 2 is the angle which
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V 2 makes with the axial direction. And just
like we have seen in
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compressors V 2 enters the rotor at an angle
which is tangential to the camber at the leading
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edge. This is to ensure that the flow, this
is obviously when the
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incidence is close to 0 to ensure that the
flow does not separate.
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At the rotor exit, we have V 3.V 3 is less
then V 2 as you can see, and of course, that
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also depends upon the type of the turbine,
whether it is impulse or reaction,
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and you also have C 3 here and this is the
blade speed U. Beta 3 is the angle which V
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3 makes with the axial direction, alpha 3
is the angle which the absolute
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velocity C 3 makes with the axial direction.
So, if you now come go back to the earlier
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slide is of lecture 2 or 3, where we had discussed
about velocity triangles
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for an axial compressor, you can quite easily
see the similarities as well as the differences.
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I would strongly urge you to compare both
these velocities triangles
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by keeping them side by side.
So, you can understand the differences between
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a compressor and a turbine; at the same time,
you can also try to figure out some similarities
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between these two
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components. And so it is very necessary that
you have understand clearly both the differences
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as well as the similarities from a very fundamental
aspect that is
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the velocity triangle point of view. So, this
is a standard velocity triangle for a typical
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turbine stage. I am not really mentioned here,
what kind of a turbine it is
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whether it is impulse or reaction, we will
come to that classification very soon, and
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you will see that there are different ways,
in which you can express the
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velocity triangle for both of these types
of turbines.
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So, let us now try to take a look at the different
types of turbines. I mentioned in the beginning
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that there are two different configurations
of axial turbines that
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are possible, the impulse and the reaction
turbine. In an impulse turbine, the entire
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pressure drop takes place in the nozzle, and
the rotor blades would simply
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deflect the flow and would have a symmetrical
shape. So, there is no acceleration or pressure
21:21.210 --> 21:27.220
drop taking place in the rotor in an impulse
turbine. So, the the
21:27.220 --> 21:32.820
rotor blades would simply deflect the flow
and guided to the next nozzle if there is
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one present.
In a reaction turbine on the other hand, the
21:37.500 --> 21:43.450
pressure drop is shared by the rotor as well
as the stator. And the amount of pressure
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drop that is shared is defined by
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the degree of reaction, which we will discuss
in detail in the next lecture. Now, which
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means that the degree of reaction of an impulse
turbine would be 0,
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because the entire pressure drop as already
taken place in the stator, the rotor does
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not contribute to any pressure drop, and so
the degree of reaction for an
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impulse turbine should be 0. So, these are
two different configurations of axial turbines
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which are possible. And what will do is that
we will take a look at their
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velocity triangles also, but before that we
need to understand the basic mechanism by
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which work is done by a turbine.
22:27.090 --> 22:34.090
Now, if you were to apply angular momentum
equation for an axial turbine, what you will
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notice is that power generated by a turbine
is a function of well three
22:41.460 --> 22:48.150
parameters; one is of course a mass flow rate,
the other parameters are the blade speed and
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the tangential component of velocity - the
absolute velocity. So, if
22:52.419 --> 22:59.419
you apply angular momentum at the inlet and
exit of the rotor, then the power generated
22:59.940 --> 23:06.210
by the turbine is equal to mass flow rate
multiplied by U 2 into C w2
23:06.210 --> 23:11.600
which is the product of the blade speed and
the tangential velocity absolute at the inlet
23:11.600 --> 23:17.580
of the rotor, minus U 3 times C w3 which is
again blade speed at rotor
23:17.580 --> 23:24.580
exit, and multiplied by the tangential component
of the absolute velocity at the rotor exit.
23:26.220 --> 23:33.220
Now, we would normally assume that the blade
speed is does not change from at a given radial
23:35.400 --> 23:40.470
plane, and therefore U 2 can be assume to
be equal to U 3, and
23:40.470 --> 23:47.470
therefore the work done per unit mass would
now be equal to blade speed that is U multiplied
23:47.559 --> 23:54.559
by C w2 minus C w3 or which is also equal
to the from the
23:56.669 --> 24:02.919
thermodynamics point of view, there is a stagnation
pressure, stagnation temperature drop taking
24:02.919 --> 24:07.200
place in a turbine, because the turbine expands
the flow, and
24:07.200 --> 24:12.150
work is extracted from the turbine, and therefore
there has to be a stagnation temperature drop
24:12.150 --> 24:16.970
taking place in a turbine.
Therefore the enthalpy difference between
24:16.970 --> 24:23.970
the inlet and exit of the turbine would basically
equal to the work done by or work developed
24:24.850 --> 24:26.110
by this particular
24:26.110 --> 24:33.110
turbine. So, work done per unit mass is also
equal to C p time T 01 minus T 03, where this
24:34.159 --> 24:38.919
is basically the enthalpy difference; C p
T 01 is enthalpy at inlet of the
24:38.919 --> 24:45.919
turbine, C p T 03 is the enthalpy at the exit
of the turbine. Let us now denote delta T
24:47.360 --> 24:52.220
0 which basically refers to the stagnation
temperature. The net change in
24:52.220 --> 24:59.220
the stagnation temperature in the turbine
delta T naught is equal to T 01 minus T 03
25:00.100 --> 25:07.059
which is also equal to T 02 minus T 03, because
1 to 2 is the stator and there
25:07.059 --> 25:13.400
cannot be any change in stagnation temperature
in the stator. Therefore, T 01 minus T 03
25:13.400 --> 25:20.400
is equal to T 02 minus T 03.
So, we now define what is known as the stage
25:20.429 --> 25:27.429
work ratio, which is basically delta T naught
by T 01 and that is equal to U times C w2
25:27.909 --> 25:31.020
minus C w3 divided by C p
25:31.020 --> 25:36.000
times T 01. So, this is basically follows
from these two equations here which correspond
25:36.000 --> 25:40.980
to the work done per unit mass; one is in
terms of the velocities and
25:40.980 --> 25:46.909
other is in terms of stagnation temperatures.
So, a similar analysis was also carried out
25:46.909 --> 25:52.940
when we were discussing about axial compressors,
and were also we had a
25:52.940 --> 25:59.710
kind of equated the work that the flow does
on well work done by the compressor on the
25:59.710 --> 26:05.950
flow as compared to the stagnation temperature
rise taking place in a
26:05.950 --> 26:12.470
compressor as a result of the work done on
the flow. So, there are also we have defined
26:12.470 --> 26:16.340
the pressure rise or pressure ratio per stage
in terms of the temperature
26:16.340 --> 26:22.220
rise across that particular stage, and the
velocity components which come from the velocity
26:22.220 --> 26:23.070
triangles.
26:23.070 --> 26:30.070
Now, what you can see here is that - the turbine
work per stage would basically be limited
26:31.890 --> 26:38.030
by two parameters; one is the pressure ratio
that is available for
26:38.030 --> 26:44.159
expansion, and of course the other aspect
is the allowable the amount of blade stress
26:44.159 --> 26:50.600
and turning that is physically possible for
one to achieve in in the case of a
26:50.600 --> 26:55.460
particular turbines. So, there are two parameters;
one being the available pressure ratio and
26:55.460 --> 26:59.720
other is allowable blade stress and turning.
That one can achieve in a
26:59.720 --> 27:05.620
particular turbine configuration.
So, in unlike in a compressor where we also
27:05.620 --> 27:12.299
had the issue of boundary layer behavior,
because the flow was always operating in an
27:12.299 --> 27:13.510
adverse pressure gradient
27:13.510 --> 27:19.270
mode in compressors, in a turbine the pressure
gradient is favorable. So, boundary layer
27:19.270 --> 27:24.640
behavior is generally something that can be
controlled, and there are
27:24.640 --> 27:30.909
normally not much issues related to boundary
layer boundary layer separation or growth
27:30.909 --> 27:34.039
of boundary layer and so on. Of course, there
are certain operating
27:34.039 --> 27:41.039
conditions, and which under which certain
the stages of turbine may undergo, local flow
27:41.220 --> 27:48.220
separation, but that is for only short durations.
In general in a favorable pressure gradient
27:48.779 --> 27:54.830
boundary layers generally, tend to be well
behaved. Now, the turbine work ratio that
27:54.830 --> 27:57.520
we had seen in the previous slide
27:57.520 --> 28:03.960
is also often defined in and as a ratio between
the work done per unit mass divided by the
28:03.960 --> 28:09.559
square of the blade speed. Therefore, W t
by u square which is also
28:09.559 --> 28:15.710
equal to the enthalpy rise or rather enthalpy
drop in the case of turbine divided by U square,
28:15.710 --> 28:22.360
which is basically equal to delta C w divided
by U or net change in
28:22.360 --> 28:26.960
the tangential velocity absolute divided by
the blade speed.
28:26.960 --> 28:33.960
Now, this is an important parameter, because
based on this we can understand or the differences
28:33.990 --> 28:37.700
between an impulse turbine and a reaction
turbine, which is
28:37.700 --> 28:43.710
what we are going to next to take a look at
what are the fundamental differences, besides
28:43.710 --> 28:48.760
of course, the fact that in an impulse turbine,
flow is the entire pressure
28:48.760 --> 28:55.470
drop takes place only in the nozzle and in
reaction turbine that is shared between the
28:55.470 --> 29:01.159
nozzle and the rotor. Let us take up an impulse
turbine first and we will take
29:01.159 --> 29:06.240
look at the velocity triangles for an impulse
turbine, and then try to find out the work
29:06.240 --> 29:11.500
ratio per stage of an impulse turbine, and
related to some parameters which
29:11.500 --> 29:14.340
we can get from the velocity triangles.
29:14.340 --> 29:21.340
So, here we have a typical impulse turbine
stage, a set of a row of nozzle blades followed
29:23.919 --> 29:30.169
by a row rotor of the blades. And... So, flow
is accelerated in the
29:30.169 --> 29:37.169
nozzle, and so the velocity that reaches the
rotor. The absolute component is C 2, and
29:40.000 --> 29:44.870
at an angle of alpha 2 with the acceleration
and as the result of the blade
29:44.870 --> 29:50.460
speed U, the relative velocity which enters
the rotor is V 2 which is at an angle of beta
29:50.460 --> 29:55.220
2 with the acceleration. And in an impulse
turbine, I mentioned that the
29:55.220 --> 30:02.010
rotor simply deflects the flow and there is
no pressure drop taking place in the rotor,
30:02.010 --> 30:07.150
and therefore, at the exit of the rotor we
have V 3 which is at an angle of
30:07.150 --> 30:14.150
beta 3 by virtue of the symmetry of the blades,
we will have beta 2 is equal to minus beta
30:15.220 --> 30:19.570
3, and velocity in magnitude v 2 would be
equal to v 3.
30:19.570 --> 30:26.140
So, which we can also see from the velocity
triangle shown here; C 2 is the absolute velocity
30:26.140 --> 30:30.049
and entering the rotor, V 2 is the relative
velocity and the
30:30.049 --> 30:37.049
corresponding angles here alpha 2 and beta
2. Now, in the rotor we have V 3 which is
30:39.200 --> 30:44.919
equal to V 2 in magnitude, but at an angle
which is different from the inlet,
30:44.919 --> 30:51.919
that is beta 3 will be negative of beta 2
in the other direction. Absolute velocity
30:52.570 --> 30:58.360
leaving the blade is C 3. Now, if you look
at the other components of a velocities
30:58.360 --> 31:05.360
like this is the actual component of the absolute
velocities C a, and the corresponding tangential
31:08.970 --> 31:11.100
components of the relative velocity which
are obviously equal
31:11.100 --> 31:16.870
and are opposite in direction like V w2 and
V w3; you can see that these are equal in
31:16.870 --> 31:21.460
magnitude, but of course the that directions
are opposite, because V 2 and
31:21.460 --> 31:28.460
V 3 are in opposite directions. And C w2 is
the absolute component of well tangential
31:29.409 --> 31:34.870
component of the absolute velocity are inlet,
C w3 that at the exit of the
31:34.870 --> 31:35.970
rotor.
31:35.970 --> 31:42.970
So, this is typical velocity triangle of of
an impulse turbine stage. And if if you take
31:48.960 --> 31:53.580
a closure look at the velocity triangles,
I have mentioned that the angles beta
31:53.580 --> 31:59.570
3 and beta 2 are equal in magnitude, but they
are different by their orientations. So, beta
31:59.570 --> 32:05.070
3 is equal to minus beta 2 which means that
we have V w3 is equal to
32:05.070 --> 32:12.070
minus V w2. And the difference in the tangential
component of the absolute velocities C w2
32:12.640 --> 32:18.440
minus C w3 will be equal to twice of V w2.
So, let us take a look at
32:18.440 --> 32:25.440
the velocity triangle again C w2 is this minus
C w3 is equal to the sum of V w2 and Vw3,
32:28.539 --> 32:35.100
and since they are equal we have that is equal
to twice of V w2, which
32:35.100 --> 32:42.100
is also equal to 2 into C w2 minus U or this
is equal to 2 U into C a by tan alpha 2 minus
32:44.299 --> 32:49.770
1.
So, that is again coming from the velocity
32:49.770 --> 32:55.450
triangles, you can see that C a tan alpha
2 is this component minus U is equal to twice
32:55.450 --> 32:58.279
of this. So, the difference
32:58.279 --> 33:05.279
between the tangential component of the absolute
velocity C w2 and C w3 that is delta C w for
33:07.130 --> 33:12.730
an impulse turbine equal to 2 U into C a by
U tan alpha 2 minus
33:12.730 --> 33:18.570
1. Therefore, the work ratio that we have
defined earlier, for an impulse turbine that
33:18.570 --> 33:23.480
is delta h naught by U square is equal to
2 U into C a by U tan alpha 2 minus
33:23.480 --> 33:30.480
1. We will now, take a look at what happens
in the case of an of a reaction turbine and
33:31.799 --> 33:37.659
calculate the work ratio as applicable for
a reaction turbine, and see the is
33:37.659 --> 33:42.770
there a difference fundamentally in the work
ratio of an impulse turbine and a reaction
33:42.770 --> 33:43.230
turbine.
33:43.230 --> 33:50.230
Now, let us take a look at a typical 50 percent
reaction turbine, just for simplicity. The
33:51.220 --> 33:55.000
reason why we took up a 50 percent reaction
turbine is, because in a 50
33:55.000 --> 34:02.000
percent reaction turbine the pressure drop
is shared equally between the nozzle and the
34:02.649 --> 34:06.529
rotor. And therefore, the velocity triangles
as you can see are mirror
34:06.529 --> 34:13.529
images of one another; the velocity triangle
at the inlet of the rotor is this, where this
34:13.609 --> 34:18.629
is C 2 the absolute velocity coming in from
the rotor from the nozzle; V 2 is
34:18.629 --> 34:22.309
a relative velocity and this is the blade
speed.
34:22.309 --> 34:29.309
And since they are mirror images and the exit
of the rotor, you have V 3 and C 3. And therefore,
34:29.719 --> 34:33.839
you can clearly see that C 2 will be equal
to V 3 and V 2 will be
34:33.839 --> 34:39.649
equal to C 3, corresponding the angles alpha
2 will be equal to beta 3, and beta 2 will
34:39.649 --> 34:44.999
be equal to alpha 3. For this is true for
only 50 percent reaction turbine, for
34:44.999 --> 34:51.940
any other reaction stages of course, the velocity
triangles need not necessarily be symmetrical,
34:51.940 --> 34:56.209
and this is also assuming that the axial velocity
is does not change
34:56.209 --> 35:03.209
across the rotor and the nozzle. Now, for
this kind of a reaction turbine which is having
35:03.509 --> 35:08.130
a degree of reaction of 0.5; since the velocity
of triangles are mirror
35:08.130 --> 35:15.130
images are symmetrically. If we assume constant
axial velocity, we have C w3 is equal to minus
35:15.950 --> 35:21.319
C a tan alpha 2 minus U. And therefore, the
turbine work ratio
35:21.319 --> 35:27.989
would basically be equal to twice into twice
of C a by U tan alpha 2 minus 1.
35:27.989 --> 35:34.989
This we can compare with that of the impulse
turbine where it was 2 U multiplied by C a
35:36.059 --> 35:41.650
by U tan alpha 2 minus 1. So, you can immediately
see that there is
35:41.650 --> 35:48.650
fundamental difference between the work ratio
as compared to a turbine which is impulse
35:48.650 --> 35:53.609
or in this case of course example was for
a 50 percent reaction of
35:53.609 --> 35:59.039
turbine. So, there is a fundamental difference
between the work ratio as applicable for an
35:59.039 --> 36:03.640
impulse turbine as compared to that of a 50
percent reaction turbine,
36:03.640 --> 36:08.509
and in general for any reaction turbine as
well.
36:08.509 --> 36:14.739
Now, this was as per as the different types
of turbine configurations were concerned and
36:14.739 --> 36:19.469
how one can analyze these turbine configurations.
And what are the
36:19.469 --> 36:24.069
fundamental differences between let see an
impulse turbine and a reaction turbine, and
36:24.069 --> 36:30.089
how one can from the velocity triangle estimate
the work ratio that or the
36:30.089 --> 36:37.089
work done by these kind of turbine stages.
So, what I was suggesting right of the beginning
36:38.309 --> 36:43.959
was that you can clearly see differences between
the compressor and
36:43.959 --> 36:49.190
turbines by looking at the velocity triangle
for these two different cases, and comparing
36:49.190 --> 36:55.839
them to understand the fundamental working
of compressors and
36:55.839 --> 37:00.289
turbines and what makes them two different
components.
37:00.289 --> 37:06.529
What we can take up next for discussion is
something we have discussed in detail for
37:06.529 --> 37:13.009
compressors as well; that is to do with a
cascade. And as you have already
37:13.009 --> 37:20.009
seen a cascade is a simplified version of
rotating machine, and you could have different
37:21.589 --> 37:25.489
versions of cascade, you could have a linear
cascade or an annular
37:25.489 --> 37:32.279
cascade. And basically a cascade would have
a set of blades which are arranged; set of
37:32.279 --> 37:36.789
similar blades which are all arranged in certain
fashion, and at a certain
37:36.789 --> 37:43.789
angle which we have referred to as the straggler
angle. And cascade analysis forms a very fundamental
37:44.950 --> 37:50.140
analysis of design of turbo machines whether
it is
37:50.140 --> 37:52.509
compressors or turbines.
37:52.509 --> 37:59.509
So, cascade basically consists of an array
of stationary blades. And constructed basically
37:59.609 --> 38:05.309
from measurement of performance parameters,
and what is usually
38:05.309 --> 38:12.309
done is that we would like to eliminate any
three-dimensional effects which are likely
38:12.549 --> 38:17.279
to come up in a cascade. And one of the sources
of three dimensionality is
38:17.279 --> 38:22.440
the presence of boundary layer .So, one would
like to remove boundary layer from the end
38:22.440 --> 38:26.339
walls of the cascade, and so that is the standard
practice one would
38:26.339 --> 38:32.269
have porous end walls through which boundary
layer fluid can be removed; to ensure two
38:32.269 --> 38:38.569
dimensionality of the flow entering into a
cascade. Now, it is also a
38:38.569 --> 38:43.769
standard assumption that radial variations
in velocity field can be kind of eliminated
38:43.769 --> 38:49.890
or ignored. And cascade analysis is primarily
meant to give us some idea
38:49.890 --> 38:56.890
about the amount of blade loading that a particular
configuration can give us, as well as the
38:57.910 --> 39:04.910
losses in total pressure that one can measure
from a cascade analysis.
39:04.969 --> 39:11.969
So, and in turbine cascades testing also involves
wind tunnels which are very similar to what
39:13.609 --> 39:18.099
we have discussed for compressors. I had shown
you cascade wind
39:18.099 --> 39:24.479
tunnels, when we are discussing about a cascades
in the context of compressors. In turbine
39:24.479 --> 39:30.170
cascades are also tested in similar wind tunnels.
And just that in a
39:30.170 --> 39:35.589
case of turbines, since they are operating
in an accelerating flow. There there is a
39:35.589 --> 39:42.589
requirement of a certain pressure drop across
a turbine. So therefore, the wind
39:42.650 --> 39:49.640
tunnel is required to generate sufficient
pressure which can be expanded through a turbine
39:49.640 --> 39:55.549
cascade. Now, turbine blades has are probably
aware would are likely
39:55.549 --> 40:02.549
to have much higher camber, than compressor
cascades or compressor blades. And turbine
40:02.999 --> 40:07.440
cascades are set at a negative stagger unlike
in compressor blades;
40:07.440 --> 40:14.440
something I will explain when we take up a
cascade, schematic in in detail.
40:14.900 --> 40:20.920
Now, cascade analysis will basically give
us as I mentioned two parameters besides the
40:20.920 --> 40:26.339
sets of other parameters, like boundary line
thickness and all and losses,
40:26.339 --> 40:31.489
etcetera. The most fundamental parameter we
would like to look at from the cascade analysis
40:31.489 --> 40:36.709
is this surface static pressure distribution
or CP distribution, which
40:36.709 --> 40:42.130
is related to the loading of the blade, and
the second aspect of the is the total pressure
40:42.130 --> 40:47.109
loss across the cascade, which is yet another
parameter that one would like
40:47.109 --> 40:50.759
to infer from the cascade analysis.
40:50.759 --> 40:57.759
Now, let us take a look at a typical cascade,
turbine cascade nomenclature. I think I mentioned
40:58.660 --> 41:03.349
in the beginning that all the terms that we
have used for
41:03.349 --> 41:09.249
compressors will it is the same nomenclature
that we apply for a turbine as well. Just
41:09.249 --> 41:12.640
that that the way the blades are set, so the
blade geometry they are quiet
41:12.640 --> 41:19.640
different between compressors and turbines.
So, if we look at a typical compressor cascade,
41:20.559 --> 41:24.410
these are the blades you can immediately see
that these blades have much higher turning
41:24.410 --> 41:25.680
for camber than
41:25.680 --> 41:31.559
compressor blades. So, it is a set of these
blades which are arranged either linearly
41:31.559 --> 41:37.999
or in an annular fashion which constitute
a cascade. So, these blades are set
41:37.999 --> 41:44.759
apart by a certain distance, which is as you
can see denoted by pitch or spacing. And these
41:44.759 --> 41:49.410
blades are set at a certain angle, which is
called the blade setting or
41:49.410 --> 41:54.650
the stagger angle. So, you can see this lambda
which you see here refers to the blade setting
41:54.650 --> 42:00.190
or stagger angle. The blades have a certain
camber which is basically
42:00.190 --> 42:07.130
the angle subtended between the tangent to
the camber line at the leading edge and that
42:07.130 --> 42:10.809
at the trailing edge. So, the difference between
that gives us the a blade
42:10.809 --> 42:17.489
camber.
Now, the flow enters the cascade at a certain
42:17.489 --> 42:23.849
angle, you can see that inlet blade angle
is given here as beta 1 and the blade outlet
42:23.849 --> 42:29.519
angle is beta 2. Now, so if there
42:29.519 --> 42:34.779
is a difference between the blade angle and
the flow angle at the inlet; that basically
42:34.779 --> 42:40.690
the incidence which is denoted by i here .So,
this is the incidence angle.
42:40.690 --> 42:47.039
Similarly, a difference between the blade
outlet angle and the outflow angle is the
42:47.039 --> 42:52.410
deviation which is denoted by delta. So, at
the exit you may have flow
42:52.410 --> 42:59.410
deviation and the inlet one may have an incidence.
And if you draw a normal rect normal to the
43:04.130 --> 43:09.650
tangent at the trailing edge and take it to
the next adjacent blade, the section surface
43:09.650 --> 43:13.119
of the adjacent blade. So, this
43:13.119 --> 43:19.029
distance that you see here is basically refer
to as the throat or opening at the turbine
43:19.029 --> 43:26.029
exit, and that is here denoted by a symbol
o. The blade called as you already
43:26.910 --> 43:33.910
know is denoted by C. And then the blades
also would have a certain finite thickness
43:35.699 --> 43:40.380
at the trailing edge. So, that is denoted
here by the trailing edge thickness.
43:40.380 --> 43:46.390
So, the blades practically will have a certain
amount of finite thickness and that is what
43:46.390 --> 43:49.549
is denoted here as the thickness at the trailing
edge.
43:49.549 --> 43:56.549
So, these are the fundamental nomenclatures
nomenclatures that used in turbine very similar
43:57.269 --> 44:01.130
aspect was also used in compressor, where
we had defined all these
44:01.130 --> 44:07.949
different parameters like incidence, deflection,
deviation and blade angle, the camber, the
44:07.949 --> 44:11.519
pitch, stagger all of them defined. Difference
is, of course, the way the
44:11.519 --> 44:16.829
blades are set, this is set at a negative
stagger as you can see, the compressor cascade
44:16.829 --> 44:21.259
if you go back you will see that the way the
blades are set is opposite to
44:21.259 --> 44:26.660
what you see in the case of a turbine. That
is basically to ensure that the flow passage
44:26.660 --> 44:31.279
gives you the required amount of flow turning,
and also the flow
44:31.279 --> 44:38.039
acceleration in the case of turbine cascades,
and in in compressor cascade, the setting
44:38.039 --> 44:44.069
is to ensure that you get a defilation in
a compressor.
44:44.069 --> 44:51.069
So, having understood the fundamental nomenclature
of a turbine cascade; we would now take a
44:54.539 --> 44:58.359
closer look at the different aspects of flow
through a cascade
44:58.359 --> 45:04.099
and I would be deriving well not really a
detailed derivation. But I would just give
45:04.099 --> 45:11.099
you some idea about how one calculates the
lift developed by a certain
45:11.469 --> 45:18.150
cascade turbine cascade. In two different
cases, one is if you do not assume any losses
45:18.150 --> 45:25.150
or if it is inviscid analysis, and followed
by an viscous analysis one of
45:25.660 --> 45:30.949
course would also get a drag in the case of
viscous analysis, how one calculate the lift
45:30.949 --> 45:36.160
and of course that is basically related to
loading of the blades eventually.
45:36.160 --> 45:43.160
So, the basic idea of cascade analysis is
that just like in case of an airfoil, because
45:43.599 --> 45:49.869
cascade is in some sense in airfoil analysis,
we can determine the lift and drag
45:49.869 --> 45:54.959
forces acting on the blades. And this analysis,
as I mention can be carried out using both
45:54.959 --> 46:00.150
these assumptions potential flow or inviscid
analysis or by considering
46:00.150 --> 46:05.579
viscous effect in a rather simplistic manner.
So, we will assume that the mean velocity
46:05.579 --> 46:10.670
which we going to denote as V subscript m,
makes an angle of alpha
46:10.670 --> 46:16.269
subscript m with axial the direction. What
we will do is to determine the circulation
46:16.269 --> 46:22.670
developed on the blades, and subsequently
the lift force. In the inviscid
46:22.670 --> 46:28.809
analysis obviously there is no drag and there
is only a lift force, which lift is only force
46:28.809 --> 46:32.729
acting on the blade. In the case of an inviscid
analysis, when you take up
46:32.729 --> 46:37.239
a viscous analysis there are two components
of a force and resultant force, they will
46:37.239 --> 46:39.699
lift and they drag.
46:39.699 --> 46:45.989
So, this is the geometry, we are considering
for an inviscid flow through turbine cascade.
46:45.989 --> 46:50.400
If, you take a look at two different stream
lines let us say, this is one
46:50.400 --> 46:57.019
stream line and another stream line which
is bounding one particular blade, that is
46:57.019 --> 47:01.690
shown here these are the two different stream
lines. What we are going to do is
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to find the circulation reduced over this
particular airfoil which is currently an airfoil
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here, and then relate that to lift developed
on this particular blade. So, the
47:14.910 --> 47:21.569
inlet flow the entering the cascade is V 1
and the flow exceeding the blade is V 2, and
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of course, we will assume mean velocity of
V m which makes an angle
47:27.059 --> 47:29.019
alpha m with the acceleration.
47:29.019 --> 47:34.859
So, if this is the case and this is how you
can take a look at this circulation axis;
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so, this is the axis along which we are calculating
the circulation, and therefore,
47:38.789 --> 47:43.130
this is the lift acting on this particular
blade. Since it is a turbine blade, you know
47:43.130 --> 47:48.420
that this is basically the direction in which
the lift is going to act. So, the mean
47:48.420 --> 47:53.779
velocity that is showed here by vector V m
acts in this direction, this is inflow velocity
47:53.779 --> 48:00.779
V 1, and this is the exit velocity V 2.
So, the circulation that is denoted by capital
48:01.469 --> 48:08.469
lambda here is equal to S multiplied by the
difference in the tangential velocities V
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w2 minus V w1. And lift is
48:12.160 --> 48:18.009
related to the circulation which is product
of density, times, the mean velocity and the
48:18.009 --> 48:25.009
circulation. Therefore, lift acting when there
are no other effects
48:25.140 --> 48:29.759
considered like viscous affects, then the
lift acting here would be simply the product
48:29.759 --> 48:36.229
of rho times V m into the circulation which
S into V w2 minus V w1.
48:36.229 --> 48:41.059
So, this is expressed in a non-dimensional
form which we referred to as the lift coefficient.
48:41.059 --> 48:47.430
So, see l here lift divided by half rho V
m square into C, and this is
48:47.430 --> 48:54.430
equal to rho into V m into S V w2 minus V
w1 by half rho V m square into C. So, this
48:55.479 --> 49:02.479
can be related to the angles, the across,
the cascade, and so we can simplify
49:02.660 --> 49:09.660
this lift coefficient as 2 into S by C into
tan alpha 2 minus tan alpha 1 multiplied by
49:10.380 --> 49:17.380
cos alpha alpha m. So, this is the this is
basically lift coefficient on a turbine
49:20.509 --> 49:25.779
blade, assuming that flow is in this inviscid.
49:25.779 --> 49:32.779
Now, what happens if there are viscous effects?
The primary effect of viscous flow on the
49:35.009 --> 49:38.789
flow through a turbine cascade is the fact
that viscous effects manifest
49:38.789 --> 49:43.829
themselves in the form of pressure losses
- total pressure losses. And therefore, the
49:43.829 --> 49:48.099
wake from the blade trailing edge will lead
to a non uniform velocity leaving
49:48.099 --> 49:52.869
the blades. In the previous analysis, we were
assuming uniform velocity entering the blades
49:52.869 --> 49:56.809
and uniform velocity leaving the blades, because
it is a potential
49:56.809 --> 49:59.589
flow.
So, here in the case of viscous analysis in
49:59.589 --> 50:05.559
addition to lift, one would also have a drag
which we will also contribute to left in some
50:05.559 --> 50:08.849
where the other. So, the
50:08.849 --> 50:13.969
effecting force acting on the blade will be
resultant of both the left as well as the
50:13.969 --> 50:19.359
drag acting on the blade. So, we now defined
what is known as total pressure
50:19.359 --> 50:24.650
loss coefficient where defined a similar parameter
for compressors as well. So, this is denoted
50:24.650 --> 50:28.430
by omega bar, because there is a total pressure
loss taking place
50:28.430 --> 50:35.430
across the blades as a result of the viscous
effects. So, omega bar is equal to P 01 minus
50:35.539 --> 50:41.410
P 02 divided by half rho V 2 square. This
is the losing total pressure
50:41.410 --> 50:43.229
across the turbine cascade.
50:43.229 --> 50:50.229
So, the schematic ahead shown earlier now
gets modified, because you have a set of uniform
50:50.949 --> 50:55.749
stream lines entering turbine cascade, but
as they leave the cascade
50:55.749 --> 51:01.239
you can see that they have became non uniform,
basically at the trailing edge where there
51:01.239 --> 51:06.559
is a wake. So this, what is shown here schematically
is the these are
51:06.559 --> 51:11.249
the different wakes of all these blades that
are present here. So, there is difference
51:11.249 --> 51:18.249
in the forces acting on the blade as a result
of this non uniformity in the
51:19.170 --> 51:24.039
velocity at the at the exit of the turbine
cascade.
51:24.039 --> 51:29.689
So, in this case, we can calculate drag as
equal to the losses, we can relate the drag
51:29.689 --> 51:34.910
to the losses total pressure losses, omega
bar into S into cos alpha m. And
51:34.910 --> 51:40.920
therefore, the effective lift will now be
equal to the sum of the lift as well as the
51:40.920 --> 51:45.869
component of drag in that effective direction
that is omega bar into S into cos
51:45.869 --> 51:52.529
alpha m. And lift we know is the product of
density and the mean velocity and the circulation.
51:52.529 --> 51:58.569
So, that is rho V m into delta plus omega
bar S cos alpha one alpha
51:58.569 --> 52:04.900
m. Therefore, the lift coefficient in this
case will get modified as twice into S by
52:04.900 --> 52:11.009
C tan alpha 2 minus tan alpha 1 cos alpha
m plus the drag components C D times
52:11.009 --> 52:17.869
tan alpha m. So, this is the manner in which
we can calculate lift coefficient for both
52:17.869 --> 52:22.989
this cases; one is for case the without viscous
effects and the second is if we
52:22.989 --> 52:24.959
consider the viscous effects.
52:24.959 --> 52:31.959
So, the basic idea for calculating these coefficients
was to calculate, also calculate the blade
52:31.969 --> 52:36.319
efficiency. So, based on the calculation of
the lift and drag
52:36.319 --> 52:41.400
coefficient, we can now calculate the blade
efficiency, which is basically the ratio of
52:41.400 --> 52:45.499
ideal static pressure drop to obtain a certain
degree of kinetic energy
52:45.499 --> 52:51.269
change to the actual static pressure drop
which will produce the same change in kinetic
52:51.269 --> 52:57.609
energy. Therefore, the blade efficiency is
in have of course skip the
52:57.609 --> 53:02.160
derivation of the blade efficiency. But it
can be related to the lift and drag coefficient
53:02.160 --> 53:08.089
like blade efficiency is 1 minus C D by C
L tan alpha m divided by 1 plus
53:08.089 --> 53:14.369
C D by C L cot alpha m. And if you want to
neglect the drag term in the lift definition,
53:14.369 --> 53:19.079
because C D - the drag term is usually much
smaller in comparison to the
53:19.079 --> 53:26.079
lift. The blade efficiency is simply 1 by
1 plus 2 into C D divided by C L sin into
53:27.019 --> 53:34.019
twice alpha m. So, this basic idea of calculating
the lift drag and coefficient was
53:35.890 --> 53:42.449
also to calculate the blade efficiency, which
is basically a function of C D C L have the
53:42.449 --> 53:43.699
mean angle alpha m.
53:43.699 --> 53:50.699
So, let me now quickly recap our discussion
in today's class. We had taken up three distinct
53:53.680 --> 53:58.029
topics for discussion; one was the different
types of turbine,
53:58.029 --> 54:03.009
configuration the axial turbine configuration,
the impulse and the reaction turbine stages
54:03.009 --> 54:07.410
and we have done. We had a look at the velocity
triangles and how we
54:07.410 --> 54:13.410
can calculate the work ratio for impulse and
reaction turbine stages. We have also carried
54:13.410 --> 54:16.910
out the work and stage dynamic, we looked
at these different
54:16.910 --> 54:23.910
components or configurations of axial turbines,
and how we can go about determining the work
54:24.239 --> 54:29.979
ratio of these two configurations of axial
turbine. And then we
54:29.979 --> 54:36.979
had some discussion on turbine cascades, and
calculation of lift and drag for a typical
54:37.999 --> 54:44.009
turbine configuration, and how we can use
that information to calculate the
54:44.009 --> 54:50.869
blade efficiency from simple turbine cascade
testing. That is simple cascade testing can
54:50.869 --> 54:56.279
actually give us some idea about the blade
efficiency that this kind of a
54:56.279 --> 55:00.390
blade configuration can give us. So, that
bring us to end of this lecture.
55:00.390 --> 55:06.849
We will continue discussion on axial turbines
in the next lecture as well, where we will
55:06.849 --> 55:10.929
primarily talking about the performance parameters,
degree of reaction
55:10.929 --> 55:17.929
losses as well as efficiency of axial turbines.
And were we also take up detailed discussion
55:18.640 --> 55:22.619
on whatever the different losses in a two-dimensional
sense. And how
55:22.619 --> 55:28.959
we can define the efficiency and you will
see that different ways of defining efficiency
55:28.959 --> 55:33.039
for a turbine. So, we will take up some of
these topics for discussion in
55:33.039 --> 55:33.480
the next class.