WEBVTT
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Hello and welcome to lecture number thirty-two
of this lecture series on turbomachinery aerodynamics.
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We have been talking about the different types
of turbo-machines and with stronger bias or
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emphasis towards the axial flow turbo-machines,
basically the axial compressors and axial
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turbines. We have already elaborated the reasons,
why we are sort of giving little more weightage
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to the axial flow turbo-machines, primarily
due to the fact, that modern day jet engines
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operate primarily with these types of turbo-machines,
the axial compressors and the axial turbines
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due to some of their inherent advantages.
Of course, it is not to say, that the radial
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flow counterparts are not being used at all.
These are also having applications in, in
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certain specific areas, primarily to do with
the smaller sized engines and, and various
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other applications and therefore, with this
in mind, we also initiated some lectures on
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the radial flow machines and we started off
with the centrifugal compressors. In fact,
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it must be kept in mind, that the earliest
jet engines actually operated with centrifugal
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compressors and for a long time, centrifugal
compressors were used in most of the jet engines.
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And of course, once the axial compressors
were developed and designed, they sort of,
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slowly replaced centrifugal compressors, especially
in the larger sized engines. But when you
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look at the smaller class engines, the thrust
class, smaller thrust class engines, centrifugal
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compressors continue to be used in such applications.
So, in, with this in mind we had started the
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previous lecture with discussion on centrifugal
compressors and lecture 31 was devoted towards
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an introduction towards centrifugal compressors.
We discussed the thermodynamics of centrifugal
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compression process and also the work done
and we had looked at power calculations and
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so on, and the governing equations, which
are involved in centrifugal compressor design
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and calculations and analysis. We also had
a quick look at the different components,
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which constitute centrifugal compressor, like
the inlet part of, or the intake of the centrifugal
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compressor, the inducer, the impeller, then
the diffuser vanes and so on. So, we also
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discussed about how one can make calculations
and analysis of these different components
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of a centrifugal compressor. So, that was
what we had discussed in the previous class.
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What we will do to, is to continue some of
our discussion, which we had, had in the last
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class and sort of wind up our discussions
on centrifugal compressor with this lecture.
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And the next lecture we would obviously, be
having a tutorial because having undergone
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two lectures and an overview of centrifugal
compressors it is essential, that we understand
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how one can make analysis and calculations
on centrifugal compressors. So, we will devote
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the next lecture towards the tutorial on centrifugal
compressor.
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In today's class, we will basically be taking
up on a few important concepts. We will start
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our discussion with, what is known as, the
Coriolis acceleration. Then, we will discuss
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about the slip factor, the performance characteristics
and also, stall surge and choking associated
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with centrifugal compressors.
Now, we will start our discussion with Coriolis
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acceleration. I am sure you must have learnt,
at least heard about this term, called Coriolis
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acceleration, or called Coriolis forces, in
your high school classes. In, in your physics
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side, guess you must have learnt about Coriolis
forces and Coriolis acceleration in sort of
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a very general perspective. We are going to
use some of those principles in, in relation
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to our current topic of discussion, that is,
the centrifugal compressors.
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Now, if you remember, in one of the slides,
which I had flashed in my previous class,
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I mentioned, that one of the aspects, that
distinguishes a centrifugal compressor from
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an axial compressor is the fact, that in a
centrifugal compressor the pressure rise mechanism
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is slightly different from that of an axial
compressor, in a sense, that in a centrifugal
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compressor, pressure rise can also occur because
of, or it primarily occurs because of displacement
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of the centrifugal, centripetal force field,
and because of that there is a pressure rise
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taking place as a result of diffusion in the
centrifugal compressor. So, there are at least
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two components, which contribute towards the
overall pressure rise in a centrifugal compressor.
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I think I mentioned that the problems associated
with the boundary layers flows are not that
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severe in a centrifugal compressor. This is
indeed true, that boundary layer separation
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is not, it is still matter of concern, but
it is not the primary matter of concern, like
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we have in axial compressors, which also partly
explains the fact, that axial compressors
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can develop much lower pressure ratio per
stage as compared to centrifugal compressor,
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primarily because of the fact, that axial
flow performance is impeded by boundary layer
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characteristics. So, we will also try to explain
in the context of centrifugal compressors.
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Let us also try to look at the pressure rise
mechanism and deceleration or diffusion in
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the passages through Coriolis acceleration
as a possible means. So, let us try to analyze
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what Coriolis acceleration does to this overall
performance of a centrifugal compressor.
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Now, what we will see very soon is the fact,
that Coriolis acceleration is going to lead
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towards a certain discrepancy in the velocity
triangle from the ideal characteristics, that
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is, the velocity triangle at the exit would
be slightly different from what it should
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have been and that is attributed to, well,
Coriolis acceleration and because of that
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it leads to certain amount of pressure loss
as we will see it little later.
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So, what we are going to do is that we will
consider a certain fluid element, which is
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travelling radially outward, in the passage
of a rotor. So, before this let me just quickly
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go through these two bullets. I have written
here one is to do with centrifugal flow field,
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which is primarily, not a result of the boundary
layer separation and basically, that the fact,
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that the pressure change due to centrifugal
force field is not really, a cause of boundary
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layer separation. We will try to explain that
with Coriolis forces.
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So, if you consider a certain fluid element,
which is, let us say travelling radially outward
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in the passage, and we will look at the velocity
triangles of this particular fluid element
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during a certain time period dt.
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So, this is the fluid element that I am referring
to. Let us consider a fluid element, which
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is passing through these straight radially
vanes. Of course, you can see that this is
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the impeller and these are straight radial
vanes. A fluid element is passing through
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the radial vanes, which are straight. So,
the fluid element, obviously, has a relative
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velocity of V and a blade speed or rotational
velocity omega r, where r is the radius at
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which this fluid element is currently located
and omega is the rotational speed.
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So, if you look at the velocity triangles
for this fluid element in addition to V and
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omega, the absolute velocity c is given by
the resultant of V and omega r. So, is the
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basic velocity triangle is shown by the solid
lines and you can see, V and omega r and its
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resultant is the absolute velocity C.
Now, after a certain time period, well, the
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impeller is rotating, that also displaces
this fluid element by a certain distance.
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If we consider the fact, that this fluid element
is, is being rotated by this center, as I
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showed here, then after certain time period
dt, the fluid element deflects and therefore,
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the new velocity triangle is shown here by
the dotted lines. So, new velocity triangle
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corresponds to a radial location, which is
equal to r plus dr, where dr is the differential
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change in the radius with time dt.
So, the new speed rotational speed becomes
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omega. Omega is unchanged; omega multiplied
by r times dr and let us assume the fact,
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that there is negligible change in the relative
velocity during this time, but because omega
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has changed, c prime also changes, that is,
the absolute velocity changes and it takes
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a new value, which is C prime. Therefore,
the net change in absolute velocity is given
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by dC, which is, as you can see, a tangential
component of, which is shown by dC subscript
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w, dC w, which has two contributions. One
contribution is because of the change in the
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radius, omega times dr and the other contribution
is because of V times d theta, where d theta
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is this angular deflection.
So, here, what you need to understand is that
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there are two distinct aspects, which has
led to this change in the absolute velocity
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or let us say the tangential component of
the absolute velocity. So, there are two contributions
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here, one of them is from the fact, that the
radius has now changed to a differential.
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There is a differential change in the radius
or radial location of the fluid element, which
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is given by dr and that leads to a change
in the peripheral velocity, which is omega
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times dr.
There is also a change, our contribution from
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the fact, that relative velocity being remaining
unchanged for the fact, that we will assume,
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that it is unchanged, because it changes much
smaller compared to the other components.
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So, V times d theta, where d theta is the
angular deflection angle through which the
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fluid element was deflected.
So, these two components put together result
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in a change in the tangential component of
the absolute velocity and at the moment we
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are going to be looking at this change in
the tangential component and its change with
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reference to time. So, dC w by dt is basically,
rate of change of the tangential component
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of velocity with time, that is acceleration,
that is basically the Coriolis acceleration,
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and that is primarily because of the fact,
that there is a rotation given to the fluid
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element.
If the fluid element was displaced simply
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radially upward without any angular deflection,
Coriolis forces would be negligible; there
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would be hardly any Coriolis force acting
on the fluid element. But because of the fact,
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that in addition to the fact, that the fluid
element is getting displaced radially outwards,
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there is also an angular displacement of the
radial element. Both of these contributions
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put together results in this acceleration,
which we will denote as the Coriolis acceleration.
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Let me recap the velocity triangles once again.
So, here, in these velocity triangles the
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default velocity triangle is this, this is
all shown by the solid lines V and omega r;
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V is the relative velocity of the fluid element,
omega r is the tangential component or the
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blade speed peripheral velocity, resultant
of that is C. When it is displaced, we will
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assume negligible change in V because this
magnitude is much smaller than any other magnitudes
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here. And so, we have a new absolute velocity,
which we have denoted by C prime, which is
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a resultant of omega time r plus dr and V.
And this change in the absolute component
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has tangential component, which is dC w, which
has two contributions omega dr and V d theta.
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So, the sum of this two gives us the dC theta.
So, let us add up these two.
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Now, the magnitude of this change in the tangential
component dC w is basically sum of omega dr
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plus Vd theta or dC w is omega into dr, we
have expressed as V times dt plus d theta
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can be expressed as omega into dt, so V into
omega into dt. So, this basically would be
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equal to 2 into omega V into dt and therefore,
dC w by dt is acceleration a theta, that is,
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acceleration in the tangential direction or
a w or a theta. As it is called, it is the
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Coriolis acceleration is equal to twice omega
V, that is, the Coriolis acceleration is directly
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a function of the relative velocity and the
rotational speed. So, higher the rotational
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speed, higher the Coriolis acceleration and
higher the relative velocity. Obviously, that
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also changes the Coriolis acceleration.
Now, this basically requires a certain amount
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of pressure gradient. Why is there a Coriolis
acceleration in the first place? Firstly,
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there is a rotation given to the fluid element,
there is also a certain amount of pressure
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gradient, which is leading to this Coriolis
acceleration. So, if you were to look at the
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pressure gradient, we can express the pressure
gradient in the tangential direction as 1
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by r del P by del theta, that is the rate
of change of pressure in the tangential direction,
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which is basically, equal to twice rho into
omega into V.
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So, the radial pressure or tangential pressure
gradient can be expressed in terms of, twice
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into, 2 into minus, is basically referring
to the fact, that the pressure gradient is
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direction dependent and depends on which direction
the rotor is rotating. So, minus 2 rho into
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omega into V. So, this is the amount of pressure
gradient, which is acting in the tangential
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direction and that leads to this much amount
of Coriolis acceleration. So, the Coriolis
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acceleration, basically, requires certain
amount of pressure gradient.
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So, what we will do is to locate the rate
of change of relative velocity in the tangential
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direction and see how it is related to the,
or whether it is indeed, related to the Coriolis
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acceleration or not? We have seen, that the
tangential velocity, well, Coriolis acceleration
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is, in fact, directly proportional to the
relative velocity. Let us see the rate of
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change, or is there a rate of change of the
relative velocity in the tangential direction?
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So, if we look at the fact, that there is
a tangential pressure gradient, then tangential
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pressure gradient needs to result in a positive
gradient of V in the tangential direction.
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Because if there is a pressure gradient in
a certain direction, that needs to also being
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the fact, that there has to be a velocity
gradient in the same direction and therefore,
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the pressure gradient in the tangential direction.
We have expressed in the previous slide, we
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can equate that with the Coriolis acceleration
and then, we will see, that 1 by rho dP by
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rd theta is basically equal to minus d into
V square by 2 by rd theta, which is minus
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V by r dV by d theta.
So, this if you look at, if you compare this
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with the Coriolis acceleration, we can basically
infer, that 1 by r dV by d theta is equal
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to 2 into omega. And what does this basically
tell us? This tells us the fact, that there
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would be a change in the relative velocity
in the tangential direction as well.
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Now, that was in, in an idealized scenario
one would not have expected any change in
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relative velocity in the tangential direction.
But, we can now see from our analysis, that
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in addition to the fact that the relative
velocity will keep changing in the radial
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direction because that is what the centrifugal
compressor does. There will also be a change
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in the relative velocity if the tangential
direction and that is of very crucial piece
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of information for us because we will very
soon see that this leads to a change in the
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velocity triangle at the exit of the impeller
from what it should have been.
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So, from our understanding of the Coriolis
acceleration we have seen, that as a result
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of the Coriolis acceleration, an eventual
outcome of the fact, that there will be a
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Coriolis acceleration due to the tangential
pressure gradient. Tangential pressure gradient,
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in turn, leads to a tangential, has to lead
to a tangential velocity gradient and this
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velocity would be affected the component of
velocity, which is affected is the relative
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velocity. So, the relative velocity would
have a tangential gradient, it will keep changing
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with the tangential direction.
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So, if you look at this aspect in a schematic
sense, I mentioned, that let us consider a
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same impeller; this is the same impeller we
were talking about. These are the two straight
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radial blades and as an outcome of the analysis,
we just have seen, there will be a tangential
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variation in the relative velocities. So,
these vectors, which are shown here, are basically
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the velocities, relative velocities. There
is a gradient of this relative velocity in
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the tangential direction.
Here, plus and minus indicate the pressure
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or higher pressure and this would be the pressure
surface and suction surface, let us say on,
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in the case of an axial compressor blade.
So, there is an increase in pressure gradient
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and correspondingly, a change in the relative
velocity in the tangential direction.
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So, what happens because of this is, that
as the fluid element begins to leave the impeller,
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that is, if you trace the fluid element, which
was let us say here, and as it leaves the
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impeller, by the time it reaches the tip of
the impeller, because of this tangential velocity
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gradient in the relative velocity, you can
see, that there relative velocity is lagging
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behind with the radial direction. So, the
relative velocity is actually, now pointing
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in this direction because it is having a gradient
in the tangential direction.
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So, because of the gradient, the vector of
the relative velocity leaving the impeller
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is inclined at an angle. It is lagging behind
the radial direction and therefore, what is
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the outcome of this? The outcome of this is
the fact, that ideally, one would have assumed
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a radial velocity; relative velocity is being
equal to radial velocity; that is no longer
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true. Therefore, one has tangential component
in the absolute velocities scale, that is,
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C w2, which is not equal to u 2.
So, if you recall velocity triangles that
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I had shown in the previous class, where I
had shown three different types of impellers:
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forward leaning, straight radial and the backward
leaning blades. For the straight radial blades,
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please go back and take a look at the velocity
triangle. You will see, that I had drawn the
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velocity triangle with the relative velocity
V or V 2 in the radial direction, which means,
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that C w2 would have been equal to u 2. That
is an ideal scenario.
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Now, here we have seen that due to the Coriolis
acceleration and its defect and the tangential
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velocity gradient, there will not be radial,
relative velocity will not really, will not
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necessarily leave the impeller radially. So,
there is a change or difference between the
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tangential component of the absolute velocity,
that is, C w2 and the blade speed at the impeller
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exit u 2.
So, this change in our difference in C w2
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and u 2 is captured by a parameter, which
is referred to as the slip factor while slip
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factor basically tells us the fact, that at
the exit of the impeller the relative velocity
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is lagging behind the radial direction, resulting
in a change or difference in the tangential
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component of absolute velocity C w2 as it
should have been equal to u 2. So, the ratio
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of C w2 to u 2 is called the slip factor.
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And this difference in the velocities is basically,
expressed as a fraction, that is, C w2 by
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u 2, which is usually denoted by sigma and
the sigma is, is a factor of actual blades.
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It will be less than 1 and one would like
to have a slip factor close to 1. So, that
21:59.960 --> 22:06.789
means, C w2 will be equal to u 2 and it will
lead to much higher, let us say, efficiency
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and pressure ratios for, for the same rotational
speeds and impeller diameter.
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Now, slip factor is also a function, a strong
function of the number of blades. You can
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see, I have shown two straight radial blades
and you have seen that from one blade to another,
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there is a tangential velocity gradient. That
means, the larger the distance between the
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blades, the velocity gradient will keep increasing
and therefore, the slip factor would be or
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the difference between C w2 and u 2 would
be higher. As we increase the blades' spacing,
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lower the number of blades, the greater would
be the difference and as you keep increasing
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the number of blades, the tangential velocity
gradient is lower and therefore, the slip
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factor is also likely to be low.
So, slip factor being strongly related to
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the number of blades, people have come up
with empirical correlations for calculating
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slip factor based on the number of blades.
One of the most commonly used empirical correlations
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is given by Stands and that known as the Stands
slip factor. It is equal to 1 minus 2 by n,
23:21.179 --> 23:26.730
where n is the number of blades, which means,
if there is, let us say, ten blades for straight
23:26.730 --> 23:32.220
radial blades, the slip factor would be, 1
by, 1 minus 2 by 10, that is, 1 minus 0.2
23:32.220 --> 23:36.049
and that is 0.8.
And if we increase the number of blades to
23:36.049 --> 23:42.029
let us say, twenty, then you can see that
the slip factor becomes 0.9, and so on. And
23:42.029 --> 23:49.029
that is, as we increase the number of blades,
the radial velocity of the, change in the
23:49.039 --> 23:55.230
tangential direction of the relative velocity
becomes lesser and therefore, that would be
23:55.230 --> 24:02.230
lesser difference between C w2 and u 2. Whereas
for larger spacing or lesser number of blades,
24:03.720 --> 24:09.110
the variation in the relative velocity in
the tangential direction would be larger and
24:09.110 --> 24:16.110
larger leading to much larger difference between
C w2 and u 2, leading to poor values of slip
24:18.380 --> 24:24.570
factor and so, slip factor being a strong
correlation, strongly correlated to the number
24:24.570 --> 24:28.779
of blades.
There are different parameters or different
24:28.779 --> 24:33.520
researches have come up with different empirical
correlation, the most common one is what one
24:33.520 --> 24:38.830
I have shown, which is strictly applicable
for a straight radial blade. For backward
24:38.830 --> 24:44.490
leaning blades, the slip factor is calculated
in a different way and there are different
24:44.490 --> 24:49.880
other correlations, which are used for such
blades and you will find in literature, there
24:49.880 --> 24:56.179
are many more empirical correlations, which
are, which are in some way or the other related
24:56.179 --> 25:02.240
to the number of blades.
So, the basic effect of slip factor is the
25:02.240 --> 25:08.169
fact, that it reduces the magnitude of the
swirl velocity or tangential component of
25:08.169 --> 25:14.419
velocity leaving the impeller and since pressure
rise is a function of this tangential velocity
25:14.419 --> 25:21.309
for, for lower values of tangential velocity,
the pressure rise also decreases. Therefore,
25:21.309 --> 25:25.779
slip factor directly affects the pressure
ratio of the centrifugal compressor, which
25:25.779 --> 25:31.760
obviously, is not a good thing, that designer
would want to keep for a given rotational
25:31.760 --> 25:38.760
speed and impeller diameter, try to maximize
the pressure ratio and presence of slip factor,
25:39.159 --> 25:45.909
can bring down the pressure ratio, but because
of the fact, that it affect the tangential
25:45.909 --> 25:49.799
velocity.
So, way out, the way out of this is to use
25:49.799 --> 25:54.700
more number of blades. Well, more number of
blades means, you would, obviously, have higher
25:54.700 --> 26:00.149
frictional losses and therefore, it is probably
not a good idea to keep increasing the number
26:00.149 --> 26:07.149
of blades or other option is to increase the
impeller diameter, which is for normal application.
26:08.820 --> 26:14.760
A land based application may not be a big
deal, you can keep increasing the diameter
26:14.760 --> 26:18.950
to some levels, but not for an aero-engineer
application, where you know, it will increase
26:18.950 --> 26:24.080
the drag as well.
But larger the diameter of the impeller, the
26:24.080 --> 26:27.760
stress on the impeller blades also goes up,
which means, then if they will have to invest
26:27.760 --> 26:34.110
in better materials, which can withstand higher
stress or the other option is to increase
26:34.110 --> 26:39.740
the rotational speed. And again, if you increase
the rotational speed for the same diameter,
26:39.740 --> 26:45.020
that also affects the stresses on the, on
the blades and therefore, that again is a
26:45.020 --> 26:48.840
constraint. So, you can see, that the all
kinds of constrains here in terms of stresses
26:48.840 --> 26:55.840
on the blades or frictional losses. So, there
is lot of scope for an optimization to be
26:56.669 --> 27:02.090
carried out here and to determine what is
the best possible configuration for number
27:02.090 --> 27:09.080
of blades versus impeller diameter versus
the rotational speed and that can give us
27:09.080 --> 27:13.940
the best possible efficiency, as well as,
the pressure ratio.
27:13.940 --> 27:20.940
So, having understood slip factor and its
effect on centrifugal compressor performance,
27:22.490 --> 27:27.870
let us now talk about the performance of a
centrifugal compressor in general. We have
27:27.870 --> 27:34.870
already discussed about the performance characteristic
of axial compressors, as well as, axial turbines.
27:35.140 --> 27:40.529
Centrifugal compressor performance characteristic
would look in some way, similar to what we
27:40.529 --> 27:47.130
have already discussed for an axial compressor,
but problems related to choking is a little
27:47.130 --> 27:54.130
more severe in a, in a centrifugal compressor
as compared to an axial compressor. Also,
27:55.850 --> 28:01.159
centrifugal compressors undergo similar problems
that we have seen for axial compressors like
28:01.159 --> 28:06.700
stall and surge. And so, let us look at the
performance characteristics of a centrifugal
28:06.700 --> 28:12.519
compressor and try to understand how we can
estimate the performance of a centrifugal
28:12.519 --> 28:13.440
compressor.
28:13.440 --> 28:19.190
So, we will evaluate the performance of a
centrifugal compressor in same way as we have
28:19.190 --> 28:23.600
done for an axial compressor. We will look
at the pressure ratio of the, dependence of
28:23.600 --> 28:28.019
the pressure ratio and efficiency on mass
flow rate, all of which of course, pressure
28:28.019 --> 28:34.240
ratio and mass flow rate, mass flow rate being
non-dimensionalized for different non-dimensionalized
28:34.240 --> 28:38.830
operating speed. And we will very soon realize
that compressors, centrifugal compressors
28:38.830 --> 28:42.950
also suffer from instability problems, like
surge and rotating stall.
28:42.950 --> 28:49.529
So, I will make this a little quick because
we have already done this for an axial compressor,
28:49.529 --> 28:56.529
it is exactly the same procedure. The non-dimensional
groups are derived from a dimensional analysis,
28:56.610 --> 29:02.269
the exit pressure ratio or exit, total pressure
P 02 and efficiency are functions of variety
29:02.269 --> 29:06.909
of parameters, like mass flow rate, inlet
stagnation pressure, inlet stagnation temperature,
29:06.909 --> 29:12.309
rotational speed, ratio of specific, it is
gamma, the gas constant or the viscosity,
29:12.309 --> 29:17.760
the design, as well as, diameter.
So, if we non-dimensionalize this, we get
29:17.760 --> 29:23.620
these many non-dimensional groups. P 02 by
P 01, that is, the pressure-ratio efficiency
29:23.620 --> 29:30.269
being functions of m dot root gamma RT 01
by P 01 D square, omega D by square root of
29:30.269 --> 29:35.590
gamma RT 01, omega D square by nu, gamma and
design.
29:35.590 --> 29:42.590
Now, for a given design and given diameter,
we can drop a lot of these terms and basically,
29:43.570 --> 29:48.149
the pressure ratio and efficiency becomes
function of mass flow rate, m dot root T 01
29:48.149 --> 29:54.580
by P 01 and N by root T 01. This is the non
dimensional speed and this is the non dimensional
29:54.580 --> 29:55.730
mass flow rate.
29:55.730 --> 30:00.980
This we will further process for a standard
day pressure and temperature. So, we get P
30:00.980 --> 30:07.980
zero two by p zero one efficiency be functions
of m dot root theta by delta and n by root
30:07.980 --> 30:13.730
theta where, theta is t zero one by t zero
one of standard day and delta is p zero one
30:13.730 --> 30:19.190
by p zero one standard day typical values
taken for this are t zero one standard day
30:19.190 --> 30:24.549
is two eighty eight point one five kelvin
which is basically twenty five degree celsius
30:24.549 --> 30:28.149
and pressure of standard day is one zero one
point three two five kilopascal.
30:28.149 --> 30:35.110
So, with this set of non dimensional parameters
pressure ratio and efficiency being functions
30:35.110 --> 30:40.980
mass flow rate non dimensionalized as well
as the non dimensional speed we will now,
30:40.980 --> 30:46.250
look at how the performance characteristics
can be plotted. Before that, let us first
30:46.250 --> 30:51.600
look at a very general characteristics which,
is applicable for any centrifugal compressor
30:51.600 --> 30:55.250
and then we will look at a typical performance
characteristics in general.
30:55.250 --> 31:00.890
So, if we look at the variation of pressure
ratio versus mass flow rate, one can trace
31:00.890 --> 31:06.330
characteristic like this; ideally, one could
trace a characteristic like this. There are
31:06.330 --> 31:12.730
several salient points, which I marked here,
point A, B, C, D and E.
31:12.730 --> 31:17.889
Now, let us take a look at what happens as,
let us say, the compressor was operating at
31:17.889 --> 31:23.799
some point E and then, as the mass flow is
reduced, as we throttled the compressor and
31:23.799 --> 31:28.559
decrease the mass flow, the pressure ratio
across the compressor increases. And as it
31:28.559 --> 31:33.100
increases, it reaches its peak and eventually,
you will see that after it reaches its peak,
31:33.100 --> 31:37.740
the pressure ratio drops and this could, this
drop, of course, could be very drastic as
31:37.740 --> 31:40.960
well in some compressors.
In the case of axial compressors we have already
31:40.960 --> 31:45.799
seen what really happens and the fact, that
when the throttle characteristics become,
31:45.799 --> 31:52.799
well, intersect the pressure ratio characteristics
beyond a certain level, the compressor undergoes,
31:53.110 --> 31:56.549
what are known as, instabilities.
So, that exact same thing happens even in
31:56.549 --> 32:01.210
the case of centrifugal compressor beyond
this point E, where the slope of the pressure
32:01.210 --> 32:08.210
ratio, mass flow characteristics is positive,
the centrifugal compressor undergoes instabilities
32:08.590 --> 32:15.590
and that is why, B is referred to as the surging
limit. And any point after the, on the left
32:15.820 --> 32:21.700
hand side of B, between A and B, let us say
point D, would be considered unstable point
32:21.700 --> 32:25.620
and the compressor cannot really operate in
a stable condition there.
32:25.620 --> 32:31.850
We have discussed the stability in reference
to axial compressors in quite detail and all
32:31.850 --> 32:37.970
those arguments are very much valid even for
a centrifugal compressor. And on the other
32:37.970 --> 32:43.899
hand, here we have the, what is known as,
the choking limit. Beyond point E and little
32:43.899 --> 32:50.659
later, the slope becomes extremely sharp and
there is a very sharp drop in the pressure
32:50.659 --> 32:55.260
ratio characteristic and mass flow remains
more or less constant and that is referred
32:55.260 --> 32:58.840
to as the choking limit.
So, we will discuss choking in little more
32:58.840 --> 33:04.450
detail in relation to centrifugal compressor
because these compressors tend to be affected
33:04.450 --> 33:11.450
more by choking and as compared to axial compressors.
And so, we will discuss choking in little
33:12.580 --> 33:17.960
more detail and limit our discussion on surge
and stall because that we have already discussed
33:17.960 --> 33:24.860
in relation to axial compressors.
So, let us now look at an actual centrifugal
33:24.860 --> 33:29.779
compressor map, performance map, in terms
of pressure ratio, non-dimensional mass flow
33:29.779 --> 33:33.289
rate and for a different non-dimensional speed.
33:33.289 --> 33:40.289
So, if we look at the pressure ratio characteristics
versus mass flow rate characteristic, so the
33:40.450 --> 33:46.570
previous slide I had shown an idealized curve.
Here, we have seen that the curve on the left
33:46.570 --> 33:51.580
hand side is not possible and beyond this
it chokes. So, the actual performance characteristic
33:51.580 --> 33:58.250
is limited between these two points B and
E and that is what is shown here by these
33:58.250 --> 34:03.299
different lines.
So, as you keep changing the speeds, the performance
34:03.299 --> 34:07.559
characteristics also change and you can say,
that as speed reaches its max of the design
34:07.559 --> 34:14.100
speeds, the performance characteristics become
sharper and sharper. For lower speeds, one
34:14.100 --> 34:20.750
can see, one can notice a shallower pressure
ratio versus mass flow characteristic, that
34:20.750 --> 34:26.930
becomes cheaper as one proceeds for higher
and higher speeds and all these lines are
34:26.930 --> 34:32.370
terminated on the left hand side by the surge
line and on the right hand side by the choking
34:32.370 --> 34:37.530
line.
And if we join all the points of maximum efficiency,
34:37.530 --> 34:43.580
we get the dotted line that is I have shown
here. And one could ideally want to operate
34:43.580 --> 34:48.580
the compressor very close to this maximum
efficiency line provided, that of course,
34:48.580 --> 34:54.160
this line is not very close to the surge line,
which of course, can put the compressor into
34:54.160 --> 34:55.500
the risk of surging.
34:55.500 --> 35:00.790
Now, if you look at the efficiency characteristic,
again very similar to the axial compressor
35:00.790 --> 35:07.790
that we discussed. Efficiency versus mass
flow rate for non-dimensional speeds and with
35:08.470 --> 35:14.280
increase in speed you can say, that the range
of high efficiency becomes narrower.
35:14.280 --> 35:19.160
For higher speeds we have a narrower band
of operation, where the efficiency is high
35:19.160 --> 35:23.430
and it becomes very sensitive to the mass
flow rate. For lower speeds, of course, efficiency
35:23.430 --> 35:29.630
is not, is relatively lesser sensitive to
the mass flow rate and there is a wider range
35:29.630 --> 35:33.510
of operation possible with higher slightly
higher efficiencies.
35:33.510 --> 35:39.140
In comparison with an axial compressor we
can see, that even centrifugal compressors
35:39.140 --> 35:46.140
have two aspects of two lines, which basically
define the performance. On left hand side
35:46.760 --> 35:52.460
we have the surge line; on the right hand
side we have the choking line. Now, between
35:52.460 --> 35:57.610
those two points on the map, that I shown,
that is, point A and B, the compressor may
35:57.610 --> 36:02.950
undergo instabilities. It could be a rotating
stall, which eventually leads to surge.
36:02.950 --> 36:09.800
Now, we have already discussed these instability
mechanisms in fairly good detail and just
36:09.800 --> 36:16.800
quickly mentioned what happens while a compressor
undergoes some of these instabilities. So,
36:17.410 --> 36:21.630
basically, the operation of a compressor in
the positive slope of pressure ratio versus
36:21.630 --> 36:28.630
mass flow rate is unstable operation and the
compressor cannot operate in that region in
36:28.710 --> 36:30.260
a stable manner.
36:30.260 --> 36:34.510
One of the instabilities, that affect the
performance is rotating stall and the more
36:34.510 --> 36:39.550
severe one is the surge, wherein there is
a sudden drop in the delivery pressure and
36:39.550 --> 36:46.390
of course, violent aerodynamics pulsations.
And in the case of centrifugal compressor
36:46.390 --> 36:51.400
it has been noticed, that in general, the
surging begins in the diffuser passages, because
36:51.400 --> 36:56.290
I think I mentioned in the last class, that
diffuser passages are significantly affected
36:56.290 --> 37:03.290
by boundary layer performance and so, that
is one of the weak links in a centrifugal
37:03.440 --> 37:10.440
compressor, where the performance is very
sensitive to boundary layer flow and therefore,
37:11.150 --> 37:17.610
surging has been, in general, observed to
initiate, get initiated in the diffuser passages.
37:17.610 --> 37:23.390
Now, in a centrifugal compressor the pressure
ratio or the performance is also a type of
37:23.390 --> 37:27.840
function of the, type of blade, that is used
in a centrifugal compressor.
37:27.840 --> 37:33.460
In the last class, we discussed three different
possible blades, blade configurations: a forward
37:33.460 --> 37:38.280
leaning type, straight radial and backward
leaning. Now, if you look at the performance
37:38.280 --> 37:42.840
of these three different types of blades and
analyze the velocity triangles at the exit,
37:42.840 --> 37:46.890
as urge you to go back to the previous lecture
and take up that slide, where I had shown
37:46.890 --> 37:52.110
the velocity triangle for these three cases,
you will notice, that theoretically, forward
37:52.110 --> 37:56.510
leaning blades produce higher pressure ratio.
Because if we look at the velocity triangle
37:56.510 --> 38:01.630
at the exit, you would appreciate this aspect
and one would expect forward leaning blades
38:01.630 --> 38:07.020
to be much better in terms of performance.
But what is interesting to notice is the fact,
38:07.020 --> 38:12.700
that forward leaning blades have an inherent
instability, because if you look at pressure
38:12.700 --> 38:17.510
ratio versus mass flow characteristic for
forward leaning blade, the characteristic
38:17.510 --> 38:22.680
always has a positive slope. And, we have
just now discussed, that operation on the
38:22.680 --> 38:28.840
positive slope of a pressure ratio mass flow
characteristic is inherently unstable and
38:28.840 --> 38:32.540
that is the reason, I think I mentioned in
the last class, that forward leaning blades
38:32.540 --> 38:37.660
are not really used in centrifugal compressors
because of the fact, that they are inherently
38:37.660 --> 38:43.580
unstable, that is why straight radial and
backward leaning blades are commonly used
38:43.580 --> 38:45.690
in modern day centrifugal compressors.
38:45.690 --> 38:50.460
So, if we look at the performance characteristics,
either in terms of pressure ratio or the temperature
38:50.460 --> 38:56.190
rise versus either mass flow rate or the flow
coefficient, a forward leaning blade would
38:56.190 --> 38:59.410
have a characteristic, which has positive
slope throughout.
38:59.410 --> 39:06.410
So, this is like the left hand side curve
of a centrifugal compressor characteristic,
39:06.460 --> 39:11.890
where of course, this is idealized, but one
would still get a positive slope throughout
39:11.890 --> 39:16.810
for a forward leaning blade, which means,
that this blade is going to have instabilities,
39:16.810 --> 39:23.810
irrespective of the mass flow rate and therefore,
this is not a favorable type of blade, that
39:24.310 --> 39:29.080
can be used even though the pressure ratio
performance is much better than straight radial
39:29.080 --> 39:34.800
or backward leaning. So, these two configurations
are the ones, which are commonly used, the
39:34.800 --> 39:41.070
straight radial and the backward leaning blades.
So, what I will do next is to discuss about
39:41.070 --> 39:47.700
a problem of choking, which is what is probably
more severe in the case of centrifugal compressors
39:47.700 --> 39:54.700
as compared to axial compressors, whereas
rotating stall and surge are still the limiting
39:54.790 --> 40:01.790
performance parameters on one side. On the
other side, we also have the choking problem
40:02.020 --> 40:08.380
associated with trying to increase mass flow
rate and beyond a certain level of mass flow
40:08.380 --> 40:14.690
rate, compressibility effects prevent us from
operating the compressor beyond a certain
40:14.690 --> 40:16.270
level of mass flow rate.
40:16.270 --> 40:20.600
So, let us take closer look at choking in
more detail because you already discussed
40:20.600 --> 40:24.100
about the other instabilities in relation
to axial compressors.
40:24.100 --> 40:29.100
So, as the mass flow increases in, in the
case of centrifugal compressors, the pressure
40:29.100 --> 40:34.460
ratio will decrease, as we have seen the performance
characteristic and therefore, that also reduces
40:34.460 --> 40:39.890
the density. After a certain point one would
not be able to increase the mass flow beyond
40:39.890 --> 40:44.930
a certain value, the compressor is then said
to have choked, that is on this, that is,
40:44.930 --> 40:48.620
probably we have reached the right hand side
of the performance characteristic, wherein
40:48.620 --> 40:52.360
the mass flow rate has reached its maxima
and we are not able to increase mass flow
40:52.360 --> 40:58.060
rate beyond that level and that is when the
compressor is said to have choked.
40:58.060 --> 41:03.620
So, in a centrifugal compressor we have seen
that different components, which constitute
41:03.620 --> 41:10.370
a centrifugal compressor. We have the inlet,
the impeller and the diffuser vanes. So, calculation
41:10.370 --> 41:15.840
of the choking mass flow is different depending
upon whether it is the stationary component
41:15.840 --> 41:20.840
or the rotating components. So, we will take
a look at the choking mass flow as applicable
41:20.840 --> 41:27.840
for inlet, the impeller and diffuses vanes
and see their dependence on the upstream parameters.
41:28.320 --> 41:33.030
So, choking behavior, because it is different
for rotating passages from that of stationary
41:33.030 --> 41:38.410
passages, so if you consider the inlet, we
know, that of course choking, irrespective
41:38.410 --> 41:42.700
of whether it is inlet or impeller, takes
places when Mach, the Mach number reaches
41:42.700 --> 41:46.510
1.
So, when Much number is unity, the ratio of
41:46.510 --> 41:52.640
static temperature to total temperature T
by T 0 is equal to 2 by gamma plus 1 because
41:52.640 --> 41:57.910
Mach number, if the isentropic relation with
the x equates M is equal to 1, we get this
41:57.910 --> 42:03.150
relation between static and stagnation temperature.
So, for the moment if you assume an isentropic
42:03.150 --> 42:09.490
flow, then the choking mass flow rate is basically,
m dot A by per unit area is equal rho naught
42:09.490 --> 42:15.360
a naught into 2 by gamma plus 1 raise to gamma
plus 1 by 2 into gamma minus 1. So, this basically
42:15.360 --> 42:19.560
comes from mass flow rate being equal to rho
a v.
42:19.560 --> 42:26.140
And we have expressed rho in terms of stagnation
density and velocity in terms of the speed
42:26.140 --> 42:32.050
of sound and here we can see, that this mass
flow rate is expressed purely in terms of
42:32.050 --> 42:38.140
parameters, which are the upstream parameters
of the inlet stagnation conditions, which
42:38.140 --> 42:42.280
remain constant. So, as we keep changing the
operating condition, the inlet conditions
42:42.280 --> 42:46.680
are still fixed, which means, that the mass
flow rate also has to remain constant and
42:46.680 --> 42:52.270
that is why, when Mach number becomes 1, what
you can see is that the right hand side of
42:52.270 --> 42:59.270
the mass flow rate equation as all the parameters,
which are basically constants. And therefore,
43:00.580 --> 43:05.970
when Mach number becomes 1, one can actually
calculate the choking mass flow based on the
43:05.970 --> 43:07.550
inlet stagnation condition.
43:07.550 --> 43:11.690
The next component that we will take a look
at is the impeller in a rotating passage.
43:11.690 --> 43:17.810
As we have seen, the flow conditions are usually
referred through the rothalpy, which I had
43:17.810 --> 43:22.250
discussed in the last class.
And during choking, in the case of an impeller,
43:22.250 --> 43:26.250
it is the relative velocity, which basically
becomes equal to this period of sound when
43:26.250 --> 43:30.740
Mach number becomes unity and so, if look
at the expression for rothalpy we have, I
43:30.740 --> 43:37.740
is equal to h plus half into V square minus
U square and so, and we know, that stagnation
43:38.170 --> 43:45.170
temperature can be expressed in terms of the
corresponding static temperature and the speed
43:46.150 --> 43:51.160
of sound.
So, here we, if we express enthalpy in terms
43:51.160 --> 43:58.160
of stagnation temperature and V square, because
when Mach number is equal to 1, V becomes
43:58.790 --> 44:04.590
equal to the speed of sound and therefore,
that becomes gamma RT divided by 2C p because
44:04.590 --> 44:11.590
enthalpy is T naught times C p minus U square
by 2C p. So, we have divided this by the speed
44:13.400 --> 44:19.560
of sound and therefore, we get T naught 1
is equal to T plus gamma RT by 2C p minus
44:19.560 --> 44:23.870
U square by 2C p.
So, if we simplify, that we get T by T naught
44:23.870 --> 44:30.870
1 is equal to 2 by gamma plus 1 into 1 plus
U square by 2C p T 01 and therefore, mass
44:32.160 --> 44:39.160
flow rate, which is rho a V, again gets expressed
as rho 01 a 01 into T by T 01 raised to gamma
44:39.470 --> 44:45.260
plus 1 by 2 into gamma minus 1, which can
be simplified further and what you see is,
44:45.260 --> 44:52.260
that mass flow rate is equal to rho 01 a 01
into 2 plus gamma minus 1 U square by a 01
44:52.610 --> 44:58.230
square by gamma plus 1 raised to gamma plus
1 by 2 into gamma minus 1. So, this is rather
44:58.230 --> 45:03.120
complex expression for the mass flow rate,
once again tell us, that in addition to the
45:03.120 --> 45:06.710
inlet condition.
In the case of impeller we also see, that
45:06.710 --> 45:11.550
it is a function of the rotational speed and
in the case of the inlet we have seen, that
45:11.550 --> 45:16.170
mass flow rate is purely a function of the
inlet parameters and that, therefore mass
45:16.170 --> 45:21.200
flow rate gets fixed once the inlet conditions
are fixed. In the case of impeller, besides
45:21.200 --> 45:28.190
the inlet conditions you also have the rotational
speed, that means, that in principle it should
45:28.190 --> 45:35.090
be possible for us to operate the compressor
at a different mass flow, at higher mass flow,
45:35.090 --> 45:40.160
than the choking mass flow for a higher rotational
speed.
45:40.160 --> 45:44.420
Of course, this will also require, that if
once you start operating the compressor at
45:44.420 --> 45:50.430
a higher rotational speed, the other components
do not choke because if the other components
45:50.430 --> 45:55.780
choke, the compressor will still remain at
the choke condition. So, other components
45:55.780 --> 46:02.780
per meeting, like the inlet or the diffuser,
the impeller performance or impeller choking
46:03.020 --> 46:09.560
because it is a function also, of the rotational
speed operating the compressor at higher rotational
46:09.560 --> 46:15.700
speed, may permit us to operate or deliver
a higher mass flow, provided all other components
46:15.700 --> 46:19.890
also, are able to operate at this new operating
point.
46:19.890 --> 46:26.090
Choking mass flow in an impeller is basically,
function of the rotational speed and compressor
46:26.090 --> 46:30.550
in principle should be able to handle a higher
mass flow with an increasing speed provided,
46:30.550 --> 46:36.660
that other components, like inlet or diffuser
does not really undergo choking for this rotational
46:36.660 --> 46:37.280
speed.
46:37.280 --> 46:42.770
Now, the last component, if the diffuser,
the choking mass flow in a diffuser has an
46:42.770 --> 46:47.520
expression the same as that of inlet, so it
is again a function of the inlet condition
46:47.520 --> 46:52.800
for the diffuser. So, mass flow rate is function
of rho naught a naught into 2 by gamma plus
46:52.800 --> 46:56.420
1 raise to gamma plus 1 by 2 into gamma minus
1.
46:56.420 --> 47:01.600
Now, here, you can see that the stagnation
conditions at the inlet of the diffuser depend
47:01.600 --> 47:07.690
upon the inlet of the diffuser itself, which
is the impeller exit and therefore, mass flow
47:07.690 --> 47:12.430
rate can basically be related to the rotational
speed of the impeller and therefore, you can
47:12.430 --> 47:17.890
see, that the operation of the diffuser and
impeller are sort of coupled because the diffuser,
47:17.890 --> 47:22.150
choking mass flow, is a function of the impeller
exit conditions. And impeller exit conditions
47:22.150 --> 47:28.530
can basically be changed by changing the rotational
speed and so, there is a certain amount of
47:28.530 --> 47:32.610
coupling between the impeller operation and
the diffuser operation.
47:32.610 --> 47:39.610
And so, so with this analysis it should be
possible for us to calculate under what conditions
47:42.380 --> 47:49.380
centrifugal compressor is likely to choke
and how is still, that we can operate, still
47:50.130 --> 47:55.300
operate the compressor at a probably, slightly
a higher mass flow by looking at the other
47:55.300 --> 48:00.970
components involved, like the inlet and the
diffuser, and ensuring, that all the three
48:00.970 --> 48:07.970
components can operate in under the new rotational
speed without undergoing a choke.
48:08.250 --> 48:15.250
So, let me now quickly recap our discussion
in today's class. We had discussed few aspects,
48:16.470 --> 48:21.540
which are an extension of what we had discussed
in the last class. We started our lecture
48:21.540 --> 48:27.790
today with detailed discussion on the Coriolis
acceleration and we have derived an expression
48:27.790 --> 48:34.030
for Corialis acceleration. We have seen, that
Corialis acceleration leads to tangential
48:34.030 --> 48:41.030
velocity gradient, velocity gradient in the
tangential direction and that is what leads
48:42.690 --> 48:49.690
to a difference in the velocity, that is,
leaving the impeller that is, it leads to
48:49.760 --> 48:55.410
certain amount of difference between C w2,
which is the tangential component of absolute
48:55.410 --> 49:01.570
velocity and U 2 and that difference is what
is referred to as the slip factor.
49:01.570 --> 49:06.410
We have seen, that slip factor is a strong
function of the number of blades. Lesser the
49:06.410 --> 49:11.450
number of blades, the greater the spacing
between the blades, the greater is the tangential
49:11.450 --> 49:18.450
velocity gradient and therefore, the slip
factor becomes lower and lower. So, this can
49:18.750 --> 49:25.380
be, that is, with higher number of blades
one can reduce the tangential variation in
49:25.380 --> 49:31.570
relative velocity and therefore, one can achieve
higher values of slip factors. So, slip factors
49:31.570 --> 49:38.090
and its dependents on the number of blades.
We have seen the formula, where one can approximate
49:38.090 --> 49:44.100
slip factor as 1 minus 2 by n, where n is
the number of blades and of course, that is
49:44.100 --> 49:48.850
true for straight radial blades.
We have then discussed about the performance
49:48.850 --> 49:54.300
characteristics and of a centrifugal compressor.
Of course, we did not discuss too much of
49:54.300 --> 49:59.620
the details of the performance characteristics
because it is very similar to that of an axial
49:59.620 --> 50:04.730
compressor. We spent much less time discussing
about surge and stall because we already discussed
50:04.730 --> 50:11.730
that and we have discussed slightly more details
about the choking conditions, which is affecting,
50:12.720 --> 50:17.330
which might affect centrifugal compressor
performance, and how one can estimate the
50:17.330 --> 50:23.960
choking mass flow for the inlet, the impeller
and the diffuser and the interrelation between
50:23.960 --> 50:26.610
choking conditions for these three different
components.
50:26.610 --> 50:33.530
So, these were the topics, that we had discussed
in today's class and so this will sort of
50:33.530 --> 50:40.460
wind up our discussion on centrifugal compressors
and therefore, the next lecture, as I mentioned
50:40.460 --> 50:44.860
in the beginning, we will basically take up
few problems for solving.
50:44.860 --> 50:50.700
So, we will have a tutorial session in the
next class. We will discuss about how we can
50:50.700 --> 50:56.620
solve problems obtaining centrifugal compressors
and at the end of the tutorial we will also
50:56.620 --> 51:01.790
have a few exercise problems for you to solve
based on the discussion during the last few
51:01.790 --> 51:07.040
lectures. So, we will take up a tutorial in
the next lecture, which will be lecture number
51:07.040 --> 51:07.329
thirty three.